Subgroups — #332×C₂₅₇ #30656201

Order 16448.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈×C₂₅₇	—	2	1	no	—
5	C₂×C₈×C₂₅₇	—	2	1	no	—
6	C₂×C₈×C₂₅₇	—	2	1	no	—
7	C₄⋊C₄×C₂₅₇	—	2	1	no	—
8	C₄⋊C₄×C₂₅₇	C₄	1	2	yes	—
9	C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
10	Q₈×C₂₅₇	—	4	1	no	—
11	Q₈×C₂₅₇	—	4	1	no	—
12	C₂×C₄×C₂₅₇	—	2	2	no	—
13	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₂₅₇	D₄	1	1	yes	characteristic
15	C₂×C₄×C₂₅₇	D₄	1	1	yes	characteristic
16	C₈×C₂₅₇	—	4	1	no	—
17	C₈×C₂₅₇	—	4	1	no	—
18	C₂²×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₂₅₇	—	2	1	no	—
20	C₄×C₂₅₇	—	4	2	no	—
21	C₄×C₂₅₇	D₈	1	1	yes	characteristic
22	C₄×C₂₅₇	SD₁₆	1	1	yes	characteristic
23	C₄×C₂₅₇	SD₁₆	1	1	yes	characteristic
24	C₄×C₂₅₇	Q₁₆	1	1	yes	characteristic
25	C₂×C₂₅₇	#101	1	1	yes	characteristic
26	C₂×C₂₅₇	#102	1	1	yes	characteristic
27	C₂×C₂₅₇	#103	1	1	yes	characteristic
28	C₂₅₇	#332	1	1	yes	characteristic, direct product factor, Sylow
29	#332	C₂₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₄⋊Q₈	C₂×C₂₅₇	1	1	yes	characteristic
31	C₄⋊C₈	C₂×C₂₅₇	1	1	yes	characteristic
32	C₄⋊C₈	C₂×C₂₅₇	1	1	yes	characteristic
33	C₂×Q₈	—	2	1	no	—
34	C₂×C₈	—	2	1	no	—
35	C₂×C₈	—	2	1	no	—
36	C₄⋊C₄	—	2	1	no	—
37	C₄⋊C₄	C₄×C₂₅₇	1	2	yes	—
38	C₄²	C₂²×C₂₅₇	1	1	yes	characteristic
39	Q₈	—	4	1	no	—
40	Q₈	—	4	1	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	C₂×C₄×C₂₅₇	1	1	yes	characteristic
43	C₂×C₄	D₄×C₂₅₇	1	1	yes	characteristic
44	C₂×C₄	D₄×C₂₅₇	1	1	yes	characteristic
45	C₈	—	4	1	no	—
46	C₈	—	4	1	no	—
47	C₂²	C₂²⋊C₄×C₂₅₇	1	1	yes	characteristic
48	C₄	—	2	1	no	—
49	C₄	—	4	2	no	—
50	C₄	D₈×C₂₅₇	1	1	yes	characteristic
51	C₄	SD₁₆×C₂₅₇	1	1	yes	characteristic
52	C₄	SD₁₆×C₂₅₇	1	1	yes	characteristic
53	C₄	Q₁₆×C₂₅₇	1	1	yes	characteristic
54	C₂	#101×C₂₅₇	1	1	yes	characteristic
55	C₂	#102×C₂₅₇	1	1	yes	characteristic
56	C₂	#103×C₂₅₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
28. C₂₅₇	#332	direct (N × H)
29. #332	C₂₅₇	direct (N × H)

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
C₄⋊C₈×C₂₅₇	2
#332	257

Subgroups — #332×C257 #30656201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #332×C₂₅₇ #30656201