Subgroups — #22562483

Order 8576.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863800	C₂	1	2	yes	maximal
2	#16863222	C₂	1	1	yes	maximal, characteristic
3	#16863221	C₂	1	2	yes	maximal
4	C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
5	#366×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#14524042	—	2	2	no	—
7	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	#14523977	C₂²	1	2	yes	—
9	#14523976	—	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×Dic₁₃₄	—	4	2	no	—
14	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
15	C₄⋊Dic₆₇	—	2	2	no	—
16	Dic₆₇⋊C₄	—	4	2	no	—
17	C₄×Dic₆₇	D₄	1	1	yes	characteristic
18	C₆₇⋊C₁₆	Q₈	1	2	yes	complemented
19	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₆₇	—	2	2	no	—
21	C₁₆×C₆₇	Q₈	1	2	yes	complemented
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	Dic₁₃₄	—	8	4	no	—
28	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
29	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
30	C₆₇⋊C₈	Q₁₆	1	2	yes	—
31	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
32	C₄×C₆₇	—	8	2	no	—
33	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
34	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
35	Dic₆₇	—	8	2	no	—
36	Dic₆₇	D₁₆	1	2	yes	—
37	Dic₆₇	Q₃₂	1	2	yes	—
38	C₂×C₆₇	#500	1	1	yes	characteristic
39	C₂×C₆₇	C₂×D₁₆	1	1	yes	characteristic
40	C₂×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
41	#2252	—	67	1	no	maximal, Sylow
42	C₆₇	#2252	1	1	yes	characteristic, complemented, Sylow
43	#500	—	67	2	no	—
44	#366	—	67	1	no	—
45	#366	—	67	2	no	—
46	#366	D₆₇	1	1	yes	characteristic
47	C₄×C₁₆	—	67	1	no	—
48	C₄⋊Q₈	—	134	2	no	—
49	C₂×C₁₆	—	67	1	no	—
50	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
51	#107	—	67	2	no	—
52	#107	—	134	2	no	—
53	#107	C₂×D₆₇	1	2	yes	—
54	C₄×C₈	—	67	1	no	—
55	C₂×Q₈	—	268	2	no	—
56	C₂×C₈	—	67	1	no	—
57	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	268	2	no	—
60	C₄⋊C₄	—	134	2	no	—
61	C₄²	—	67	1	no	—
62	C₁₆	—	67	2	no	—
63	C₁₆	Dic₁₃₄	1	2	yes	complemented
64	Q₈	—	536	4	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	268	2	no	—
67	C₂×C₄	—	67	1	no	—
68	C₂×C₄	—	67	1	no	—
69	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
70	C₈	—	67	2	no	—
71	C₈	Q₈×D₆₇	1	1	yes	characteristic
72	C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
73	C₂²	D₈×D₆₇	1	1	yes	characteristic
74	C₄	—	8	2	no	—
75	C₄	—	536	2	no	—
76	C₄	—	67	2	no	—
77	C₄	—	67	2	no	—
78	C₄	#14524042	1	1	yes	characteristic
79	C₄	Q₁₆×D₆₇	1	1	yes	characteristic
80	C₂	#16863800	1	1	yes	characteristic
81	C₂	D₁₆×D₆₇	1	1	yes	characteristic
82	C₂	Q₃₂×D₆₇	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	#2252
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
18. C₆₇⋊C₁₆	Q₈	semidirect
21. C₁₆×C₆₇	Q₈	semidirect
42. C₆₇	#2252	semidirect
63. C₁₆	Dic₁₃₄	semidirect