Subgroups — #45662354

Order 32896.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30657106	C₂	1	1	yes	maximal, characteristic
2	#30656515	C₂	1	1	yes	maximal, characteristic
3	#347×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	#22551592	C₄	1	1	yes	characteristic
5	#22551584	C₄	1	1	yes	characteristic
6	#22551532	—	2	1	no	—
7	#22551517	—	2	1	no	—
8	#22551517	—	2	1	no	—
9	M₃₂×C₂₅₇	—	2	1	no	—
10	#97×C₂₅₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₅₁₄	—	2	1	no	—
12	C₄⋊Dic₂₅₇	—	2	1	no	—
13	C₄⋊Dic₂₅₇	—	4	1	no	—
14	#16861597	—	4	1	no	—
15	#16861593	—	2	2	no	—
16	C₂×C₈×C₂₅₇	D₄	1	1	yes	characteristic
17	C₂×C₈×C₂₅₇	D₄	1	1	yes	characteristic
18	C₄²×C₂₅₇	C₂×C₄	1	1	yes	characteristic
19	C₁₆×C₂₅₇	—	2	2	no	—
20	Dic₅₁₄	—	4	1	no	—
21	Dic₅₁₄	—	4	1	no	—
22	C₂×Dic₂₅₇	—	4	1	no	—
23	C₂×Dic₂₅₇	—	4	1	no	—
24	C₂×C₄×C₂₅₇	—	2	1	no	—
25	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₂₅₇	—	2	1	no	—
27	C₈×C₂₅₇	—	2	1	no	—
28	C₂²×C₂₅₇	#100	1	1	yes	characteristic
29	C₄×C₂₅₇	—	4	1	no	—
30	C₄×C₂₅₇	#104	1	1	yes	characteristic
31	C₄×C₂₅₇	#104	1	1	yes	characteristic
32	Dic₂₅₇	—	8	1	no	—
33	Dic₂₅₇	—	8	1	no	—
34	C₂×C₂₅₇	—	2	1	no	—
35	C₂×C₂₅₇	#329	1	1	yes	characteristic
36	C₂₅₇	#1356	1	1	yes	characteristic, complemented, Sylow
37	#1356	—	257	1	no	maximal, Sylow
38	#491	—	257	1	no	—
39	#347	—	257	1	no	—
40	#347	D₂₅₇	1	1	yes	characteristic
41	C₄⋊Q₈	—	257	1	no	—
42	#125	—	257	1	no	—
43	M₃₂	—	2	1	no	—
44	M₃₂	—	514	1	no	—
45	#103	—	514	1	no	—
46	#103	—	514	1	no	—
47	#97	C₂×D₂₅₇	1	1	yes	characteristic
48	C₂×Q₈	—	514	1	no	—
49	C₂×C₈	C₂²⋊D₂₅₇	1	1	yes	characteristic
50	C₂×C₈	D₁₀₂₈	1	1	yes	characteristic
51	C₄⋊C₄	—	514	1	no	—
52	C₄⋊C₄	—	1028	1	no	—
53	C₄⋊C₄	—	1028	1	no	—
54	C₄²	C₄×D₂₅₇	1	1	yes	characteristic
55	C₁₆	—	2	2	no	—
56	C₁₆	—	514	2	no	—
57	Q₈	—	1028	1	no	—
58	Q₈	—	1028	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	1028	1	no	—
61	C₂×C₄	—	1028	1	no	—
62	C₂×C₄	#16861595	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	2	1	no	—
65	C₂²	#22551508	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	—	2056	1	no	—
68	C₄	—	2056	1	no	—
69	C₄	#22551518	1	1	yes	characteristic
70	C₄	#22551520	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#30656476	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30656476
commutator	C₄²×C₂₅₇	C₂×C₄
Frattini	#97	C₂×D₂₅₇
Fitting	#347×C₂₅₇	C₂
socle	C₂×C₂₅₇	#329
prime radical	C₂²×C₂₅₇	#100
cosocle	#97×C₂₅₇	C₂²
hypercenter	#347	D₂₅₇
hypocenter	C₂₅₇	#1356
LCS	G ▷ C₄²×C₂₅₇ ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
36. C₂₅₇	#1356	semidirect

Maximal subgroups

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

subgroup	index
#30657106	2
#30656515	2
#347×C₂₅₇	2
#1356	257