Group #45663646 — subgroups

Subgroups — #45663646

Order 32896.

172 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₅₇⋊#564	C₂	1	1	yes	maximal, characteristic
2	#30657143	C₂	1	2	yes	maximal
3	#30656479	C₂	1	1	yes	maximal, characteristic
4	#30656478	C₂	1	2	yes	maximal
5	#491×C₂₅₇	C₂	1	1	yes	maximal, characteristic
6	#22551584	C₂²	1	2	yes	—
7	#22551583	—	2	2	no	—
8	#22551582	—	2	2	no	—
9	#22551582	—	2	2	no	—
10	#22551525	—	2	2	no	—
11	#22551523	—	2	2	no	—
12	#22551503	C₂²	1	2	yes	—
13	C₄⋊Q₈×C₂₅₇	C₂²	1	1	yes	characteristic
14	#125×C₂₅₇	C₂²	1	1	yes	characteristic
15	#103×C₂₅₇	—	2	2	no	—
16	#97×C₂₅₇	C₂²	1	1	yes	characteristic
17	C₂×#14523339	—	2	2	no	—
18	C₄⋊Dic₂₅₇	—	2	2	no	—
19	C₄⋊Dic₂₅₇	D₄	1	2	yes	—
20	C₄⋊Dic₂₅₇	D₄	1	2	yes	—
21	#16861597	—	2	2	no	—
22	#16861597	—	2	2	no	—
23	#16861597	—	2	2	no	—
24	#16861597	—	2	2	no	—
25	C₄×Dic₂₅₇	—	2	2	no	—
26	C₂×Q₈×C₂₅₇	—	2	1	no	—
27	C₂×C₈×C₂₅₇	—	2	1	no	—
28	C₄⋊C₄×C₂₅₇	—	2	2	no	—
29	C₄⋊C₄×C₂₅₇	—	2	2	no	—
30	C₄⋊C₄×C₂₅₇	D₄	1	2	yes	—
31	C₄²×C₂₅₇	C₂³	1	1	yes	characteristic
32	C₂×Dic₂₅₇	—	2	2	no	—
33	C₂×Dic₂₅₇	—	2	2	no	—
34	#14523339	—	2	4	no	—
35	Q₈×C₂₅₇	—	4	2	no	—
36	C₂×C₄×C₂₅₇	—	2	2	no	—
37	C₂×C₄×C₂₅₇	—	4	1	no	—
38	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
40	C₈×C₂₅₇	—	4	1	no	—
41	C₂²×C₂₅₇	#120	1	1	yes	characteristic
42	C₄×C₂₅₇	—	2	1	no	—
43	C₄×C₂₅₇	—	2	2	no	—
44	C₄×C₂₅₇	—	4	2	no	—
45	C₄×C₂₅₇	—	8	1	no	—
46	Dic₂₅₇	—	4	2	no	—
47	Dic₂₅₇	—	4	2	no	—
48	C₂×C₂₅₇	#454	1	1	yes	characteristic
49	C₂×C₂₅₇	#456	1	2	yes	—
50	C₂₅₇	#1663	1	1	yes	characteristic, complemented, Sylow
51	#1663	—	257	1	no	maximal, Sylow
52	#564	—	257	1	no	—
53	#501	—	257	2	no	—
54	#491	D₂₅₇	1	1	yes	characteristic
55	#330	—	257	1	no	—
56	#330	—	257	2	no	—
57	C₄⋊Q₈	C₂×D₂₅₇	1	1	yes	characteristic
58	#125	—	257	2	no	—
59	#125	—	514	2	no	—
60	#125	—	514	2	no	—
61	#125	—	514	2	no	—
62	#125	C₂×D₂₅₇	1	1	yes	characteristic
63	#107	—	514	2	no	—
64	#106	—	514	2	no	—
65	#103	—	2	2	no	—
66	#97	—	257	2	no	—
67	#97	C₂×D₂₅₇	1	1	yes	characteristic
68	C₂×Q₈	—	2	1	no	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	514	2	no	—
71	C₄⋊C₄	—	257	2	no	—
72	C₄⋊C₄	—	257	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄⋊C₄	—	514	2	no	—
76	C₄⋊C₄	—	514	2	no	—
77	C₄⋊C₄	—	514	2	no	—
78	C₄⋊C₄	—	514	2	no	—
79	C₄⋊C₄	—	514	2	no	—
80	C₄⋊C₄	C₂²⋊D₂₅₇	1	2	yes	—
81	C₄²	—	514	2	no	—
82	C₄²	C₂²×D₂₅₇	1	1	yes	characteristic
83	Q₈	—	4	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	514	2	no	—
86	C₂×C₄	—	514	2	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	D₄×D₂₅₇	1	2	yes	—
89	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
90	C₈	—	514	4	no	—
91	C₈	—	4	1	no	—
92	C₂²	#22551560	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	4	2	no	—
96	C₄	—	1028	2	no	—
97	C₄	—	1028	2	no	—
98	C₄	—	8	1	no	—
99	C₂	#30656909	1	1	yes	characteristic
100	C₂	#30656918	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₅₇⋊#564

C₂

yes

maximal, characteristic

#30657143

C₂

yes

maximal

#30656479

C₂

yes

maximal, characteristic

#30656478

C₂

yes

maximal

#491×C₂₅₇

C₂

yes

maximal, characteristic

#22551584

C₂²

yes

—

—

—

—

—

—

C₂²

yes

—

C₄⋊Q₈×C₂₅₇

C₂²

yes

characteristic

#125×C₂₅₇

C₂²

yes

characteristic

#103×C₂₅₇

—

#97×C₂₅₇

C₂²

yes

characteristic

C₂×#14523339

—

C₄⋊Dic₂₅₇

—

C₄⋊Dic₂₅₇

D₄

yes

—

C₄⋊Dic₂₅₇

D₄

yes

—

—

—

—

—

C₄×Dic₂₅₇

—

C₂×Q₈×C₂₅₇

—

C₂×C₈×C₂₅₇

—

C₄⋊C₄×C₂₅₇

—

C₄⋊C₄×C₂₅₇

—

C₄⋊C₄×C₂₅₇

D₄

yes

—

C₄²×C₂₅₇

C₂³

yes

characteristic

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

#14523339

—

Q₈×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₈×C₂₅₇

—

C₂²×C₂₅₇

#120

yes

characteristic

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

Dic₂₅₇

—

Dic₂₅₇

—

C₂×C₂₅₇

#454

yes

characteristic

C₂×C₂₅₇

#456

yes

—

C₂₅₇

#1663

yes

characteristic, complemented, Sylow

#1663

—

257

maximal, Sylow

#564

—

257

—

#501

—

257

—

#491

D₂₅₇

yes

characteristic

#330

—

257

—

#330

—

257

—

C₄⋊Q₈

C₂×D₂₅₇

yes

characteristic

#125

—

257

—

#125

—

514

—

#125

—

514

—

#125

—

514

—

#125

C₂×D₂₅₇

yes

characteristic

#107

—

514

—

#106

—

514

—

#103

—

#97

—

257

—

#97

C₂×D₂₅₇

yes

characteristic

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

514

—

C₄⋊C₄

—

257

—

C₄⋊C₄

—

257

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

514

—

C₄⋊C₄

—

514

—

C₄⋊C₄

—

514

—

C₄⋊C₄

—

514

—

C₄⋊C₄

—

514

—

C₄⋊C₄

C₂²⋊D₂₅₇

yes

—

C₄²

—

514

—

C₄²

C₂²×D₂₅₇

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

514

—

C₂×C₄

—

514

—

C₂×C₄

—

C₂×C₄

D₄×D₂₅₇

yes

—

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

characteristic

C₈

—

514

—

C₈

—

C₂²

#22551560

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

1028

—

C₄

—

1028

—

C₄

—

C₂

#30656909

yes

characteristic

100

C₂

#30656918

yes

—

	subgroup	G/H
center	C₂²	#22551560
commutator, cosocle	C₄²×C₂₅₇	C₂³
Frattini	C₄²	C₂²×D₂₅₇
Fitting	#491×C₂₅₇	C₂
socle, prime radical	C₂²×C₂₅₇	#120
hypercenter	#491	D₂₅₇
hypocenter	C₂₅₇	#1663
LCS	G ▷ C₄²×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #491

subgroup

G/H

center

C₂²

#22551560

commutator, cosocle

C₄²×C₂₅₇

C₂³

Frattini

C₄²

C₂²×D₂₅₇

Fitting

#491×C₂₅₇

C₂

C₂²×C₂₅₇

D₂₅₇

C₂₅₇

G ▷ C₄²×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #491

N	H ≅ G/N	type
50. C₂₅₇	#1663	semidirect

H ≅ G/N

type

50. C₂₅₇

#1663

semidirect

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

subgroup	index
C₂₅₇⋊#564	2
#30657143	2
#30656479	2
#30656478	2
#491×C₂₅₇	2
#1663	257

subgroup

index

C₂₅₇⋊#564

#491×C₂₅₇

257

Subgroups — #45663646

Conjugacy classes of subgroups

Complements

Maximal subgroups