Group #15761538 — subgroups

Subgroups — #15761538

Order 3328.

294 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₄×#88059	C₂	1	1	yes	maximal, characteristic
2	#13113586	C₂	1	2	yes	maximal
3	#13113128	C₂	1	1	yes	maximal, characteristic
4	#13112686	C₂	1	2	yes	maximal
5	C₂×C₁₃×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88059	C₄	1	2	yes	—
9	#11718586	C₂²	1	1	yes	characteristic
10	#11718586	C₂²	1	1	yes	characteristic
11	C₄×#6099	C₂²	1	2	yes	—
12	C₁₃×#334	—	2	2	no	—
13	#11718144	C₄	1	1	yes	characteristic
14	#11718144	C₄	1	1	yes	characteristic
15	#11718072	—	2	2	no	—
16	#11718071	—	2	2	no	—
17	#11718046	—	2	2	no	—
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×#6099	C₂×C₄	1	2	yes	—
23	C₂×#6099	C₂×C₄	1	2	yes	—
24	C₁₃×C₄⋊C₈	—	2	2	no	—
25	C₁₃×C₄⋊C₈	—	2	2	no	—
26	C₄×C₈×C₁₃	D₄	1	2	yes	—
27	C₄×C₈×C₁₃	Q₈	1	2	yes	—
28	C₂²⋊#1055	—	2	1	no	—
29	C₂²⋊#1055	Q₈	1	2	yes	—
30	#88059	—	2	4	no	—
31	C₄⋊#1055	—	2	1	no	—
32	C₄⋊#1055	D₄	1	2	yes	—
33	#88043	—	2	2	no	—
34	C₄×#1055	—	2	1	no	—
35	C₄×#1055	—	2	1	no	—
36	C₄×#1055	—	2	2	no	—
37	#9609	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
41	C₂×C₈×C₁₃	—	2	2	no	—
42	C₂×C₈×C₁₃	—	2	2	no	—
43	C₂×C₈×C₁₃	—	4	2	no	—
44	C₂×C₈×C₁₃	—	8	2	no	—
45	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
47	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₁₃	#41	1	1	yes	characteristic
51	#6099	—	2	2	no	—
52	#6099	—	2	2	no	—
53	#6099	—	2	2	no	—
54	#6099	M₁₆	1	4	yes	—
55	C₂×#1055	—	2	2	no	—
56	C₂×#1055	—	2	2	no	—
57	C₂×#1055	—	2	2	no	—
58	C₂×#1055	—	2	2	no	—
59	C₄×#108	—	13	1	no	—
60	#1565	—	13	2	no	—
61	C₂×#334	D₁₃	1	1	yes	characteristic
62	#1452	—	13	1	no	—
63	#1283	—	13	2	no	—
64	C₂³×C₁₃	C₄×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	4	1	no	—
67	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₁₃	#131	1	2	yes	—
75	C₈×C₁₃	—	2	2	no	—
76	C₈×C₁₃	—	2	2	no	—
77	C₈×C₁₃	—	8	2	no	—
78	#1055	—	2	4	no	—
79	#1055	—	4	2	no	—
80	#1055	—	4	2	no	—
81	#1055	—	4	2	no	—
82	#431	—	13	1	no	—
83	#431	—	13	1	no	—
84	C₂×#108	—	13	2	no	—
85	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
86	C₄×M₁₆	—	13	2	no	—
87	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
88	#341	—	13	1	no	—
89	#341	—	13	1	no	—
90	#335	—	26	2	no	—
91	#334	—	2	2	no	—
92	#334	—	26	2	no	—
93	#322	—	26	2	no	—
94	C₂²×C₁₃	—	2	1	no	—
95	C₂²×C₁₃	—	2	1	no	—
96	C₂²×C₁₃	#389	1	1	yes	characteristic
97	C₂²×C₁₃	#446	1	2	yes	—
98	C₄×C₁₃	—	2	1	no	—
99	C₄×C₁₃	—	2	1	no	—
100	C₄×C₁₃	—	2	1	no	—
101	C₄×C₁₃	#423	1	1	yes	characteristic
102	C₄×C₁₃	#424	1	1	yes	characteristic
103	C₄×C₁₃	#435	1	2	yes	—
104	C₄×C₁₃	#471	1	2	yes	—
105	C₂×M₁₆	—	13	2	no	—
106	C₂×M₁₆	—	13	2	no	—
107	C₂²×C₈	—	2	2	no	—
108	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
109	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
110	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
111	#108	—	26	4	no	—
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	13	2	no	—
115	C₄⋊C₈	—	26	1	no	—
116	C₂²⋊C₈	—	13	2	no	—
117	C₂²⋊C₈	—	26	1	no	—
118	#97	—	26	2	no	—
119	C₄×C₈	—	26	2	no	—
120	C₄×C₈	—	26	1	no	—
121	C₄×C₈	—	26	1	no	—
122	C₄×C₈	Dic₂₆	1	2	yes	—
123	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
124	C₂×C₁₃	—	2	2	no	—
125	C₂×C₁₃	#1940	1	1	yes	characteristic
126	C₂×C₁₃	#1951	1	1	yes	characteristic
127	C₂×C₁₃	#1953	1	1	yes	characteristic
128	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
129	C₂²×C₄	#6129	1	1	yes	characteristic
130	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
131	M₁₆	—	13	4	no	—
132	M₁₆	—	26	2	no	—
133	M₁₆	—	26	2	no	—
134	M₁₆	—	26	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	8	2	no	—
139	C₂×C₈	—	26	2	no	—
140	C₂×C₈	—	26	2	no	—
141	C₂×C₈	—	26	2	no	—
142	C₂×C₈	—	26	2	no	—
143	C₂×C₈	#6101	1	2	yes	—
144	C₂×C₈	#6101	1	2	yes	—
145	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
146	C₄²	D₄×D₁₃	1	1	yes	characteristic
147	C₄²	#6131	1	1	yes	characteristic
148	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
149	C₁₃	#9609	1	1	yes	characteristic, complemented, Sylow
150	C₂³	#88141	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	#88142	1	1	yes	characteristic
154	C₂×C₄	#88144	1	1	yes	characteristic
155	C₂×C₄	#88147	1	1	yes	characteristic
156	C₂×C₄	#88155	1	2	yes	—
157	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
158	C₂×C₄	C₂×#6101	1	1	yes	characteristic
159	C₂×C₄	#88192	1	1	yes	characteristic
160	C₂×C₄	#88200	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	—	8	2	no	—
164	C₈	—	26	4	no	—
165	C₈	—	52	2	no	—
166	C₈	—	52	2	no	—
167	C₈	—	52	2	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#11718489	1	2	yes	—
171	C₂²	#11718634	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	#11718497	1	2	yes	—
176	C₄	#11718763	1	1	yes	characteristic
177	C₄	#11718779	1	1	yes	characteristic
178	C₄	#11718838	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	#13115557	1	1	yes	characteristic
181	C₂	#13115614	1	1	yes	characteristic
182	C₂	#13115624	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#88059

C₂

yes

maximal, characteristic

#13113586

C₂

yes

maximal

#13113128

C₂

yes

maximal, characteristic

#13112686

C₂

yes

maximal

C₂×C₁₃×#334

C₂

yes

maximal, characteristic

C₂×C₁₃×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88059

C₄

yes

—

#11718586

C₂²

yes

characteristic

#11718586

C₂²

yes

characteristic

C₄×#6099

C₂²

yes

—

C₁₃×#334

—

#11718144

C₄

yes

characteristic

#11718144

C₄

yes

—

—

—

C₂²×C₈×C₁₃

—

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×#6099

C₂×C₄

yes

—

C₂×#6099

C₂×C₄

yes

—

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

C₂²⋊#1055

—

C₂²⋊#1055

Q₈

yes

—

#88059

—

C₄⋊#1055

—

C₄⋊#1055

D₄

yes

—

—

—

—

—

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

C₄²×C₁₃

yes

—

—

—

M₁₆

yes

—

—

—

—

—

—

—

D₁₃

yes

characteristic

#1452

—

#1283

—

C₂³×C₁₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

#125

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

—

—

—

—

—

—

—

C₂×C₄⋊C₈

C₂×D₁₃

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

—

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#389

yes

characteristic

C₂²×C₁₃

#446

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

100

C₄×C₁₃

—

101

C₄×C₁₃

#423

yes

characteristic

102

C₄×C₁₃

#424

yes

characteristic

103

C₄×C₁₃

#435

yes

—

104

C₄×C₁₃

#471

yes

—

105

C₂×M₁₆

—

106

C₂×M₁₆

—

107

C₂²×C₈

—

108

C₂²×C₈

C₄×D₁₃

yes

characteristic

109

C₂²×C₈

C₄×D₁₃

yes

characteristic

110

C₂×C₄²

C₂²×D₁₃

yes

characteristic

111

#108

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

C₂²⋊C₈

—

117

C₂²⋊C₈

—

118

#97

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

Dic₂₆

yes

—

123

C₄×C₈

C₂²⋊D₁₃

yes

—

124

C₂×C₁₃

—

125

C₂×C₁₃

#1940

yes

characteristic

126

C₂×C₁₃

#1951

yes

characteristic

127

C₂×C₁₃

#1953

yes

characteristic

128

C₂²×C₄

C₂×C₄×D₁₃

yes

characteristic

129

C₂²×C₄

#6129

yes

characteristic

130

C₂²×C₄

Q₈×D₁₃

yes

characteristic

131

M₁₆

—

132

M₁₆

—

133

M₁₆

—

134

M₁₆

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

#6101

yes

—

144

C₂×C₈

#6101

yes

—

145

C₄²

C₂×Dic₂₆

yes

characteristic

146

C₄²

D₄×D₁₃

yes

characteristic

147

C₄²

#6131

yes

characteristic

148

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

149

C₁₃

#9609

yes

characteristic, complemented, Sylow

150

C₂³

#88141

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

#88142

yes

characteristic

154

C₂×C₄

#88144

yes

characteristic

155

C₂×C₄

#88147

yes

characteristic

156

C₂×C₄

#88155

yes

—

157

C₂×C₄

M₁₆×D₁₃

yes

—

158

C₂×C₄

C₂×#6101

yes

characteristic

159

C₂×C₄

#88192

yes

characteristic

160

C₂×C₄

#88200

yes

characteristic

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

166

C₈

—

167

C₈

—

168

C₂²

—

169

C₂²

—

170

C₂²

#11718489

yes

—

171

C₂²

#11718634

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

#11718497

yes

—

176

C₄

#11718763

yes

characteristic

177

C₄

#11718779

yes

characteristic

178

C₄

#11718838

yes

—

179

C₂

—

180

C₂

#13115557

yes

characteristic

181

C₂

#13115614

yes

characteristic

182

C₂

#13115624

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#88192
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#334	C₂
socle	C₂²×C₁₃	#389
prime radical	C₂³×C₁₃	C₄×Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#334	D₁₃
hypocenter	C₁₃	#9609
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

subgroup

G/H

center

C₂×C₄

#88192

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₂×C₁₃×#334

C₂

socle

C₂²×C₁₃

#389

prime radical

C₂³×C₁₃

C₄×Q₈

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

N	H ≅ G/N	type
149. C₁₃	#9609	semidirect

H ≅ G/N

type

149. C₁₃

#9609

semidirect

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

subgroup	index
C₄×#88059	2
#13113586	2
#13113128	2
#13112686	2
C₂×C₁₃×#334	2
#9609	13

subgroup

index

C₂×C₁₃×#334

Subgroups — #15761538

Conjugacy classes of subgroups

Complements

Maximal subgroups