Subgroups — C₂×C₆₇×#2622 #30713807

Order 17152.

4378 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#120×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2622	C₂	1	16	yes	maximal, direct product factor
4	C₂×#397×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂×#394×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#386×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#385×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#379×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁶×C₆₇	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
12	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
13	C₂×#120×C₆₇	—	2	8	no	—
14	C₂×#120×C₆₇	C₂²	1	8	yes	—
15	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	complemented
16	C₂²×C₂²⋊C₄×C₆₇	—	2	8	no	—
17	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
20	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
23	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
24	#397×C₆₇	C₂²	1	32	yes	complemented
25	#394×C₆₇	C₂²	1	32	yes	complemented
26	#386×C₆₇	C₂²	1	16	yes	complemented
27	#385×C₆₇	C₂²	1	16	yes	complemented
28	#379×C₆₇	C₂²	1	8	yes	complemented
29	C₂×#95×C₆₇	C₂²	1	4	yes	complemented
30	C₂×#95×C₆₇	C₂²	1	4	yes	complemented
31	C₂⁵×C₆₇	—	2	8	no	—
32	C₂⁵×C₆₇	—	4	8	no	—
33	C₂⁵×C₆₇	D₄	1	8	yes	—
34	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
35	C₂⁵×C₆₇	C₂³	1	2	yes	—
36	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
37	C₂⁵×C₆₇	C₂³	1	4	yes	—
38	C₂²×D₄×C₆₇	—	2	8	no	—
39	C₂²×D₄×C₆₇	—	2	4	no	—
40	C₂²×D₄×C₆₇	—	2	8	no	—
41	C₂²×D₄×C₆₇	—	4	8	no	—
42	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
43	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
44	C₂³×C₄×C₆₇	—	2	8	no	—
45	C₂³×C₄×C₆₇	—	4	8	no	—
46	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
47	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
48	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
49	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
51	#120×C₆₇	—	2	32	no	—
52	C₂×C₄⋊C₄×C₆₇	—	4	8	no	—
53	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	complemented
54	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
55	C₂×C₂²⋊C₄×C₆₇	—	2	32	no	—
56	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
57	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
58	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
59	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
60	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
61	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
62	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
63	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
64	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
65	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
66	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	complemented
67	#95×C₆₇	C₂³	1	16	yes	complemented
68	#95×C₆₇	C₂³	1	16	yes	complemented
69	C₂⁴×C₆₇	—	2	8	no	—
70	C₂⁴×C₆₇	—	2	32	no	—
71	C₂⁴×C₆₇	—	2	16	no	—
72	C₂⁴×C₆₇	—	2	8	no	—
73	C₂⁴×C₆₇	—	2	8	no	—
74	C₂⁴×C₆₇	—	2	8	no	—
75	C₂⁴×C₆₇	—	2	2	no	—
76	C₂⁴×C₆₇	—	2	4	no	—
77	C₂⁴×C₆₇	—	2	8	no	—
78	C₂⁴×C₆₇	—	4	32	no	—
79	C₂⁴×C₆₇	—	4	16	no	—
80	C₂⁴×C₆₇	—	4	64	no	—
81	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
82	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
83	C₂⁴×C₆₇	C₂×D₄	1	16	yes	—
84	C₂⁴×C₆₇	#41	1	8	yes	—
85	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
86	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
87	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
88	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
89	C₂×D₄×C₆₇	—	2	16	no	—
90	C₂×D₄×C₆₇	—	2	8	no	—
91	C₂×D₄×C₆₇	—	2	16	no	—
92	C₂×D₄×C₆₇	—	4	16	no	—
93	C₂×D₄×C₆₇	—	4	32	no	—
94	C₂×D₄×C₆₇	—	4	16	no	—
95	C₂²×C₄×C₆₇	—	2	16	no	—
96	C₂²×C₄×C₆₇	—	2	8	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	—	2	4	no	—
99	C₂²×C₄×C₆₇	—	2	8	no	—
100	C₂²×C₄×C₆₇	—	2	4	no	—
101	C₂²×C₄×C₆₇	—	2	8	no	—
102	C₂²×C₄×C₆₇	—	2	8	no	—
103	C₂²×C₄×C₆₇	—	2	8	no	—
104	C₂²×C₄×C₆₇	—	2	8	no	—
105	C₂²×C₄×C₆₇	—	2	8	no	—
106	C₂²×C₄×C₆₇	—	4	16	no	—
107	C₂²×C₄×C₆₇	—	4	32	no	—
108	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
109	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
110	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
111	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
112	C₄⋊C₄×C₆₇	—	4	16	no	—
113	C₂²⋊C₄×C₆₇	—	2	32	no	—
114	C₂²⋊C₄×C₆₇	—	2	16	no	—
115	C₂²⋊C₄×C₆₇	—	2	8	no	—
116	C₂²⋊C₄×C₆₇	—	2	16	no	—
117	C₂²⋊C₄×C₆₇	—	4	16	no	—
118	C₂²⋊C₄×C₆₇	—	4	16	no	—
119	C₂³×C₆₇	—	2	16	no	—
120	C₂³×C₆₇	—	2	32	no	—
121	C₂³×C₆₇	—	2	8	no	—
122	C₂³×C₆₇	—	2	32	no	—
123	C₂³×C₆₇	—	2	16	no	—
124	C₂³×C₆₇	—	2	16	no	—
125	C₂³×C₆₇	—	2	16	no	—
126	C₂³×C₆₇	—	2	8	no	—
127	C₂³×C₆₇	—	2	8	no	—
128	C₂³×C₆₇	—	2	8	no	—
129	C₂³×C₆₇	—	2	2	no	—
130	C₂³×C₆₇	—	2	4	no	—
131	C₂³×C₆₇	—	2	8	no	—
132	C₂³×C₆₇	—	4	32	no	—
133	C₂³×C₆₇	—	4	128	no	—
134	C₂³×C₆₇	—	4	64	no	—
135	C₂³×C₆₇	—	4	32	no	—
136	C₂³×C₆₇	—	4	4	no	—
137	C₂³×C₆₇	—	4	8	no	—
138	C₂³×C₆₇	—	4	16	no	—
139	C₂³×C₆₇	#123	1	16	yes	—
140	C₂³×C₆₇	C₂²×D₄	1	8	yes	—
141	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
142	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
143	C₂³×C₆₇	C₂×#41	1	8	yes	—
144	C₂³×C₆₇	C₂×#41	1	4	yes	—
145	C₂³×C₆₇	H(5,𝔽₂)	1	8	yes	—
146	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
147	D₄×C₆₇	—	4	32	no	—
148	D₄×C₆₇	—	4	16	no	—
149	C₂×C₄×C₆₇	—	2	8	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	4	no	—
152	C₂×C₄×C₆₇	—	2	8	no	—
153	C₂×C₄×C₆₇	—	2	4	no	—
154	C₂×C₄×C₆₇	—	2	8	no	—
155	C₂×C₄×C₆₇	—	2	8	no	—
156	C₂×C₄×C₆₇	—	2	8	no	—
157	C₂×C₄×C₆₇	—	2	8	no	—
158	C₂×C₄×C₆₇	—	2	8	no	—
159	C₂×C₄×C₆₇	—	4	32	no	—
160	C₂×C₄×C₆₇	—	4	4	no	—
161	C₂×C₄×C₆₇	—	4	16	no	—
162	C₂×C₄×C₆₇	—	4	8	no	—
163	C₂×C₄×C₆₇	—	4	16	no	—
164	C₂²×C₆₇	—	2	32	no	—
165	C₂²×C₆₇	—	2	8	no	—
166	C₂²×C₆₇	—	2	16	no	—
167	C₂²×C₆₇	—	2	8	no	—
168	C₂²×C₆₇	—	2	8	no	—
169	C₂²×C₆₇	—	2	8	no	—
170	C₂²×C₆₇	—	4	64	no	—
171	C₂²×C₆₇	—	4	32	no	—
172	C₂²×C₆₇	—	4	16	no	—
173	C₂²×C₆₇	—	4	2	no	—
174	C₂²×C₆₇	—	4	4	no	—
175	C₂²×C₆₇	—	4	8	no	—
176	C₂²×C₆₇	—	8	8	no	—
177	C₂²×C₆₇	C₂×#123	1	8	yes	—
178	C₂²×C₆₇	C₂×#123	1	4	yes	—
179	C₂²×C₆₇	#534	1	8	yes	—
180	C₂²×C₆₇	#538	1	16	yes	—
181	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
182	C₂²×C₆₇	C₂²×#41	1	2	yes	—
183	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	—
184	C₄×C₆₇	—	4	2	no	—
185	C₄×C₆₇	—	4	8	no	—
186	C₄×C₆₇	—	4	4	no	—
187	C₄×C₆₇	—	4	8	no	—
188	C₂×#2622	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂×C₆₇	—	2	8	no	—
190	C₂×C₆₇	—	4	8	no	—
191	C₂×C₆₇	—	8	2	no	—
192	C₂×C₆₇	#2622	1	8	yes	direct product factor
193	C₂×C₆₇	C₂²×#123	1	1	yes	characteristic
194	C₂×C₆₇	C₂×#534	1	2	yes	—
195	C₂×C₆₇	C₂×#538	1	4	yes	—
196	C₂²×#120	C₂×C₆₇	1	1	yes	characteristic
197	C₂³×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
198	#2622	C₂×C₆₇	1	16	yes	direct product factor
199	C₂×#397	C₂×C₆₇	1	4	yes	complemented
200	C₂×#394	C₂×C₆₇	1	4	yes	complemented
201	C₂×#386	C₂×C₆₇	1	2	yes	complemented
202	C₂×#385	C₂×C₆₇	1	2	yes	complemented
203	C₂×#379	C₂×C₆₇	1	1	yes	characteristic, complemented
204	C₆₇	C₂×#2622	1	1	yes	characteristic, direct product factor, Sylow
205	C₂⁶	C₂²×C₆₇	1	1	yes	characteristic
206	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
207	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
208	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
209	C₂×#120	—	2	8	no	—
210	C₂×#120	C₂²×C₆₇	1	8	yes	—
211	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	complemented
212	C₂²×C₂²⋊C₄	—	2	8	no	—
213	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
214	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
215	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
216	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
217	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
218	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
219	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
220	#397	C₂²×C₆₇	1	32	yes	complemented
221	#394	C₂²×C₆₇	1	32	yes	complemented
222	#386	C₂²×C₆₇	1	16	yes	complemented
223	#385	C₂²×C₆₇	1	16	yes	complemented
224	#379	C₂²×C₆₇	1	8	yes	complemented
225	C₂×#95	C₂²×C₆₇	1	4	yes	complemented
226	C₂×#95	C₂²×C₆₇	1	4	yes	complemented
227	C₂⁵	—	2	8	no	—
228	C₂⁵	—	4	8	no	—
229	C₂⁵	D₄×C₆₇	1	8	yes	—
230	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
231	C₂⁵	C₂³×C₆₇	1	2	yes	—
232	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
233	C₂⁵	C₂³×C₆₇	1	4	yes	—
234	C₂²×D₄	—	2	8	no	—
235	C₂²×D₄	—	2	4	no	—
236	C₂²×D₄	—	2	8	no	—
237	C₂²×D₄	—	4	8	no	—
238	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
239	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
240	C₂³×C₄	—	2	8	no	—
241	C₂³×C₄	—	4	8	no	—
242	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
243	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
244	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
245	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
246	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
247	#120	—	2	32	no	—
248	C₂×C₄⋊C₄	—	4	8	no	—
249	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	complemented
250	C₂×C₂²⋊C₄	—	2	16	no	—
251	C₂×C₂²⋊C₄	—	2	32	no	—
252	C₂×C₂²⋊C₄	—	2	8	no	—
253	C₂×C₂²⋊C₄	—	2	4	no	—
254	C₂×C₂²⋊C₄	—	2	8	no	—
255	C₂×C₂²⋊C₄	—	4	8	no	—
256	C₂×C₂²⋊C₄	—	4	8	no	—
257	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
258	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
259	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
260	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
261	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
262	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	complemented
263	#95	C₂³×C₆₇	1	16	yes	complemented
264	#95	C₂³×C₆₇	1	16	yes	complemented
265	C₂⁴	—	2	8	no	—
266	C₂⁴	—	2	32	no	—
267	C₂⁴	—	2	16	no	—
268	C₂⁴	—	2	8	no	—
269	C₂⁴	—	2	8	no	—
270	C₂⁴	—	2	8	no	—
271	C₂⁴	—	2	2	no	—
272	C₂⁴	—	2	4	no	—
273	C₂⁴	—	2	8	no	—
274	C₂⁴	—	4	32	no	—
275	C₂⁴	—	4	16	no	—
276	C₂⁴	—	4	64	no	—
277	C₂⁴	C₂×D₄×C₆₇	1	8	yes	—
278	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
279	C₂⁴	C₂×D₄×C₆₇	1	16	yes	—
280	C₂⁴	#41×C₆₇	1	8	yes	—
281	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
282	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
283	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
284	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
285	C₂×D₄	—	2	16	no	—
286	C₂×D₄	—	2	8	no	—
287	C₂×D₄	—	2	16	no	—
288	C₂×D₄	—	4	16	no	—
289	C₂×D₄	—	4	32	no	—
290	C₂×D₄	—	4	16	no	—
291	C₂²×C₄	—	2	16	no	—
292	C₂²×C₄	—	2	8	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	4	no	—
295	C₂²×C₄	—	2	8	no	—
296	C₂²×C₄	—	2	4	no	—
297	C₂²×C₄	—	2	8	no	—
298	C₂²×C₄	—	2	8	no	—
299	C₂²×C₄	—	2	8	no	—
300	C₂²×C₄	—	2	8	no	—
301	C₂²×C₄	—	2	8	no	—
302	C₂²×C₄	—	4	16	no	—
303	C₂²×C₄	—	4	32	no	—
304	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
305	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
306	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
307	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
308	C₄⋊C₄	—	4	16	no	—
309	C₂²⋊C₄	—	2	32	no	—
310	C₂²⋊C₄	—	2	16	no	—
311	C₂²⋊C₄	—	2	8	no	—
312	C₂²⋊C₄	—	2	16	no	—
313	C₂²⋊C₄	—	4	16	no	—
314	C₂²⋊C₄	—	4	16	no	—
315	C₂³	—	2	16	no	—
316	C₂³	—	2	32	no	—
317	C₂³	—	2	8	no	—
318	C₂³	—	2	32	no	—
319	C₂³	—	2	16	no	—
320	C₂³	—	2	16	no	—
321	C₂³	—	2	16	no	—
322	C₂³	—	2	8	no	—
323	C₂³	—	2	8	no	—
324	C₂³	—	2	8	no	—
325	C₂³	—	2	2	no	—
326	C₂³	—	2	4	no	—
327	C₂³	—	2	8	no	—
328	C₂³	—	4	32	no	—
329	C₂³	—	4	128	no	—
330	C₂³	—	4	64	no	—
331	C₂³	—	4	32	no	—
332	C₂³	—	4	4	no	—
333	C₂³	—	4	8	no	—
334	C₂³	—	4	16	no	—
335	C₂³	#123×C₆₇	1	16	yes	—
336	C₂³	C₂²×D₄×C₆₇	1	8	yes	—
337	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
338	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
339	C₂³	C₂×#41×C₆₇	1	8	yes	—
340	C₂³	C₂×#41×C₆₇	1	4	yes	—
341	C₂³	H(5,𝔽₂)×C₆₇	1	8	yes	—
342	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
343	D₄	—	4	32	no	—
344	D₄	—	4	16	no	—
345	C₂×C₄	—	2	8	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	4	no	—
348	C₂×C₄	—	2	8	no	—
349	C₂×C₄	—	2	4	no	—
350	C₂×C₄	—	2	8	no	—
351	C₂×C₄	—	2	8	no	—
352	C₂×C₄	—	2	8	no	—
353	C₂×C₄	—	2	8	no	—
354	C₂×C₄	—	2	8	no	—
355	C₂×C₄	—	4	32	no	—
356	C₂×C₄	—	4	4	no	—
357	C₂×C₄	—	4	16	no	—
358	C₂×C₄	—	4	8	no	—
359	C₂×C₄	—	4	16	no	—
360	C₂²	—	2	32	no	—
361	C₂²	—	2	8	no	—
362	C₂²	—	2	16	no	—
363	C₂²	—	2	8	no	—
364	C₂²	—	2	8	no	—
365	C₂²	—	2	8	no	—
366	C₂²	—	4	64	no	—
367	C₂²	—	4	32	no	—
368	C₂²	—	4	16	no	—
369	C₂²	—	4	2	no	—
370	C₂²	—	4	4	no	—
371	C₂²	—	4	8	no	—
372	C₂²	—	8	8	no	—
373	C₂²	C₂×#123×C₆₇	1	8	yes	—
374	C₂²	C₂×#123×C₆₇	1	4	yes	—
375	C₂²	#534×C₆₇	1	8	yes	—
376	C₂²	#538×C₆₇	1	16	yes	—
377	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
378	C₂²	C₂²×#41×C₆₇	1	2	yes	—
379	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	4	yes	—
380	C₄	—	4	2	no	—
381	C₄	—	4	8	no	—
382	C₄	—	4	4	no	—
383	C₄	—	4	8	no	—
384	C₂	—	2	8	no	—
385	C₂	—	4	8	no	—
386	C₂	—	8	2	no	—
387	C₂	C₆₇×#2622	1	8	yes	direct product factor
388	C₂	C₂²×#123×C₆₇	1	1	yes	characteristic
389	C₂	C₂×#534×C₆₇	1	2	yes	—
390	C₂	C₂×#538×C₆₇	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂²×#120×C₆₇	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₂²⋊C₄×C₆₇	C₂	semidirect
3. C₆₇×#2622	C₂	direct (N × H)
4. C₂×#397×C₆₇	C₂	semidirect
5. C₂×#394×C₆₇	C₂	semidirect
6. C₂×#386×C₆₇	C₂	semidirect
7. C₂×#385×C₆₇	C₂	semidirect
8. C₂×#379×C₆₇	C₂	semidirect
15. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
22. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
23. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
24. #397×C₆₇	C₂²	semidirect
25. #394×C₆₇	C₂²	semidirect
26. #386×C₆₇	C₂²	semidirect
27. #385×C₆₇	C₂²	semidirect
28. #379×C₆₇	C₂²	semidirect
29. C₂×#95×C₆₇	C₂²	semidirect
30. C₂×#95×C₆₇	C₂²	semidirect
53. C₂×C₄⋊C₄×C₆₇	C₂³	semidirect
66. C₂×C₂²⋊C₄×C₆₇	C₂³	semidirect
67. #95×C₆₇	C₂³	semidirect
68. #95×C₆₇	C₂³	semidirect
188. C₂×#2622	C₆₇	direct (N × H)
192. C₂×C₆₇	#2622	direct (N × H)
197. C₂³×C₂²⋊C₄	C₂×C₆₇	semidirect
198. #2622	C₂×C₆₇	direct (N × H)
199. C₂×#397	C₂×C₆₇	semidirect
200. C₂×#394	C₂×C₆₇	semidirect
201. C₂×#386	C₂×C₆₇	semidirect
202. C₂×#385	C₂×C₆₇	semidirect
203. C₂×#379	C₂×C₆₇	semidirect
204. C₆₇	C₂×#2622	direct (N × H)
211. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
218. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
219. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
220. #397	C₂²×C₆₇	semidirect
221. #394	C₂²×C₆₇	semidirect
222. #386	C₂²×C₆₇	semidirect
223. #385	C₂²×C₆₇	semidirect
224. #379	C₂²×C₆₇	semidirect
225. C₂×#95	C₂²×C₆₇	semidirect
226. C₂×#95	C₂²×C₆₇	semidirect
249. C₂×C₄⋊C₄	C₂³×C₆₇	semidirect
262. C₂×C₂²⋊C₄	C₂³×C₆₇	semidirect
263. #95	C₂³×C₆₇	semidirect
264. #95	C₂³×C₆₇	semidirect
387. C₂	C₆₇×#2622	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#120×C₆₇	2
C₂³×C₂²⋊C₄×C₆₇	2
C₆₇×#2622	2
C₂×#397×C₆₇	2
C₂×#394×C₆₇	2
C₂×#386×C₆₇	2
C₂×#385×C₆₇	2
C₂×#379×C₆₇	2
C₂×#2622	67

Subgroups — C2×C67×#2622 #30713807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2622 #30713807