Subgroups — #31725495

Order 17152.

1890 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863953	C₂	1	2	yes	maximal
2	#22570761	C₂	1	8	yes	maximal, complemented
3	#22570755	C₂	1	8	yes	maximal, complemented
4	#22570733	C₂	1	4	yes	maximal, complemented
5	#22570723	C₂	1	4	yes	maximal, complemented
6	#22568799	C₂	1	4	yes	maximal
7	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#16863953	—	2	4	no	—
9	#16863953	—	2	4	no	—
10	#16863953	C₂²	1	4	yes	—
11	#16863953	C₂²	1	4	yes	—
12	#16863953	C₂²	1	4	yes	—
13	#16863953	C₂²	1	4	yes	—
14	C₂×#14524043	C₂²	1	4	yes	—
15	C₂×#14524035	C₂²	1	4	yes	—
16	C₂×#14524022	—	2	4	no	—
17	C₂×C₄×Dic₁₃₄	—	2	4	no	—
18	C₂×#14524002	C₂²	1	4	yes	—
19	#16863805	C₂²	1	32	yes	complemented
20	#16863801	C₂²	1	8	yes	complemented
21	#16863798	C₂²	1	16	yes	complemented
22	#16863794	C₂²	1	8	yes	complemented
23	#16863438	C₂²	1	32	yes	—
24	#16863434	C₂²	1	16	yes	—
25	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₆₇	C₂²	1	4	yes	—
28	C₄×M₁₆×C₆₇	C₂²	1	4	yes	—
29	C₂×#97×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
31	C₂²×Dic₁₃₄	—	2	4	no	—
32	#14524043	—	2	4	no	—
33	#14524043	—	2	4	no	—
34	#14524043	C₂³	1	8	yes	—
35	#14524043	C₂³	1	8	yes	—
36	#14524035	—	2	8	no	—
37	#14524035	C₂³	1	8	yes	—
38	#14524035	C₂³	1	8	yes	—
39	#14524022	—	2	8	no	—
40	#14524022	—	2	8	no	—
41	#14524022	—	2	8	no	—
42	#14524022	—	2	8	no	—
43	C₄×Dic₁₃₄	—	2	8	no	—
44	C₄×Dic₁₃₄	—	2	8	no	—
45	C₄×Dic₁₃₄	—	2	8	no	—
46	C₄×Dic₁₃₄	—	2	8	no	—
47	#14524002	—	2	4	no	—
48	#14524002	—	2	4	no	—
49	#14524002	C₂³	1	8	yes	—
50	#14524002	C₂³	1	8	yes	—
51	C₂×C₄⋊Dic₆₇	—	2	4	no	—
52	C₂×C₄⋊Dic₆₇	C₂³	1	8	yes	—
53	C₂×Dic₆₇⋊C₄	—	2	8	no	—
54	C₂×C₂²⋊Dic₆₇	—	2	4	no	—
55	C₂×C₄×Dic₆₇	—	2	4	no	—
56	#14523977	C₂³	1	32	yes	—
57	#14523975	C₂³	1	32	yes	—
58	C₂³×C₄×C₆₇	D₄	1	2	yes	—
59	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₆₇	Q₈	1	16	yes	complemented
61	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
62	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
63	C₂×C₄²×C₆₇	D₄	1	4	yes	—
64	C₂×C₄²×C₆₇	D₄	1	2	yes	—
65	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
66	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
67	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
69	#97×C₆₇	C₂³	1	8	yes	—
70	C₄×C₈×C₆₇	C₂³	1	4	yes	—
71	C₄×C₈×C₆₇	C₂³	1	4	yes	—
72	C₂×Dic₁₃₄	—	2	8	no	—
73	C₂×Dic₁₃₄	—	2	8	no	—
74	C₂×Dic₁₃₄	—	4	16	no	—
75	C₂²×Dic₆₇	—	2	8	no	—
76	C₄⋊Dic₆₇	—	2	8	no	—
77	C₄⋊Dic₆₇	—	2	8	no	—
78	C₄⋊Dic₆₇	—	2	8	no	—
79	C₄⋊Dic₆₇	C₂⁴	1	16	yes	—
80	Dic₆₇⋊C₄	—	2	16	no	—
81	Dic₆₇⋊C₄	—	2	16	no	—
82	C₂²⋊Dic₆₇	—	2	8	no	—
83	C₂²⋊Dic₆₇	—	2	8	no	—
84	C₄×Dic₆₇	—	2	8	no	—
85	C₄×Dic₆₇	—	2	8	no	—
86	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	—	4	4	no	—
88	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
89	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
91	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂×Q₈	1	8	yes	—
95	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
97	M₁₆×C₆₇	C₂×Q₈	1	32	yes	complemented
98	C₂×C₈×C₆₇	C₂×Q₈	1	16	yes	complemented
99	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
100	C₄²×C₆₇	C₂×D₄	1	2	yes	—
101	C₄²×C₆₇	C₂×D₄	1	2	yes	—
102	C₄²×C₆₇	C₂×D₄	1	4	yes	—
103	C₄²×C₆₇	C₂×D₄	1	4	yes	—
104	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
105	C₄²×C₆₇	C₂⁴	1	2	yes	—
106	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
107	C₂³×C₆₇	—	4	2	no	—
108	C₂³×C₆₇	C₄⋊Q₈	1	4	yes	—
109	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
110	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
111	C₂×C₄×C₆₇	—	2	4	no	—
112	C₂×C₄×C₆₇	—	2	8	no	—
113	C₂×C₄×C₆₇	—	4	2	no	—
114	C₂×C₄×C₆₇	—	4	4	no	—
115	C₂×C₄×C₆₇	C₄⋊Q₈	1	8	yes	—
116	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
117	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
118	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
119	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
120	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
121	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
123	C₂×C₄×C₆₇	C₂²×Q₈	1	8	yes	—
124	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
125	C₂×Dic₆₇	—	2	16	no	—
126	C₂×Dic₆₇	—	4	16	no	—
127	Dic₁₃₄	—	4	16	no	—
128	Dic₁₃₄	—	4	16	no	—
129	C₈×C₆₇	C₂²×Q₈	1	16	yes	—
130	C₂²×C₆₇	—	2	4	no	—
131	C₂²×C₆₇	—	2	2	no	—
132	C₂²×C₆₇	—	4	4	no	—
133	C₂²×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
134	C₂²×C₆₇	C₂×C₄⋊Q₈	1	4	yes	—
135	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
136	C₄×C₆₇	—	2	4	no	—
137	C₄×C₆₇	—	2	2	no	—
138	C₄×C₆₇	C₂×C₄⋊Q₈	1	4	yes	—
139	C₄×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
140	C₄×C₆₇	#575	1	4	yes	—
141	C₄×C₆₇	C₂³×Q₈	1	2	yes	—
142	Dic₆₇	—	4	16	no	—
143	#60663	—	67	1	no	maximal, Sylow
144	C₂×C₆₇	—	2	4	no	—
145	C₂×C₆₇	—	2	2	no	—
146	C₂×C₆₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
147	C₂×C₆₇	C₂×#575	1	2	yes	—
148	C₂×#533	—	67	2	no	—
149	#3166	—	67	8	no	—
150	#3165	—	67	8	no	—
151	#3162	—	67	4	no	—
152	#3160	—	67	4	no	—
153	#2911	—	67	4	no	—
154	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
155	C₆₇	#60663	1	1	yes	characteristic, complemented, Sylow
156	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
157	#533	—	67	4	no	—
158	#533	—	67	4	no	—
159	#533	—	67	4	no	—
160	#533	—	67	4	no	—
161	#533	—	134	4	no	—
162	#533	—	134	4	no	—
163	C₂×C₄⋊Q₈	—	67	4	no	—
164	C₂×#125	—	67	4	no	—
165	C₂×C₂²⋊Q₈	—	134	4	no	—
166	C₂×C₄×Q₈	—	134	4	no	—
167	C₂×#117	—	67	4	no	—
168	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
169	#501	—	67	32	no	—
170	#500	—	67	8	no	—
171	#499	—	67	16	no	—
172	#498	—	67	8	no	—
173	#428	—	67	32	no	—
174	#427	—	67	16	no	—
175	C₄×M₁₆	C₂×D₆₇	1	4	yes	—
176	C₄×M₁₆	C₂×D₆₇	1	4	yes	—
177	C₂×#97	C₂×D₆₇	1	2	yes	—
178	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
179	C₂²×Q₈	—	134	4	no	—
180	C₂³×C₄	D₂₆₈	1	2	yes	—
181	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
182	C₂×M₁₆	Dic₁₃₄	1	16	yes	complemented
183	C₂×M₁₆	C₂²×D₆₇	1	8	yes	—
184	C₂²×C₈	C₂²×D₆₇	1	4	yes	—
185	C₄⋊Q₈	—	67	8	no	—
186	C₄⋊Q₈	—	67	8	no	—
187	C₄⋊Q₈	—	134	4	no	—
188	C₄⋊Q₈	—	134	4	no	—
189	#125	—	67	8	no	—
190	#125	—	67	8	no	—
191	#125	—	134	8	no	—
192	C₂²⋊Q₈	—	134	8	no	—
193	C₂²⋊Q₈	—	134	8	no	—
194	C₂²⋊Q₈	—	134	8	no	—
195	C₂²⋊Q₈	—	134	8	no	—
196	C₄×Q₈	—	134	8	no	—
197	C₄×Q₈	—	134	8	no	—
198	C₄×Q₈	—	134	8	no	—
199	C₄×Q₈	—	134	8	no	—
200	#117	—	67	8	no	—
201	#117	—	67	8	no	—
202	#117	—	134	4	no	—
203	#117	—	134	4	no	—
204	C₂×C₄⋊C₄	—	67	8	no	—
205	C₂×C₄⋊C₄	—	134	8	no	—
206	C₂×C₄⋊C₄	—	134	4	no	—
207	C₂×C₂²⋊C₄	—	134	4	no	—
208	C₂×C₄²	—	134	4	no	—
209	C₂×C₄²	D₂₆₈	1	4	yes	—
210	C₂×C₄²	D₂₆₈	1	2	yes	—
211	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
212	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
213	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
214	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
215	#107	—	67	32	no	—
216	#106	—	67	32	no	—
217	#97	C₂²×D₆₇	1	8	yes	—
218	C₄×C₈	C₂²×D₆₇	1	4	yes	—
219	C₄×C₈	C₂²×D₆₇	1	4	yes	—
220	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
221	C₂×Q₈	—	268	16	no	—
222	C₂×Q₈	—	134	8	no	—
223	C₂×Q₈	—	134	8	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	—	134	8	no	—
226	C₂²×C₄	C₂×D₂₆₈	1	4	yes	—
227	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
228	C₂²×C₄	C₂×D₂₆₈	1	4	yes	—
229	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
230	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
232	C₂²×C₄	C₂×Dic₁₃₄	1	8	yes	—
233	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
234	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
235	M₁₆	C₂×Dic₁₃₄	1	32	yes	complemented
236	C₂×C₈	C₂×Dic₁₃₄	1	16	yes	complemented
237	C₂×C₈	C₂³×D₆₇	1	8	yes	—
238	C₄⋊C₄	—	67	16	no	—
239	C₄⋊C₄	—	134	8	no	—
240	C₄⋊C₄	—	134	16	no	—
241	C₄⋊C₄	—	134	16	no	—
242	C₄⋊C₄	—	134	8	no	—
243	C₄⋊C₄	—	134	8	no	—
244	C₂²⋊C₄	—	134	8	no	—
245	C₂²⋊C₄	—	134	8	no	—
246	C₄²	—	134	8	no	—
247	C₄²	—	134	8	no	—
248	C₄²	C₂×D₂₆₈	1	4	yes	—
249	C₄²	C₂×D₂₆₈	1	4	yes	—
250	C₄²	C₂×D₂₆₈	1	2	yes	—
251	C₄²	C₂×D₂₆₈	1	2	yes	—
252	C₄²	C₂³×D₆₇	1	1	yes	characteristic
253	C₄²	C₂³×D₆₇	1	1	yes	characteristic
254	C₄²	C₂³×D₆₇	1	2	yes	—
255	C₂³	—	4	2	no	—
256	C₂³	#14524043	1	4	yes	—
257	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
258	C₂³	C₂²×D₂₆₈	1	2	yes	—
259	Q₈	—	268	16	no	—
260	Q₈	—	268	16	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	268	16	no	—
266	C₂×C₄	—	134	16	no	—
267	C₂×C₄	#14524043	1	4	yes	—
268	C₂×C₄	#14524043	1	8	yes	—
269	C₂×C₄	C₂²×D₂₆₈	1	2	yes	—
270	C₂×C₄	C₂²×D₂₆₈	1	4	yes	—
271	C₂×C₄	C₂²×D₂₆₈	1	2	yes	—
272	C₂×C₄	C₂²×D₂₆₈	1	2	yes	—
273	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
274	C₂×C₄	C₂²×Dic₁₃₄	1	4	yes	—
275	C₂×C₄	C₂²×Dic₁₃₄	1	8	yes	—
276	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
277	C₈	C₂²×Dic₁₃₄	1	16	yes	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	4	no	—
281	C₂²	C₂×#14524043	1	2	yes	—
282	C₂²	C₂×#14524043	1	4	yes	—
283	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
284	C₄	—	2	4	no	—
285	C₄	—	2	2	no	—
286	C₄	—	268	16	no	—
287	C₄	C₂×#14524043	1	4	yes	—
288	C₄	C₂×#14524043	1	2	yes	—
289	C₄	#16864196	1	4	yes	—
290	C₄	C₂³×Dic₁₃₄	1	2	yes	—
291	C₂	—	2	4	no	—
292	C₂	—	2	2	no	—
293	C₂	C₂²×#14524043	1	1	yes	characteristic
294	C₂	C₂×#16864196	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂³×D₄
prime radical	C₂⁴×C₆₇	C₂×D₄
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	#60663
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #22570761	C₂	semidirect
3. #22570755	C₂	semidirect
4. #22570733	C₂	semidirect
5. #22570723	C₂	semidirect
19. #16863805	C₂²	semidirect
20. #16863801	C₂²	semidirect
21. #16863798	C₂²	semidirect
22. #16863794	C₂²	semidirect
60. C₂×M₁₆×C₆₇	Q₈	semidirect
97. M₁₆×C₆₇	C₂×Q₈	semidirect
98. C₂×C₈×C₆₇	C₂×Q₈	semidirect
155. C₆₇	#60663	semidirect
182. C₂×M₁₆	Dic₁₃₄	semidirect
235. M₁₆	C₂×Dic₁₃₄	semidirect
236. C₂×C₈	C₂×Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863953	2
#22570761	2
#22570755	2
#22570733	2
#22570723	2
#22568799	2
C₂×C₄×M₁₆×C₆₇	2
#60663	67