Subgroups — #30657096

Order 16448.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551592	C₂	1	1	yes	maximal, characteristic, complemented
2	#22551516	C₂	1	2	yes	maximal, complemented
3	#22551513	C₂	1	2	yes	maximal
4	#22551503	C₂	1	1	yes	maximal, characteristic, complemented
5	#124×C₂₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₁₄	—	2	1	no	—
7	C₂×#14523339	C₂²	1	2	yes	—
8	C₄⋊Dic₂₅₇	—	2	1	no	—
9	C₄⋊Dic₂₅₇	C₂²	1	2	yes	—
10	C₂×Q₈×C₂₅₇	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₂₅₇	C₂²	1	1	yes	characteristic
12	C₂²⋊C₄×C₂₅₇	—	2	1	no	—
13	C₂²⋊C₄×C₂₅₇	—	2	1	no	—
14	C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
15	Dic₅₁₄	—	4	2	no	—
16	C₂×Dic₂₅₇	—	2	2	no	—
17	#14523339	—	2	2	no	—
18	C₂³×C₂₅₇	—	2	1	no	—
19	Q₈×C₂₅₇	—	2	2	no	—
20	D₄×C₂₅₇	—	2	2	no	—
21	C₂×C₄×C₂₅₇	—	2	1	no	—
22	C₂×C₄×C₂₅₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₅₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
25	C₂²×C₂₅₇	—	4	1	no	—
26	C₂²×C₂₅₇	—	4	1	no	—
27	C₂²×C₂₅₇	—	4	1	no	—
28	C₂²×C₂₅₇	C₂×D₄	1	1	yes	characteristic
29	C₄×C₂₅₇	—	2	1	no	—
30	C₄×C₂₅₇	—	2	1	no	—
31	C₄×C₂₅₇	—	4	1	no	—
32	C₄×C₂₅₇	#41	1	2	yes	—
33	Dic₂₅₇	—	4	2	no	—
34	C₂×C₂₅₇	—	8	1	no	—
35	C₂×C₂₅₇	#124	1	1	yes	characteristic
36	C₂×C₂₅₇	AGL(1,ℤ/8)	1	1	yes	characteristic
37	C₂×C₂₅₇	#137	1	1	yes	characteristic
38	C₂₅₇	#489	1	1	yes	characteristic, complemented, Sylow
39	#489	—	257	1	no	maximal, Sylow
40	C₄⋊Q₈	—	257	1	no	—
41	#124	D₂₅₇	1	1	yes	characteristic
42	#103	—	257	2	no	—
43	#102	—	257	2	no	—
44	#97	—	257	1	no	—
45	C₂×Q₈	—	514	1	no	—
46	C₂×Q₈	C₂×D₂₅₇	1	1	yes	characteristic
47	C₂×D₄	C₂×D₂₅₇	1	1	yes	characteristic
48	C₂×C₈	—	257	2	no	—
49	C₄⋊C₄	—	257	2	no	—
50	C₄⋊C₄	—	514	1	no	—
51	C₂²⋊C₄	—	2	1	no	—
52	C₂²⋊C₄	—	2	1	no	—
53	C₄²	C₂×D₂₅₇	1	1	yes	characteristic
54	C₂³	—	2	1	no	—
55	Q₈	—	2	2	no	—
56	Q₈	—	1028	2	no	—
57	D₄	—	2	2	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	514	2	no	—
60	C₂×C₄	C₂²⋊D₂₅₇	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₂₅₇	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
63	C₈	—	514	2	no	—
64	C₂²	—	4	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	4	1	no	—
71	C₄	—	1028	2	no	—
72	C₄	#16861617	1	2	yes	—
73	C₂	—	8	1	no	—
74	C₂	#22551579	1	1	yes	characteristic
75	C₂	#22551617	1	1	yes	characteristic
76	C₂	#22551622	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₅₇
commutator, cosocle	C₂×C₄×C₂₅₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₅₇
Fitting	#124×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×D₄
prime radical	C₂×D₄×C₂₅₇	C₂²
hypercenter	#124	D₂₅₇
hypocenter	C₂₅₇	#489
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
1. #22551592	C₂	semidirect
2. #22551516	C₂	semidirect
4. #22551503	C₂	semidirect
38. C₂₅₇	#489	semidirect

Maximal subgroups

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

subgroup	index
#22551592	2
#22551516	2
#22551513	2
#22551503	2
#124×C₂₅₇	2
#489	257