Group #45664404 — subgroups

Subgroups — #45664404

Order 32896.

344 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 117 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×C₄⋊Dic₂₅₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#14523339	C₂	1	1	yes	maximal, characteristic
3	#30656499	C₂	1	4	yes	maximal, complemented
4	#380×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₂₅₇	—	2	1	no	—
6	C₂²×#14523339	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₅₇	—	2	1	no	—
8	C₂×C₄⋊Dic₂₅₇	—	2	2	no	—
9	C₂×C₄⋊Dic₂₅₇	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₂₅₇	C₂²	1	2	yes	—
11	C₂²⋊#14523339	—	2	2	no	—
12	C₂³×C₄×C₂₅₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₅₇	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₄×C₂₅₇	—	2	1	no	—
15	#95×C₂₅₇	—	2	1	no	—
16	C₂²×Dic₂₅₇	—	2	4	no	—
17	C₂²×Dic₂₅₇	—	2	2	no	—
18	C₂²×Dic₂₅₇	—	4	2	no	—
19	C₂²×Dic₂₅₇	—	4	2	no	—
20	C₂×#14523339	—	2	2	no	—
21	C₂×#14523339	—	2	4	no	—
22	C₄⋊Dic₂₅₇	—	2	4	no	—
23	C₄⋊Dic₂₅₇	—	2	2	no	—
24	C₄⋊Dic₂₅₇	C₂×C₄	1	4	yes	—
25	C₂⁴×C₂₅₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₅₇	—	2	4	no	—
27	C₂²×C₄×C₂₅₇	—	2	2	no	—
28	C₂²×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₂₅₇	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₅₇	D₄	1	2	yes	—
31	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₂₅₇	—	2	4	no	—
33	C₂²⋊C₄×C₂₅₇	—	4	2	no	—
34	C₂×Dic₂₅₇	—	2	4	no	—
35	C₂×Dic₂₅₇	—	4	8	no	—
36	C₂×Dic₂₅₇	—	4	2	no	—
37	C₂×Dic₂₅₇	—	4	2	no	—
38	#14523339	—	4	2	no	—
39	C₂³×C₂₅₇	—	2	2	no	—
40	C₂³×C₂₅₇	—	2	2	no	—
41	C₂³×C₂₅₇	C₂²⋊C₄	1	2	yes	—
42	C₂³×C₂₅₇	D₈	1	2	yes	—
43	C₂³×C₂₅₇	SD₁₆	1	2	yes	—
44	C₂³×C₂₅₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₅₇	—	2	2	no	—
46	C₂×C₄×C₂₅₇	—	2	4	no	—
47	C₂×C₄×C₂₅₇	—	2	4	no	—
48	C₂×C₄×C₂₅₇	—	4	2	no	—
49	C₂×C₄×C₂₅₇	—	4	2	no	—
50	C₂×C₄×C₂₅₇	—	4	2	no	—
51	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₅₇	#41	1	4	yes	—
55	C₂²×C₂₅₇	—	2	2	no	—
56	C₂²×C₂₅₇	—	2	2	no	—
57	C₂²×C₂₅₇	—	2	4	no	—
58	C₂²×C₂₅₇	—	2	4	no	—
59	C₂²×C₂₅₇	#102	1	4	yes	—
60	C₂²×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₂₅₇	#123	1	2	yes	—
62	C₂²×C₂₅₇	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₂₅₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₂₅₇	#137	1	2	yes	—
65	C₄×C₂₅₇	—	2	2	no	—
66	C₄×C₂₅₇	—	8	2	no	—
67	C₄×C₂₅₇	#117	1	2	yes	—
68	C₄×C₂₅₇	#123	1	2	yes	—
69	Dic₂₅₇	—	4	4	no	—
70	C₂×C₂₅₇	—	2	4	no	—
71	C₂×C₂₅₇	#381	1	1	yes	characteristic
72	C₂×C₂₅₇	C₂×#102	1	1	yes	characteristic
73	C₂×C₂₅₇	#419	1	1	yes	characteristic
74	C₂×C₂₅₇	#480	1	2	yes	—
75	C₂×C₂₅₇	#483	1	2	yes	—
76	C₂₅₇	#1823	1	1	yes	characteristic, complemented, Sylow
77	#1823	—	257	1	no	maximal, Sylow
78	C₂²×C₄⋊C₄	—	257	1	no	—
79	C₂×C₂²⋊C₈	—	257	1	no	—
80	#380	D₂₅₇	1	1	yes	characteristic
81	#340	—	257	4	no	—
82	C₂³×C₄	—	514	1	no	—
83	C₂³×C₄	C₂×D₂₅₇	1	1	yes	characteristic
84	C₂²×C₈	—	257	2	no	—
85	C₂×C₄⋊C₄	—	257	4	no	—
86	C₂×C₄⋊C₄	—	257	2	no	—
87	C₂×C₄⋊C₄	—	514	1	no	—
88	C₂×C₄⋊C₄	—	514	2	no	—
89	C₂×C₄⋊C₄	C₂×D₂₅₇	1	2	yes	—
90	C₂×C₂²⋊C₄	—	2	1	no	—
91	C₂²⋊C₈	—	514	2	no	—
92	#95	—	2	1	no	—
93	C₂⁴	C₂²⋊D₂₅₇	1	1	yes	characteristic
94	C₂²×C₄	—	2	4	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	514	4	no	—
97	C₂²×C₄	—	514	2	no	—
98	C₂²×C₄	—	1028	2	no	—
99	C₂²×C₄	—	1028	2	no	—
100	C₂²×C₄	C₄×D₂₅₇	1	2	yes	—
101	C₂²×C₄	C₂²⋊D₂₅₇	1	1	yes	characteristic
102	C₂²×C₄	D₁₀₂₈	1	2	yes	—
103	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
104	C₂×C₈	—	514	2	no	—
105	C₂×C₈	—	514	4	no	—
106	C₄⋊C₄	—	257	4	no	—
107	C₄⋊C₄	—	2	4	no	—
108	C₄⋊C₄	—	514	4	no	—
109	C₄⋊C₄	—	514	2	no	—
110	C₂²⋊C₄	—	4	2	no	—
111	C₂³	—	2	2	no	—
112	C₂³	—	2	2	no	—
113	C₂³	#16861595	1	2	yes	—
114	C₂³	#16861603	1	2	yes	—
115	C₂³	#16861605	1	2	yes	—
116	C₂³	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	514	4	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	1028	2	no	—
125	C₂×C₄	—	1028	2	no	—
126	C₂×C₄	—	1028	8	no	—
127	C₂×C₄	#16861595	1	2	yes	—
128	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
129	C₂×C₄	C₂×D₁₀₂₈	1	1	yes	characteristic
130	C₂×C₄	#16861617	1	4	yes	—
131	C₈	—	1028	2	no	—
132	C₂²	—	2	4	no	—
133	C₂²	—	2	4	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	#22551512	1	4	yes	—
137	C₂²	C₂×#16861595	1	1	yes	characteristic
138	C₂²	#22551573	1	2	yes	—
139	C₂²	C₂×#16861603	1	1	yes	characteristic
140	C₂²	C₂×#16861605	1	1	yes	characteristic
141	C₂²	#22551622	1	2	yes	—
142	C₄	—	2	2	no	—
143	C₄	—	1028	4	no	—
144	C₄	—	8	2	no	—
145	C₄	#22551548	1	2	yes	—
146	C₄	#22551576	1	2	yes	—
147	C₂	—	2	4	no	—
148	C₂	#30656598	1	1	yes	characteristic
149	C₂	C₂×#22551512	1	1	yes	characteristic
150	C₂	#30656745	1	1	yes	characteristic
151	C₂	#30657048	1	2	yes	—
152	C₂	#30657065	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₂₅₇

C₂

yes

maximal, characteristic

C₂×C₂²⋊#14523339

C₂

yes

maximal, characteristic

#30656499

C₂

yes

maximal, complemented

#380×C₂₅₇

C₂

yes

maximal, characteristic

C₂³×Dic₂₅₇

—

C₂²×#14523339

C₂²

yes

—

C₂×C₄⋊Dic₂₅₇

—

C₂×C₄⋊Dic₂₅₇

—

C₂×C₄⋊Dic₂₅₇

C₄

yes

complemented

C₂×C₄⋊Dic₂₅₇

C₂²

yes

—

C₂²⋊#14523339

—

C₂³×C₄×C₂₅₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₅₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₅₇

—

#95×C₂₅₇

—

C₂²×Dic₂₅₇

—

C₂²×Dic₂₅₇

—

C₂²×Dic₂₅₇

—

C₂²×Dic₂₅₇

—

C₂×#14523339

—

C₂×#14523339

—

C₄⋊Dic₂₅₇

—

C₄⋊Dic₂₅₇

—

C₄⋊Dic₂₅₇

C₂×C₄

yes

—

C₂⁴×C₂₅₇

D₄

yes

characteristic

C₂²×C₄×C₂₅₇

—

C₂²×C₄×C₂₅₇

—

C₂²×C₄×C₂₅₇

C₂×C₄

yes

—

C₂²×C₄×C₂₅₇

D₄

yes

characteristic

C₂²×C₄×C₂₅₇

D₄

yes

—

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₅₇

—

C₂²⋊C₄×C₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

#14523339

—

C₂³×C₂₅₇

—

C₂³×C₂₅₇

—

C₂³×C₂₅₇

C₂²⋊C₄

yes

—

C₂³×C₂₅₇

D₈

yes

—

C₂³×C₂₅₇

SD₁₆

yes

—

C₂³×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

#41

yes

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

#102

yes

—

C₂²×C₂₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₅₇

#123

yes

—

C₂²×C₂₅₇

C₂×D₈

yes

characteristic

C₂²×C₂₅₇

C₂×SD₁₆

yes

characteristic

C₂²×C₂₅₇

#137

yes

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

#117

yes

—

C₄×C₂₅₇

#123

yes

—

Dic₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

—

C₂×C₂₅₇

#483

yes

—

C₂₅₇

#1823

yes

characteristic, complemented, Sylow

#1823

—

257

maximal, Sylow

C₂²×C₄⋊C₄

—

257

—

C₂×C₂²⋊C₈

—

257

—

#380

D₂₅₇

yes

characteristic

#340

—

257

—

C₂³×C₄

—

514

—

C₂³×C₄

C₂×D₂₅₇

yes

characteristic

C₂²×C₈

—

257

—

C₂×C₄⋊C₄

—

257

—

C₂×C₄⋊C₄

—

257

—

C₂×C₄⋊C₄

—

514

—

C₂×C₄⋊C₄

—

514

—

C₂×C₄⋊C₄

C₂×D₂₅₇

yes

—

C₂×C₂²⋊C₄

—

C₂²⋊C₈

—

514

—

#95

—

C₂⁴

C₂²⋊D₂₅₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

514

—

C₂²×C₄

—

514

—

C₂²×C₄

—

1028

—

C₂²×C₄

—

1028

—

100

C₂²×C₄

C₄×D₂₅₇

yes

—

101

C₂²×C₄

C₂²⋊D₂₅₇

yes

characteristic

102

C₂²×C₄

D₁₀₂₈

yes

—

103

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

104

C₂×C₈

—

514

—

105

C₂×C₈

—

514

—

106

C₄⋊C₄

—

257

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

514

—

109

C₄⋊C₄

—

514

—

110

C₂²⋊C₄

—

111

C₂³

—

112

C₂³

—

113

C₂³

#16861595

yes

—

114

C₂³

#16861603

yes

—

115

C₂³

#16861605

yes

—

116

C₂³

C₂×C₂²⋊D₂₅₇

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

514

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

1028

—

125

C₂×C₄

—

1028

—

126

C₂×C₄

—

1028

—

127

C₂×C₄

#16861595

yes

—

128

C₂×C₄

C₂×C₄×D₂₅₇

yes

characteristic

129

C₂×C₄

C₂×D₁₀₂₈

yes

characteristic

130

C₂×C₄

#16861617

yes

—

131

C₈

—

1028

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

#22551512

yes

—

137

C₂²

C₂×#16861595

yes

characteristic

138

C₂²

#22551573

yes

—

139

C₂²

C₂×#16861603

yes

characteristic

140

C₂²

C₂×#16861605

yes

characteristic

141

C₂²

#22551622

yes

—

142

C₄

—

143

C₄

—

1028

—

144

C₄

—

145

C₄

#22551548

yes

—

146

C₄

#22551576

yes

—

147

C₂

—

148

C₂

#30656598

yes

characteristic

149

C₂

C₂×#22551512

yes

characteristic

150

C₂

#30656745

yes

characteristic

151

C₂

#30657048

yes

—

152

C₂

#30657065

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₅₇
commutator	C₂×C₄×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	#380×C₂₅₇	C₂
socle	C₂³×C₂₅₇	C₂×D₄
prime radical	C₂⁴×C₂₅₇	D₄
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	#380	D₂₅₇
hypocenter	C₂₅₇	#1823
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₅₇

commutator

C₂×C₄×C₂₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

#380×C₂₅₇

C₂

socle

C₂³×C₂₅₇

C₂×D₄

prime radical

C₂⁴×C₂₅₇

D₄

cosocle

C₂²×C₄×C₂₅₇

C₂³

D₂₅₇

C₂₅₇

G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

N	H ≅ G/N	type
3. #30656499	C₂	semidirect
9. C₂×C₄⋊Dic₂₅₇	C₄	semidirect
76. C₂₅₇	#1823	semidirect

H ≅ G/N

type

3. #30656499

C₂

semidirect

9. C₂×C₄⋊Dic₂₅₇

C₄

semidirect

76. C₂₅₇

#1823

semidirect

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

subgroup	index
C₂²×C₄⋊Dic₂₅₇	2
C₂×C₂²⋊#14523339	2
#30656499	2
#380×C₂₅₇	2
#1823	257

subgroup

index

C₂²×C₄⋊Dic₂₅₇

C₂×C₂²⋊#14523339

#30656499

#380×C₂₅₇

#1823

257

Subgroups — #45664404

Conjugacy classes of subgroups

Complements

Maximal subgroups