Group #15751237 — subgroups

Subgroups — #15751237

Order 3328.

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

normal subgroups only

This group has 384 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	#13114979	C₂	1	2	yes	maximal
2	#13112659	C₂	1	4	yes	maximal
3	C₄×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#125	C₄	1	1	yes	characteristic
5	C₁₃×#389	C₂²	1	2	yes	—
6	C₁₃×#387	—	2	2	no	—
7	C₁₃×#382	C₄	1	1	yes	characteristic
8	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
9	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
10	C₄³×C₁₃	C₂²	1	1	yes	characteristic
11	#11718116	—	2	2	no	—
12	#11718116	—	2	2	no	—
13	#11718116	C₂²	1	4	yes	—
14	C₁₃×#125	—	2	4	no	—
15	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
16	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
17	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
18	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₁₃	—	2	2	no	—
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	D₄	1	2	yes	—
23	C₂×C₄²×C₁₃	D₄	1	2	yes	—
24	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
25	C₂²×#1055	—	2	4	no	—
26	C₁₃×#95	—	2	2	no	—
27	C₁₃×#95	—	2	2	no	—
28	C₁₃×#95	—	2	2	no	—
29	#7650	—	13	1	no	maximal, Sylow
30	C₂²×C₄×C₁₃	—	2	2	no	—
31	C₂²×C₄×C₁₃	—	2	2	no	—
32	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
37	C₁₃×C₄⋊C₄	—	2	8	no	—
38	C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₁₃×C₄⋊C₄	—	4	2	no	—
40	C₁₃×C₄⋊C₄	—	4	4	no	—
41	C₁₃×C₄⋊C₄	M₁₆	1	4	yes	—
42	C₄²×C₁₃	—	2	2	no	—
43	C₄²×C₁₃	—	2	2	no	—
44	C₄²×C₁₃	—	2	4	no	—
45	C₄²×C₁₃	—	4	2	no	—
46	C₄²×C₁₃	—	4	1	no	—
47	C₄²×C₁₃	—	4	4	no	—
48	C₂×#1055	—	4	4	no	—
49	C₂×#1055	—	4	4	no	—
50	C₂×#1055	—	4	4	no	—
51	C₄×#125	D₁₃	1	1	yes	characteristic
52	#1839	—	13	2	no	—
53	#1271	—	13	4	no	—
54	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	—	2	1	no	—
57	C₂×C₄×C₁₃	—	2	1	no	—
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	—	2	2	no	—
60	C₂×C₄×C₁₃	—	2	2	no	—
61	C₂×C₄×C₁₃	—	2	4	no	—
62	C₂×C₄×C₁₃	—	2	1	no	—
63	C₂×C₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	4	2	no	—
67	C₂×C₄×C₁₃	—	4	2	no	—
68	C₂×C₄×C₁₃	—	4	2	no	—
69	C₂×C₄×C₁₃	—	4	2	no	—
70	C₂×C₄×C₁₃	—	4	2	no	—
71	C₂×C₄×C₁₃	#101	1	2	yes	—
72	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
73	C₂×C₄×C₁₃	#135	1	4	yes	—
74	#1055	—	8	4	no	—
75	C₂×#125	Dic₁₃	1	1	yes	characteristic
76	#389	C₂×D₁₃	1	2	yes	—
77	#387	—	2	2	no	—
78	#382	Dic₁₃	1	1	yes	characteristic
79	C₄×C₄⋊C₄	—	2	2	no	—
80	C₄×C₄⋊C₄	Dic₁₃	1	2	yes	—
81	C₄³	C₂×D₁₃	1	1	yes	characteristic
82	#336	—	13	4	no	—
83	#336	—	26	2	no	—
84	#336	—	26	2	no	—
85	C₂²×C₁₃	#407	1	1	yes	characteristic
86	C₂²×C₁₃	#410	1	1	yes	characteristic
87	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
88	C₂²×C₁₃	#416	1	2	yes	—
89	C₂²×C₁₃	#421	1	2	yes	—
90	C₄×C₁₃	—	2	2	no	—
91	C₄×C₁₃	—	2	4	no	—
92	C₄×C₁₃	—	2	2	no	—
93	C₄×C₁₃	—	4	1	no	—
94	C₄×C₁₃	—	4	2	no	—
95	C₄×C₁₃	—	4	4	no	—
96	C₄×C₁₃	—	8	2	no	—
97	C₂²×C₈	—	26	4	no	—
98	#125	—	2	4	no	—
99	C₂×C₄⋊C₄	—	2	2	no	—
100	C₂×C₄⋊C₄	—	2	2	no	—
101	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
102	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
103	C₂×C₄²	—	2	2	no	—
104	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
105	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
106	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
107	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
108	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
109	#95	—	2	2	no	—
110	#95	—	2	2	no	—
111	#95	—	2	2	no	—
112	C₂×C₁₃	#1467	1	1	yes	characteristic
113	C₂×C₁₃	#1488	1	2	yes	—
114	C₂×C₁₃	#1521	1	2	yes	—
115	C₂×C₁₃	#1534	1	1	yes	characteristic
116	C₂×C₁₃	#1544	1	1	yes	characteristic
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
120	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
121	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
122	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
124	C₂×C₈	—	52	4	no	—
125	C₂×C₈	—	52	4	no	—
126	C₂×C₈	—	52	4	no	—
127	C₄⋊C₄	—	2	8	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	4	4	no	—
131	C₄⋊C₄	#6099	1	4	yes	—
132	C₄²	—	2	2	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	4	no	—
135	C₄²	—	4	2	no	—
136	C₄²	—	4	1	no	—
137	C₄²	—	4	4	no	—
138	C₁₃	#7650	1	1	yes	characteristic, complemented, Sylow
139	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	#88076	1	2	yes	—
157	C₂×C₄	C₂×#6099	1	2	yes	—
158	C₂×C₄	#88204	1	4	yes	—
159	C₈	—	104	4	no	—
160	C₂²	#11718868	1	1	yes	characteristic
161	C₂²	C₂×#88076	1	1	yes	characteristic
162	C₂²	#11718894	1	2	yes	—
163	C₂²	#11718897	1	2	yes	—
164	C₂²	#11718901	1	1	yes	characteristic
165	C₄	—	2	4	no	—
166	C₄	—	2	2	no	—
167	C₄	—	2	2	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	4	no	—
171	C₄	—	8	2	no	—
172	C₂	#13113180	1	1	yes	characteristic
173	C₂	#13113270	1	2	yes	—
174	C₂	#13113413	1	2	yes	—
175	C₂	#13113463	1	1	yes	characteristic
176	C₂	#13113501	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114979

C₂

yes

maximal

#13112659

C₂

yes

maximal

C₄×C₁₃×#125

C₂

yes

maximal, characteristic

C₂×C₁₃×#125

C₄

yes

characteristic

C₁₃×#389

C₂²

yes

—

C₁₃×#387

—

C₁₃×#382

C₄

yes

characteristic

C₄×C₁₃×C₄⋊C₄

—

C₄×C₁₃×C₄⋊C₄

C₄

yes

—

C₄³×C₁₃

C₂²

yes

—

—

C₂²

yes

—

C₁₃×#125

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂²×#1055

—

C₁₃×#95

—

C₁₃×#95

—

C₁₃×#95

—

#7650

—

maximal, Sylow

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

M₁₆

yes

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

—

—

—

D₁₃

yes

characteristic

#1839

—

#1271

—

C₂³×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#101

yes

—

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₂×C₄×C₁₃

#135

yes

—

#1055

—

C₂×#125

Dic₁₃

yes

characteristic

#389

C₂×D₁₃

yes

—

#387

—

#382

Dic₁₃

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

Dic₁₃

yes

—

C₄³

C₂×D₁₃

yes

—

—

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

#421

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

C₂×Dic₁₃

yes

—

102

C₂×C₄⋊C₄

C₂×Dic₁₃

yes

—

103

C₂×C₄²

—

104

C₂×C₄²

C₂×Dic₁₃

yes

characteristic

105

C₂×C₄²

C₂×Dic₁₃

yes

characteristic

106

C₂×C₄²

C₂²⋊D₁₃

yes

—

107

C₂×C₄²

C₂²⋊D₁₃

yes

—

108

C₂×C₄²

C₂²×D₁₃

yes

characteristic

109

#95

—

110

#95

—

111

#95

—

112

C₂×C₁₃

#1467

yes

characteristic

113

C₂×C₁₃

#1488

yes

—

114

C₂×C₁₃

#1521

yes

—

115

C₂×C₁₃

#1534

yes

characteristic

116

C₂×C₁₃

#1544

yes

characteristic

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂²⋊Dic₁₃

yes

characteristic

120

C₂²×C₄

C₂²⋊Dic₁₃

yes

—

121

C₂²×C₄

C₂²⋊Dic₁₃

yes

characteristic

122

C₂²×C₄

C₂²×Dic₁₃

yes

characteristic

123

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

#6099

yes

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₁₃

#7650

yes

characteristic, complemented, Sylow

139

C₂³

C₂×C₂²⋊Dic₁₃

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

#88076

yes

—

157

C₂×C₄

C₂×#6099

yes

—

158

C₂×C₄

#88204

yes

—

159

C₈

—

104

—

160

C₂²

#11718868

yes

characteristic

161

C₂²

C₂×#88076

yes

characteristic

162

C₂²

#11718894

yes

—

163

C₂²

#11718897

yes

—

164

C₂²

#11718901

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₂

#13113180

yes

characteristic

173

C₂

#13113270

yes

—

174

C₂

#13113413

yes

—

175

C₂

#13113463

yes

characteristic

176

C₂

#13113501

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#125	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#125	D₁₃
hypocenter	C₁₃	#7650
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₁₃

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₄×C₁₃×#125

C₂

socle, prime radical

C₂³×C₁₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
138. C₁₃	#7650	semidirect

H ≅ G/N

type

138. C₁₃

#7650

semidirect

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

subgroup	index
#13114979	2
#13112659	2
C₄×C₁₃×#125	2
#7650	13

subgroup

index

#13114979

#13112659

C₄×C₁₃×#125

#7650

Subgroups — #15751237

Conjugacy classes of subgroups

Complements

Maximal subgroups