Subgroups — #30657596

Order 16448.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551659	C₂	1	1	yes	maximal, characteristic
2	#22551659	C₂	1	1	yes	maximal, characteristic
3	#22551591	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523346	—	2	1	no	—
5	C₂×#14523346	—	2	1	no	—
6	C₂×Dic₅₁₄	—	2	1	no	—
7	C₄⋊Dic₂₅₇	C₄	1	1	yes	characteristic
8	#16861597	—	2	1	no	—
9	C₄×Dic₂₅₇	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₂₅₇	C₄	1	1	yes	characteristic
11	#14523346	—	4	1	no	—
12	#14523346	—	4	1	no	—
13	Dic₅₁₄	—	4	1	no	—
14	Dic₅₁₄	—	4	1	no	—
15	C₂×Dic₂₅₇	—	2	1	no	—
16	C₂×Dic₂₅₇	D₄	1	1	yes	characteristic
17	C₂×Dic₂₅₇	D₄	1	1	yes	characteristic
18	C₂×C₄×C₂₅₇	—	2	1	no	—
19	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₂₅₇	—	2	1	no	—
22	C₄×C₂₅₇	—	4	1	no	—
23	Dic₂₅₇	—	4	1	no	—
24	Dic₂₅₇	D₈	1	1	yes	characteristic
25	Dic₂₅₇	SD₁₆	1	1	yes	characteristic
26	Dic₂₅₇	SD₁₆	1	1	yes	characteristic
27	Dic₂₅₇	Q₁₆	1	1	yes	characteristic
28	C₂×C₂₅₇	#101	1	1	yes	characteristic
29	C₂×C₂₅₇	#102	1	1	yes	characteristic
30	C₂×C₂₅₇	#103	1	1	yes	characteristic
31	C₂₅₇	#332	1	1	yes	characteristic, complemented, Sylow
32	#332	—	257	1	no	maximal, Sylow
33	C₄⋊Q₈	—	257	1	no	—
34	C₄⋊C₈	—	257	1	no	—
35	C₄⋊C₈	—	257	1	no	—
36	C₂×Q₈	—	514	1	no	—
37	C₂×C₈	—	514	1	no	—
38	C₂×C₈	—	514	1	no	—
39	C₄⋊C₄	—	257	1	no	—
40	C₄⋊C₄	—	514	1	no	—
41	C₄⋊C₄	#11759967	1	1	yes	characteristic
42	C₄²	—	257	1	no	—
43	Q₈	—	1028	1	no	—
44	Q₈	—	1028	1	no	—
45	C₂×C₄	—	257	1	no	—
46	C₂×C₄	—	257	1	no	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	—	514	1	no	—
49	C₂×C₄	C₂×#11759967	1	1	yes	characteristic
50	C₈	—	1028	1	no	—
51	C₈	—	1028	1	no	—
52	C₂²	C₂²⋊#11759967	1	1	yes	characteristic
53	C₄	—	257	1	no	—
54	C₄	—	257	1	no	—
55	C₄	—	257	1	no	—
56	C₄	—	257	1	no	—
57	C₄	—	2	1	no	—
58	C₄	—	4	1	no	—
59	C₄	—	1028	1	no	—
60	C₂	#22551653	1	1	yes	characteristic
61	C₂	#22551654	1	1	yes	characteristic
62	C₂	#22551655	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11759967
commutator	C₂×C₄×C₂₅₇	C₂×C₄
Frattini	C₂×C₄	C₂×#11759967
Fitting	C₄⋊C₄×C₂₅₇	C₄
socle, prime radical	C₂²×C₂₅₇	C₂²⋊C₄
cosocle	C₄×Dic₂₅₇	C₂²
hypercenter	C₄⋊C₄	#11759967
hypocenter	C₂₅₇	#332
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
31. C₂₅₇	#332	semidirect

Maximal subgroups

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

subgroup	index
#22551659	2
#22551591	2
#332	257