Group #45664361 — subgroups

Subgroups — #45664361

Order 32896.

272 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂²×#16861601	C₂	1	1	yes	maximal, characteristic
2	C₂×#22551527	C₂	1	2	yes	maximal, complemented
3	#30656700	C₂	1	2	yes	maximal
4	#30656502	C₂	1	1	yes	maximal, characteristic, complemented
5	#341×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₂₅₇	C₂²	1	1	yes	characteristic
7	C₂×#16861602	C₂²	1	2	yes	—
8	C₂×#16861601	—	2	4	no	—
9	C₂×#16861601	C₄	1	4	yes	—
10	C₂²×#14523339	C₂²	1	1	yes	characteristic
11	#22551527	—	2	4	no	—
12	#22551504	—	2	4	no	—
13	C₂×M₁₆×C₂₅₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
15	C₂³×D₂₅₇	Q₈	1	1	yes	characteristic
16	C₂²×Dic₂₅₇	Q₈	1	1	yes	characteristic
17	C₂×C₄×D₂₅₇	—	2	2	no	—
18	C₂×C₄×D₂₅₇	—	2	2	no	—
19	C₂×C₄×D₂₅₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×D₂₅₇	D₄	1	2	yes	—
21	#16861602	—	2	2	no	—
22	#16861602	—	2	2	no	—
23	#16861601	—	2	8	no	—
24	C₂×#14523339	—	2	2	no	—
25	C₂×#14523339	C₂×C₄	1	2	yes	—
26	C₂×#14523339	D₄	1	4	yes	—
27	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
28	M₁₆×C₂₅₇	—	2	2	no	—
29	M₁₆×C₂₅₇	—	2	2	no	—
30	C₂×C₈×C₂₅₇	—	2	2	no	—
31	C₂×C₈×C₂₅₇	—	2	2	no	—
32	C₂×C₈×C₂₅₇	C₂×C₄	1	2	yes	—
33	C₂²×D₂₅₇	—	2	2	no	—
34	C₂²×D₂₅₇	—	4	2	no	—
35	C₂²×D₂₅₇	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₂₅₇	—	2	2	no	—
37	C₂×Dic₂₅₇	C₄⋊C₄	1	2	yes	—
38	C₄×D₂₅₇	—	2	2	no	—
39	C₄×D₂₅₇	—	2	2	no	—
40	C₄×D₂₅₇	—	2	4	no	—
41	#14523339	—	4	2	no	—
42	#14523339	C₂²⋊C₄	1	4	yes	—
43	C₂³×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₅₇	#41	1	2	yes	—
48	C₈×C₂₅₇	—	2	2	no	—
49	C₈×C₂₅₇	—	4	2	no	—
50	C₂²×C₂₅₇	—	2	2	no	—
51	C₂²×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₂₅₇	C₂²⋊Q₈	1	2	yes	—
53	C₂×D₂₅₇	—	2	2	no	—
54	C₂×D₂₅₇	—	2	4	no	—
55	C₂×D₂₅₇	—	4	4	no	—
56	C₄×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₂₅₇	#117	1	1	yes	characteristic
58	C₄×C₂₅₇	#121	1	2	yes	—
59	Dic₂₅₇	—	2	2	no	—
60	C₂×C₂₅₇	—	2	2	no	—
61	C₂×C₂₅₇	#380	1	1	yes	characteristic
62	C₂×C₂₅₇	#430	1	2	yes	—
63	D₂₅₇	—	4	2	no	—
64	C₂₅₇	#1811	1	1	yes	characteristic, complemented, Sylow
65	#1811	—	257	1	no	maximal, Sylow
66	C₂²×M₁₆	—	257	1	no	—
67	C₂×#108	—	257	2	no	—
68	#407	—	257	2	no	—
69	#341	—	257	1	no	—
70	#341	D₂₅₇	1	1	yes	characteristic, complemented
71	C₂³×C₄	—	257	1	no	—
72	C₂×M₁₆	—	257	4	no	—
73	C₂×M₁₆	—	257	2	no	—
74	C₂×M₁₆	—	514	4	no	—
75	C₂×M₁₆	C₂×D₂₅₇	1	2	yes	—
76	C₂²×C₈	—	257	1	no	—
77	C₂²×C₈	C₂×D₂₅₇	1	1	yes	characteristic
78	#108	—	514	4	no	—
79	C₂²⋊C₈	—	514	4	no	—
80	C₂⁴	—	257	1	no	—
81	C₂²×C₄	—	257	1	no	—
82	C₂²×C₄	—	257	2	no	—
83	C₂²×C₄	—	257	2	no	—
84	C₂²×C₄	—	514	2	no	—
85	C₂²×C₄	—	514	2	no	—
86	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	514	8	no	—
90	M₁₆	—	514	2	no	—
91	M₁₆	—	514	2	no	—
92	C₂×C₈	—	257	2	no	—
93	C₂×C₈	—	257	4	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	514	2	no	—
97	C₂×C₈	C₄×D₂₅₇	1	2	yes	—
98	C₂³	—	257	2	no	—
99	C₂³	—	514	2	no	—
100	C₂³	—	1028	2	no	—
101	C₂³	Q₈×D₂₅₇	1	1	yes	characteristic
102	C₂×C₄	—	257	2	no	—
103	C₂×C₄	—	514	2	no	—
104	C₂×C₄	—	514	4	no	—
105	C₂×C₄	—	514	2	no	—
106	C₂×C₄	—	514	2	no	—
107	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
108	C₂×C₄	D₄×D₂₅₇	1	2	yes	—
109	C₂×C₄	D₄×D₂₅₇	1	1	yes	characteristic
110	C₂×C₄	#16861618	1	2	yes	—
111	C₈	—	257	4	no	—
112	C₈	—	2	2	no	—
113	C₈	—	4	2	no	—
114	C₈	—	1028	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	514	4	no	—
117	C₂²	—	514	2	no	—
118	C₂²	—	1028	4	no	—
119	C₂²	C₄⋊C₄×D₂₅₇	1	1	yes	characteristic
120	C₂²	#22551570	1	2	yes	—
121	C₄	—	514	2	no	—
122	C₄	C₂²⋊C₄×D₂₅₇	1	1	yes	characteristic
123	C₄	#22551549	1	1	yes	characteristic
124	C₄	#22551565	1	2	yes	—
125	C₂	—	2	2	no	—
126	C₂	—	1028	2	no	—
127	C₂	#30656596	1	1	yes	characteristic
128	C₂	#30656784	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#16861601

C₂

yes

maximal, characteristic

C₂×#22551527

C₂

yes

maximal, complemented

#30656700

C₂

yes

maximal

#30656502

C₂

yes

maximal, characteristic, complemented

#341×C₂₅₇

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₂₅₇

C₂²

yes

characteristic

C₂×#16861602

C₂²

yes

—

C₂×#16861601

—

C₂×#16861601

C₄

yes

—

C₂²×#14523339

C₂²

yes

characteristic

#22551527

—

#22551504

—

C₂×M₁₆×C₂₅₇

C₂²

yes

—

C₂²×C₈×C₂₅₇

C₂²

yes

characteristic

C₂³×D₂₅₇

Q₈

yes

characteristic

C₂²×Dic₂₅₇

Q₈

yes

characteristic

C₂×C₄×D₂₅₇

—

C₂×C₄×D₂₅₇

—

C₂×C₄×D₂₅₇

C₂×C₄

yes

—

C₂×C₄×D₂₅₇

D₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂×#14523339

D₄

yes

—

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

M₁₆×C₂₅₇

—

M₁₆×C₂₅₇

—

C₂×C₈×C₂₅₇

—

C₂×C₈×C₂₅₇

—

C₂×C₈×C₂₅₇

C₂×C₄

yes

—

C₂²×D₂₅₇

—

C₂²×D₂₅₇

—

C₂²×D₂₅₇

C₄⋊C₄

yes

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

C₄⋊C₄

yes

—

C₄×D₂₅₇

—

C₄×D₂₅₇

—

C₄×D₂₅₇

—

#14523339

—

#14523339

C₂²⋊C₄

yes

—

C₂³×C₂₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×D₄

yes

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

#41

yes

—

C₈×C₂₅₇

—

C₈×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₅₇

C₂²⋊Q₈

yes

—

C₂×D₂₅₇

—

C₂×D₂₅₇

—

C₂×D₂₅₇

—

C₄×C₂₅₇

C₂×C₂²⋊C₄

yes

C₄×C₂₅₇

yes

C₄×C₂₅₇

yes

—

Dic₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

#380

yes

characteristic

C₂×C₂₅₇

#430

yes

—

D₂₅₇

—

C₂₅₇

#1811

yes

characteristic, complemented, Sylow

#1811

—

257

maximal, Sylow

C₂²×M₁₆

—

257

—

C₂×#108

—

257

—

#407

—

257

—

#341

—

257

—

#341

D₂₅₇

yes

characteristic, complemented

C₂³×C₄

—

257

—

C₂×M₁₆

—

257

—

C₂×M₁₆

—

257

—

C₂×M₁₆

—

514

—

C₂×M₁₆

C₂×D₂₅₇

yes

—

C₂²×C₈

—

257

—

C₂²×C₈

C₂×D₂₅₇

yes

characteristic

#108

—

514

—

C₂²⋊C₈

—

514

—

C₂⁴

—

257

—

C₂²×C₄

—

257

—

C₂²×C₄

—

257

—

C₂²×C₄

—

257

—

C₂²×C₄

—

514

—

C₂²×C₄

—

514

—

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

514

—

M₁₆

—

514

—

M₁₆

—

514

—

C₂×C₈

—

257

—

C₂×C₈

—

257

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

514

—

C₂×C₈

C₄×D₂₅₇

yes

—

C₂³

—

257

—

C₂³

—

514

—

100

C₂³

—

1028

—

101

C₂³

Q₈×D₂₅₇

yes

characteristic

102

C₂×C₄

—

257

—

103

C₂×C₄

—

514

—

104

C₂×C₄

—

514

—

105

C₂×C₄

—

514

—

106

C₂×C₄

—

514

—

107

C₂×C₄

C₂×C₄×D₂₅₇

yes

characteristic

108

C₂×C₄

D₄×D₂₅₇

yes

—

109

C₂×C₄

D₄×D₂₅₇

yes

characteristic

110

C₂×C₄

#16861618

yes

—

111

C₈

—

257

—

112

C₈

—

113

C₈

—

114

C₈

—

1028

—

115

C₂²

—

116

C₂²

—

514

—

117

C₂²

—

514

—

118

C₂²

—

1028

—

119

C₂²

C₄⋊C₄×D₂₅₇

yes

characteristic

120

C₂²

#22551570

yes

—

121

C₄

—

514

—

122

C₄

C₂²⋊C₄×D₂₅₇

yes

characteristic

123

C₄

#22551549

yes

characteristic

124

C₄

#22551565

yes

—

125

C₂

—

126

C₂

—

1028

—

127

C₂

#30656596

yes

characteristic

128

C₂

#30656784

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₅₇
commutator	C₂×C₄×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	#341×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×C₄⋊C₄
prime radical	C₂³×D₂₅₇	Q₈
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	#341	D₂₅₇
hypocenter	C₂₅₇	#1811
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₂₅₇

commutator

C₂×C₄×C₂₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

#341×C₂₅₇

C₂

socle

C₂²×C₂₅₇

C₂×C₄⋊C₄

prime radical

C₂³×D₂₅₇

Q₈

cosocle

C₂²×C₄×C₂₅₇

C₂³

D₂₅₇

C₂₅₇

G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. C₂×#22551527	C₂	semidirect
4. #30656502	C₂	semidirect
5. #341×C₂₅₇	C₂	semidirect
64. C₂₅₇	#1811	semidirect
70. #341	D₂₅₇	semidirect

H ≅ G/N

type

2. C₂×#22551527

C₂

semidirect

4. #30656502

C₂

semidirect

5. #341×C₂₅₇

C₂

semidirect

64. C₂₅₇

#1811

semidirect

70. #341

D₂₅₇

semidirect

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

subgroup	index
C₂²×#16861601	2
C₂×#22551527	2
#30656700	2
#30656502	2
#341×C₂₅₇	2
#1811	257

subgroup

index

#341×C₂₅₇

257

Subgroups — #45664361

Conjugacy classes of subgroups

Complements

Maximal subgroups