Subgroups — #45663930

Order 32896.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30657142	C₂	1	2	yes	maximal
2	#30657088	C₂	1	4	yes	maximal
3	C₈²×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	#22551592	C₂²	1	4	yes	—
5	#22551526	—	2	4	no	—
6	#22551517	—	2	8	no	—
7	C₄×C₈×C₂₅₇	C₂²	1	2	yes	—
8	C₄×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₅₁₄	—	2	4	no	—
10	C₄⋊Dic₂₅₇	—	2	4	no	—
11	C₄⋊Dic₂₅₇	—	2	4	no	—
12	C₂×C₈×C₂₅₇	D₄	1	4	yes	—
13	C₂×C₈×C₂₅₇	D₄	1	2	yes	—
14	C₄²×C₂₅₇	C₂³	1	1	yes	characteristic
15	Dic₅₁₄	—	4	8	no	—
16	C₂×Dic₂₅₇	—	4	4	no	—
17	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
18	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
19	C₈×C₂₅₇	—	2	4	no	—
20	C₈×C₂₅₇	Q₁₆	1	4	yes	—
21	C₂²×C₂₅₇	#127	1	1	yes	characteristic
22	C₄×C₂₅₇	C₂×Q₁₆	1	2	yes	—
23	C₄×C₂₅₇	#135	1	4	yes	—
24	Dic₂₅₇	—	8	4	no	—
25	C₂×C₂₅₇	#494	1	1	yes	characteristic
26	C₂×C₂₅₇	#495	1	2	yes	—
27	C₂₅₇	#1710	1	1	yes	characteristic, complemented, Sylow
28	#1710	—	257	1	no	maximal, Sylow
29	#500	—	257	2	no	—
30	#487	—	257	4	no	—
31	C₈²	D₂₅₇	1	1	yes	characteristic
32	C₄⋊Q₈	—	257	4	no	—
33	#107	—	514	4	no	—
34	#103	—	514	8	no	—
35	C₄×C₈	C₂×D₂₅₇	1	2	yes	—
36	C₄×C₈	C₂×D₂₅₇	1	1	yes	characteristic
37	C₂×Q₈	—	514	4	no	—
38	C₂×C₈	D₁₀₂₈	1	4	yes	—
39	C₂×C₈	D₁₀₂₈	1	2	yes	—
40	C₄⋊C₄	—	514	4	no	—
41	C₄⋊C₄	—	514	4	no	—
42	C₄²	C₂²×D₂₅₇	1	1	yes	characteristic
43	Q₈	—	1028	8	no	—
44	C₂×C₄	—	1028	4	no	—
45	C₂×C₄	C₂×D₁₀₂₈	1	1	yes	characteristic
46	C₂×C₄	C₂×D₁₀₂₈	1	2	yes	—
47	C₈	—	2	4	no	—
48	C₈	Dic₁₀₂₈	1	4	yes	—
49	C₂²	#22551589	1	1	yes	characteristic
50	C₄	—	2056	4	no	—
51	C₄	C₂×Dic₁₀₂₈	1	2	yes	—
52	C₄	#22551614	1	4	yes	—
53	C₂	#30657117	1	1	yes	characteristic
54	C₂	#30657121	1	2	yes	—

	subgroup	G/H
center	C₂²	#22551589
commutator, cosocle	C₄²×C₂₅₇	C₂³
Frattini	C₄²	C₂²×D₂₅₇
Fitting	C₈²×C₂₅₇	C₂
socle, prime radical	C₂²×C₂₅₇	#127
hypercenter	C₈²	D₂₅₇
hypocenter	C₂₅₇	#1710
LCS	G ▷ C₄²×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
27. C₂₅₇	#1710	semidirect

Maximal subgroups

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

subgroup	index
#30657142	2
#30657088	2
C₈²×C₂₅₇	2
#1710	257