Subgroups — #46854754

Order 35008.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₅₄₇⋊C₈)	C₂	1	2	yes	maximal
2	C₄×C₈×C₅₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₄₇⋊C₈	—	2	2	no	—
4	C₂×C₈×C₅₄₇	C₄	1	1	yes	characteristic
5	C₂×C₈×C₅₄₇	C₄	1	1	yes	characteristic
6	C₄²×C₅₄₇	C₂²	1	1	yes	characteristic
7	C₅₄₇⋊C₈	—	4	2	no	—
8	C₂×C₄×C₅₄₇	C₂×C₄	1	1	yes	characteristic
9	C₂×C₄×C₅₄₇	D₄	1	1	yes	characteristic
10	C₂×C₄×C₅₄₇	Q₈	1	1	yes	characteristic
11	C₈×C₅₄₇	—	2	1	no	—
12	C₈×C₅₄₇	C₈	1	2	yes	complemented
13	C₂²×C₅₄₇	C₄⋊C₄	1	1	yes	characteristic
14	C₄×C₅₄₇	—	2	1	no	—
15	C₄×C₅₄₇	C₂×C₈	1	1	yes	characteristic
16	C₄×C₅₄₇	M₁₆	1	1	yes	characteristic
17	C₄×C₅₄₇	D₈	1	1	yes	characteristic
18	C₄×C₅₄₇	Q₁₆	1	1	yes	characteristic
19	C₂×C₅₄₇	C₄⋊C₈	1	1	yes	characteristic
20	C₂×C₅₄₇	#107	1	1	yes	characteristic
21	C₂×C₅₄₇	#108	1	1	yes	characteristic
22	C₅₄₇	#335	1	1	yes	characteristic, complemented, Sylow
23	#335	—	547	1	no	maximal, Sylow
24	C₄⋊C₈	—	547	2	no	—
25	C₄×C₈	D₅₄₇	1	1	yes	characteristic
26	C₂×C₈	—	1094	2	no	—
27	C₂×C₈	Dic₅₄₇	1	1	yes	characteristic
28	C₂×C₈	Dic₅₄₇	1	1	yes	characteristic
29	C₄²	C₂×D₅₄₇	1	1	yes	characteristic
30	C₂×C₄	C₂×Dic₅₄₇	1	1	yes	characteristic
31	C₂×C₄	D₂₁₈₈	1	1	yes	characteristic
32	C₂×C₄	Dic₁₀₉₄	1	1	yes	characteristic
33	C₈	—	2	1	no	—
34	C₈	—	2188	2	no	—
35	C₈	C₅₄₇⋊C₈	1	2	yes	complemented
36	C₂²	C₄⋊Dic₅₄₇	1	1	yes	characteristic
37	C₄	—	2	1	no	—
38	C₄	C₂×C₅₄₇⋊C₈	1	1	yes	characteristic
39	C₄	#22576760	1	1	yes	characteristic
40	C₄	D₄₃₇₆	1	1	yes	characteristic
41	C₄	Dic₂₁₈₈	1	1	yes	characteristic
42	C₂	C₄⋊(C₅₄₇⋊C₈)	1	1	yes	characteristic
43	C₂	#31775936	1	1	yes	characteristic
44	C₂	#31775938	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₂₁₈₈
commutator	C₄×C₅₄₇	C₂×C₈
Frattini	C₄²	C₂×D₅₄₇
Fitting	C₄×C₈×C₅₄₇	C₂
socle, prime radical	C₂²×C₅₄₇	C₄⋊C₄
cosocle	C₄²×C₅₄₇	C₂²
hypercenter	C₄×C₈	D₅₄₇
hypocenter	C₅₄₇	#335
LCS	G ▷ C₄×C₅₄₇ ▷ C₂×C₅₄₇ ▷ C₅₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
12. C₈×C₅₄₇	C₈	semidirect
22. C₅₄₇	#335	semidirect
35. C₈	C₅₄₇⋊C₈	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄⋊(C₅₄₇⋊C₈)	2
C₄×C₈×C₅₄₇	2
#335	547