Group #15760442 — subgroups

Subgroups — #15760442

Order 3328.

384 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 103 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	#13123942	C₂	1	1	yes	maximal, characteristic
2	#13113553	C₂	1	2	yes	maximal
3	#13113474	C₂	1	1	yes	maximal, characteristic
4	#13112708	C₂	1	2	yes	maximal
5	C₁₃×#3139	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#533	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₁₃×#491	—	2	2	no	—
9	C₁₃×#487	C₄	1	4	yes	—
10	C₁₃×#419	—	2	2	no	—
11	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
12	#11718685	—	2	4	no	—
13	#11718661	—	2	2	no	—
14	#11718587	C₂²	1	2	yes	—
15	#11718587	C₂²	1	2	yes	—
16	#11718139	—	2	4	no	—
17	#11718135	—	2	2	no	—
18	C₂²×Q₈×C₁₃	—	2	1	no	—
19	C₂×C₁₃×M₁₆	—	2	1	no	—
20	C₁₃×C₄⋊Q₈	—	2	2	no	—
21	C₁₃×C₄⋊Q₈	—	2	1	no	—
22	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
23	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
24	C₁₃×#125	D₄	1	2	yes	—
25	C₁₃×C₂²⋊Q₈	—	2	2	no	—
26	C₄×Q₈×C₁₃	—	2	2	no	—
27	C₁₃×#117	D₄	1	2	yes	—
28	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
29	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
30	C₁₃×#103	—	2	8	no	—
31	C₁₃×#97	—	2	1	no	—
32	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
33	C₂²⋊#1055	—	2	2	no	—
34	C₂²⋊#1055	—	2	2	no	—
35	C₄⋊#1055	—	2	4	no	—
36	C₄⋊#1055	—	2	2	no	—
37	C₄⋊#1055	D₄	1	4	yes	—
38	#88043	—	2	2	no	—
39	#88043	—	2	2	no	—
40	#88043	—	2	2	no	—
41	C₄×#1055	—	2	2	no	—
42	#9422	—	13	1	no	maximal, Sylow
43	C₂×Q₈×C₁₃	—	2	2	no	—
44	C₂×Q₈×C₁₃	—	2	2	no	—
45	C₂×Q₈×C₁₃	—	4	4	no	—
46	C₂×Q₈×C₁₃	—	4	1	no	—
47	C₂²×C₄×C₁₃	—	4	1	no	—
48	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	#41	1	2	yes	—
50	C₁₃×M₁₆	—	4	2	no	—
51	C₂×C₈×C₁₃	—	2	2	no	—
52	C₁₃×C₄⋊C₄	—	2	2	no	—
53	C₁₃×C₄⋊C₄	—	2	4	no	—
54	C₁₃×C₄⋊C₄	—	2	4	no	—
55	C₁₃×C₄⋊C₄	—	4	2	no	—
56	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	4	yes	—
57	C₁₃×C₂²⋊C₄	—	2	2	no	—
58	C₄²×C₁₃	—	2	2	no	—
59	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₁₃	C₂×D₄	1	2	yes	—
61	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂×#1055	—	2	4	no	—
63	C₂×#1055	—	4	2	no	—
64	#3139	D₁₃	1	1	yes	characteristic
65	#2987	—	13	1	no	—
66	#1556	—	13	2	no	—
67	#1537	—	13	1	no	—
68	#1291	—	13	2	no	—
69	C₂³×C₁₃	#123	1	1	yes	characteristic
70	Q₈×C₁₃	—	4	4	no	—
71	Q₈×C₁₃	—	8	2	no	—
72	Q₈×C₁₃	—	8	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	4	no	—
75	C₂×C₄×C₁₃	—	4	2	no	—
76	C₂×C₄×C₁₃	—	4	2	no	—
77	C₂×C₄×C₁₃	—	8	2	no	—
78	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#121	1	2	yes	—
82	C₈×C₁₃	—	4	2	no	—
83	#1055	—	4	4	no	—
84	#1055	—	8	2	no	—
85	#533	C₂×D₁₃	1	1	yes	characteristic
86	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
87	#491	—	2	2	no	—
88	#487	Dic₁₃	1	4	yes	—
89	#446	—	26	4	no	—
90	#433	—	26	2	no	—
91	#432	—	13	2	no	—
92	#432	—	13	2	no	—
93	#419	—	2	2	no	—
94	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
95	#332	—	26	4	no	—
96	#330	—	26	2	no	—
97	C₂²×C₁₃	—	2	1	no	—
98	C₂²×C₁₃	—	2	2	no	—
99	C₂²×C₁₃	#386	1	1	yes	characteristic
100	C₄×C₁₃	—	2	1	no	—
101	C₄×C₁₃	—	2	2	no	—
102	C₄×C₁₃	—	2	2	no	—
103	C₄×C₁₃	—	4	4	no	—
104	C₄×C₁₃	—	8	2	no	—
105	C₄×C₁₃	#454	1	2	yes	—
106	C₂²×Q₈	—	2	1	no	—
107	C₂×M₁₆	—	2	1	no	—
108	C₄⋊Q₈	—	2	2	no	—
109	C₄⋊Q₈	—	2	1	no	—
110	C₄⋊Q₈	C₂×Dic₁₃	1	2	yes	—
111	C₄⋊Q₈	C₂×Dic₁₃	1	2	yes	—
112	#125	C₂²⋊D₁₃	1	2	yes	—
113	C₂²⋊Q₈	—	2	2	no	—
114	C₄×Q₈	—	2	2	no	—
115	#117	C₂²⋊D₁₃	1	2	yes	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
118	C₄⋊C₈	—	13	4	no	—
119	C₄⋊C₈	—	26	4	no	—
120	C₄⋊C₈	—	26	2	no	—
121	#103	—	2	8	no	—
122	C₂²⋊C₈	—	26	2	no	—
123	C₂²⋊C₈	—	26	2	no	—
124	#97	—	2	1	no	—
125	#97	—	26	2	no	—
126	#97	—	26	2	no	—
127	#97	—	26	2	no	—
128	C₄×C₈	—	26	2	no	—
129	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
130	C₂×C₁₃	—	4	1	no	—
131	C₂×C₁₃	#1888	1	1	yes	characteristic
132	C₂×C₁₃	#1902	1	2	yes	—
133	C₂×Q₈	—	2	2	no	—
134	C₂×Q₈	—	2	2	no	—
135	C₂×Q₈	—	4	4	no	—
136	C₂×Q₈	—	4	1	no	—
137	C₂²×C₄	—	4	1	no	—
138	C₂²×C₄	#6129	1	2	yes	—
139	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
140	M₁₆	—	4	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	26	4	no	—
143	C₂×C₈	—	52	2	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	C₂²⋊Dic₁₃	1	4	yes	—
149	C₂²⋊C₄	—	2	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	D₄×D₁₃	1	2	yes	—
152	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
153	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
154	C₁₃	#9422	1	1	yes	characteristic, complemented, Sylow
155	C₂³	#88189	1	1	yes	characteristic
156	Q₈	—	4	4	no	—
157	Q₈	—	8	2	no	—
158	Q₈	—	8	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
165	C₂×C₄	#88191	1	1	yes	characteristic
166	C₂×C₄	#88193	1	2	yes	—
167	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
168	C₈	—	4	2	no	—
169	C₈	—	52	4	no	—
170	C₈	—	104	2	no	—
171	C₂²	—	2	1	no	—
172	C₂²	—	2	2	no	—
173	C₂²	#11718881	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	4	4	no	—
178	C₄	—	8	2	no	—
179	C₄	#11718723	1	2	yes	—
180	C₂	—	4	1	no	—
181	C₂	#13115240	1	1	yes	characteristic
182	C₂	#13115311	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13123942

C₂

yes

maximal, characteristic

#13113553

C₂

yes

maximal

#13113474

C₂

yes

maximal, characteristic

#13112708

C₂

yes

maximal

C₁₃×#3139

C₂

yes

maximal, characteristic

C₁₃×#533

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊Q₈

C₂²

yes

characteristic

C₁₃×#491

—

C₁₃×#487

C₄

yes

—

C₁₃×#419

—

C₄×C₁₃×M₁₆

C₂²

yes

—

—

C₂²

yes

—

#11718587

C₂²

yes

—

#11718139

—

#11718135

—

C₂²×Q₈×C₁₃

—

C₂×C₁₃×M₁₆

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

C₂×C₄

yes

—

C₁₃×C₄⋊Q₈

C₂×C₄

yes

—

C₁₃×#125

D₄

yes

—

C₁₃×C₂²⋊Q₈

—

C₄×Q₈×C₁₃

—

C₁₃×#117

D₄

yes

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₁₃×#103

—

C₁₃×#97

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₂²⋊#1055

—

C₂²⋊#1055

—

C₄⋊#1055

—

C₄⋊#1055

—

C₄⋊#1055

D₄

yes

—

—

—

—

—

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂²⋊C₄

yes

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×D₄

yes

—

—

D₁₃

yes

—

—

—

—

C₂³×C₁₃

#123

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

—

C₂×C₄×C₁₃

#120

yes

characteristic

C₂×C₄×C₁₃

#121

yes

—

C₈×C₁₃

—

#1055

—

#1055

—

#533

C₂×D₁₃

yes

characteristic

C₂×C₄⋊Q₈

C₂×D₁₃

yes

characteristic

#491

—

#487

Dic₁₃

yes

—

—

—

—

—

—

C₄×M₁₆

C₂×D₁₃

yes

characteristic

#332

—

#330

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#386

yes

characteristic

100

C₄×C₁₃

—

101

C₄×C₁₃

—

102

C₄×C₁₃

—

103

C₄×C₁₃

—

104

C₄×C₁₃

—

105

C₄×C₁₃

#454

yes

—

106

C₂²×Q₈

—

107

C₂×M₁₆

—

108

C₄⋊Q₈

—

109

C₄⋊Q₈

—

110

C₄⋊Q₈

C₂×Dic₁₃

yes

—

111

C₄⋊Q₈

C₂×Dic₁₃

yes

—

112

#125

C₂²⋊D₁₃

yes

—

113

C₂²⋊Q₈

—

114

C₄×Q₈

—

115

#117

C₂²⋊D₁₃

yes

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄²

C₂²×D₁₃

yes

characteristic

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

#103

—

122

C₂²⋊C₈

—

123

C₂²⋊C₈

—

124

#97

—

125

#97

—

126

#97

—

127

#97

—

128

C₄×C₈

—

129

C₄×C₈

C₂×Dic₁₃

yes

—

130

C₂×C₁₃

—

131

C₂×C₁₃

#1888

yes

characteristic

132

C₂×C₁₃

#1902

yes

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂²×C₄

—

138

C₂²×C₄

#6129

yes

—

139

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

140

M₁₆

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

C₂²⋊Dic₁₃

yes

—

149

C₂²⋊C₄

—

150

C₄²

—

151

C₄²

D₄×D₁₃

yes

—

152

C₄²

C₂²×Dic₁₃

yes

characteristic

153

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

154

C₁₃

#9422

yes

characteristic, complemented, Sylow

155

C₂³

#88189

yes

characteristic

156

Q₈

—

157

Q₈

—

158

Q₈

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

D₄×Dic₁₃

yes

—

165

C₂×C₄

#88191

yes

characteristic

166

C₂×C₄

#88193

yes

—

167

C₂×C₄

C₂×C₂²⋊Dic₁₃

yes

characteristic

168

C₈

—

169

C₈

—

170

C₈

—

104

—

171

C₂²

—

172

C₂²

—

173

C₂²

#11718881

yes

characteristic

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

#11718723

yes

—

180

C₂

—

181

C₂

#13115240

yes

characteristic

182

C₂

#13115311

yes

—

	subgroup	G/H
center	C₂²	#11718881
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#3139	C₂
socle	C₂²×C₁₃	#386
prime radical	C₂³×C₁₃	#123
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#3139	D₁₃
hypocenter	C₁₃	#9422
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3139

subgroup

G/H

center

C₂²

#11718881

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#3139

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3139

N	H ≅ G/N	type
154. C₁₃	#9422	semidirect

H ≅ G/N

type

154. C₁₃

#9422

semidirect

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

subgroup	index
#13123942	2
#13113553	2
#13113474	2
#13112708	2
C₁₃×#3139	2
#9422	13

subgroup

index

C₁₃×#3139

Subgroups — #15760442

Conjugacy classes of subgroups

Complements

Maximal subgroups