Group #45663621 — subgroups

Subgroups — #45663621

Order 32896.

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

normal subgroups only

This group has 182 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, complemented
2	#30657101	C₂	1	2	yes	maximal
3	#30656479	C₂	1	1	yes	maximal, characteristic, complemented
4	#30656478	C₂	1	2	yes	maximal, complemented
5	#490×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	#22551513	—	2	2	no	—
11	#22551513	—	2	2	no	—
12	#22551503	C₂²	1	2	yes	—
13	#127×C₂₅₇	C₂²	1	1	yes	characteristic
14	#125×C₂₅₇	C₂²	1	1	yes	characteristic
15	#102×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×D₄×C₂₅₇	—	2	1	no	—
27	C₂×D₄×C₂₅₇	—	2	2	no	—
28	C₂×C₈×C₂₅₇	—	2	1	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	—	4	2	no	—
35	C₂³×C₂₅₇	—	4	1	no	—
36	D₄×C₂₅₇	—	4	2	no	—
37	D₄×C₂₅₇	—	4	4	no	—
38	C₂×C₄×C₂₅₇	—	2	2	no	—
39	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
41	C₈×C₂₅₇	—	4	1	no	—
42	C₂²×C₂₅₇	—	8	2	no	—
43	C₂²×C₂₅₇	—	8	1	no	—
44	C₂²×C₂₅₇	#120	1	1	yes	characteristic
45	C₄×C₂₅₇	—	2	1	no	—
46	C₄×C₂₅₇	—	2	2	no	—
47	C₄×C₂₅₇	—	4	2	no	—
48	Dic₂₅₇	—	4	2	no	—
49	Dic₂₅₇	—	4	2	no	—
50	C₂×C₂₅₇	—	16	1	no	—
51	C₂×C₂₅₇	#454	1	2	yes	—
52	C₂×C₂₅₇	#454	1	1	yes	characteristic
53	C₂₅₇	#1658	1	1	yes	characteristic, complemented, Sylow
54	#1658	—	257	1	no	maximal, Sylow
55	#564	—	257	1	no	—
56	#490	—	257	2	no	—
57	#490	D₂₅₇	1	1	yes	characteristic
58	#330	—	257	1	no	—
59	#330	—	257	2	no	—
60	#127	C₂×D₂₅₇	1	1	yes	characteristic
61	#125	—	257	2	no	—
62	#125	—	514	2	no	—
63	#125	—	514	2	no	—
64	#125	—	514	2	no	—
65	#125	C₂×D₂₅₇	1	1	yes	characteristic
66	#102	—	2	2	no	—
67	#102	—	514	2	no	—
68	#102	—	514	2	no	—
69	#97	—	257	2	no	—
70	#97	C₂×D₂₅₇	1	1	yes	characteristic
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	1	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	514	2	no	—
75	C₄⋊C₄	—	257	2	no	—
76	C₄⋊C₄	—	257	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	514	2	no	—
79	C₄⋊C₄	—	514	2	no	—
80	C₄⋊C₄	—	514	2	no	—
81	C₄⋊C₄	—	514	2	no	—
82	C₄⋊C₄	—	514	2	no	—
83	C₄⋊C₄	C₂²⋊D₂₅₇	1	2	yes	—
84	C₄²	—	514	2	no	—
85	C₄²	C₂²×D₂₅₇	1	1	yes	characteristic
86	C₂³	—	4	1	no	—
87	D₄	—	4	4	no	—
88	D₄	—	4	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	514	2	no	—
91	C₂×C₄	—	514	2	no	—
92	C₂×C₄	D₄×D₂₅₇	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
94	C₈	—	4	1	no	—
95	C₈	—	1028	2	no	—
96	C₂²	—	8	2	no	—
97	C₂²	—	8	1	no	—
98	C₂²	#22551560	1	1	yes	characteristic
99	C₄	—	2	2	no	—
100	C₄	—	2	1	no	—
101	C₄	—	4	2	no	—
102	C₄	—	1028	2	no	—
103	C₄	—	1028	2	no	—
104	C₂	—	16	1	no	—
105	C₂	#30656908	1	2	yes	—
106	C₂	#30656909	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₅₇⋊#564

C₂

yes

maximal, characteristic, complemented

#30657101

C₂

yes

maximal

#30656479

C₂

yes

maximal, characteristic, complemented

#30656478

C₂

yes

maximal, complemented

#490×C₂₅₇

C₂

yes

maximal, characteristic

#22551584

C₂²

yes

—

—

—

—

—

—

C₂²

yes

—

#127×C₂₅₇

C₂²

yes

characteristic

#125×C₂₅₇

C₂²

yes

characteristic

#102×C₂₅₇

—

#97×C₂₅₇

C₂²

yes

characteristic

C₂×#14523339

—

C₄⋊Dic₂₅₇

—

C₄⋊Dic₂₅₇

D₄

yes

—

C₄⋊Dic₂₅₇

D₄

yes

—

—

—

—

—

C₄×Dic₂₅₇

—

C₂×D₄×C₂₅₇

—

C₂×D₄×C₂₅₇

—

C₂×C₈×C₂₅₇

—

C₄⋊C₄×C₂₅₇

—

C₄⋊C₄×C₂₅₇

D₄

yes

—

C₄²×C₂₅₇

C₂³

yes

characteristic

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

#14523339

—

C₂³×C₂₅₇

—

D₄×C₂₅₇

—

D₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₈×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

#120

yes

characteristic

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

Dic₂₅₇

—

Dic₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

#454

yes

—

C₂×C₂₅₇

#454

yes

characteristic

C₂₅₇

#1658

yes

characteristic, complemented, Sylow

#1658

—

257

maximal, Sylow

#564

—

257

—

#490

—

257

—

#490

D₂₅₇

yes

characteristic

#330

—

257

—

#330

—

257

—

#127

C₂×D₂₅₇

yes

characteristic

#125

—

257

—

#125

—

514

—

#125

—

514

—

#125

—

514

—

#125

C₂×D₂₅₇

yes

characteristic

#102

—

#102

—

514

—

#102

—

514

—

#97

—

257

—

#97

C₂×D₂₅₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×C₈

—

C₂×C₈

—

514

—

C₄⋊C₄

—

257

—

C₄⋊C₄

—

257

—

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

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

514

—

C₂×C₄

—

514

—

C₂×C₄

D₄×D₂₅₇

yes

—

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

characteristic

C₈

—

C₈

—

1028

—

C₂²

—

C₂²

—

C₂²

#22551560

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

1028

—

103

C₄

—

1028

—

104

C₂

—

105

C₂

#30656908

yes

—

106

C₂

#30656909

yes

characteristic

	subgroup	G/H
center	C₂²	#22551560
commutator, cosocle	C₄²×C₂₅₇	C₂³
Frattini	C₄²	C₂²×D₂₅₇
Fitting	#490×C₂₅₇	C₂
socle	C₂²×C₂₅₇	#120
prime radical	#127×C₂₅₇	C₂²
hypercenter	#490	D₂₅₇
hypocenter	C₂₅₇	#1658
LCS	G ▷ C₄²×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #490

subgroup

G/H

center

C₂²

#22551560

commutator, cosocle

C₄²×C₂₅₇

C₂³

Frattini

C₄²

C₂²×D₂₅₇

Fitting

#490×C₂₅₇

C₂

socle

C₂²×C₂₅₇

#120

prime radical

#127×C₂₅₇

C₂²

D₂₅₇

C₂₅₇

G ▷ C₄²×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #490

N	H ≅ G/N	type
1. C₂₅₇⋊#564	C₂	semidirect
3. #30656479	C₂	semidirect
4. #30656478	C₂	semidirect
53. C₂₅₇	#1658	semidirect

H ≅ G/N

type

1. C₂₅₇⋊#564

C₂

semidirect

3. #30656479

C₂

semidirect

4. #30656478

C₂

semidirect

53. C₂₅₇

#1658

semidirect

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

subgroup	index
C₂₅₇⋊#564	2
#30657101	2
#30656479	2
#30656478	2
#490×C₂₅₇	2
#1658	257

subgroup

index

C₂₅₇⋊#564

#490×C₂₅₇

257

Subgroups — #45663621

Conjugacy classes of subgroups

Complements

Maximal subgroups