Subgroups — #45662296

Order 32896.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30656558	C₂	1	1	yes	maximal, characteristic, complemented
2	#30656458	C₂	1	1	yes	maximal, characteristic, complemented
3	#443×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	#22551532	—	2	1	no	—
5	C₄×#14523339	—	2	1	no	—
6	#135×C₂₅₇	C₄	1	1	yes	characteristic
7	#131×C₂₅₇	—	2	1	no	—
8	#108×C₂₅₇	C₄	1	1	yes	characteristic
9	#104×C₂₅₇	—	2	1	no	—
10	C₄×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
11	C₂×#14523339	—	2	2	no	—
12	#16861593	—	4	1	no	—
13	#41×C₂₅₇	—	2	1	no	—
14	Q₁₆×C₂₅₇	C₈	1	1	yes	characteristic, complemented
15	SD₁₆×C₂₅₇	—	2	1	no	—
16	D₈×C₂₅₇	C₈	1	1	yes	characteristic, complemented
17	M₁₆×C₂₅₇	—	2	1	no	—
18	M₁₆×C₂₅₇	—	4	1	no	—
19	C₂×C₈×C₂₅₇	—	4	1	no	—
20	C₂×C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₂₅₇	D₄	1	1	yes	characteristic
22	C₄²×C₂₅₇	D₄	1	1	yes	characteristic
23	#14523339	—	4	2	no	—
24	Q₈×C₂₅₇	—	2	1	no	—
25	D₄×C₂₅₇	—	2	1	no	—
26	D₄×C₂₅₇	—	4	1	no	—
27	C₂×C₄×C₂₅₇	—	2	1	no	—
28	C₂×C₄×C₂₅₇	—	4	1	no	—
29	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₈×C₂₅₇	—	4	1	no	—
31	C₈×C₂₅₇	C₂×C₈	1	1	yes	characteristic
32	C₈×C₂₅₇	M₁₆	1	1	yes	characteristic
33	C₈×C₂₅₇	D₈	1	1	yes	characteristic
34	C₈×C₂₅₇	SD₁₆	1	1	yes	characteristic
35	C₂²×C₂₅₇	—	4	1	no	—
36	C₂²×C₂₅₇	#104	1	1	yes	characteristic
37	C₄×C₂₅₇	—	2	2	no	—
38	C₄×C₂₅₇	—	4	1	no	—
39	C₄×C₂₅₇	C₂²⋊C₈	1	1	yes	characteristic
40	C₄×C₂₅₇	#102	1	1	yes	characteristic
41	C₂×C₂₅₇	—	2	1	no	—
42	C₂×C₂₅₇	—	8	1	no	—
43	C₂×C₂₅₇	D₄⋊C₈	1	1	yes	characteristic
44	C₂₅₇	#1336	1	1	yes	characteristic, complemented, Sylow
45	#1336	—	257	1	no	maximal, Sylow
46	#443	D₂₅₇	1	1	yes	characteristic
47	#364	—	257	1	no	—
48	#322	—	257	1	no	—
49	#135	Dic₂₅₇	1	1	yes	characteristic
50	#131	—	2	1	no	—
51	M₃₂	—	514	1	no	—
52	#108	Dic₂₅₇	1	1	yes	characteristic
53	#104	—	2	1	no	—
54	C₄×C₈	—	514	1	no	—
55	C₄×C₈	C₂×D₂₅₇	1	1	yes	characteristic
56	#41	—	2	1	no	—
57	Q₁₆	#14523339	1	1	yes	characteristic, complemented
58	SD₁₆	—	2	1	no	—
59	D₈	#14523339	1	1	yes	characteristic, complemented
60	M₁₆	—	2	1	no	—
61	M₁₆	—	4	1	no	—
62	C₂×C₈	—	514	2	no	—
63	C₂×C₈	—	4	1	no	—
64	C₂×C₈	C₂×Dic₂₅₇	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₂₅₇	1	1	yes	characteristic
66	C₄²	C₂²⋊D₂₅₇	1	1	yes	characteristic
67	C₁₆	—	1028	1	no	—
68	Q₈	—	2	1	no	—
69	D₄	—	2	1	no	—
70	D₄	—	4	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	4	1	no	—
73	C₂×C₄	C₂²⋊Dic₂₅₇	1	1	yes	characteristic
74	C₈	—	4	1	no	—
75	C₈	—	1028	2	no	—
76	C₈	C₂×#14523339	1	1	yes	characteristic
77	C₈	#16861602	1	1	yes	characteristic
78	C₈	#16861603	1	1	yes	characteristic
79	C₈	#16861606	1	1	yes	characteristic
80	C₂²	—	4	1	no	—
81	C₂²	#22551519	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	4	1	no	—
84	C₄	C₂²⋊#14523339	1	1	yes	characteristic
85	C₄	#22551513	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	8	1	no	—
88	C₂	D₄⋊#14523339	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#22551513
commutator	C₈×C₂₅₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₅₇
Fitting	#443×C₂₅₇	C₂
socle	C₂×C₂₅₇	D₄⋊C₈
prime radical	#135×C₂₅₇	C₄
cosocle	C₄×C₈×C₂₅₇	C₂²
hypercenter	#443	D₂₅₇
hypocenter	C₂₅₇	#1336
LCS	G ▷ C₈×C₂₅₇ ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
1. #30656558	C₂	semidirect
2. #30656458	C₂	semidirect
14. Q₁₆×C₂₅₇	C₈	semidirect
16. D₈×C₂₅₇	C₈	semidirect
44. C₂₅₇	#1336	semidirect
57. Q₁₆	#14523339	semidirect
59. D₈	#14523339	semidirect

Maximal subgroups

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

subgroup	index
#30656558	2
#30656458	2
#443×C₂₅₇	2
#1336	257