Group #13115650 — subgroups

Subgroups — #13115650

Order 1664.

344 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×#88129	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88050	C₂	1	2	yes	maximal, complemented
4	C₂×#88049	C₂	1	2	yes	maximal
5	C₂×#88043	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₅₂	C₂²	1	1	yes	characteristic
7	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
9	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#6103	—	2	2	no	—
12	C₂²×#1055	C₂²	1	2	yes	—
13	#88129	—	2	2	no	—
14	#88129	C₄	1	4	yes	complemented
15	#88050	—	2	4	no	—
16	#88049	—	2	4	no	—
17	#88043	—	2	2	no	—
18	C₂³×D₁₃	—	2	1	no	—
19	C₂²×C₄×C₁₃	—	2	2	no	—
20	C₂²×C₄×C₁₃	D₄	1	2	yes	—
21	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₁₃	—	2	1	no	—
23	C₂×D₅₂	—	2	2	no	—
24	C₂×D₅₂	—	2	2	no	—
25	C₂×D₅₂	—	2	1	no	—
26	C₂×D₅₂	C₂×C₄	1	2	yes	—
27	C₂×Dic₂₆	—	2	2	no	—
28	C₂×Dic₂₆	—	2	2	no	—
29	C₂×Dic₂₆	—	2	1	no	—
30	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
31	C₁₃×C₄⋊C₄	—	2	2	no	—
32	C₁₃×C₄⋊C₄	—	2	2	no	—
33	C₁₃×C₄⋊C₄	—	4	4	no	—
34	C₄²×C₁₃	C₂×C₄	1	2	yes	—
35	C₄²×C₁₃	D₄	1	2	yes	—
36	#6103	—	2	8	no	—
37	C₂×#1055	—	2	2	no	—
38	C₂×#1055	—	2	2	no	—
39	C₂×#1055	D₄	1	4	yes	—
40	#1959	—	13	1	no	maximal, Sylow
41	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₁₃	—	2	2	no	—
43	C₂²×D₁₃	—	4	1	no	—
44	C₂²×D₁₃	—	4	1	no	—
45	C₂²×D₁₃	—	4	2	no	—
46	C₂²×D₁₃	—	4	2	no	—
47	C₂×C₄×C₁₃	—	2	2	no	—
48	C₂×C₄×C₁₃	—	2	2	no	—
49	C₂×C₄×C₁₃	—	4	2	no	—
50	C₂×C₄×C₁₃	—	4	4	no	—
51	C₂×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
52	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃	#41	1	2	yes	—
56	C₂×Dic₁₃	—	2	2	no	—
57	C₂×Dic₁₃	—	4	1	no	—
58	C₂×Dic₁₃	—	4	1	no	—
59	D₅₂	—	2	4	no	—
60	D₅₂	—	4	2	no	—
61	Dic₂₆	—	2	4	no	—
62	Dic₂₆	—	4	2	no	—
63	#1055	—	2	4	no	—
64	C₂×#124	—	13	1	no	—
65	C₂×#103	—	13	2	no	—
66	C₂×#102	—	13	2	no	—
67	C₂×#97	—	13	1	no	—
68	#384	D₁₃	1	1	yes	characteristic, complemented
69	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₁₃	#124	1	1	yes	characteristic
71	C₂²×C₁₃	#127	1	1	yes	characteristic
72	C₂²×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
73	C₂²×C₁₃	#137	1	2	yes	—
74	C₂×D₁₃	—	4	2	no	—
75	C₂×D₁₃	—	4	4	no	—
76	C₂×D₁₃	—	8	2	no	—
77	C₂×D₁₃	—	8	1	no	—
78	C₂×D₁₃	—	8	1	no	—
79	C₄×C₁₃	—	2	4	no	—
80	C₄×C₁₃	—	8	2	no	—
81	C₄×C₁₃	C₄×D₄	1	2	yes	—
82	C₄×C₁₃	#121	1	2	yes	—
83	Dic₁₃	—	4	2	no	—
84	C₂²×Q₈	—	13	1	no	—
85	C₂²×D₄	—	13	1	no	—
86	C₂²×C₈	—	13	2	no	—
87	#124	—	13	4	no	—
88	#124	—	26	2	no	—
89	C₂×C₄⋊C₄	—	2	2	no	—
90	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
91	C₂×C₂²⋊C₄	—	26	2	no	—
92	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
93	#103	—	26	4	no	—
94	#102	—	26	4	no	—
95	#97	—	26	2	no	—
96	C₂×C₁₃	#390	1	1	yes	characteristic
97	C₂×C₁₃	#417	1	2	yes	—
98	C₂×C₁₃	#489	1	2	yes	—
99	C₂×C₁₃	#496	1	1	yes	characteristic
100	C₂×C₁₃	#497	1	1	yes	characteristic
101	D₁₃	—	8	2	no	—
102	C₂⁴	—	26	1	no	—
103	C₂×Q₈	—	13	2	no	—
104	C₂×Q₈	—	26	1	no	—
105	C₂×Q₈	—	26	2	no	—
106	C₂×Q₈	—	26	2	no	—
107	C₂×D₄	—	13	2	no	—
108	C₂×D₄	—	26	1	no	—
109	C₂×D₄	—	26	2	no	—
110	C₂×D₄	—	26	2	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	26	1	no	—
113	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
114	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
115	C₂×C₈	—	13	4	no	—
116	C₂×C₈	—	26	2	no	—
117	C₂×C₈	—	26	2	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	4	4	no	—
121	C₂²⋊C₄	—	26	8	no	—
122	C₄²	C₄×D₁₃	1	2	yes	—
123	C₄²	D₅₂	1	2	yes	—
124	C₁₃	#1959	1	1	yes	characteristic, complemented, Sylow
125	C₂³	—	26	2	no	—
126	C₂³	—	52	2	no	—
127	C₂³	—	52	1	no	—
128	C₂³	—	52	2	no	—
129	C₂³	—	52	1	no	—
130	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
131	Q₈	—	26	4	no	—
132	Q₈	—	52	2	no	—
133	D₄	—	26	4	no	—
134	D₄	—	52	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	4	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	26	2	no	—
140	C₂×C₄	—	52	1	no	—
141	C₂×C₄	—	52	1	no	—
142	C₂×C₄	#6103	1	4	yes	—
143	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
144	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
145	C₂×C₄	D₄×D₁₃	1	2	yes	—
146	C₂×C₄	#6131	1	2	yes	—
147	C₈	—	26	4	no	—
148	C₂²	—	52	4	no	—
149	C₂²	—	52	2	no	—
150	C₂²	—	104	1	no	—
151	C₂²	—	104	1	no	—
152	C₂²	—	104	2	no	—
153	C₂²	C₂×#6103	1	1	yes	characteristic
154	C₂²	#88186	1	2	yes	—
155	C₂²	#88194	1	1	yes	characteristic
156	C₂²	#88196	1	2	yes	—
157	C₂²	#88201	1	1	yes	characteristic
158	C₄	—	2	4	no	—
159	C₄	—	8	2	no	—
160	C₄	—	52	2	no	—
161	C₄	#88147	1	2	yes	—
162	C₄	#88149	1	2	yes	—
163	C₂	—	104	2	no	—
164	C₂	#11718627	1	2	yes	—
165	C₂	#11718649	1	1	yes	characteristic
166	C₂	#11718735	1	1	yes	characteristic
167	C₂	#11718747	1	1	yes	characteristic
168	C₂	#11718750	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88129

C₂

yes

maximal, characteristic

C₁₃×#384

C₂

yes

maximal, characteristic, complemented

C₂×#88050

C₂

yes

maximal, complemented

C₂×#88049

C₂

yes

maximal

C₂×#88043

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₂

C₂²

yes

characteristic

C₂²×Dic₂₆

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×#6103

—

C₂²×#1055

C₂²

yes

—

#88129

—

#88129

C₄

yes

—

—

—

C₂³×D₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

C₂×C₄

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

C₂×C₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×C₄

yes

—

C₄²×C₁₃

D₄

yes

—

—

—

—

D₄

yes

—

#1959

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

Dic₂₆

—

Dic₂₆

—

—

—

—

—

C₂×#97

—

#384

D₁₃

yes

characteristic, complemented

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

#137

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×D₄

yes

—

C₄×C₁₃

#121

yes

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×D₄

—

C₂²×C₈

—

#124

—

#124

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

—

—

—

C₂×C₁₃

#390

yes

characteristic

C₂×C₁₃

#417

yes

—

C₂×C₁₃

#489

yes

—

C₂×C₁₃

#496

yes

characteristic

100

C₂×C₁₃

#497

yes

characteristic

101

D₁₃

—

102

C₂⁴

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂²⋊D₁₃

yes

—

114

C₂²×C₄

C₂²×D₁₃

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

C₄×D₁₃

yes

—

123

C₄²

D₅₂

yes

—

124

C₁₃

#1959

yes

characteristic, complemented, Sylow

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

131

Q₈

—

132

Q₈

—

133

D₄

—

134

D₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

#6103

yes

—

143

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

144

C₂×C₄

C₂×D₅₂

yes

characteristic

145

C₂×C₄

D₄×D₁₃

yes

—

146

C₂×C₄

#6131

yes

—

147

C₈

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

104

—

151

C₂²

—

104

—

152

C₂²

—

104

—

153

C₂²

C₂×#6103

yes

characteristic

154

C₂²

#88186

yes

—

155

C₂²

#88194

yes

characteristic

156

C₂²

#88196

yes

—

157

C₂²

#88201

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#88147

yes

—

162

C₄

#88149

yes

—

163

C₂

—

104

—

164

C₂

#11718627

yes

—

165

C₂

#11718649

yes

characteristic

166

C₂

#11718735

yes

characteristic

167

C₂

#11718747

yes

characteristic

168

C₂

#11718750

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#384	C₂
socle	C₂³×C₁₃	C₂×D₄
prime radical	C₂²×D₅₂	C₂²
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	#384	D₁₃
hypocenter	C₁₃	#1959
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#384

C₂

socle

C₂³×C₁₃

C₂×D₄

prime radical

C₂²×D₅₂

C₂²

cosocle

C₂²×C₄×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
2. C₁₃×#384	C₂	semidirect
3. C₂×#88050	C₂	semidirect
5. C₂×#88043	C₂	semidirect
14. #88129	C₄	semidirect
68. #384	D₁₃	semidirect
124. C₁₃	#1959	semidirect

H ≅ G/N

type

2. C₁₃×#384

C₂

semidirect

3. C₂×#88050

C₂

semidirect

5. C₂×#88043

C₂

semidirect

14. #88129

C₄

semidirect

68. #384

D₁₃

semidirect

124. C₁₃

#1959

semidirect

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

subgroup	index
C₂×#88129	2
C₁₃×#384	2
C₂×#88050	2
C₂×#88049	2
C₂×#88043	2
#1959	13

subgroup

index

C₁₃×#384

Subgroups — #13115650

Conjugacy classes of subgroups

Complements

Maximal subgroups