Subgroups — C₆₇×#35102 #30714353

Order 17152.

2902 conjugacy classes of subgroups.

The subgroup lattice has 624 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, complemented
2	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2848	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2847	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2623	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2613	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2603	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2598	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2562	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2540	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2536	C₂	1	2	yes	maximal, complemented
12	C₆₇×#2535	C₂	1	2	yes	maximal, complemented
13	C₆₇×#2321	C₂	1	4	yes	maximal, complemented
14	C₆₇×#2283	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
17	#529×C₆₇	—	2	8	no	—
18	C₂×#126×C₆₇	C₂²	1	2	yes	complemented
19	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic, complemented
20	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic, complemented
21	C₂×#123×C₆₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
24	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#121×C₆₇	C₂²	1	4	yes	complemented
26	C₂×#120×C₆₇	C₂²	1	4	yes	—
27	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
28	C₂×C₄×D₄×C₆₇	—	2	8	no	—
29	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
34	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
35	C₂×#117×C₆₇	C₂²	1	2	yes	complemented
36	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
37	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
38	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
39	#400×C₆₇	C₂²	1	4	yes	complemented
40	#399×C₆₇	C₂²	1	4	yes	complemented
41	#398×C₆₇	C₂²	1	4	yes	complemented
42	#397×C₆₇	C₂²	1	4	yes	complemented
43	#394×C₆₇	C₂²	1	4	yes	—
44	#394×C₆₇	C₂²	1	4	yes	—
45	#392×C₆₇	C₂²	1	4	yes	—
46	#392×C₆₇	C₂²	1	4	yes	—
47	#390×C₆₇	C₂²	1	8	yes	complemented
48	#388×C₆₇	C₂²	1	8	yes	complemented
49	#388×C₆₇	C₂²	1	8	yes	complemented
50	#386×C₆₇	C₂²	1	4	yes	—
51	#386×C₆₇	C₂²	1	8	yes	complemented
52	#386×C₆₇	C₂²	1	8	yes	complemented
53	#385×C₆₇	C₂²	1	4	yes	—
54	#385×C₆₇	C₂²	1	8	yes	complemented
55	#383×C₆₇	C₂²	1	2	yes	complemented
56	#382×C₆₇	C₂²	1	2	yes	complemented
57	#381×C₆₇	C₂²	1	2	yes	—
58	#380×C₆₇	C₂²	1	2	yes	—
59	#380×C₆₇	C₂²	1	4	yes	complemented
60	#380×C₆₇	C₂²	1	4	yes	complemented
61	#379×C₆₇	C₂²	1	4	yes	—
62	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
63	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
64	#376×C₆₇	C₂²	1	2	yes	complemented
65	C₂⁵×C₆₇	C₂³	1	2	yes	—
66	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₆₇	—	2	8	no	—
68	C₂²×D₄×C₆₇	—	2	4	no	—
69	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
70	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
71	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
72	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
74	C₂³×C₄×C₆₇	—	2	8	no	—
75	C₂³×C₄×C₆₇	—	4	8	no	—
76	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
77	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
80	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
81	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
82	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
83	#126×C₆₇	—	2	8	no	—
84	#125×C₆₇	—	2	4	no	—
85	#124×C₆₇	—	2	4	no	—
86	#123×C₆₇	—	2	8	no	—
87	C₂²⋊Q₈×C₆₇	—	2	4	no	—
88	C₂²⋊Q₈×C₆₇	—	4	8	no	—
89	#121×C₆₇	—	2	4	no	—
90	#121×C₆₇	—	4	8	no	—
91	#120×C₆₇	—	2	16	no	—
92	C₄×Q₈×C₆₇	—	2	4	no	—
93	C₄×D₄×C₆₇	—	2	8	no	—
94	C₄×D₄×C₆₇	—	2	8	no	—
95	C₄×D₄×C₆₇	—	2	8	no	—
96	C₄×D₄×C₆₇	—	2	8	no	—
97	C₄×D₄×C₆₇	—	2	8	no	—
98	C₄×D₄×C₆₇	—	2	4	no	—
99	#117×C₆₇	—	2	8	no	—
100	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
111	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
112	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
118	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
120	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
121	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
122	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
123	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
124	C₂×C₄²×C₆₇	D₄	1	8	yes	complemented
125	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
126	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
127	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
131	#95×C₆₇	C₂³	1	4	yes	—
132	#95×C₆₇	C₂³	1	4	yes	—
133	#95×C₆₇	C₂³	1	4	yes	—
134	#95×C₆₇	C₂³	1	4	yes	—
135	#95×C₆₇	C₂³	1	8	yes	—
136	#95×C₆₇	C₂³	1	8	yes	—
137	C₂⁴×C₆₇	—	2	4	no	—
138	C₂⁴×C₆₇	—	2	4	no	—
139	C₂⁴×C₆₇	—	4	4	no	—
140	C₂⁴×C₆₇	—	4	4	no	—
141	C₂⁴×C₆₇	#41	1	8	yes	—
142	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
143	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
144	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
145	C₂×Q₈×C₆₇	—	2	2	no	—
146	C₂×Q₈×C₆₇	—	2	4	no	—
147	C₂×D₄×C₆₇	—	2	32	no	—
148	C₂×D₄×C₆₇	—	2	8	no	—
149	C₂×D₄×C₆₇	—	2	8	no	—
150	C₂×D₄×C₆₇	—	2	4	no	—
151	C₂×D₄×C₆₇	—	2	8	no	—
152	C₂×D₄×C₆₇	—	2	16	no	—
153	C₂×D₄×C₆₇	—	2	4	no	—
154	C₂×D₄×C₆₇	—	2	4	no	—
155	C₂×D₄×C₆₇	—	2	2	no	—
156	C₂×D₄×C₆₇	—	2	4	no	—
157	C₂×D₄×C₆₇	—	4	8	no	—
158	C₂×D₄×C₆₇	—	4	8	no	—
159	C₂²×C₄×C₆₇	—	2	4	no	—
160	C₂²×C₄×C₆₇	—	2	4	no	—
161	C₂²×C₄×C₆₇	—	2	4	no	—
162	C₂²×C₄×C₆₇	—	2	4	no	—
163	C₂²×C₄×C₆₇	—	2	8	no	—
164	C₂²×C₄×C₆₇	—	2	8	no	—
165	C₂²×C₄×C₆₇	—	2	8	no	—
166	C₂²×C₄×C₆₇	—	2	8	no	—
167	C₂²×C₄×C₆₇	—	4	4	no	—
168	C₂²×C₄×C₆₇	—	4	4	no	—
169	C₂²×C₄×C₆₇	—	4	4	no	—
170	C₂²×C₄×C₆₇	—	4	2	no	—
171	C₂²×C₄×C₆₇	—	4	4	no	—
172	C₂²×C₄×C₆₇	—	4	32	no	—
173	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
174	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
175	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
176	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
179	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
181	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
185	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
186	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
187	C₄⋊C₄×C₆₇	—	2	4	no	—
188	C₄⋊C₄×C₆₇	—	2	4	no	—
189	C₄⋊C₄×C₆₇	—	2	4	no	—
190	C₄⋊C₄×C₆₇	—	2	4	no	—
191	C₄⋊C₄×C₆₇	—	2	2	no	—
192	C₄⋊C₄×C₆₇	—	2	4	no	—
193	C₄⋊C₄×C₆₇	—	2	2	no	—
194	C₄⋊C₄×C₆₇	—	2	4	no	—
195	C₄⋊C₄×C₆₇	—	2	8	no	—
196	C₄⋊C₄×C₆₇	—	4	8	no	—
197	C₂²⋊C₄×C₆₇	—	2	4	no	—
198	C₂²⋊C₄×C₆₇	—	2	16	no	—
199	C₂²⋊C₄×C₆₇	—	2	8	no	—
200	C₂²⋊C₄×C₆₇	—	2	8	no	—
201	C₂²⋊C₄×C₆₇	—	2	8	no	—
202	C₂²⋊C₄×C₆₇	—	2	4	no	—
203	C₂²⋊C₄×C₆₇	—	2	4	no	—
204	C₂²⋊C₄×C₆₇	—	2	4	no	—
205	C₂²⋊C₄×C₆₇	—	2	4	no	—
206	C₂²⋊C₄×C₆₇	—	4	8	no	—
207	C₂²⋊C₄×C₆₇	—	4	8	no	—
208	C₂²⋊C₄×C₆₇	—	4	8	no	—
209	C₄²×C₆₇	—	2	4	no	—
210	C₄²×C₆₇	—	4	4	no	—
211	C₄²×C₆₇	C₂×D₄	1	4	yes	—
212	C₄²×C₆₇	C₂×D₄	1	4	yes	—
213	C₄²×C₆₇	C₂×D₄	1	4	yes	—
214	C₄²×C₆₇	C₂×D₄	1	4	yes	—
215	C₂³×C₆₇	—	2	2	no	—
216	C₂³×C₆₇	—	2	2	no	—
217	C₂³×C₆₇	—	2	16	no	—
218	C₂³×C₆₇	—	2	4	no	—
219	C₂³×C₆₇	—	2	4	no	—
220	C₂³×C₆₇	—	2	4	no	—
221	C₂³×C₆₇	—	2	4	no	—
222	C₂³×C₆₇	—	2	4	no	—
223	C₂³×C₆₇	—	2	4	no	—
224	C₂³×C₆₇	—	2	2	no	—
225	C₂³×C₆₇	—	2	4	no	—
226	C₂³×C₆₇	—	2	4	no	—
227	C₂³×C₆₇	—	2	4	no	—
228	C₂³×C₆₇	—	4	16	no	—
229	C₂³×C₆₇	—	4	8	no	—
230	C₂³×C₆₇	—	4	8	no	—
231	C₂³×C₆₇	—	4	16	no	—
232	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
233	C₂³×C₆₇	C₂×#41	1	8	yes	—
234	C₂³×C₆₇	C₂×#41	1	4	yes	—
235	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
236	Q₈×C₆₇	—	4	4	no	—
237	D₄×C₆₇	—	2	16	no	—
238	D₄×C₆₇	—	2	16	no	—
239	D₄×C₆₇	—	4	8	no	—
240	D₄×C₆₇	—	4	4	no	—
241	D₄×C₆₇	—	8	16	no	—
242	C₂×C₄×C₆₇	—	2	8	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	2	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	8	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	2	2	no	—
251	C₂×C₄×C₆₇	—	2	2	no	—
252	C₂×C₄×C₆₇	—	2	4	no	—
253	C₂×C₄×C₆₇	—	2	4	no	—
254	C₂×C₄×C₆₇	—	2	8	no	—
255	C₂×C₄×C₆₇	—	2	8	no	—
256	C₂×C₄×C₆₇	—	2	4	no	—
257	C₂×C₄×C₆₇	—	2	4	no	—
258	C₂×C₄×C₆₇	—	2	4	no	—
259	C₂×C₄×C₆₇	—	2	8	no	—
260	C₂×C₄×C₆₇	—	2	4	no	—
261	C₂×C₄×C₆₇	—	4	4	no	—
262	C₂×C₄×C₆₇	—	4	8	no	—
263	C₂×C₄×C₆₇	—	4	16	no	—
264	C₂×C₄×C₆₇	—	4	8	no	—
265	C₂×C₄×C₆₇	—	4	16	no	—
266	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
267	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
268	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
269	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
270	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
271	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
272	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
273	C₂²×C₆₇	—	2	2	no	—
274	C₂²×C₆₇	—	2	2	no	—
275	C₂²×C₆₇	—	2	8	no	—
276	C₂²×C₆₇	—	2	8	no	—
277	C₂²×C₆₇	—	2	8	no	—
278	C₂²×C₆₇	—	2	8	no	—
279	C₂²×C₆₇	—	4	32	no	—
280	C₂²×C₆₇	—	4	4	no	—
281	C₂²×C₆₇	—	4	4	no	—
282	C₂²×C₆₇	—	4	4	no	—
283	C₂²×C₆₇	—	4	4	no	—
284	C₂²×C₆₇	—	4	2	no	—
285	C₂²×C₆₇	—	4	2	no	—
286	C₂²×C₆₇	—	4	2	no	—
287	C₂²×C₆₇	—	4	8	no	—
288	C₂²×C₆₇	#525	1	4	yes	—
289	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
290	C₂²×C₆₇	C₂²×#41	1	2	yes	—
291	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
292	C₂²×C₆₇	#585	1	2	yes	—
293	C₄×C₆₇	—	2	2	no	—
294	C₄×C₆₇	—	2	2	no	—
295	C₄×C₆₇	—	2	2	no	—
296	C₄×C₆₇	—	2	2	no	—
297	C₄×C₆₇	—	2	2	no	—
298	C₄×C₆₇	—	4	2	no	—
299	C₄×C₆₇	—	4	4	no	—
300	C₄×C₆₇	—	4	2	no	—
301	C₄×C₆₇	—	4	2	no	—
302	C₄×C₆₇	—	4	8	no	—
303	C₄×C₆₇	#535	1	4	yes	—
304	#35102	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
305	C₂×C₆₇	—	2	4	no	—
306	C₂×C₆₇	—	4	8	no	—
307	C₂×C₆₇	—	8	2	no	—
308	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
309	C₂×C₆₇	C₂×#535	1	1	yes	characteristic
310	C₂×C₆₇	#3449	1	1	yes	characteristic
311	C₂×C₆₇	#3505	1	4	yes	—
312	C₂×#529	C₂×C₆₇	1	1	yes	characteristic, complemented
313	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
314	#2848	C₂×C₆₇	1	2	yes	complemented
315	#2847	C₂×C₆₇	1	2	yes	complemented
316	#2623	C₂×C₆₇	1	2	yes	complemented
317	#2613	C₂×C₆₇	1	2	yes	complemented
318	#2603	C₂×C₆₇	1	4	yes	complemented
319	#2598	C₂×C₆₇	1	4	yes	complemented
320	#2562	C₂×C₆₇	1	2	yes	complemented
321	#2540	C₂×C₆₇	1	2	yes	complemented
322	#2536	C₂×C₆₇	1	2	yes	complemented
323	#2535	C₂×C₆₇	1	2	yes	complemented
324	#2321	C₂×C₆₇	1	4	yes	complemented
325	#2283	C₂×C₆₇	1	1	yes	characteristic, complemented
326	C₆₇	#35102	1	1	yes	characteristic, direct product factor, Sylow
327	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
328	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
329	#529	—	2	8	no	—
330	C₂×#126	C₂²×C₆₇	1	2	yes	complemented
331	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic, complemented
332	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic, complemented
333	C₂×#123	C₂²×C₆₇	1	2	yes	—
334	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
335	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
336	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
337	C₂×#121	C₂²×C₆₇	1	4	yes	complemented
338	C₂×#120	C₂²×C₆₇	1	4	yes	—
339	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
340	C₂×C₄×D₄	—	2	8	no	—
341	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
342	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
343	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
344	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
345	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
346	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
347	C₂×#117	C₂²×C₆₇	1	2	yes	complemented
348	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
349	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
350	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
351	#400	C₂²×C₆₇	1	4	yes	complemented
352	#399	C₂²×C₆₇	1	4	yes	complemented
353	#398	C₂²×C₆₇	1	4	yes	complemented
354	#397	C₂²×C₆₇	1	4	yes	complemented
355	#394	C₂²×C₆₇	1	4	yes	—
356	#394	C₂²×C₆₇	1	4	yes	—
357	#392	C₂²×C₆₇	1	4	yes	—
358	#392	C₂²×C₆₇	1	4	yes	—
359	#390	C₂²×C₆₇	1	8	yes	complemented
360	#388	C₂²×C₆₇	1	8	yes	complemented
361	#388	C₂²×C₆₇	1	8	yes	complemented
362	#386	C₂²×C₆₇	1	4	yes	—
363	#386	C₂²×C₆₇	1	8	yes	complemented
364	#386	C₂²×C₆₇	1	8	yes	complemented
365	#385	C₂²×C₆₇	1	4	yes	—
366	#385	C₂²×C₆₇	1	8	yes	complemented
367	#383	C₂²×C₆₇	1	2	yes	complemented
368	#382	C₂²×C₆₇	1	2	yes	complemented
369	#381	C₂²×C₆₇	1	2	yes	—
370	#380	C₂²×C₆₇	1	2	yes	—
371	#380	C₂²×C₆₇	1	4	yes	complemented
372	#380	C₂²×C₆₇	1	4	yes	complemented
373	#379	C₂²×C₆₇	1	4	yes	—
374	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
375	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
376	#376	C₂²×C₆₇	1	2	yes	complemented
377	C₂⁵	C₂³×C₆₇	1	2	yes	—
378	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
379	C₂²×D₄	—	2	8	no	—
380	C₂²×D₄	—	2	4	no	—
381	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
382	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
383	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
384	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
385	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
386	C₂³×C₄	—	2	8	no	—
387	C₂³×C₄	—	4	8	no	—
388	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
389	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
390	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
391	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
392	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
393	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
394	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
395	#126	—	2	8	no	—
396	#125	—	2	4	no	—
397	#124	—	2	4	no	—
398	#123	—	2	8	no	—
399	C₂²⋊Q₈	—	2	4	no	—
400	C₂²⋊Q₈	—	4	8	no	—
401	#121	—	2	4	no	—
402	#121	—	4	8	no	—
403	#120	—	2	16	no	—
404	C₄×Q₈	—	2	4	no	—
405	C₄×D₄	—	2	8	no	—
406	C₄×D₄	—	2	8	no	—
407	C₄×D₄	—	2	8	no	—
408	C₄×D₄	—	2	8	no	—
409	C₄×D₄	—	2	8	no	—
410	C₄×D₄	—	2	4	no	—
411	#117	—	2	8	no	—
412	C₂×C₄⋊C₄	—	2	4	no	—
413	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
414	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
415	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
416	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
417	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
418	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
419	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
420	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
421	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
422	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
423	C₂×C₂²⋊C₄	—	2	8	no	—
424	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
425	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
426	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
427	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
428	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
431	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
432	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
433	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
434	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
435	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
436	C₂×C₄²	D₄×C₆₇	1	8	yes	complemented
437	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
438	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
439	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
440	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
441	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
442	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
443	#95	C₂³×C₆₇	1	4	yes	—
444	#95	C₂³×C₆₇	1	4	yes	—
445	#95	C₂³×C₆₇	1	4	yes	—
446	#95	C₂³×C₆₇	1	4	yes	—
447	#95	C₂³×C₆₇	1	8	yes	—
448	#95	C₂³×C₆₇	1	8	yes	—
449	C₂⁴	—	2	4	no	—
450	C₂⁴	—	2	4	no	—
451	C₂⁴	—	4	4	no	—
452	C₂⁴	—	4	4	no	—
453	C₂⁴	#41×C₆₇	1	8	yes	—
454	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
455	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
456	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
457	C₂×Q₈	—	2	2	no	—
458	C₂×Q₈	—	2	4	no	—
459	C₂×D₄	—	2	32	no	—
460	C₂×D₄	—	2	8	no	—
461	C₂×D₄	—	2	8	no	—
462	C₂×D₄	—	2	4	no	—
463	C₂×D₄	—	2	8	no	—
464	C₂×D₄	—	2	16	no	—
465	C₂×D₄	—	2	4	no	—
466	C₂×D₄	—	2	4	no	—
467	C₂×D₄	—	2	2	no	—
468	C₂×D₄	—	2	4	no	—
469	C₂×D₄	—	4	8	no	—
470	C₂×D₄	—	4	8	no	—
471	C₂²×C₄	—	2	4	no	—
472	C₂²×C₄	—	2	4	no	—
473	C₂²×C₄	—	2	4	no	—
474	C₂²×C₄	—	2	4	no	—
475	C₂²×C₄	—	2	8	no	—
476	C₂²×C₄	—	2	8	no	—
477	C₂²×C₄	—	2	8	no	—
478	C₂²×C₄	—	2	8	no	—
479	C₂²×C₄	—	4	4	no	—
480	C₂²×C₄	—	4	4	no	—
481	C₂²×C₄	—	4	4	no	—
482	C₂²×C₄	—	4	2	no	—
483	C₂²×C₄	—	4	4	no	—
484	C₂²×C₄	—	4	32	no	—
485	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
486	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
487	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
488	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
489	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
490	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
491	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
492	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
493	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
494	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
495	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
496	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
497	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
498	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
499	C₄⋊C₄	—	2	4	no	—
500	C₄⋊C₄	—	2	4	no	—
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	—	2	2	no	—
504	C₄⋊C₄	—	2	4	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	2	8	no	—
508	C₄⋊C₄	—	4	8	no	—
509	C₂²⋊C₄	—	2	4	no	—
510	C₂²⋊C₄	—	2	16	no	—
511	C₂²⋊C₄	—	2	8	no	—
512	C₂²⋊C₄	—	2	8	no	—
513	C₂²⋊C₄	—	2	8	no	—
514	C₂²⋊C₄	—	2	4	no	—
515	C₂²⋊C₄	—	2	4	no	—
516	C₂²⋊C₄	—	2	4	no	—
517	C₂²⋊C₄	—	2	4	no	—
518	C₂²⋊C₄	—	4	8	no	—
519	C₂²⋊C₄	—	4	8	no	—
520	C₂²⋊C₄	—	4	8	no	—
521	C₄²	—	2	4	no	—
522	C₄²	—	4	4	no	—
523	C₄²	C₂×D₄×C₆₇	1	4	yes	—
524	C₄²	C₂×D₄×C₆₇	1	4	yes	—
525	C₄²	C₂×D₄×C₆₇	1	4	yes	—
526	C₄²	C₂×D₄×C₆₇	1	4	yes	—
527	C₂³	—	2	2	no	—
528	C₂³	—	2	2	no	—
529	C₂³	—	2	16	no	—
530	C₂³	—	2	4	no	—
531	C₂³	—	2	4	no	—
532	C₂³	—	2	4	no	—
533	C₂³	—	2	4	no	—
534	C₂³	—	2	4	no	—
535	C₂³	—	2	4	no	—
536	C₂³	—	2	2	no	—
537	C₂³	—	2	4	no	—
538	C₂³	—	2	4	no	—
539	C₂³	—	2	4	no	—
540	C₂³	—	4	16	no	—
541	C₂³	—	4	8	no	—
542	C₂³	—	4	8	no	—
543	C₂³	—	4	16	no	—
544	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
545	C₂³	C₂×#41×C₆₇	1	8	yes	—
546	C₂³	C₂×#41×C₆₇	1	4	yes	—
547	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
548	Q₈	—	4	4	no	—
549	D₄	—	2	16	no	—
550	D₄	—	2	16	no	—
551	D₄	—	4	8	no	—
552	D₄	—	4	4	no	—
553	D₄	—	8	16	no	—
554	C₂×C₄	—	2	8	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	8	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	2	8	no	—
567	C₂×C₄	—	2	8	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	8	no	—
572	C₂×C₄	—	2	4	no	—
573	C₂×C₄	—	4	4	no	—
574	C₂×C₄	—	4	16	no	—
575	C₂×C₄	—	4	8	no	—
576	C₂×C₄	—	4	8	no	—
577	C₂×C₄	—	4	16	no	—
578	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
579	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
580	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
581	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
582	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
583	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
584	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
585	C₂²	—	2	2	no	—
586	C₂²	—	2	2	no	—
587	C₂²	—	2	8	no	—
588	C₂²	—	2	8	no	—
589	C₂²	—	2	8	no	—
590	C₂²	—	2	8	no	—
591	C₂²	—	4	32	no	—
592	C₂²	—	4	4	no	—
593	C₂²	—	4	4	no	—
594	C₂²	—	4	4	no	—
595	C₂²	—	4	4	no	—
596	C₂²	—	4	2	no	—
597	C₂²	—	4	2	no	—
598	C₂²	—	4	2	no	—
599	C₂²	—	4	8	no	—
600	C₂²	#525×C₆₇	1	4	yes	—
601	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
602	C₂²	C₂²×#41×C₆₇	1	2	yes	—
603	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
604	C₂²	#585×C₆₇	1	2	yes	—
605	C₄	—	2	2	no	—
606	C₄	—	2	2	no	—
607	C₄	—	2	2	no	—
608	C₄	—	2	2	no	—
609	C₄	—	2	2	no	—
610	C₄	—	4	2	no	—
611	C₄	—	4	4	no	—
612	C₄	—	4	2	no	—
613	C₄	—	4	2	no	—
614	C₄	—	4	8	no	—
615	C₄	#535×C₆₇	1	4	yes	—
616	C₂	—	2	4	no	—
617	C₂	—	4	8	no	—
618	C₂	—	8	2	no	—
619	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
620	C₂	C₂×#535×C₆₇	1	1	yes	characteristic
621	C₂	C₆₇×#3449	1	1	yes	characteristic
622	C₂	C₆₇×#3505	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#529×C₆₇	C₂	semidirect
2. C₂²×C₄×D₄×C₆₇	C₂	semidirect
3. C₆₇×#2848	C₂	semidirect
4. C₆₇×#2847	C₂	semidirect
5. C₆₇×#2623	C₂	semidirect
6. C₆₇×#2613	C₂	semidirect
7. C₆₇×#2603	C₂	semidirect
8. C₆₇×#2598	C₂	semidirect
9. C₆₇×#2562	C₂	semidirect
10. C₆₇×#2540	C₂	semidirect
11. C₆₇×#2536	C₂	semidirect
12. C₆₇×#2535	C₂	semidirect
13. C₆₇×#2321	C₂	semidirect
14. C₆₇×#2283	C₂	semidirect
18. C₂×#126×C₆₇	C₂²	semidirect
19. C₂×#125×C₆₇	C₂²	semidirect
20. C₂×#124×C₆₇	C₂²	semidirect
23. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
25. C₂×#121×C₆₇	C₂²	semidirect
27. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
34. C₂×C₄×D₄×C₆₇	C₂²	semidirect
35. C₂×#117×C₆₇	C₂²	semidirect
39. #400×C₆₇	C₂²	semidirect
40. #399×C₆₇	C₂²	semidirect
41. #398×C₆₇	C₂²	semidirect
42. #397×C₆₇	C₂²	semidirect
47. #390×C₆₇	C₂²	semidirect
48. #388×C₆₇	C₂²	semidirect
49. #388×C₆₇	C₂²	semidirect
51. #386×C₆₇	C₂²	semidirect
52. #386×C₆₇	C₂²	semidirect
54. #385×C₆₇	C₂²	semidirect
55. #383×C₆₇	C₂²	semidirect
56. #382×C₆₇	C₂²	semidirect
59. #380×C₆₇	C₂²	semidirect
60. #380×C₆₇	C₂²	semidirect
62. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
64. #376×C₆₇	C₂²	semidirect
124. C₂×C₄²×C₆₇	D₄	semidirect
304. #35102	C₆₇	direct (N × H)
312. C₂×#529	C₂×C₆₇	semidirect
313. C₂²×C₄×D₄	C₂×C₆₇	semidirect
314. #2848	C₂×C₆₇	semidirect
315. #2847	C₂×C₆₇	semidirect
316. #2623	C₂×C₆₇	semidirect
317. #2613	C₂×C₆₇	semidirect
318. #2603	C₂×C₆₇	semidirect
319. #2598	C₂×C₆₇	semidirect
320. #2562	C₂×C₆₇	semidirect
321. #2540	C₂×C₆₇	semidirect
322. #2536	C₂×C₆₇	semidirect
323. #2535	C₂×C₆₇	semidirect
324. #2321	C₂×C₆₇	semidirect
325. #2283	C₂×C₆₇	semidirect
326. C₆₇	#35102	direct (N × H)
330. C₂×#126	C₂²×C₆₇	semidirect
331. C₂×#125	C₂²×C₆₇	semidirect
332. C₂×#124	C₂²×C₆₇	semidirect
335. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
337. C₂×#121	C₂²×C₆₇	semidirect
339. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
346. C₂×C₄×D₄	C₂²×C₆₇	semidirect
347. C₂×#117	C₂²×C₆₇	semidirect
351. #400	C₂²×C₆₇	semidirect
352. #399	C₂²×C₆₇	semidirect
353. #398	C₂²×C₆₇	semidirect
354. #397	C₂²×C₆₇	semidirect
359. #390	C₂²×C₆₇	semidirect
360. #388	C₂²×C₆₇	semidirect
361. #388	C₂²×C₆₇	semidirect
363. #386	C₂²×C₆₇	semidirect
364. #386	C₂²×C₆₇	semidirect
366. #385	C₂²×C₆₇	semidirect
367. #383	C₂²×C₆₇	semidirect
368. #382	C₂²×C₆₇	semidirect
371. #380	C₂²×C₆₇	semidirect
372. #380	C₂²×C₆₇	semidirect
374. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
376. #376	C₂²×C₆₇	semidirect
436. C₂×C₄²	D₄×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
C₂²×C₄×D₄×C₆₇	2
C₆₇×#2848	2
C₆₇×#2847	2
C₆₇×#2623	2
C₆₇×#2613	2
C₆₇×#2603	2
C₆₇×#2598	2
C₆₇×#2562	2
C₆₇×#2540	2
C₆₇×#2536	2
C₆₇×#2535	2
C₆₇×#2321	2
C₆₇×#2283	2
#35102	67

Subgroups — C67×#35102 #30714353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#35102 #30714353