Subgroups — C₆₇×#60264 #30739515

Order 17152.

1872 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2995	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2994	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2993	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2992	C₂	1	4	yes	maximal, complemented
6	Q₈×M₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
7	D₄×M₁₆×C₆₇	C₂	1	4	yes	maximal, complemented
8	D₄×M₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2921	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2920	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2905	C₂	1	2	yes	maximal, complemented
12	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₆₇	C₂²	1	4	yes	—
15	#529×C₆₇	C₄	1	2	yes	—
16	#529×C₆₇	C₄	1	2	yes	—
17	#529×C₆₇	C₄	1	2	yes	—
18	#529×C₆₇	C₄	1	2	yes	—
19	#529×C₆₇	C₂²	1	2	yes	—
20	#529×C₆₇	C₂²	1	2	yes	—
21	#529×C₆₇	C₂²	1	2	yes	—
22	#529×C₆₇	C₂²	1	2	yes	—
23	C₂×#126×C₆₇	C₄	1	2	yes	—
24	C₂×#125×C₆₇	C₄	1	1	yes	characteristic
25	C₂×#124×C₆₇	C₄	1	1	yes	characteristic
26	C₂×#123×C₆₇	C₄	1	2	yes	—
27	C₂×C₂²⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
28	C₂×#121×C₆₇	C₄	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
32	C₂×#117×C₆₇	C₂²	1	2	yes	—
33	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
34	#446×C₆₇	C₂²	1	8	yes	complemented
35	#446×C₆₇	C₂²	1	4	yes	complemented
36	C₈×Q₈×C₆₇	C₂²	1	4	yes	complemented
37	#436×C₆₇	C₂²	1	4	yes	complemented
38	#436×C₆₇	C₂²	1	8	yes	complemented
39	#435×C₆₇	C₂²	1	8	yes	complemented
40	#435×C₆₇	C₂²	1	8	yes	complemented
41	#435×C₆₇	C₂²	1	8	yes	complemented
42	C₈×D₄×C₆₇	C₂²	1	8	yes	complemented
43	C₈×D₄×C₆₇	C₂²	1	4	yes	complemented
44	#433×C₆₇	C₂²	1	8	yes	complemented
45	#433×C₆₇	C₂²	1	8	yes	complemented
46	#432×C₆₇	C₂²	1	2	yes	complemented
47	#432×C₆₇	C₂²	1	2	yes	complemented
48	#431×C₆₇	C₂²	1	2	yes	complemented
49	#431×C₆₇	C₂²	1	2	yes	complemented
50	#424×C₆₇	C₂²	1	8	yes	complemented
51	#424×C₆₇	C₂²	1	8	yes	complemented
52	#423×C₆₇	C₂²	1	2	yes	—
53	#423×C₆₇	C₂²	1	2	yes	—
54	#423×C₆₇	C₂²	1	1	yes	characteristic
55	#423×C₆₇	C₂²	1	4	yes	—
56	#423×C₆₇	C₂²	1	1	yes	characteristic
57	#423×C₆₇	C₂²	1	2	yes	complemented
58	#423×C₆₇	C₂²	1	2	yes	complemented
59	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
60	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
61	#407×C₆₇	C₂²	1	2	yes	—
62	#407×C₆₇	C₂²	1	4	yes	—
63	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
64	C₄×M₁₆×C₆₇	C₂²	1	4	yes	—
65	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
67	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
72	C₂×M₁₆×C₆₇	—	4	4	no	—
73	C₂×M₁₆×C₆₇	—	4	8	no	—
74	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
75	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
76	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
77	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
78	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
79	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
80	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
81	#126×C₆₇	C₂×C₄	1	4	yes	—
82	#126×C₆₇	C₂×C₄	1	4	yes	—
83	#126×C₆₇	C₂×C₄	1	4	yes	—
84	#126×C₆₇	C₂×C₄	1	4	yes	—
85	#125×C₆₇	C₂×C₄	1	2	yes	—
86	#125×C₆₇	C₂×C₄	1	2	yes	—
87	#125×C₆₇	C₂×C₄	1	2	yes	—
88	#125×C₆₇	C₂×C₄	1	2	yes	—
89	#124×C₆₇	C₂×C₄	1	2	yes	—
90	#124×C₆₇	C₂×C₄	1	2	yes	—
91	#124×C₆₇	C₂×C₄	1	2	yes	—
92	#124×C₆₇	C₂×C₄	1	2	yes	—
93	#123×C₆₇	C₂×C₄	1	4	yes	—
94	#123×C₆₇	C₂×C₄	1	4	yes	—
95	#123×C₆₇	C₂×C₄	1	4	yes	—
96	#123×C₆₇	C₂×C₄	1	4	yes	—
97	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
98	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
99	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
100	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
101	#121×C₆₇	C₂×C₄	1	2	yes	—
102	#121×C₆₇	C₂×C₄	1	2	yes	—
103	#121×C₆₇	C₂×C₄	1	2	yes	—
104	#121×C₆₇	C₂×C₄	1	2	yes	—
105	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
106	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
107	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
108	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
109	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
110	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
111	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
112	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
113	C₄×D₄×C₆₇	C₂³	1	4	yes	—
114	C₄×D₄×C₆₇	C₂³	1	2	yes	—
115	C₄×D₄×C₆₇	C₂³	1	2	yes	—
116	C₄×D₄×C₆₇	C₂³	1	4	yes	—
117	#117×C₆₇	C₂×C₄	1	4	yes	—
118	#117×C₆₇	C₂×C₄	1	4	yes	—
119	#117×C₆₇	C₂³	1	4	yes	—
120	#117×C₆₇	C₂³	1	4	yes	—
121	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
122	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
123	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
124	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
125	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
126	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
127	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
128	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
129	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
130	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
131	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
132	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
133	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
134	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
135	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
136	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
137	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
138	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
139	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
140	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
141	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
142	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
143	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
144	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
145	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
146	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
147	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
148	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
149	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
150	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
151	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
152	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
153	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
154	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
155	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
156	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
157	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
158	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
159	#97×C₆₇	C₂³	1	4	yes	—
160	#97×C₆₇	C₂³	1	8	yes	—
161	C₄×C₈×C₆₇	C₂³	1	4	yes	—
162	C₄×C₈×C₆₇	C₂³	1	8	yes	—
163	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
164	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
165	C₂×Q₈×C₆₇	—	4	2	no	—
166	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
167	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
168	C₂×D₄×C₆₇	—	4	2	no	—
169	C₂×D₄×C₆₇	—	4	4	no	—
170	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
171	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
172	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
173	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
174	C₂²×C₄×C₆₇	—	2	2	no	—
175	C₂²×C₄×C₆₇	—	2	4	no	—
176	C₂²×C₄×C₆₇	—	2	2	no	—
177	C₂²×C₄×C₆₇	—	2	4	no	—
178	C₂²×C₄×C₆₇	—	4	2	no	—
179	C₂²×C₄×C₆₇	—	4	2	no	—
180	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
181	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
182	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
183	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
184	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
185	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
186	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
187	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
188	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
189	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
190	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
191	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
192	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
193	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
194	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
195	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
196	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
197	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
198	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
199	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
200	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
201	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
202	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
203	M₁₆×C₆₇	—	2	8	no	—
204	M₁₆×C₆₇	—	2	4	no	—
205	M₁₆×C₆₇	—	2	4	no	—
206	M₁₆×C₆₇	—	4	8	no	—
207	M₁₆×C₆₇	—	4	4	no	—
208	C₂×C₈×C₆₇	—	4	4	no	—
209	C₂×C₈×C₆₇	—	4	8	no	—
210	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
211	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
212	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
213	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
214	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
215	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
216	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
217	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
218	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
219	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
220	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
221	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
222	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
223	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
224	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
225	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
226	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
227	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
228	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
229	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
230	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
231	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
232	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
233	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
234	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
235	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
236	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
237	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
238	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
239	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
240	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
241	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
242	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
243	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
244	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
245	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
246	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
247	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
248	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
249	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
250	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
251	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
252	C₄²×C₆₇	C₂⁴	1	4	yes	—
253	C₂³×C₆₇	—	2	2	no	—
254	C₂³×C₆₇	—	2	2	no	—
255	C₂³×C₆₇	—	2	4	no	—
256	C₂³×C₆₇	—	4	1	no	—
257	C₂³×C₆₇	—	4	2	no	—
258	C₂³×C₆₇	—	4	2	no	—
259	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
260	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
261	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
262	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
263	Q₈×C₆₇	—	2	4	no	—
264	Q₈×C₆₇	—	4	2	no	—
265	D₄×C₆₇	—	2	8	no	—
266	D₄×C₆₇	—	2	4	no	—
267	D₄×C₆₇	—	4	2	no	—
268	D₄×C₆₇	—	4	4	no	—
269	C₂×C₄×C₆₇	—	2	2	no	—
270	C₂×C₄×C₆₇	—	2	4	no	—
271	C₂×C₄×C₆₇	—	2	2	no	—
272	C₂×C₄×C₆₇	—	2	2	no	—
273	C₂×C₄×C₆₇	—	2	2	no	—
274	C₂×C₄×C₆₇	—	2	2	no	—
275	C₂×C₄×C₆₇	—	2	8	no	—
276	C₂×C₄×C₆₇	—	2	2	no	—
277	C₂×C₄×C₆₇	—	2	2	no	—
278	C₂×C₄×C₆₇	—	2	2	no	—
279	C₂×C₄×C₆₇	—	2	1	no	—
280	C₂×C₄×C₆₇	—	2	4	no	—
281	C₂×C₄×C₆₇	—	2	1	no	—
282	C₂×C₄×C₆₇	—	2	1	no	—
283	C₂×C₄×C₆₇	—	2	1	no	—
284	C₂×C₄×C₆₇	—	2	4	no	—
285	C₂×C₄×C₆₇	—	4	1	no	—
286	C₂×C₄×C₆₇	—	4	1	no	—
287	C₂×C₄×C₆₇	—	4	1	no	—
288	C₂×C₄×C₆₇	—	4	1	no	—
289	C₂×C₄×C₆₇	—	4	1	no	—
290	C₂×C₄×C₆₇	—	4	1	no	—
291	C₂×C₄×C₆₇	—	4	1	no	—
292	C₂×C₄×C₆₇	—	4	2	no	—
293	C₂×C₄×C₆₇	—	4	2	no	—
294	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
295	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
296	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
297	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
298	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
299	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
300	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
301	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
302	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
303	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
304	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
305	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
306	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
307	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
308	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
309	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
310	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
311	C₈×C₆₇	—	2	8	no	—
312	C₈×C₆₇	—	2	4	no	—
313	C₈×C₆₇	—	2	4	no	—
314	C₂²×C₆₇	—	2	1	no	—
315	C₂²×C₆₇	—	2	1	no	—
316	C₂²×C₆₇	—	2	1	no	—
317	C₂²×C₆₇	—	2	1	no	—
318	C₂²×C₆₇	—	2	1	no	—
319	C₂²×C₆₇	—	2	4	no	—
320	C₂²×C₆₇	—	2	1	no	—
321	C₂²×C₆₇	—	2	1	no	—
322	C₂²×C₆₇	—	2	4	no	—
323	C₂²×C₆₇	—	2	4	no	—
324	C₂²×C₆₇	—	4	4	no	—
325	C₂²×C₆₇	—	8	4	no	—
326	C₂²×C₆₇	#520	1	2	yes	—
327	C₂²×C₆₇	#568	1	2	yes	—
328	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
329	C₄×C₆₇	—	2	1	no	—
330	C₄×C₆₇	—	2	4	no	—
331	C₄×C₆₇	—	2	1	no	—
332	C₄×C₆₇	—	2	1	no	—
333	C₄×C₆₇	—	2	1	no	—
334	C₄×C₆₇	—	2	1	no	—
335	C₄×C₆₇	—	2	1	no	—
336	C₄×C₆₇	—	2	1	no	—
337	C₄×C₆₇	—	2	1	no	—
338	C₄×C₆₇	—	2	1	no	—
339	C₄×C₆₇	—	2	4	no	—
340	C₄×C₆₇	—	2	4	no	—
341	C₄×C₆₇	#520	1	2	yes	—
342	C₄×C₆₇	#568	1	2	yes	—
343	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
344	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
345	#60264	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
346	C₂×C₆₇	—	2	2	no	—
347	C₂×C₆₇	—	2	2	no	—
348	C₂×C₆₇	—	4	1	no	—
349	C₂×C₆₇	—	4	4	no	—
350	C₂×C₆₇	C₂×#520	1	1	yes	characteristic
351	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
352	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
353	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
354	#2995	C₂×C₆₇	1	4	yes	complemented
355	#2994	C₂×C₆₇	1	4	yes	complemented
356	#2993	C₂×C₆₇	1	4	yes	complemented
357	#2992	C₂×C₆₇	1	4	yes	complemented
358	Q₈×M₁₆	C₂×C₆₇	1	2	yes	complemented
359	D₄×M₁₆	C₂×C₆₇	1	4	yes	complemented
360	D₄×M₁₆	C₂×C₆₇	1	2	yes	complemented
361	#2921	C₂×C₆₇	1	2	yes	complemented
362	#2920	C₂×C₆₇	1	1	yes	characteristic, complemented
363	#2905	C₂×C₆₇	1	2	yes	complemented
364	C₂×#423	C₂×C₆₇	1	1	yes	characteristic, complemented
365	C₆₇	#60264	1	1	yes	characteristic, direct product factor, Sylow
366	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
367	C₂²×M₁₆	C₂²×C₆₇	1	4	yes	—
368	#529	C₄×C₆₇	1	2	yes	—
369	#529	C₄×C₆₇	1	2	yes	—
370	#529	C₄×C₆₇	1	2	yes	—
371	#529	C₄×C₆₇	1	2	yes	—
372	#529	C₂²×C₆₇	1	2	yes	—
373	#529	C₂²×C₆₇	1	2	yes	—
374	#529	C₂²×C₆₇	1	2	yes	—
375	#529	C₂²×C₆₇	1	2	yes	—
376	C₂×#126	C₄×C₆₇	1	2	yes	—
377	C₂×#125	C₄×C₆₇	1	1	yes	characteristic
378	C₂×#124	C₄×C₆₇	1	1	yes	characteristic
379	C₂×#123	C₄×C₆₇	1	2	yes	—
380	C₂×C₂²⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
381	C₂×#121	C₄×C₆₇	1	1	yes	characteristic
382	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
383	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
384	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
385	C₂×#117	C₂²×C₆₇	1	2	yes	—
386	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
387	#446	C₂²×C₆₇	1	8	yes	complemented
388	#446	C₂²×C₆₇	1	4	yes	complemented
389	C₈×Q₈	C₂²×C₆₇	1	4	yes	complemented
390	#436	C₂²×C₆₇	1	4	yes	complemented
391	#436	C₂²×C₆₇	1	8	yes	complemented
392	#435	C₂²×C₆₇	1	8	yes	complemented
393	#435	C₂²×C₆₇	1	8	yes	complemented
394	#435	C₂²×C₆₇	1	8	yes	complemented
395	C₈×D₄	C₂²×C₆₇	1	8	yes	complemented
396	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
397	#433	C₂²×C₆₇	1	8	yes	complemented
398	#433	C₂²×C₆₇	1	8	yes	complemented
399	#432	C₂²×C₆₇	1	2	yes	complemented
400	#432	C₂²×C₆₇	1	2	yes	complemented
401	#431	C₂²×C₆₇	1	2	yes	complemented
402	#431	C₂²×C₆₇	1	2	yes	complemented
403	#424	C₂²×C₆₇	1	8	yes	complemented
404	#424	C₂²×C₆₇	1	8	yes	complemented
405	#423	C₂²×C₆₇	1	2	yes	—
406	#423	C₂²×C₆₇	1	2	yes	—
407	#423	C₂²×C₆₇	1	1	yes	characteristic
408	#423	C₂²×C₆₇	1	4	yes	—
409	#423	C₂²×C₆₇	1	1	yes	characteristic
410	#423	C₂²×C₆₇	1	2	yes	complemented
411	#423	C₂²×C₆₇	1	2	yes	complemented
412	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
413	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
414	#407	C₂²×C₆₇	1	2	yes	—
415	#407	C₂²×C₆₇	1	4	yes	—
416	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
417	C₄×M₁₆	C₂²×C₆₇	1	4	yes	—
418	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
419	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
420	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
421	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
422	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
423	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
424	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
425	C₂×M₁₆	—	4	4	no	—
426	C₂×M₁₆	—	4	8	no	—
427	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
428	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
429	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
430	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
431	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
432	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
433	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
434	#126	C₂×C₄×C₆₇	1	4	yes	—
435	#126	C₂×C₄×C₆₇	1	4	yes	—
436	#126	C₂×C₄×C₆₇	1	4	yes	—
437	#126	C₂×C₄×C₆₇	1	4	yes	—
438	#125	C₂×C₄×C₆₇	1	2	yes	—
439	#125	C₂×C₄×C₆₇	1	2	yes	—
440	#125	C₂×C₄×C₆₇	1	2	yes	—
441	#125	C₂×C₄×C₆₇	1	2	yes	—
442	#124	C₂×C₄×C₆₇	1	2	yes	—
443	#124	C₂×C₄×C₆₇	1	2	yes	—
444	#124	C₂×C₄×C₆₇	1	2	yes	—
445	#124	C₂×C₄×C₆₇	1	2	yes	—
446	#123	C₂×C₄×C₆₇	1	4	yes	—
447	#123	C₂×C₄×C₆₇	1	4	yes	—
448	#123	C₂×C₄×C₆₇	1	4	yes	—
449	#123	C₂×C₄×C₆₇	1	4	yes	—
450	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
451	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
452	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
453	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
454	#121	C₂×C₄×C₆₇	1	2	yes	—
455	#121	C₂×C₄×C₆₇	1	2	yes	—
456	#121	C₂×C₄×C₆₇	1	2	yes	—
457	#121	C₂×C₄×C₆₇	1	2	yes	—
458	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
459	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
460	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
461	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
462	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
463	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
464	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
465	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
466	C₄×D₄	C₂³×C₆₇	1	4	yes	—
467	C₄×D₄	C₂³×C₆₇	1	2	yes	—
468	C₄×D₄	C₂³×C₆₇	1	2	yes	—
469	C₄×D₄	C₂³×C₆₇	1	4	yes	—
470	#117	C₂×C₄×C₆₇	1	4	yes	—
471	#117	C₂×C₄×C₆₇	1	4	yes	—
472	#117	C₂³×C₆₇	1	4	yes	—
473	#117	C₂³×C₆₇	1	4	yes	—
474	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
475	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
476	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
477	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
478	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
479	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
480	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
481	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
482	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
483	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
484	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
485	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
486	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
487	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
488	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
489	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
490	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
491	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
492	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
493	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
494	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
495	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
496	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
497	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
498	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
499	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
500	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
501	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
502	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
503	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
504	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
505	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
506	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
507	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
508	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
509	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
510	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
511	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
512	#97	C₂³×C₆₇	1	4	yes	—
513	#97	C₂³×C₆₇	1	8	yes	—
514	C₄×C₈	C₂³×C₆₇	1	4	yes	—
515	C₄×C₈	C₂³×C₆₇	1	8	yes	—
516	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
517	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
518	C₂×Q₈	—	4	2	no	—
519	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
520	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
521	C₂×D₄	—	4	2	no	—
522	C₂×D₄	—	4	4	no	—
523	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
524	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
525	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
526	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
527	C₂²×C₄	—	2	2	no	—
528	C₂²×C₄	—	2	4	no	—
529	C₂²×C₄	—	2	2	no	—
530	C₂²×C₄	—	2	4	no	—
531	C₂²×C₄	—	4	2	no	—
532	C₂²×C₄	—	4	2	no	—
533	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
534	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
535	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
536	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
537	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
538	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
539	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
540	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
541	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
542	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
543	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
544	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
545	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
546	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
547	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
548	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
549	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
550	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
551	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
552	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
553	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
554	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
555	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
556	M₁₆	—	2	8	no	—
557	M₁₆	—	2	4	no	—
558	M₁₆	—	2	4	no	—
559	M₁₆	—	4	8	no	—
560	M₁₆	—	4	4	no	—
561	C₂×C₈	—	4	4	no	—
562	C₂×C₈	—	4	8	no	—
563	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
564	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
565	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
566	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
567	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
568	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
569	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
570	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
571	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
572	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
573	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
574	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
575	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
576	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
577	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
578	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
579	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
580	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
581	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
582	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
583	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
584	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
585	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
586	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
587	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
588	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
589	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
590	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
591	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
592	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
593	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
594	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
595	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
596	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
597	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
598	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
599	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
600	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
601	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
602	C₄²	C₂⁴×C₆₇	1	4	yes	—
603	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
604	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
605	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
606	C₂³	—	2	2	no	—
607	C₂³	—	2	2	no	—
608	C₂³	—	2	4	no	—
609	C₂³	—	4	1	no	—
610	C₂³	—	4	2	no	—
611	C₂³	—	4	2	no	—
612	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
613	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
614	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
615	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
616	Q₈	—	2	4	no	—
617	Q₈	—	4	2	no	—
618	D₄	—	2	8	no	—
619	D₄	—	2	4	no	—
620	D₄	—	4	2	no	—
621	D₄	—	4	4	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	4	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	2	no	—
627	C₂×C₄	—	2	2	no	—
628	C₂×C₄	—	2	8	no	—
629	C₂×C₄	—	2	2	no	—
630	C₂×C₄	—	2	2	no	—
631	C₂×C₄	—	2	2	no	—
632	C₂×C₄	—	2	1	no	—
633	C₂×C₄	—	2	4	no	—
634	C₂×C₄	—	2	1	no	—
635	C₂×C₄	—	2	1	no	—
636	C₂×C₄	—	2	1	no	—
637	C₂×C₄	—	2	4	no	—
638	C₂×C₄	—	4	1	no	—
639	C₂×C₄	—	4	1	no	—
640	C₂×C₄	—	4	1	no	—
641	C₂×C₄	—	4	1	no	—
642	C₂×C₄	—	4	1	no	—
643	C₂×C₄	—	4	1	no	—
644	C₂×C₄	—	4	1	no	—
645	C₂×C₄	—	4	2	no	—
646	C₂×C₄	—	4	2	no	—
647	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
648	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
649	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
650	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
651	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
652	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
653	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
654	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
655	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
656	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
657	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
658	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
659	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
660	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
661	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
662	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
663	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
664	C₈	—	2	8	no	—
665	C₈	—	2	4	no	—
666	C₈	—	2	4	no	—
667	C₂²	—	2	1	no	—
668	C₂²	—	2	1	no	—
669	C₂²	—	2	1	no	—
670	C₂²	—	2	1	no	—
671	C₂²	—	2	1	no	—
672	C₂²	—	2	1	no	—
673	C₂²	—	2	4	no	—
674	C₂²	—	2	4	no	—
675	C₂²	—	2	1	no	—
676	C₂²	—	2	4	no	—
677	C₂²	—	4	4	no	—
678	C₂²	—	8	4	no	—
679	C₂²	#520×C₆₇	1	2	yes	—
680	C₂²	#568×C₆₇	1	2	yes	—
681	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
682	C₄	—	2	1	no	—
683	C₄	—	2	1	no	—
684	C₄	—	2	1	no	—
685	C₄	—	2	1	no	—
686	C₄	—	2	1	no	—
687	C₄	—	2	1	no	—
688	C₄	—	2	1	no	—
689	C₄	—	2	4	no	—
690	C₄	—	2	4	no	—
691	C₄	—	2	1	no	—
692	C₄	—	2	4	no	—
693	C₄	—	2	1	no	—
694	C₄	#520×C₆₇	1	2	yes	—
695	C₄	#568×C₆₇	1	2	yes	—
696	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
697	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
698	C₂	—	2	2	no	—
699	C₂	—	2	2	no	—
700	C₂	—	4	1	no	—
701	C₂	—	4	4	no	—
702	C₂	C₂×#520×C₆₇	1	1	yes	characteristic
703	C₂	C₂×#568×C₆₇	1	1	yes	characteristic
704	C₂	C₂×#568×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂²×C₆₇	C₂⁴×C₄
prime radical	C₂×#121×C₆₇	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₆₇×#2995	C₂	semidirect
3. C₆₇×#2994	C₂	semidirect
4. C₆₇×#2993	C₂	semidirect
5. C₆₇×#2992	C₂	semidirect
6. Q₈×M₁₆×C₆₇	C₂	semidirect
7. D₄×M₁₆×C₆₇	C₂	semidirect
8. D₄×M₁₆×C₆₇	C₂	semidirect
9. C₆₇×#2921	C₂	semidirect
10. C₆₇×#2920	C₂	semidirect
11. C₆₇×#2905	C₂	semidirect
12. C₂×#423×C₆₇	C₂	semidirect
34. #446×C₆₇	C₂²	semidirect
35. #446×C₆₇	C₂²	semidirect
36. C₈×Q₈×C₆₇	C₂²	semidirect
37. #436×C₆₇	C₂²	semidirect
38. #436×C₆₇	C₂²	semidirect
39. #435×C₆₇	C₂²	semidirect
40. #435×C₆₇	C₂²	semidirect
41. #435×C₆₇	C₂²	semidirect
42. C₈×D₄×C₆₇	C₂²	semidirect
43. C₈×D₄×C₆₇	C₂²	semidirect
44. #433×C₆₇	C₂²	semidirect
45. #433×C₆₇	C₂²	semidirect
46. #432×C₆₇	C₂²	semidirect
47. #432×C₆₇	C₂²	semidirect
48. #431×C₆₇	C₂²	semidirect
49. #431×C₆₇	C₂²	semidirect
50. #424×C₆₇	C₂²	semidirect
51. #424×C₆₇	C₂²	semidirect
57. #423×C₆₇	C₂²	semidirect
58. #423×C₆₇	C₂²	semidirect
59. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
60. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
345. #60264	C₆₇	direct (N × H)
354. #2995	C₂×C₆₇	semidirect
355. #2994	C₂×C₆₇	semidirect
356. #2993	C₂×C₆₇	semidirect
357. #2992	C₂×C₆₇	semidirect
358. Q₈×M₁₆	C₂×C₆₇	semidirect
359. D₄×M₁₆	C₂×C₆₇	semidirect
360. D₄×M₁₆	C₂×C₆₇	semidirect
361. #2921	C₂×C₆₇	semidirect
362. #2920	C₂×C₆₇	semidirect
363. #2905	C₂×C₆₇	semidirect
364. C₂×#423	C₂×C₆₇	semidirect
365. C₆₇	#60264	direct (N × H)
387. #446	C₂²×C₆₇	semidirect
388. #446	C₂²×C₆₇	semidirect
389. C₈×Q₈	C₂²×C₆₇	semidirect
390. #436	C₂²×C₆₇	semidirect
391. #436	C₂²×C₆₇	semidirect
392. #435	C₂²×C₆₇	semidirect
393. #435	C₂²×C₆₇	semidirect
394. #435	C₂²×C₆₇	semidirect
395. C₈×D₄	C₂²×C₆₇	semidirect
396. C₈×D₄	C₂²×C₆₇	semidirect
397. #433	C₂²×C₆₇	semidirect
398. #433	C₂²×C₆₇	semidirect
399. #432	C₂²×C₆₇	semidirect
400. #432	C₂²×C₆₇	semidirect
401. #431	C₂²×C₆₇	semidirect
402. #431	C₂²×C₆₇	semidirect
403. #424	C₂²×C₆₇	semidirect
404. #424	C₂²×C₆₇	semidirect
410. #423	C₂²×C₆₇	semidirect
411. #423	C₂²×C₆₇	semidirect
412. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
413. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#529×C₆₇	2
C₆₇×#2995	2
C₆₇×#2994	2
C₆₇×#2993	2
C₆₇×#2992	2
Q₈×M₁₆×C₆₇	2
D₄×M₁₆×C₆₇	2
C₆₇×#2921	2
C₆₇×#2920	2
C₆₇×#2905	2
C₂×#423×C₆₇	2
#60264	67

Subgroups — C67×#60264 #30739515

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60264 #30739515