Subgroups — #45662384

Order 32896.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#16861593	C₂	1	2	yes	maximal
2	C₈²×C₂₅₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#16861593	—	2	2	no	—
4	C₄×C₈×C₂₅₇	C₄	1	1	yes	characteristic
5	C₄×C₈×C₂₅₇	C₄	1	1	yes	characteristic
6	C₄×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
7	#16861593	—	4	2	no	—
8	C₂×C₈×C₂₅₇	—	2	1	no	—
9	C₂×C₈×C₂₅₇	C₈	1	1	yes	characteristic
10	C₂×C₈×C₂₅₇	C₈	1	1	yes	characteristic
11	C₂×C₈×C₂₅₇	D₄	1	1	yes	characteristic
12	C₂×C₈×C₂₅₇	Q₈	1	1	yes	characteristic
13	C₄²×C₂₅₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₂₅₇	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₂₅₇	C₂×C₈	1	1	yes	characteristic
16	C₂×C₄×C₂₅₇	M₁₆	1	1	yes	characteristic
17	C₈×C₂₅₇	—	2	2	no	—
18	C₈×C₂₅₇	—	2	1	no	—
19	C₈×C₂₅₇	—	2	1	no	—
20	C₈×C₂₅₇	C₁₆	1	2	yes	complemented
21	C₈×C₂₅₇	SD₁₆	1	2	yes	—
22	C₂²×C₂₅₇	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₂₅₇	—	2	1	no	—
24	C₄×C₂₅₇	#106	1	1	yes	characteristic
25	C₄×C₂₅₇	#108	1	1	yes	characteristic
26	C₄×C₂₅₇	C₂×C₁₆	1	1	yes	characteristic
27	C₄×C₂₅₇	M₃₂	1	1	yes	characteristic
28	C₂×C₂₅₇	#334	1	1	yes	characteristic
29	C₂×C₂₅₇	C₄⋊C₁₆	1	1	yes	characteristic
30	C₂×C₂₅₇	#364	1	1	yes	characteristic
31	C₂₅₇	#1367	1	1	yes	characteristic, complemented, Sylow
32	#1367	—	257	1	no	maximal, Sylow
33	C₄⋊C₁₆	—	257	2	no	—
34	C₈²	D₂₅₇	1	1	yes	characteristic
35	C₂×C₁₆	—	514	2	no	—
36	C₄×C₈	Dic₂₅₇	1	1	yes	characteristic
37	C₄×C₈	Dic₂₅₇	1	1	yes	characteristic
38	C₄×C₈	C₂×D₂₅₇	1	1	yes	characteristic
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	#14523339	1	1	yes	characteristic
41	C₂×C₈	#14523339	1	1	yes	characteristic
42	C₂×C₈	D₁₀₂₈	1	1	yes	characteristic
43	C₂×C₈	Dic₅₁₄	1	1	yes	characteristic
44	C₄²	C₂×Dic₂₅₇	1	1	yes	characteristic
45	C₁₆	—	1028	2	no	—
46	C₂×C₄	C₄⋊Dic₂₅₇	1	1	yes	characteristic
47	C₂×C₄	C₂×#14523339	1	1	yes	characteristic
48	C₂×C₄	#16861602	1	1	yes	characteristic
49	C₈	—	2	2	no	—
50	C₈	—	2	1	no	—
51	C₈	—	2	1	no	—
52	C₈	#16861593	1	2	yes	complemented
53	C₈	#16861607	1	2	yes	—
54	C₂²	C₄⋊#14523339	1	1	yes	characteristic
55	C₄	—	2	1	no	—
56	C₄	#22551524	1	1	yes	characteristic
57	C₄	#22551528	1	1	yes	characteristic
58	C₄	C₂×#16861593	1	1	yes	characteristic
59	C₄	#22551532	1	1	yes	characteristic
60	C₂	#30656488	1	1	yes	characteristic
61	C₂	C₄⋊#16861593	1	1	yes	characteristic
62	C₂	#30656558	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₁₀₂₈
commutator	C₄×C₂₅₇	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₂₅₇
Fitting	C₈²×C₂₅₇	C₂
socle, prime radical	C₂²×C₂₅₇	C₄⋊C₈
cosocle	C₄×C₈×C₂₅₇	C₂²
hypercenter	C₈²	D₂₅₇
hypocenter	C₂₅₇	#1367
LCS	G ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
20. C₈×C₂₅₇	C₁₆	semidirect
31. C₂₅₇	#1367	semidirect
52. C₈	#16861593	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#16861593	2
C₈²×C₂₅₇	2
#1367	257