Group #45665181 — subgroups

Subgroups — #45665181

Order 32896.

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

normal subgroups only

This group has 312 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×Dic₁₀₂₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#22551526	C₂	1	1	yes	maximal, characteristic
3	C₂×#22551517	C₂	1	2	yes	maximal
4	#30656643	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₁₄	C₂²	1	2	yes	—
7	C₂×Dic₁₀₂₈	—	2	2	no	—
8	C₂×Dic₁₀₂₈	—	2	2	no	—
9	C₂×Dic₁₀₂₈	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₂₅₇	C₂²	1	2	yes	—
11	#22551526	—	2	2	no	—
12	#22551517	—	2	4	no	—
13	#22551499	—	2	4	no	—
14	C₂²×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
17	C₄×C₈×C₂₅₇	—	2	2	no	—
18	C₂×Dic₅₁₄	—	2	2	no	—
19	C₂×Dic₅₁₄	—	2	2	no	—
20	C₂×Dic₅₁₄	—	2	2	no	—
21	C₂×Dic₅₁₄	—	2	2	no	—
22	C₂×Dic₅₁₄	—	2	2	no	—
23	C₂×Dic₅₁₄	C₂×C₄	1	4	yes	—
24	C₂²×Dic₂₅₇	—	2	2	no	—
25	C₂²×Dic₂₅₇	—	2	2	no	—
26	Dic₁₀₂₈	—	2	8	no	—
27	C₄⋊Dic₂₅₇	—	2	2	no	—
28	C₄⋊Dic₂₅₇	—	2	2	no	—
29	C₂²×C₄×C₂₅₇	D₄	1	2	yes	—
30	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₂₅₇	—	2	1	no	—
32	C₂×C₈×C₂₅₇	—	2	1	no	—
33	C₂×C₈×C₂₅₇	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₂₅₇	D₄	1	2	yes	—
35	C₂×C₈×C₂₅₇	D₄	1	2	yes	—
36	C₂×C₈×C₂₅₇	D₄	1	2	yes	—
37	C₄²×C₂₅₇	—	2	1	no	—
38	C₄²×C₂₅₇	—	2	1	no	—
39	Dic₅₁₄	—	2	4	no	—
40	Dic₅₁₄	—	2	4	no	—
41	Dic₅₁₄	—	4	2	no	—
42	Dic₅₁₄	—	4	2	no	—
43	C₂×Dic₂₅₇	—	2	4	no	—
44	C₂×Dic₂₅₇	—	4	2	no	—
45	C₂×Dic₂₅₇	—	4	2	no	—
46	C₂×Dic₂₅₇	—	4	2	no	—
47	C₂×Dic₂₅₇	—	4	4	no	—
48	C₂³×C₂₅₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₅₇	—	2	2	no	—
50	C₂×C₄×C₂₅₇	—	2	2	no	—
51	C₂×C₄×C₂₅₇	Q₁₆	1	4	yes	complemented
52	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₅₇	#41	1	1	yes	characteristic
57	C₂×C₄×C₂₅₇	#41	1	1	yes	characteristic
58	C₈×C₂₅₇	—	2	2	no	—
59	C₈×C₂₅₇	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₂₅₇	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₂₅₇	#121	1	1	yes	characteristic
62	C₂²×C₂₅₇	#127	1	1	yes	characteristic
63	C₂²×C₂₅₇	C₂×Q₁₆	1	2	yes	—
64	C₂²×C₂₅₇	#135	1	2	yes	—
65	C₄×C₂₅₇	—	2	4	no	—
66	C₄×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₂₅₇	C₄×D₄	1	1	yes	characteristic
68	C₄×C₂₅₇	#121	1	1	yes	characteristic
69	C₄×C₂₅₇	#124	1	1	yes	characteristic
70	Dic₂₅₇	—	4	4	no	—
71	Dic₂₅₇	—	8	2	no	—
72	C₂×C₂₅₇	#390	1	1	yes	characteristic
73	C₂×C₂₅₇	C₄×Q₁₆	1	2	yes	—
74	C₂×C₂₅₇	#467	1	2	yes	—
75	C₂×C₂₅₇	#494	1	1	yes	characteristic
76	C₂×C₂₅₇	#495	1	1	yes	characteristic
77	C₂₅₇	#1969	1	1	yes	characteristic, complemented, Sylow
78	#1969	—	257	1	no	maximal, Sylow
79	C₂²×Q₁₆	—	257	1	no	—
80	C₂×#107	—	257	1	no	—
81	C₂×#103	—	257	2	no	—
82	C₂×C₄×C₈	D₂₅₇	1	1	yes	characteristic
83	#391	—	257	2	no	—
84	C₂²×Q₈	—	257	2	no	—
85	C₂×Q₁₆	—	257	4	no	—
86	C₂×Q₁₆	—	514	2	no	—
87	C₂×Q₁₆	—	514	2	no	—
88	C₂²×C₈	C₂×D₂₅₇	1	1	yes	characteristic
89	C₂²×C₈	C₂×D₂₅₇	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	257	2	no	—
91	C₂×C₄²	C₂×D₂₅₇	1	1	yes	characteristic
92	#107	—	514	2	no	—
93	#103	—	514	4	no	—
94	C₄×C₈	—	2	2	no	—
95	#95	—	514	4	no	—
96	C₂×Q₈	—	257	4	no	—
97	C₂×Q₈	—	514	2	no	—
98	C₂×Q₈	—	514	2	no	—
99	C₂×Q₈	—	514	2	no	—
100	C₂×Q₈	—	514	2	no	—
101	C₂×Q₈	—	514	2	no	—
102	C₂²×C₄	—	514	2	no	—
103	C₂²×C₄	—	514	2	no	—
104	C₂²×C₄	D₁₀₂₈	1	2	yes	—
105	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
106	Q₁₆	—	514	8	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₄×D₂₅₇	1	2	yes	—
110	C₂×C₈	C₂²⋊D₂₅₇	1	2	yes	—
111	C₂×C₈	D₁₀₂₈	1	2	yes	—
112	C₂×C₈	D₁₀₂₈	1	2	yes	—
113	C₄⋊C₄	—	514	2	no	—
114	C₄⋊C₄	—	514	2	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₂³	C₂×D₁₀₂₈	1	1	yes	characteristic
118	Q₈	—	514	4	no	—
119	Q₈	—	514	4	no	—
120	Q₈	—	1028	2	no	—
121	Q₈	—	1028	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	514	4	no	—
125	C₂×C₄	—	1028	2	no	—
126	C₂×C₄	—	1028	2	no	—
127	C₂×C₄	—	1028	2	no	—
128	C₂×C₄	—	1028	4	no	—
129	C₂×C₄	Dic₁₀₂₈	1	4	yes	complemented
130	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
132	C₂×C₄	C₂×D₁₀₂₈	1	1	yes	characteristic
133	C₂×C₄	C₂×D₁₀₂₈	1	1	yes	characteristic
134	C₂×C₄	#16861618	1	1	yes	characteristic
135	C₂×C₄	#16861618	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	#16861595	1	4	yes	—
138	C₂²	C₄×D₁₀₂₈	1	1	yes	characteristic
139	C₂²	#22551566	1	1	yes	characteristic
140	C₂²	#22551589	1	1	yes	characteristic
141	C₂²	C₂×Dic₁₀₂₈	1	2	yes	—
142	C₂²	#22551614	1	2	yes	—
143	C₄	—	2	4	no	—
144	C₄	—	1028	4	no	—
145	C₄	—	2056	2	no	—
146	C₄	C₂×#16861595	1	1	yes	characteristic
147	C₄	C₄×D₁₀₂₈	1	1	yes	characteristic
148	C₄	#22551566	1	1	yes	characteristic
149	C₄	#22551580	1	1	yes	characteristic
150	C₂	#30656639	1	1	yes	characteristic
151	C₂	C₄×Dic₁₀₂₈	1	2	yes	—
152	C₂	#30656978	1	2	yes	—
153	C₂	#30657117	1	1	yes	characteristic
154	C₂	#30657121	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₀₂₈

C₂

yes

maximal, characteristic

C₂×#22551526

C₂

yes

maximal, characteristic

C₂×#22551517

C₂

yes

maximal

#30656643

C₂

yes

maximal

C₂×C₄×C₈×C₂₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₅₁₄

C₂²

yes

—

C₂×Dic₁₀₂₈

—

C₂×Dic₁₀₂₈

—

C₂×Dic₁₀₂₈

C₄

yes

complemented

C₂×C₄⋊Dic₂₅₇

C₂²

yes

—

#22551526

—

#22551517

—

#22551499

—

C₂²×C₈×C₂₅₇

C₂²

yes

characteristic

C₂²×C₈×C₂₅₇

C₂²

yes

characteristic

C₂×C₄²×C₂₅₇

C₂²

yes

characteristic

C₄×C₈×C₂₅₇

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

C₂×C₄

yes

—

C₂²×Dic₂₅₇

—

C₂²×Dic₂₅₇

—

Dic₁₀₂₈

—

C₄⋊Dic₂₅₇

—

C₄⋊Dic₂₅₇

—

C₂²×C₄×C₂₅₇

D₄

yes

—

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

C₂×C₈×C₂₅₇

—

C₂×C₈×C₂₅₇

—

C₂×C₈×C₂₅₇

C₂×C₄

yes

—

C₂×C₈×C₂₅₇

D₄

yes

—

C₂×C₈×C₂₅₇

D₄

yes

—

C₂×C₈×C₂₅₇

D₄

yes

—

C₄²×C₂₅₇

—

C₄²×C₂₅₇

—

Dic₅₁₄

—

Dic₅₁₄

—

Dic₅₁₄

—

Dic₅₁₄

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂³×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

Q₁₆

yes

complemented

C₂×C₄×C₂₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×D₄

yes

C₂×C₄×C₂₅₇

yes

C₂×C₄×C₂₅₇

yes

C₈×C₂₅₇

—

C₈×C₂₅₇

C₂²⋊C₄

yes

—

C₂²×C₂₅₇

C₄×D₄

yes

C₂²×C₂₅₇

yes

C₂²×C₂₅₇

yes

C₂²×C₂₅₇

C₂×Q₁₆

yes

—

C₂²×C₂₅₇

#135

yes

—

C₄×C₂₅₇

—

C₄×C₂₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₅₇

C₄×D₄

yes

C₄×C₂₅₇

yes

C₄×C₂₅₇

yes

Dic₂₅₇

—

Dic₂₅₇

—

C₂×C₂₅₇

#390

yes

characteristic

C₂×C₂₅₇

C₄×Q₁₆

yes

—

C₂×C₂₅₇

#467

yes

—

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

C₂₅₇

yes

characteristic, complemented, Sylow

#1969

—

257

maximal, Sylow

C₂²×Q₁₆

—

257

—

C₂×#107

—

257

—

C₂×#103

—

257

—

C₂×C₄×C₈

D₂₅₇

yes

characteristic

#391

—

257

—

C₂²×Q₈

—

257

—

C₂×Q₁₆

—

257

—

C₂×Q₁₆

—

514

—

C₂×Q₁₆

—

514

—

C₂²×C₈

C₂×D₂₅₇

yes

characteristic

C₂²×C₈

C₂×D₂₅₇

yes

characteristic

C₂×C₄⋊C₄

—

257

—

C₂×C₄²

C₂×D₂₅₇

yes

characteristic

#107

—

514

—

#103

—

514

—

C₄×C₈

—

#95

—

514

—

C₂×Q₈

—

257

—

C₂×Q₈

—

514

—

C₂×Q₈

—

514

—

C₂×Q₈

—

514

—

100

C₂×Q₈

—

514

—

101

C₂×Q₈

—

514

—

102

C₂²×C₄

—

514

—

103

C₂²×C₄

—

514

—

104

C₂²×C₄

D₁₀₂₈

yes

—

105

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

106

Q₁₆

—

514

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₄×D₂₅₇

yes

—

110

C₂×C₈

C₂²⋊D₂₅₇

yes

—

111

C₂×C₈

D₁₀₂₈

yes

—

112

C₂×C₈

D₁₀₂₈

yes

—

113

C₄⋊C₄

—

514

—

114

C₄⋊C₄

—

514

—

115

C₄²

—

116

C₄²

—

117

C₂³

C₂×D₁₀₂₈

yes

characteristic

118

Q₈

—

514

—

119

Q₈

—

514

—

120

Q₈

—

1028

—

121

Q₈

—

1028

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

514

—

125

C₂×C₄

—

1028

—

126

C₂×C₄

—

1028

—

127

C₂×C₄

—

1028

—

128

C₂×C₄

—

1028

—

129

C₂×C₄

Dic₁₀₂₈

yes

complemented

130

C₂×C₄

C₂×C₄×D₂₅₇

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

characteristic

132

C₂×C₄

C₂×D₁₀₂₈

yes

characteristic

133

C₂×C₄

C₂×D₁₀₂₈

yes

characteristic

134

C₂×C₄

#16861618

yes

characteristic

135

C₂×C₄

#16861618

yes

characteristic

136

C₈

—

137

C₈

#16861595

yes

—

138

C₂²

C₄×D₁₀₂₈

yes

characteristic

139

C₂²

#22551566

yes

characteristic

140

C₂²

#22551589

yes

characteristic

141

C₂²

C₂×Dic₁₀₂₈

yes

—

142

C₂²

#22551614

yes

—

143

C₄

—

144

C₄

—

1028

—

145

C₄

—

2056

—

146

C₄

C₂×#16861595

yes

characteristic

147

C₄

C₄×D₁₀₂₈

yes

characteristic

148

C₄

#22551566

yes

characteristic

149

C₄

#22551580

yes

characteristic

150

C₂

#30656639

yes

characteristic

151

C₂

C₄×Dic₁₀₂₈

yes

—

152

C₂

#30656978

yes

—

153

C₂

#30657117

yes

characteristic

154

C₂

#30657121

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₀₂₈
commutator	C₂×C₄×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	C₂×C₄×C₈×C₂₅₇	C₂
socle, prime radical	C₂³×C₂₅₇	C₂×D₄
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	C₂×C₄×C₈	D₂₅₇
hypocenter	C₂₅₇	#1969
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₁₀₂₈

commutator

C₂×C₄×C₂₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

C₂×C₄×C₈×C₂₅₇

C₂

socle, prime radical

C₂³×C₂₅₇

C₂×D₄

cosocle

C₂²×C₄×C₂₅₇

C₂³

hypercenter

C₂×C₄×C₈

D₂₅₇

hypocenter

C₂₅₇

#1969

LCS

G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
9. C₂×Dic₁₀₂₈	C₄	semidirect
51. C₂×C₄×C₂₅₇	Q₁₆	semidirect
77. C₂₅₇	#1969	semidirect
129. C₂×C₄	Dic₁₀₂₈	semidirect

H ≅ G/N

type

9. C₂×Dic₁₀₂₈

C₄

semidirect

51. C₂×C₄×C₂₅₇

Q₁₆

semidirect

77. C₂₅₇

#1969

semidirect

129. C₂×C₄

Dic₁₀₂₈

semidirect

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

subgroup	index
C₂²×Dic₁₀₂₈	2
C₂×#22551526	2
C₂×#22551517	2
#30656643	2
C₂×C₄×C₈×C₂₅₇	2
#1969	257

subgroup

index

C₂²×Dic₁₀₂₈

C₂×C₄×C₈×C₂₅₇

257

Subgroups — #45665181

Conjugacy classes of subgroups

Complements

Maximal subgroups