Subgroups — #45662379

Order 32896.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30657142	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#16861593	C₂	1	1	yes	maximal, characteristic
3	#335×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	#22551592	—	2	1	no	—
5	C₂×#16861593	—	2	1	no	—
6	#22551526	—	2	1	no	—
7	#22551526	C₄	1	2	yes	—
8	C₄⋊C₈×C₂₅₇	—	2	1	no	—
9	C₄×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₅₁₄	—	4	1	no	—
11	C₄⋊Dic₂₅₇	—	2	2	no	—
12	C₄⋊Dic₂₅₇	—	4	1	no	—
13	#16861593	—	4	1	no	—
14	C₂×C₈×C₂₅₇	—	4	1	no	—
15	C₂×C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₂₅₇	D₄	1	1	yes	characteristic
17	C₄²×C₂₅₇	D₄	1	1	yes	characteristic
18	Dic₅₁₄	—	8	2	no	—
19	C₂×Dic₂₅₇	—	4	2	no	—
20	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₅₇	D₈	1	1	yes	characteristic
22	C₂×C₄×C₂₅₇	SD₁₆	1	1	yes	characteristic
23	C₈×C₂₅₇	—	2	1	no	—
24	C₈×C₂₅₇	—	8	1	no	—
25	C₈×C₂₅₇	SD₁₆	1	1	yes	characteristic
26	C₈×C₂₅₇	Q₁₆	1	1	yes	characteristic
27	C₂²×C₂₅₇	#102	1	1	yes	characteristic
28	C₄×C₂₅₇	—	2	1	no	—
29	C₄×C₂₅₇	#101	1	1	yes	characteristic
30	C₄×C₂₅₇	#103	1	1	yes	characteristic
31	C₄×C₂₅₇	SD₃₂	1	1	yes	characteristic
32	C₄×C₂₅₇	Q₃₂	1	1	yes	characteristic
33	Dic₂₅₇	—	8	2	no	—
34	C₂×C₂₅₇	#332	1	1	yes	characteristic
35	C₂×C₂₅₇	#358	1	1	yes	characteristic
36	C₂×C₂₅₇	#361	1	1	yes	characteristic
37	C₂₅₇	#1365	1	1	yes	characteristic, complemented, Sylow
38	#1365	—	257	1	no	maximal, Sylow
39	#500	—	257	1	no	—
40	C₄⋊C₁₆	—	257	1	no	—
41	#335	D₂₅₇	1	1	yes	characteristic
42	C₄⋊Q₈	—	514	1	no	—
43	C₂×C₁₆	—	514	1	no	—
44	#107	—	257	2	no	—
45	#107	—	514	1	no	—
46	C₄⋊C₈	—	2	1	no	—
47	C₄×C₈	C₂×D₂₅₇	1	1	yes	characteristic
48	C₂×Q₈	—	1028	1	no	—
49	C₂×C₈	—	4	1	no	—
50	C₂×C₈	C₄×D₂₅₇	1	1	yes	characteristic
51	C₂×C₈	D₁₀₂₈	1	1	yes	characteristic
52	C₄⋊C₄	—	514	2	no	—
53	C₄⋊C₄	—	1028	1	no	—
54	C₄²	C₂²⋊D₂₅₇	1	1	yes	characteristic
55	C₁₆	—	1028	1	no	—
56	Q₈	—	2056	2	no	—
57	C₂×C₄	—	1028	2	no	—
58	C₂×C₄	#16861595	1	1	yes	characteristic
59	C₂×C₄	#16861603	1	1	yes	characteristic
60	C₂×C₄	#16861605	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	8	1	no	—
63	C₈	#16861607	1	1	yes	characteristic
64	C₈	Dic₁₀₂₈	1	1	yes	characteristic
65	C₂²	#22551512	1	1	yes	characteristic
66	C₄	—	2	1	no	—
67	C₄	—	2056	2	no	—
68	C₄	#22551510	1	1	yes	characteristic
69	C₄	#22551517	1	1	yes	characteristic
70	C₄	#22551536	1	1	yes	characteristic
71	C₄	#22551538	1	1	yes	characteristic
72	C₂	#30656484	1	1	yes	characteristic
73	C₂	#30656540	1	1	yes	characteristic
74	C₂	#30656549	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22551512
commutator	C₂×C₈×C₂₅₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₅₇
Fitting	#335×C₂₅₇	C₂
socle, prime radical	C₂²×C₂₅₇	#102
cosocle	C₄×C₈×C₂₅₇	C₂²
hypercenter	#335	D₂₅₇
hypocenter	C₂₅₇	#1365
LCS	G ▷ C₂×C₈×C₂₅₇ ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
37. C₂₅₇	#1365	semidirect

Maximal subgroups

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

subgroup	index
#30657142	2
C₄⋊#16861593	2
#335×C₂₅₇	2
#1365	257