Subgroups — #30657122

Order 16448.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861605	C₂	1	2	yes	maximal, complemented
2	C₂×D₂₀₅₆	C₂	1	1	yes	maximal, characteristic
3	#22551581	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₁₆×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₀₂₈	C₂²	1	2	yes	—
7	#16861605	—	2	4	no	—
8	D₂₀₅₆	—	2	2	no	—
9	C₂×#14523339	C₂²	1	1	yes	characteristic
10	#16861595	—	2	4	no	—
11	C₄×Dic₂₅₇	C₂²	1	1	yes	characteristic
12	C₂×Q₈×C₂₅₇	C₂²	1	2	yes	—
13	Q₁₆×C₂₅₇	—	2	2	no	—
14	C₂×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
15	C₂²×D₂₅₇	—	2	2	no	—
16	D₁₀₂₈	—	2	2	no	—
17	D₁₀₂₈	—	2	2	no	—
18	C₂×Dic₂₅₇	D₄	1	2	yes	—
19	#14523339	D₄	1	2	yes	—
20	Q₈×C₂₅₇	—	2	2	no	—
21	Q₈×C₂₅₇	—	2	2	no	—
22	C₂×C₄×C₂₅₇	—	2	2	no	—
23	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
24	C₈×C₂₅₇	D₄	1	2	yes	—
25	C₂²×C₂₅₇	C₂×D₄	1	1	yes	characteristic
26	C₂×D₂₅₇	—	4	2	no	—
27	C₂×D₂₅₇	—	4	4	no	—
28	C₄×C₂₅₇	—	4	2	no	—
29	C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
30	C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
31	Dic₂₅₇	—	2	2	no	—
32	C₂×C₂₅₇	#127	1	1	yes	characteristic
33	C₂×C₂₅₇	#135	1	2	yes	—
34	D₂₅₇	—	8	2	no	—
35	C₂₅₇	#495	1	1	yes	characteristic, complemented, Sylow
36	#495	—	257	1	no	maximal, Sylow
37	C₂×Q₁₆	D₂₅₇	1	1	yes	characteristic, complemented
38	C₂×SD₁₆	—	257	2	no	—
39	C₂×D₈	—	257	1	no	—
40	#124	—	257	2	no	—
41	C₄×C₈	—	257	1	no	—
42	C₂×Q₈	C₂×D₂₅₇	1	2	yes	—
43	C₂×D₄	—	257	2	no	—
44	Q₁₆	—	2	2	no	—
45	SD₁₆	—	514	4	no	—
46	D₈	—	514	2	no	—
47	C₂×C₈	—	257	1	no	—
48	C₂×C₈	C₂×D₂₅₇	1	1	yes	characteristic
49	C₂²⋊C₄	—	514	4	no	—
50	C₄²	—	257	1	no	—
51	C₂³	—	514	2	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	2	2	no	—
54	D₄	—	514	2	no	—
55	D₄	—	514	2	no	—
56	C₂×C₄	—	257	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
59	C₈	—	257	2	no	—
60	C₈	C₂²⋊D₂₅₇	1	2	yes	—
61	C₂²	—	1028	2	no	—
62	C₂²	—	1028	4	no	—
63	C₂²	D₄×D₂₅₇	1	1	yes	characteristic
64	C₄	—	514	2	no	—
65	C₄	—	4	2	no	—
66	C₄	D₄×D₂₅₇	1	1	yes	characteristic
67	C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
68	C₂	—	2056	2	no	—
69	C₂	#22551588	1	1	yes	characteristic
70	C₂	#22551615	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₅₇
commutator, cosocle	C₂×C₄×C₂₅₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₅₇
Fitting	C₂×Q₁₆×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×D₄
prime radical	C₂×D₂₀₅₆	C₂
hypercenter	C₂×Q₁₆	D₂₅₇
hypocenter	C₂₅₇	#495
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16861605	C₂	semidirect
3. #22551581	C₂	semidirect
4. C₈×Dic₂₅₇	C₂	semidirect
5. C₂×Q₁₆×C₂₅₇	C₂	semidirect
35. C₂₅₇	#495	semidirect
37. C₂×Q₁₆	D₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861605	2
C₂×D₂₀₅₆	2
#22551581	2
C₈×Dic₂₅₇	2
C₂×Q₁₆×C₂₅₇	2
#495	257