Subgroups — #22562486

Order 8576.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863797	C₂	1	2	yes	maximal
2	#16863223	C₂	1	2	yes	maximal
3	#16863222	C₂	1	1	yes	maximal, characteristic
4	C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
5	#366×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#14524034	—	2	2	no	—
7	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	#14523977	C₂²	1	2	yes	—
9	#14523974	—	2	2	no	—
10	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	#107×C₆₇	C₂²	1	2	yes	—
13	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
14	C₄⋊Dic₆₇	—	2	2	no	—
15	C₄⋊Dic₆₇	—	4	2	no	—
16	Dic₆₇⋊C₄	—	4	2	no	—
17	C₄×Dic₆₇	D₄	1	1	yes	characteristic
18	C₆₇⋊C₁₆	Q₈	1	2	yes	—
19	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₆₇	—	2	2	no	—
21	C₁₆×C₆₇	Q₈	1	2	yes	—
22	C₂×C₄×C₆₇	—	4	2	no	—
23	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
24	C₂×Dic₆₇	—	4	2	no	—
25	C₂×Dic₆₇	D₈	1	1	yes	characteristic
26	C₂×Dic₆₇	D₈	1	1	yes	characteristic
27	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
28	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
29	C₆₇⋊C₈	Q₁₆	1	2	yes	—
30	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
31	C₄×C₆₇	—	8	2	no	—
32	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
33	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
34	Dic₆₇	—	2	1	no	—
35	Dic₆₇	—	2	1	no	—
36	Dic₆₇	—	8	2	no	—
37	C₂×C₆₇	#500	1	1	yes	characteristic
38	C₂×C₆₇	#508	1	1	yes	characteristic
39	C₂×C₆₇	#508	1	1	yes	characteristic
40	#2253	—	67	1	no	maximal, Sylow
41	C₆₇	#2253	1	1	yes	characteristic, complemented, Sylow
42	#499	—	67	2	no	—
43	#367	—	67	2	no	—
44	#366	—	67	1	no	—
45	#366	D₆₇	1	1	yes	characteristic
46	C₄×C₁₆	—	67	1	no	—
47	#125	—	134	2	no	—
48	C₂×C₁₆	—	67	1	no	—
49	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
50	#107	—	67	2	no	—
51	#107	C₂×D₆₇	1	2	yes	—
52	#106	—	134	2	no	—
53	C₄×C₈	—	67	1	no	—
54	C₂×C₈	—	67	1	no	—
55	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
56	C₄⋊C₄	—	2	2	no	—
57	C₄⋊C₄	—	268	2	no	—
58	C₄⋊C₄	—	268	2	no	—
59	C₄⋊C₄	—	134	2	no	—
60	C₄²	—	67	1	no	—
61	C₁₆	—	67	2	no	—
62	C₁₆	Dic₁₃₄	1	2	yes	—
63	C₂×C₄	—	4	2	no	—
64	C₂×C₄	—	268	2	no	—
65	C₂×C₄	—	67	1	no	—
66	C₂×C₄	—	67	1	no	—
67	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
68	C₈	—	67	2	no	—
69	C₈	Q₈×D₆₇	1	1	yes	characteristic
70	C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
71	C₂²	D₈×D₆₇	1	1	yes	characteristic
72	C₄	—	8	2	no	—
73	C₄	—	536	2	no	—
74	C₄	—	134	1	no	—
75	C₄	—	134	1	no	—
76	C₄	#14524042	1	1	yes	characteristic
77	C₄	Q₁₆×D₆₇	1	1	yes	characteristic
78	C₂	#16863800	1	1	yes	characteristic
79	C₂	#16863825	1	1	yes	characteristic
80	C₂	#16863829	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₆₇
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#366×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	#366	D₆₇
hypocenter	C₆₇	#2253
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
41. C₆₇	#2253	semidirect