Subgroups — #30657175

Order 16448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551614	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₂₀₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#22551537	C₂	1	2	yes	maximal, complemented
4	D₄₁₁₂	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861618	—	2	1	no	—
7	C₂×D₁₀₂₈	—	2	1	no	—
8	Dic₁₀₂₈	C₂²	1	1	yes	characteristic, complemented
9	#16861607	—	2	1	no	—
10	D₂₀₅₆	—	2	1	no	—
11	D₂₀₅₆	C₂²	1	2	yes	—
12	D₂₀₅₆	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
14	C₁₆×C₂₅₇	C₂²	1	2	yes	complemented
15	C₂²×D₂₅₇	—	4	1	no	—
16	Dic₅₁₄	—	2	1	no	—
17	D₁₀₂₈	—	2	2	no	—
18	D₁₀₂₈	—	2	1	no	—
19	D₁₀₂₈	—	4	1	no	—
20	C₂²⋊D₂₅₇	—	4	1	no	—
21	C₄×D₂₅₇	—	4	1	no	—
22	C₂×C₄×C₂₅₇	D₄	1	1	yes	characteristic
23	C₈×C₂₅₇	D₄	1	1	yes	characteristic
24	C₈×C₂₅₇	C₂³	1	1	yes	characteristic
25	C₂²×C₂₅₇	D₈	1	1	yes	characteristic
26	C₂×D₂₅₇	—	4	1	no	—
27	C₂×D₂₅₇	—	4	2	no	—
28	C₂×D₂₅₇	—	8	2	no	—
29	C₄×C₂₅₇	D₈	1	1	yes	characteristic
30	C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
31	Dic₂₅₇	—	4	1	no	—
32	C₂×C₂₅₇	—	2	1	no	—
33	C₂×C₂₅₇	C₂×D₈	1	1	yes	characteristic
34	D₂₅₇	—	8	1	no	—
35	D₂₅₇	—	8	2	no	—
36	C₂₅₇	#509	1	1	yes	characteristic, complemented, Sylow
37	#509	—	257	1	no	maximal, Sylow
38	#135	—	257	1	no	—
39	C₂×D₈	—	257	1	no	—
40	SD₃₂	—	257	2	no	—
41	D₁₆	—	257	2	no	—
42	M₃₂	D₂₅₇	1	1	yes	characteristic, complemented
43	#41	—	514	1	no	—
44	C₂×D₄	—	514	1	no	—
45	Q₁₆	—	257	1	no	—
46	SD₁₆	—	514	1	no	—
47	D₈	—	257	1	no	—
48	D₈	—	257	2	no	—
49	D₈	—	514	1	no	—
50	C₂×C₈	C₂×D₂₅₇	1	1	yes	characteristic
51	C₁₆	C₂×D₂₅₇	1	2	yes	complemented
52	C₂³	—	1028	1	no	—
53	Q₈	—	514	1	no	—
54	D₄	—	514	2	no	—
55	D₄	—	514	1	no	—
56	D₄	—	1028	1	no	—
57	D₄	—	1028	1	no	—
58	C₂×C₄	—	1028	1	no	—
59	C₂×C₄	D₁₀₂₈	1	1	yes	characteristic
60	C₈	D₁₀₂₈	1	1	yes	characteristic
61	C₈	C₂²×D₂₅₇	1	1	yes	characteristic
62	C₂²	—	1028	1	no	—
63	C₂²	—	1028	2	no	—
64	C₂²	—	2056	2	no	—
65	C₂²	D₂₀₅₆	1	1	yes	characteristic
66	C₄	—	1028	1	no	—
67	C₄	D₂₀₅₆	1	1	yes	characteristic
68	C₄	C₂×D₁₀₂₈	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	2056	2	no	—
71	C₂	—	2056	1	no	—
72	C₂	C₂×D₂₀₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₂₀₅₆
commutator, cosocle	C₈×C₂₅₇	C₂³
Frattini	C₈	C₂²×D₂₅₇
Fitting	M₃₂×C₂₅₇	C₂
socle	C₂×C₂₅₇	C₂×D₈
hypercenter	M₃₂	D₂₅₇
hypocenter	C₂₅₇	#509
LCS	G ▷ C₈×C₂₅₇ ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
1. #22551614	C₂	semidirect
2. C₂×D₂₀₅₆	C₂	semidirect
3. #22551537	C₂	semidirect
4. D₄₁₁₂	C₂	semidirect
5. M₃₂×C₂₅₇	C₂	semidirect
8. Dic₁₀₂₈	C₂²	semidirect
12. D₂₀₅₆	C₂²	semidirect
14. C₁₆×C₂₅₇	C₂²	semidirect
36. C₂₅₇	#509	semidirect
42. M₃₂	D₂₅₇	semidirect
51. C₁₆	C₂×D₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
#22551614	2
C₂×D₂₀₅₆	2
#22551537	2
D₄₁₁₂	2
M₃₂×C₂₅₇	2
#509	257