Subgroups — #45662370

Order 32896.

98 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	#30656490	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₁₆×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₀₂₈	—	2	1	no	—
5	C₂×Dic₁₀₂₈	C₄	1	2	yes	—
6	#22551592	—	2	1	no	—
7	C₄⋊#14523339	—	2	1	no	—
8	C₂×C₁₆×C₂₅₇	—	2	1	no	—
9	C₄×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₅₁₄	—	2	2	no	—
11	Dic₁₀₂₈	—	4	2	no	—
12	Dic₁₀₂₈	—	4	1	no	—
13	Dic₁₀₂₈	—	4	1	no	—
14	C₂×#14523339	—	4	1	no	—
15	C₄⋊Dic₂₅₇	—	4	1	no	—
16	C₄⋊Dic₂₅₇	—	4	1	no	—
17	C₂×C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₂₅₇	D₄	1	1	yes	characteristic
19	C₄²×C₂₅₇	D₄	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	#14523339	—	8	1	no	—
25	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₂₅₇	D₈	1	1	yes	characteristic
27	C₂×C₄×C₂₅₇	SD₁₆	1	1	yes	characteristic
28	C₈×C₂₅₇	—	2	1	no	—
29	C₈×C₂₅₇	D₈	1	1	yes	characteristic
30	C₈×C₂₅₇	SD₁₆	1	1	yes	characteristic
31	C₂²×C₂₅₇	#102	1	1	yes	characteristic
32	C₄×C₂₅₇	—	2	1	no	—
33	C₄×C₂₅₇	#100	1	1	yes	characteristic
34	C₄×C₂₅₇	#102	1	1	yes	characteristic
35	C₄×C₂₅₇	SD₃₂	1	1	yes	characteristic
36	C₄×C₂₅₇	Q₃₂	1	1	yes	characteristic
37	Dic₂₅₇	—	8	2	no	—
38	C₂×C₂₅₇	#331	1	1	yes	characteristic
39	C₂×C₂₅₇	#358	1	1	yes	characteristic
40	C₂×C₂₅₇	#362	1	1	yes	characteristic
41	C₂₅₇	#1362	1	1	yes	characteristic, complemented, Sylow
42	#1362	—	257	1	no	maximal, Sylow
43	#494	—	257	1	no	—
44	C₄⋊C₁₆	D₂₅₇	1	1	yes	characteristic
45	#335	—	257	1	no	—
46	C₂×Q₁₆	—	257	2	no	—
47	C₂×Q₁₆	—	514	1	no	—
48	C₄⋊Q₈	—	514	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₄⋊C₈	—	514	1	no	—
51	C₄×C₈	C₂×D₂₅₇	1	1	yes	characteristic
52	C₂×Q₈	—	514	2	no	—
53	Q₁₆	—	1028	2	no	—
54	Q₁₆	—	1028	1	no	—
55	Q₁₆	—	1028	1	no	—
56	C₂×C₈	—	1028	1	no	—
57	C₂×C₈	C₄×D₂₅₇	1	1	yes	characteristic
58	C₂×C₈	C₂²⋊D₂₅₇	1	1	yes	characteristic
59	C₄⋊C₄	—	1028	1	no	—
60	C₄⋊C₄	—	1028	1	no	—
61	C₄²	D₁₀₂₈	1	1	yes	characteristic
62	C₁₆	—	4	1	no	—
63	Q₈	—	1028	2	no	—
64	Q₈	—	1028	2	no	—
65	C₂×C₄	—	1028	2	no	—
66	C₂×C₄	#16861595	1	1	yes	characteristic
67	C₂×C₄	D₂₀₅₆	1	1	yes	characteristic
68	C₂×C₄	#16861607	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	2056	1	no	—
71	C₈	#16861603	1	1	yes	characteristic
72	C₈	#16861605	1	1	yes	characteristic
73	C₂²	#22551514	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2056	2	no	—
76	C₄	#22551508	1	1	yes	characteristic
77	C₄	#22551512	1	1	yes	characteristic
78	C₄	#22551537	1	1	yes	characteristic
79	C₄	Dic₂₀₅₆	1	1	yes	characteristic
80	C₂	#30656480	1	1	yes	characteristic
81	C₂	#30656542	1	1	yes	characteristic
82	C₂	#30656554	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22551514
commutator	C₂×C₈×C₂₅₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₅₇
Fitting	C₄⋊C₁₆×C₂₅₇	C₂
socle, prime radical	C₂²×C₂₅₇	#102
cosocle	C₄×C₈×C₂₅₇	C₂²
hypercenter	C₄⋊C₁₆	D₂₅₇
hypocenter	C₂₅₇	#1362
LCS	G ▷ C₂×C₈×C₂₅₇ ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
41. C₂₅₇	#1362	semidirect

Maximal subgroups

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

subgroup	index
#30657117	2
#30656490	2
C₄⋊C₁₆×C₂₅₇	2
#1362	257