Subgroups — #45662385

Order 32896.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30656513	C₂	1	1	yes	maximal, characteristic
2	#30656488	C₂	1	1	yes	maximal, characteristic
3	#334×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#16861593	C₄	1	1	yes	characteristic
5	C₂×#16861593	C₄	1	1	yes	characteristic
6	C₄⋊#14523339	—	2	1	no	—
7	C₄⋊C₈×C₂₅₇	—	2	1	no	—
8	C₄×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
9	C₂×#14523339	—	4	1	no	—
10	#16861593	—	2	1	no	—
11	#16861593	C₈	1	2	yes	complemented
12	C₂×C₈×C₂₅₇	—	4	1	no	—
13	C₂×C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₂₅₇	Q₈	1	1	yes	characteristic
15	C₄²×C₂₅₇	D₄	1	1	yes	characteristic
16	#14523339	—	8	1	no	—
17	C₂×C₄×C₂₅₇	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₅₇	SD₁₆	1	1	yes	characteristic
19	C₂×C₄×C₂₅₇	SD₁₆	1	1	yes	characteristic
20	C₈×C₂₅₇	—	2	1	no	—
21	C₈×C₂₅₇	—	8	1	no	—
22	C₈×C₂₅₇	C₂×C₈	1	1	yes	characteristic
23	C₈×C₂₅₇	M₁₆	1	1	yes	characteristic
24	C₂²×C₂₅₇	#106	1	1	yes	characteristic
25	C₄×C₂₅₇	—	2	1	no	—
26	C₄×C₂₅₇	—	2	1	no	—
27	C₄×C₂₅₇	C₄⋊C₈	1	1	yes	characteristic
28	C₄×C₂₅₇	#108	1	1	yes	characteristic
29	C₂×C₂₅₇	#334	1	1	yes	characteristic
30	C₂×C₂₅₇	#365	1	1	yes	characteristic
31	C₂×C₂₅₇	#365	1	1	yes	characteristic
32	C₂₅₇	#1368	1	1	yes	characteristic, complemented, Sylow
33	#1368	—	257	1	no	maximal, Sylow
34	#346	—	257	1	no	—
35	#334	—	257	1	no	—
36	#334	D₂₅₇	1	1	yes	characteristic
37	C₂×C₁₆	—	257	1	no	—
38	C₂×C₁₆	—	257	1	no	—
39	C₄⋊C₈	—	2	1	no	—
40	C₄⋊C₈	—	514	1	no	—
41	C₄×C₈	C₂×D₂₅₇	1	1	yes	characteristic
42	C₂×C₈	—	4	1	no	—
43	C₂×C₈	—	1028	1	no	—
44	C₂×C₈	C₄×D₂₅₇	1	1	yes	characteristic
45	C₂×C₈	Dic₅₁₄	1	1	yes	characteristic
46	C₄²	C₂²⋊D₂₅₇	1	1	yes	characteristic
47	C₁₆	—	257	2	no	—
48	C₁₆	—	514	1	no	—
49	C₂×C₄	#16861597	1	1	yes	characteristic
50	C₂×C₄	#16861605	1	1	yes	characteristic
51	C₂×C₄	#16861606	1	1	yes	characteristic
52	C₈	—	2	1	no	—
53	C₈	—	8	1	no	—
54	C₈	—	2056	1	no	—
55	C₈	C₈×D₂₅₇	1	1	yes	characteristic
56	C₈	#16861601	1	1	yes	characteristic
57	C₂²	#22551523	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	#22551521	1	1	yes	characteristic
61	C₄	#22551527	1	1	yes	characteristic
62	C₂	#30656487	1	1	yes	characteristic
63	C₂	#30656559	1	1	yes	characteristic
64	C₂	#30656561	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22551523
commutator	C₈×C₂₅₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₅₇
Fitting	#334×C₂₅₇	C₂
socle, prime radical	C₂²×C₂₅₇	#106
cosocle	C₄×C₈×C₂₅₇	C₂²
hypercenter	#334	D₂₅₇
hypocenter	C₂₅₇	#1368
LCS	G ▷ C₈×C₂₅₇ ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

Complements

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

N	H ≅ G/N	type
11. #16861593	C₈	semidirect
32. C₂₅₇	#1368	semidirect

Maximal subgroups

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

subgroup	index
#30656513	2
#30656488	2
#334×C₂₅₇	2
#1368	257