Subgroups — C₆₇×#60269 #30739520

Order 17152.

1864 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3444	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2995	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2992	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2990	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2989	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2961	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2933	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2932	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2925	C₂	1	2	yes	maximal, complemented
10	C₂×#432×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2906	C₂	1	2	yes	maximal, complemented
12	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#131×C₆₇	C₂²	1	4	yes	—
14	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
15	#533×C₆₇	C₄	1	1	yes	characteristic
16	#533×C₆₇	C₄	1	1	yes	characteristic
17	#533×C₆₇	C₂²	1	1	yes	characteristic
18	#532×C₆₇	C₄	1	1	yes	characteristic
19	#532×C₆₇	C₄	1	1	yes	characteristic
20	#532×C₆₇	C₂²	1	1	yes	characteristic
21	#529×C₆₇	C₄	1	2	yes	—
22	#529×C₆₇	C₄	1	2	yes	—
23	#529×C₆₇	C₄	1	2	yes	—
24	#529×C₆₇	C₄	1	2	yes	—
25	#529×C₆₇	C₂²	1	2	yes	—
26	#529×C₆₇	C₂²	1	2	yes	—
27	#525×C₆₇	C₄	1	2	yes	—
28	#525×C₆₇	C₄	1	2	yes	—
29	#525×C₆₇	C₂²	1	2	yes	—
30	C₄×#41×C₆₇	C₂²	1	2	yes	—
31	C₄×#41×C₆₇	C₂²	1	2	yes	—
32	C₄×#41×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	8	yes	complemented
36	#436×C₆₇	C₂²	1	8	yes	—
37	#435×C₆₇	C₂²	1	8	yes	—
38	#435×C₆₇	C₂²	1	8	yes	—
39	#435×C₆₇	C₂²	1	8	yes	complemented
40	#435×C₆₇	C₂²	1	8	yes	complemented
41	C₈×D₄×C₆₇	C₂²	1	8	yes	—
42	#433×C₆₇	C₂²	1	8	yes	complemented
43	#432×C₆₇	C₂²	1	2	yes	—
44	#432×C₆₇	C₂²	1	2	yes	—
45	#432×C₆₇	C₂²	1	2	yes	—
46	#432×C₆₇	C₂²	1	2	yes	—
47	#432×C₆₇	C₂²	1	4	yes	complemented
48	#431×C₆₇	C₂²	1	4	yes	complemented
49	#424×C₆₇	C₂²	1	4	yes	—
50	#424×C₆₇	C₂²	1	8	yes	complemented
51	#423×C₆₇	C₂²	1	2	yes	—
52	#423×C₆₇	C₂²	1	2	yes	—
53	#423×C₆₇	C₂²	1	2	yes	—
54	#423×C₆₇	C₂²	1	2	yes	—
55	#423×C₆₇	C₂²	1	4	yes	complemented
56	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
57	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
58	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
59	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
60	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	complemented
61	#408×C₆₇	C₂²	1	4	yes	—
62	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
63	#405×C₆₇	C₂²	1	4	yes	—
64	C₂×#97×C₆₇	C₂²	1	2	yes	—
65	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
66	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
67	C₂×#41×C₆₇	C₂³	1	2	yes	—
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	#131×C₆₇	—	4	8	no	—
72	C₂×M₁₆×C₆₇	—	4	4	no	—
73	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
74	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
75	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
76	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
77	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
78	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
79	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
80	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
81	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
83	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
84	#126×C₆₇	C₂×C₄	1	4	yes	—
85	#126×C₆₇	C₂×C₄	1	4	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	#123×C₆₇	C₂×C₄	1	4	yes	—
93	#123×C₆₇	C₂×C₄	1	4	yes	—
94	#123×C₆₇	C₂×C₄	1	4	yes	—
95	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
96	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	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	#121×C₆₇	C₂×C₄	1	2	yes	—
106	#121×C₆₇	C₂×C₄	1	2	yes	—
107	#120×C₆₇	C₂×C₄	1	4	yes	—
108	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
109	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
110	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
111	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
112	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
113	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
114	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
115	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
116	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
117	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
118	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
119	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
120	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
121	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
122	C₄×D₄×C₆₇	C₂³	1	4	yes	—
123	C₄×D₄×C₆₇	C₂³	1	4	yes	—
124	C₄×D₄×C₆₇	C₂³	1	2	yes	—
125	C₄×D₄×C₆₇	C₂³	1	2	yes	—
126	#117×C₆₇	C₂×C₄	1	2	yes	—
127	#117×C₆₇	C₂×C₄	1	2	yes	—
128	#117×C₆₇	C₂×C₄	1	4	yes	—
129	#117×C₆₇	C₂×C₄	1	4	yes	—
130	#117×C₆₇	C₂³	1	2	yes	—
131	#117×C₆₇	C₂³	1	4	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	4	yes	—
145	C₄⋊C₈×C₆₇	C₂³	1	2	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	4	yes	—
150	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
151	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
152	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
153	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
154	C₄⋊C₈×C₆₇	C₂³	1	8	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	4	yes	—
161	#97×C₆₇	C₂³	1	8	yes	—
162	C₄×C₈×C₆₇	C₂³	1	8	yes	—
163	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
164	#41×C₆₇	—	2	4	no	—
165	#41×C₆₇	—	2	4	no	—
166	#41×C₆₇	—	4	4	no	—
167	#41×C₆₇	—	4	4	no	—
168	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
169	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
170	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
171	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
172	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
173	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
174	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
175	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
176	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
177	C₂²×C₄×C₆₇	—	2	2	no	—
178	C₂²×C₄×C₆₇	—	2	2	no	—
179	C₂²×C₄×C₆₇	—	4	2	no	—
180	C₂²×C₄×C₆₇	—	4	2	no	—
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	1	yes	characteristic
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	4	yes	—
198	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
199	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
200	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
201	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
202	M₁₆×C₆₇	—	2	8	no	—
203	M₁₆×C₆₇	—	4	4	no	—
204	M₁₆×C₆₇	—	4	8	no	—
205	C₂×C₈×C₆₇	—	2	4	no	—
206	C₂×C₈×C₆₇	—	2	4	no	—
207	C₂×C₈×C₆₇	—	4	4	no	—
208	C₂×C₈×C₆₇	—	4	8	no	—
209	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
210	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
211	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
212	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
213	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	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	2	yes	—
217	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
218	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
219	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
220	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
221	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
222	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
223	C₄⋊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	C₂²×C₄	1	4	yes	—
247	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
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	4	yes	—
252	C₂³×C₆₇	—	2	2	no	—
253	C₂³×C₆₇	—	2	2	no	—
254	C₂³×C₆₇	—	4	1	no	—
255	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
256	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
257	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
258	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
259	Q₈×C₆₇	—	2	4	no	—
260	Q₈×C₆₇	—	4	2	no	—
261	Q₈×C₆₇	—	4	2	no	—
262	D₄×C₆₇	—	2	8	no	—
263	D₄×C₆₇	—	2	8	no	—
264	D₄×C₆₇	—	2	4	no	—
265	D₄×C₆₇	—	4	2	no	—
266	D₄×C₆₇	—	4	2	no	—
267	D₄×C₆₇	—	4	4	no	—
268	C₂×C₄×C₆₇	—	2	2	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	2	no	—
274	C₂×C₄×C₆₇	—	2	2	no	—
275	C₂×C₄×C₆₇	—	2	2	no	—
276	C₂×C₄×C₆₇	—	2	8	no	—
277	C₂×C₄×C₆₇	—	2	1	no	—
278	C₂×C₄×C₆₇	—	2	4	no	—
279	C₂×C₄×C₆₇	—	2	1	no	—
280	C₂×C₄×C₆₇	—	2	1	no	—
281	C₂×C₄×C₆₇	—	2	1	no	—
282	C₂×C₄×C₆₇	—	2	4	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	1	no	—
289	C₂×C₄×C₆₇	—	4	1	no	—
290	C₂×C₄×C₆₇	—	4	4	no	—
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	1	yes	characteristic
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	4	yes	—
303	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
304	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
305	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
306	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
307	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
308	C₈×C₆₇	—	2	4	no	—
309	C₈×C₆₇	—	2	8	no	—
310	C₈×C₆₇	—	2	4	no	—
311	C₂²×C₆₇	—	2	1	no	—
312	C₂²×C₆₇	—	2	1	no	—
313	C₂²×C₆₇	—	2	1	no	—
314	C₂²×C₆₇	—	2	1	no	—
315	C₂²×C₆₇	—	2	1	no	—
316	C₂²×C₆₇	—	2	4	no	—
317	C₂²×C₆₇	—	2	1	no	—
318	C₂²×C₆₇	—	2	4	no	—
319	C₂²×C₆₇	—	2	1	no	—
320	C₂²×C₆₇	—	2	4	no	—
321	C₂²×C₆₇	—	4	4	no	—
322	C₂²×C₆₇	#520	1	2	yes	—
323	C₂²×C₆₇	#568	1	2	yes	—
324	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
325	C₄×C₆₇	—	2	1	no	—
326	C₄×C₆₇	—	2	1	no	—
327	C₄×C₆₇	—	2	4	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	1	no	—
335	C₄×C₆₇	—	2	4	no	—
336	C₄×C₆₇	—	2	4	no	—
337	C₄×C₆₇	#520	1	2	yes	—
338	C₄×C₆₇	#568	1	2	yes	—
339	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
340	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
341	#60269	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
342	C₂×C₆₇	—	2	2	no	—
343	C₂×C₆₇	—	2	2	no	—
344	C₂×C₆₇	—	4	4	no	—
345	C₂×C₆₇	—	4	1	no	—
346	C₂×C₆₇	C₂×#520	1	1	yes	characteristic
347	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
348	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
349	#3444	C₂×C₆₇	1	1	yes	characteristic
350	#2995	C₂×C₆₇	1	4	yes	complemented
351	#2992	C₂×C₆₇	1	4	yes	complemented
352	#2990	C₂×C₆₇	1	4	yes	complemented
353	#2989	C₂×C₆₇	1	4	yes	complemented
354	#2961	C₂×C₆₇	1	2	yes	complemented
355	#2933	C₂×C₆₇	1	4	yes	complemented
356	#2932	C₂×C₆₇	1	2	yes	complemented
357	#2925	C₂×C₆₇	1	2	yes	complemented
358	C₂×#432	C₂×C₆₇	1	1	yes	characteristic, complemented
359	#2906	C₂×C₆₇	1	2	yes	complemented
360	C₂×#423	C₂×C₆₇	1	1	yes	characteristic, complemented
361	C₆₇	#60269	1	1	yes	characteristic, direct product factor, Sylow
362	C₂×#131	C₂²×C₆₇	1	4	yes	—
363	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
364	#533	C₄×C₆₇	1	1	yes	characteristic
365	#533	C₄×C₆₇	1	1	yes	characteristic
366	#533	C₂²×C₆₇	1	1	yes	characteristic
367	#532	C₄×C₆₇	1	1	yes	characteristic
368	#532	C₄×C₆₇	1	1	yes	characteristic
369	#532	C₂²×C₆₇	1	1	yes	characteristic
370	#529	C₄×C₆₇	1	2	yes	—
371	#529	C₄×C₆₇	1	2	yes	—
372	#529	C₄×C₆₇	1	2	yes	—
373	#529	C₄×C₆₇	1	2	yes	—
374	#529	C₂²×C₆₇	1	2	yes	—
375	#529	C₂²×C₆₇	1	2	yes	—
376	#525	C₄×C₆₇	1	2	yes	—
377	#525	C₄×C₆₇	1	2	yes	—
378	#525	C₂²×C₆₇	1	2	yes	—
379	C₄×#41	C₂²×C₆₇	1	2	yes	—
380	C₄×#41	C₂²×C₆₇	1	2	yes	—
381	C₄×#41	C₂²×C₆₇	1	2	yes	—
382	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
383	#446	C₂²×C₆₇	1	8	yes	complemented
384	#446	C₂²×C₆₇	1	8	yes	complemented
385	#436	C₂²×C₆₇	1	8	yes	—
386	#435	C₂²×C₆₇	1	8	yes	—
387	#435	C₂²×C₆₇	1	8	yes	—
388	#435	C₂²×C₆₇	1	8	yes	complemented
389	#435	C₂²×C₆₇	1	8	yes	complemented
390	C₈×D₄	C₂²×C₆₇	1	8	yes	—
391	#433	C₂²×C₆₇	1	8	yes	complemented
392	#432	C₂²×C₆₇	1	2	yes	—
393	#432	C₂²×C₆₇	1	2	yes	—
394	#432	C₂²×C₆₇	1	2	yes	—
395	#432	C₂²×C₆₇	1	2	yes	—
396	#432	C₂²×C₆₇	1	4	yes	complemented
397	#431	C₂²×C₆₇	1	4	yes	complemented
398	#424	C₂²×C₆₇	1	4	yes	—
399	#424	C₂²×C₆₇	1	8	yes	complemented
400	#423	C₂²×C₆₇	1	2	yes	—
401	#423	C₂²×C₆₇	1	2	yes	—
402	#423	C₂²×C₆₇	1	2	yes	—
403	#423	C₂²×C₆₇	1	2	yes	—
404	#423	C₂²×C₆₇	1	4	yes	complemented
405	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
406	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
407	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
408	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
409	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	complemented
410	#408	C₂²×C₆₇	1	4	yes	—
411	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
412	#405	C₂²×C₆₇	1	4	yes	—
413	C₂×#97	C₂²×C₆₇	1	2	yes	—
414	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
415	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
416	C₂×#41	C₂³×C₆₇	1	2	yes	—
417	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
418	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
419	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
420	#131	—	4	8	no	—
421	C₂×M₁₆	—	4	4	no	—
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	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
426	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
427	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
428	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
429	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
430	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
431	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
432	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
433	#126	C₂×C₄×C₆₇	1	4	yes	—
434	#126	C₂×C₄×C₆₇	1	4	yes	—
435	#125	C₂×C₄×C₆₇	1	2	yes	—
436	#125	C₂×C₄×C₆₇	1	2	yes	—
437	#125	C₂×C₄×C₆₇	1	2	yes	—
438	#124	C₂×C₄×C₆₇	1	2	yes	—
439	#124	C₂×C₄×C₆₇	1	2	yes	—
440	#124	C₂×C₄×C₆₇	1	2	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	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
449	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
450	#121	C₂×C₄×C₆₇	1	2	yes	—
451	#121	C₂×C₄×C₆₇	1	2	yes	—
452	#121	C₂×C₄×C₆₇	1	2	yes	—
453	#121	C₂×C₄×C₆₇	1	2	yes	—
454	#121	C₂×C₄×C₆₇	1	2	yes	—
455	#121	C₂×C₄×C₆₇	1	2	yes	—
456	#120	C₂×C₄×C₆₇	1	4	yes	—
457	C₄×Q₈	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	C₂×C₄×C₆₇	1	2	yes	—
461	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
462	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
463	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
464	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
465	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
466	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
467	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
468	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
469	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
470	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
471	C₄×D₄	C₂³×C₆₇	1	4	yes	—
472	C₄×D₄	C₂³×C₆₇	1	4	yes	—
473	C₄×D₄	C₂³×C₆₇	1	2	yes	—
474	C₄×D₄	C₂³×C₆₇	1	2	yes	—
475	#117	C₂×C₄×C₆₇	1	2	yes	—
476	#117	C₂×C₄×C₆₇	1	2	yes	—
477	#117	C₂×C₄×C₆₇	1	4	yes	—
478	#117	C₂×C₄×C₆₇	1	4	yes	—
479	#117	C₂³×C₆₇	1	2	yes	—
480	#117	C₂³×C₆₇	1	4	yes	—
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	C₂×C₄×C₆₇	1	1	yes	characteristic
486	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
487	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
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	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
493	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
494	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
495	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
496	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
497	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
498	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
499	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
500	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
501	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
502	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
503	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
504	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
505	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
506	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
507	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
508	#97	C₂³×C₆₇	1	4	yes	—
509	#97	C₂³×C₆₇	1	4	yes	—
510	#97	C₂³×C₆₇	1	8	yes	—
511	C₄×C₈	C₂³×C₆₇	1	8	yes	—
512	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
513	#41	—	2	4	no	—
514	#41	—	2	4	no	—
515	#41	—	4	4	no	—
516	#41	—	4	4	no	—
517	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
518	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
519	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
520	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
521	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
522	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
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	2	yes	—
526	C₂²×C₄	—	2	2	no	—
527	C₂²×C₄	—	2	2	no	—
528	C₂²×C₄	—	4	2	no	—
529	C₂²×C₄	—	4	2	no	—
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	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	4	yes	—
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	1	yes	characteristic
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	4	yes	—
547	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
548	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
549	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
550	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
551	M₁₆	—	2	8	no	—
552	M₁₆	—	4	4	no	—
553	M₁₆	—	4	8	no	—
554	C₂×C₈	—	2	4	no	—
555	C₂×C₈	—	2	4	no	—
556	C₂×C₈	—	4	4	no	—
557	C₂×C₈	—	4	8	no	—
558	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
559	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
560	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
561	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
562	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
563	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
564	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
565	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
566	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	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	2	yes	—
570	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
571	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
572	C₄⋊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	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
581	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
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	1	yes	characteristic
593	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
594	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
595	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
596	C₄²	C₂⁴×C₆₇	1	4	yes	—
597	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
598	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
599	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
600	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
601	C₂³	—	2	2	no	—
602	C₂³	—	2	2	no	—
603	C₂³	—	4	1	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	Q₈	—	4	2	no	—
611	D₄	—	2	8	no	—
612	D₄	—	2	8	no	—
613	D₄	—	2	4	no	—
614	D₄	—	4	2	no	—
615	D₄	—	4	2	no	—
616	D₄	—	4	4	no	—
617	C₂×C₄	—	2	2	no	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	2	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	8	no	—
626	C₂×C₄	—	2	1	no	—
627	C₂×C₄	—	2	4	no	—
628	C₂×C₄	—	2	1	no	—
629	C₂×C₄	—	2	1	no	—
630	C₂×C₄	—	2	1	no	—
631	C₂×C₄	—	2	4	no	—
632	C₂×C₄	—	4	1	no	—
633	C₂×C₄	—	4	1	no	—
634	C₂×C₄	—	4	1	no	—
635	C₂×C₄	—	4	1	no	—
636	C₂×C₄	—	4	1	no	—
637	C₂×C₄	—	4	1	no	—
638	C₂×C₄	—	4	1	no	—
639	C₂×C₄	—	4	4	no	—
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	1	yes	characteristic
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	4	yes	—
650	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
651	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
652	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
653	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
654	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
655	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
656	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
657	C₈	—	2	4	no	—
658	C₈	—	2	8	no	—
659	C₈	—	2	4	no	—
660	C₂²	—	2	1	no	—
661	C₂²	—	2	1	no	—
662	C₂²	—	2	1	no	—
663	C₂²	—	2	1	no	—
664	C₂²	—	2	1	no	—
665	C₂²	—	2	4	no	—
666	C₂²	—	2	1	no	—
667	C₂²	—	2	4	no	—
668	C₂²	—	2	1	no	—
669	C₂²	—	2	4	no	—
670	C₂²	—	4	4	no	—
671	C₂²	#520×C₆₇	1	2	yes	—
672	C₂²	#568×C₆₇	1	2	yes	—
673	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
674	C₄	—	2	1	no	—
675	C₄	—	2	1	no	—
676	C₄	—	2	1	no	—
677	C₄	—	2	1	no	—
678	C₄	—	2	1	no	—
679	C₄	—	2	1	no	—
680	C₄	—	2	1	no	—
681	C₄	—	2	4	no	—
682	C₄	—	2	4	no	—
683	C₄	—	2	1	no	—
684	C₄	—	2	1	no	—
685	C₄	—	2	4	no	—
686	C₄	#520×C₆₇	1	2	yes	—
687	C₄	#568×C₆₇	1	2	yes	—
688	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
689	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
690	C₂	—	2	2	no	—
691	C₂	—	2	2	no	—
692	C₂	—	4	1	no	—
693	C₂	—	4	4	no	—
694	C₂	C₂×#520×C₆₇	1	1	yes	characteristic
695	C₂	C₂×#568×C₆₇	1	1	yes	characteristic
696	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	C₆₇×#3444	C₂
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×#2992	C₂	semidirect
4. C₆₇×#2990	C₂	semidirect
5. C₆₇×#2989	C₂	semidirect
6. C₆₇×#2961	C₂	semidirect
7. C₆₇×#2933	C₂	semidirect
8. C₆₇×#2932	C₂	semidirect
9. C₆₇×#2925	C₂	semidirect
10. C₂×#432×C₆₇	C₂	semidirect
11. C₆₇×#2906	C₂	semidirect
12. C₂×#423×C₆₇	C₂	semidirect
34. #446×C₆₇	C₂²	semidirect
35. #446×C₆₇	C₂²	semidirect
39. #435×C₆₇	C₂²	semidirect
40. #435×C₆₇	C₂²	semidirect
42. #433×C₆₇	C₂²	semidirect
47. #432×C₆₇	C₂²	semidirect
48. #431×C₆₇	C₂²	semidirect
50. #424×C₆₇	C₂²	semidirect
55. #423×C₆₇	C₂²	semidirect
60. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
341. #60269	C₆₇	direct (N × H)
350. #2995	C₂×C₆₇	semidirect
351. #2992	C₂×C₆₇	semidirect
352. #2990	C₂×C₆₇	semidirect
353. #2989	C₂×C₆₇	semidirect
354. #2961	C₂×C₆₇	semidirect
355. #2933	C₂×C₆₇	semidirect
356. #2932	C₂×C₆₇	semidirect
357. #2925	C₂×C₆₇	semidirect
358. C₂×#432	C₂×C₆₇	semidirect
359. #2906	C₂×C₆₇	semidirect
360. C₂×#423	C₂×C₆₇	semidirect
361. C₆₇	#60269	direct (N × H)
383. #446	C₂²×C₆₇	semidirect
384. #446	C₂²×C₆₇	semidirect
388. #435	C₂²×C₆₇	semidirect
389. #435	C₂²×C₆₇	semidirect
391. #433	C₂²×C₆₇	semidirect
396. #432	C₂²×C₆₇	semidirect
397. #431	C₂²×C₆₇	semidirect
399. #424	C₂²×C₆₇	semidirect
404. #423	C₂²×C₆₇	semidirect
409. 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
C₆₇×#3444	2
C₆₇×#2995	2
C₆₇×#2992	2
C₆₇×#2990	2
C₆₇×#2989	2
C₆₇×#2961	2
C₆₇×#2933	2
C₆₇×#2932	2
C₆₇×#2925	2
C₂×#432×C₆₇	2
C₆₇×#2906	2
C₂×#423×C₆₇	2
#60269	67

Subgroups — C67×#60269 #30739520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60269 #30739520