Subgroups — #30657823

Order 16448.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#14523348	C₂	1	2	yes	maximal
2	C₂×C₄⋊#11759967	C₂	1	1	yes	maximal, characteristic
3	C₂×#14523348	—	2	2	no	—
4	C₂²×#11759967	C₄	1	1	yes	characteristic
5	C₂²×#11759967	C₂²	1	1	yes	characteristic
6	C₄⋊#11759967	—	2	2	no	—
7	C₂×C₄×D₂₅₇	C₄	1	1	yes	characteristic
8	#14523348	—	4	2	no	—
9	C₂×#11759967	—	2	1	no	—
10	C₂×#11759967	—	4	1	no	—
11	C₂×#11759967	—	4	1	no	—
12	C₂×#11759967	D₄	1	2	yes	—
13	C₂²×D₂₅₇	C₂×C₄	1	1	yes	characteristic
14	C₂×Dic₂₅₇	C₈	1	1	yes	characteristic, complemented
15	C₄×D₂₅₇	—	2	1	no	—
16	C₄×D₂₅₇	—	2	1	no	—
17	C₂×C₄×C₂₅₇	C₈	1	1	yes	characteristic, complemented
18	C₂²×C₂₅₇	C₂×C₈	1	1	yes	characteristic
19	C₂×D₂₅₇	—	2	1	no	—
20	C₂×D₂₅₇	—	2	1	no	—
21	C₂×D₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×D₂₅₇	M₁₆	1	1	yes	characteristic
23	#11759967	—	2	2	no	—
24	#11759967	—	4	1	no	—
25	C₄×C₂₅₇	—	2	1	no	—
26	Dic₂₅₇	—	2	1	no	—
27	C₂×C₂₅₇	—	2	1	no	—
28	C₂×C₂₅₇	C₂²⋊C₈	1	1	yes	characteristic
29	D₂₅₇	—	2	1	no	—
30	D₂₅₇	#99	1	1	yes	characteristic
31	D₂₅₇	#101	1	1	yes	characteristic
32	C₂₅₇	#324	1	1	yes	characteristic, complemented, Sylow
33	#324	—	257	1	no	maximal, Sylow
34	C₂×C₄⋊C₄	—	257	1	no	—
35	C₂²⋊C₈	—	257	2	no	—
36	C₂²×C₄	—	257	1	no	—
37	C₂²×C₄	—	257	1	no	—
38	C₂²×C₄	—	257	1	no	—
39	C₂×C₈	—	514	2	no	—
40	C₄⋊C₄	—	514	2	no	—
41	C₂³	—	257	1	no	—
42	C₂×C₄	—	257	2	no	—
43	C₂×C₄	—	257	1	no	—
44	C₂×C₄	—	514	1	no	—
45	C₂×C₄	—	514	1	no	—
46	C₂×C₄	—	514	1	no	—
47	C₂×C₄	—	1028	1	no	—
48	C₂×C₄	—	1028	1	no	—
49	C₂×C₄	#14523348	1	1	yes	characteristic, complemented
50	C₈	—	1028	2	no	—
51	C₂²	—	257	1	no	—
52	C₂²	—	257	1	no	—
53	C₂²	—	514	1	no	—
54	C₂²	—	514	1	no	—
55	C₂²	C₂×#14523348	1	1	yes	characteristic
56	C₄	—	2	1	no	—
57	C₄	—	514	2	no	—
58	C₄	—	514	1	no	—
59	C₄	—	1028	1	no	—
60	C₂	—	257	1	no	—
61	C₂	—	257	1	no	—
62	C₂	—	2	1	no	—
63	C₂	—	514	1	no	—
64	C₂	C₂²⋊#14523348	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊#14523348
commutator	C₂²×C₂₅₇	C₂×C₈
Frattini	C₂²	C₂×#14523348
Fitting	C₂×C₄×C₂₅₇	C₈
socle	C₂×C₂₅₇	C₂²⋊C₈
prime radical	C₂²×D₂₅₇	C₂×C₄
cosocle	C₂²×#11759967	C₂²
hypercenter	C₂×C₄	#14523348
hypocenter	C₂₅₇	#324
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
14. C₂×Dic₂₅₇	C₈	semidirect
17. C₂×C₄×C₂₅₇	C₈	semidirect
32. C₂₅₇	#324	semidirect
49. C₂×C₄	#14523348	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊#14523348	2
C₂×C₄⋊#11759967	2
#324	257