Subgroups — #30657173

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	#22551536	C₂	1	2	yes	maximal, complemented
3	#22551533	C₂	1	2	yes	maximal, complemented
4	#22551532	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861618	—	2	1	no	—
7	Dic₁₀₂₈	C₂²	1	1	yes	characteristic, complemented
8	#16861607	—	2	1	no	—
9	D₂₀₅₆	C₂²	1	1	yes	characteristic, complemented
10	#16861593	C₂²	1	2	yes	complemented
11	C₂×D₄×C₂₅₇	—	2	1	no	—
12	D₈×C₂₅₇	—	2	1	no	—
13	D₈×C₂₅₇	C₂²	1	2	yes	—
14	C₂×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
15	Dic₅₁₄	—	2	1	no	—
16	D₁₀₂₈	—	2	1	no	—
17	C₂²⋊D₂₅₇	—	4	1	no	—
18	C₄×D₂₅₇	—	4	1	no	—
19	C₂³×C₂₅₇	—	4	1	no	—
20	D₄×C₂₅₇	—	2	2	no	—
21	D₄×C₂₅₇	—	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	—	4	2	no	—
26	C₂²×C₂₅₇	—	8	2	no	—
27	C₂²×C₂₅₇	D₈	1	1	yes	characteristic
28	C₂×D₂₅₇	—	4	1	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	—	8	2	no	—
34	C₂×C₂₅₇	C₂×D₈	1	1	yes	characteristic
35	D₂₅₇	—	8	1	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	D₂₅₇	1	1	yes	characteristic, complemented
40	SD₃₂	—	257	2	no	—
41	D₁₆	—	257	2	no	—
42	M₃₂	—	257	1	no	—
43	#41	—	514	1	no	—
44	C₂×D₄	—	2	1	no	—
45	Q₁₆	—	257	1	no	—
46	SD₁₆	—	514	1	no	—
47	D₈	—	257	1	no	—
48	D₈	—	2	1	no	—
49	D₈	C₂×D₂₅₇	1	2	yes	—
50	C₂×C₈	C₂×D₂₅₇	1	1	yes	characteristic
51	C₁₆	—	257	2	no	—
52	C₂³	—	4	1	no	—
53	Q₈	—	514	1	no	—
54	D₄	—	2	2	no	—
55	D₄	—	514	1	no	—
56	D₄	—	4	1	no	—
57	D₄	—	1028	1	no	—
58	C₂×C₄	—	1028	1	no	—
59	C₂×C₄	C₂²⋊D₂₅₇	1	1	yes	characteristic
60	C₈	C₂²⋊D₂₅₇	1	1	yes	characteristic
61	C₈	C₂²×D₂₅₇	1	1	yes	characteristic
62	C₂²	—	4	2	no	—
63	C₂²	—	1028	1	no	—
64	C₂²	—	8	2	no	—
65	C₂²	#16861603	1	1	yes	characteristic
66	C₄	—	1028	1	no	—
67	C₄	#16861603	1	1	yes	characteristic
68	C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	8	2	no	—
71	C₂	—	2056	1	no	—
72	C₂	C₂×#16861603	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16861603
commutator, cosocle	C₈×C₂₅₇	C₂³
Frattini	C₈	C₂²×D₂₅₇
Fitting	C₂×D₈×C₂₅₇	C₂
socle	C₂×C₂₅₇	C₂×D₈
hypercenter	C₂×D₈	D₂₅₇
hypocenter	C₂₅₇	#509
LCS	G ▷ C₈×C₂₅₇ ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. #22551614	C₂	semidirect
2. #22551536	C₂	semidirect
3. #22551533	C₂	semidirect
4. #22551532	C₂	semidirect
5. C₂×D₈×C₂₅₇	C₂	semidirect
7. Dic₁₀₂₈	C₂²	semidirect
9. D₂₀₅₆	C₂²	semidirect
10. #16861593	C₂²	semidirect
36. C₂₅₇	#509	semidirect
39. C₂×D₈	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
#22551536	2
#22551533	2
#22551532	2
C₂×D₈×C₂₅₇	2
#509	257