Subgroups — #45664458

Order 32896.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30656761	C₂	1	2	yes	maximal
2	#30656502	C₂	1	4	yes	maximal
3	C₂×C₄×C₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#16861602	C₂²	1	4	yes	—
5	C₄⋊#14523339	—	2	4	no	—
6	C₂²×C₈×C₂₅₇	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₂₅₇	C₄	1	2	yes	—
9	C₄×C₈×C₂₅₇	C₄	1	2	yes	—
10	#16861602	—	2	8	no	—
11	C₂×#14523339	—	2	4	no	—
12	C₂²×C₄×C₂₅₇	Q₈	1	2	yes	—
13	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
14	C₂×C₈×C₂₅₇	—	2	4	no	—
15	C₂×C₈×C₂₅₇	C₂×C₄	1	4	yes	—
16	C₄²×C₂₅₇	C₂×C₄	1	1	yes	characteristic
17	C₄²×C₂₅₇	C₂×C₄	1	1	yes	characteristic
18	C₄²×C₂₅₇	D₄	1	2	yes	—
19	#14523339	—	4	4	no	—
20	C₂³×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
21	C₂×C₄×C₂₅₇	—	2	4	no	—
22	C₂×C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
23	C₂×C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
24	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
26	C₂×C₄×C₂₅₇	#41	1	4	yes	—
27	C₈×C₂₅₇	—	2	4	no	—
28	C₂²×C₂₅₇	—	2	2	no	—
29	C₂²×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₂₅₇	#125	1	2	yes	—
31	C₄×C₂₅₇	—	2	2	no	—
32	C₄×C₂₅₇	#108	1	4	yes	—
33	C₄×C₂₅₇	#117	1	2	yes	—
34	C₄×C₂₅₇	#124	1	2	yes	—
35	C₂×C₂₅₇	—	2	2	no	—
36	C₂×C₂₅₇	#382	1	1	yes	characteristic
37	C₂×C₂₅₇	C₂×#108	1	2	yes	—
38	C₂₅₇	#1837	1	1	yes	characteristic, complemented, Sylow
39	#1837	—	257	1	no	maximal, Sylow
40	#423	—	257	2	no	—
41	C₂×C₄×C₈	D₂₅₇	1	1	yes	characteristic
42	#341	—	257	4	no	—
43	C₂×M₁₆	—	257	4	no	—
44	C₂²×C₈	C₂×D₂₅₇	1	2	yes	—
45	C₂×C₄²	C₂×D₂₅₇	1	1	yes	characteristic
46	C₄⋊C₈	—	514	4	no	—
47	C₄×C₈	Dic₂₅₇	1	2	yes	—
48	C₄×C₈	Dic₂₅₇	1	2	yes	—
49	C₂²×C₄	Dic₅₁₄	1	2	yes	—
50	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
51	M₁₆	—	514	8	no	—
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	514	4	no	—
54	C₂×C₈	C₂×Dic₂₅₇	1	4	yes	—
55	C₄²	C₂×Dic₂₅₇	1	1	yes	characteristic
56	C₄²	C₂×Dic₂₅₇	1	1	yes	characteristic
57	C₄²	D₁₀₂₈	1	2	yes	—
58	C₂³	C₂×Dic₅₁₄	1	1	yes	characteristic
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	C₄⋊Dic₂₅₇	1	2	yes	—
61	C₂×C₄	C₄⋊Dic₂₅₇	1	2	yes	—
62	C₂×C₄	C₂²×Dic₂₅₇	1	1	yes	characteristic
63	C₂×C₄	C₂×D₁₀₂₈	1	1	yes	characteristic
64	C₂×C₄	#16861618	1	4	yes	—
65	C₈	—	2	4	no	—
66	C₈	—	1028	4	no	—
67	C₂²	—	2	2	no	—
68	C₂²	C₂×C₄⋊Dic₂₅₇	1	1	yes	characteristic
69	C₂²	#22551584	1	2	yes	—
70	C₄	—	2	2	no	—
71	C₄	#22551528	1	4	yes	—
72	C₄	#22551551	1	2	yes	—
73	C₄	#22551580	1	2	yes	—
74	C₂	—	2	2	no	—
75	C₂	#30656603	1	1	yes	characteristic
76	C₂	C₂×#22551528	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₀₂₈
commutator	C₂×C₄×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	C₂×C₄×C₈×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×C₄⋊C₄
prime radical	C₂³×C₂₅₇	C₂×Q₈
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	C₂×C₄×C₈	D₂₅₇
hypocenter	C₂₅₇	#1837
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
38. C₂₅₇	#1837	semidirect

Maximal subgroups

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

subgroup	index
#30656761	2
#30656502	2
C₂×C₄×C₈×C₂₅₇	2
#1837	257