Subgroups — #22574921

Order 8576.

1424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16864163	C₂	1	15	yes	maximal, complemented
2	C₂×#14524050	C₂	1	15	yes	maximal, complemented
3	#585×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#14524050	C₂²	1	20	yes	complemented
5	#14524050	C₂²	1	60	yes	complemented
6	C₂×#11761612	C₂²	1	45	yes	complemented
7	C₂²×C₆₇⋊C₈	C₂²	1	15	yes	complemented
8	2₋¹⁺⁴×C₆₇	C₄	1	6	yes	—
9	H(5,𝔽₂)×C₆₇	C₄	1	10	yes	—
10	C₂×#41×C₆₇	C₂²	1	15	yes	—
11	#11761612	C₂³	1	60	yes	—
12	C₂×C₆₇⋊C₈	C₂³	1	15	yes	—
13	C₂×C₆₇⋊C₈	C₂³	1	45	yes	—
14	#41×C₆₇	C₂×C₄	1	60	yes	—
15	#41×C₆₇	C₂³	1	20	yes	—
16	C₂×Q₈×C₆₇	C₂×C₄	1	15	yes	—
17	C₂×D₄×C₆₇	C₂×C₄	1	45	yes	—
18	C₂²×C₄×C₆₇	C₂³	1	15	yes	—
19	C₂³×C₆₇	C₂²×C₄	1	15	yes	—
20	Q₈×C₆₇	C₂²×C₄	1	20	yes	—
21	D₄×C₆₇	C₂²×C₄	1	60	yes	—
22	C₂×C₄×C₆₇	C₂²×C₄	1	45	yes	—
23	C₂×C₄×C₆₇	C₂⁴	1	15	yes	—
24	C₆₇⋊C₈	C₂⁴	1	1	yes	characteristic
25	C₆₇⋊C₈	C₂⁴	1	15	yes	—
26	C₂²×C₆₇	—	4	15	no	—
27	C₂²×C₆₇	C₂³×C₄	1	15	yes	—
28	C₄×C₆₇	C₂³×C₄	1	15	yes	—
29	C₄×C₆₇	C₂⁵	1	1	yes	characteristic
30	C₂×C₆₇	—	2	15	no	—
31	C₂×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
32	#3573	—	67	1	no	maximal, Sylow
33	C₆₇	#3573	1	1	yes	characteristic, complemented, Sylow
34	#585	D₆₇	1	1	yes	characteristic
35	#568	—	67	15	no	—
36	C₂×#131	—	67	15	no	—
37	2₋¹⁺⁴	Dic₆₇	1	6	yes	—
38	H(5,𝔽₂)	Dic₆₇	1	10	yes	—
39	C₂×#41	C₂×D₆₇	1	15	yes	—
40	#131	—	67	20	no	—
41	#131	—	67	60	no	—
42	C₂×M₁₆	—	67	45	no	—
43	C₂²×C₈	—	67	15	no	—
44	#41	C₂×Dic₆₇	1	60	yes	—
45	#41	C₂²×D₆₇	1	20	yes	—
46	C₂×Q₈	C₂×Dic₆₇	1	15	yes	—
47	C₂×D₄	C₂×Dic₆₇	1	45	yes	—
48	C₂²×C₄	C₂²×D₆₇	1	15	yes	—
49	M₁₆	—	67	60	no	—
50	C₂×C₈	—	67	15	no	—
51	C₂×C₈	—	67	45	no	—
52	C₂³	C₂²×Dic₆₇	1	15	yes	—
53	Q₈	C₂²×Dic₆₇	1	20	yes	—
54	D₄	C₂²×Dic₆₇	1	60	yes	—
55	C₂×C₄	C₂²×Dic₆₇	1	45	yes	—
56	C₂×C₄	C₂³×D₆₇	1	15	yes	—
57	C₈	—	67	1	no	—
58	C₈	—	67	15	no	—
59	C₂²	—	4	15	no	—
60	C₂²	C₂³×Dic₆₇	1	15	yes	—
61	C₄	C₂³×Dic₆₇	1	15	yes	—
62	C₄	C₂⁴×D₆₇	1	1	yes	characteristic
63	C₂	—	2	15	no	—
64	C₂	C₂⁴×Dic₆₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂⁴×D₆₇
commutator, socle	C₂×C₆₇	C₂⁴×C₄
Fitting	#585×C₆₇	C₂
hypercenter	#585	D₆₇
hypocenter	C₆₇	#3573
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ #585

Complements

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

N	H ≅ G/N	type
1. #16864163	C₂	semidirect
2. C₂×#14524050	C₂	semidirect
4. #14524050	C₂²	semidirect
5. #14524050	C₂²	semidirect
6. C₂×#11761612	C₂²	semidirect
7. C₂²×C₆₇⋊C₈	C₂²	semidirect
33. C₆₇	#3573	semidirect