Subgroups — C₂×#22573818 #31762916

Order 17152.

8722 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573818	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16864106	C₂	1	2	yes	maximal, complemented
3	C₂×#16864096	C₂	1	4	yes	maximal, complemented
4	C₂×#16864074	C₂	1	4	yes	maximal, complemented
5	C₂×#16864039	C₂	1	4	yes	maximal, complemented
6	C₂×#16864031	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16863944	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16863925	C₂	1	4	yes	maximal, complemented
9	C₂×#16863904	C₂	1	4	yes	maximal, complemented
10	C₂×C₄×#11761629	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×D₄×D₆₇	C₂	1	4	yes	maximal, complemented
12	C₂×#559×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#11761629	C₂²	1	2	yes	—
14	C₂²×#11761627	C₂²	1	2	yes	complemented
15	C₂²×D₄×D₆₇	C₂²	1	4	yes	—
16	C₂³×C₄×D₆₇	C₂²	1	4	yes	complemented
17	#16864106	C₂²	1	32	yes	complemented
18	#16864096	C₂²	1	64	yes	complemented
19	#16864074	C₂²	1	64	yes	complemented
20	#16864039	C₂²	1	64	yes	complemented
21	#16864031	C₂²	1	16	yes	complemented
22	#16863944	C₂²	1	16	yes	complemented
23	C₂×#14524043	C₂²	1	1	yes	characteristic, complemented
24	#16863925	C₂²	1	64	yes	complemented
25	C₂×#14524036	C₂²	1	8	yes	complemented
26	C₂×#14524035	C₂²	1	4	yes	complemented
27	C₂×#14524030	C₂²	1	2	yes	complemented
28	#16863904	C₂²	1	64	yes	complemented
29	C₂×#14524026	C₂²	1	4	yes	complemented
30	C₂×#14524025	C₂²	1	8	yes	complemented
31	C₂×#14524021	C₂²	1	8	yes	complemented
32	C₂×#14524019	C₂²	1	4	yes	complemented
33	C₂×#14524016	C₂²	1	8	yes	complemented
34	C₂×#14524014	C₂²	1	4	yes	—
35	C₂×#14524014	C₂²	1	4	yes	complemented
36	C₂×#14524013	C₂²	1	4	yes	complemented
37	C₂×#14524011	C₂²	1	4	yes	—
38	C₄×#11761629	C₂²	1	32	yes	complemented
39	C₂×Q₈×Dic₆₇	C₂²	1	2	yes	complemented
40	C₂×C₄×D₂₆₈	C₂²	1	4	yes	—
41	C₂×C₄×D₂₆₈	C₂²	1	2	yes	—
42	C₂×C₄×D₂₆₈	C₂²	1	4	yes	complemented
43	C₂×D₄×Dic₆₇	C₂²	1	4	yes	complemented
44	C₂×#14524005	C₂²	1	2	yes	—
45	C₂×#14524005	C₂²	1	4	yes	complemented
46	C₂×C₄×C₂²⋊D₆₇	C₂²	1	8	yes	complemented
47	C₂×(C₂²⋊D₆₇)⋊C₄	C₂²	1	8	yes	complemented
48	C₄×D₄×D₆₇	C₂²	1	64	yes	complemented
49	C₂×#14524001	C₂²	1	4	yes	complemented
50	C₂×#14524001	C₂²	1	2	yes	complemented
51	C₂×#14523999	C₂²	1	4	yes	complemented
52	C₂×C₄⋊C₄×D₆₇	C₂²	1	4	yes	complemented
53	C₂×C₂²⋊C₄×D₆₇	C₂²	1	8	yes	complemented
54	C₂×C₄²×D₆₇	C₂²	1	4	yes	complemented
55	C₂×C₄²×D₆₇	C₂²	1	1	yes	characteristic, complemented
56	#559×C₆₇	C₂²	1	16	yes	complemented
57	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
58	C₂×#124×C₆₇	C₂²	1	4	yes	complemented
59	C₂×#123×C₆₇	C₂²	1	4	yes	complemented
60	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
61	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
62	C₂⁴×D₆₇	C₂³	1	4	yes	—
63	C₂×#11761629	—	2	16	no	—
64	C₂×#11761629	C₂³	1	16	yes	—
65	C₂×#11761627	—	4	8	no	—
66	C₂×#11761627	C₂³	1	16	yes	complemented
67	C₂²×Dic₁₃₄	C₂³	1	2	yes	—
68	C₂²×D₂₆₈	C₂³	1	4	yes	—
69	C₂²×D₂₆₈	C₂³	1	2	yes	—
70	C₂²×C₂²⋊D₆₇	C₂³	1	8	yes	—
71	C₂²×C₂²⋊D₆₇	C₂³	1	4	yes	—
72	C₂×D₄×D₆₇	—	2	32	no	—
73	C₂×D₄×D₆₇	C₂³	1	32	yes	—
74	C₂³×Dic₆₇	C₂³	1	4	yes	—
75	C₂²×C₄×D₆₇	—	4	16	no	—
76	C₂²×C₄×D₆₇	C₂³	1	4	yes	—
77	C₂²×C₄×D₆₇	C₂³	1	4	yes	—
78	C₂²×C₄×D₆₇	C₂³	1	2	yes	—
79	C₂²×C₄×D₆₇	C₂³	1	4	yes	—
80	C₂²×C₄×D₆₇	C₂³	1	1	yes	characteristic
81	C₂²×C₄×D₆₇	C₂³	1	8	yes	—
82	C₂²×C₄×D₆₇	C₂³	1	32	yes	complemented
83	#14524043	C₂³	1	8	yes	complemented
84	#14524036	C₂³	1	64	yes	complemented
85	#14524035	C₂³	1	32	yes	complemented
86	#14524030	C₂³	1	16	yes	complemented
87	#14524026	C₂³	1	32	yes	complemented
88	#14524025	C₂³	1	64	yes	complemented
89	#14524021	C₂³	1	64	yes	complemented
90	#14524019	C₂³	1	32	yes	complemented
91	#14524016	C₂³	1	64	yes	complemented
92	#14524014	C₂³	1	32	yes	—
93	#14524014	C₂³	1	32	yes	complemented
94	#14524013	C₂³	1	32	yes	complemented
95	#14524011	C₂³	1	32	yes	—
96	Q₈×Dic₆₇	C₂³	1	16	yes	complemented
97	C₄×D₂₆₈	C₂³	1	16	yes	—
98	C₄×D₂₆₈	C₂³	1	32	yes	—
99	C₄×D₂₆₈	C₂³	1	32	yes	complemented
100	D₄×Dic₆₇	C₂³	1	32	yes	complemented
101	#14524005	C₂³	1	16	yes	—
102	#14524005	C₂³	1	32	yes	complemented
103	C₄×C₂²⋊D₆₇	C₂³	1	64	yes	complemented
104	(C₂²⋊D₆₇)⋊C₄	C₂³	1	64	yes	complemented
105	#14524001	C₂³	1	32	yes	complemented
106	#14524001	C₂³	1	16	yes	complemented
107	#14523999	C₂³	1	32	yes	complemented
108	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
109	C₂×C₄⋊Dic₆₇	C₂³	1	2	yes	—
110	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
111	C₂×Dic₆₇⋊C₄	C₂³	1	8	yes	—
112	C₂×Dic₆₇⋊C₄	C₂³	1	8	yes	—
113	C₄⋊C₄×D₆₇	C₂³	1	32	yes	complemented
114	C₂×C₂²⋊Dic₆₇	C₂³	1	8	yes	—
115	C₂×C₂²⋊Dic₆₇	C₂³	1	4	yes	—
116	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	4	yes	—
117	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
118	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	4	yes	—
119	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	4	yes	—
120	C₂²⋊C₄×D₆₇	C₂³	1	64	yes	complemented
121	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
122	C₂×C₄×Dic₆₇	C₂³	1	4	yes	—
123	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
124	C₂×C₄×Dic₆₇	C₂³	1	4	yes	—
125	C₄²×D₆₇	C₂³	1	32	yes	complemented
126	C₄²×D₆₇	C₂³	1	8	yes	complemented
127	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
128	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
129	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
130	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
131	#127×C₆₇	C₂³	1	8	yes	—
132	#124×C₆₇	C₂³	1	32	yes	complemented
133	#123×C₆₇	C₂³	1	32	yes	complemented
134	C₄×Q₈×C₆₇	C₂³	1	16	yes	complemented
135	C₄×D₄×C₆₇	C₂³	1	32	yes	complemented
136	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
137	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
138	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
139	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
140	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
141	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
142	C₂³×D₆₇	—	2	32	no	—
143	C₂³×D₆₇	—	4	8	no	—
144	C₂³×D₆₇	C₂⁴	1	1	yes	characteristic
145	C₂³×D₆₇	C₂⁴	1	16	yes	—
146	C₂³×D₆₇	C₂⁴	1	4	yes	—
147	#11761629	—	2	64	no	—
148	#11761627	—	4	32	no	—
149	C₂×Dic₁₃₄	—	4	4	no	—
150	C₂×Dic₁₃₄	C₂⁴	1	8	yes	—
151	C₂×D₂₆₈	—	2	16	no	—
152	C₂×D₂₆₈	#41	1	16	yes	complemented
153	C₂×D₂₆₈	C₂⁴	1	16	yes	—
154	C₂×D₂₆₈	C₂⁴	1	8	yes	—
155	C₂×C₂²⋊D₆₇	—	2	32	no	—
156	C₂×C₂²⋊D₆₇	—	4	8	no	—
157	C₂×C₂²⋊D₆₇	C₂⁴	1	32	yes	—
158	C₂×C₂²⋊D₆₇	C₂⁴	1	16	yes	—
159	D₄×D₆₇	—	2	128	no	—
160	C₂²×Dic₆₇	—	2	16	no	—
161	C₂²×Dic₆₇	C₂⁴	1	2	yes	—
162	C₂²×Dic₆₇	C₂⁴	1	4	yes	—
163	C₂²×Dic₆₇	C₂⁴	1	4	yes	—
164	C₂²×Dic₆₇	C₂⁴	1	16	yes	—
165	C₂²×Dic₆₇	C₂⁴	1	1	yes	characteristic
166	C₂×C₄×D₆₇	—	2	16	no	—
167	C₂×C₄×D₆₇	—	2	16	no	—
168	C₂×C₄×D₆₇	—	2	8	no	—
169	C₂×C₄×D₆₇	—	2	16	no	—
170	C₂×C₄×D₆₇	—	2	4	no	—
171	C₂×C₄×D₆₇	—	4	64	no	—
172	C₂×C₄×D₆₇	—	4	16	no	—
173	C₂×C₄×D₆₇	C₂⁴	1	16	yes	—
174	C₂×C₄×D₆₇	C₂⁴	1	16	yes	—
175	C₂×C₄×D₆₇	C₂⁴	1	8	yes	—
176	C₂×C₄×D₆₇	C₂⁴	1	16	yes	—
177	C₂×C₄×D₆₇	C₂⁴	1	4	yes	—
178	C₂×C₄×D₆₇	C₂⁴	1	32	yes	—
179	C₄⋊Dic₆₇	C₂⁴	1	16	yes	—
180	C₄⋊Dic₆₇	C₂⁴	1	8	yes	—
181	C₄⋊Dic₆₇	C₂⁴	1	16	yes	—
182	Dic₆₇⋊C₄	C₂⁴	1	32	yes	—
183	Dic₆₇⋊C₄	C₂⁴	1	32	yes	—
184	C₂²⋊Dic₆₇	C₂⁴	1	32	yes	—
185	C₂²⋊Dic₆₇	C₂⁴	1	16	yes	—
186	(C₂×D₆₇)⋊C₄	C₂⁴	1	16	yes	—
187	(C₂×D₆₇)⋊C₄	C₂⁴	1	32	yes	—
188	(C₂×D₆₇)⋊C₄	C₂⁴	1	16	yes	—
189	(C₂×D₆₇)⋊C₄	C₂⁴	1	16	yes	—
190	C₄×Dic₆₇	C₂⁴	1	8	yes	—
191	C₄×Dic₆₇	C₂⁴	1	16	yes	—
192	C₄×Dic₆₇	C₂⁴	1	4	yes	—
193	C₄×Dic₆₇	C₂⁴	1	16	yes	—
194	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
195	C₂×Q₈×C₆₇	—	2	8	no	—
196	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
197	C₂×D₄×C₆₇	—	2	16	no	—
198	C₂×D₄×C₆₇	—	4	4	no	—
199	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
200	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
201	C₂²×C₄×C₆₇	—	4	8	no	—
202	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
203	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
204	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
205	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
206	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
207	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
208	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
209	C₂²⋊C₄×C₆₇	C₂⁴	1	32	yes	—
210	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
211	C₄²×C₆₇	C₂⁴	1	16	yes	—
212	C₄²×C₆₇	C₂⁴	1	4	yes	—
213	C₂³×C₆₇	—	2	16	no	—
214	C₂³×C₆₇	—	2	8	no	—
215	C₂³×C₆₇	C₂⁵	1	8	yes	—
216	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
217	C₂²×D₆₇	—	2	8	no	—
218	C₂²×D₆₇	—	2	8	no	—
219	C₂²×D₆₇	—	2	4	no	—
220	C₂²×D₆₇	—	2	2	no	—
221	C₂²×D₆₇	—	2	64	no	—
222	C₂²×D₆₇	—	4	16	no	—
223	C₂²×D₆₇	—	4	64	no	—
224	C₂²×D₆₇	C₂×#41	1	16	yes	—
225	C₂²×D₆₇	C₂⁵	1	2	yes	—
226	C₂²×D₆₇	C₂⁵	1	8	yes	—
227	Q₈×C₆₇	—	2	16	no	—
228	D₄×C₆₇	—	2	32	no	—
229	D₄×C₆₇	—	4	8	no	—
230	C₂×C₄×C₆₇	—	2	8	no	—
231	C₂×C₄×C₆₇	—	2	8	no	—
232	C₂×C₄×C₆₇	—	2	2	no	—
233	C₂×C₄×C₆₇	—	4	16	no	—
234	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
235	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
236	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
237	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
238	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
239	C₂²⋊D₆₇	—	2	64	no	—
240	C₂²⋊D₆₇	—	4	16	no	—
241	C₂×Dic₆₇	—	2	4	no	—
242	C₂×Dic₆₇	—	2	8	no	—
243	C₂×Dic₆₇	—	2	8	no	—
244	C₂×Dic₆₇	—	2	32	no	—
245	C₂×Dic₆₇	H(5,𝔽₂)	1	4	yes	—
246	C₂×Dic₆₇	C₂⁵	1	4	yes	—
247	C₂×Dic₆₇	C₂⁵	1	8	yes	—
248	C₂×Dic₆₇	C₂⁵	1	8	yes	—
249	C₂×Dic₆₇	C₂⁵	1	2	yes	—
250	D₂₆₈	—	2	32	no	—
251	D₂₆₈	C₂×#41	1	32	yes	complemented
252	C₄×D₆₇	—	2	32	no	—
253	C₄×D₆₇	—	2	32	no	—
254	C₄×D₆₇	—	2	16	no	—
255	C₄×D₆₇	—	2	32	no	—
256	C₄×D₆₇	—	2	8	no	—
257	C₄×D₆₇	—	4	32	no	—
258	Dic₁₃₄	—	4	8	no	—
259	C₂²×C₆₇	—	2	16	no	—
260	C₂²×C₆₇	—	2	8	no	—
261	C₂²×C₆₇	—	4	16	no	—
262	C₂²×C₆₇	C₂²×#41	1	4	yes	—
263	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
264	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
265	C₂×D₆₇	—	2	8	no	—
266	C₂×D₆₇	—	2	8	no	—
267	C₂×D₆₇	—	2	32	no	—
268	C₂×D₆₇	—	2	4	no	—
269	C₂×D₆₇	—	2	2	no	—
270	C₂×D₆₇	—	4	64	no	—
271	C₂×D₆₇	—	4	4	no	—
272	C₂×D₆₇	C₂²×#41	1	16	yes	—
273	C₄×C₆₇	—	2	8	no	—
274	C₄×C₆₇	—	2	2	no	—
275	C₄×C₆₇	—	2	8	no	—
276	C₄×C₆₇	C₂²×#41	1	8	yes	—
277	Dic₆₇	—	2	4	no	—
278	Dic₆₇	—	2	8	no	—
279	Dic₆₇	—	2	8	no	—
280	Dic₆₇	C₂×H(5,𝔽₂)	1	4	yes	—
281	C₂×#3481	—	67	1	no	maximal, Sylow
282	C₂×C₆₇	—	4	8	no	—
283	C₂×C₆₇	#3481	1	4	yes	complemented
284	C₂×C₆₇	C₂³×#41	1	2	yes	—
285	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
286	D₆₇	—	2	16	no	—
287	D₆₇	—	4	2	no	—
288	#3481	—	67	32	no	—
289	C₂×#559	D₆₇	1	1	yes	characteristic, complemented
290	C₂×#554	—	67	2	no	—
291	C₂×#553	—	67	4	no	—
292	C₂×#551	—	67	4	no	—
293	C₂×#546	—	67	4	no	—
294	C₂×D₄²	—	67	1	no	—
295	C₂×#532	—	67	1	no	—
296	C₂×#529	—	67	4	no	—
297	C₂×#525	—	67	4	no	—
298	C₂×C₄×#41	—	67	2	no	—
299	C₂²×C₄×D₄	—	67	4	no	—
300	C₆₇	C₂×#3481	1	1	yes	characteristic, complemented, Sylow
301	C₂²×#41	—	67	2	no	—
302	C₂²×#41	—	67	2	no	—
303	C₂³×D₄	—	67	4	no	—
304	C₂⁴×C₄	—	67	4	no	—
305	#559	C₂×D₆₇	1	16	yes	complemented
306	#554	—	67	32	no	—
307	#553	—	67	64	no	—
308	#551	—	67	64	no	—
309	#546	—	67	64	no	—
310	D₄²	—	67	16	no	—
311	#532	—	67	16	no	—
312	C₂×C₄⋊Q₈	—	67	1	no	—
313	C₂×#127	C₂×D₆₇	1	1	yes	characteristic
314	#529	—	67	64	no	—
315	C₂×#126	—	67	8	no	—
316	C₂×#125	—	67	4	no	—
317	C₂×#124	—	67	2	no	—
318	C₂×#124	C₂×D₆₇	1	4	yes	complemented
319	#525	—	67	64	no	—
320	C₂×#123	—	67	4	no	—
321	C₂×#123	—	67	8	no	—
322	C₂×#123	C₂×D₆₇	1	4	yes	complemented
323	C₂×C₂²⋊Q₈	—	67	4	no	—
324	C₂×C₂²⋊Q₈	—	67	8	no	—
325	C₂×#121	—	67	4	no	—
326	C₂×#121	—	67	4	no	—
327	C₂×#121	—	67	8	no	—
328	C₂×#120	—	67	4	no	—
329	C₂×#120	—	67	4	no	—
330	C₄×#41	—	67	32	no	—
331	C₂×C₄×Q₈	—	67	2	no	—
332	C₂×C₄×Q₈	C₂×D₆₇	1	2	yes	complemented
333	C₂×C₄×D₄	—	67	4	no	—
334	C₂×C₄×D₄	—	67	8	no	—
335	C₂×C₄×D₄	—	67	64	no	—
336	C₂×C₄×D₄	—	67	2	no	—
337	C₂×C₄×D₄	—	67	4	no	—
338	C₂×C₄×D₄	—	67	8	no	—
339	C₂×C₄×D₄	—	67	4	no	—
340	C₂×C₄×D₄	—	67	2	no	—
341	C₂×C₄×D₄	—	67	4	no	—
342	C₂×C₄×D₄	C₂×D₆₇	1	4	yes	complemented
343	C₂×#117	—	67	4	no	—
344	C₂×#117	—	67	2	no	—
345	C₂×#117	—	67	4	no	—
346	C₂²×C₄⋊C₄	—	67	4	no	—
347	C₂²×C₂²⋊C₄	—	67	8	no	—
348	C₂²×C₄²	—	67	4	no	—
349	C₂²×C₄²	—	67	1	no	—
350	C₂⁵	—	67	4	no	—
351	C₂×#41	—	268	8	no	—
352	C₂×#41	—	67	16	no	—
353	C₂×#41	—	67	16	no	—
354	C₂×#41	—	134	16	no	—
355	C₂²×Q₈	—	67	2	no	—
356	C₂²×Q₈	C₂²×D₆₇	1	2	yes	—
357	C₂²×D₄	—	67	4	no	—
358	C₂²×D₄	—	67	8	no	—
359	C₂²×D₄	—	67	32	no	—
360	C₂²×D₄	—	67	2	no	—
361	C₂²×D₄	—	67	4	no	—
362	C₂²×D₄	—	134	32	no	—
363	C₂²×D₄	C₂²×D₆₇	1	4	yes	—
364	C₂²×D₄	C₂²×D₆₇	1	2	yes	—
365	C₂³×C₄	—	268	16	no	—
366	C₂³×C₄	—	67	8	no	—
367	C₂³×C₄	—	67	32	no	—
368	C₂³×C₄	—	67	4	no	—
369	C₂³×C₄	—	67	4	no	—
370	C₂³×C₄	—	67	4	no	—
371	C₂³×C₄	—	67	2	no	—
372	C₂³×C₄	—	67	4	no	—
373	C₂³×C₄	—	67	1	no	—
374	C₂³×C₄	C₂²×D₆₇	1	4	yes	—
375	C₄⋊Q₈	—	67	8	no	—
376	#127	C₂²×D₆₇	1	8	yes	—
377	#126	—	67	64	no	—
378	#125	—	67	32	no	—
379	#124	—	67	16	no	—
380	#124	C₂²×D₆₇	1	32	yes	complemented
381	#123	—	67	32	no	—
382	#123	—	67	64	no	—
383	#123	C₂²×D₆₇	1	32	yes	complemented
384	C₂²⋊Q₈	—	67	32	no	—
385	C₂²⋊Q₈	—	67	64	no	—
386	#121	—	67	32	no	—
387	#121	—	67	32	no	—
388	#121	—	67	64	no	—
389	#120	—	67	32	no	—
390	#120	—	67	32	no	—
391	C₄×Q₈	—	67	16	no	—
392	C₄×Q₈	C₂²×D₆₇	1	16	yes	complemented
393	C₄×D₄	—	67	32	no	—
394	C₄×D₄	—	67	64	no	—
395	C₄×D₄	—	67	16	no	—
396	C₄×D₄	—	67	32	no	—
397	C₄×D₄	—	67	64	no	—
398	C₄×D₄	—	67	32	no	—
399	C₄×D₄	—	67	16	no	—
400	C₄×D₄	—	67	32	no	—
401	C₄×D₄	C₂²×D₆₇	1	32	yes	complemented
402	#117	—	67	32	no	—
403	#117	—	67	16	no	—
404	#117	—	67	32	no	—
405	C₂×C₄⋊C₄	—	67	4	no	—
406	C₂×C₄⋊C₄	—	67	8	no	—
407	C₂×C₄⋊C₄	—	67	2	no	—
408	C₂×C₄⋊C₄	—	67	4	no	—
409	C₂×C₄⋊C₄	—	67	8	no	—
410	C₂×C₄⋊C₄	—	67	32	no	—
411	C₂×C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
412	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
413	C₂×C₂²⋊C₄	—	67	8	no	—
414	C₂×C₂²⋊C₄	—	67	4	no	—
415	C₂×C₂²⋊C₄	—	67	4	no	—
416	C₂×C₂²⋊C₄	—	67	8	no	—
417	C₂×C₂²⋊C₄	—	67	4	no	—
418	C₂×C₂²⋊C₄	—	67	4	no	—
419	C₂×C₂²⋊C₄	—	67	64	no	—
420	C₂×C₂²⋊C₄	C₂²×D₆₇	1	8	yes	—
421	C₂×C₂²⋊C₄	C₂²×D₆₇	1	4	yes	—
422	C₂×C₄²	—	67	1	no	—
423	C₂×C₄²	—	67	4	no	—
424	C₂×C₄²	—	67	4	no	—
425	C₂×C₄²	—	67	2	no	—
426	C₂×C₄²	—	67	32	no	—
427	C₂×C₄²	—	67	8	no	—
428	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
429	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
430	C₂⁴	—	268	8	no	—
431	C₂⁴	—	67	16	no	—
432	C₂⁴	—	67	4	no	—
433	C₂⁴	—	67	1	no	—
434	C₂⁴	—	134	32	no	—
435	C₂⁴	C₂³×D₆₇	1	4	yes	—
436	#41	—	268	32	no	—
437	#41	—	134	64	no	—
438	C₂×Q₈	—	2	8	no	—
439	C₂×Q₈	—	268	4	no	—
440	C₂×Q₈	—	67	8	no	—
441	C₂×Q₈	C₂³×D₆₇	1	8	yes	—
442	C₂×D₄	—	2	16	no	—
443	C₂×D₄	—	4	4	no	—
444	C₂×D₄	—	268	8	no	—
445	C₂×D₄	—	67	16	no	—
446	C₂×D₄	—	67	32	no	—
447	C₂×D₄	—	67	16	no	—
448	C₂×D₄	—	67	8	no	—
449	C₂×D₄	—	67	16	no	—
450	C₂×D₄	—	134	16	no	—
451	C₂×D₄	—	134	128	no	—
452	C₂×D₄	—	134	32	no	—
453	C₂×D₄	C₂³×D₆₇	1	16	yes	—
454	C₂×D₄	C₂³×D₆₇	1	8	yes	—
455	C₂²×C₄	—	4	8	no	—
456	C₂²×C₄	—	268	16	no	—
457	C₂²×C₄	—	268	64	no	—
458	C₂²×C₄	—	67	32	no	—
459	C₂²×C₄	—	67	1	no	—
460	C₂²×C₄	—	67	16	no	—
461	C₂²×C₄	—	67	2	no	—
462	C₂²×C₄	—	67	4	no	—
463	C₂²×C₄	—	67	4	no	—
464	C₂²×C₄	—	67	16	no	—
465	C₂²×C₄	—	67	16	no	—
466	C₂²×C₄	—	67	8	no	—
467	C₂²×C₄	—	67	16	no	—
468	C₂²×C₄	—	67	4	no	—
469	C₂²×C₄	—	134	16	no	—
470	C₂²×C₄	—	134	4	no	—
471	C₂²×C₄	—	134	16	no	—
472	C₂²×C₄	—	134	16	no	—
473	C₂²×C₄	—	134	16	no	—
474	C₂²×C₄	—	134	8	no	—
475	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
476	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
477	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
478	C₂²×C₄	C₂³×D₆₇	1	16	yes	—
479	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
480	C₄⋊C₄	—	67	16	no	—
481	C₄⋊C₄	—	67	32	no	—
482	C₄⋊C₄	—	67	8	no	—
483	C₄⋊C₄	—	67	16	no	—
484	C₄⋊C₄	—	67	32	no	—
485	C₄⋊C₄	C₂³×D₆₇	1	8	yes	—
486	C₄⋊C₄	C₂³×D₆₇	1	16	yes	—
487	C₂²⋊C₄	—	67	32	no	—
488	C₂²⋊C₄	—	67	16	no	—
489	C₂²⋊C₄	—	67	16	no	—
490	C₂²⋊C₄	—	67	32	no	—
491	C₂²⋊C₄	—	67	16	no	—
492	C₂²⋊C₄	—	67	16	no	—
493	C₂²⋊C₄	C₂³×D₆₇	1	32	yes	—
494	C₂²⋊C₄	C₂³×D₆₇	1	16	yes	—
495	C₄²	—	67	8	no	—
496	C₄²	—	67	4	no	—
497	C₄²	—	67	16	no	—
498	C₄²	—	67	16	no	—
499	C₄²	C₂³×D₆₇	1	16	yes	—
500	C₄²	C₂³×D₆₇	1	4	yes	—
501	C₂³	—	2	16	no	—
502	C₂³	—	2	8	no	—
503	C₂³	—	268	16	no	—
504	C₂³	—	268	64	no	—
505	C₂³	—	67	16	no	—
506	C₂³	—	67	8	no	—
507	C₂³	—	67	2	no	—
508	C₂³	—	134	64	no	—
509	C₂³	—	134	4	no	—
510	C₂³	—	134	2	no	—
511	C₂³	—	134	8	no	—
512	C₂³	—	134	8	no	—
513	C₂³	C₂⁴×D₆₇	1	8	yes	—
514	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
515	Q₈	—	2	16	no	—
516	Q₈	—	268	8	no	—
517	D₄	—	2	32	no	—
518	D₄	—	4	8	no	—
519	D₄	—	268	16	no	—
520	D₄	—	67	32	no	—
521	D₄	—	134	32	no	—
522	D₄	—	134	64	no	—
523	C₂×C₄	—	2	8	no	—
524	C₂×C₄	—	2	8	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	4	16	no	—
527	C₂×C₄	—	268	32	no	—
528	C₂×C₄	—	67	4	no	—
529	C₂×C₄	—	67	2	no	—
530	C₂×C₄	—	67	4	no	—
531	C₂×C₄	—	67	8	no	—
532	C₂×C₄	—	67	8	no	—
533	C₂×C₄	—	134	32	no	—
534	C₂×C₄	—	134	8	no	—
535	C₂×C₄	—	134	32	no	—
536	C₂×C₄	—	134	32	no	—
537	C₂×C₄	—	134	4	no	—
538	C₂×C₄	—	134	8	no	—
539	C₂×C₄	—	134	8	no	—
540	C₂×C₄	—	134	32	no	—
541	C₂×C₄	—	134	16	no	—
542	C₂×C₄	#41×D₆₇	1	8	yes	—
543	C₂×C₄	C₂⁴×D₆₇	1	2	yes	—
544	C₂×C₄	C₂⁴×D₆₇	1	8	yes	—
545	C₂×C₄	C₂⁴×D₆₇	1	8	yes	—
546	C₂×C₄	C₂⁴×D₆₇	1	4	yes	—
547	C₂²	—	2	16	no	—
548	C₂²	—	2	8	no	—
549	C₂²	—	4	16	no	—
550	C₂²	—	268	64	no	—
551	C₂²	—	268	4	no	—
552	C₂²	—	67	16	no	—
553	C₂²	—	134	4	no	—
554	C₂²	—	134	2	no	—
555	C₂²	—	134	8	no	—
556	C₂²	—	134	8	no	—
557	C₂²	—	134	32	no	—
558	C₂²	C₂×#41×D₆₇	1	4	yes	—
559	C₂²	H(5,𝔽₂)×D₆₇	1	2	yes	—
560	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
561	C₄	—	2	8	no	—
562	C₄	—	2	8	no	—
563	C₄	—	2	2	no	—
564	C₄	—	67	4	no	—
565	C₄	—	134	4	no	—
566	C₄	—	134	8	no	—
567	C₄	—	134	8	no	—
568	C₄	C₂×#41×D₆₇	1	8	yes	—
569	C₂	—	4	8	no	—
570	C₂	—	268	2	no	—
571	C₂	—	134	16	no	—
572	C₂	#22573818	1	4	yes	direct product factor
573	C₂	C₂²×#41×D₆₇	1	2	yes	—
574	C₂	C₂×H(5,𝔽₂)×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₆₇
commutator, cosocle	C₂²×C₆₇	C₂⁶
Frattini	C₂²	C₂⁵×D₆₇
Fitting	C₂×#559×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁵
hypercenter	C₂×#559	D₆₇
hypocenter	C₆₇	C₂×#3481
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×#559

Complements

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

N	H ≅ G/N	type
1. #22573818	C₂	direct (N × H)
2. C₂×#16864106	C₂	semidirect
3. C₂×#16864096	C₂	semidirect
4. C₂×#16864074	C₂	semidirect
5. C₂×#16864039	C₂	semidirect
6. C₂×#16864031	C₂	semidirect
7. C₂×#16863944	C₂	semidirect
8. C₂×#16863925	C₂	semidirect
9. C₂×#16863904	C₂	semidirect
10. C₂×C₄×#11761629	C₂	semidirect
11. C₂×C₄×D₄×D₆₇	C₂	semidirect
12. C₂×#559×C₆₇	C₂	semidirect
14. C₂²×#11761627	C₂²	semidirect
16. C₂³×C₄×D₆₇	C₂²	semidirect
17. #16864106	C₂²	semidirect
18. #16864096	C₂²	semidirect
19. #16864074	C₂²	semidirect
20. #16864039	C₂²	semidirect
21. #16864031	C₂²	semidirect
22. #16863944	C₂²	semidirect
23. C₂×#14524043	C₂²	semidirect
24. #16863925	C₂²	semidirect
25. C₂×#14524036	C₂²	semidirect
26. C₂×#14524035	C₂²	semidirect
27. C₂×#14524030	C₂²	semidirect
28. #16863904	C₂²	semidirect
29. C₂×#14524026	C₂²	semidirect
30. C₂×#14524025	C₂²	semidirect
31. C₂×#14524021	C₂²	semidirect
32. C₂×#14524019	C₂²	semidirect
33. C₂×#14524016	C₂²	semidirect
35. C₂×#14524014	C₂²	semidirect
36. C₂×#14524013	C₂²	semidirect
38. C₄×#11761629	C₂²	semidirect
39. C₂×Q₈×Dic₆₇	C₂²	semidirect
42. C₂×C₄×D₂₆₈	C₂²	semidirect
43. C₂×D₄×Dic₆₇	C₂²	semidirect
45. C₂×#14524005	C₂²	semidirect
46. C₂×C₄×C₂²⋊D₆₇	C₂²	semidirect
47. C₂×(C₂²⋊D₆₇)⋊C₄	C₂²	semidirect
48. C₄×D₄×D₆₇	C₂²	semidirect
49. C₂×#14524001	C₂²	semidirect
50. C₂×#14524001	C₂²	semidirect
51. C₂×#14523999	C₂²	semidirect
52. C₂×C₄⋊C₄×D₆₇	C₂²	semidirect
53. C₂×C₂²⋊C₄×D₆₇	C₂²	semidirect
54. C₂×C₄²×D₆₇	C₂²	semidirect
55. C₂×C₄²×D₆₇	C₂²	semidirect
56. #559×C₆₇	C₂²	semidirect
58. C₂×#124×C₆₇	C₂²	semidirect
59. C₂×#123×C₆₇	C₂²	semidirect
60. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
61. C₂×C₄×D₄×C₆₇	C₂²	semidirect
66. C₂×#11761627	C₂³	semidirect
82. C₂²×C₄×D₆₇	C₂³	semidirect
83. #14524043	C₂³	semidirect
84. #14524036	C₂³	semidirect
85. #14524035	C₂³	semidirect
86. #14524030	C₂³	semidirect
87. #14524026	C₂³	semidirect
88. #14524025	C₂³	semidirect
89. #14524021	C₂³	semidirect
90. #14524019	C₂³	semidirect
91. #14524016	C₂³	semidirect
93. #14524014	C₂³	semidirect
94. #14524013	C₂³	semidirect
96. Q₈×Dic₆₇	C₂³	semidirect
99. C₄×D₂₆₈	C₂³	semidirect
100. D₄×Dic₆₇	C₂³	semidirect
102. #14524005	C₂³	semidirect
103. C₄×C₂²⋊D₆₇	C₂³	semidirect
104. (C₂²⋊D₆₇)⋊C₄	C₂³	semidirect
105. #14524001	C₂³	semidirect
106. #14524001	C₂³	semidirect
107. #14523999	C₂³	semidirect
113. C₄⋊C₄×D₆₇	C₂³	semidirect
120. C₂²⋊C₄×D₆₇	C₂³	semidirect
125. C₄²×D₆₇	C₂³	semidirect
126. C₄²×D₆₇	C₂³	semidirect
132. #124×C₆₇	C₂³	semidirect
133. #123×C₆₇	C₂³	semidirect
134. C₄×Q₈×C₆₇	C₂³	semidirect
135. C₄×D₄×C₆₇	C₂³	semidirect
152. C₂×D₂₆₈	#41	semidirect
251. D₂₆₈	C₂×#41	semidirect
283. C₂×C₆₇	#3481	semidirect
289. C₂×#559	D₆₇	semidirect
300. C₆₇	C₂×#3481	semidirect
305. #559	C₂×D₆₇	semidirect
318. C₂×#124	C₂×D₆₇	semidirect
322. C₂×#123	C₂×D₆₇	semidirect
332. C₂×C₄×Q₈	C₂×D₆₇	semidirect
342. C₂×C₄×D₄	C₂×D₆₇	semidirect
380. #124	C₂²×D₆₇	semidirect
383. #123	C₂²×D₆₇	semidirect
392. C₄×Q₈	C₂²×D₆₇	semidirect
401. C₄×D₄	C₂²×D₆₇	semidirect
572. C₂	#22573818	direct (N × H)

Maximal subgroups

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

subgroup	index
#22573818	2
C₂×#16864106	2
C₂×#16864096	2
C₂×#16864074	2
C₂×#16864039	2
C₂×#16864031	2
C₂×#16863944	2
C₂×#16863925	2
C₂×#16863904	2
C₂×C₄×#11761629	2
C₂×C₄×D₄×D₆₇	2
C₂×#559×C₆₇	2
C₂×#3481	67

Subgroups — C2×#22573818 #31762916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22573818 #31762916