Subgroups — #45664461

Order 32896.

180 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×#97×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	#97×C₂₅₇	C₄	1	4	yes	—
9	#16861602	—	2	8	no	—
10	C₂×#14523339	—	2	4	no	—
11	C₂²×C₄×C₂₅₇	Q₈	1	2	yes	—
12	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
13	C₂×C₈×C₂₅₇	—	2	4	no	—
14	C₂×C₈×C₂₅₇	C₂×C₄	1	4	yes	—
15	C₄²×C₂₅₇	C₂×C₄	1	2	yes	—
16	C₄²×C₂₅₇	D₄	1	2	yes	—
17	#14523339	—	4	4	no	—
18	C₂³×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
19	C₂×C₄×C₂₅₇	—	4	2	no	—
20	C₂×C₄×C₂₅₇	C₄⋊C₄	1	4	yes	—
21	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₅₇	#41	1	4	yes	—
24	C₈×C₂₅₇	—	2	4	no	—
25	C₂²×C₂₅₇	—	2	2	no	—
26	C₂²×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₂₅₇	#125	1	2	yes	—
28	C₄×C₂₅₇	—	2	4	no	—
29	C₄×C₂₅₇	#117	1	2	yes	—
30	C₄×C₂₅₇	#124	1	2	yes	—
31	C₂×C₂₅₇	—	2	2	no	—
32	C₂×C₂₅₇	#382	1	1	yes	characteristic
33	C₂×C₂₅₇	#430	1	2	yes	—
34	C₂₅₇	#1838	1	1	yes	characteristic, complemented, Sylow
35	#1838	—	257	1	no	maximal, Sylow
36	#423	—	257	2	no	—
37	C₂×#97	D₂₅₇	1	1	yes	characteristic
38	#341	—	257	4	no	—
39	C₂×M₁₆	—	257	4	no	—
40	C₂²×C₈	C₂×D₂₅₇	1	2	yes	—
41	C₂×C₄²	C₂×D₂₅₇	1	1	yes	characteristic
42	C₄⋊C₈	—	514	4	no	—
43	#97	Dic₂₅₇	1	4	yes	—
44	C₂²×C₄	Dic₅₁₄	1	2	yes	—
45	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
46	M₁₆	—	514	8	no	—
47	C₂×C₈	—	2	4	no	—
48	C₂×C₈	—	514	4	no	—
49	C₂×C₈	C₂×Dic₂₅₇	1	4	yes	—
50	C₄²	C₂×Dic₂₅₇	1	2	yes	—
51	C₄²	D₁₀₂₈	1	2	yes	—
52	C₂³	C₂×Dic₅₁₄	1	1	yes	characteristic
53	C₂×C₄	—	4	2	no	—
54	C₂×C₄	C₄⋊Dic₂₅₇	1	4	yes	—
55	C₂×C₄	C₂²×Dic₂₅₇	1	1	yes	characteristic
56	C₂×C₄	C₂×D₁₀₂₈	1	1	yes	characteristic
57	C₂×C₄	#16861618	1	4	yes	—
58	C₈	—	2	4	no	—
59	C₈	—	1028	4	no	—
60	C₂²	—	2	2	no	—
61	C₂²	C₂×C₄⋊Dic₂₅₇	1	1	yes	characteristic
62	C₂²	#22551584	1	2	yes	—
63	C₄	—	2	4	no	—
64	C₄	#22551551	1	2	yes	—
65	C₄	#22551580	1	2	yes	—
66	C₂	—	2	2	no	—
67	C₂	#30656603	1	1	yes	characteristic
68	C₂	#30656786	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₀₂₈
commutator	C₂×C₄×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	C₂×#97×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×C₄⋊C₄
prime radical	C₂³×C₂₅₇	C₂×Q₈
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	C₂×#97	D₂₅₇
hypocenter	C₂₅₇	#1838
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
34. C₂₅₇	#1838	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×#97×C₂₅₇	2
#1838	257