Group #15767697 — subgroups

Subgroups — #15767697

Order 3328.

310 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 87 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	#13114956	C₂	1	1	yes	maximal, characteristic, complemented
2	#13113485	C₂	1	2	yes	maximal
3	#13112731	C₂	1	2	yes	maximal, complemented
4	#13112665	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1963	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#532	C₂²	1	1	yes	characteristic
7	C₁₃×#423	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#104	—	2	2	no	—
9	C₁₃×#376	C₂²	1	1	yes	characteristic
10	#11718586	C₂²	1	2	yes	—
11	C₂×C₄⋊#1055	C₂²	1	2	yes	—
12	#11718139	—	2	2	no	—
13	#11718136	—	2	2	no	—
14	#11718116	—	2	2	no	—
15	C₂×C₁₃×#41	—	2	1	no	—
16	C₂×C₁₃×M₁₆	—	2	1	no	—
17	C₁₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
18	C₁₃×#127	D₄	1	1	yes	characteristic
19	C₁₃×#124	—	2	1	no	—
20	C₁₃×#121	—	2	2	no	—
21	C₄×D₄×C₁₃	—	2	2	no	—
22	C₂×C₄²×C₁₃	D₄	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂²×#1055	—	2	2	no	—
25	C₁₃×C₄⋊C₈	—	2	2	no	—
26	C₁₃×#104	—	4	4	no	—
27	C₁₃×#95	—	2	2	no	—
28	C₂²⋊#1055	—	2	2	no	—
29	C₄⋊#1055	—	2	2	no	—
30	C₄⋊#1055	—	2	2	no	—
31	C₄⋊#1055	—	2	2	no	—
32	C₄×#1055	—	2	2	no	—
33	#10629	—	13	1	no	maximal, Sylow
34	C₁₃×#41	—	4	2	no	—
35	C₁₃×#41	—	4	2	no	—
36	C₂×Q₈×C₁₃	—	2	1	no	—
37	C₂×D₄×C₁₃	—	2	1	no	—
38	C₂×D₄×C₁₃	—	2	2	no	—
39	C₂×D₄×C₁₃	—	4	1	no	—
40	C₂²×C₄×C₁₃	—	2	2	no	—
41	C₂²×C₄×C₁₃	—	4	1	no	—
42	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	#41	1	2	yes	—
44	C₁₃×M₁₆	—	4	2	no	—
45	C₂×C₈×C₁₃	—	4	1	no	—
46	C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₁₃×C₂²⋊C₄	—	4	2	no	—
48	C₄²×C₁₃	—	2	2	no	—
49	C₄²×C₁₃	—	2	2	no	—
50	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₁₃	#41	1	1	yes	characteristic
52	C₄²×C₁₃	#41	1	2	yes	—
53	C₂×#1055	—	4	2	no	—
54	C₂×#1055	—	4	2	no	—
55	C₂×#1055	—	4	2	no	—
56	C₂×#1055	—	4	2	no	—
57	#1963	D₁₃	1	1	yes	characteristic
58	#1832	—	13	1	no	—
59	#1540	—	13	2	no	—
60	#1301	—	13	2	no	—
61	#1274	—	13	1	no	—
62	C₂³×C₁₃	—	4	1	no	—
63	C₂³×C₁₃	#123	1	1	yes	characteristic
64	Q₈×C₁₃	—	4	2	no	—
65	D₄×C₁₃	—	4	2	no	—
66	D₄×C₁₃	—	8	2	no	—
67	D₄×C₁₃	—	8	2	no	—
68	C₂×C₄×C₁₃	—	4	2	no	—
69	C₂×C₄×C₁₃	—	4	4	no	—
70	C₂×C₄×C₁₃	—	4	1	no	—
71	C₂×C₄×C₁₃	—	4	2	no	—
72	C₂×C₄×C₁₃	—	8	1	no	—
73	C₂×C₄×C₁₃	—	8	1	no	—
74	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₁₃	#123	1	2	yes	—
76	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
77	C₂×C₄×C₁₃	#126	1	2	yes	—
78	C₂×C₄×C₁₃	#135	1	2	yes	—
79	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
80	C₈×C₁₃	—	8	1	no	—
81	#1055	—	8	2	no	—
82	#1055	—	8	2	no	—
83	#532	C₂×D₁₃	1	1	yes	characteristic
84	#431	—	13	2	no	—
85	#423	C₂×D₁₃	1	1	yes	characteristic
86	C₂×C₄⋊C₈	—	13	2	no	—
87	C₂×#104	—	2	2	no	—
88	#376	C₂×D₁₃	1	1	yes	characteristic
89	#336	—	26	2	no	—
90	#332	—	26	2	no	—
91	#331	—	26	2	no	—
92	C₂²×C₁₃	—	2	2	no	—
93	C₂²×C₁₃	—	8	1	no	—
94	C₂²×C₁₃	—	8	2	no	—
95	C₂²×C₁₃	#397	1	1	yes	characteristic
96	C₂²×C₁₃	#482	1	2	yes	—
97	C₄×C₁₃	—	2	2	no	—
98	C₄×C₁₃	—	2	2	no	—
99	C₄×C₁₃	—	8	2	no	—
100	C₄×C₁₃	—	8	1	no	—
101	C₄×C₁₃	#463	1	2	yes	—
102	C₄×C₁₃	#488	1	1	yes	characteristic
103	C₄×C₁₃	#490	1	1	yes	characteristic
104	C₂×#41	—	2	1	no	—
105	C₂×M₁₆	—	2	1	no	—
106	C₂²×C₈	—	26	2	no	—
107	C₄⋊Q₈	D₅₂	1	1	yes	characteristic
108	#127	D₅₂	1	1	yes	characteristic
109	#124	—	2	1	no	—
110	#121	—	2	2	no	—
111	C₄×D₄	—	2	2	no	—
112	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
113	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	26	2	no	—
116	C₄⋊C₈	—	26	2	no	—
117	C₄⋊C₈	—	26	2	no	—
118	#104	—	4	4	no	—
119	C₂²⋊C₈	—	26	2	no	—
120	C₄×C₈	—	26	2	no	—
121	#95	—	2	2	no	—
122	C₂×C₁₃	—	2	2	no	—
123	C₂×C₁₃	—	16	1	no	—
124	C₂×C₁₃	#2073	1	1	yes	characteristic
125	C₂×C₁₃	#2080	1	2	yes	—
126	#41	—	4	2	no	—
127	#41	—	4	2	no	—
128	C₂×Q₈	—	2	1	no	—
129	C₂×D₄	—	2	2	no	—
130	C₂×D₄	—	2	1	no	—
131	C₂×D₄	—	4	1	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	—	4	1	no	—
134	C₂²×C₄	#6129	1	2	yes	—
135	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
136	M₁₆	—	4	2	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	—	52	2	no	—
139	C₂×C₈	—	52	2	no	—
140	C₂×C₈	—	52	2	no	—
141	C₂×C₈	—	52	2	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₂²⋊C₄	—	4	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	C₂×D₅₂	1	1	yes	characteristic
147	C₄²	#6127	1	1	yes	characteristic
148	C₄²	#6129	1	2	yes	—
149	C₁₃	#10629	1	1	yes	characteristic, complemented, Sylow
150	C₂³	—	4	1	no	—
151	C₂³	#88189	1	1	yes	characteristic
152	Q₈	—	4	2	no	—
153	D₄	—	4	2	no	—
154	D₄	—	8	2	no	—
155	D₄	—	8	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	#88133	1	2	yes	—
163	C₂×C₄	#88140	1	2	yes	—
164	C₂×C₄	#88184	1	1	yes	characteristic
165	C₂×C₄	#88186	1	2	yes	—
166	C₂×C₄	#88190	1	1	yes	characteristic
167	C₂×C₄	#88204	1	2	yes	—
168	C₈	—	8	1	no	—
169	C₈	—	104	2	no	—
170	C₈	—	104	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	8	1	no	—
173	C₂²	—	8	2	no	—
174	C₂²	#11718614	1	1	yes	characteristic
175	C₂²	#11718696	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	8	2	no	—
179	C₄	—	8	1	no	—
180	C₄	#11718677	1	2	yes	—
181	C₄	#11718717	1	1	yes	characteristic
182	C₄	#11718732	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	16	1	no	—
185	C₂	#13116249	1	1	yes	characteristic
186	C₂	#13116286	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13114956

C₂

yes

maximal, characteristic, complemented

#13113485

C₂

yes

maximal

#13112731

C₂

yes

maximal, complemented

#13112665

C₂

yes

maximal, characteristic, complemented

C₁₃×#1963

C₂

yes

maximal, characteristic

C₁₃×#532

C₂²

yes

characteristic

C₁₃×#423

C₂²

yes

characteristic

C₂×C₁₃×#104

—

C₁₃×#376

C₂²

yes

characteristic

#11718586

C₂²

yes

—

C₂×C₄⋊#1055

C₂²

yes

—

#11718139

—

#11718136

—

#11718116

—

C₂×C₁₃×#41

—

C₂×C₁₃×M₁₆

—

C₁₃×C₄⋊Q₈

D₄

yes

characteristic

C₁₃×#127

D₄

yes

characteristic

C₁₃×#124

—

C₁₃×#121

—

C₄×D₄×C₁₃

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂²×#1055

—

C₁₃×C₄⋊C₈

—

C₁₃×#104

—

C₁₃×#95

—

C₂²⋊#1055

—

—

—

—

—

—

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×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×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×D₄

yes

C₄²×C₁₃

yes

C₄²×C₁₃

yes

—

—

—

—

—

D₁₃

yes

—

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

#123

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

#123

yes

—

C₂×C₄×C₁₃

#124

yes

characteristic

C₂×C₄×C₁₃

#126

yes

—

C₂×C₄×C₁₃

#135

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₈×C₁₃

—

#1055

—

#1055

—

#532

C₂×D₁₃

yes

characteristic

#431

—

#423

C₂×D₁₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#104

—

#376

C₂×D₁₃

yes

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#397

yes

characteristic

C₂²×C₁₃

#482

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

100

C₄×C₁₃

—

101

C₄×C₁₃

#463

yes

—

102

C₄×C₁₃

#488

yes

characteristic

103

C₄×C₁₃

#490

yes

characteristic

104

C₂×#41

—

105

C₂×M₁₆

—

106

C₂²×C₈

—

107

C₄⋊Q₈

D₅₂

yes

characteristic

108

#127

D₅₂

yes

characteristic

109

#124

—

110

#121

—

111

C₄×D₄

—

112

C₂×C₄²

C₂²⋊D₁₃

yes

—

113

C₂×C₄²

C₂²×D₁₃

yes

characteristic

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

#104

—

119

C₂²⋊C₈

—

120

C₄×C₈

—

121

#95

—

122

C₂×C₁₃

—

123

C₂×C₁₃

—

124

C₂×C₁₃

#2073

yes

characteristic

125

C₂×C₁₃

#2080

yes

—

126

#41

—

127

#41

—

128

C₂×Q₈

—

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

#6129

yes

—

135

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

136

M₁₆

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₄⋊C₄

—

143

C₂²⋊C₄

—

144

C₄²

—

145

C₄²

—

146

C₄²

C₂×D₅₂

yes

characteristic

147

C₄²

#6127

yes

characteristic

148

C₄²

#6129

yes

—

149

C₁₃

#10629

yes

characteristic, complemented, Sylow

150

C₂³

—

151

C₂³

#88189

yes

characteristic

152

Q₈

—

153

D₄

—

154

D₄

—

155

D₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

#88133

yes

—

163

C₂×C₄

#88140

yes

—

164

C₂×C₄

#88184

yes

characteristic

165

C₂×C₄

#88186

yes

—

166

C₂×C₄

#88190

yes

characteristic

167

C₂×C₄

#88204

yes

—

168

C₈

—

169

C₈

—

104

—

170

C₈

—

104

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

#11718614

yes

characteristic

175

C₂²

#11718696

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#11718677

yes

—

181

C₄

#11718717

yes

characteristic

182

C₄

#11718732

yes

characteristic

183

C₂

—

184

C₂

—

185

C₂

#13116249

yes

characteristic

186

C₂

#13116286

yes

—

	subgroup	G/H
center	C₂×C₄	#88190
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1963	C₂
socle	C₂²×C₁₃	#397
prime radical	C₁₃×#532	C₂²
hypercenter	#1963	D₁₃
hypocenter	C₁₃	#10629
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1963

subgroup

G/H

center

C₂×C₄

#88190

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1963

C₂

socle

C₂²×C₁₃

#397

prime radical

C₁₃×#532

C₂²

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1963

N	H ≅ G/N	type
1. #13114956	C₂	semidirect
3. #13112731	C₂	semidirect
4. #13112665	C₂	semidirect
149. C₁₃	#10629	semidirect

H ≅ G/N

type

1. #13114956

C₂

semidirect

3. #13112731

C₂

semidirect

4. #13112665

C₂

semidirect

149. C₁₃

#10629

semidirect

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

subgroup	index
#13114956	2
#13113485	2
#13112731	2
#13112665	2
C₁₃×#1963	2
#10629	13

subgroup

index

C₁₃×#1963

Subgroups — #15767697

Conjugacy classes of subgroups

Complements

Maximal subgroups