Subgroups — #22562485

Order 8576.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863798	C₂	1	1	yes	maximal, characteristic
2	#16863223	C₂	1	2	yes	maximal
3	#16863221	C₂	1	2	yes	maximal
4	C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#499×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#14524035	—	2	1	no	—
7	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	#14523977	C₂²	1	2	yes	—
10	#14523975	—	2	1	no	—
11	#125×C₆₇	—	2	1	no	—
12	#107×C₆₇	C₂²	1	2	yes	—
13	#106×C₆₇	—	2	1	no	—
14	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₆₇	—	2	2	no	—
16	Dic₆₇⋊C₄	—	4	2	no	—
17	C₆₇⋊C₁₆	Q₈	1	2	yes	—
18	C₆₇⋊C₁₆	Q₈	1	2	yes	—
19	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
20	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₆₇	—	2	2	no	—
22	C₄⋊C₄×C₆₇	—	4	2	no	—
23	C₄²×C₆₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₆₇	—	4	2	no	—
25	C₂×C₄×C₆₇	D₈	1	2	yes	—
26	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₆₇	—	4	2	no	—
28	C₈×C₆₇	Q₁₆	1	2	yes	—
29	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
30	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
31	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
32	C₄×C₆₇	—	2	2	no	—
33	C₄×C₆₇	—	8	2	no	—
34	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
35	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
36	Dic₆₇	—	8	2	no	—
37	C₂×C₆₇	#500	1	1	yes	characteristic
38	C₂×C₆₇	#508	1	2	yes	—
39	#2253	—	67	1	no	maximal, Sylow
40	C₆₇	#2253	1	1	yes	characteristic, complemented, Sylow
41	#499	—	67	1	no	—
42	#499	D₆₇	1	1	yes	characteristic
43	#367	—	67	2	no	—
44	#366	—	67	2	no	—
45	C₄×C₁₆	—	67	1	no	—
46	#125	—	2	1	no	—
47	#125	—	134	1	no	—
48	C₂×C₁₆	—	67	1	no	—
49	C₂×C₁₆	—	67	1	no	—
50	#107	—	67	2	no	—
51	#107	C₂×D₆₇	1	2	yes	—
52	#106	—	2	1	no	—
53	#106	—	134	1	no	—
54	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
55	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
56	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
57	C₄⋊C₄	—	2	2	no	—
58	C₄⋊C₄	—	4	2	no	—
59	C₄⋊C₄	—	268	2	no	—
60	C₄⋊C₄	—	134	2	no	—
61	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
62	C₁₆	—	67	2	no	—
63	C₁₆	—	67	2	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	268	2	no	—
66	C₂×C₄	#11761613	1	2	yes	—
67	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
68	C₈	#11761618	1	2	yes	—
69	C₈	Q₈×D₆₇	1	1	yes	characteristic
70	C₈	Q₈×D₆₇	1	1	yes	characteristic
71	C₂²	C₂×#11761613	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	8	2	no	—
74	C₄	—	536	2	no	—
75	C₄	#14524041	1	1	yes	characteristic
76	C₄	C₂×#11761618	1	1	yes	characteristic
77	C₂	#16863799	1	1	yes	characteristic
78	C₂	#16863826	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#499×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	#499	D₆₇
hypocenter	C₆₇	#2253
LCS	G ▷ C₂×C₈×C₆₇ ▷ 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	#2253	semidirect