Subgroups — #30657174

Order 16448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551615	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×D₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#22551535	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄₁₁₂	C₂	1	1	yes	maximal, characteristic, complemented
5	#22551533	C₂	1	1	yes	maximal, characteristic, complemented
6	#22551531	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₃₂×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#16861619	—	2	1	no	—
9	D₄×D₂₅₇	—	2	1	no	—
10	#16861605	—	2	1	no	—
11	D₂₀₅₆	C₂²	1	1	yes	characteristic
12	D₂₀₅₆	C₂²	1	1	yes	characteristic, complemented
13	#16861603	—	2	1	no	—
14	C₈×D₂₅₇	C₂²	1	1	yes	characteristic
15	#16861593	C₂²	1	1	yes	characteristic, complemented
16	Q₁₆×C₂₅₇	C₂²	1	1	yes	characteristic, complemented
17	D₈×C₂₅₇	C₂²	1	1	yes	characteristic
18	C₁₆×C₂₅₇	C₂²	1	1	yes	characteristic, complemented
19	C₂²×D₂₅₇	—	4	1	no	—
20	D₁₀₂₈	—	2	1	no	—
21	D₁₀₂₈	—	2	1	no	—
22	D₁₀₂₈	—	4	1	no	—
23	C₂²⋊D₂₅₇	—	4	1	no	—
24	C₄×D₂₅₇	—	4	1	no	—
25	C₄×D₂₅₇	D₄	1	1	yes	characteristic
26	#14523339	D₄	1	1	yes	characteristic
27	Q₈×C₂₅₇	—	2	1	no	—
28	D₄×C₂₅₇	—	2	1	no	—
29	C₈×C₂₅₇	C₂³	1	1	yes	characteristic
30	C₂²×C₂₅₇	—	4	1	no	—
31	C₂×D₂₅₇	—	4	1	no	—
32	C₂×D₂₅₇	—	4	1	no	—
33	C₂×D₂₅₇	—	8	2	no	—
34	C₂×D₂₅₇	D₈	1	1	yes	characteristic
35	C₄×C₂₅₇	—	4	1	no	—
36	C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
37	Dic₂₅₇	D₈	1	1	yes	characteristic
38	C₂×C₂₅₇	—	8	1	no	—
39	C₂×C₂₅₇	C₂×D₈	1	1	yes	characteristic
40	D₂₅₇	—	2	1	no	—
41	D₂₅₇	—	8	1	no	—
42	D₂₅₇	—	8	1	no	—
43	C₂₅₇	#509	1	1	yes	characteristic, complemented, Sylow
44	#509	—	257	1	no	maximal, Sylow
45	#135	—	257	1	no	—
46	C₂×D₈	—	257	1	no	—
47	SD₃₂	—	257	1	no	—
48	SD₃₂	D₂₅₇	1	1	yes	characteristic, complemented
49	D₁₆	—	257	1	no	—
50	D₁₆	—	257	1	no	—
51	M₃₂	—	257	1	no	—
52	#41	—	514	1	no	—
53	C₂×D₄	—	514	1	no	—
54	Q₁₆	C₂×D₂₅₇	1	1	yes	characteristic, complemented
55	SD₁₆	—	514	1	no	—
56	D₈	—	257	1	no	—
57	D₈	—	257	1	no	—
58	D₈	—	514	1	no	—
59	D₈	C₂×D₂₅₇	1	1	yes	characteristic
60	C₂×C₈	—	257	1	no	—
61	C₁₆	—	257	1	no	—
62	C₁₆	C₂×D₂₅₇	1	1	yes	characteristic, complemented
63	C₂³	—	1028	1	no	—
64	Q₈	—	2	1	no	—
65	D₄	—	2	1	no	—
66	D₄	—	514	1	no	—
67	D₄	—	514	1	no	—
68	D₄	—	1028	1	no	—
69	D₄	—	1028	1	no	—
70	C₂×C₄	—	257	1	no	—
71	C₂×C₄	—	1028	1	no	—
72	C₈	—	257	1	no	—
73	C₈	C₂²×D₂₅₇	1	1	yes	characteristic
74	C₂²	—	257	1	no	—
75	C₂²	—	4	1	no	—
76	C₂²	—	1028	1	no	—
77	C₂²	—	1028	1	no	—
78	C₂²	—	2056	2	no	—
79	C₄	—	257	1	no	—
80	C₄	—	4	1	no	—
81	C₄	D₄×D₂₅₇	1	1	yes	characteristic
82	C₂	—	514	1	no	—
83	C₂	—	8	1	no	—
84	C₂	—	2056	1	no	—
85	C₂	—	2056	1	no	—
86	C₂	D₈×D₂₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₂₅₇
commutator, cosocle	C₈×C₂₅₇	C₂³
Frattini	C₈	C₂²×D₂₅₇
Fitting	SD₃₂×C₂₅₇	C₂
socle	C₂×C₂₅₇	C₂×D₈
hypercenter	SD₃₂	D₂₅₇
hypocenter	C₂₅₇	#509
LCS	G ▷ C₈×C₂₅₇ ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

Complements

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

N	H ≅ G/N	type
1. #22551615	C₂	semidirect
2. D₈×D₂₅₇	C₂	semidirect
3. #22551535	C₂	semidirect
4. D₄₁₁₂	C₂	semidirect
5. #22551533	C₂	semidirect
6. #22551531	C₂	semidirect
7. SD₃₂×C₂₅₇	C₂	semidirect
12. D₂₀₅₆	C₂²	semidirect
15. #16861593	C₂²	semidirect
16. Q₁₆×C₂₅₇	C₂²	semidirect
18. C₁₆×C₂₅₇	C₂²	semidirect
43. C₂₅₇	#509	semidirect
48. SD₃₂	D₂₅₇	semidirect
54. Q₁₆	C₂×D₂₅₇	semidirect
62. 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
#22551615	2
D₈×D₂₅₇	2
#22551535	2
D₄₁₁₂	2
#22551533	2
#22551531	2
SD₃₂×C₂₅₇	2
#509	257