Subgroups — #30657170

Order 16448.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551615	C₂	1	2	yes	maximal, complemented
2	#22551535	C₂	1	2	yes	maximal, complemented
3	D₄₁₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×D₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₃₂×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861619	—	2	2	no	—
7	#16861605	—	2	2	no	—
8	D₂₀₅₆	C₂²	1	2	yes	—
9	C₈×D₂₅₇	C₂²	1	1	yes	characteristic
10	#16861593	C₂²	1	1	yes	characteristic
11	Q₁₆×C₂₅₇	C₂²	1	2	yes	—
12	C₁₆×C₂₅₇	C₂²	1	1	yes	characteristic
13	D₁₀₂₈	—	2	2	no	—
14	D₁₀₂₈	—	4	2	no	—
15	C₄×D₂₅₇	—	4	2	no	—
16	C₄×D₂₅₇	D₄	1	1	yes	characteristic
17	#14523339	D₄	1	1	yes	characteristic
18	Q₈×C₂₅₇	—	2	2	no	—
19	C₈×C₂₅₇	C₂³	1	1	yes	characteristic
20	C₂×D₂₅₇	—	4	2	no	—
21	C₂×D₂₅₇	D₈	1	1	yes	characteristic
22	C₄×C₂₅₇	—	4	2	no	—
23	C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
24	Dic₂₅₇	D₈	1	1	yes	characteristic
25	C₂×C₂₅₇	C₂×D₈	1	1	yes	characteristic
26	D₂₅₇	—	2	1	no	—
27	D₂₅₇	—	8	2	no	—
28	C₂₅₇	#508	1	1	yes	characteristic, complemented, Sylow
29	#508	—	257	1	no	maximal, Sylow
30	#135	—	257	2	no	—
31	Q₃₂	D₂₅₇	1	1	yes	characteristic, complemented
32	SD₃₂	—	257	2	no	—
33	D₁₆	—	257	1	no	—
34	C₂×C₁₆	—	257	1	no	—
35	#41	—	514	2	no	—
36	Q₁₆	C₂×D₂₅₇	1	2	yes	—
37	SD₁₆	—	514	2	no	—
38	D₈	—	257	2	no	—
39	C₂×C₈	—	257	1	no	—
40	C₁₆	—	257	1	no	—
41	C₁₆	C₂×D₂₅₇	1	1	yes	characteristic
42	Q₈	—	2	2	no	—
43	D₄	—	514	2	no	—
44	D₄	—	1028	2	no	—
45	C₂×C₄	—	257	1	no	—
46	C₂×C₄	—	1028	2	no	—
47	C₈	—	257	1	no	—
48	C₈	C₂²×D₂₅₇	1	1	yes	characteristic
49	C₂²	—	257	1	no	—
50	C₂²	—	1028	2	no	—
51	C₄	—	257	1	no	—
52	C₄	—	4	2	no	—
53	C₄	D₄×D₂₅₇	1	1	yes	characteristic
54	C₂	—	514	1	no	—
55	C₂	—	2056	2	no	—
56	C₂	D₈×D₂₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₂₅₇
commutator, cosocle	C₈×C₂₅₇	C₂³
Frattini	C₈	C₂²×D₂₅₇
Fitting	Q₃₂×C₂₅₇	C₂
socle	C₂×C₂₅₇	C₂×D₈
hypercenter	Q₃₂	D₂₅₇
hypocenter	C₂₅₇	#508
LCS	G ▷ C₈×C₂₅₇ ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
1. #22551615	C₂	semidirect
2. #22551535	C₂	semidirect
3. D₄₁₁₂	C₂	semidirect
4. C₁₆×D₂₅₇	C₂	semidirect
5. Q₃₂×C₂₅₇	C₂	semidirect
28. C₂₅₇	#508	semidirect
31. Q₃₂	D₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
#22551615	2
#22551535	2
D₄₁₁₂	2
C₁₆×D₂₅₇	2
Q₃₂×C₂₅₇	2
#508	257