Subgroups — #30657822

Order 16448.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#14523348	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊#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	C₂²⋊#11759967	—	2	2	no	—
7	C₂³×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	—	2	1	no	—
14	C₂²×D₂₅₇	—	2	1	no	—
15	C₂²×D₂₅₇	—	4	1	no	—
16	C₂²×D₂₅₇	—	4	1	no	—
17	C₂²×D₂₅₇	C₈	1	1	yes	characteristic, complemented
18	C₂²×D₂₅₇	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₂₅₇	C₈	1	1	yes	characteristic, complemented
20	C₂²×C₂₅₇	—	2	1	no	—
21	C₂²×C₂₅₇	—	4	1	no	—
22	C₂²×C₂₅₇	C₂×C₈	1	1	yes	characteristic
23	C₂×D₂₅₇	—	2	1	no	—
24	C₂×D₂₅₇	—	2	1	no	—
25	C₂×D₂₅₇	—	2	1	no	—
26	C₂×D₂₅₇	—	4	2	no	—
27	C₂×D₂₅₇	—	4	1	no	—
28	C₂×D₂₅₇	—	4	1	no	—
29	C₂×D₂₅₇	—	4	1	no	—
30	C₂×D₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×D₂₅₇	M₁₆	1	1	yes	characteristic
32	#11759967	—	2	2	no	—
33	#11759967	—	4	1	no	—
34	C₂×C₂₅₇	—	2	1	no	—
35	C₂×C₂₅₇	—	4	1	no	—
36	C₂×C₂₅₇	C₂²⋊C₈	1	1	yes	characteristic
37	D₂₅₇	—	2	1	no	—
38	D₂₅₇	—	4	1	no	—
39	D₂₅₇	#99	1	1	yes	characteristic
40	D₂₅₇	#100	1	1	yes	characteristic
41	C₂₅₇	#323	1	1	yes	characteristic, complemented, Sylow
42	#323	—	257	1	no	maximal, Sylow
43	C₂×C₂²⋊C₄	—	257	1	no	—
44	C₂²⋊C₈	—	257	2	no	—
45	C₂⁴	—	257	1	no	—
46	C₂²×C₄	—	257	1	no	—
47	C₂²×C₄	—	257	1	no	—
48	C₂×C₈	—	514	2	no	—
49	C₂²⋊C₄	—	514	2	no	—
50	C₂³	—	257	1	no	—
51	C₂³	—	257	1	no	—
52	C₂³	—	514	1	no	—
53	C₂³	—	514	1	no	—
54	C₂³	—	1028	1	no	—
55	C₂³	—	1028	1	no	—
56	C₂³	#14523348	1	1	yes	characteristic, complemented
57	C₂×C₄	—	257	2	no	—
58	C₂×C₄	—	514	1	no	—
59	C₂×C₄	—	1028	1	no	—
60	C₂×C₄	—	1028	1	no	—
61	C₈	—	1028	2	no	—
62	C₂²	—	257	1	no	—
63	C₂²	—	257	1	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	514	1	no	—
66	C₂²	—	514	1	no	—
67	C₂²	—	514	1	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	1028	1	no	—
70	C₂²	—	1028	1	no	—
71	C₂²	—	1028	1	no	—
72	C₂²	—	1028	2	no	—
73	C₂²	C₂×#14523348	1	1	yes	characteristic
74	C₄	—	514	2	no	—
75	C₄	—	1028	1	no	—
76	C₂	—	257	1	no	—
77	C₂	—	257	1	no	—
78	C₂	—	2	1	no	—
79	C₂	—	514	1	no	—
80	C₂	—	4	1	no	—
81	C₂	—	1028	1	no	—
82	C₂	C₂²⋊#14523348	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊#14523348
commutator	C₂²×C₂₅₇	C₂×C₈
Frattini	C₂²	C₂×#14523348
Fitting	C₂³×C₂₅₇	C₈
socle	C₂×C₂₅₇	C₂²⋊C₈
prime radical	C₂³×D₂₅₇	C₄
cosocle	C₂²×#11759967	C₂²
hypercenter	C₂³	#14523348
hypocenter	C₂₅₇	#323
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
1. C₂²⋊#14523348	C₂	semidirect
17. C₂²×D₂₅₇	C₈	semidirect
19. C₂³×C₂₅₇	C₈	semidirect
41. C₂₅₇	#323	semidirect
56. C₂³	#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×C₂²⋊#11759967	2
#323	257