Subgroups — C₂×#22574448 #31763066

Order 17152.

8890 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22574448	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16864136	C₂	1	4	yes	maximal, complemented
3	C₂×#16864092	C₂	1	4	yes	maximal, complemented
4	C₂×D₄×D₂₆₈	C₂	1	4	yes	maximal, complemented
5	C₂×#16864028	C₂	1	4	yes	maximal, complemented
6	C₂×#16864001	C₂	1	2	yes	maximal, complemented
7	C₂×#16863986	C₂	1	4	yes	maximal, complemented
8	C₂×#16863960	C₂	1	2	yes	maximal, complemented
9	C₂×#16863954	C₂	1	4	yes	maximal, complemented
10	C₂²×#14524017	C₂	1	2	yes	maximal, complemented
11	C₂×#525×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₂₆₈	C₂²	1	2	yes	complemented
13	C₂³×D₂₆₈	C₂²	1	2	yes	complemented
14	C₂³×C₂²⋊D₆₇	C₂²	1	4	yes	complemented
15	C₂²×D₄×D₆₇	C₂²	1	8	yes	complemented
16	#16864136	C₂²	1	64	yes	complemented
17	#16864092	C₂²	1	64	yes	complemented
18	D₄×D₂₆₈	C₂²	1	64	yes	complemented
19	#16864028	C₂²	1	64	yes	complemented
20	#16864001	C₂²	1	32	yes	complemented
21	#16863986	C₂²	1	64	yes	complemented
22	#16863960	C₂²	1	32	yes	complemented
23	#16863954	C₂²	1	64	yes	complemented
24	C₂×#14524040	C₂²	1	2	yes	complemented
25	C₂×#14524039	C₂²	1	4	yes	complemented
26	C₂×#14524038	C₂²	1	8	yes	complemented
27	C₂×#14524035	C₂²	1	2	yes	complemented
28	C₂×#14524032	C₂²	1	4	yes	complemented
29	C₂×#14524028	C₂²	1	8	yes	complemented
30	C₂×#14524025	C₂²	1	8	yes	complemented
31	C₂×#14524018	C₂²	1	8	yes	complemented
32	C₂×#14524017	C₂²	1	4	yes	complemented
33	C₂×#14524017	C₂²	1	4	yes	complemented
34	C₂×#14524017	C₂²	1	4	yes	complemented
35	C₂×#14524017	C₂²	1	4	yes	complemented
36	C₂×#14524017	C₂²	1	4	yes	complemented
37	C₂×#14524017	C₂²	1	32	yes	complemented
38	C₂×#14524016	C₂²	1	8	yes	complemented
39	C₂×#14524015	C₂²	1	8	yes	complemented
40	C₂×#14524013	C₂²	1	8	yes	complemented
41	C₂×#14524013	C₂²	1	8	yes	complemented
42	C₂×#14524011	C₂²	1	4	yes	complemented
43	C₂×#14524011	C₂²	1	4	yes	complemented
44	C₂×C₄×D₂₆₈	C₂²	1	4	yes	complemented
45	C₂×(C₂²⋊D₆₇)⋊C₄	C₂²	1	8	yes	complemented
46	C₂×#14524002	C₂²	1	2	yes	complemented
47	C₂²×C₄⋊Dic₆₇	C₂²	1	2	yes	complemented
48	C₂²×(C₂×D₆₇)⋊C₄	C₂²	1	4	yes	complemented
49	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic, complemented
50	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
51	#525×C₆₇	C₂²	1	16	yes	complemented
52	C₂×#123×C₆₇	C₂²	1	2	yes	complemented
53	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
54	C₂×#121×C₆₇	C₂²	1	2	yes	complemented
55	C₂×#120×C₆₇	C₂²	1	2	yes	complemented
56	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
57	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic, complemented
58	C₂⁴×D₆₇	C₂³	1	4	yes	—
59	C₂⁴×D₆₇	C₂³	1	4	yes	—
60	C₂²×D₂₆₈	—	4	8	no	—
61	C₂²×D₂₆₈	—	4	8	no	—
62	C₂²×D₂₆₈	C₂³	1	4	yes	—
63	C₂²×D₂₆₈	C₂³	1	8	yes	—
64	C₂²×D₂₆₈	C₂³	1	4	yes	—
65	C₂²×D₂₆₈	C₂³	1	4	yes	—
66	C₂²×D₂₆₈	C₂³	1	16	yes	complemented
67	C₂²×D₂₆₈	C₂³	1	16	yes	complemented
68	C₂²×C₂²⋊D₆₇	—	4	16	no	—
69	C₂²×C₂²⋊D₆₇	C₂³	1	8	yes	—
70	C₂²×C₂²⋊D₆₇	C₂³	1	8	yes	—
71	C₂²×C₂²⋊D₆₇	C₂³	1	8	yes	—
72	C₂²×C₂²⋊D₆₇	C₂³	1	32	yes	complemented
73	C₂×D₄×D₆₇	—	4	32	no	—
74	C₂×D₄×D₆₇	C₂³	1	64	yes	complemented
75	C₂³×Dic₆₇	C₂³	1	4	yes	—
76	C₂²×C₄×D₆₇	C₂³	1	8	yes	—
77	#14524040	C₂³	1	16	yes	complemented
78	#14524039	C₂³	1	32	yes	complemented
79	#14524038	C₂³	1	64	yes	complemented
80	#14524035	C₂³	1	16	yes	complemented
81	#14524032	C₂³	1	32	yes	complemented
82	#14524028	C₂³	1	64	yes	complemented
83	#14524025	C₂³	1	64	yes	complemented
84	#14524018	C₂³	1	64	yes	complemented
85	#14524017	C₂³	1	32	yes	complemented
86	#14524017	C₂³	1	32	yes	complemented
87	#14524017	C₂³	1	32	yes	complemented
88	#14524017	C₂³	1	32	yes	complemented
89	#14524017	C₂³	1	32	yes	complemented
90	#14524016	C₂³	1	64	yes	complemented
91	#14524015	C₂³	1	64	yes	complemented
92	#14524013	C₂³	1	64	yes	complemented
93	#14524013	C₂³	1	64	yes	complemented
94	#14524011	C₂³	1	32	yes	complemented
95	#14524011	C₂³	1	32	yes	complemented
96	C₄×D₂₆₈	C₂³	1	32	yes	complemented
97	(C₂²⋊D₆₇)⋊C₄	C₂³	1	64	yes	complemented
98	#14524002	C₂³	1	16	yes	complemented
99	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
100	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
101	C₂×C₄⋊Dic₆₇	C₂³	1	16	yes	complemented
102	C₂×Dic₆₇⋊C₄	C₂³	1	8	yes	—
103	C₂×C₂²⋊Dic₆₇	C₂³	1	4	yes	—
104	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
105	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
106	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
107	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
108	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	32	yes	complemented
109	C₂×C₄×Dic₆₇	C₂³	1	4	yes	—
110	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
111	C₂×#41×C₆₇	—	4	4	no	—
112	C₂×#41×C₆₇	C₂³	1	8	yes	complemented
113	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
114	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
115	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
116	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
117	C₂³×C₄×C₆₇	—	4	4	no	—
118	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
119	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
120	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
121	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
122	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
123	C₂³×C₄×C₆₇	C₂³	1	8	yes	complemented
124	#123×C₆₇	C₂³	1	16	yes	complemented
125	C₂²⋊Q₈×C₆₇	C₂³	1	16	yes	complemented
126	#121×C₆₇	C₂³	1	16	yes	complemented
127	#120×C₆₇	C₂³	1	16	yes	complemented
128	C₄×D₄×C₆₇	C₂³	1	32	yes	complemented
129	#117×C₆₇	C₂³	1	8	yes	complemented
130	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
131	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
132	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
133	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
134	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
135	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
136	C₂³×D₆₇	—	2	16	no	—
137	C₂³×D₆₇	—	4	16	no	—
138	C₂³×D₆₇	—	4	8	no	—
139	C₂³×D₆₇	—	4	8	no	—
140	C₂³×D₆₇	—	4	8	no	—
141	C₂³×D₆₇	C₂⁴	1	8	yes	—
142	C₂³×D₆₇	C₂⁴	1	16	yes	—
143	C₂³×D₆₇	C₂⁴	1	16	yes	—
144	C₂×D₂₆₈	—	2	16	no	—
145	C₂×D₂₆₈	—	4	32	no	—
146	C₂×D₂₆₈	—	4	32	no	—
147	C₂×D₂₆₈	—	4	16	no	—
148	C₂×D₂₆₈	—	4	8	no	—
149	C₂×D₂₆₈	—	4	8	no	—
150	C₂×D₂₆₈	C₂⁴	1	16	yes	—
151	C₂×D₂₆₈	C₂⁴	1	32	yes	—
152	C₂×D₂₆₈	C₂⁴	1	16	yes	—
153	C₂×D₂₆₈	C₂⁴	1	16	yes	—
154	C₂×C₂²⋊D₆₇	—	2	32	no	—
155	C₂×C₂²⋊D₆₇	—	4	16	no	—
156	C₂×C₂²⋊D₆₇	—	4	16	no	—
157	C₂×C₂²⋊D₆₇	—	4	64	no	—
158	C₂×C₂²⋊D₆₇	C₂⁴	1	32	yes	—
159	C₂×C₂²⋊D₆₇	C₂⁴	1	32	yes	—
160	C₂×C₂²⋊D₆₇	C₂⁴	1	32	yes	—
161	D₄×D₆₇	—	4	128	no	—
162	C₂²×Dic₆₇	—	4	8	no	—
163	C₂²×Dic₆₇	C₂⁴	1	8	yes	—
164	C₂²×Dic₆₇	C₂⁴	1	16	yes	—
165	C₂×C₄×D₆₇	—	4	16	no	—
166	C₂×C₄×D₆₇	C₂⁴	1	32	yes	—
167	C₄⋊Dic₆₇	C₂⁴	1	16	yes	—
168	C₄⋊Dic₆₇	C₂⁴	1	16	yes	—
169	Dic₆₇⋊C₄	C₂⁴	1	32	yes	—
170	C₂²⋊Dic₆₇	C₂⁴	1	16	yes	—
171	(C₂×D₆₇)⋊C₄	C₂⁴	1	32	yes	—
172	(C₂×D₆₇)⋊C₄	C₂⁴	1	32	yes	—
173	(C₂×D₆₇)⋊C₄	C₂⁴	1	32	yes	—
174	(C₂×D₆₇)⋊C₄	C₂⁴	1	32	yes	—
175	C₄×Dic₆₇	C₂⁴	1	16	yes	—
176	C₂⁴×C₆₇	—	2	8	no	—
177	C₂⁴×C₆₇	—	4	2	no	—
178	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
179	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
180	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
181	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
182	#41×C₆₇	—	4	16	no	—
183	C₂×Q₈×C₆₇	—	4	2	no	—
184	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
185	C₂×D₄×C₆₇	—	2	16	no	—
186	C₂×D₄×C₆₇	—	4	2	no	—
187	C₂×D₄×C₆₇	—	4	4	no	—
188	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
189	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
190	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
191	C₂²×C₄×C₆₇	—	2	4	no	—
192	C₂²×C₄×C₆₇	—	2	8	no	—
193	C₂²×C₄×C₆₇	—	2	8	no	—
194	C₂²×C₄×C₆₇	—	4	16	no	—
195	C₂²×C₄×C₆₇	—	4	4	no	—
196	C₂²×C₄×C₆₇	—	4	2	no	—
197	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
198	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
199	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
200	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
201	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
202	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
203	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
204	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
205	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
206	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
207	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
208	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
209	C₂²⋊C₄×C₆₇	C₂⁴	1	8	yes	—
210	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
211	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
212	C₄²×C₆₇	C₂⁴	1	8	yes	—
213	C₂³×C₆₇	—	2	4	no	—
214	C₂³×C₆₇	—	2	4	no	—
215	C₂³×C₆₇	—	2	2	no	—
216	C₂³×C₆₇	—	2	4	no	—
217	C₂³×C₆₇	—	2	8	no	—
218	C₂³×C₆₇	—	2	16	no	—
219	C₂³×C₆₇	—	2	4	no	—
220	C₂³×C₆₇	—	4	4	no	—
221	C₂³×C₆₇	—	4	16	no	—
222	C₂³×C₆₇	H(5,𝔽₂)	1	8	yes	—
223	C₂³×C₆₇	C₂⁵	1	2	yes	—
224	C₂³×C₆₇	C₂⁵	1	4	yes	—
225	C₂³×C₆₇	C₂⁵	1	4	yes	—
226	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
227	C₂²×D₆₇	—	2	16	no	—
228	C₂²×D₆₇	—	2	16	no	—
229	C₂²×D₆₇	—	2	16	no	—
230	C₂²×D₆₇	—	2	32	no	—
231	C₂²×D₆₇	—	4	32	no	—
232	C₂²×D₆₇	—	4	16	no	—
233	C₂²×D₆₇	—	4	16	no	—
234	C₂²×D₆₇	—	4	16	no	—
235	C₂²×D₆₇	—	8	32	no	—
236	C₂²×D₆₇	—	8	32	no	—
237	C₂²×D₆₇	C₂⁵	1	16	yes	—
238	Q₈×C₆₇	—	4	4	no	—
239	D₄×C₆₇	—	2	32	no	—
240	D₄×C₆₇	—	4	8	no	—
241	D₄×C₆₇	—	4	4	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	4	no	—
244	C₂×C₄×C₆₇	—	2	4	no	—
245	C₂×C₄×C₆₇	—	2	8	no	—
246	C₂×C₄×C₆₇	—	2	8	no	—
247	C₂×C₄×C₆₇	—	2	16	no	—
248	C₂×C₄×C₆₇	—	2	16	no	—
249	C₂×C₄×C₆₇	—	4	4	no	—
250	C₂×C₄×C₆₇	—	4	8	no	—
251	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
252	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
253	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
254	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
255	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
256	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
257	C₂²⋊D₆₇	—	2	64	no	—
258	C₂²⋊D₆₇	—	4	32	no	—
259	C₂²⋊D₆₇	—	4	32	no	—
260	C₂×Dic₆₇	—	2	16	no	—
261	C₂×Dic₆₇	—	4	16	no	—
262	C₂×Dic₆₇	C₂⁵	1	16	yes	—
263	D₂₆₈	—	2	32	no	—
264	D₂₆₈	—	4	32	no	—
265	D₂₆₈	—	4	16	no	—
266	D₂₆₈	—	4	16	no	—
267	C₄×D₆₇	—	4	32	no	—
268	C₂²×C₆₇	—	2	4	no	—
269	C₂²×C₆₇	—	2	4	no	—
270	C₂²×C₆₇	—	2	16	no	—
271	C₂²×C₆₇	—	2	4	no	—
272	C₂²×C₆₇	—	2	8	no	—
273	C₂²×C₆₇	—	2	2	no	—
274	C₂²×C₆₇	—	2	4	no	—
275	C₂²×C₆₇	—	4	16	no	—
276	C₂²×C₆₇	—	4	8	no	—
277	C₂²×C₆₇	—	4	4	no	—
278	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
279	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	8	yes	—
280	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	—
281	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
282	C₂×D₆₇	—	2	16	no	—
283	C₂×D₆₇	—	2	16	no	—
284	C₂×D₆₇	—	2	16	no	—
285	C₂×D₆₇	—	4	32	no	—
286	C₂×D₆₇	—	8	32	no	—
287	C₂×D₆₇	—	8	32	no	—
288	C₄×C₆₇	—	2	8	no	—
289	C₄×C₆₇	—	2	4	no	—
290	C₄×C₆₇	—	2	4	no	—
291	C₄×C₆₇	—	2	2	no	—
292	C₄×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	—
293	Dic₆₇	—	2	16	no	—
294	C₂×#3537	—	67	1	no	maximal, Sylow
295	C₂×C₆₇	—	2	8	no	—
296	C₂×C₆₇	—	4	4	no	—
297	C₂×C₆₇	—	4	2	no	—
298	C₂×C₆₇	#3537	1	4	yes	complemented
299	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
300	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	2	yes	—
301	D₆₇	—	4	16	no	—
302	#3537	—	67	32	no	—
303	C₂×#560	—	67	4	no	—
304	C₂×#553	—	67	4	no	—
305	C₂×D₄²	—	67	4	no	—
306	C₂×D₄²	—	67	4	no	—
307	C₂×#540	—	67	2	no	—
308	C₂×#538	—	67	4	no	—
309	C₂×#535	—	67	2	no	—
310	C₂×#534	—	67	4	no	—
311	C₂×#525	D₆₇	1	1	yes	characteristic, complemented
312	C₂²×#121	—	67	2	no	—
313	C₆₇	C₂×#3537	1	1	yes	characteristic, complemented, Sylow
314	C₂²×#41	C₂×D₆₇	1	1	yes	characteristic, complemented
315	C₂³×D₄	—	67	4	no	—
316	C₂³×D₄	—	67	8	no	—
317	C₂³×D₄	—	67	2	no	—
318	C₂³×D₄	—	67	2	no	—
319	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic, complemented
320	#560	—	67	64	no	—
321	#553	—	67	64	no	—
322	D₄²	—	67	64	no	—
323	D₄²	—	67	64	no	—
324	#540	—	67	32	no	—
325	#538	—	67	64	no	—
326	#535	—	67	32	no	—
327	#534	—	67	64	no	—
328	C₂×#127	—	67	4	no	—
329	C₂×#127	—	67	2	no	—
330	C₂×#126	—	67	8	no	—
331	C₂×#125	—	67	2	no	—
332	C₂×#124	—	67	4	no	—
333	#525	C₂×D₆₇	1	16	yes	complemented
334	C₂×#123	—	67	8	no	—
335	C₂×#123	—	67	8	no	—
336	C₂×#123	C₂×D₆₇	1	2	yes	complemented
337	C₂×C₂²⋊Q₈	C₂×D₆₇	1	2	yes	complemented
338	C₂×#121	—	67	8	no	—
339	C₂×#121	—	67	4	no	—
340	C₂×#121	—	67	4	no	—
341	C₂×#121	—	67	4	no	—
342	C₂×#121	—	67	4	no	—
343	C₂×#121	—	67	4	no	—
344	C₂×#121	—	67	32	no	—
345	C₂×#121	—	67	8	no	—
346	C₂×#121	—	67	8	no	—
347	C₂×#121	C₂×D₆₇	1	2	yes	complemented
348	C₂×#120	—	67	4	no	—
349	C₂×#120	—	67	4	no	—
350	C₂×#120	—	67	8	no	—
351	C₂×#120	—	67	8	no	—
352	C₂×#120	C₂×D₆₇	1	2	yes	complemented
353	C₂×C₄×D₄	—	67	8	no	—
354	C₂×C₄×D₄	—	67	4	no	—
355	C₂×C₄×D₄	C₂×D₆₇	1	4	yes	complemented
356	C₂×#117	—	67	2	no	—
357	C₂×#117	C₂×D₆₇	1	1	yes	characteristic, complemented
358	C₂²×C₄⋊C₄	—	67	2	no	—
359	C₂²×C₂²⋊C₄	—	67	4	no	—
360	C₂⁵	—	67	4	no	—
361	C₂⁵	—	67	4	no	—
362	C₂⁵	C₂²×D₆₇	1	1	yes	characteristic
363	C₂×#41	—	4	4	no	—
364	C₂×#41	C₂²×D₆₇	1	8	yes	complemented
365	C₂²×Q₈	C₂²×D₆₇	1	1	yes	characteristic
366	C₂²×D₄	—	268	8	no	—
367	C₂²×D₄	—	268	8	no	—
368	C₂²×D₄	—	268	32	no	—
369	C₂²×D₄	—	268	16	no	—
370	C₂²×D₄	—	67	8	no	—
371	C₂²×D₄	—	67	8	no	—
372	C₂²×D₄	—	67	32	no	—
373	C₂²×D₄	—	67	64	no	—
374	C₂²×D₄	—	67	8	no	—
375	C₂²×D₄	—	67	4	no	—
376	C₂²×D₄	—	67	8	no	—
377	C₂²×D₄	—	67	4	no	—
378	C₂²×D₄	—	67	16	no	—
379	C₂²×D₄	—	67	16	no	—
380	C₂²×D₄	—	67	4	no	—
381	C₂²×D₄	C₂²×D₆₇	1	4	yes	—
382	C₂²×D₄	C₂²×D₆₇	1	1	yes	characteristic
383	C₂²×D₄	C₂²×D₆₇	1	2	yes	—
384	C₂³×C₄	—	4	4	no	—
385	C₂³×C₄	—	67	4	no	—
386	C₂³×C₄	—	67	8	no	—
387	C₂³×C₄	C₂²×D₆₇	1	2	yes	—
388	C₂³×C₄	C₂²×D₆₇	1	2	yes	—
389	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
390	C₂³×C₄	C₂²×D₆₇	1	2	yes	—
391	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
392	C₂³×C₄	C₂²×D₆₇	1	8	yes	complemented
393	#127	—	67	32	no	—
394	#127	—	67	16	no	—
395	#126	—	67	64	no	—
396	#125	—	67	16	no	—
397	#124	—	67	32	no	—
398	#123	—	67	64	no	—
399	#123	—	67	64	no	—
400	#123	C₂²×D₆₇	1	16	yes	complemented
401	C₂²⋊Q₈	C₂²×D₆₇	1	16	yes	complemented
402	#121	—	67	64	no	—
403	#121	—	67	32	no	—
404	#121	—	67	32	no	—
405	#121	—	67	32	no	—
406	#121	—	67	32	no	—
407	#121	—	67	32	no	—
408	#121	—	67	64	no	—
409	#121	—	67	64	no	—
410	#121	C₂²×D₆₇	1	16	yes	complemented
411	#120	—	67	32	no	—
412	#120	—	67	32	no	—
413	#120	—	67	64	no	—
414	#120	—	67	64	no	—
415	#120	C₂²×D₆₇	1	16	yes	complemented
416	C₄×D₄	—	67	64	no	—
417	C₄×D₄	—	67	32	no	—
418	C₄×D₄	C₂²×D₆₇	1	32	yes	complemented
419	#117	—	67	16	no	—
420	#117	C₂²×D₆₇	1	8	yes	complemented
421	C₂×C₄⋊C₄	—	67	4	no	—
422	C₂×C₄⋊C₄	—	67	4	no	—
423	C₂×C₄⋊C₄	—	67	16	no	—
424	C₂×C₄⋊C₄	—	67	8	no	—
425	C₂×C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
426	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
427	C₂×C₂²⋊C₄	—	67	4	no	—
428	C₂×C₂²⋊C₄	—	67	8	no	—
429	C₂×C₂²⋊C₄	—	67	8	no	—
430	C₂×C₂²⋊C₄	—	67	32	no	—
431	C₂×C₂²⋊C₄	—	67	8	no	—
432	C₂×C₂²⋊C₄	—	67	8	no	—
433	C₂×C₂²⋊C₄	C₂²×D₆₇	1	2	yes	—
434	C₂×C₂²⋊C₄	C₂²×D₆₇	1	4	yes	—
435	C₂×C₂²⋊C₄	C₂²×D₆₇	1	4	yes	—
436	C₂×C₄²	—	67	4	no	—
437	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
438	C₂⁴	—	2	8	no	—
439	C₂⁴	—	4	2	no	—
440	C₂⁴	—	268	8	no	—
441	C₂⁴	—	268	8	no	—
442	C₂⁴	—	268	8	no	—
443	C₂⁴	—	268	16	no	—
444	C₂⁴	—	67	16	no	—
445	C₂⁴	—	67	16	no	—
446	C₂⁴	—	67	8	no	—
447	C₂⁴	—	134	16	no	—
448	C₂⁴	C₂³×D₆₇	1	2	yes	—
449	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
450	C₂⁴	C₂³×D₆₇	1	4	yes	—
451	C₂⁴	C₂³×D₆₇	1	2	yes	—
452	#41	—	4	16	no	—
453	C₂×Q₈	—	4	2	no	—
454	C₂×Q₈	C₂³×D₆₇	1	4	yes	—
455	C₂×D₄	—	2	16	no	—
456	C₂×D₄	—	4	2	no	—
457	C₂×D₄	—	4	4	no	—
458	C₂×D₄	—	268	8	no	—
459	C₂×D₄	—	268	8	no	—
460	C₂×D₄	—	268	16	no	—
461	C₂×D₄	—	268	32	no	—
462	C₂×D₄	—	268	32	no	—
463	C₂×D₄	—	268	128	no	—
464	C₂×D₄	—	268	64	no	—
465	C₂×D₄	—	268	16	no	—
466	C₂×D₄	—	268	16	no	—
467	C₂×D₄	—	67	32	no	—
468	C₂×D₄	—	67	32	no	—
469	C₂×D₄	—	67	32	no	—
470	C₂×D₄	—	67	16	no	—
471	C₂×D₄	—	67	32	no	—
472	C₂×D₄	—	67	16	no	—
473	C₂×D₄	—	67	16	no	—
474	C₂×D₄	—	134	32	no	—
475	C₂×D₄	—	134	16	no	—
476	C₂×D₄	C₂³×D₆₇	1	16	yes	—
477	C₂×D₄	C₂³×D₆₇	1	4	yes	—
478	C₂×D₄	C₂³×D₆₇	1	8	yes	—
479	C₂²×C₄	—	2	8	no	—
480	C₂²×C₄	—	2	4	no	—
481	C₂²×C₄	—	2	8	no	—
482	C₂²×C₄	—	4	16	no	—
483	C₂²×C₄	—	4	4	no	—
484	C₂²×C₄	—	4	2	no	—
485	C₂²×C₄	—	268	16	no	—
486	C₂²×C₄	—	268	8	no	—
487	C₂²×C₄	—	67	16	no	—
488	C₂²×C₄	—	67	32	no	—
489	C₂²×C₄	—	67	8	no	—
490	C₂²×C₄	C₂³×D₆₇	1	8	yes	—
491	C₂²×C₄	C₂³×D₆₇	1	8	yes	—
492	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
493	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
494	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
495	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
496	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
497	C₂²×C₄	C₂³×D₆₇	1	8	yes	—
498	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
499	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
500	C₄⋊C₄	—	67	16	no	—
501	C₄⋊C₄	—	67	16	no	—
502	C₄⋊C₄	—	67	32	no	—
503	C₄⋊C₄	C₂³×D₆₇	1	8	yes	—
504	C₄⋊C₄	C₂³×D₆₇	1	16	yes	—
505	C₂²⋊C₄	—	67	16	no	—
506	C₂²⋊C₄	—	67	32	no	—
507	C₂²⋊C₄	—	67	32	no	—
508	C₂²⋊C₄	—	67	32	no	—
509	C₂²⋊C₄	—	67	32	no	—
510	C₂²⋊C₄	C₂³×D₆₇	1	8	yes	—
511	C₂²⋊C₄	C₂³×D₆₇	1	16	yes	—
512	C₂²⋊C₄	C₂³×D₆₇	1	16	yes	—
513	C₄²	—	67	16	no	—
514	C₄²	C₂³×D₆₇	1	8	yes	—
515	C₂³	—	2	4	no	—
516	C₂³	—	2	4	no	—
517	C₂³	—	2	2	no	—
518	C₂³	—	2	4	no	—
519	C₂³	—	2	8	no	—
520	C₂³	—	2	16	no	—
521	C₂³	—	2	4	no	—
522	C₂³	—	4	4	no	—
523	C₂³	—	4	16	no	—
524	C₂³	—	268	16	no	—
525	C₂³	—	268	16	no	—
526	C₂³	—	268	16	no	—
527	C₂³	—	268	32	no	—
528	C₂³	—	536	32	no	—
529	C₂³	—	536	32	no	—
530	C₂³	—	67	16	no	—
531	C₂³	—	134	16	no	—
532	C₂³	—	134	16	no	—
533	C₂³	—	134	16	no	—
534	C₂³	—	134	32	no	—
535	C₂³	#14524089	1	8	yes	—
536	C₂³	C₂⁴×D₆₇	1	4	yes	—
537	C₂³	C₂⁴×D₆₇	1	2	yes	—
538	C₂³	C₂⁴×D₆₇	1	4	yes	—
539	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
540	Q₈	—	4	4	no	—
541	D₄	—	2	32	no	—
542	D₄	—	4	4	no	—
543	D₄	—	4	8	no	—
544	D₄	—	268	16	no	—
545	D₄	—	268	16	no	—
546	D₄	—	268	32	no	—
547	D₄	—	268	32	no	—
548	D₄	—	268	32	no	—
549	D₄	—	134	64	no	—
550	D₄	—	134	32	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	8	no	—
555	C₂×C₄	—	2	16	no	—
556	C₂×C₄	—	2	8	no	—
557	C₂×C₄	—	2	16	no	—
558	C₂×C₄	—	4	8	no	—
559	C₂×C₄	—	4	4	no	—
560	C₂×C₄	—	268	32	no	—
561	C₂×C₄	—	268	16	no	—
562	C₂×C₄	—	67	16	no	—
563	C₂×C₄	—	134	16	no	—
564	C₂×C₄	#14524088	1	4	yes	—
565	C₂×C₄	C₂⁴×D₆₇	1	4	yes	—
566	C₂×C₄	C₂⁴×D₆₇	1	8	yes	—
567	C₂×C₄	C₂⁴×D₆₇	1	2	yes	—
568	C₂×C₄	C₂⁴×D₆₇	1	4	yes	—
569	C₂×C₄	C₂⁴×D₆₇	1	2	yes	—
570	C₂²	—	2	16	no	—
571	C₂²	—	2	4	no	—
572	C₂²	—	2	4	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	2	4	no	—
575	C₂²	—	2	8	no	—
576	C₂²	—	2	4	no	—
577	C₂²	—	4	16	no	—
578	C₂²	—	4	4	no	—
579	C₂²	—	4	8	no	—
580	C₂²	—	268	32	no	—
581	C₂²	—	536	32	no	—
582	C₂²	—	536	32	no	—
583	C₂²	—	134	16	no	—
584	C₂²	—	134	16	no	—
585	C₂²	—	134	16	no	—
586	C₂²	C₂×#14524088	1	2	yes	—
587	C₂²	C₂×#14524089	1	8	yes	—
588	C₂²	C₂×#14524089	1	4	yes	—
589	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
590	C₄	—	2	2	no	—
591	C₄	—	2	4	no	—
592	C₄	—	2	4	no	—
593	C₄	—	2	8	no	—
594	C₄	—	134	16	no	—
595	C₄	C₂×#14524088	1	4	yes	—
596	C₂	—	2	8	no	—
597	C₂	—	4	2	no	—
598	C₂	—	4	4	no	—
599	C₂	—	268	16	no	—
600	C₂	#22574448	1	4	yes	direct product factor
601	C₂	C₂²×#14524088	1	1	yes	characteristic
602	C₂	C₂²×#14524089	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₆₇
commutator, cosocle	C₂²×C₆₇	C₂⁶
Frattini	C₂²	C₂⁵×D₆₇
Fitting	C₂×#525×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁵
hypercenter	C₂×#525	D₆₇
hypocenter	C₆₇	C₂×#3537
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×#525

Complements

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

N	H ≅ G/N	type
1. #22574448	C₂	direct (N × H)
2. C₂×#16864136	C₂	semidirect
3. C₂×#16864092	C₂	semidirect
4. C₂×D₄×D₂₆₈	C₂	semidirect
5. C₂×#16864028	C₂	semidirect
6. C₂×#16864001	C₂	semidirect
7. C₂×#16863986	C₂	semidirect
8. C₂×#16863960	C₂	semidirect
9. C₂×#16863954	C₂	semidirect
10. C₂²×#14524017	C₂	semidirect
11. C₂×#525×C₆₇	C₂	semidirect
12. C₂³×D₂₆₈	C₂²	semidirect
13. C₂³×D₂₆₈	C₂²	semidirect
14. C₂³×C₂²⋊D₆₇	C₂²	semidirect
15. C₂²×D₄×D₆₇	C₂²	semidirect
16. #16864136	C₂²	semidirect
17. #16864092	C₂²	semidirect
18. D₄×D₂₆₈	C₂²	semidirect
19. #16864028	C₂²	semidirect
20. #16864001	C₂²	semidirect
21. #16863986	C₂²	semidirect
22. #16863960	C₂²	semidirect
23. #16863954	C₂²	semidirect
24. C₂×#14524040	C₂²	semidirect
25. C₂×#14524039	C₂²	semidirect
26. C₂×#14524038	C₂²	semidirect
27. C₂×#14524035	C₂²	semidirect
28. C₂×#14524032	C₂²	semidirect
29. C₂×#14524028	C₂²	semidirect
30. C₂×#14524025	C₂²	semidirect
31. C₂×#14524018	C₂²	semidirect
32. C₂×#14524017	C₂²	semidirect
33. C₂×#14524017	C₂²	semidirect
34. C₂×#14524017	C₂²	semidirect
35. C₂×#14524017	C₂²	semidirect
36. C₂×#14524017	C₂²	semidirect
37. C₂×#14524017	C₂²	semidirect
38. C₂×#14524016	C₂²	semidirect
39. C₂×#14524015	C₂²	semidirect
40. C₂×#14524013	C₂²	semidirect
41. C₂×#14524013	C₂²	semidirect
42. C₂×#14524011	C₂²	semidirect
43. C₂×#14524011	C₂²	semidirect
44. C₂×C₄×D₂₆₈	C₂²	semidirect
45. C₂×(C₂²⋊D₆₇)⋊C₄	C₂²	semidirect
46. C₂×#14524002	C₂²	semidirect
47. C₂²×C₄⋊Dic₆₇	C₂²	semidirect
48. C₂²×(C₂×D₆₇)⋊C₄	C₂²	semidirect
49. C₂²×#41×C₆₇	C₂²	semidirect
50. C₂⁴×C₄×C₆₇	C₂²	semidirect
51. #525×C₆₇	C₂²	semidirect
52. C₂×#123×C₆₇	C₂²	semidirect
53. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
54. C₂×#121×C₆₇	C₂²	semidirect
55. C₂×#120×C₆₇	C₂²	semidirect
56. C₂×C₄×D₄×C₆₇	C₂²	semidirect
57. C₂×#117×C₆₇	C₂²	semidirect
66. C₂²×D₂₆₈	C₂³	semidirect
67. C₂²×D₂₆₈	C₂³	semidirect
72. C₂²×C₂²⋊D₆₇	C₂³	semidirect
74. C₂×D₄×D₆₇	C₂³	semidirect
77. #14524040	C₂³	semidirect
78. #14524039	C₂³	semidirect
79. #14524038	C₂³	semidirect
80. #14524035	C₂³	semidirect
81. #14524032	C₂³	semidirect
82. #14524028	C₂³	semidirect
83. #14524025	C₂³	semidirect
84. #14524018	C₂³	semidirect
85. #14524017	C₂³	semidirect
86. #14524017	C₂³	semidirect
87. #14524017	C₂³	semidirect
88. #14524017	C₂³	semidirect
89. #14524017	C₂³	semidirect
90. #14524016	C₂³	semidirect
91. #14524015	C₂³	semidirect
92. #14524013	C₂³	semidirect
93. #14524013	C₂³	semidirect
94. #14524011	C₂³	semidirect
95. #14524011	C₂³	semidirect
96. C₄×D₂₆₈	C₂³	semidirect
97. (C₂²⋊D₆₇)⋊C₄	C₂³	semidirect
98. #14524002	C₂³	semidirect
101. C₂×C₄⋊Dic₆₇	C₂³	semidirect
108. C₂×(C₂×D₆₇)⋊C₄	C₂³	semidirect
112. C₂×#41×C₆₇	C₂³	semidirect
123. C₂³×C₄×C₆₇	C₂³	semidirect
124. #123×C₆₇	C₂³	semidirect
125. C₂²⋊Q₈×C₆₇	C₂³	semidirect
126. #121×C₆₇	C₂³	semidirect
127. #120×C₆₇	C₂³	semidirect
128. C₄×D₄×C₆₇	C₂³	semidirect
129. #117×C₆₇	C₂³	semidirect
298. C₂×C₆₇	#3537	semidirect
311. C₂×#525	D₆₇	semidirect
313. C₆₇	C₂×#3537	semidirect
314. C₂²×#41	C₂×D₆₇	semidirect
319. C₂⁴×C₄	C₂×D₆₇	semidirect
333. #525	C₂×D₆₇	semidirect
336. C₂×#123	C₂×D₆₇	semidirect
337. C₂×C₂²⋊Q₈	C₂×D₆₇	semidirect
347. C₂×#121	C₂×D₆₇	semidirect
352. C₂×#120	C₂×D₆₇	semidirect
355. C₂×C₄×D₄	C₂×D₆₇	semidirect
357. C₂×#117	C₂×D₆₇	semidirect
364. C₂×#41	C₂²×D₆₇	semidirect
392. C₂³×C₄	C₂²×D₆₇	semidirect
400. #123	C₂²×D₆₇	semidirect
401. C₂²⋊Q₈	C₂²×D₆₇	semidirect
410. #121	C₂²×D₆₇	semidirect
415. #120	C₂²×D₆₇	semidirect
418. C₄×D₄	C₂²×D₆₇	semidirect
420. #117	C₂²×D₆₇	semidirect
600. C₂	#22574448	direct (N × H)

Maximal subgroups

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

subgroup	index
#22574448	2
C₂×#16864136	2
C₂×#16864092	2
C₂×D₄×D₂₆₈	2
C₂×#16864028	2
C₂×#16864001	2
C₂×#16863986	2
C₂×#16863960	2
C₂×#16863954	2
C₂²×#14524017	2
C₂×#525×C₆₇	2
C₂×#3537	67

Subgroups — C2×#22574448 #31763066

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22574448 #31763066