Subgroups — #45662380

Order 32896.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30657139	C₂	1	1	yes	maximal, characteristic
2	#30656558	C₂	1	1	yes	maximal, characteristic
3	#364×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	#22551584	C₄	1	2	yes	—
5	#22551532	—	2	1	no	—
6	#22551526	—	2	1	no	—
7	#22551524	—	2	1	no	—
8	M₃₂×C₂₅₇	—	2	1	no	—
9	C₄×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₅₇	—	2	2	no	—
11	#16861597	—	4	2	no	—
12	#16861593	—	4	1	no	—
13	C₂×C₈×C₂₅₇	D₄	1	1	yes	characteristic
14	C₂×C₈×C₂₅₇	D₄	1	1	yes	characteristic
15	C₄²×C₂₅₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₂₅₇	—	4	1	no	—
17	C₂×Dic₂₅₇	—	4	2	no	—
18	C₂×C₄×C₂₅₇	—	2	1	no	—
19	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₈×C₂₅₇	SD₁₆	1	1	yes	characteristic
21	C₈×C₂₅₇	SD₁₆	1	1	yes	characteristic
22	C₈×C₂₅₇	Q₁₆	1	1	yes	characteristic
23	C₈×C₂₅₇	Q₁₆	1	1	yes	characteristic
24	C₂²×C₂₅₇	#100	1	1	yes	characteristic
25	C₄×C₂₅₇	—	4	1	no	—
26	C₄×C₂₅₇	#103	1	1	yes	characteristic
27	C₄×C₂₅₇	#103	1	1	yes	characteristic
28	Dic₂₅₇	—	8	2	no	—
29	C₂×C₂₅₇	—	2	1	no	—
30	C₂×C₂₅₇	#333	1	1	yes	characteristic
31	C₂₅₇	#1366	1	1	yes	characteristic, complemented, Sylow
32	#1366	—	257	1	no	maximal, Sylow
33	#499	—	257	1	no	—
34	#364	—	257	1	no	—
35	#364	D₂₅₇	1	1	yes	characteristic
36	#125	—	257	2	no	—
37	M₃₂	—	2	1	no	—
38	M₃₂	—	514	1	no	—
39	#107	—	514	1	no	—
40	#106	—	514	1	no	—
41	C₄×C₈	C₂×D₂₅₇	1	1	yes	characteristic
42	C₂×C₈	C₂²⋊D₂₅₇	1	1	yes	characteristic
43	C₂×C₈	D₁₀₂₈	1	1	yes	characteristic
44	C₄⋊C₄	—	514	2	no	—
45	C₄⋊C₄	—	1028	2	no	—
46	C₄²	C₄×D₂₅₇	1	1	yes	characteristic
47	C₁₆	—	4	1	no	—
48	C₁₆	—	1028	1	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	1028	2	no	—
51	C₂×C₄	#16861595	1	1	yes	characteristic
52	C₈	#16861606	1	1	yes	characteristic
53	C₈	#16861607	1	1	yes	characteristic
54	C₈	#16861608	1	1	yes	characteristic
55	C₈	Dic₁₀₂₈	1	1	yes	characteristic
56	C₂²	#22551508	1	1	yes	characteristic
57	C₄	—	4	1	no	—
58	C₄	—	2056	2	no	—
59	C₄	#22551515	1	1	yes	characteristic
60	C₄	#22551517	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	#30656485	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30656485
commutator	C₄²×C₂₅₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₅₇
Fitting	#364×C₂₅₇	C₂
socle	C₂×C₂₅₇	#333
prime radical	C₂²×C₂₅₇	#100
cosocle	C₄×C₈×C₂₅₇	C₂²
hypercenter	#364	D₂₅₇
hypocenter	C₂₅₇	#1366
LCS	G ▷ C₄²×C₂₅₇ ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
31. C₂₅₇	#1366	semidirect

Maximal subgroups

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

subgroup	index
#30657139	2
#30656558	2
#364×C₂₅₇	2
#1366	257