Subgroups — #45666178

Order 32896.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₀₅₆	C₂	1	1	yes	maximal, characteristic
2	#30656831	C₂	1	2	yes	maximal
3	#30656540	C₂	1	2	yes	maximal
4	C₁₆×Dic₂₅₇	C₂	1	1	yes	maximal, characteristic
5	#366×C₂₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₂₈	C₂²	1	2	yes	—
7	#22551557	—	2	2	no	—
8	Dic₂₀₅₆	C₄	1	4	yes	complemented
9	C₂×#16861593	C₂²	1	1	yes	characteristic
10	#22551515	—	2	2	no	—
11	C₈×Dic₂₅₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₂₅₇	C₂²	1	1	yes	characteristic
13	#107×C₂₅₇	C₂²	1	2	yes	—
14	C₂×Dic₅₁₄	—	2	2	no	—
15	Dic₁₀₂₈	—	2	2	no	—
16	Dic₁₀₂₈	C₂×C₄	1	4	yes	—
17	C₂×#14523339	D₄	1	1	yes	characteristic
18	#16861597	—	4	2	no	—
19	C₄×Dic₂₅₇	—	4	2	no	—
20	C₄×Dic₂₅₇	D₄	1	1	yes	characteristic
21	#16861593	—	2	1	no	—
22	C₂×C₈×C₂₅₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₂₅₇	—	2	2	no	—
24	C₁₆×C₂₅₇	C₂×C₄	1	2	yes	—
25	Dic₅₁₄	—	2	4	no	—
26	Dic₅₁₄	—	4	2	no	—
27	C₂×Dic₂₅₇	—	4	2	no	—
28	C₂×Dic₂₅₇	D₈	1	1	yes	characteristic
29	C₂×Dic₂₅₇	D₈	1	1	yes	characteristic
30	#14523339	—	2	1	no	—
31	C₂×C₄×C₂₅₇	—	4	2	no	—
32	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
33	C₈×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₂₅₇	#41	1	1	yes	characteristic
35	C₂²×C₂₅₇	C₂×D₈	1	1	yes	characteristic
36	C₄×C₂₅₇	—	8	2	no	—
37	C₄×C₂₅₇	C₄×D₄	1	1	yes	characteristic
38	C₄×C₂₅₇	#135	1	1	yes	characteristic
39	Dic₂₅₇	—	2	1	no	—
40	Dic₂₅₇	—	4	4	no	—
41	Dic₂₅₇	Q₃₂	1	2	yes	complemented
42	C₂×C₂₅₇	C₄×D₈	1	1	yes	characteristic
43	C₂×C₂₅₇	C₂×Q₃₂	1	1	yes	characteristic
44	C₂×C₂₅₇	#508	1	1	yes	characteristic
45	C₂₅₇	C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
46	C₄×Q₃₂	—	257	1	no	maximal, Sylow
47	C₂×Q₃₂	—	257	1	no	—
48	C₄×Q₁₆	—	257	2	no	—
49	#366	D₂₅₇	1	1	yes	characteristic
50	#358	—	257	2	no	—
51	C₄×C₁₆	—	257	1	no	—
52	C₂×Q₁₆	—	257	2	no	—
53	C₄×Q₈	—	514	2	no	—
54	Q₃₂	—	257	4	no	—
55	C₂×C₁₆	—	257	1	no	—
56	C₂×C₁₆	C₂×D₂₅₇	1	1	yes	characteristic
57	#107	C₂×D₂₅₇	1	2	yes	—
58	#103	—	514	2	no	—
59	C₄×C₈	—	257	1	no	—
60	C₂×Q₈	—	514	2	no	—
61	Q₁₆	—	257	4	no	—
62	Q₁₆	—	514	2	no	—
63	C₂×C₈	—	257	1	no	—
64	C₂×C₈	C₂²×D₂₅₇	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	1028	2	no	—
67	C₄²	—	257	1	no	—
68	C₄²	—	1028	2	no	—
69	C₁₆	—	514	1	no	—
70	C₁₆	C₄×D₂₅₇	1	2	yes	—
71	Q₈	—	514	4	no	—
72	Q₈	—	1028	2	no	—
73	C₂×C₄	—	257	1	no	—
74	C₂×C₄	—	257	1	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	—	1028	2	no	—
77	C₂×C₄	D₄×D₂₅₇	1	1	yes	characteristic
78	C₈	—	514	1	no	—
79	C₈	C₂×C₄×D₂₅₇	1	1	yes	characteristic
80	C₈	#16861619	1	1	yes	characteristic
81	C₂²	D₈×D₂₅₇	1	1	yes	characteristic
82	C₄	—	257	2	no	—
83	C₄	—	514	1	no	—
84	C₄	—	1028	4	no	—
85	C₄	—	8	2	no	—
86	C₄	#22551554	1	1	yes	characteristic
87	C₄	#22551615	1	1	yes	characteristic
88	C₂	#30656819	1	1	yes	characteristic
89	C₂	Q₃₂×D₂₅₇	1	1	yes	characteristic
90	C₂	#30657166	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₂₅₇
commutator	C₈×C₂₅₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₅₇
Fitting	#366×C₂₅₇	C₂
socle, prime radical	C₂²×C₂₅₇	C₂×D₈
cosocle	C₂×C₈×C₂₅₇	C₂³
hypercenter	#366	D₂₅₇
hypocenter	C₂₅₇	C₄×Q₃₂
LCS	G ▷ C₈×C₂₅₇ ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
8. Dic₂₀₅₆	C₄	semidirect
41. Dic₂₅₇	Q₃₂	semidirect
45. C₂₅₇	C₄×Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₀₅₆	2
#30656831	2
#30656540	2
C₁₆×Dic₂₅₇	2
#366×C₂₅₇	2
C₄×Q₃₂	257