Subgroups — #45662272

Order 32896.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30656765	C₂	1	1	yes	maximal, characteristic
2	#30656519	C₂	1	1	yes	maximal, characteristic
3	#349×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#14523339	C₄	1	1	yes	characteristic
5	#22551551	C₄	1	1	yes	characteristic
6	#22551532	—	2	1	no	—
7	#22551521	—	2	1	no	—
8	#22551521	—	2	1	no	—
9	C₂×M₁₆×C₂₅₇	C₂²	1	1	yes	characteristic
10	M₃₂×C₂₅₇	—	2	1	no	—
11	C₂×#14523339	—	2	1	no	—
12	C₂×#14523339	—	4	1	no	—
13	C₄⋊Dic₂₅₇	—	2	1	no	—
14	C₂²⋊Dic₂₅₇	C₈	1	2	yes	—
15	C₄×Dic₂₅₇	—	2	1	no	—
16	#16861593	—	4	1	no	—
17	C₂²×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
18	M₁₆×C₂₅₇	—	4	1	no	—
19	C₂×C₈×C₂₅₇	D₄	1	1	yes	characteristic
20	C₂×C₈×C₂₅₇	D₄	1	1	yes	characteristic
21	C₁₆×C₂₅₇	—	4	1	no	—
22	C₂×Dic₂₅₇	—	2	2	no	—
23	#14523339	—	4	1	no	—
24	C₂³×C₂₅₇	C₂×C₈	1	1	yes	characteristic
25	C₂×C₄×C₂₅₇	—	2	1	no	—
26	C₂×C₄×C₂₅₇	—	2	1	no	—
27	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₂₅₇	M₁₆	1	1	yes	characteristic
29	C₈×C₂₅₇	—	2	1	no	—
30	C₈×C₂₅₇	—	2	1	no	—
31	C₂²×C₂₅₇	—	2	1	no	—
32	C₂²×C₂₅₇	—	4	1	no	—
33	C₂²×C₂₅₇	C₂²⋊C₈	1	1	yes	characteristic
34	C₄×C₂₅₇	—	2	1	no	—
35	C₄×C₂₅₇	#99	1	1	yes	characteristic
36	C₄×C₂₅₇	#101	1	1	yes	characteristic
37	Dic₂₅₇	—	4	2	no	—
38	C₂×C₂₅₇	—	2	1	no	—
39	C₂×C₂₅₇	—	4	1	no	—
40	C₂×C₂₅₇	#324	1	1	yes	characteristic
41	C₂₅₇	#1328	1	1	yes	characteristic, complemented, Sylow
42	#1328	—	257	1	no	maximal, Sylow
43	#424	—	257	1	no	—
44	#349	—	257	1	no	—
45	#349	D₂₅₇	1	1	yes	characteristic
46	C₂×M₁₆	C₂×D₂₅₇	1	1	yes	characteristic
47	C₂²×C₈	—	257	1	no	—
48	#117	—	257	1	no	—
49	M₃₂	—	2	1	no	—
50	M₃₂	—	514	1	no	—
51	C₄⋊C₈	—	514	1	no	—
52	C₄⋊C₈	—	514	1	no	—
53	C₂²×C₄	C₄×D₂₅₇	1	1	yes	characteristic
54	M₁₆	—	4	1	no	—
55	C₂×C₈	—	514	1	no	—
56	C₂×C₈	—	1028	1	no	—
57	C₂×C₈	C₂²⋊D₂₅₇	1	1	yes	characteristic
58	C₂×C₈	D₁₀₂₈	1	1	yes	characteristic
59	C₄⋊C₄	—	514	1	no	—
60	C₂²⋊C₄	—	257	2	no	—
61	C₄²	—	514	1	no	—
62	C₁₆	—	4	1	no	—
63	C₁₆	—	1028	1	no	—
64	C₂³	C₈×D₂₅₇	1	1	yes	characteristic
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	514	2	no	—
68	C₂×C₄	#16861595	1	1	yes	characteristic
69	C₂×C₄	#16861601	1	1	yes	characteristic
70	C₈	—	2	1	no	—
71	C₈	—	2	1	no	—
72	C₈	—	1028	1	no	—
73	C₂²	—	2	1	no	—
74	C₂²	—	4	1	no	—
75	C₂²	#22551504	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	1028	2	no	—
78	C₄	#22551506	1	1	yes	characteristic
79	C₄	#22551510	1	1	yes	characteristic
80	C₂	—	2	1	no	—
81	C₂	—	4	1	no	—
82	C₂	#30656462	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#22551506
commutator, prime radical	C₂³×C₂₅₇	C₂×C₈
Frattini	C₂×M₁₆	C₂×D₂₅₇
Fitting	#349×C₂₅₇	C₂
socle	C₂×C₂₅₇	#324
cosocle	C₂×M₁₆×C₂₅₇	C₂²
hypercenter	#349	D₂₅₇
hypocenter	C₂₅₇	#1328
LCS	G ▷ C₂³×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

Complements

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

N	H ≅ G/N	type
41. C₂₅₇	#1328	semidirect

Maximal subgroups

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

subgroup	index
#30656765	2
#30656519	2
#349×C₂₅₇	2
#1328	257