Group #15750140 — subgroups

Subgroups — #15750140

Order 3328.

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

normal subgroups only

This group has 322 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	#13113154	C₂	1	2	yes	maximal
2	C₁₃×#1955	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
4	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
5	C₂×C₁₃×C₂²⋊C₈	—	2	2	no	—
6	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
7	C₂×C₂²⋊#1055	—	2	2	no	—
8	#11718120	—	2	2	no	—
9	#11718117	C₄	1	4	yes	—
10	C₂²×D₄×C₁₃	Q₈	1	1	yes	characteristic
11	C₂³×C₄×C₁₃	D₄	1	2	yes	—
12	C₂²×C₈×C₁₃	—	2	1	no	—
13	C₂²×C₈×C₁₃	—	2	1	no	—
14	C₄×D₄×C₁₃	—	4	1	no	—
15	C₄×D₄×C₁₃	—	4	1	no	—
16	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
17	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×#1055	—	4	2	no	—
20	C₁₃×C₄⋊C₈	—	4	1	no	—
21	C₁₃×C₂²⋊C₈	—	4	4	no	—
22	C₁₃×#97	—	4	1	no	—
23	C₂²⋊#1055	—	2	4	no	—
24	C₂²⋊#1055	—	4	2	no	—
25	#7287	—	13	1	no	maximal, Sylow
26	C₂⁴×C₁₃	C₄⋊C₄	1	2	yes	—
27	C₂×D₄×C₁₃	—	2	4	no	—
28	C₂×D₄×C₁₃	—	4	1	no	—
29	C₂²×C₄×C₁₃	—	2	2	no	—
30	C₂²×C₄×C₁₃	—	2	2	no	—
31	C₂²×C₄×C₁₃	—	4	2	no	—
32	C₂²×C₄×C₁₃	—	4	2	no	—
33	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃	—	2	2	no	—
38	C₂×C₈×C₁₃	—	4	1	no	—
39	C₂×C₈×C₁₃	—	4	1	no	—
40	C₂×C₈×C₁₃	—	4	2	no	—
41	C₂×C₈×C₁₃	—	4	1	no	—
42	C₁₃×C₄⋊C₄	—	4	1	no	—
43	C₁₃×C₂²⋊C₄	—	4	2	no	—
44	C₄²×C₁₃	—	2	1	no	—
45	C₄²×C₁₃	—	2	1	no	—
46	C₂×#1055	—	4	4	no	—
47	C₂×#1055	—	8	2	no	—
48	C₂×#1055	—	8	2	no	—
49	#1955	D₁₃	1	1	yes	characteristic
50	#1459	—	13	2	no	—
51	C₂³×C₁₃	—	2	2	no	—
52	C₂³×C₁₃	—	2	2	no	—
53	C₂³×C₁₃	—	4	2	no	—
54	C₂³×C₁₃	—	4	2	no	—
55	C₂³×C₁₃	#95	1	1	yes	characteristic
56	C₂³×C₁₃	#108	1	4	yes	—
57	D₄×C₁₃	—	4	4	no	—
58	C₂×C₄×C₁₃	—	2	1	no	—
59	C₂×C₄×C₁₃	—	2	1	no	—
60	C₂×C₄×C₁₃	—	2	2	no	—
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	1	no	—
63	C₂×C₄×C₁₃	—	2	1	no	—
64	C₂×C₄×C₁₃	—	4	2	no	—
65	C₂×C₄×C₁₃	—	4	2	no	—
66	C₂×C₄×C₁₃	—	4	2	no	—
67	C₂×C₄×C₁₃	—	4	1	no	—
68	C₂×C₄×C₁₃	—	8	2	no	—
69	C₂×C₄×C₁₃	#99	1	1	yes	characteristic
70	C₂×C₄×C₁₃	#99	1	1	yes	characteristic
71	C₈×C₁₃	—	4	2	no	—
72	C₈×C₁₃	—	8	1	no	—
73	#1055	—	8	4	no	—
74	C₂×C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
75	C₂×C₄⋊C₈	Dic₁₃	1	1	yes	characteristic
76	C₂×C₂²⋊C₈	—	2	2	no	—
77	C₂×C₂²⋊C₈	—	26	2	no	—
78	C₂×#97	Dic₁₃	1	1	yes	characteristic
79	#324	—	26	2	no	—
80	#323	—	13	4	no	—
81	C₂²×C₁₃	—	2	1	no	—
82	C₂²×C₁₃	—	2	1	no	—
83	C₂²×C₁₃	—	2	4	no	—
84	C₂²×C₁₃	—	4	4	no	—
85	C₂²×C₁₃	—	4	2	no	—
86	C₂²×C₁₃	—	4	2	no	—
87	C₂²×C₁₃	—	8	2	no	—
88	C₂²×C₁₃	#341	1	2	yes	—
89	C₂²×C₁₃	#342	1	1	yes	characteristic
90	C₄×C₁₃	—	2	1	no	—
91	C₄×C₁₃	—	4	1	no	—
92	C₄×C₁₃	—	4	1	no	—
93	C₄×C₁₃	—	4	2	no	—
94	C₄×C₁₃	#353	1	1	yes	characteristic
95	C₄×C₁₃	#354	1	1	yes	characteristic
96	C₂²×D₄	Dic₂₆	1	1	yes	characteristic
97	C₂³×C₄	C₂²⋊D₁₃	1	2	yes	—
98	C₂²×C₈	—	2	1	no	—
99	C₂²×C₈	—	2	1	no	—
100	C₂²×C₈	—	52	2	no	—
101	C₄×D₄	—	4	1	no	—
102	C₄×D₄	—	4	1	no	—
103	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
104	C₂×C₂²⋊C₄	C₄×D₁₃	1	2	yes	—
105	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
106	C₄⋊C₈	—	4	1	no	—
107	C₂²⋊C₈	—	4	4	no	—
108	C₂²⋊C₈	—	26	4	no	—
109	C₂²⋊C₈	—	52	2	no	—
110	#97	—	4	1	no	—
111	C₂×C₁₃	—	2	2	no	—
112	C₂×C₁₃	—	4	4	no	—
113	C₂×C₁₃	#1298	1	1	yes	characteristic
114	C₂×C₁₃	#1392	1	1	yes	characteristic
115	C₂×C₁₃	#1393	1	1	yes	characteristic
116	C₂⁴	#6101	1	2	yes	—
117	C₂×D₄	—	2	4	no	—
118	C₂×D₄	—	4	1	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
124	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
125	C₂²×C₄	#6103	1	2	yes	—
126	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	4	1	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	52	4	no	—
133	C₂×C₈	—	104	2	no	—
134	C₂×C₈	—	104	2	no	—
135	C₄⋊C₄	—	4	1	no	—
136	C₂²⋊C₄	—	4	2	no	—
137	C₄²	—	2	1	no	—
138	C₄²	—	2	1	no	—
139	C₁₃	#7287	1	1	yes	characteristic, complemented, Sylow
140	C₂³	—	2	2	no	—
141	C₂³	—	2	2	no	—
142	C₂³	—	4	2	no	—
143	C₂³	—	4	2	no	—
144	C₂³	#88062	1	4	yes	—
145	C₂³	#88071	1	1	yes	characteristic
146	D₄	—	4	4	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	#88074	1	1	yes	characteristic
159	C₂×C₄	#88074	1	1	yes	characteristic
160	C₈	—	4	2	no	—
161	C₈	—	8	1	no	—
162	C₈	—	104	4	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	4	no	—
166	C₂²	—	4	4	no	—
167	C₂²	—	4	2	no	—
168	C₂²	—	4	2	no	—
169	C₂²	—	8	2	no	—
170	C₂²	#11718118	1	1	yes	characteristic
171	C₂²	#11718150	1	2	yes	—
172	C₄	—	2	1	no	—
173	C₄	—	4	1	no	—
174	C₄	—	4	2	no	—
175	C₄	—	4	1	no	—
176	C₄	#11718133	1	1	yes	characteristic
177	C₄	#11718137	1	1	yes	characteristic
178	C₂	—	2	2	no	—
179	C₂	—	4	4	no	—
180	C₂	#13112723	1	1	yes	characteristic
181	C₂	#13112959	1	1	yes	characteristic
182	C₂	#13112961	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13113154

C₂

yes

maximal

C₁₃×#1955

C₂

yes

maximal, characteristic

C₂×C₄×D₄×C₁₃

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₈

C₄

yes

characteristic

C₂×C₁₃×C₂²⋊C₈

—

C₂×C₁₃×#97

C₄

yes

characteristic

C₂×C₂²⋊#1055

—

#11718120

—

#11718117

C₄

yes

—

C₂²×D₄×C₁₃

Q₈

yes

characteristic

C₂³×C₄×C₁₃

D₄

yes

—

C₂²×C₈×C₁₃

—

C₂²×C₈×C₁₃

—

C₄×D₄×C₁₃

—

C₄×D₄×C₁₃

—

C₂×C₁₃×C₄⋊C₄

D₄

yes

characteristic

C₂×C₁₃×C₂²⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂²×#1055

—

C₁₃×C₄⋊C₈

—

C₁₃×C₂²⋊C₈

—

C₁₃×#97

—

C₂²⋊#1055

—

C₂²⋊#1055

—

#7287

—

maximal, Sylow

C₂⁴×C₁₃

C₄⋊C₄

yes

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₄²

yes

characteristic

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

—

—

—

—

D₁₃

yes

characteristic

#1459

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

#95

yes

characteristic

C₂³×C₁₃

#108

yes

—

D₄×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

yes

C₂×C₄×C₁₃

yes

C₈×C₁₃

—

C₈×C₁₃

—

#1055

—

C₂×C₄×D₄

C₂×D₁₃

yes

characteristic

C₂×C₄⋊C₈

Dic₁₃

yes

characteristic

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×#97

Dic₁₃

yes

characteristic

#324

—

#323

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#341

yes

—

C₂²×C₁₃

#342

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₂²×D₄

Dic₂₆

yes

characteristic

C₂³×C₄

C₂²⋊D₁₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

100

C₂²×C₈

—

101

C₄×D₄

—

102

C₄×D₄

—

103

C₂×C₄⋊C₄

D₅₂

yes

characteristic

104

C₂×C₂²⋊C₄

C₄×D₁₃

yes

—

105

C₂×C₄²

C₂×Dic₁₃

yes

characteristic

106

C₄⋊C₈

—

107

C₂²⋊C₈

—

108

C₂²⋊C₈

—

109

C₂²⋊C₈

—

110

#97

—

111

C₂×C₁₃

—

112

C₂×C₁₃

—

113

C₂×C₁₃

#1298

yes

characteristic

114

C₂×C₁₃

#1392

yes

characteristic

115

C₂×C₁₃

#1393

yes

characteristic

116

C₂⁴

#6101

yes

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₄×Dic₁₃

yes

characteristic

124

C₂²×C₄

C₄⋊Dic₁₃

yes

characteristic

125

C₂²×C₄

#6103

yes

—

126

C₂²×C₄

C₂²⋊Dic₁₃

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

104

—

134

C₂×C₈

—

104

—

135

C₄⋊C₄

—

136

C₂²⋊C₄

—

137

C₄²

—

138

C₄²

—

139

C₁₃

#7287

yes

characteristic, complemented, Sylow

140

C₂³

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

#88062

yes

—

145

C₂³

#88071

yes

characteristic

146

D₄

—

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

—

157

C₂×C₄

—

158

C₂×C₄

#88074

yes

characteristic

159

C₂×C₄

#88074

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

104

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

#11718118

yes

characteristic

171

C₂²

#11718150

yes

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

#11718133

yes

characteristic

177

C₄

#11718137

yes

characteristic

178

C₂

—

179

C₂

—

180

C₂

#13112723

yes

characteristic

181

C₂

#13112959

yes

characteristic

182

C₂

#13112961

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#88074
commutator	C₂²×C₄×C₁₃	C₄²
Frattini	C₂×C₄×D₄	C₂×D₁₃
Fitting	C₁₃×#1955	C₂
socle	C₂²×C₁₃	#342
prime radical	C₂²×D₄×C₁₃	Q₈
cosocle	C₂×C₄×D₄×C₁₃	C₂²
hypercenter	#1955	D₁₃
hypocenter	C₁₃	#7287
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ #1955

subgroup

G/H

center

C₂×C₄

#88074

commutator

C₂²×C₄×C₁₃

C₄²

Frattini

C₂×C₄×D₄

C₂×D₁₃

Fitting

C₁₃×#1955

C₂

socle

C₂²×C₁₃

#342

prime radical

C₂²×D₄×C₁₃

Q₈

cosocle

C₂×C₄×D₄×C₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ #1955

N	H ≅ G/N	type
139. C₁₃	#7287	semidirect

H ≅ G/N

type

139. C₁₃

#7287

semidirect

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

subgroup	index
#13113154	2
C₁₃×#1955	2
#7287	13

subgroup

index

#13113154

C₁₃×#1955

#7287

Subgroups — #15750140

Conjugacy classes of subgroups

Complements

Maximal subgroups