Subgroups — #28410583

Order 14624.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20300741	C₂	1	1	yes	maximal, characteristic
2	#20300731	C₂	1	2	yes	maximal, complemented
3	#20300728	C₂	1	2	yes	maximal, complemented
4	#20300725	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₈×C₄₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₉₁₄	C₂²	1	1	yes	characteristic
7	D₁₈₂₈	C₂²	1	1	yes	characteristic
8	C₂²⋊D₄₅₇	—	2	1	no	—
9	C₄×D₄₅₇	—	2	1	no	—
10	#16814532	C₂²	1	2	yes	—
11	Q₈×C₄₅₇	—	2	1	no	—
12	Q₈×C₄₅₇	C₂²	1	2	yes	—
13	C₂×C₄×C₄₅₇	—	2	1	no	—
14	C₂×C₄×C₄₅₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄₅₇	D₄	1	1	yes	characteristic
16	C₂×D₄₅₇	—	2	1	no	—
17	C₄×C₄₅₇	—	2	2	no	—
18	C₄×C₄₅₇	D₄	1	1	yes	characteristic
19	C₄×C₄₅₇	C₂³	1	1	yes	characteristic
20	Dic₄₅₇	—	2	1	no	—
21	C₂×C₄₅₇	—	2	1	no	—
22	C₂×C₄₅₇	C₂×D₄	1	1	yes	characteristic
23	D₄₅₇	—	4	1	no	—
24	C₄₅₇	#137	1	1	yes	characteristic, complemented, Sylow
25	#137	—	457	1	no	maximal, Sylow
26	#41	—	457	1	no	—
27	C₂×Q₈	D₄₅₇	1	1	yes	characteristic, complemented
28	Q₁₆	—	457	2	no	—
29	SD₁₆	—	457	2	no	—
30	M₁₆	—	457	1	no	—
31	Q₈	—	2	1	no	—
32	Q₈	—	457	1	no	—
33	Q₈	C₂×D₄₅₇	1	2	yes	—
34	D₄	—	914	1	no	—
35	D₄	—	457	1	no	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	—	914	1	no	—
38	C₂×C₄	C₂×D₄₅₇	1	1	yes	characteristic
39	C₈	—	457	2	no	—
40	C₂²	—	914	1	no	—
41	C₂²	C₂²⋊D₄₅₇	1	1	yes	characteristic
42	C₄	—	2	2	no	—
43	C₄	—	914	1	no	—
44	C₄	C₂²⋊D₄₅₇	1	1	yes	characteristic
45	C₄	C₂²×D₄₅₇	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	1828	1	no	—
48	C₂	C₂×C₂²⋊D₄₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊D₄₅₇
commutator, cosocle	C₄×C₄₅₇	C₂³
Frattini	C₄	C₂²×D₄₅₇
Fitting	C₂×Q₈×C₄₅₇	C₂
socle	C₂×C₄₅₇	C₂×D₄
prime radical	#20300741	C₂
hypercenter	C₂×Q₈	D₄₅₇
hypocenter	C₄₅₇	#137
LCS	G ▷ C₄×C₄₅₇ ▷ C₂×C₄₅₇ ▷ C₄₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
2. #20300731	C₂	semidirect
3. #20300728	C₂	semidirect
4. #20300725	C₂	semidirect
5. C₂×Q₈×C₄₅₇	C₂	semidirect
24. C₄₅₇	#137	semidirect
27. C₂×Q₈	D₄₅₇	semidirect

Maximal subgroups

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

subgroup	index
#20300741	2
#20300731	2
#20300728	2
#20300725	2
C₂×Q₈×C₄₅₇	2
#137	457