Subgroups — #30657113

Order 16448.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861603	C₂	1	4	yes	maximal, complemented
2	#22551589	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14523339	C₂	1	1	yes	maximal, characteristic, complemented
4	#127×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₀₂₈	—	2	2	no	—
6	C₂×D₁₀₂₈	C₂²	1	2	yes	—
7	#16861603	—	2	8	no	—
8	C₂×#14523339	C₂²	1	2	yes	—
9	C₂×D₄×C₂₅₇	—	2	2	no	—
10	C₂×D₄×C₂₅₇	C₂²	1	2	yes	—
11	C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
12	C₂²×D₂₅₇	—	2	2	no	—
13	D₁₀₂₈	—	2	4	no	—
14	D₁₀₂₈	—	4	4	no	—
15	#14523339	D₄	1	4	yes	complemented
16	C₂³×C₂₅₇	—	2	2	no	—
17	D₄×C₂₅₇	—	2	4	no	—
18	D₄×C₂₅₇	—	4	4	no	—
19	C₂×C₄×C₂₅₇	D₄	1	2	yes	—
20	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
21	C₂²×C₂₅₇	—	4	2	no	—
22	C₂²×C₂₅₇	—	4	4	no	—
23	C₂²×C₂₅₇	C₂×D₄	1	1	yes	characteristic
24	C₂×D₂₅₇	—	4	2	no	—
25	C₂×D₂₅₇	—	4	4	no	—
26	C₄×C₂₅₇	D₈	1	4	yes	complemented
27	C₄×C₂₅₇	C₂×D₄	1	2	yes	—
28	C₂×C₂₅₇	—	8	2	no	—
29	C₂×C₂₅₇	#127	1	1	yes	characteristic
30	C₂×C₂₅₇	C₂×D₈	1	2	yes	—
31	D₂₅₇	—	8	2	no	—
32	C₂₅₇	#493	1	1	yes	characteristic, complemented, Sylow
33	#493	—	257	1	no	maximal, Sylow
34	C₂×D₈	—	257	4	no	—
35	#127	—	257	1	no	—
36	#127	D₂₅₇	1	1	yes	characteristic, complemented
37	C₄×C₈	—	257	1	no	—
38	C₂×D₄	—	257	2	no	—
39	C₂×D₄	—	2	2	no	—
40	C₂×D₄	—	514	2	no	—
41	C₂×D₄	C₂×D₂₅₇	1	2	yes	—
42	D₈	—	514	8	no	—
43	C₂×C₈	—	257	2	no	—
44	C₄²	C₂×D₂₅₇	1	1	yes	characteristic
45	C₂³	—	2	2	no	—
46	C₂³	—	514	2	no	—
47	D₄	—	2	4	no	—
48	D₄	—	514	4	no	—
49	D₄	—	4	4	no	—
50	D₄	—	1028	4	no	—
51	C₂×C₄	C₂²⋊D₂₅₇	1	2	yes	—
52	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
53	C₈	—	257	4	no	—
54	C₂²	—	4	2	no	—
55	C₂²	—	4	4	no	—
56	C₂²	—	1028	2	no	—
57	C₂²	—	1028	4	no	—
58	C₂²	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
59	C₄	#16861603	1	4	yes	complemented
60	C₄	D₄×D₂₅₇	1	2	yes	—
61	C₂	—	8	2	no	—
62	C₂	—	2056	2	no	—
63	C₂	#22551588	1	1	yes	characteristic
64	C₂	C₂×#16861603	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₅₇
commutator, cosocle	C₂×C₄×C₂₅₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₅₇
Fitting	#127×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×D₄
hypercenter	#127	D₂₅₇
hypocenter	C₂₅₇	#493
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×#16861603	C₂	semidirect
2. #22551589	C₂	semidirect
3. C₄×#14523339	C₂	semidirect
4. #127×C₂₅₇	C₂	semidirect
15. #14523339	D₄	semidirect
26. C₄×C₂₅₇	D₈	semidirect
32. C₂₅₇	#493	semidirect
36. #127	D₂₅₇	semidirect
59. C₄	#16861603	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861603	2
#22551589	2
C₄×#14523339	2
#127×C₂₅₇	2
#493	257