Subgroups — #30657641

Order 16448.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551651	C₂	1	2	yes	maximal, complemented
2	#22551588	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#11759967	—	2	2	no	—
4	C₂×D₁₀₂₈	C₂²	1	1	yes	characteristic
5	C₂×C₂²⋊D₂₅₇	—	2	2	no	—
6	C₄×Dic₂₅₇	C₄	1	1	yes	characteristic, complemented
7	C₂×D₄×C₂₅₇	C₄	1	1	yes	characteristic, complemented
8	C₂×#11759967	—	4	2	no	—
9	C₂²×D₂₅₇	D₄	1	2	yes	—
10	D₁₀₂₈	—	4	1	no	—
11	C₂²⋊D₂₅₇	—	4	4	no	—
12	C₂×Dic₂₅₇	—	2	1	no	—
13	C₂³×C₂₅₇	—	2	1	no	—
14	D₄×C₂₅₇	—	4	1	no	—
15	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₂₅₇	—	4	2	no	—
17	C₂²×C₂₅₇	—	4	1	no	—
18	C₂²×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×D₂₅₇	—	2	2	no	—
20	C₂×D₂₅₇	—	4	2	no	—
21	#11759967	—	8	2	no	—
22	C₄×C₂₅₇	—	2	1	no	—
23	Dic₂₅₇	—	2	2	no	—
24	C₂×C₂₅₇	—	2	1	no	—
25	C₂×C₂₅₇	—	8	1	no	—
26	C₂×C₂₅₇	#99	1	1	yes	characteristic
27	D₂₅₇	—	4	2	no	—
28	C₂₅₇	#353	1	1	yes	characteristic, complemented, Sylow
29	#353	—	257	1	no	maximal, Sylow
30	#127	—	257	1	no	—
31	#99	—	257	2	no	—
32	C₂×D₄	—	257	1	no	—
33	C₂×D₄	—	514	2	no	—
34	C₂×D₄	#11759967	1	1	yes	characteristic, complemented
35	C₂²⋊C₄	—	514	2	no	—
36	C₄²	—	257	1	no	—
37	C₂³	—	257	2	no	—
38	C₂³	—	2	1	no	—
39	D₄	—	4	1	no	—
40	D₄	—	1028	1	no	—
41	D₄	—	1028	4	no	—
42	C₂×C₄	—	514	1	no	—
43	C₂×C₄	—	1028	2	no	—
44	C₂×C₄	C₂×#11759967	1	1	yes	characteristic
45	C₂²	—	514	2	no	—
46	C₂²	—	4	2	no	—
47	C₂²	—	4	1	no	—
48	C₂²	—	1028	2	no	—
49	C₂²	C₂²⋊#11759967	1	1	yes	characteristic
50	C₄	—	2	1	no	—
51	C₄	—	514	2	no	—
52	C₄	—	2056	2	no	—
53	C₂	—	2	1	no	—
54	C₂	—	1028	2	no	—
55	C₂	—	8	1	no	—
56	C₂	#22551650	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22551650
commutator	C₂×C₄×C₂₅₇	C₂×C₄
Frattini	C₂×C₄	C₂×#11759967
Fitting	C₂×D₄×C₂₅₇	C₄
socle	C₂×C₂₅₇	#99
prime radical	#22551588	C₂
cosocle	C₂×D₁₀₂₈	C₂²
hypercenter	C₂×D₄	#11759967
hypocenter	C₂₅₇	#353
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. #22551651	C₂	semidirect
6. C₄×Dic₂₅₇	C₄	semidirect
7. C₂×D₄×C₂₅₇	C₄	semidirect
28. C₂₅₇	#353	semidirect
34. C₂×D₄	#11759967	semidirect

Maximal subgroups

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

subgroup	index
#22551651	2
#22551588	2
#353	257