Group #45665439 — subgroups

Subgroups — #45665439

Order 32896.

342 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₂×#22551623	C₂	1	2	yes	maximal
2	C₂×#22551511	C₂	1	2	yes	maximal, complemented
3	#30656703	C₂	1	1	yes	maximal, characteristic
4	#30656502	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Q₁₆×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16861618	C₂²	1	1	yes	characteristic
7	#22551623	—	2	4	no	—
8	#22551623	—	2	4	no	—
9	C₂×#16861608	—	2	2	no	—
10	C₂×#16861605	—	2	2	no	—
11	C₂×#16861602	C₂²	1	1	yes	characteristic
12	C₂×#16861602	C₂²	1	2	yes	—
13	#22551511	—	2	4	no	—
14	#22551504	—	2	2	no	—
15	C₂²×Q₈×C₂₅₇	C₂²	1	2	yes	—
16	C₂×Q₁₆×C₂₅₇	—	2	4	no	—
17	C₂×Q₁₆×C₂₅₇	—	2	2	no	—
18	C₂²×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
19	#16861618	—	2	2	no	—
20	#16861618	—	2	2	no	—
21	C₂×Dic₅₁₄	D₄	1	1	yes	characteristic
22	C₂×D₁₀₂₈	D₄	1	1	yes	characteristic
23	C₂×C₂²⋊D₂₅₇	—	2	1	no	—
24	C₂×C₄×D₂₅₇	—	2	1	no	—
25	#16861608	—	2	8	no	—
26	#16861605	—	2	8	no	—
27	#16861602	—	2	2	no	—
28	#16861602	—	2	2	no	—
29	#16861602	D₄	1	4	yes	—
30	C₂×#14523339	—	2	2	no	—
31	C₂×#14523339	—	2	1	no	—
32	C₂×Q₈×C₂₅₇	—	2	2	no	—
33	C₂×Q₈×C₂₅₇	—	2	2	no	—
34	C₂×Q₈×C₂₅₇	—	2	2	no	—
35	C₂×Q₈×C₂₅₇	—	2	2	no	—
36	C₂×Q₈×C₂₅₇	—	2	2	no	—
37	C₂×Q₈×C₂₅₇	D₄	1	4	yes	—
38	C₂²×C₄×C₂₅₇	—	2	2	no	—
39	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
40	Q₁₆×C₂₅₇	—	2	8	no	—
41	C₂×C₈×C₂₅₇	—	2	2	no	—
42	C₂×C₈×C₂₅₇	D₄	1	2	yes	—
43	C₂²×D₂₅₇	—	2	1	no	—
44	Dic₅₁₄	—	2	2	no	—
45	D₁₀₂₈	—	2	2	no	—
46	C₂²⋊D₂₅₇	—	4	2	no	—
47	C₂×Dic₂₅₇	—	2	1	no	—
48	C₄×D₂₅₇	—	4	1	no	—
49	C₄×D₂₅₇	—	4	1	no	—
50	#14523339	—	2	4	no	—
51	#14523339	—	4	1	no	—
52	C₂³×C₂₅₇	#41	1	1	yes	characteristic
53	Q₈×C₂₅₇	—	2	4	no	—
54	Q₈×C₂₅₇	—	2	4	no	—
55	Q₈×C₂₅₇	—	4	2	no	—
56	Q₈×C₂₅₇	—	4	2	no	—
57	C₂×C₄×C₂₅₇	—	2	4	no	—
58	C₂×C₄×C₂₅₇	—	4	2	no	—
59	C₂×C₄×C₂₅₇	—	4	2	no	—
60	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
64	C₈×C₂₅₇	—	2	2	no	—
65	C₂²×C₂₅₇	—	2	2	no	—
66	C₂²×C₂₅₇	#121	1	1	yes	characteristic
67	C₂²×C₂₅₇	#121	1	2	yes	—
68	C₂×D₂₅₇	—	4	1	no	—
69	C₂×D₂₅₇	—	4	2	no	—
70	C₄×C₂₅₇	—	4	4	no	—
71	C₄×C₂₅₇	#120	1	2	yes	—
72	C₄×C₂₅₇	#120	1	1	yes	characteristic
73	C₄×C₂₅₇	#127	1	1	yes	characteristic
74	Dic₂₅₇	—	4	1	no	—
75	C₂×C₂₅₇	—	2	2	no	—
76	C₂×C₂₅₇	#392	1	1	yes	characteristic
77	C₂×C₂₅₇	#473	1	2	yes	—
78	D₂₅₇	—	8	1	no	—
79	C₂₅₇	#2020	1	1	yes	characteristic, complemented, Sylow
80	#2020	—	257	1	no	maximal, Sylow
81	C₂×#137	—	257	2	no	—
82	C₂²×Q₁₆	D₂₅₇	1	1	yes	characteristic, complemented
83	C₂×#101	—	257	2	no	—
84	#408	—	257	1	no	—
85	#341	—	257	1	no	—
86	C₂×#41	—	257	1	no	—
87	C₂²×Q₈	C₂×D₂₅₇	1	2	yes	—
88	#137	—	514	4	no	—
89	#137	—	514	4	no	—
90	C₂×Q₁₆	—	2	4	no	—
91	C₂×Q₁₆	—	2	2	no	—
92	C₂×Q₁₆	—	514	2	no	—
93	C₂×SD₁₆	—	514	2	no	—
94	C₂×M₁₆	—	257	2	no	—
95	C₂×M₁₆	—	257	1	no	—
96	C₂²×C₈	C₂×D₂₅₇	1	1	yes	characteristic
97	#101	—	514	4	no	—
98	C₂²⋊C₈	—	514	2	no	—
99	#41	—	514	2	no	—
100	#41	—	514	2	no	—
101	C₂×Q₈	—	257	1	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	C₂²⋊D₂₅₇	1	4	yes	—
108	C₂×D₄	—	257	1	no	—
109	C₂×D₄	—	514	1	no	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	—	514	1	no	—
112	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
113	Q₁₆	—	2	8	no	—
114	Q₁₆	—	514	8	no	—
115	SD₁₆	—	514	8	no	—
116	M₁₆	—	257	4	no	—
117	M₁₆	—	514	2	no	—
118	M₁₆	—	514	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	514	2	no	—
121	C₂×C₈	—	514	1	no	—
122	C₂×C₈	C₂²⋊D₂₅₇	1	2	yes	—
123	C₂³	—	514	1	no	—
124	C₂³	#16861617	1	1	yes	characteristic
125	Q₈	—	2	4	no	—
126	Q₈	—	2	4	no	—
127	Q₈	—	514	2	no	—
128	Q₈	—	4	2	no	—
129	Q₈	—	4	2	no	—
130	D₄	—	514	2	no	—
131	D₄	—	1028	2	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	514	1	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	1028	1	no	—
137	C₂×C₄	—	1028	1	no	—
138	C₂×C₄	D₄×D₂₅₇	1	1	yes	characteristic
139	C₂×C₄	D₄×D₂₅₇	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	514	4	no	—
144	C₈	—	1028	1	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	1028	1	no	—
147	C₂²	—	1028	2	no	—
148	C₂²	#22551563	1	1	yes	characteristic
149	C₂²	#22551564	1	2	yes	—
150	C₄	—	4	4	no	—
151	C₄	—	1028	1	no	—
152	C₄	#22551560	1	2	yes	—
153	C₄	#22551561	1	1	yes	characteristic
154	C₄	#22551588	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	2056	1	no	—
157	C₂	#30656645	1	1	yes	characteristic
158	C₂	#30657009	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22551623

C₂

yes

maximal

C₂×#22551511

C₂

yes

maximal, complemented

#30656703

C₂

yes

maximal, characteristic

#30656502

C₂

yes

maximal, characteristic, complemented

C₂²×Q₁₆×C₂₅₇

C₂

yes

maximal, characteristic, complemented

C₂×#16861618

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#16861602

C₂²

yes

—

#22551511

—

#22551504

—

C₂²×Q₈×C₂₅₇

C₂²

yes

—

C₂×Q₁₆×C₂₅₇

—

C₂×Q₁₆×C₂₅₇

—

C₂²×C₈×C₂₅₇

C₂²

yes

characteristic

#16861618

—

#16861618

—

C₂×Dic₅₁₄

D₄

yes

characteristic

C₂×D₁₀₂₈

D₄

yes

characteristic

C₂×C₂²⋊D₂₅₇

—

C₂×C₄×D₂₅₇

—

—

—

—

—

D₄

yes

—

C₂×#14523339

—

C₂×#14523339

—

C₂×Q₈×C₂₅₇

—

C₂×Q₈×C₂₅₇

—

C₂×Q₈×C₂₅₇

—

C₂×Q₈×C₂₅₇

—

C₂×Q₈×C₂₅₇

—

C₂×Q₈×C₂₅₇

D₄

yes

—

C₂²×C₄×C₂₅₇

—

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

Q₁₆×C₂₅₇

—

C₂×C₈×C₂₅₇

—

C₂×C₈×C₂₅₇

D₄

yes

—

C₂²×D₂₅₇

—

Dic₅₁₄

—

D₁₀₂₈

—

C₂²⋊D₂₅₇

—

C₂×Dic₂₅₇

—

C₄×D₂₅₇

—

C₄×D₂₅₇

—

#14523339

—

#14523339

—

C₂³×C₂₅₇

#41

yes

characteristic

Q₈×C₂₅₇

—

Q₈×C₂₅₇

—

Q₈×C₂₅₇

—

Q₈×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×D₄

yes

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₈×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

#121

yes

characteristic

C₂²×C₂₅₇

#121

yes

—

C₂×D₂₅₇

—

C₂×D₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

#120

yes

—

C₄×C₂₅₇

yes

C₄×C₂₅₇

yes

Dic₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

#392

yes

characteristic

C₂×C₂₅₇

#473

yes

—

D₂₅₇

—

C₂₅₇

#2020

yes

characteristic, complemented, Sylow

#2020

—

257

maximal, Sylow

C₂×#137

—

257

—

C₂²×Q₁₆

D₂₅₇

yes

characteristic, complemented

C₂×#101

—

257

—

#408

—

257

—

#341

—

257

—

C₂×#41

—

257

—

C₂²×Q₈

C₂×D₂₅₇

yes

—

#137

—

514

—

#137

—

514

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

514

—

C₂×SD₁₆

—

514

—

C₂×M₁₆

—

257

—

C₂×M₁₆

—

257

—

C₂²×C₈

C₂×D₂₅₇

yes

characteristic

#101

—

514

—

C₂²⋊C₈

—

514

—

#41

—

514

—

100

#41

—

514

—

101

C₂×Q₈

—

257

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

C₂²⋊D₂₅₇

yes

—

108

C₂×D₄

—

257

—

109

C₂×D₄

—

514

—

110

C₂²×C₄

—

111

C₂²×C₄

—

514

—

112

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

113

Q₁₆

—

114

Q₁₆

—

514

—

115

SD₁₆

—

514

—

116

M₁₆

—

257

—

117

M₁₆

—

514

—

118

M₁₆

—

514

—

119

C₂×C₈

—

120

C₂×C₈

—

514

—

121

C₂×C₈

—

514

—

122

C₂×C₈

C₂²⋊D₂₅₇

yes

—

123

C₂³

—

514

—

124

C₂³

#16861617

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

514

—

128

Q₈

—

129

Q₈

—

130

D₄

—

514

—

131

D₄

—

1028

—

132

C₂×C₄

—

133

C₂×C₄

—

514

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

1028

—

137

C₂×C₄

—

1028

—

138

C₂×C₄

D₄×D₂₅₇

yes

characteristic

139

C₂×C₄

D₄×D₂₅₇

yes

—

140

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

—

141

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

characteristic

142

C₈

—

143

C₈

—

514

—

144

C₈

—

1028

—

145

C₂²

—

146

C₂²

—

1028

—

147

C₂²

—

1028

—

148

C₂²

#22551563

yes

characteristic

149

C₂²

#22551564

yes

—

150

C₄

—

151

C₄

—

1028

—

152

C₄

#22551560

yes

—

153

C₄

#22551561

yes

characteristic

154

C₄

#22551588

yes

characteristic

155

C₂

—

156

C₂

—

2056

—

157

C₂

#30656645

yes

characteristic

158

C₂

#30657009

yes

—

	subgroup	G/H
center	C₂²	#22551563
commutator, cosocle	C₂²×C₄×C₂₅₇	C₂³
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	C₂²×Q₁₆×C₂₅₇	C₂
socle	C₂²×C₂₅₇	#121
prime radical	C₂×#16861618	C₂²
hypercenter	C₂²×Q₁₆	D₂₅₇
hypocenter	C₂₅₇	#2020
LCS	G ▷ C₂²×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

#22551563

commutator, cosocle

C₂²×C₄×C₂₅₇

C₂³

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

C₂²×Q₁₆×C₂₅₇

C₂

C₂²×C₂₅₇

C₂²

C₂²×Q₁₆

D₂₅₇

hypocenter

C₂₅₇

#2020

LCS

G ▷ C₂²×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
2. C₂×#22551511	C₂	semidirect
4. #30656502	C₂	semidirect
5. C₂²×Q₁₆×C₂₅₇	C₂	semidirect
79. C₂₅₇	#2020	semidirect
82. C₂²×Q₁₆	D₂₅₇	semidirect

H ≅ G/N

type

2. C₂×#22551511

C₂

semidirect

4. #30656502

C₂

semidirect

5. C₂²×Q₁₆×C₂₅₇

C₂

semidirect

79. C₂₅₇

#2020

semidirect

82. C₂²×Q₁₆

D₂₅₇

semidirect

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

subgroup	index
C₂×#22551623	2
C₂×#22551511	2
#30656703	2
#30656502	2
C₂²×Q₁₆×C₂₅₇	2
#2020	257

subgroup

index

C₂²×Q₁₆×C₂₅₇

257

Subgroups — #45665439

Conjugacy classes of subgroups

Complements

Maximal subgroups