Subgroups — #45662387

Order 32896.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30656558	C₂	1	1	yes	maximal, characteristic
2	#30656513	C₂	1	1	yes	maximal, characteristic
3	#364×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	#22551532	—	2	1	no	—
5	C₂×#16861593	C₄	1	1	yes	characteristic
6	C₂×#16861593	C₄	1	1	yes	characteristic
7	M₃₂×C₂₅₇	—	2	1	no	—
8	C₄×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
9	#16861593	—	2	1	no	—
10	#16861593	—	4	1	no	—
11	#16861593	C₈	1	2	yes	—
12	C₂×C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₂₅₇	D₄	1	1	yes	characteristic
14	C₄²×C₂₅₇	Q₈	1	1	yes	characteristic
15	C₁₆×C₂₅₇	—	4	1	no	—
16	C₂×C₄×C₂₅₇	—	2	1	no	—
17	C₂×C₄×C₂₅₇	C₄⋊C₄	1	1	yes	characteristic
18	C₈×C₂₅₇	C₂×C₈	1	1	yes	characteristic
19	C₈×C₂₅₇	M₁₆	1	1	yes	characteristic
20	C₈×C₂₅₇	SD₁₆	1	1	yes	characteristic
21	C₈×C₂₅₇	SD₁₆	1	1	yes	characteristic
22	C₂²×C₂₅₇	#108	1	1	yes	characteristic
23	C₄×C₂₅₇	—	4	1	no	—
24	C₄×C₂₅₇	C₄⋊C₈	1	1	yes	characteristic
25	C₄×C₂₅₇	#106	1	1	yes	characteristic
26	C₂×C₂₅₇	—	2	1	no	—
27	C₂×C₂₅₇	#334	1	1	yes	characteristic
28	C₂₅₇	#1369	1	1	yes	characteristic, complemented, Sylow
29	#1369	—	257	1	no	maximal, Sylow
30	#364	—	257	1	no	—
31	#364	D₂₅₇	1	1	yes	characteristic
32	#346	—	257	1	no	—
33	M₃₂	—	2	1	no	—
34	M₃₂	—	514	1	no	—
35	C₂×C₁₆	—	257	1	no	—
36	C₂×C₁₆	—	257	1	no	—
37	C₄×C₈	C₂×D₂₅₇	1	1	yes	characteristic
38	C₂×C₈	C₄×D₂₅₇	1	1	yes	characteristic
39	C₂×C₈	C₂²⋊D₂₅₇	1	1	yes	characteristic
40	C₄²	Dic₅₁₄	1	1	yes	characteristic
41	C₁₆	—	257	2	no	—
42	C₁₆	—	514	1	no	—
43	C₁₆	—	4	1	no	—
44	C₁₆	—	1028	1	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	#16861597	1	1	yes	characteristic
47	C₈	C₈×D₂₅₇	1	1	yes	characteristic
48	C₈	#16861601	1	1	yes	characteristic
49	C₈	#16861605	1	1	yes	characteristic
50	C₈	#16861606	1	1	yes	characteristic
51	C₂²	#22551527	1	1	yes	characteristic
52	C₄	—	4	1	no	—
53	C₄	#22551521	1	1	yes	characteristic
54	C₄	#22551523	1	1	yes	characteristic
55	C₂	—	2	1	no	—
56	C₂	#30656487	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#22551523
commutator	C₈×C₂₅₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₅₇
Fitting	#364×C₂₅₇	C₂
socle	C₂×C₂₅₇	#334
prime radical	C₂²×C₂₅₇	#108
cosocle	C₄×C₈×C₂₅₇	C₂²
hypercenter	#364	D₂₅₇
hypocenter	C₂₅₇	#1369
LCS	G ▷ C₈×C₂₅₇ ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
28. C₂₅₇	#1369	semidirect

Maximal subgroups

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

subgroup	index
#30656558	2
#30656513	2
#364×C₂₅₇	2
#1369	257