Subgroups — #45662358

Order 32896.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30657117	C₂	1	1	yes	maximal, characteristic
2	#30656558	C₂	1	1	yes	maximal, characteristic
3	#364×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₀₂₈	—	2	1	no	—
5	C₂×Dic₁₀₂₈	—	2	1	no	—
6	#22551592	C₄	1	2	yes	—
7	#22551532	—	2	1	no	—
8	M₃₂×C₂₅₇	—	2	1	no	—
9	C₄×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₅₁₄	—	2	2	no	—
11	Dic₁₀₂₈	—	4	1	no	—
12	Dic₁₀₂₈	—	4	1	no	—
13	Dic₁₀₂₈	—	4	1	no	—
14	Dic₁₀₂₈	—	4	1	no	—
15	C₄⋊Dic₂₅₇	—	4	2	no	—
16	#16861593	—	4	1	no	—
17	C₂×C₈×C₂₅₇	D₄	1	1	yes	characteristic
18	C₂×C₈×C₂₅₇	D₄	1	1	yes	characteristic
19	C₄²×C₂₅₇	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₂₅₇	—	4	1	no	—
21	Dic₅₁₄	—	4	2	no	—
22	Dic₅₁₄	—	4	2	no	—
23	C₂×Dic₂₅₇	—	4	2	no	—
24	C₂×C₄×C₂₅₇	—	2	1	no	—
25	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₂₅₇	D₈	1	1	yes	characteristic
27	C₈×C₂₅₇	D₈	1	1	yes	characteristic
28	C₈×C₂₅₇	SD₁₆	1	1	yes	characteristic
29	C₈×C₂₅₇	SD₁₆	1	1	yes	characteristic
30	C₂²×C₂₅₇	#100	1	1	yes	characteristic
31	C₄×C₂₅₇	—	2	2	no	—
32	C₄×C₂₅₇	#102	1	1	yes	characteristic
33	C₄×C₂₅₇	#102	1	1	yes	characteristic
34	Dic₂₅₇	—	8	2	no	—
35	C₂×C₂₅₇	—	2	1	no	—
36	C₂×C₂₅₇	#331	1	1	yes	characteristic
37	C₂₅₇	#1358	1	1	yes	characteristic, complemented, Sylow
38	#1358	—	257	1	no	maximal, Sylow
39	#494	—	257	1	no	—
40	#364	—	257	1	no	—
41	#364	D₂₅₇	1	1	yes	characteristic
42	C₂×Q₁₆	—	514	1	no	—
43	C₂×Q₁₆	—	514	1	no	—
44	C₄⋊Q₈	—	257	2	no	—
45	M₃₂	—	2	1	no	—
46	M₃₂	—	514	1	no	—
47	C₄×C₈	C₂×D₂₅₇	1	1	yes	characteristic
48	C₂×Q₈	—	514	2	no	—
49	Q₁₆	—	1028	1	no	—
50	Q₁₆	—	1028	1	no	—
51	Q₁₆	—	1028	1	no	—
52	Q₁₆	—	1028	1	no	—
53	C₂×C₈	C₂²⋊D₂₅₇	1	1	yes	characteristic
54	C₂×C₈	D₁₀₂₈	1	1	yes	characteristic
55	C₄⋊C₄	—	1028	2	no	—
56	C₄²	C₄×D₂₅₇	1	1	yes	characteristic
57	C₁₆	—	4	1	no	—
58	C₁₆	—	1028	1	no	—
59	Q₈	—	1028	2	no	—
60	Q₈	—	1028	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	1028	2	no	—
63	C₂×C₄	#16861595	1	1	yes	characteristic
64	C₈	#16861603	1	1	yes	characteristic
65	C₈	D₂₀₅₆	1	1	yes	characteristic
66	C₈	#16861605	1	1	yes	characteristic
67	C₈	#16861607	1	1	yes	characteristic
68	C₂²	#22551508	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	2056	2	no	—
71	C₄	#22551512	1	1	yes	characteristic
72	C₄	#22551514	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	#30656480	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30656480
commutator	C₄²×C₂₅₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₅₇
Fitting	#364×C₂₅₇	C₂
socle	C₂×C₂₅₇	#331
prime radical	C₂²×C₂₅₇	#100
cosocle	C₄×C₈×C₂₅₇	C₂²
hypercenter	#364	D₂₅₇
hypocenter	C₂₅₇	#1358
LCS	G ▷ C₄²×C₂₅₇ ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
37. C₂₅₇	#1358	semidirect

Maximal subgroups

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

subgroup	index
#30657117	2
#30656558	2
#364×C₂₅₇	2
#1358	257