Subgroups — #30657778

Order 16448.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#11759967	C₂	1	2	yes	maximal, complemented
2	#22551660	C₂	1	1	yes	maximal, characteristic, complemented
3	#22551654	C₂	1	2	yes	maximal, complemented
4	#22551646	C₂	1	1	yes	maximal, characteristic, complemented
5	D₈×D₂₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11759967	—	2	2	no	—
7	#16861626	C₂²	1	1	yes	characteristic
8	C₄⋊#11759967	C₂²	1	2	yes	—
9	C₂²⋊#11759967	—	2	2	no	—
10	C₄×#11759967	C₂²	1	1	yes	characteristic
11	D₄×D₂₅₇	C₂²	1	2	yes	—
12	D₂₀₅₆	C₄	1	1	yes	characteristic, complemented
13	#16861603	C₄	1	2	yes	complemented
14	C₈×D₂₅₇	C₂²	1	1	yes	characteristic
15	D₈×C₂₅₇	C₄	1	1	yes	characteristic, complemented
16	C₂×#11759967	—	2	2	no	—
17	C₂×#11759967	—	4	4	no	—
18	C₂×#11759967	D₄	1	2	yes	—
19	#14523346	—	2	1	no	—
20	C₂²×D₂₅₇	—	2	2	no	—
21	D₁₀₂₈	C₂×C₄	1	2	yes	complemented
22	C₂²⋊D₂₅₇	—	2	2	no	—
23	C₄×D₂₅₇	C₂³	1	1	yes	characteristic
24	#14523339	C₂×C₄	1	1	yes	characteristic, complemented
25	D₄×C₂₅₇	C₂×C₄	1	2	yes	complemented
26	C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic, complemented
27	C₂²×C₂₅₇	—	2	2	no	—
28	C₂×D₂₅₇	—	2	2	no	—
29	C₂×D₂₅₇	—	4	2	no	—
30	C₂×D₂₅₇	—	4	2	no	—
31	C₂×D₂₅₇	C₂×D₄	1	1	yes	characteristic
32	#11759967	—	2	2	no	—
33	#11759967	—	4	2	no	—
34	C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
35	Dic₂₅₇	#41	1	1	yes	characteristic
36	C₂×C₂₅₇	—	4	2	no	—
37	C₂×C₂₅₇	C₄×D₄	1	1	yes	characteristic
38	D₂₅₇	—	4	2	no	—
39	D₂₅₇	AGL(1,ℤ/8)	1	2	yes	—
40	C₂₅₇	#442	1	1	yes	characteristic, complemented, Sylow
41	#442	—	257	1	no	maximal, Sylow
42	C₂×D₈	—	257	1	no	—
43	C₄×D₄	—	257	2	no	—
44	#106	—	257	1	no	—
45	#102	—	257	2	no	—
46	#97	—	257	1	no	—
47	C₂×D₄	—	257	2	no	—
48	C₂²×C₄	—	514	2	no	—
49	D₈	—	257	2	no	—
50	D₈	—	257	1	no	—
51	D₈	#11759967	1	1	yes	characteristic, complemented
52	C₂×C₈	—	257	1	no	—
53	C₂×C₈	—	257	1	no	—
54	C₄⋊C₄	—	257	2	no	—
55	C₂²⋊C₄	—	514	2	no	—
56	C₄²	—	257	1	no	—
57	C₂³	—	514	2	no	—
58	D₄	—	257	2	no	—
59	D₄	—	514	2	no	—
60	D₄	C₂×#11759967	1	2	yes	complemented
61	C₂×C₄	—	257	1	no	—
62	C₂×C₄	—	257	2	no	—
63	C₂×C₄	—	514	2	no	—
64	C₂×C₄	—	1028	4	no	—
65	C₈	—	257	1	no	—
66	C₈	—	514	1	no	—
67	C₈	C₂×#11759967	1	1	yes	characteristic, complemented
68	C₂²	—	257	1	no	—
69	C₂²	—	2	2	no	—
70	C₂²	—	514	2	no	—
71	C₂²	—	1028	2	no	—
72	C₂²	—	1028	2	no	—
73	C₄	—	257	1	no	—
74	C₄	—	514	2	no	—
75	C₄	—	1028	2	no	—
76	C₄	C₂²×#11759967	1	1	yes	characteristic
77	C₂	—	257	2	no	—
78	C₂	—	4	2	no	—
79	C₂	—	1028	2	no	—
80	C₂	D₄×#11759967	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#11759967
commutator	C₄×C₂₅₇	C₂²×C₄
Frattini	C₄	C₂²×#11759967
Fitting	D₈×C₂₅₇	C₄
socle	C₂×C₂₅₇	C₄×D₄
prime radical	D₈×D₂₅₇	C₂
cosocle	C₄×D₂₅₇	C₂³
hypercenter	D₈	#11759967
hypocenter	C₂₅₇	#442
LCS	G ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. D₄×#11759967	C₂	semidirect
2. #22551660	C₂	semidirect
3. #22551654	C₂	semidirect
4. #22551646	C₂	semidirect
12. D₂₀₅₆	C₄	semidirect
13. #16861603	C₄	semidirect
15. D₈×C₂₅₇	C₄	semidirect
21. D₁₀₂₈	C₂×C₄	semidirect
24. #14523339	C₂×C₄	semidirect
25. D₄×C₂₅₇	C₂×C₄	semidirect
26. C₈×C₂₅₇	C₂×C₄	semidirect
40. C₂₅₇	#442	semidirect
51. D₈	#11759967	semidirect
60. D₄	C₂×#11759967	semidirect
67. C₈	C₂×#11759967	semidirect

Maximal subgroups

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

subgroup	index
D₄×#11759967	2
#22551660	2
#22551654	2
#22551646	2
D₈×D₂₅₇	2
#442	257