Subgroups — #45663936

Order 32896.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30657140	C₂	1	4	yes	maximal
2	#30657117	C₂	1	1	yes	maximal, characteristic
3	C₈×#14523339	C₂	1	1	yes	maximal, characteristic
4	#494×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₁₀₂₈	—	2	2	no	—
6	#22551592	C₂²	1	2	yes	—
7	#22551525	—	2	8	no	—
8	C₄×#14523339	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₂₅₇	—	2	2	no	—
10	C₄⋊Q₈×C₂₅₇	C₂²	1	2	yes	—
11	C₄×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₅₁₄	—	2	2	no	—
13	Dic₁₀₂₈	—	4	4	no	—
14	C₂×#14523339	D₄	1	4	yes	—
15	C₄⋊Dic₂₅₇	—	2	4	no	—
16	C₂×Q₈×C₂₅₇	—	2	2	no	—
17	Q₁₆×C₂₅₇	—	4	4	no	—
18	C₂×C₈×C₂₅₇	D₄	1	2	yes	—
19	C₄⋊C₄×C₂₅₇	—	2	4	no	—
20	C₄²×C₂₅₇	C₂³	1	1	yes	characteristic
21	Dic₅₁₄	—	4	4	no	—
22	C₂×Dic₂₅₇	—	4	2	no	—
23	#14523339	Q₁₆	1	8	yes	complemented
24	Q₈×C₂₅₇	—	4	4	no	—
25	C₂×C₄×C₂₅₇	—	4	2	no	—
26	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
27	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
28	C₈×C₂₅₇	D₈	1	4	yes	—
29	C₂²×C₂₅₇	#127	1	1	yes	characteristic
30	C₄×C₂₅₇	—	8	2	no	—
31	C₄×C₂₅₇	C₂×D₈	1	2	yes	—
32	C₄×C₂₅₇	C₂×Q₁₆	1	4	yes	—
33	Dic₂₅₇	—	8	2	no	—
34	C₂×C₂₅₇	#493	1	1	yes	characteristic
35	C₂×C₂₅₇	#494	1	2	yes	—
36	C₂₅₇	#1713	1	1	yes	characteristic, complemented, Sylow
37	#1713	—	257	1	no	maximal, Sylow
38	#500	—	257	4	no	—
39	#494	—	257	1	no	—
40	#494	D₂₅₇	1	1	yes	characteristic
41	C₈²	—	257	1	no	—
42	C₂×Q₁₆	—	2	2	no	—
43	C₂×Q₁₆	—	514	2	no	—
44	C₄⋊Q₈	—	257	2	no	—
45	C₄⋊Q₈	C₂×D₂₅₇	1	2	yes	—
46	#107	—	514	8	no	—
47	C₄×C₈	—	257	2	no	—
48	C₄×C₈	C₂×D₂₅₇	1	1	yes	characteristic
49	C₂×Q₈	—	2	2	no	—
50	C₂×Q₈	—	514	2	no	—
51	Q₁₆	—	4	4	no	—
52	Q₁₆	—	1028	4	no	—
53	C₂×C₈	—	257	4	no	—
54	C₂×C₈	C₂²⋊D₂₅₇	1	2	yes	—
55	C₄⋊C₄	—	2	4	no	—
56	C₄⋊C₄	—	514	4	no	—
57	C₄²	C₂²×D₂₅₇	1	1	yes	characteristic
58	Q₈	—	4	4	no	—
59	Q₈	—	1028	4	no	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	—	1028	2	no	—
62	C₂×C₄	D₄×D₂₅₇	1	2	yes	—
63	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
64	C₈	—	257	8	no	—
65	C₈	#16861603	1	4	yes	—
66	C₂²	#22551588	1	1	yes	characteristic
67	C₄	—	8	2	no	—
68	C₄	—	2056	2	no	—
69	C₄	C₂×#16861603	1	2	yes	—
70	C₄	Q₁₆×D₂₅₇	1	4	yes	—
71	C₂	#30657113	1	1	yes	characteristic
72	C₂	#30657115	1	2	yes	—

	subgroup	G/H
center	C₂²	#22551588
commutator, cosocle	C₄²×C₂₅₇	C₂³
Frattini	C₄²	C₂²×D₂₅₇
Fitting	#494×C₂₅₇	C₂
socle, prime radical	C₂²×C₂₅₇	#127
hypercenter	#494	D₂₅₇
hypocenter	C₂₅₇	#1713
LCS	G ▷ C₄²×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
23. #14523339	Q₁₆	semidirect
36. C₂₅₇	#1713	semidirect

Maximal subgroups

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

subgroup	index
#30657140	2
#30657117	2
C₈×#14523339	2
#494×C₂₅₇	2
#1713	257