Subgroups — C₂×#22573897 #31763743

Order 17152.

9502 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573897	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16864069	C₂	1	8	yes	maximal, complemented
3	C₂×#16864033	C₂	1	8	yes	maximal, complemented
4	C₂×#16864022	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863983	C₂	1	4	yes	maximal, complemented
6	C₂×#16863906	C₂	1	4	yes	maximal, complemented
7	C₂²×#14524011	C₂	1	2	yes	maximal, complemented
8	C₂×#16863871	C₂	1	2	yes	maximal, complemented
9	C₂²×C₂²⋊C₄×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#534×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁵×D₆₇	C₂²	1	1	yes	characteristic
12	C₂²×#11761627	C₂²	1	4	yes	complemented
13	C₂³×C₂²⋊D₆₇	C₂²	1	4	yes	complemented
14	C₂³×C₄×D₆₇	C₂²	1	2	yes	complemented
15	#16864069	C₂²	1	128	yes	complemented
16	#16864033	C₂²	1	128	yes	complemented
17	#16864022	C₂²	1	16	yes	complemented
18	#16863983	C₂²	1	64	yes	complemented
19	C₂×#14524038	C₂²	1	8	yes	complemented
20	C₂×#14524029	C₂²	1	8	yes	complemented
21	#16863906	C₂²	1	64	yes	complemented
22	C₂×#14524027	C₂²	1	8	yes	complemented
23	C₂×#14524025	C₂²	1	8	yes	complemented
24	C₂×#14524023	C₂²	1	4	yes	complemented
25	C₂×#14524022	C₂²	1	4	yes	complemented
26	C₂×#14524020	C₂²	1	8	yes	complemented
27	C₂×#14524015	C₂²	1	8	yes	complemented
28	C₂×#14524014	C₂²	1	4	yes	complemented
29	C₂×#14524011	C₂²	1	8	yes	—
30	C₂×#14524011	C₂²	1	4	yes	complemented
31	C₂×#14524011	C₂²	1	4	yes	complemented
32	C₂×#14524011	C₂²	1	32	yes	complemented
33	#16863871	C₂²	1	32	yes	complemented
34	C₂×D₄×Dic₆₇	C₂²	1	8	yes	complemented
35	C₂×D₄×Dic₆₇	C₂²	1	8	yes	complemented
36	C₂×(C₂²⋊D₆₇)⋊C₄	C₂²	1	8	yes	complemented
37	C₂×#14524002	C₂²	1	4	yes	complemented
38	C₂×#14524001	C₂²	1	4	yes	complemented
39	C₂²×C₂²⋊Dic₆₇	C₂²	1	2	yes	—
40	C₂²×C₂²⋊Dic₆₇	C₂²	1	1	yes	characteristic
41	C₂²×C₂²⋊Dic₆₇	C₂²	1	4	yes	complemented
42	C₂²×(C₂×D₆₇)⋊C₄	C₂²	1	2	yes	complemented
43	C₂²×(C₂×D₆₇)⋊C₄	C₂²	1	4	yes	complemented
44	C₂×C₂²⋊C₄×D₆₇	C₂²	1	8	yes	complemented
45	C₂×C₂²⋊C₄×D₆₇	C₂²	1	16	yes	complemented
46	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
47	#534×C₆₇	C₂²	1	16	yes	complemented
48	C₂×#123×C₆₇	C₂²	1	4	yes	complemented
49	C₂×#121×C₆₇	C₂²	1	4	yes	complemented
50	C₂×#120×C₆₇	C₂²	1	4	yes	complemented
51	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
52	C₂⁴×D₆₇	—	2	8	no	—
53	C₂⁴×D₆₇	—	4	8	no	—
54	C₂⁴×D₆₇	C₂³	1	1	yes	characteristic
55	C₂⁴×D₆₇	C₂³	1	4	yes	—
56	C₂⁴×D₆₇	C₂³	1	8	yes	—
57	C₂⁴×D₆₇	C₂³	1	1	yes	characteristic
58	C₂×#11761627	—	4	16	no	—
59	C₂×#11761627	C₂³	1	32	yes	complemented
60	C₂²×Dic₁₃₄	C₂³	1	4	yes	complemented
61	C₂²×C₂²⋊D₆₇	—	4	16	no	—
62	C₂²×C₂²⋊D₆₇	C₂³	1	8	yes	—
63	C₂²×C₂²⋊D₆₇	C₂³	1	8	yes	—
64	C₂²×C₂²⋊D₆₇	C₂³	1	32	yes	complemented
65	C₂³×Dic₆₇	C₂³	1	2	yes	—
66	C₂³×Dic₆₇	C₂³	1	4	yes	—
67	C₂³×Dic₆₇	C₂³	1	8	yes	complemented
68	C₂²×C₄×D₆₇	—	4	8	no	—
69	C₂²×C₄×D₆₇	C₂³	1	4	yes	—
70	C₂²×C₄×D₆₇	C₂³	1	4	yes	—
71	C₂²×C₄×D₆₇	C₂³	1	16	yes	complemented
72	#14524038	C₂³	1	64	yes	complemented
73	#14524029	C₂³	1	64	yes	complemented
74	#14524027	C₂³	1	64	yes	complemented
75	#14524025	C₂³	1	64	yes	complemented
76	#14524023	C₂³	1	32	yes	complemented
77	#14524022	C₂³	1	32	yes	complemented
78	#14524020	C₂³	1	64	yes	complemented
79	#14524015	C₂³	1	64	yes	complemented
80	#14524014	C₂³	1	32	yes	complemented
81	#14524011	C₂³	1	64	yes	—
82	#14524011	C₂³	1	32	yes	complemented
83	#14524011	C₂³	1	32	yes	complemented
84	D₄×Dic₆₇	C₂³	1	64	yes	complemented
85	D₄×Dic₆₇	C₂³	1	64	yes	complemented
86	(C₂²⋊D₆₇)⋊C₄	C₂³	1	64	yes	complemented
87	#14524002	C₂³	1	32	yes	complemented
88	#14524001	C₂³	1	32	yes	complemented
89	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	complemented
90	C₂×C₄⋊Dic₆₇	C₂³	1	8	yes	complemented
91	C₂×Dic₆₇⋊C₄	C₂³	1	8	yes	complemented
92	C₂×C₂²⋊Dic₆₇	C₂³	1	16	yes	—
93	C₂×C₂²⋊Dic₆₇	C₂³	1	4	yes	—
94	C₂×C₂²⋊Dic₆₇	C₂³	1	8	yes	—
95	C₂×C₂²⋊Dic₆₇	C₂³	1	4	yes	—
96	C₂×C₂²⋊Dic₆₇	C₂³	1	4	yes	—
97	C₂×C₂²⋊Dic₆₇	C₂³	1	32	yes	complemented
98	C₂×C₂²⋊Dic₆₇	C₂³	1	8	yes	complemented
99	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
100	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
101	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
102	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	32	yes	complemented
103	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	16	yes	complemented
104	C₂²⋊C₄×D₆₇	C₂³	1	64	yes	complemented
105	C₂×C₄×Dic₆₇	C₂³	1	8	yes	complemented
106	C₂⁵×C₆₇	C₂³	1	2	yes	—
107	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
108	C₂²×D₄×C₆₇	—	2	16	no	—
109	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
110	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
111	C₂²×D₄×C₆₇	C₂³	1	16	yes	—
112	C₂²×D₄×C₆₇	C₂³	1	4	yes	complemented
113	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
114	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
115	#123×C₆₇	C₂³	1	32	yes	complemented
116	#121×C₆₇	C₂³	1	32	yes	complemented
117	#120×C₆₇	C₂³	1	32	yes	complemented
118	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	complemented
119	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
120	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
121	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	complemented
122	C₂³×D₆₇	—	2	4	no	—
123	C₂³×D₆₇	—	2	8	no	—
124	C₂³×D₆₇	—	2	16	no	—
125	C₂³×D₆₇	—	2	4	no	—
126	C₂³×D₆₇	—	2	32	no	—
127	C₂³×D₆₇	—	2	16	no	—
128	C₂³×D₆₇	—	4	32	no	—
129	C₂³×D₆₇	—	4	8	no	—
130	C₂³×D₆₇	—	4	4	no	—
131	C₂³×D₆₇	—	4	64	no	—
132	C₂³×D₆₇	C₂⁴	1	2	yes	—
133	C₂³×D₆₇	C₂⁴	1	4	yes	—
134	C₂³×D₆₇	C₂⁴	1	16	yes	—
135	C₂³×D₆₇	C₂⁴	1	2	yes	—
136	C₂³×D₆₇	C₂⁴	1	4	yes	—
137	#11761627	—	4	64	no	—
138	C₂×Dic₁₃₄	—	4	8	no	—
139	C₂×Dic₁₃₄	C₂⁴	1	16	yes	complemented
140	C₂×C₂²⋊D₆₇	—	2	32	no	—
141	C₂×C₂²⋊D₆₇	—	4	16	no	—
142	C₂×C₂²⋊D₆₇	—	4	64	no	—
143	C₂×C₂²⋊D₆₇	C₂⁴	1	32	yes	—
144	C₂×C₂²⋊D₆₇	C₂⁴	1	32	yes	—
145	C₂²×Dic₆₇	—	4	4	no	—
146	C₂²×Dic₆₇	—	4	8	no	—
147	C₂²×Dic₆₇	—	4	16	no	—
148	C₂²×Dic₆₇	C₂⁴	1	4	yes	—
149	C₂²×Dic₆₇	C₂⁴	1	8	yes	—
150	C₂²×Dic₆₇	C₂⁴	1	8	yes	—
151	C₂²×Dic₆₇	C₂⁴	1	16	yes	—
152	C₂²×Dic₆₇	C₂⁴	1	32	yes	complemented
153	C₂×C₄×D₆₇	—	2	16	no	—
154	C₂×C₄×D₆₇	—	4	32	no	—
155	C₂×C₄×D₆₇	—	4	8	no	—
156	C₂×C₄×D₆₇	C₂⁴	1	16	yes	—
157	C₂×C₄×D₆₇	C₂⁴	1	16	yes	—
158	C₄⋊Dic₆₇	C₂⁴	1	16	yes	complemented
159	C₄⋊Dic₆₇	C₂⁴	1	32	yes	complemented
160	Dic₆₇⋊C₄	C₂⁴	1	32	yes	complemented
161	C₂²⋊Dic₆₇	C₂⁴	1	16	yes	—
162	C₂²⋊Dic₆₇	C₂⁴	1	16	yes	—
163	C₂²⋊Dic₆₇	C₂⁴	1	16	yes	—
164	C₂²⋊Dic₆₇	C₂⁴	1	32	yes	complemented
165	(C₂×D₆₇)⋊C₄	C₂⁴	1	32	yes	—
166	(C₂×D₆₇)⋊C₄	C₂⁴	1	32	yes	—
167	(C₂×D₆₇)⋊C₄	C₂⁴	1	32	yes	—
168	C₄×Dic₆₇	C₂⁴	1	32	yes	complemented
169	C₂⁴×C₆₇	—	2	8	no	—
170	C₂⁴×C₆₇	—	4	4	no	—
171	C₂⁴×C₆₇	—	4	8	no	—
172	C₂⁴×C₆₇	#41	1	8	yes	complemented
173	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
174	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
175	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
176	C₂⁴×C₆₇	C₂⁴	1	8	yes	—
177	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
178	C₂×D₄×C₆₇	—	2	16	no	—
179	C₂×D₄×C₆₇	—	2	16	no	—
180	C₂×D₄×C₆₇	—	2	64	no	—
181	C₂×D₄×C₆₇	—	4	8	no	—
182	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
183	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
184	C₂×D₄×C₆₇	C₂⁴	1	16	yes	complemented
185	C₂²×C₄×C₆₇	—	2	8	no	—
186	C₂²×C₄×C₆₇	—	4	4	no	—
187	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
188	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
189	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
190	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
191	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	complemented
192	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
193	C₂²⋊C₄×C₆₇	C₂⁴	1	32	yes	complemented
194	C₂³×C₆₇	—	2	4	no	—
195	C₂³×C₆₇	—	2	8	no	—
196	C₂³×C₆₇	—	2	8	no	—
197	C₂³×C₆₇	—	2	16	no	—
198	C₂³×C₆₇	—	2	16	no	—
199	C₂³×C₆₇	—	4	16	no	—
200	C₂³×C₆₇	—	4	8	no	—
201	C₂³×C₆₇	—	4	16	no	—
202	C₂³×C₆₇	—	8	16	no	—
203	C₂³×C₆₇	C₂×#41	1	8	yes	—
204	C₂³×C₆₇	C₂×#41	1	4	yes	—
205	C₂³×C₆₇	C₂×#41	1	16	yes	complemented
206	C₂³×C₆₇	C₂⁵	1	8	yes	—
207	C₂³×C₆₇	C₂⁵	1	4	yes	—
208	C₂³×C₆₇	C₂⁵	1	2	yes	—
209	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
210	C₂²×D₆₇	—	2	32	no	—
211	C₂²×D₆₇	—	2	8	no	—
212	C₂²×D₆₇	—	2	4	no	—
213	C₂²×D₆₇	—	2	4	no	—
214	C₂²×D₆₇	—	2	16	no	—
215	C₂²×D₆₇	—	2	32	no	—
216	C₂²×D₆₇	—	2	8	no	—
217	C₂²×D₆₇	—	2	4	no	—
218	C₂²×D₆₇	—	2	8	no	—
219	C₂²×D₆₇	—	2	32	no	—
220	C₂²×D₆₇	—	4	16	no	—
221	C₂²×D₆₇	—	4	8	no	—
222	C₂²×D₆₇	—	4	64	no	—
223	C₂²×D₆₇	—	4	128	no	—
224	C₂²×D₆₇	—	4	16	no	—
225	C₂²×D₆₇	H(5,𝔽₂)	1	8	yes	—
226	C₂²×D₆₇	C₂⁵	1	4	yes	—
227	C₂²×D₆₇	C₂⁵	1	4	yes	—
228	D₄×C₆₇	—	2	32	no	—
229	D₄×C₆₇	—	2	32	no	—
230	D₄×C₆₇	—	4	16	no	—
231	C₂×C₄×C₆₇	—	2	16	no	—
232	C₂×C₄×C₆₇	—	2	8	no	—
233	C₂×C₄×C₆₇	—	2	8	no	—
234	C₂×C₄×C₆₇	—	4	8	no	—
235	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
236	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
237	C₂²⋊D₆₇	—	2	64	no	—
238	C₂²⋊D₆₇	—	4	32	no	—
239	C₂×Dic₆₇	—	2	8	no	—
240	C₂×Dic₆₇	—	2	16	no	—
241	C₂×Dic₆₇	—	4	8	no	—
242	C₂×Dic₆₇	—	4	16	no	—
243	C₂×Dic₆₇	—	4	32	no	—
244	C₂×Dic₆₇	C₂⁵	1	8	yes	—
245	C₂×Dic₆₇	C₂⁵	1	16	yes	—
246	C₄×D₆₇	—	2	32	no	—
247	C₄×D₆₇	—	4	16	no	—
248	Dic₁₃₄	—	4	16	no	—
249	C₂²×C₆₇	—	2	4	no	—
250	C₂²×C₆₇	—	2	8	no	—
251	C₂²×C₆₇	—	2	8	no	—
252	C₂²×C₆₇	—	2	16	no	—
253	C₂²×C₆₇	—	2	8	no	—
254	C₂²×C₆₇	—	2	16	no	—
255	C₂²×C₆₇	—	4	16	no	—
256	C₂²×C₆₇	—	4	16	no	—
257	C₂²×C₆₇	—	8	16	no	—
258	C₂²×C₆₇	C₂²×#41	1	8	yes	—
259	C₂²×C₆₇	C₂²×#41	1	4	yes	—
260	C₂²×C₆₇	C₂²×#41	1	2	yes	—
261	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	—
262	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
263	C₂×D₆₇	—	2	32	no	—
264	C₂×D₆₇	—	2	4	no	—
265	C₂×D₆₇	—	2	4	no	—
266	C₂×D₆₇	—	2	16	no	—
267	C₂×D₆₇	—	2	4	no	—
268	C₂×D₆₇	—	2	8	no	—
269	C₂×D₆₇	—	4	64	no	—
270	C₂×D₆₇	—	4	16	no	—
271	C₂×D₆₇	—	4	8	no	—
272	C₂×D₆₇	C₂×H(5,𝔽₂)	1	8	yes	—
273	C₄×C₆₇	—	2	8	no	—
274	C₄×C₆₇	—	2	8	no	—
275	Dic₆₇	—	2	16	no	—
276	Dic₆₇	—	2	8	no	—
277	C₂×#3490	—	67	1	no	maximal, Sylow
278	C₂×C₆₇	—	2	4	no	—
279	C₂×C₆₇	—	2	8	no	—
280	C₂×C₆₇	—	4	8	no	—
281	C₂×C₆₇	#3490	1	4	yes	complemented
282	C₂×C₆₇	C₂³×#41	1	1	yes	characteristic
283	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	2	yes	—
284	D₆₇	—	2	8	no	—
285	D₆₇	—	4	4	no	—
286	#3490	—	67	32	no	—
287	C₂×#551	—	67	8	no	—
288	C₂×#546	—	67	8	no	—
289	C₂×#543	—	67	1	no	—
290	C₂×#538	—	67	4	no	—
291	C₂×#534	D₆₇	1	1	yes	characteristic, complemented
292	C₂×#525	—	67	4	no	—
293	C₂²×#120	—	67	2	no	—
294	C₂×#518	—	67	2	no	—
295	C₂³×C₂²⋊C₄	—	67	1	no	—
296	C₆₇	C₂×#3490	1	1	yes	characteristic, complemented, Sylow
297	C₂⁶	—	67	1	no	—
298	C₂²×#41	—	67	4	no	—
299	C₂³×D₄	—	67	4	no	—
300	C₂³×D₄	C₂×D₆₇	1	2	yes	—
301	C₂⁴×C₄	—	67	2	no	—
302	#551	—	67	128	no	—
303	#546	—	67	128	no	—
304	#543	—	67	16	no	—
305	#538	—	67	64	no	—
306	#534	C₂×D₆₇	1	16	yes	complemented
307	C₂×#126	—	67	8	no	—
308	C₂×#124	—	67	8	no	—
309	#525	—	67	64	no	—
310	C₂×#123	—	67	8	no	—
311	C₂×#123	—	67	8	no	—
312	C₂×#123	C₂×D₆₇	1	4	yes	complemented
313	C₂×C₂²⋊Q₈	—	67	4	no	—
314	C₂×C₂²⋊Q₈	—	67	4	no	—
315	C₂×C₂²⋊Q₈	—	67	8	no	—
316	C₂×#121	—	67	4	no	—
317	C₂×#121	—	67	8	no	—
318	C₂×#121	C₂×D₆₇	1	4	yes	complemented
319	C₂×#120	—	67	4	no	—
320	C₂×#120	—	67	4	no	—
321	C₂×#120	—	67	8	no	—
322	C₂×#120	—	67	32	no	—
323	C₂×#120	C₂×D₆₇	1	4	yes	complemented
324	#518	—	67	32	no	—
325	C₂×C₄×D₄	—	67	8	no	—
326	C₂×C₄×D₄	—	67	8	no	—
327	C₂×C₄×D₄	—	67	8	no	—
328	C₂×#117	—	67	4	no	—
329	C₂×#117	—	67	4	no	—
330	C₂²×C₂²⋊C₄	—	67	1	no	—
331	C₂²×C₂²⋊C₄	—	67	2	no	—
332	C₂²×C₂²⋊C₄	—	67	4	no	—
333	C₂²×C₂²⋊C₄	—	67	2	no	—
334	C₂²×C₂²⋊C₄	—	67	4	no	—
335	C₂²×C₂²⋊C₄	—	67	8	no	—
336	C₂²×C₂²⋊C₄	—	67	16	no	—
337	C₂²×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
338	C₂⁵	—	268	8	no	—
339	C₂⁵	—	67	1	no	—
340	C₂⁵	—	67	8	no	—
341	C₂⁵	—	67	4	no	—
342	C₂⁵	—	67	1	no	—
343	C₂⁵	—	134	8	no	—
344	C₂⁵	C₂²×D₆₇	1	2	yes	—
345	C₂⁵	C₂²×D₆₇	1	1	yes	characteristic
346	C₂×#41	—	268	16	no	—
347	C₂×#41	—	67	32	no	—
348	C₂²×Q₈	—	67	4	no	—
349	C₂²×D₄	—	2	16	no	—
350	C₂²×D₄	—	268	16	no	—
351	C₂²×D₄	—	67	8	no	—
352	C₂²×D₄	—	67	32	no	—
353	C₂²×D₄	—	67	8	no	—
354	C₂²×D₄	C₂²×D₆₇	1	4	yes	—
355	C₂²×D₄	C₂²×D₆₇	1	4	yes	—
356	C₂²×D₄	C₂²×D₆₇	1	16	yes	—
357	C₂²×D₄	C₂²×D₆₇	1	4	yes	complemented
358	C₂³×C₄	—	268	8	no	—
359	C₂³×C₄	—	67	2	no	—
360	C₂³×C₄	—	67	4	no	—
361	C₂³×C₄	—	67	8	no	—
362	C₂³×C₄	—	67	4	no	—
363	C₂³×C₄	—	67	16	no	—
364	C₂³×C₄	—	67	4	no	—
365	C₂³×C₄	C₂²×D₆₇	1	2	yes	—
366	C₂³×C₄	C₂²×D₆₇	1	2	yes	—
367	#126	—	67	64	no	—
368	#124	—	67	64	no	—
369	#123	—	67	64	no	—
370	#123	—	67	64	no	—
371	#123	C₂²×D₆₇	1	32	yes	complemented
372	C₂²⋊Q₈	—	67	32	no	—
373	C₂²⋊Q₈	—	67	32	no	—
374	C₂²⋊Q₈	—	67	64	no	—
375	#121	—	67	32	no	—
376	#121	—	67	64	no	—
377	#121	C₂²×D₆₇	1	32	yes	complemented
378	#120	—	67	32	no	—
379	#120	—	67	32	no	—
380	#120	—	67	64	no	—
381	#120	C₂²×D₆₇	1	32	yes	complemented
382	C₄×D₄	—	67	64	no	—
383	C₄×D₄	—	67	64	no	—
384	C₄×D₄	—	67	64	no	—
385	#117	—	67	32	no	—
386	#117	—	67	32	no	—
387	C₂×C₄⋊C₄	—	67	8	no	—
388	C₂×C₄⋊C₄	—	67	4	no	—
389	C₂×C₄⋊C₄	—	67	8	no	—
390	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	complemented
391	C₂×C₂²⋊C₄	—	67	4	no	—
392	C₂×C₂²⋊C₄	—	67	8	no	—
393	C₂×C₂²⋊C₄	—	67	4	no	—
394	C₂×C₂²⋊C₄	—	67	4	no	—
395	C₂×C₂²⋊C₄	—	67	8	no	—
396	C₂×C₂²⋊C₄	—	67	16	no	—
397	C₂×C₂²⋊C₄	—	67	32	no	—
398	C₂×C₂²⋊C₄	—	67	8	no	—
399	C₂×C₂²⋊C₄	—	67	16	no	—
400	C₂×C₂²⋊C₄	—	67	8	no	—
401	C₂×C₂²⋊C₄	—	67	8	no	—
402	C₂×C₂²⋊C₄	—	67	32	no	—
403	C₂×C₂²⋊C₄	—	67	64	no	—
404	C₂×C₂²⋊C₄	C₂²×D₆₇	1	4	yes	—
405	C₂×C₂²⋊C₄	C₂²×D₆₇	1	8	yes	—
406	C₂×C₂²⋊C₄	C₂²×D₆₇	1	8	yes	complemented
407	C₂×C₄²	—	67	8	no	—
408	C₂⁴	—	2	8	no	—
409	C₂⁴	—	4	4	no	—
410	C₂⁴	—	4	8	no	—
411	C₂⁴	—	268	4	no	—
412	C₂⁴	—	268	8	no	—
413	C₂⁴	—	268	64	no	—
414	C₂⁴	—	268	32	no	—
415	C₂⁴	—	67	4	no	—
416	C₂⁴	—	67	16	no	—
417	C₂⁴	—	67	2	no	—
418	C₂⁴	—	67	4	no	—
419	C₂⁴	—	67	2	no	—
420	C₂⁴	—	134	16	no	—
421	C₂⁴	—	134	8	no	—
422	C₂⁴	—	134	4	no	—
423	C₂⁴	—	134	16	no	—
424	C₂⁴	—	134	32	no	—
425	C₂⁴	—	134	4	no	—
426	C₂⁴	#11761627	1	8	yes	complemented
427	C₂⁴	C₂³×D₆₇	1	4	yes	—
428	C₂⁴	C₂³×D₆₇	1	2	yes	—
429	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
430	C₂⁴	C₂³×D₆₇	1	8	yes	—
431	C₂⁴	C₂³×D₆₇	1	4	yes	—
432	#41	—	268	64	no	—
433	C₂×Q₈	—	268	8	no	—
434	C₂×Q₈	—	67	16	no	—
435	C₂×D₄	—	2	16	no	—
436	C₂×D₄	—	2	16	no	—
437	C₂×D₄	—	2	64	no	—
438	C₂×D₄	—	4	8	no	—
439	C₂×D₄	—	268	64	no	—
440	C₂×D₄	—	268	16	no	—
441	C₂×D₄	—	67	32	no	—
442	C₂×D₄	—	67	32	no	—
443	C₂×D₄	—	134	32	no	—
444	C₂×D₄	C₂³×D₆₇	1	16	yes	—
445	C₂×D₄	C₂³×D₆₇	1	16	yes	—
446	C₂×D₄	C₂³×D₆₇	1	16	yes	complemented
447	C₂²×C₄	—	2	8	no	—
448	C₂²×C₄	—	4	4	no	—
449	C₂²×C₄	—	268	8	no	—
450	C₂²×C₄	—	268	32	no	—
451	C₂²×C₄	—	268	4	no	—
452	C₂²×C₄	—	268	8	no	—
453	C₂²×C₄	—	268	16	no	—
454	C₂²×C₄	—	67	8	no	—
455	C₂²×C₄	—	67	16	no	—
456	C₂²×C₄	—	67	32	no	—
457	C₂²×C₄	—	67	16	no	—
458	C₂²×C₄	—	67	4	no	—
459	C₂²×C₄	—	67	8	no	—
460	C₂²×C₄	—	67	16	no	—
461	C₂²×C₄	—	134	16	no	—
462	C₂²×C₄	C₂³×D₆₇	1	8	yes	—
463	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
464	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
465	C₂²×C₄	C₂³×D₆₇	1	8	yes	—
466	C₄⋊C₄	—	67	32	no	—
467	C₄⋊C₄	—	67	16	no	—
468	C₄⋊C₄	—	67	32	no	—
469	C₄⋊C₄	C₂³×D₆₇	1	16	yes	complemented
470	C₂²⋊C₄	—	67	16	no	—
471	C₂²⋊C₄	—	67	16	no	—
472	C₂²⋊C₄	—	67	16	no	—
473	C₂²⋊C₄	—	67	32	no	—
474	C₂²⋊C₄	—	67	32	no	—
475	C₂²⋊C₄	—	67	32	no	—
476	C₂²⋊C₄	—	67	32	no	—
477	C₂²⋊C₄	C₂³×D₆₇	1	16	yes	—
478	C₂²⋊C₄	C₂³×D₆₇	1	32	yes	complemented
479	C₄²	—	67	32	no	—
480	C₂³	—	2	4	no	—
481	C₂³	—	2	8	no	—
482	C₂³	—	2	8	no	—
483	C₂³	—	2	16	no	—
484	C₂³	—	2	16	no	—
485	C₂³	—	4	16	no	—
486	C₂³	—	4	8	no	—
487	C₂³	—	4	16	no	—
488	C₂³	—	8	16	no	—
489	C₂³	—	268	16	no	—
490	C₂³	—	268	64	no	—
491	C₂³	—	268	8	no	—
492	C₂³	—	268	16	no	—
493	C₂³	—	268	128	no	—
494	C₂³	—	67	8	no	—
495	C₂³	—	67	4	no	—
496	C₂³	—	67	4	no	—
497	C₂³	—	134	32	no	—
498	C₂³	—	134	16	no	—
499	C₂³	—	134	32	no	—
500	C₂³	—	134	8	no	—
501	C₂³	—	134	32	no	—
502	C₂³	—	134	4	no	—
503	C₂³	—	134	8	no	—
504	C₂³	—	134	8	no	—
505	C₂³	—	134	4	no	—
506	C₂³	—	134	4	no	—
507	C₂³	C₂×#11761627	1	8	yes	—
508	C₂³	C₂×#11761627	1	4	yes	—
509	C₂³	C₂×#11761627	1	16	yes	complemented
510	C₂³	C₂⁴×D₆₇	1	8	yes	—
511	C₂³	C₂⁴×D₆₇	1	4	yes	—
512	C₂³	C₂⁴×D₆₇	1	2	yes	—
513	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
514	Q₈	—	268	16	no	—
515	D₄	—	2	32	no	—
516	D₄	—	2	32	no	—
517	D₄	—	4	16	no	—
518	D₄	—	268	32	no	—
519	D₄	—	134	64	no	—
520	C₂×C₄	—	2	16	no	—
521	C₂×C₄	—	2	8	no	—
522	C₂×C₄	—	2	8	no	—
523	C₂×C₄	—	4	8	no	—
524	C₂×C₄	—	268	16	no	—
525	C₂×C₄	—	268	8	no	—
526	C₂×C₄	—	268	16	no	—
527	C₂×C₄	—	268	32	no	—
528	C₂×C₄	—	67	8	no	—
529	C₂×C₄	—	67	16	no	—
530	C₂×C₄	—	134	32	no	—
531	C₂×C₄	—	134	8	no	—
532	C₂×C₄	—	134	16	no	—
533	C₂×C₄	C₂⁴×D₆₇	1	8	yes	—
534	C₂×C₄	C₂⁴×D₆₇	1	8	yes	—
535	C₂²	—	2	16	no	—
536	C₂²	—	2	8	no	—
537	C₂²	—	2	4	no	—
538	C₂²	—	2	8	no	—
539	C₂²	—	2	8	no	—
540	C₂²	—	2	16	no	—
541	C₂²	—	4	16	no	—
542	C₂²	—	4	16	no	—
543	C₂²	—	8	16	no	—
544	C₂²	—	268	16	no	—
545	C₂²	—	268	64	no	—
546	C₂²	—	268	8	no	—
547	C₂²	—	67	8	no	—
548	C₂²	—	134	32	no	—
549	C₂²	—	134	4	no	—
550	C₂²	—	134	8	no	—
551	C₂²	—	134	16	no	—
552	C₂²	—	134	4	no	—
553	C₂²	—	134	4	no	—
554	C₂²	C₂²×#11761627	1	8	yes	—
555	C₂²	C₂²×#11761627	1	4	yes	—
556	C₂²	C₂²×#11761627	1	2	yes	—
557	C₂²	H(5,𝔽₂)×D₆₇	1	4	yes	—
558	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
559	C₄	—	2	8	no	—
560	C₄	—	2	8	no	—
561	C₄	—	134	8	no	—
562	C₄	—	134	16	no	—
563	C₂	—	2	8	no	—
564	C₂	—	2	4	no	—
565	C₂	—	4	8	no	—
566	C₂	—	268	4	no	—
567	C₂	—	134	8	no	—
568	C₂	#22573897	1	4	yes	direct product factor
569	C₂	C₂³×#11761627	1	1	yes	characteristic
570	C₂	C₂×H(5,𝔽₂)×D₆₇	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₆₇
commutator, cosocle	C₂²×C₆₇	C₂⁶
Frattini	C₂²	C₂⁵×D₆₇
Fitting	C₂×#534×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁵
hypercenter	C₂×#534	D₆₇
hypocenter	C₆₇	C₂×#3490
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×#534

Complements

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

N	H ≅ G/N	type
1. #22573897	C₂	direct (N × H)
2. C₂×#16864069	C₂	semidirect
3. C₂×#16864033	C₂	semidirect
4. C₂×#16864022	C₂	semidirect
5. C₂×#16863983	C₂	semidirect
6. C₂×#16863906	C₂	semidirect
7. C₂²×#14524011	C₂	semidirect
8. C₂×#16863871	C₂	semidirect
9. C₂²×C₂²⋊C₄×D₆₇	C₂	semidirect
10. C₂×#534×C₆₇	C₂	semidirect
12. C₂²×#11761627	C₂²	semidirect
13. C₂³×C₂²⋊D₆₇	C₂²	semidirect
14. C₂³×C₄×D₆₇	C₂²	semidirect
15. #16864069	C₂²	semidirect
16. #16864033	C₂²	semidirect
17. #16864022	C₂²	semidirect
18. #16863983	C₂²	semidirect
19. C₂×#14524038	C₂²	semidirect
20. C₂×#14524029	C₂²	semidirect
21. #16863906	C₂²	semidirect
22. C₂×#14524027	C₂²	semidirect
23. C₂×#14524025	C₂²	semidirect
24. C₂×#14524023	C₂²	semidirect
25. C₂×#14524022	C₂²	semidirect
26. C₂×#14524020	C₂²	semidirect
27. C₂×#14524015	C₂²	semidirect
28. C₂×#14524014	C₂²	semidirect
30. C₂×#14524011	C₂²	semidirect
31. C₂×#14524011	C₂²	semidirect
32. C₂×#14524011	C₂²	semidirect
33. #16863871	C₂²	semidirect
34. C₂×D₄×Dic₆₇	C₂²	semidirect
35. C₂×D₄×Dic₆₇	C₂²	semidirect
36. C₂×(C₂²⋊D₆₇)⋊C₄	C₂²	semidirect
37. C₂×#14524002	C₂²	semidirect
38. C₂×#14524001	C₂²	semidirect
41. C₂²×C₂²⋊Dic₆₇	C₂²	semidirect
42. C₂²×(C₂×D₆₇)⋊C₄	C₂²	semidirect
43. C₂²×(C₂×D₆₇)⋊C₄	C₂²	semidirect
44. C₂×C₂²⋊C₄×D₆₇	C₂²	semidirect
45. C₂×C₂²⋊C₄×D₆₇	C₂²	semidirect
47. #534×C₆₇	C₂²	semidirect
48. C₂×#123×C₆₇	C₂²	semidirect
49. C₂×#121×C₆₇	C₂²	semidirect
50. C₂×#120×C₆₇	C₂²	semidirect
59. C₂×#11761627	C₂³	semidirect
60. C₂²×Dic₁₃₄	C₂³	semidirect
64. C₂²×C₂²⋊D₆₇	C₂³	semidirect
67. C₂³×Dic₆₇	C₂³	semidirect
71. C₂²×C₄×D₆₇	C₂³	semidirect
72. #14524038	C₂³	semidirect
73. #14524029	C₂³	semidirect
74. #14524027	C₂³	semidirect
75. #14524025	C₂³	semidirect
76. #14524023	C₂³	semidirect
77. #14524022	C₂³	semidirect
78. #14524020	C₂³	semidirect
79. #14524015	C₂³	semidirect
80. #14524014	C₂³	semidirect
82. #14524011	C₂³	semidirect
83. #14524011	C₂³	semidirect
84. D₄×Dic₆₇	C₂³	semidirect
85. D₄×Dic₆₇	C₂³	semidirect
86. (C₂²⋊D₆₇)⋊C₄	C₂³	semidirect
87. #14524002	C₂³	semidirect
88. #14524001	C₂³	semidirect
89. C₂×C₄⋊Dic₆₇	C₂³	semidirect
90. C₂×C₄⋊Dic₆₇	C₂³	semidirect
91. C₂×Dic₆₇⋊C₄	C₂³	semidirect
97. C₂×C₂²⋊Dic₆₇	C₂³	semidirect
98. C₂×C₂²⋊Dic₆₇	C₂³	semidirect
102. C₂×(C₂×D₆₇)⋊C₄	C₂³	semidirect
103. C₂×(C₂×D₆₇)⋊C₄	C₂³	semidirect
104. C₂²⋊C₄×D₆₇	C₂³	semidirect
105. C₂×C₄×Dic₆₇	C₂³	semidirect
112. C₂²×D₄×C₆₇	C₂³	semidirect
115. #123×C₆₇	C₂³	semidirect
116. #121×C₆₇	C₂³	semidirect
117. #120×C₆₇	C₂³	semidirect
118. C₂×C₄⋊C₄×C₆₇	C₂³	semidirect
121. C₂×C₂²⋊C₄×C₆₇	C₂³	semidirect
139. C₂×Dic₁₃₄	C₂⁴	semidirect
152. C₂²×Dic₆₇	C₂⁴	semidirect
158. C₄⋊Dic₆₇	C₂⁴	semidirect
159. C₄⋊Dic₆₇	C₂⁴	semidirect
160. Dic₆₇⋊C₄	C₂⁴	semidirect
164. C₂²⋊Dic₆₇	C₂⁴	semidirect
168. C₄×Dic₆₇	C₂⁴	semidirect
172. C₂⁴×C₆₇	#41	semidirect
184. C₂×D₄×C₆₇	C₂⁴	semidirect
191. C₄⋊C₄×C₆₇	C₂⁴	semidirect
193. C₂²⋊C₄×C₆₇	C₂⁴	semidirect
205. C₂³×C₆₇	C₂×#41	semidirect
281. C₂×C₆₇	#3490	semidirect
291. C₂×#534	D₆₇	semidirect
296. C₆₇	C₂×#3490	semidirect
306. #534	C₂×D₆₇	semidirect
312. C₂×#123	C₂×D₆₇	semidirect
318. C₂×#121	C₂×D₆₇	semidirect
323. C₂×#120	C₂×D₆₇	semidirect
357. C₂²×D₄	C₂²×D₆₇	semidirect
371. #123	C₂²×D₆₇	semidirect
377. #121	C₂²×D₆₇	semidirect
381. #120	C₂²×D₆₇	semidirect
390. C₂×C₄⋊C₄	C₂²×D₆₇	semidirect
406. C₂×C₂²⋊C₄	C₂²×D₆₇	semidirect
426. C₂⁴	#11761627	semidirect
446. C₂×D₄	C₂³×D₆₇	semidirect
469. C₄⋊C₄	C₂³×D₆₇	semidirect
478. C₂²⋊C₄	C₂³×D₆₇	semidirect
509. C₂³	C₂×#11761627	semidirect
568. C₂	#22573897	direct (N × H)

Maximal subgroups

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

subgroup	index
#22573897	2
C₂×#16864069	2
C₂×#16864033	2
C₂×#16864022	2
C₂×#16863983	2
C₂×#16863906	2
C₂²×#14524011	2
C₂×#16863871	2
C₂²×C₂²⋊C₄×D₆₇	2
C₂×#534×C₆₇	2
C₂×#3490	67

Subgroups — C2×#22573897 #31763743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22573897 #31763743