Subgroups — C₆₇×#60214 #30739465

Order 17152.

1960 conjugacy classes of subgroups.

The subgroup lattice has 694 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×#2969	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2968	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2967	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2966	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×#2873	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×M₁₆×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	4	yes	complemented
35	#446×C₆₇	C₂²	1	8	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	#423×C₆₇	C₂²	1	2	yes	—
51	#423×C₆₇	C₂²	1	4	yes	—
52	#407×C₆₇	C₂²	1	2	yes	—
53	#407×C₆₇	C₂²	1	4	yes	—
54	#405×C₆₇	C₂²	1	8	yes	complemented
55	#405×C₆₇	C₂²	1	8	yes	complemented
56	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
57	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
58	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
59	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
60	C₄×M₁₆×C₆₇	C₂²	1	4	yes	—
61	C₄×M₁₆×C₆₇	C₂²	1	2	yes	complemented
62	C₄×M₁₆×C₆₇	C₂²	1	2	yes	complemented
63	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
64	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
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	—	2	8	no	—
73	C₂×M₁₆×C₆₇	—	2	16	no	—
74	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
75	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
76	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
77	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
78	C₂×M₁₆×C₆₇	C₂³	1	2	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	8	yes	—
148	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
149	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
150	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
151	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
152	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
153	#97×C₆₇	C₂³	1	2	yes	—
154	#97×C₆₇	C₂³	1	2	yes	—
155	#97×C₆₇	C₂³	1	4	yes	—
156	#97×C₆₇	C₂³	1	4	yes	—
157	#97×C₆₇	C₂³	1	8	yes	—
158	C₄×C₈×C₆₇	C₂³	1	2	yes	—
159	C₄×C₈×C₆₇	C₂³	1	2	yes	—
160	C₄×C₈×C₆₇	C₂³	1	4	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	4	no	—
175	C₂²×C₄×C₆₇	—	2	8	no	—
176	C₂²×C₄×C₆₇	—	4	2	no	—
177	C₂²×C₄×C₆₇	—	4	2	no	—
178	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
179	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
187	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
188	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
189	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
196	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
197	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
198	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
199	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
200	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
201	M₁₆×C₆₇	—	2	4	no	—
202	M₁₆×C₆₇	—	2	8	no	—
203	M₁₆×C₆₇	—	2	8	no	—
204	M₁₆×C₆₇	—	2	16	no	—
205	M₁₆×C₆₇	#41	1	8	yes	—
206	C₂×C₈×C₆₇	—	2	8	no	—
207	C₂×C₈×C₆₇	—	2	16	no	—
208	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
209	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
210	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
211	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
212	C₄⋊C₄×C₆₇	C₂²×C₄	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	2	yes	—
216	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
217	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
218	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
219	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
220	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
221	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
222	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
223	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
224	C₂²⋊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	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
234	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	4	yes	—
246	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
247	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
248	C₄²×C₆₇	C₂⁴	1	4	yes	—
249	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
250	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
251	C₂³×C₆₇	—	2	2	no	—
252	C₂³×C₆₇	—	2	2	no	—
253	C₂³×C₆₇	—	2	4	no	—
254	C₂³×C₆₇	—	4	1	no	—
255	C₂³×C₆₇	—	4	2	no	—
256	C₂³×C₆₇	—	4	2	no	—
257	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
258	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
259	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
260	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
261	Q₈×C₆₇	—	2	4	no	—
262	Q₈×C₆₇	—	4	2	no	—
263	D₄×C₆₇	—	2	8	no	—
264	D₄×C₆₇	—	2	4	no	—
265	D₄×C₆₇	—	4	2	no	—
266	D₄×C₆₇	—	4	4	no	—
267	C₂×C₄×C₆₇	—	2	2	no	—
268	C₂×C₄×C₆₇	—	2	4	no	—
269	C₂×C₄×C₆₇	—	2	2	no	—
270	C₂×C₄×C₆₇	—	2	2	no	—
271	C₂×C₄×C₆₇	—	2	2	no	—
272	C₂×C₄×C₆₇	—	2	2	no	—
273	C₂×C₄×C₆₇	—	2	8	no	—
274	C₂×C₄×C₆₇	—	2	2	no	—
275	C₂×C₄×C₆₇	—	2	2	no	—
276	C₂×C₄×C₆₇	—	2	2	no	—
277	C₂×C₄×C₆₇	—	2	2	no	—
278	C₂×C₄×C₆₇	—	2	4	no	—
279	C₂×C₄×C₆₇	—	2	2	no	—
280	C₂×C₄×C₆₇	—	2	4	no	—
281	C₂×C₄×C₆₇	—	4	1	no	—
282	C₂×C₄×C₆₇	—	4	1	no	—
283	C₂×C₄×C₆₇	—	4	1	no	—
284	C₂×C₄×C₆₇	—	4	1	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	2	no	—
289	C₂×C₄×C₆₇	—	4	2	no	—
290	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
291	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
292	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
293	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
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	4	yes	—
300	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
301	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
302	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
303	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
304	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
305	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
306	C₈×C₆₇	—	2	4	no	—
307	C₈×C₆₇	—	2	8	no	—
308	C₈×C₆₇	C₂×#41	1	8	yes	—
309	C₂²×C₆₇	—	2	1	no	—
310	C₂²×C₆₇	—	2	1	no	—
311	C₂²×C₆₇	—	2	1	no	—
312	C₂²×C₆₇	—	2	1	no	—
313	C₂²×C₆₇	—	2	1	no	—
314	C₂²×C₆₇	—	2	4	no	—
315	C₂²×C₆₇	—	2	1	no	—
316	C₂²×C₆₇	—	2	1	no	—
317	C₂²×C₆₇	—	2	4	no	—
318	C₂²×C₆₇	—	2	4	no	—
319	C₂²×C₆₇	—	4	4	no	—
320	C₂²×C₆₇	—	8	4	no	—
321	C₂²×C₆₇	C₄×#41	1	2	yes	—
322	C₂²×C₆₇	#568	1	2	yes	—
323	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
324	C₄×C₆₇	—	2	1	no	—
325	C₄×C₆₇	—	2	4	no	—
326	C₄×C₆₇	—	2	1	no	—
327	C₄×C₆₇	—	2	1	no	—
328	C₄×C₆₇	—	2	1	no	—
329	C₄×C₆₇	—	2	1	no	—
330	C₄×C₆₇	—	2	1	no	—
331	C₄×C₆₇	—	2	1	no	—
332	C₄×C₆₇	—	2	1	no	—
333	C₄×C₆₇	—	2	1	no	—
334	C₄×C₆₇	—	2	4	no	—
335	C₄×C₆₇	—	2	4	no	—
336	C₄×C₆₇	C₄×#41	1	2	yes	—
337	C₄×C₆₇	#568	1	2	yes	—
338	C₄×C₆₇	C₂²×#41	1	2	yes	—
339	#60214	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
340	C₂×C₆₇	—	2	2	no	—
341	C₂×C₆₇	—	2	2	no	—
342	C₂×C₆₇	—	4	1	no	—
343	C₂×C₆₇	—	4	4	no	—
344	C₂×C₆₇	C₂×C₄×#41	1	1	yes	characteristic
345	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
346	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
347	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
348	#2969	C₂×C₆₇	1	4	yes	complemented
349	#2968	C₂×C₆₇	1	4	yes	complemented
350	#2967	C₂×C₆₇	1	4	yes	complemented
351	#2966	C₂×C₆₇	1	4	yes	complemented
352	Q₈×M₁₆	C₂×C₆₇	1	2	yes	complemented
353	D₄×M₁₆	C₂×C₆₇	1	4	yes	complemented
354	D₄×M₁₆	C₂×C₆₇	1	2	yes	complemented
355	#2921	C₂×C₆₇	1	2	yes	complemented
356	#2920	C₂×C₆₇	1	1	yes	characteristic, complemented
357	#2873	C₂×C₆₇	1	2	yes	complemented
358	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
359	C₆₇	#60214	1	1	yes	characteristic, direct product factor, Sylow
360	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
361	C₂²×M₁₆	C₂²×C₆₇	1	4	yes	—
362	#529	C₄×C₆₇	1	2	yes	—
363	#529	C₄×C₆₇	1	2	yes	—
364	#529	C₄×C₆₇	1	2	yes	—
365	#529	C₄×C₆₇	1	2	yes	—
366	#529	C₂²×C₆₇	1	2	yes	—
367	#529	C₂²×C₆₇	1	2	yes	—
368	#529	C₂²×C₆₇	1	2	yes	—
369	#529	C₂²×C₆₇	1	2	yes	—
370	C₂×#126	C₄×C₆₇	1	2	yes	—
371	C₂×#125	C₄×C₆₇	1	1	yes	characteristic
372	C₂×#124	C₄×C₆₇	1	1	yes	characteristic
373	C₂×#123	C₄×C₆₇	1	2	yes	—
374	C₂×C₂²⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
375	C₂×#121	C₄×C₆₇	1	1	yes	characteristic
376	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
377	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
378	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
379	C₂×#117	C₂²×C₆₇	1	2	yes	—
380	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
381	#446	C₂²×C₆₇	1	4	yes	complemented
382	#446	C₂²×C₆₇	1	8	yes	complemented
383	C₈×Q₈	C₂²×C₆₇	1	4	yes	complemented
384	#436	C₂²×C₆₇	1	4	yes	complemented
385	#436	C₂²×C₆₇	1	8	yes	complemented
386	#435	C₂²×C₆₇	1	8	yes	complemented
387	#435	C₂²×C₆₇	1	8	yes	complemented
388	#435	C₂²×C₆₇	1	8	yes	complemented
389	C₈×D₄	C₂²×C₆₇	1	8	yes	complemented
390	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
391	#433	C₂²×C₆₇	1	8	yes	complemented
392	#433	C₂²×C₆₇	1	8	yes	complemented
393	#432	C₂²×C₆₇	1	2	yes	complemented
394	#432	C₂²×C₆₇	1	2	yes	complemented
395	#431	C₂²×C₆₇	1	2	yes	complemented
396	#431	C₂²×C₆₇	1	2	yes	complemented
397	#423	C₂²×C₆₇	1	2	yes	—
398	#423	C₂²×C₆₇	1	4	yes	—
399	#407	C₂²×C₆₇	1	2	yes	—
400	#407	C₂²×C₆₇	1	4	yes	—
401	#405	C₂²×C₆₇	1	8	yes	complemented
402	#405	C₂²×C₆₇	1	8	yes	complemented
403	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
404	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
405	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
406	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
407	C₄×M₁₆	C₂²×C₆₇	1	4	yes	—
408	C₄×M₁₆	C₂²×C₆₇	1	2	yes	complemented
409	C₄×M₁₆	C₂²×C₆₇	1	2	yes	complemented
410	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
411	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
412	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
413	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
414	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
415	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
416	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
417	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
418	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
419	C₂×M₁₆	—	2	8	no	—
420	C₂×M₁₆	—	2	16	no	—
421	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
422	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
423	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
424	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
425	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
426	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
427	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
428	#126	C₂×C₄×C₆₇	1	4	yes	—
429	#126	C₂×C₄×C₆₇	1	4	yes	—
430	#126	C₂×C₄×C₆₇	1	4	yes	—
431	#126	C₂×C₄×C₆₇	1	4	yes	—
432	#125	C₂×C₄×C₆₇	1	2	yes	—
433	#125	C₂×C₄×C₆₇	1	2	yes	—
434	#125	C₂×C₄×C₆₇	1	2	yes	—
435	#125	C₂×C₄×C₆₇	1	2	yes	—
436	#124	C₂×C₄×C₆₇	1	2	yes	—
437	#124	C₂×C₄×C₆₇	1	2	yes	—
438	#124	C₂×C₄×C₆₇	1	2	yes	—
439	#124	C₂×C₄×C₆₇	1	2	yes	—
440	#123	C₂×C₄×C₆₇	1	4	yes	—
441	#123	C₂×C₄×C₆₇	1	4	yes	—
442	#123	C₂×C₄×C₆₇	1	4	yes	—
443	#123	C₂×C₄×C₆₇	1	4	yes	—
444	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
445	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
446	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
447	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
448	#121	C₂×C₄×C₆₇	1	2	yes	—
449	#121	C₂×C₄×C₆₇	1	2	yes	—
450	#121	C₂×C₄×C₆₇	1	2	yes	—
451	#121	C₂×C₄×C₆₇	1	2	yes	—
452	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
453	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
454	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
455	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
456	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
457	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
458	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
459	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
460	C₄×D₄	C₂³×C₆₇	1	4	yes	—
461	C₄×D₄	C₂³×C₆₇	1	2	yes	—
462	C₄×D₄	C₂³×C₆₇	1	2	yes	—
463	C₄×D₄	C₂³×C₆₇	1	4	yes	—
464	#117	C₂×C₄×C₆₇	1	4	yes	—
465	#117	C₂×C₄×C₆₇	1	4	yes	—
466	#117	C₂³×C₆₇	1	4	yes	—
467	#117	C₂³×C₆₇	1	4	yes	—
468	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
469	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
470	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
471	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
472	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
473	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
474	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
475	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
476	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
477	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
478	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
479	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
480	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
481	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
482	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
483	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
484	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
485	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
486	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
487	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
488	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
489	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
490	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
491	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
492	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
493	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
494	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
495	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
496	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
497	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
498	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
499	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
500	#97	C₂³×C₆₇	1	2	yes	—
501	#97	C₂³×C₆₇	1	2	yes	—
502	#97	C₂³×C₆₇	1	4	yes	—
503	#97	C₂³×C₆₇	1	4	yes	—
504	#97	C₂³×C₆₇	1	8	yes	—
505	C₄×C₈	C₂³×C₆₇	1	2	yes	—
506	C₄×C₈	C₂³×C₆₇	1	2	yes	—
507	C₄×C₈	C₂³×C₆₇	1	4	yes	—
508	C₄×C₈	C₂³×C₆₇	1	4	yes	—
509	C₄×C₈	C₂³×C₆₇	1	8	yes	—
510	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
511	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
512	C₂×Q₈	—	4	2	no	—
513	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
514	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
515	C₂×D₄	—	4	2	no	—
516	C₂×D₄	—	4	4	no	—
517	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
518	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
519	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
520	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
521	C₂²×C₄	—	2	4	no	—
522	C₂²×C₄	—	2	8	no	—
523	C₂²×C₄	—	4	2	no	—
524	C₂²×C₄	—	4	2	no	—
525	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
526	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
527	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
528	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
529	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
530	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
531	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
532	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
533	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
534	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
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	1	yes	characteristic
542	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
543	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
544	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
545	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
546	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
547	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
548	M₁₆	—	2	4	no	—
549	M₁₆	—	2	8	no	—
550	M₁₆	—	2	8	no	—
551	M₁₆	—	2	16	no	—
552	M₁₆	#41×C₆₇	1	8	yes	—
553	C₂×C₈	—	2	8	no	—
554	C₂×C₈	—	2	16	no	—
555	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
556	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
557	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
558	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
559	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
560	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
561	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
562	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
563	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
564	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
565	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
566	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
567	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
568	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
569	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
570	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
571	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
572	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
573	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
574	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
575	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
576	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
577	C₂²⋊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	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
581	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
582	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
583	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
584	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
585	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
586	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
587	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
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	4	yes	—
593	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
594	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
595	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
596	C₄²	C₂⁴×C₆₇	1	4	yes	—
597	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
598	C₂³	—	2	2	no	—
599	C₂³	—	2	2	no	—
600	C₂³	—	2	4	no	—
601	C₂³	—	4	1	no	—
602	C₂³	—	4	2	no	—
603	C₂³	—	4	2	no	—
604	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
605	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
606	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
607	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
608	Q₈	—	2	4	no	—
609	Q₈	—	4	2	no	—
610	D₄	—	2	8	no	—
611	D₄	—	2	4	no	—
612	D₄	—	4	2	no	—
613	D₄	—	4	4	no	—
614	C₂×C₄	—	2	2	no	—
615	C₂×C₄	—	2	4	no	—
616	C₂×C₄	—	2	2	no	—
617	C₂×C₄	—	2	2	no	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	8	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	2	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	4	no	—
626	C₂×C₄	—	2	2	no	—
627	C₂×C₄	—	2	4	no	—
628	C₂×C₄	—	4	1	no	—
629	C₂×C₄	—	4	1	no	—
630	C₂×C₄	—	4	1	no	—
631	C₂×C₄	—	4	1	no	—
632	C₂×C₄	—	4	1	no	—
633	C₂×C₄	—	4	1	no	—
634	C₂×C₄	—	4	1	no	—
635	C₂×C₄	—	4	2	no	—
636	C₂×C₄	—	4	2	no	—
637	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
638	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
639	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
640	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
641	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
642	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
643	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
644	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
645	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
646	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
647	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
648	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
649	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
650	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
651	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
652	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
653	C₈	—	2	4	no	—
654	C₈	—	2	8	no	—
655	C₈	C₂×#41×C₆₇	1	8	yes	—
656	C₂²	—	2	1	no	—
657	C₂²	—	2	1	no	—
658	C₂²	—	2	1	no	—
659	C₂²	—	2	1	no	—
660	C₂²	—	2	1	no	—
661	C₂²	—	2	1	no	—
662	C₂²	—	2	4	no	—
663	C₂²	—	2	4	no	—
664	C₂²	—	2	1	no	—
665	C₂²	—	2	4	no	—
666	C₂²	—	4	4	no	—
667	C₂²	—	8	4	no	—
668	C₂²	C₄×#41×C₆₇	1	2	yes	—
669	C₂²	#568×C₆₇	1	2	yes	—
670	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
671	C₄	—	2	1	no	—
672	C₄	—	2	1	no	—
673	C₄	—	2	1	no	—
674	C₄	—	2	1	no	—
675	C₄	—	2	1	no	—
676	C₄	—	2	1	no	—
677	C₄	—	2	1	no	—
678	C₄	—	2	4	no	—
679	C₄	—	2	4	no	—
680	C₄	—	2	1	no	—
681	C₄	—	2	4	no	—
682	C₄	—	2	1	no	—
683	C₄	C₄×#41×C₆₇	1	2	yes	—
684	C₄	#568×C₆₇	1	2	yes	—
685	C₄	C₂²×#41×C₆₇	1	2	yes	—
686	C₂	—	2	2	no	—
687	C₂	—	2	2	no	—
688	C₂	—	4	1	no	—
689	C₂	—	4	4	no	—
690	C₂	C₂×C₄×#41×C₆₇	1	1	yes	characteristic
691	C₂	C₂×#568×C₆₇	1	1	yes	characteristic
692	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×#2969	C₂	semidirect
3. C₆₇×#2968	C₂	semidirect
4. C₆₇×#2967	C₂	semidirect
5. C₆₇×#2966	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×#2873	C₂	semidirect
12. C₂×C₄×M₁₆×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
54. #405×C₆₇	C₂²	semidirect
55. #405×C₆₇	C₂²	semidirect
61. C₄×M₁₆×C₆₇	C₂²	semidirect
62. C₄×M₁₆×C₆₇	C₂²	semidirect
63. C₂×#97×C₆₇	C₂²	semidirect
64. C₂×C₄×C₈×C₆₇	C₂²	semidirect
339. #60214	C₆₇	direct (N × H)
348. #2969	C₂×C₆₇	semidirect
349. #2968	C₂×C₆₇	semidirect
350. #2967	C₂×C₆₇	semidirect
351. #2966	C₂×C₆₇	semidirect
352. Q₈×M₁₆	C₂×C₆₇	semidirect
353. D₄×M₁₆	C₂×C₆₇	semidirect
354. D₄×M₁₆	C₂×C₆₇	semidirect
355. #2921	C₂×C₆₇	semidirect
356. #2920	C₂×C₆₇	semidirect
357. #2873	C₂×C₆₇	semidirect
358. C₂×C₄×M₁₆	C₂×C₆₇	semidirect
359. C₆₇	#60214	direct (N × H)
381. #446	C₂²×C₆₇	semidirect
382. #446	C₂²×C₆₇	semidirect
383. C₈×Q₈	C₂²×C₆₇	semidirect
384. #436	C₂²×C₆₇	semidirect
385. #436	C₂²×C₆₇	semidirect
386. #435	C₂²×C₆₇	semidirect
387. #435	C₂²×C₆₇	semidirect
388. #435	C₂²×C₆₇	semidirect
389. C₈×D₄	C₂²×C₆₇	semidirect
390. C₈×D₄	C₂²×C₆₇	semidirect
391. #433	C₂²×C₆₇	semidirect
392. #433	C₂²×C₆₇	semidirect
393. #432	C₂²×C₆₇	semidirect
394. #432	C₂²×C₆₇	semidirect
395. #431	C₂²×C₆₇	semidirect
396. #431	C₂²×C₆₇	semidirect
401. #405	C₂²×C₆₇	semidirect
402. #405	C₂²×C₆₇	semidirect
408. C₄×M₁₆	C₂²×C₆₇	semidirect
409. C₄×M₁₆	C₂²×C₆₇	semidirect
410. C₂×#97	C₂²×C₆₇	semidirect
411. 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×#2969	2
C₆₇×#2968	2
C₆₇×#2967	2
C₆₇×#2966	2
Q₈×M₁₆×C₆₇	2
D₄×M₁₆×C₆₇	2
C₆₇×#2921	2
C₆₇×#2920	2
C₆₇×#2873	2
C₂×C₄×M₁₆×C₆₇	2
#60214	67

Subgroups — C67×#60214 #30739465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60214 #30739465