Subgroups — #30656848

Order 16448.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₀₅₆	C₂	1	1	yes	maximal, characteristic
2	#22551554	C₂	1	2	yes	maximal, complemented
3	#22551512	C₂	1	2	yes	maximal, complemented
4	#22551502	C₂	1	1	yes	maximal, characteristic, complemented
5	#106×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₀₂₈	C₂²	1	2	yes	—
7	C₂×C₄×D₂₅₇	—	2	2	no	—
8	D₂₀₅₆	C₄	1	4	yes	complemented
9	C₂×#14523339	C₂²	1	1	yes	characteristic
10	#16861595	—	2	2	no	—
11	C₄×Dic₂₅₇	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₂₅₇	C₂²	1	2	yes	—
14	C₂²×D₂₅₇	—	2	2	no	—
15	D₁₀₂₈	—	2	2	no	—
16	D₁₀₂₈	C₂×C₄	1	4	yes	complemented
17	C₂×Dic₂₅₇	D₄	1	2	yes	—
18	C₄×D₂₅₇	—	4	4	no	—
19	#14523339	—	2	1	no	—
20	C₂×C₄×C₂₅₇	—	2	2	no	—
21	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
22	C₈×C₂₅₇	C₂×C₄	1	2	yes	complemented
23	C₂²×C₂₅₇	C₂×D₄	1	1	yes	characteristic
24	C₂×D₂₅₇	—	2	4	no	—
25	C₂×D₂₅₇	—	4	2	no	—
26	C₂×D₂₅₇	—	4	2	no	—
27	C₄×C₂₅₇	—	4	2	no	—
28	C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₂₅₇	#41	1	1	yes	characteristic
30	Dic₂₅₇	—	2	2	no	—
31	C₂×C₂₅₇	C₄×D₄	1	1	yes	characteristic
32	C₂×C₂₅₇	AGL(1,ℤ/8)	1	2	yes	—
33	D₂₅₇	—	4	4	no	—
34	C₂₅₇	#442	1	1	yes	characteristic, complemented, Sylow
35	#442	—	257	1	no	maximal, Sylow
36	C₂×D₈	—	257	1	no	—
37	C₄×D₄	—	257	2	no	—
38	#106	D₂₅₇	1	1	yes	characteristic, complemented
39	#102	—	257	2	no	—
40	#97	—	257	1	no	—
41	C₂×D₄	—	257	2	no	—
42	C₂²×C₄	—	514	2	no	—
43	D₈	—	257	4	no	—
44	C₂×C₈	—	257	1	no	—
45	C₂×C₈	C₂×D₂₅₇	1	1	yes	characteristic
46	C₄⋊C₄	C₂×D₂₅₇	1	2	yes	—
47	C₂²⋊C₄	—	514	2	no	—
48	C₄²	—	257	1	no	—
49	C₂³	—	514	2	no	—
50	D₄	—	257	4	no	—
51	D₄	—	514	2	no	—
52	C₂×C₄	—	257	2	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	1028	4	no	—
55	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
56	C₈	—	514	1	no	—
57	C₈	C₄×D₂₅₇	1	2	yes	complemented
58	C₂²	—	514	4	no	—
59	C₂²	—	1028	2	no	—
60	C₂²	—	1028	2	no	—
61	C₂²	D₄×D₂₅₇	1	1	yes	characteristic
62	C₄	—	514	2	no	—
63	C₄	—	4	2	no	—
64	C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
65	C₄	#16861619	1	1	yes	characteristic
66	C₂	—	1028	4	no	—
67	C₂	#22551554	1	1	yes	characteristic
68	C₂	#22551619	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₅₇
commutator	C₄×C₂₅₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₅₇
Fitting	#106×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×D₄
prime radical	C₂×D₂₀₅₆	C₂
cosocle	C₂×C₄×C₂₅₇	C₂³
hypercenter	#106	D₂₅₇
hypocenter	C₂₅₇	#442
LCS	G ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
2. #22551554	C₂	semidirect
3. #22551512	C₂	semidirect
4. #22551502	C₂	semidirect
5. #106×C₂₅₇	C₂	semidirect
8. D₂₀₅₆	C₄	semidirect
16. D₁₀₂₈	C₂×C₄	semidirect
22. C₈×C₂₅₇	C₂×C₄	semidirect
34. C₂₅₇	#442	semidirect
38. #106	D₂₅₇	semidirect
57. C₈	C₄×D₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₀₅₆	2
#22551554	2
#22551512	2
#22551502	2
#106×C₂₅₇	2
#442	257