Subgroups — #500×C₂₅₇ #30656369

Order 16448.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×C₂₅₇	C₂	1	2	yes	maximal
2	#107×C₂₅₇	C₂	1	4	yes	maximal
3	C₄×C₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈×C₂₅₇	—	2	2	no	—
5	C₂×C₈×C₂₅₇	C₂²	1	2	yes	—
6	C₄⋊C₄×C₂₅₇	—	2	2	no	—
7	C₄⋊C₄×C₂₅₇	C₂²	1	4	yes	—
8	C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
9	Q₈×C₂₅₇	—	4	4	no	—
10	C₂×C₄×C₂₅₇	—	2	4	no	—
11	C₂×C₄×C₂₅₇	D₄	1	1	yes	characteristic
12	C₂×C₄×C₂₅₇	D₄	1	1	yes	characteristic
13	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
14	C₈×C₂₅₇	Q₈	1	4	yes	complemented
15	C₂²×C₂₅₇	C₂×D₄	1	1	yes	characteristic
16	C₄×C₂₅₇	—	4	4	no	—
17	C₄×C₂₅₇	D₈	1	2	yes	—
18	C₄×C₂₅₇	Q₁₆	1	2	yes	—
19	C₄×C₂₅₇	C₂×Q₈	1	2	yes	—
20	C₂×C₂₅₇	C₄⋊Q₈	1	1	yes	characteristic
21	C₂×C₂₅₇	C₂×D₈	1	1	yes	characteristic
22	C₂×C₂₅₇	C₂×Q₁₆	1	1	yes	characteristic
23	C₂₅₇	#500	1	1	yes	characteristic, direct product factor, Sylow
24	#500	C₂₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₄⋊Q₈	C₂×C₂₅₇	1	2	yes	—
26	#107	C₂×C₂₅₇	1	4	yes	—
27	C₄×C₈	C₂×C₂₅₇	1	1	yes	characteristic
28	C₂×Q₈	—	2	2	no	—
29	C₂×C₈	C₂²×C₂₅₇	1	2	yes	—
30	C₄⋊C₄	—	2	2	no	—
31	C₄⋊C₄	C₂²×C₂₅₇	1	4	yes	—
32	C₄²	C₂²×C₂₅₇	1	1	yes	characteristic
33	Q₈	—	4	4	no	—
34	C₂×C₄	—	2	4	no	—
35	C₂×C₄	D₄×C₂₅₇	1	1	yes	characteristic
36	C₂×C₄	D₄×C₂₅₇	1	1	yes	characteristic
37	C₂×C₄	C₂³×C₂₅₇	1	1	yes	characteristic
38	C₈	Q₈×C₂₅₇	1	4	yes	complemented
39	C₂²	C₂×D₄×C₂₅₇	1	1	yes	characteristic
40	C₄	—	4	4	no	—
41	C₄	D₈×C₂₅₇	1	2	yes	—
42	C₄	Q₁₆×C₂₅₇	1	2	yes	—
43	C₄	C₂×Q₈×C₂₅₇	1	2	yes	—
44	C₂	C₄⋊Q₈×C₂₅₇	1	1	yes	characteristic
45	C₂	C₂×D₈×C₂₅₇	1	1	yes	characteristic
46	C₂	C₂×Q₁₆×C₂₅₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₅₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₂₅₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₅₇ ◁ C₄²×C₂₅₇ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₈×C₂₅₇	Q₈	semidirect
23. C₂₅₇	#500	direct (N × H)
24. #500	C₂₅₇	direct (N × H)
38. C₈	Q₈×C₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×C₂₅₇	2
#107×C₂₅₇	2
C₄×C₈×C₂₅₇	2
#500	257

Subgroups — #500×C257 #30656369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #500×C₂₅₇ #30656369