Group #45664870 — subgroups

Subgroups — #45664870

Order 32896.

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

normal subgroups only

This group has 308 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	#30657504	C₂	1	1	yes	maximal, characteristic
2	#30657308	C₂	1	1	yes	maximal, characteristic
3	C₂×#22551510	C₂	1	2	yes	maximal, complemented
4	#30656508	C₂	1	2	yes	maximal, complemented
5	#413×C₂₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₁₄	C₂²	1	1	yes	characteristic
7	#22551599	—	2	4	no	—
8	C₂×#16861602	—	2	2	no	—
9	C₂×#16861602	C₂²	1	2	yes	—
10	#22551590	—	2	1	no	—
11	#22551579	—	2	1	no	—
12	#22551573	C₄	1	4	yes	—
13	#22551571	—	2	2	no	—
14	#22551551	C₂²	1	1	yes	characteristic
15	#22551510	—	2	4	no	—
16	C₂×#41×C₂₅₇	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₂₅₇	C₂²	1	2	yes	—
18	#101×C₂₅₇	—	2	1	no	—
19	#100×C₂₅₇	—	2	1	no	—
20	C₂×Dic₅₁₄	—	2	1	no	—
21	C₂×Dic₅₁₄	—	2	1	no	—
22	C₂×Dic₅₁₄	—	2	2	no	—
23	C₂×Dic₅₁₄	—	2	2	no	—
24	C₂²×Dic₂₅₇	C₂×C₄	1	2	yes	—
25	#16861602	—	2	4	no	—
26	#16861602	—	2	2	no	—
27	#16861602	D₄	1	4	yes	—
28	C₂×#14523339	—	2	4	no	—
29	C₂×#14523339	—	2	2	no	—
30	C₄⋊Dic₂₅₇	—	2	1	no	—
31	#16861597	—	2	4	no	—
32	C₂²⋊Dic₂₅₇	—	2	4	no	—
33	C₂²⋊Dic₂₅₇	C₂×C₄	1	2	yes	—
34	C₄×Dic₂₅₇	—	2	1	no	—
35	#41×C₂₅₇	—	2	2	no	—
36	#41×C₂₅₇	—	2	2	no	—
37	C₂×Q₈×C₂₅₇	D₄	1	1	yes	characteristic
38	C₂×D₄×C₂₅₇	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₂₅₇	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₅₇	D₄	1	2	yes	—
41	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
42	M₁₆×C₂₅₇	—	2	2	no	—
43	M₁₆×C₂₅₇	—	2	2	no	—
44	C₂×C₈×C₂₅₇	—	2	2	no	—
45	Dic₅₁₄	—	4	1	no	—
46	Dic₅₁₄	—	4	2	no	—
47	Dic₅₁₄	—	4	1	no	—
48	C₂×Dic₂₅₇	—	2	2	no	—
49	C₂×Dic₂₅₇	—	2	2	no	—
50	C₂×Dic₂₅₇	—	2	4	no	—
51	#14523339	—	2	4	no	—
52	C₂³×C₂₅₇	C₂²⋊C₄	1	2	yes	—
53	C₂³×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
54	Q₈×C₂₅₇	—	2	2	no	—
55	D₄×C₂₅₇	—	2	4	no	—
56	D₄×C₂₅₇	—	2	2	no	—
57	C₂×C₄×C₂₅₇	—	2	2	no	—
58	C₂×C₄×C₂₅₇	—	2	2	no	—
59	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₂₅₇	#41	1	2	yes	—
64	C₈×C₂₅₇	—	4	2	no	—
65	C₂²×C₂₅₇	—	2	2	no	—
66	C₂²×C₂₅₇	—	4	1	no	—
67	C₂²×C₂₅₇	—	4	2	no	—
68	C₂²×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₂₅₇	C₄×D₄	1	2	yes	—
70	C₄×C₂₅₇	—	2	2	no	—
71	C₄×C₂₅₇	#120	1	1	yes	characteristic
72	C₄×C₂₅₇	#121	1	2	yes	—
73	C₄×C₂₅₇	#123	1	1	yes	characteristic
74	Dic₂₅₇	—	4	2	no	—
75	Dic₂₅₇	—	4	2	no	—
76	C₂×C₂₅₇	—	2	1	no	—
77	C₂×C₂₅₇	—	2	2	no	—
78	C₂×C₂₅₇	—	4	2	no	—
79	C₂×C₂₅₇	#386	1	1	yes	characteristic
80	C₂₅₇	#1915	1	1	yes	characteristic, complemented, Sylow
81	#1915	—	257	1	no	maximal, Sylow
82	#568	—	257	1	no	—
83	#536	—	257	1	no	—
84	#413	D₂₅₇	1	1	yes	characteristic
85	C₂×#101	—	257	2	no	—
86	#344	—	257	2	no	—
87	C₂×#41	C₂×D₂₅₇	1	1	yes	characteristic
88	C₂²×Q₈	—	257	1	no	—
89	#131	—	514	4	no	—
90	C₂×M₁₆	—	257	2	no	—
91	C₂×M₁₆	—	514	2	no	—
92	C₂×M₁₆	C₂×D₂₅₇	1	2	yes	—
93	C₄⋊Q₈	—	514	1	no	—
94	#124	—	514	1	no	—
95	#123	—	257	4	no	—
96	C₂²⋊Q₈	—	514	2	no	—
97	#117	—	257	1	no	—
98	#101	—	2	1	no	—
99	#101	—	514	4	no	—
100	#100	—	2	1	no	—
101	#41	—	2	2	no	—
102	#41	—	2	2	no	—
103	C₂×Q₈	—	514	2	no	—
104	C₂×Q₈	—	514	1	no	—
105	C₂×Q₈	—	514	1	no	—
106	C₂×Q₈	—	514	2	no	—
107	C₂×Q₈	C₂²⋊D₂₅₇	1	1	yes	characteristic
108	C₂×D₄	C₄×D₂₅₇	1	2	yes	—
109	C₂×D₄	C₂²⋊D₂₅₇	1	1	yes	characteristic
110	C₂²×C₄	—	257	2	no	—
111	C₂²×C₄	D₁₀₂₈	1	2	yes	—
112	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
113	M₁₆	—	257	4	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	—	2	2	no	—
116	M₁₆	—	514	4	no	—
117	M₁₆	—	514	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	514	4	no	—
120	C₂×C₈	—	514	2	no	—
121	C₄⋊C₄	—	514	4	no	—
122	C₄⋊C₄	—	514	1	no	—
123	C₂²⋊C₄	—	257	2	no	—
124	C₂²⋊C₄	—	514	4	no	—
125	C₄²	—	514	1	no	—
126	C₂³	#16861595	1	2	yes	—
127	C₂³	C₂×C₄×D₂₅₇	1	1	yes	characteristic
128	Q₈	—	2	2	no	—
129	Q₈	—	1028	1	no	—
130	Q₈	—	1028	2	no	—
131	Q₈	—	1028	1	no	—
132	D₄	—	2	4	no	—
133	D₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	514	4	no	—
137	C₂×C₄	—	514	2	no	—
138	C₂×C₄	—	514	2	no	—
139	C₂×C₄	#16861595	1	2	yes	—
140	C₂×C₄	D₄×D₂₅₇	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
142	C₂×C₄	C₂×D₁₀₂₈	1	1	yes	characteristic
143	C₂×C₄	#16861617	1	2	yes	—
144	C₈	—	514	4	no	—
145	C₈	—	4	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	2	no	—
149	C₂²	C₂×#16861595	1	1	yes	characteristic
150	C₂²	#22551552	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	1028	2	no	—
153	C₄	—	1028	2	no	—
154	C₄	#22551562	1	1	yes	characteristic
155	C₄	#22551564	1	2	yes	—
156	C₄	#22551576	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#30656614	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30657504

C₂

yes

maximal, characteristic

#30657308

C₂

yes

maximal, characteristic

C₂×#22551510

C₂

yes

maximal, complemented

#30656508

C₂

yes

maximal, complemented

#413×C₂₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₅₁₄

C₂²

yes

—

—

C₂²

yes

—

#22551590

—

#22551579

—

#22551573

C₄

yes

—

#22551571

—

#22551551

C₂²

yes

characteristic

#22551510

—

C₂×#41×C₂₅₇

C₂²

yes

characteristic

C₂×M₁₆×C₂₅₇

C₂²

yes

—

#101×C₂₅₇

—

#100×C₂₅₇

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂²×Dic₂₅₇

C₂×C₄

yes

—

#16861602

—

#16861602

—

#16861602

D₄

yes

—

C₂×#14523339

—

C₂×#14523339

—

C₄⋊Dic₂₅₇

—

#16861597

—

C₂²⋊Dic₂₅₇

—

C₂²⋊Dic₂₅₇

C₂×C₄

yes

—

C₄×Dic₂₅₇

—

#41×C₂₅₇

—

#41×C₂₅₇

—

C₂×Q₈×C₂₅₇

D₄

yes

characteristic

C₂×D₄×C₂₅₇

C₂×C₄

yes

—

C₂×D₄×C₂₅₇

D₄

yes

characteristic

C₂²×C₄×C₂₅₇

D₄

yes

—

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

M₁₆×C₂₅₇

—

M₁₆×C₂₅₇

—

C₂×C₈×C₂₅₇

—

Dic₅₁₄

—

Dic₅₁₄

—

Dic₅₁₄

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

#14523339

—

C₂³×C₂₅₇

C₂²⋊C₄

yes

—

C₂³×C₂₅₇

C₂²×C₄

yes

characteristic

Q₈×C₂₅₇

—

D₄×C₂₅₇

—

D₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

#41

yes

—

C₈×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₅₇

C₄×D₄

yes

—

C₄×C₂₅₇

—

C₄×C₂₅₇

#120

yes

characteristic

C₄×C₂₅₇

#121

yes

—

C₄×C₂₅₇

#123

yes

characteristic

Dic₂₅₇

—

Dic₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

#386

yes

characteristic

C₂₅₇

#1915

yes

characteristic, complemented, Sylow

#1915

—

257

maximal, Sylow

#568

—

257

—

#536

—

257

—

#413

D₂₅₇

yes

characteristic

C₂×#101

—

257

—

#344

—

257

—

C₂×#41

C₂×D₂₅₇

yes

characteristic

C₂²×Q₈

—

257

—

#131

—

514

—

C₂×M₁₆

—

257

—

C₂×M₁₆

—

514

—

C₂×M₁₆

C₂×D₂₅₇

yes

—

C₄⋊Q₈

—

514

—

#124

—

514

—

#123

—

257

—

C₂²⋊Q₈

—

514

—

#117

—

257

—

#101

—

#101

—

514

—

100

#100

—

101

#41

—

102

#41

—

103

C₂×Q₈

—

514

—

104

C₂×Q₈

—

514

—

105

C₂×Q₈

—

514

—

106

C₂×Q₈

—

514

—

107

C₂×Q₈

C₂²⋊D₂₅₇

yes

characteristic

108

C₂×D₄

C₄×D₂₅₇

yes

—

109

C₂×D₄

C₂²⋊D₂₅₇

yes

characteristic

110

C₂²×C₄

—

257

—

111

C₂²×C₄

D₁₀₂₈

yes

—

112

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

113

M₁₆

—

257

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

514

—

117

M₁₆

—

514

—

118

C₂×C₈

—

119

C₂×C₈

—

514

—

120

C₂×C₈

—

514

—

121

C₄⋊C₄

—

514

—

122

C₄⋊C₄

—

514

—

123

C₂²⋊C₄

—

257

—

124

C₂²⋊C₄

—

514

—

125

C₄²

—

514

—

126

C₂³

#16861595

yes

—

127

C₂³

C₂×C₄×D₂₅₇

yes

characteristic

128

Q₈

—

129

Q₈

—

1028

—

130

Q₈

—

1028

—

131

Q₈

—

1028

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

514

—

137

C₂×C₄

—

514

—

138

C₂×C₄

—

514

—

139

C₂×C₄

#16861595

yes

—

140

C₂×C₄

D₄×D₂₅₇

yes

—

141

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

characteristic

142

C₂×C₄

C₂×D₁₀₂₈

yes

characteristic

143

C₂×C₄

#16861617

yes

—

144

C₈

—

514

—

145

C₈

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×#16861595

yes

characteristic

150

C₂²

#22551552

yes

—

151

C₄

—

152

C₄

—

1028

—

153

C₄

—

1028

—

154

C₄

#22551562

yes

characteristic

155

C₄

#22551564

yes

—

156

C₄

#22551576

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#30656614

yes

characteristic

	subgroup	G/H
center	C₂	#30656614
commutator	C₂³×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	#413×C₂₅₇	C₂
socle	C₂×C₂₅₇	#386
prime radical	C₂×#41×C₂₅₇	C₂²
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	#413	D₂₅₇
hypocenter	C₂₅₇	#1915
LCS	G ▷ C₂³×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₂

#30656614

commutator

C₂³×C₂₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

#413×C₂₅₇

C₂

socle

C₂×C₂₅₇

#386

prime radical

C₂×#41×C₂₅₇

C₂²

cosocle

C₂²×C₄×C₂₅₇

C₂³

D₂₅₇

C₂₅₇

G ▷ C₂³×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
3. C₂×#22551510	C₂	semidirect
4. #30656508	C₂	semidirect
80. C₂₅₇	#1915	semidirect

H ≅ G/N

type

3. C₂×#22551510

C₂

semidirect

4. #30656508

C₂

semidirect

80. C₂₅₇

#1915

semidirect

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

subgroup	index
#30657504	2
#30657308	2
C₂×#22551510	2
#30656508	2
#413×C₂₅₇	2
#1915	257

subgroup

index

#413×C₂₅₇

257

Subgroups — #45664870

Conjugacy classes of subgroups

Complements

Maximal subgroups