Subgroups — #28410581

Order 14624.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20300740	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₄₅₇	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₁₈₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#20300731	C₂	1	1	yes	maximal, characteristic, complemented
5	#20300729	C₂	1	1	yes	maximal, characteristic, complemented
6	#20300724	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₁₆×C₄₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₉₁₄	—	2	1	no	—
9	Dic₉₁₄	C₂²	1	1	yes	characteristic
10	Dic₉₁₄	C₂²	1	1	yes	characteristic
11	C₂²⋊D₄₅₇	—	2	1	no	—
12	C₂×Dic₄₅₇	—	2	1	no	—
13	C₄×D₄₅₇	—	2	1	no	—
14	C₄×D₄₅₇	C₂²	1	1	yes	characteristic
15	#16814532	C₂²	1	1	yes	characteristic
16	Q₈×C₄₅₇	C₂²	1	1	yes	characteristic
17	D₄×C₄₅₇	C₂²	1	1	yes	characteristic
18	C₈×C₄₅₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄₅₇	—	2	1	no	—
20	C₂×D₄₅₇	D₄	1	1	yes	characteristic
21	C₄×C₄₅₇	—	2	1	no	—
22	C₄×C₄₅₇	C₂³	1	1	yes	characteristic
23	Dic₄₅₇	—	2	1	no	—
24	Dic₄₅₇	—	2	1	no	—
25	Dic₄₅₇	D₄	1	1	yes	characteristic
26	C₂×C₄₅₇	—	4	1	no	—
27	C₂×C₄₅₇	C₂×D₄	1	1	yes	characteristic
28	D₄₅₇	—	2	1	no	—
29	C₄₅₇	#137	1	1	yes	characteristic, complemented, Sylow
30	#137	—	457	1	no	maximal, Sylow
31	#41	—	457	1	no	—
32	C₂×Q₈	—	457	1	no	—
33	Q₁₆	—	457	1	no	—
34	Q₁₆	—	457	1	no	—
35	SD₁₆	—	457	1	no	—
36	SD₁₆	D₄₅₇	1	1	yes	characteristic, complemented
37	M₁₆	—	457	1	no	—
38	Q₈	—	914	1	no	—
39	Q₈	—	457	1	no	—
40	Q₈	—	457	1	no	—
41	Q₈	C₂×D₄₅₇	1	1	yes	characteristic
42	D₄	—	914	1	no	—
43	D₄	C₂×D₄₅₇	1	1	yes	characteristic
44	C₂×C₄	—	914	1	no	—
45	C₂×C₄	—	914	1	no	—
46	C₂×C₄	—	457	1	no	—
47	C₈	—	457	1	no	—
48	C₈	C₂×D₄₅₇	1	1	yes	characteristic
49	C₂²	—	2	1	no	—
50	C₂²	—	457	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	914	1	no	—
53	C₄	—	914	1	no	—
54	C₄	—	457	1	no	—
55	C₄	C₂²×D₄₅₇	1	1	yes	characteristic
56	C₂	—	4	1	no	—
57	C₂	—	914	1	no	—
58	C₂	D₄×D₄₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₄₅₇
commutator, cosocle	C₄×C₄₅₇	C₂³
Frattini	C₄	C₂²×D₄₅₇
Fitting	SD₁₆×C₄₅₇	C₂
socle	C₂×C₄₅₇	C₂×D₄
prime radical	#20300740	C₂
hypercenter	SD₁₆	D₄₅₇
hypocenter	C₄₅₇	#137
LCS	G ▷ C₄×C₄₅₇ ▷ C₂×C₄₅₇ ▷ C₄₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₄₅₇	C₂	semidirect
3. Dic₁₈₂₈	C₂	semidirect
4. #20300731	C₂	semidirect
5. #20300729	C₂	semidirect
6. #20300724	C₂	semidirect
7. SD₁₆×C₄₅₇	C₂	semidirect
29. C₄₅₇	#137	semidirect
36. SD₁₆	D₄₅₇	semidirect

Maximal subgroups

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

subgroup	index
#20300740	2
Q₈×D₄₅₇	2
Dic₁₈₂₈	2
#20300731	2
#20300729	2
#20300724	2
SD₁₆×C₄₅₇	2
#137	457