Subgroups — #30657129

Order 16448.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861608	C₂	1	2	yes	maximal, complemented
2	C₂×#16861605	C₂	1	2	yes	maximal
3	#22551580	C₂	1	1	yes	maximal, characteristic
4	#22551503	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₅₁₄	C₂²	1	1	yes	characteristic
7	C₂×D₁₀₂₈	C₂²	1	1	yes	characteristic
8	#16861608	—	2	4	no	—
9	#16861605	—	2	4	no	—
10	C₂×#14523339	C₂²	1	2	yes	—
11	#16861595	—	2	2	no	—
12	C₂×Q₈×C₂₅₇	C₂²	1	2	yes	—
13	C₄⋊C₄×C₂₅₇	—	2	2	no	—
14	C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
15	C₂²×D₂₅₇	—	2	1	no	—
16	Dic₅₁₄	—	2	2	no	—
17	D₁₀₂₈	—	2	2	no	—
18	C₂×Dic₂₅₇	—	2	1	no	—
19	#14523339	D₄	1	4	yes	—
20	Q₈×C₂₅₇	—	2	2	no	—
21	Q₈×C₂₅₇	—	2	2	no	—
22	C₂×C₄×C₂₅₇	—	2	2	no	—
23	C₂×C₄×C₂₅₇	D₄	1	2	yes	—
24	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
25	C₂²×C₂₅₇	C₂×D₄	1	1	yes	characteristic
26	C₂×D₂₅₇	—	4	1	no	—
27	C₂×D₂₅₇	—	4	2	no	—
28	C₄×C₂₅₇	—	2	2	no	—
29	C₄×C₂₅₇	—	4	2	no	—
30	C₄×C₂₅₇	C₂×D₄	1	2	yes	—
31	Dic₂₅₇	—	4	1	no	—
32	C₂×C₂₅₇	#127	1	1	yes	characteristic
33	C₂×C₂₅₇	#137	1	2	yes	—
34	D₂₅₇	—	8	1	no	—
35	C₂₅₇	#497	1	1	yes	characteristic, complemented, Sylow
36	#497	—	257	1	no	maximal, Sylow
37	C₂×Q₁₆	—	257	2	no	—
38	C₂×SD₁₆	—	257	2	no	—
39	C₄⋊Q₈	D₂₅₇	1	1	yes	characteristic, complemented
40	#124	—	257	1	no	—
41	#97	—	257	1	no	—
42	C₂×Q₈	—	257	1	no	—
43	C₂×Q₈	C₂×D₂₅₇	1	2	yes	—
44	C₂×D₄	—	257	1	no	—
45	Q₁₆	—	514	4	no	—
46	SD₁₆	—	514	4	no	—
47	C₂×C₈	—	257	2	no	—
48	C₄⋊C₄	—	2	2	no	—
49	C₂²⋊C₄	—	514	2	no	—
50	C₄²	C₂×D₂₅₇	1	1	yes	characteristic
51	C₂³	—	514	1	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	2	2	no	—
54	Q₈	—	514	2	no	—
55	D₄	—	514	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	514	1	no	—
58	C₂×C₄	C₂²⋊D₂₅₇	1	2	yes	—
59	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
60	C₈	—	257	4	no	—
61	C₂²	—	1028	1	no	—
62	C₂²	—	1028	2	no	—
63	C₂²	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	4	2	no	—
66	C₄	—	1028	1	no	—
67	C₄	D₄×D₂₅₇	1	2	yes	—
68	C₂	—	2056	1	no	—
69	C₂	#22551588	1	1	yes	characteristic
70	C₂	#22551623	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₅₇
commutator, cosocle	C₂×C₄×C₂₅₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₅₇
Fitting	C₄⋊Q₈×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×D₄
prime radical	C₂×D₁₀₂₈	C₂²
hypercenter	C₄⋊Q₈	D₂₅₇
hypocenter	C₂₅₇	#497
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16861608	C₂	semidirect
4. #22551503	C₂	semidirect
5. C₄⋊Q₈×C₂₅₇	C₂	semidirect
35. C₂₅₇	#497	semidirect
39. C₄⋊Q₈	D₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861608	2
C₂×#16861605	2
#22551580	2
#22551503	2
C₄⋊Q₈×C₂₅₇	2
#497	257