Subgroups — C₂×C₆₇×#2564 #30714082

Order 17152.

3374 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2564	C₂	1	16	yes	maximal, direct product factor
4	C₂×#400×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#399×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#395×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#393×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#387×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×#386×C₆₇	C₂	1	2	yes	maximal
10	C₂×#385×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₂×#380×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂²×#95×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈×C₆₇	—	2	8	no	—
19	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	8	yes	complemented
20	C₂×C₄×D₄×C₆₇	—	2	8	no	—
21	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	—
22	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
29	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
30	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
31	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
32	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
33	#400×C₆₇	C₂²	1	8	yes	complemented
34	#399×C₆₇	C₂²	1	8	yes	complemented
35	#395×C₆₇	C₂²	1	8	yes	complemented
36	#393×C₆₇	C₂²	1	8	yes	complemented
37	#387×C₆₇	C₂²	1	16	yes	complemented
38	#386×C₆₇	C₂²	1	16	yes	—
39	#385×C₆₇	C₂²	1	16	yes	complemented
40	#380×C₆₇	C₂²	1	16	yes	complemented
41	C₂×#95×C₆₇	C₂²	1	2	yes	—
42	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×#95×C₆₇	C₂²	1	2	yes	—
45	C₂×#95×C₆₇	C₂²	1	2	yes	—
46	C₂×#95×C₆₇	C₂²	1	2	yes	—
47	C₂×#95×C₆₇	C₂²	1	2	yes	—
48	C₂×#95×C₆₇	C₂²	1	8	yes	complemented
49	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
50	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×Q₈×C₆₇	—	2	4	no	—
52	C₂²×Q₈×C₆₇	—	4	4	no	—
53	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
54	C₂²×D₄×C₆₇	—	2	4	no	—
55	C₂²×D₄×C₆₇	—	2	4	no	—
56	C₂²×D₄×C₆₇	Q₈	1	8	yes	complemented
57	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
58	C₂³×C₄×C₆₇	—	2	8	no	—
59	C₂³×C₄×C₆₇	—	2	4	no	—
60	C₂³×C₄×C₆₇	—	2	8	no	—
61	C₂³×C₄×C₆₇	—	4	4	no	—
62	C₂³×C₄×C₆₇	—	4	4	no	—
63	C₂³×C₄×C₆₇	—	4	4	no	—
64	C₂³×C₄×C₆₇	D₄	1	8	yes	—
65	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
67	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
68	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
70	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
71	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
75	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
76	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
77	C₂²⋊Q₈×C₆₇	—	2	32	no	—
78	C₄×D₄×C₆₇	—	2	32	no	—
79	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
80	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
81	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
82	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
83	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
84	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
89	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
90	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
91	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
92	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
93	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
94	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
97	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
98	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
99	C₂×C₄²×C₆₇	—	2	4	no	—
100	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
101	#95×C₆₇	C₂³	1	8	yes	—
102	#95×C₆₇	C₂³	1	4	yes	—
103	#95×C₆₇	C₂³	1	4	yes	—
104	#95×C₆₇	C₂³	1	8	yes	—
105	#95×C₆₇	C₂³	1	8	yes	—
106	#95×C₆₇	C₂³	1	8	yes	—
107	#95×C₆₇	C₂³	1	8	yes	—
108	C₂⁴×C₆₇	—	2	4	no	—
109	C₂⁴×C₆₇	—	2	4	no	—
110	C₂⁴×C₆₇	—	2	2	no	—
111	C₂⁴×C₆₇	—	2	4	no	—
112	C₂⁴×C₆₇	—	2	2	no	—
113	C₂⁴×C₆₇	—	2	4	no	—
114	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
115	C₂⁴×C₆₇	C₂×Q₈	1	4	yes	—
116	C₂⁴×C₆₇	C₂×Q₈	1	4	yes	—
117	C₂⁴×C₆₇	#41	1	4	yes	—
118	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
119	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
120	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
121	C₂×Q₈×C₆₇	—	2	8	no	—
122	C₂×Q₈×C₆₇	—	4	8	no	—
123	C₂×Q₈×C₆₇	—	4	16	no	—
124	C₂×D₄×C₆₇	—	2	32	no	—
125	C₂×D₄×C₆₇	—	2	8	no	—
126	C₂×D₄×C₆₇	—	2	8	no	—
127	C₂×D₄×C₆₇	C₂×Q₈	1	16	yes	complemented
128	C₂²×C₄×C₆₇	—	2	2	no	—
129	C₂²×C₄×C₆₇	—	2	4	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	4	no	—
132	C₂²×C₄×C₆₇	—	2	4	no	—
133	C₂²×C₄×C₆₇	—	2	4	no	—
134	C₂²×C₄×C₆₇	—	2	2	no	—
135	C₂²×C₄×C₆₇	—	2	4	no	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	—	2	4	no	—
138	C₂²×C₄×C₆₇	—	2	4	no	—
139	C₂²×C₄×C₆₇	—	2	4	no	—
140	C₂²×C₄×C₆₇	—	2	4	no	—
141	C₂²×C₄×C₆₇	—	2	16	no	—
142	C₂²×C₄×C₆₇	—	2	4	no	—
143	C₂²×C₄×C₆₇	—	2	4	no	—
144	C₂²×C₄×C₆₇	—	2	4	no	—
145	C₂²×C₄×C₆₇	—	2	4	no	—
146	C₂²×C₄×C₆₇	—	2	4	no	—
147	C₂²×C₄×C₆₇	—	2	4	no	—
148	C₂²×C₄×C₆₇	—	2	8	no	—
149	C₂²×C₄×C₆₇	—	2	16	no	—
150	C₂²×C₄×C₆₇	—	4	16	no	—
151	C₂²×C₄×C₆₇	—	4	8	no	—
152	C₂²×C₄×C₆₇	—	4	8	no	—
153	C₂²×C₄×C₆₇	—	4	8	no	—
154	C₂²×C₄×C₆₇	—	4	32	no	—
155	C₂²×C₄×C₆₇	—	4	16	no	—
156	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
157	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
158	C₂²×C₄×C₆₇	C₂×D₄	1	16	yes	—
159	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
160	C₂²×C₄×C₆₇	#41	1	8	yes	—
161	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
163	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
170	C₄⋊C₄×C₆₇	—	2	8	no	—
171	C₄⋊C₄×C₆₇	—	2	8	no	—
172	C₄⋊C₄×C₆₇	—	2	16	no	—
173	C₄⋊C₄×C₆₇	—	2	16	no	—
174	C₄⋊C₄×C₆₇	—	4	8	no	—
175	C₂²⋊C₄×C₆₇	—	2	8	no	—
176	C₂²⋊C₄×C₆₇	—	2	16	no	—
177	C₂²⋊C₄×C₆₇	—	2	8	no	—
178	C₂²⋊C₄×C₆₇	—	2	16	no	—
179	C₂²⋊C₄×C₆₇	—	4	8	no	—
180	C₄²×C₆₇	—	2	8	no	—
181	C₂³×C₆₇	—	2	8	no	—
182	C₂³×C₆₇	—	2	8	no	—
183	C₂³×C₆₇	—	2	16	no	—
184	C₂³×C₆₇	—	2	16	no	—
185	C₂³×C₆₇	—	2	4	no	—
186	C₂³×C₆₇	—	2	4	no	—
187	C₂³×C₆₇	—	2	8	no	—
188	C₂³×C₆₇	—	2	16	no	—
189	C₂³×C₆₇	—	2	2	no	—
190	C₂³×C₆₇	—	2	4	no	—
191	C₂³×C₆₇	—	2	2	no	—
192	C₂³×C₆₇	—	2	4	no	—
193	C₂³×C₆₇	—	4	4	no	—
194	C₂³×C₆₇	—	4	4	no	—
195	C₂³×C₆₇	—	4	8	no	—
196	C₂³×C₆₇	C₂²⋊Q₈	1	16	yes	—
197	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
198	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
199	C₂³×C₆₇	C₂²×Q₈	1	4	yes	—
200	C₂³×C₆₇	C₂²×Q₈	1	4	yes	—
201	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
202	C₂³×C₆₇	C₂×#41	1	4	yes	—
203	C₂³×C₆₇	C₂×#41	1	2	yes	—
204	C₂³×C₆₇	C₂×#41	1	4	yes	—
205	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
206	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
207	Q₈×C₆₇	—	4	16	no	—
208	D₄×C₆₇	—	2	32	no	—
209	C₂×C₄×C₆₇	—	2	8	no	—
210	C₂×C₄×C₆₇	—	2	8	no	—
211	C₂×C₄×C₆₇	—	2	8	no	—
212	C₂×C₄×C₆₇	—	2	16	no	—
213	C₂×C₄×C₆₇	—	2	2	no	—
214	C₂×C₄×C₆₇	—	2	4	no	—
215	C₂×C₄×C₆₇	—	2	2	no	—
216	C₂×C₄×C₆₇	—	2	4	no	—
217	C₂×C₄×C₆₇	—	2	4	no	—
218	C₂×C₄×C₆₇	—	2	4	no	—
219	C₂×C₄×C₆₇	—	2	2	no	—
220	C₂×C₄×C₆₇	—	2	4	no	—
221	C₂×C₄×C₆₇	—	2	2	no	—
222	C₂×C₄×C₆₇	—	2	4	no	—
223	C₂×C₄×C₆₇	—	2	4	no	—
224	C₂×C₄×C₆₇	—	2	4	no	—
225	C₂×C₄×C₆₇	—	2	4	no	—
226	C₂×C₄×C₆₇	—	2	4	no	—
227	C₂×C₄×C₆₇	—	2	4	no	—
228	C₂×C₄×C₆₇	—	2	4	no	—
229	C₂×C₄×C₆₇	—	2	4	no	—
230	C₂×C₄×C₆₇	—	2	4	no	—
231	C₂×C₄×C₆₇	—	2	4	no	—
232	C₂×C₄×C₆₇	—	4	16	no	—
233	C₂×C₄×C₆₇	—	4	32	no	—
234	C₂×C₄×C₆₇	—	4	16	no	—
235	C₂×C₄×C₆₇	—	4	4	no	—
236	C₂×C₄×C₆₇	—	4	8	no	—
237	C₂×C₄×C₆₇	—	4	4	no	—
238	C₂×C₄×C₆₇	—	4	8	no	—
239	C₂×C₄×C₆₇	—	4	8	no	—
240	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
241	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
242	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
243	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
244	C₂²×C₆₇	—	2	32	no	—
245	C₂²×C₆₇	—	2	4	no	—
246	C₂²×C₆₇	—	2	4	no	—
247	C₂²×C₆₇	—	2	8	no	—
248	C₂²×C₆₇	—	2	8	no	—
249	C₂²×C₆₇	—	2	4	no	—
250	C₂²×C₆₇	—	2	8	no	—
251	C₂²×C₆₇	—	4	16	no	—
252	C₂²×C₆₇	—	4	2	no	—
253	C₂²×C₆₇	—	4	2	no	—
254	C₂²×C₆₇	—	4	4	no	—
255	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	8	yes	—
256	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	4	yes	—
257	C₂²×C₆₇	#532	1	4	yes	—
258	C₂²×C₆₇	#534	1	4	yes	—
259	C₂²×C₆₇	#551	1	8	yes	—
260	C₂²×C₆₇	#554	1	8	yes	—
261	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
262	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
263	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
264	C₂²×C₆₇	C₂²×#41	1	2	yes	—
265	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
266	C₄×C₆₇	—	2	4	no	—
267	C₄×C₆₇	—	2	4	no	—
268	C₄×C₆₇	—	2	4	no	—
269	C₄×C₆₇	—	2	8	no	—
270	C₄×C₆₇	—	4	2	no	—
271	C₄×C₆₇	—	4	4	no	—
272	C₄×C₆₇	—	4	2	no	—
273	C₄×C₆₇	—	4	4	no	—
274	C₄×C₆₇	—	4	4	no	—
275	C₂×#2564	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
276	C₂×C₆₇	—	2	8	no	—
277	C₂×C₆₇	—	4	4	no	—
278	C₂×C₆₇	#2564	1	8	yes	direct product factor
279	C₂×C₆₇	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
280	C₂×C₆₇	C₂×#532	1	1	yes	characteristic
281	C₂×C₆₇	C₂×#534	1	1	yes	characteristic
282	C₂×C₆₇	C₂×#551	1	2	yes	—
283	C₂×C₆₇	C₂×#554	1	2	yes	—
284	C₂²×C₂²⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
285	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
286	#2564	C₂×C₆₇	1	16	yes	direct product factor
287	C₂×#400	C₂×C₆₇	1	1	yes	characteristic, complemented
288	C₂×#399	C₂×C₆₇	1	1	yes	characteristic, complemented
289	C₂×#395	C₂×C₆₇	1	1	yes	characteristic, complemented
290	C₂×#393	C₂×C₆₇	1	1	yes	characteristic, complemented
291	C₂×#387	C₂×C₆₇	1	2	yes	complemented
292	C₂×#386	C₂×C₆₇	1	2	yes	—
293	C₂×#385	C₂×C₆₇	1	2	yes	complemented
294	C₂×#380	C₂×C₆₇	1	2	yes	complemented
295	C₂²×#95	C₂×C₆₇	1	1	yes	characteristic, complemented
296	C₆₇	C₂×#2564	1	1	yes	characteristic, direct product factor, Sylow
297	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
298	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
299	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
300	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
301	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
302	C₂×C₂²⋊Q₈	—	2	8	no	—
303	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	8	yes	complemented
304	C₂×C₄×D₄	—	2	8	no	—
305	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
306	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
307	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
308	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
309	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
310	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
311	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
312	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
313	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
314	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
315	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
316	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
317	#400	C₂²×C₆₇	1	8	yes	complemented
318	#399	C₂²×C₆₇	1	8	yes	complemented
319	#395	C₂²×C₆₇	1	8	yes	complemented
320	#393	C₂²×C₆₇	1	8	yes	complemented
321	#387	C₂²×C₆₇	1	16	yes	complemented
322	#386	C₂²×C₆₇	1	16	yes	—
323	#385	C₂²×C₆₇	1	16	yes	complemented
324	#380	C₂²×C₆₇	1	16	yes	complemented
325	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
326	C₂×#95	C₂²×C₆₇	1	2	yes	—
327	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
328	C₂×#95	C₂²×C₆₇	1	2	yes	—
329	C₂×#95	C₂²×C₆₇	1	2	yes	—
330	C₂×#95	C₂²×C₆₇	1	2	yes	—
331	C₂×#95	C₂²×C₆₇	1	2	yes	—
332	C₂×#95	C₂²×C₆₇	1	8	yes	complemented
333	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
334	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
335	C₂²×Q₈	—	2	4	no	—
336	C₂²×Q₈	—	4	4	no	—
337	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
338	C₂²×D₄	—	2	4	no	—
339	C₂²×D₄	—	2	4	no	—
340	C₂²×D₄	Q₈×C₆₇	1	8	yes	complemented
341	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
342	C₂³×C₄	—	2	8	no	—
343	C₂³×C₄	—	2	4	no	—
344	C₂³×C₄	—	2	8	no	—
345	C₂³×C₄	—	4	4	no	—
346	C₂³×C₄	—	4	4	no	—
347	C₂³×C₄	—	4	4	no	—
348	C₂³×C₄	D₄×C₆₇	1	8	yes	—
349	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
350	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
351	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
352	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
353	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
354	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
355	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
357	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
358	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
359	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
360	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
361	C₂²⋊Q₈	—	2	32	no	—
362	C₄×D₄	—	2	32	no	—
363	C₂×C₄⋊C₄	—	2	4	no	—
364	C₂×C₄⋊C₄	—	2	4	no	—
365	C₂×C₄⋊C₄	—	2	8	no	—
366	C₂×C₄⋊C₄	—	2	8	no	—
367	C₂×C₄⋊C₄	—	4	4	no	—
368	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
369	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
370	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
371	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
372	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
373	C₂×C₂²⋊C₄	—	2	4	no	—
374	C₂×C₂²⋊C₄	—	2	8	no	—
375	C₂×C₂²⋊C₄	—	2	4	no	—
376	C₂×C₂²⋊C₄	—	2	8	no	—
377	C₂×C₂²⋊C₄	—	4	4	no	—
378	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
383	C₂×C₄²	—	2	4	no	—
384	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
385	#95	C₂³×C₆₇	1	8	yes	—
386	#95	C₂³×C₆₇	1	4	yes	—
387	#95	C₂³×C₆₇	1	4	yes	—
388	#95	C₂³×C₆₇	1	8	yes	—
389	#95	C₂³×C₆₇	1	8	yes	—
390	#95	C₂³×C₆₇	1	8	yes	—
391	#95	C₂³×C₆₇	1	8	yes	—
392	C₂⁴	—	2	4	no	—
393	C₂⁴	—	2	4	no	—
394	C₂⁴	—	2	2	no	—
395	C₂⁴	—	2	4	no	—
396	C₂⁴	—	2	2	no	—
397	C₂⁴	—	2	4	no	—
398	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
399	C₂⁴	C₂×Q₈×C₆₇	1	4	yes	—
400	C₂⁴	C₂×Q₈×C₆₇	1	4	yes	—
401	C₂⁴	#41×C₆₇	1	4	yes	—
402	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
403	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
404	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
405	C₂×Q₈	—	2	8	no	—
406	C₂×Q₈	—	4	8	no	—
407	C₂×Q₈	—	4	16	no	—
408	C₂×D₄	—	2	32	no	—
409	C₂×D₄	—	2	8	no	—
410	C₂×D₄	—	2	8	no	—
411	C₂×D₄	C₂×Q₈×C₆₇	1	16	yes	complemented
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	16	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	4	no	—
432	C₂²×C₄	—	2	8	no	—
433	C₂²×C₄	—	2	16	no	—
434	C₂²×C₄	—	4	16	no	—
435	C₂²×C₄	—	4	8	no	—
436	C₂²×C₄	—	4	8	no	—
437	C₂²×C₄	—	4	8	no	—
438	C₂²×C₄	—	4	32	no	—
439	C₂²×C₄	—	4	16	no	—
440	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
441	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
442	C₂²×C₄	C₂×D₄×C₆₇	1	16	yes	—
443	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
444	C₂²×C₄	#41×C₆₇	1	8	yes	—
445	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
446	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
447	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
448	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
449	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
450	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
451	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
452	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
453	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
454	C₄⋊C₄	—	2	8	no	—
455	C₄⋊C₄	—	2	8	no	—
456	C₄⋊C₄	—	2	16	no	—
457	C₄⋊C₄	—	2	16	no	—
458	C₄⋊C₄	—	4	8	no	—
459	C₂²⋊C₄	—	2	8	no	—
460	C₂²⋊C₄	—	2	16	no	—
461	C₂²⋊C₄	—	2	8	no	—
462	C₂²⋊C₄	—	2	16	no	—
463	C₂²⋊C₄	—	4	8	no	—
464	C₄²	—	2	8	no	—
465	C₂³	—	2	8	no	—
466	C₂³	—	2	8	no	—
467	C₂³	—	2	16	no	—
468	C₂³	—	2	16	no	—
469	C₂³	—	2	4	no	—
470	C₂³	—	2	4	no	—
471	C₂³	—	2	8	no	—
472	C₂³	—	2	16	no	—
473	C₂³	—	2	2	no	—
474	C₂³	—	2	4	no	—
475	C₂³	—	2	2	no	—
476	C₂³	—	2	4	no	—
477	C₂³	—	4	4	no	—
478	C₂³	—	4	4	no	—
479	C₂³	—	4	8	no	—
480	C₂³	C₂²⋊Q₈×C₆₇	1	16	yes	—
481	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
482	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
483	C₂³	C₂²×Q₈×C₆₇	1	4	yes	—
484	C₂³	C₂²×Q₈×C₆₇	1	4	yes	—
485	C₂³	C₂²×Q₈×C₆₇	1	2	yes	—
486	C₂³	C₂×#41×C₆₇	1	4	yes	—
487	C₂³	C₂×#41×C₆₇	1	2	yes	—
488	C₂³	C₂×#41×C₆₇	1	4	yes	—
489	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
490	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
491	Q₈	—	4	16	no	—
492	D₄	—	2	32	no	—
493	C₂×C₄	—	2	8	no	—
494	C₂×C₄	—	2	8	no	—
495	C₂×C₄	—	2	8	no	—
496	C₂×C₄	—	2	16	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	4	16	no	—
517	C₂×C₄	—	4	32	no	—
518	C₂×C₄	—	4	16	no	—
519	C₂×C₄	—	4	4	no	—
520	C₂×C₄	—	4	8	no	—
521	C₂×C₄	—	4	4	no	—
522	C₂×C₄	—	4	8	no	—
523	C₂×C₄	—	4	8	no	—
524	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
525	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
526	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
527	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
528	C₂²	—	2	32	no	—
529	C₂²	—	2	4	no	—
530	C₂²	—	2	4	no	—
531	C₂²	—	2	8	no	—
532	C₂²	—	2	8	no	—
533	C₂²	—	2	4	no	—
534	C₂²	—	2	8	no	—
535	C₂²	—	4	16	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	4	2	no	—
538	C₂²	—	4	4	no	—
539	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	8	yes	—
540	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	4	yes	—
541	C₂²	#532×C₆₇	1	4	yes	—
542	C₂²	#534×C₆₇	1	4	yes	—
543	C₂²	#551×C₆₇	1	8	yes	—
544	C₂²	#554×C₆₇	1	8	yes	—
545	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
546	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
547	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
548	C₂²	C₂²×#41×C₆₇	1	2	yes	—
549	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
550	C₄	—	2	4	no	—
551	C₄	—	2	4	no	—
552	C₄	—	2	4	no	—
553	C₄	—	2	8	no	—
554	C₄	—	4	2	no	—
555	C₄	—	4	4	no	—
556	C₄	—	4	2	no	—
557	C₄	—	4	4	no	—
558	C₄	—	4	4	no	—
559	C₂	—	2	8	no	—
560	C₂	—	4	4	no	—
561	C₂	C₆₇×#2564	1	8	yes	direct product factor
562	C₂	C₂²×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
563	C₂	C₂×#532×C₆₇	1	1	yes	characteristic
564	C₂	C₂×#534×C₆₇	1	1	yes	characteristic
565	C₂	C₂×#551×C₆₇	1	2	yes	—
566	C₂	C₂×#554×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
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
1. C₂²×C₂²⋊Q₈×C₆₇	C₂	semidirect
3. C₆₇×#2564	C₂	direct (N × H)
4. C₂×#400×C₆₇	C₂	semidirect
5. C₂×#399×C₆₇	C₂	semidirect
6. C₂×#395×C₆₇	C₂	semidirect
7. C₂×#393×C₆₇	C₂	semidirect
8. C₂×#387×C₆₇	C₂	semidirect
10. C₂×#385×C₆₇	C₂	semidirect
11. C₂×#380×C₆₇	C₂	semidirect
12. C₂²×#95×C₆₇	C₂	semidirect
19. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
33. #400×C₆₇	C₂²	semidirect
34. #399×C₆₇	C₂²	semidirect
35. #395×C₆₇	C₂²	semidirect
36. #393×C₆₇	C₂²	semidirect
37. #387×C₆₇	C₂²	semidirect
39. #385×C₆₇	C₂²	semidirect
40. #380×C₆₇	C₂²	semidirect
48. C₂×#95×C₆₇	C₂²	semidirect
56. C₂²×D₄×C₆₇	Q₈	semidirect
127. C₂×D₄×C₆₇	C₂×Q₈	semidirect
275. C₂×#2564	C₆₇	direct (N × H)
278. C₂×C₆₇	#2564	direct (N × H)
284. C₂²×C₂²⋊Q₈	C₂×C₆₇	semidirect
286. #2564	C₂×C₆₇	direct (N × H)
287. C₂×#400	C₂×C₆₇	semidirect
288. C₂×#399	C₂×C₆₇	semidirect
289. C₂×#395	C₂×C₆₇	semidirect
290. C₂×#393	C₂×C₆₇	semidirect
291. C₂×#387	C₂×C₆₇	semidirect
293. C₂×#385	C₂×C₆₇	semidirect
294. C₂×#380	C₂×C₆₇	semidirect
295. C₂²×#95	C₂×C₆₇	semidirect
296. C₆₇	C₂×#2564	direct (N × H)
303. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
317. #400	C₂²×C₆₇	semidirect
318. #399	C₂²×C₆₇	semidirect
319. #395	C₂²×C₆₇	semidirect
320. #393	C₂²×C₆₇	semidirect
321. #387	C₂²×C₆₇	semidirect
323. #385	C₂²×C₆₇	semidirect
324. #380	C₂²×C₆₇	semidirect
332. C₂×#95	C₂²×C₆₇	semidirect
340. C₂²×D₄	Q₈×C₆₇	semidirect
411. C₂×D₄	C₂×Q₈×C₆₇	semidirect
561. C₂	C₆₇×#2564	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×C₂²⋊Q₈×C₆₇	2
C₂²×C₄×D₄×C₆₇	2
C₆₇×#2564	2
C₂×#400×C₆₇	2
C₂×#399×C₆₇	2
C₂×#395×C₆₇	2
C₂×#393×C₆₇	2
C₂×#387×C₆₇	2
C₂×#386×C₆₇	2
C₂×#385×C₆₇	2
C₂×#380×C₆₇	2
C₂²×#95×C₆₇	2
C₂×#2564	67

Subgroups — C2×C67×#2564 #30714082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2564 #30714082