Subgroups — #30657051

Order 16448.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551566	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#22551526	C₂	1	2	yes	maximal, complemented
4	#22551514	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₀₂₈	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₂₅₇	—	2	1	no	—
8	C₂²×Dic₂₅₇	—	2	1	no	—
9	C₄⋊Dic₂₅₇	—	2	1	no	—
10	C₄⋊Dic₂₅₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₅₇	C₂²	1	2	yes	—
12	#16861595	—	2	1	no	—
13	C₂²×C₄×C₂₅₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₂₅₇	C₂²	1	2	yes	—
15	C₂²×D₂₅₇	—	2	1	no	—
16	D₁₀₂₈	—	2	2	no	—
17	C₂²⋊D₂₅₇	—	4	2	no	—
18	C₂×Dic₂₅₇	—	2	1	no	—
19	C₂×Dic₂₅₇	—	2	2	no	—
20	C₂×Dic₂₅₇	—	4	2	no	—
21	C₂³×C₂₅₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₅₇	—	2	2	no	—
23	C₂×C₄×C₂₅₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
25	C₈×C₂₅₇	—	2	2	no	—
26	C₂²×C₂₅₇	—	2	1	no	—
27	C₂²×C₂₅₇	—	2	1	no	—
28	C₂²×C₂₅₇	D₈	1	2	yes	—
29	C₂²×C₂₅₇	C₂×D₄	1	1	yes	characteristic
30	C₂×D₂₅₇	—	4	1	no	—
31	C₂×D₂₅₇	—	4	1	no	—
32	C₂×D₂₅₇	—	4	1	no	—
33	C₄×C₂₅₇	—	2	1	no	—
34	C₄×C₂₅₇	#41	1	2	yes	—
35	Dic₂₅₇	—	4	2	no	—
36	Dic₂₅₇	—	4	1	no	—
37	C₂×C₂₅₇	—	2	2	no	—
38	C₂×C₂₅₇	#123	1	1	yes	characteristic
39	C₂×C₂₅₇	C₂×D₈	1	1	yes	characteristic
40	C₂×C₂₅₇	#137	1	1	yes	characteristic
41	D₂₅₇	—	8	1	no	—
42	C₂₅₇	#480	1	1	yes	characteristic, complemented, Sylow
43	#480	—	257	1	no	maximal, Sylow
44	#121	—	257	1	no	—
45	C₂×C₄⋊C₄	—	257	1	no	—
46	#107	—	257	2	no	—
47	#102	—	257	2	no	—
48	C₂²⋊C₈	D₂₅₇	1	1	yes	characteristic, complemented
49	C₂×D₄	—	257	1	no	—
50	C₂×D₄	—	514	1	no	—
51	C₂²×C₄	—	514	1	no	—
52	C₂²×C₄	C₂×D₂₅₇	1	1	yes	characteristic
53	C₂×C₈	C₂×D₂₅₇	1	2	yes	—
54	C₄⋊C₄	—	257	2	no	—
55	C₄⋊C₄	—	257	1	no	—
56	C₄⋊C₄	—	514	1	no	—
57	C₂²⋊C₄	—	514	1	no	—
58	C₂³	—	514	1	no	—
59	C₂³	D₁₀₂₈	1	1	yes	characteristic
60	D₄	—	514	2	no	—
61	D₄	—	1028	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	514	1	no	—
64	C₂×C₄	—	514	2	no	—
65	C₂×C₄	—	1028	2	no	—
66	C₂×C₄	D₁₀₂₈	1	1	yes	characteristic
67	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
68	C₈	—	2	2	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	1028	1	no	—
72	C₂²	—	1028	1	no	—
73	C₂²	—	1028	1	no	—
74	C₂²	D₂₀₅₆	1	2	yes	—
75	C₂²	C₂×D₁₀₂₈	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	1028	2	no	—
78	C₄	—	1028	1	no	—
79	C₄	#16861617	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	—	2056	1	no	—
82	C₂	#22551576	1	1	yes	characteristic
83	C₂	C₂×D₂₀₅₆	1	1	yes	characteristic
84	C₂	#22551625	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₀₂₈
commutator, cosocle	C₂×C₄×C₂₅₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₅₇
Fitting	C₂²⋊C₈×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×D₄
prime radical	#22551566	C₂
hypercenter	C₂²⋊C₈	D₂₅₇
hypocenter	C₂₅₇	#480
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊Dic₂₅₇	C₂	semidirect
3. #22551526	C₂	semidirect
4. #22551514	C₂	semidirect
5. C₂²⋊C₈×C₂₅₇	C₂	semidirect
42. C₂₅₇	#480	semidirect
48. C₂²⋊C₈	D₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
#22551566	2
C₂×C₄⋊Dic₂₅₇	2
#22551526	2
#22551514	2
C₂²⋊C₈×C₂₅₇	2
#480	257