Subgroups — #45666076

Order 32896.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22551532	C₂	1	2	yes	maximal
2	C₄⋊#16861593	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	#22551532	—	2	4	no	—
5	C₂×#16861593	C₂²	1	4	yes	—
6	C₂²×C₈×C₂₅₇	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₂₅₇	C₄	1	1	yes	characteristic
9	C₄×C₈×C₂₅₇	C₄	1	2	yes	—
10	C₄×C₈×C₂₅₇	C₂²	1	2	yes	—
11	#16861593	—	2	4	no	—
12	C₂²×C₄×C₂₅₇	C₈	1	2	yes	—
13	C₂²×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₂₅₇	—	2	2	no	—
15	C₂×C₈×C₂₅₇	C₂×C₄	1	2	yes	—
16	C₂×C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₂₅₇	D₄	1	2	yes	—
18	C₂×C₈×C₂₅₇	Q₈	1	2	yes	—
19	C₂×C₈×C₂₅₇	C₂³	1	1	yes	characteristic
20	C₄²×C₂₅₇	C₈	1	1	yes	characteristic
21	C₄²×C₂₅₇	C₈	1	1	yes	characteristic
22	C₄²×C₂₅₇	C₂×C₄	1	2	yes	—
23	C₂³×C₂₅₇	C₂×C₈	1	1	yes	characteristic
24	C₂×C₄×C₂₅₇	—	2	4	no	—
25	C₂×C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₂₅₇	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₂₅₇	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₂₅₇	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₂₅₇	M₁₆	1	2	yes	—
30	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
31	C₈×C₂₅₇	—	2	2	no	—
32	C₈×C₂₅₇	C₄⋊C₄	1	2	yes	—
33	C₈×C₂₅₇	C₂×D₄	1	1	yes	characteristic
34	C₈×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₂₅₇	—	2	2	no	—
36	C₂²×C₂₅₇	C₄⋊C₈	1	2	yes	—
37	C₂²×C₂₅₇	C₂²×C₈	1	1	yes	characteristic
38	C₄×C₂₅₇	—	2	2	no	—
39	C₄×C₂₅₇	C₄⋊C₈	1	2	yes	—
40	C₄×C₂₅₇	M₃₂	1	4	yes	complemented
41	C₄×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₂₅₇	C₂×M₁₆	1	1	yes	characteristic
43	C₂×C₂₅₇	—	2	2	no	—
44	C₂×C₂₅₇	C₂×C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₂₅₇	C₂×M₃₂	1	2	yes	—
46	C₂₅₇	#2150	1	1	yes	characteristic, complemented, Sylow
47	#2150	—	257	1	no	maximal, Sylow
48	C₂×M₃₂	—	257	2	no	—
49	C₂×C₄×C₈	D₂₅₇	1	1	yes	characteristic
50	C₄⋊C₁₆	—	257	4	no	—
51	C₂²×C₈	Dic₂₅₇	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₂₅₇	1	1	yes	characteristic
53	C₂×C₄²	Dic₂₅₇	1	1	yes	characteristic
54	M₃₂	—	514	4	no	—
55	C₂×C₁₆	—	257	4	no	—
56	C₄×C₈	Dic₂₅₇	1	2	yes	—
57	C₄×C₈	C₂×D₂₅₇	1	2	yes	—
58	C₂²×C₄	#14523339	1	2	yes	—
59	C₂²×C₄	C₂×Dic₂₅₇	1	1	yes	characteristic
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	C₂×Dic₂₅₇	1	2	yes	—
62	C₂×C₈	C₂×Dic₂₅₇	1	1	yes	characteristic
63	C₂×C₈	D₁₀₂₈	1	2	yes	—
64	C₂×C₈	Dic₅₁₄	1	2	yes	—
65	C₂×C₈	C₂²×D₂₅₇	1	1	yes	characteristic
66	C₄²	#14523339	1	1	yes	characteristic
67	C₄²	#14523339	1	1	yes	characteristic
68	C₄²	C₂×Dic₂₅₇	1	2	yes	—
69	C₁₆	—	514	4	no	—
70	C₂³	C₂×#14523339	1	1	yes	characteristic
71	C₂×C₄	—	2	4	no	—
72	C₂×C₄	C₄⋊Dic₂₅₇	1	2	yes	—
73	C₂×C₄	C₂×#14523339	1	2	yes	—
74	C₂×C₄	C₂×#14523339	1	1	yes	characteristic
75	C₂×C₄	C₂×#14523339	1	2	yes	—
76	C₂×C₄	#16861602	1	2	yes	—
77	C₂×C₄	C₂²×Dic₂₅₇	1	1	yes	characteristic
78	C₈	—	2	2	no	—
79	C₈	C₄⋊Dic₂₅₇	1	2	yes	—
80	C₈	C₂×D₁₀₂₈	1	1	yes	characteristic
81	C₈	C₂×Dic₅₁₄	1	1	yes	characteristic
82	C₂²	—	2	2	no	—
83	C₂²	C₄⋊#14523339	1	2	yes	—
84	C₂²	C₂²×#14523339	1	1	yes	characteristic
85	C₄	—	2	2	no	—
86	C₄	C₄⋊#14523339	1	2	yes	—
87	C₄	#22551532	1	4	yes	complemented
88	C₄	C₂×C₄⋊Dic₂₅₇	1	1	yes	characteristic
89	C₄	C₂×#16861602	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×C₄⋊#14523339	1	1	yes	characteristic
92	C₂	C₂×#22551532	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₅₇
commutator, socle	C₂²×C₂₅₇	C₂²×C₈
Fitting	C₂×C₄×C₈×C₂₅₇	C₂
prime radical	C₂³×C₂₅₇	C₂×C₈
cosocle	C₂×C₈×C₂₅₇	C₂³
hypercenter	C₂×C₄×C₈	D₂₅₇
hypocenter	C₂₅₇	#2150
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₄⋊#16861593	C₂	semidirect
40. C₄×C₂₅₇	M₃₂	semidirect
46. C₂₅₇	#2150	semidirect
87. C₄	#22551532	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22551532	2
C₄⋊#16861593	2
C₂×C₄×C₈×C₂₅₇	2
#2150	257