Subgroups — #22558261

Order 8576.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#16863147	C₂	1	1	yes	maximal, characteristic
3	#499×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₆₇⋊C₁₆	—	2	1	no	—
5	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
6	#125×C₆₇	—	2	1	no	—
7	#107×C₆₇	C₄	1	1	yes	characteristic
8	#107×C₆₇	C₄	1	1	yes	characteristic
9	#106×C₆₇	—	2	1	no	—
10	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₆₇⋊C₈	—	4	1	no	—
12	C₆₇⋊C₁₆	—	4	1	no	—
13	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
15	C₄⋊C₄×C₆₇	—	2	1	no	—
16	C₄⋊C₄×C₆₇	—	2	1	no	—
17	C₄⋊C₄×C₆₇	—	4	1	no	—
18	C₄⋊C₄×C₆₇	—	4	1	no	—
19	C₄²×C₆₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₆₇	—	4	1	no	—
21	C₂×C₄×C₆₇	—	4	1	no	—
22	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
24	C₂×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
25	C₈×C₆₇	—	2	1	no	—
26	C₈×C₆₇	SD₁₆	1	1	yes	characteristic
27	C₈×C₆₇	Q₁₆	1	1	yes	characteristic
28	C₆₇⋊C₈	—	8	1	no	—
29	C₂²×C₆₇	#102	1	1	yes	characteristic
30	C₄×C₆₇	—	2	1	no	—
31	C₄×C₆₇	—	2	1	no	—
32	C₄×C₆₇	—	8	1	no	—
33	C₄×C₆₇	—	8	1	no	—
34	C₄×C₆₇	#101	1	1	yes	characteristic
35	C₄×C₆₇	#103	1	1	yes	characteristic
36	C₂×C₆₇	#332	1	1	yes	characteristic
37	C₂×C₆₇	#359	1	1	yes	characteristic
38	C₂×C₆₇	#360	1	1	yes	characteristic
39	#1363	—	67	1	no	maximal, Sylow
40	C₆₇	#1363	1	1	yes	characteristic, complemented, Sylow
41	#499	D₆₇	1	1	yes	characteristic
42	C₄⋊C₁₆	—	67	1	no	—
43	#334	—	67	1	no	—
44	#125	—	2	1	no	—
45	C₂×C₁₆	—	134	1	no	—
46	#107	Dic₆₇	1	1	yes	characteristic
47	#107	Dic₆₇	1	1	yes	characteristic
48	#106	—	2	1	no	—
49	C₄⋊C₈	—	134	1	no	—
50	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
51	C₂×C₈	—	268	1	no	—
52	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
53	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	2	1	no	—
56	C₄⋊C₄	—	4	1	no	—
57	C₄⋊C₄	—	4	1	no	—
58	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
59	C₁₆	—	268	1	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	—	4	1	no	—
62	C₂×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
63	C₂×C₄	#11761613	1	1	yes	characteristic
64	C₂×C₄	#11761616	1	1	yes	characteristic
65	C₈	—	2	1	no	—
66	C₈	—	536	1	no	—
67	C₈	#11761615	1	1	yes	characteristic
68	C₈	#11761618	1	1	yes	characteristic
69	C₂²	#14523964	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	2	1	no	—
72	C₄	—	8	1	no	—
73	C₄	—	8	1	no	—
74	C₄	#14523962	1	1	yes	characteristic
75	C₄	#14523967	1	1	yes	characteristic
76	C₂	#16863142	1	1	yes	characteristic
77	C₂	#16863203	1	1	yes	characteristic
78	C₂	#16863206	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14523964
commutator	C₂×C₈×C₆₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₆₇
Fitting	#499×C₆₇	C₂
socle	C₂²×C₆₇	#102
hypercenter	#499	D₆₇
hypocenter	C₆₇	#1363
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

Complements

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

N	H ≅ G/N	type
40. C₆₇	#1363	semidirect