Group #13115924 — subgroups

Subgroups — #13115924

Order 1664.

474 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 99 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×C₁₃×#127	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6105	C₂	1	2	yes	maximal, complemented
3	C₂×#88072	C₂	1	2	yes	maximal, complemented
4	#11718590	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718116	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₁₃	—	2	2	no	—
7	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
8	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
9	C₁₃×#127	—	2	4	no	—
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#6105	—	2	4	no	—
12	C₂×#6105	—	2	4	no	—
13	C₂×#6105	—	2	4	no	—
14	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂²×#1055	C₂²	1	2	yes	—
16	#88072	—	2	4	no	—
17	#88071	—	2	2	no	—
18	C₂⁴×C₁₃	—	2	2	no	—
19	C₂×D₄×C₁₃	—	2	8	no	—
20	C₂×D₄×C₁₃	—	2	2	no	—
21	C₂×D₄×C₁₃	—	2	2	no	—
22	C₂×D₄×C₁₃	—	2	2	no	—
23	C₂×D₄×C₁₃	—	2	2	no	—
24	C₂×D₄×C₁₃	—	2	2	no	—
25	C₂×D₄×C₁₃	—	4	4	no	—
26	C₂×D₄×C₁₃	—	4	4	no	—
27	C₂×D₄×C₁₃	D₄	1	4	yes	—
28	C₂²×C₄×C₁₃	D₄	1	2	yes	—
29	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₁₃	—	2	1	no	—
31	C₂²×Dic₁₃	—	2	1	no	—
32	C₂×Dic₂₆	—	2	2	no	—
33	C₂×Dic₂₆	—	2	2	no	—
34	C₂×Dic₂₆	—	2	1	no	—
35	C₂×Dic₂₆	D₄	1	2	yes	complemented
36	C₄²×C₁₃	—	2	2	no	—
37	#6105	—	2	16	no	—
38	C₄⋊Dic₁₃	—	2	2	no	—
39	C₂×#1055	—	2	2	no	—
40	C₂×#1055	—	2	2	no	—
41	C₂×#1055	D₄	1	4	yes	complemented
42	#2014	—	13	1	no	maximal, Sylow
43	C₂³×C₁₃	—	2	4	no	—
44	C₂³×C₁₃	—	4	2	no	—
45	C₂³×C₁₃	—	4	2	no	—
46	C₂³×C₁₃	—	4	4	no	—
47	C₂³×C₁₃	—	4	4	no	—
48	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
49	D₄×C₁₃	—	2	4	no	—
50	D₄×C₁₃	—	2	4	no	—
51	D₄×C₁₃	—	4	16	no	—
52	D₄×C₁₃	—	4	2	no	—
53	D₄×C₁₃	—	4	2	no	—
54	C₂×C₄×C₁₃	—	2	2	no	—
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	SD₁₆	1	4	yes	complemented
57	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
61	C₂×Dic₁₃	—	2	2	no	—
62	C₂×Dic₁₃	—	4	1	no	—
63	C₂×Dic₁₃	—	4	1	no	—
64	C₂×Dic₁₃	—	4	1	no	—
65	C₂×Dic₁₃	—	4	2	no	—
66	Dic₂₆	—	2	4	no	—
67	Dic₂₆	—	4	2	no	—
68	#1055	—	2	4	no	—
69	C₂²×SD₁₆	—	13	2	no	—
70	C₂×#127	D₁₃	1	1	yes	characteristic
71	C₂×#102	—	13	2	no	—
72	#391	—	13	1	no	—
73	#336	—	13	1	no	—
74	C₂²×C₁₃	—	4	4	no	—
75	C₂²×C₁₃	—	4	8	no	—
76	C₂²×C₁₃	—	8	4	no	—
77	C₂²×C₁₃	—	8	2	no	—
78	C₂²×C₁₃	—	8	2	no	—
79	C₂²×C₁₃	#120	1	1	yes	characteristic
80	C₂²×C₁₃	#121	1	1	yes	characteristic
81	C₂²×C₁₃	#127	1	1	yes	characteristic
82	C₂²×C₁₃	C₂×SD₁₆	1	2	yes	—
83	C₂²×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
84	C₄×C₁₃	—	2	4	no	—
85	C₄×C₁₃	#120	1	2	yes	—
86	C₄×C₁₃	#121	1	2	yes	—
87	Dic₁₃	—	4	2	no	—
88	Dic₁₃	—	8	1	no	—
89	C₂²×Q₈	—	13	1	no	—
90	C₂²×D₄	—	2	2	no	—
91	C₂²×D₄	C₂×D₁₃	1	2	yes	—
92	C₂×SD₁₆	—	26	4	no	—
93	C₂×SD₁₆	—	26	4	no	—
94	C₂×SD₁₆	—	26	4	no	—
95	C₂²×C₈	—	13	2	no	—
96	#127	—	2	4	no	—
97	C₂×C₄⋊C₄	—	13	1	no	—
98	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
99	#102	—	26	4	no	—
100	#95	—	26	2	no	—
101	C₂×C₁₃	—	8	4	no	—
102	C₂×C₁₃	#392	1	1	yes	characteristic
103	C₂×C₁₃	#450	1	2	yes	—
104	C₂×C₁₃	#460	1	2	yes	—
105	C₂×C₁₃	#492	1	1	yes	characteristic
106	C₂×C₁₃	#496	1	1	yes	characteristic
107	C₂⁴	—	2	2	no	—
108	C₂×Q₈	—	13	2	no	—
109	C₂×Q₈	—	26	1	no	—
110	C₂×Q₈	—	26	2	no	—
111	C₂×Q₈	—	26	2	no	—
112	C₂×D₄	—	2	8	no	—
113	C₂×D₄	—	2	2	no	—
114	C₂×D₄	—	2	2	no	—
115	C₂×D₄	—	2	2	no	—
116	C₂×D₄	—	2	2	no	—
117	C₂×D₄	—	2	2	no	—
118	C₂×D₄	—	4	4	no	—
119	C₂×D₄	—	4	4	no	—
120	C₂×D₄	C₂²⋊D₁₃	1	4	yes	—
121	C₂²×C₄	—	26	1	no	—
122	C₂²×C₄	—	26	1	no	—
123	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
124	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
125	SD₁₆	—	26	16	no	—
126	C₂×C₈	—	13	4	no	—
127	C₂×C₈	—	26	2	no	—
128	C₂×C₈	—	26	2	no	—
129	C₄⋊C₄	—	26	2	no	—
130	C₄²	—	2	2	no	—
131	C₁₃	#2014	1	1	yes	characteristic, complemented, Sylow
132	C₂³	—	2	4	no	—
133	C₂³	—	4	4	no	—
134	C₂³	—	4	4	no	—
135	C₂³	—	4	2	no	—
136	C₂³	—	4	2	no	—
137	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
138	Q₈	—	26	4	no	—
139	Q₈	—	52	2	no	—
140	D₄	—	2	4	no	—
141	D₄	—	2	4	no	—
142	D₄	—	4	16	no	—
143	D₄	—	4	2	no	—
144	D₄	—	4	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	26	2	no	—
148	C₂×C₄	—	52	1	no	—
149	C₂×C₄	—	52	1	no	—
150	C₂×C₄	—	52	2	no	—
151	C₂×C₄	—	52	1	no	—
152	C₂×C₄	#6105	1	4	yes	complemented
153	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
154	C₂×C₄	D₄×D₁₃	1	2	yes	—
155	C₂×C₄	#6129	1	1	yes	characteristic
156	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
157	C₈	—	26	4	no	—
158	C₂²	—	4	8	no	—
159	C₂²	—	4	4	no	—
160	C₂²	—	8	2	no	—
161	C₂²	—	8	4	no	—
162	C₂²	—	8	2	no	—
163	C₂²	#88186	1	2	yes	—
164	C₂²	C₂×#6105	1	2	yes	—
165	C₂²	#88192	1	1	yes	characteristic
166	C₂²	#88194	1	1	yes	characteristic
167	C₂²	#88207	1	1	yes	characteristic
168	C₄	—	2	4	no	—
169	C₄	—	52	2	no	—
170	C₄	—	104	1	no	—
171	C₄	#88191	1	2	yes	—
172	C₄	#88193	1	2	yes	—
173	C₂	—	8	4	no	—
174	C₂	#11718735	1	1	yes	characteristic
175	C₂	#11718736	1	2	yes	—
176	C₂	#11718737	1	1	yes	characteristic
177	C₂	#11718879	1	2	yes	—
178	C₂	#11718884	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#127

C₂

yes

maximal, characteristic

C₂²×#6105

C₂

yes

maximal, complemented

C₂×#88072

C₂

yes

maximal, complemented

#11718590

C₂

yes

maximal, characteristic, complemented

#11718116

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₃

—

C₂²×D₄×C₁₃

C₂²

yes

—

C₂²×Dic₂₆

C₂²

yes

characteristic

C₁₃×#127

—

C₂×C₄²×C₁₃

C₂²

yes

—

—

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂²×#1055

C₂²

yes

—

#88072

—

#88071

—

C₂⁴×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

D₄

yes

complemented

C₄²×C₁₃

—

#6105

—

C₄⋊Dic₁₃

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

D₄

yes

complemented

#2014

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂×D₄

yes

characteristic

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

SD₁₆

yes

complemented

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

#41

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

#1055

—

C₂²×SD₁₆

—

C₂×#127

D₁₃

yes

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

C₂×SD₁₆

yes

—

C₂²×C₁₃

AGL(1,ℤ/8)

yes

—

C₄×C₁₃

—

C₄×C₁₃

#120

yes

—

C₄×C₁₃

#121

yes

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×D₄

—

C₂²×D₄

C₂×D₁₃

yes

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂²×C₈

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#102

—

100

#95

—

101

C₂×C₁₃

—

102

C₂×C₁₃

#392

yes

characteristic

103

C₂×C₁₃

#450

yes

—

104

C₂×C₁₃

#460

yes

—

105

C₂×C₁₃

#492

yes

characteristic

106

C₂×C₁₃

#496

yes

characteristic

107

C₂⁴

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

C₂²⋊D₁₃

yes

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂²⋊D₁₃

yes

—

124

C₂²×C₄

C₂²×D₁₃

yes

characteristic

125

SD₁₆

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₄⋊C₄

—

130

C₄²

—

131

C₁₃

#2014

yes

characteristic, complemented, Sylow

132

C₂³

—

133

C₂³

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

138

Q₈

—

139

Q₈

—

140

D₄

—

141

D₄

—

142

D₄

—

143

D₄

—

144

D₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

#6105

yes

complemented

153

C₂×C₄

D₄×D₁₃

yes

characteristic

154

C₂×C₄

D₄×D₁₃

yes

—

155

C₂×C₄

#6129

yes

characteristic

156

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

157

C₈

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

#88186

yes

—

164

C₂²

C₂×#6105

yes

—

165

C₂²

#88192

yes

characteristic

166

C₂²

#88194

yes

characteristic

167

C₂²

#88207

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

104

—

171

C₄

#88191

yes

—

172

C₄

#88193

yes

—

173

C₂

—

174

C₂

#11718735

yes

characteristic

175

C₂

#11718736

yes

—

176

C₂

#11718737

yes

characteristic

177

C₂

#11718879

yes

—

178

C₂

#11718884

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃
commutator, cosocle	C₂²×C₄×C₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting, prime radical	C₂×C₁₃×#127	C₂
socle	C₂³×C₁₃	C₂×D₄
hypercenter	C₂×#127	D₁₃
hypocenter	C₁₃	#2014
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃

commutator, cosocle

C₂²×C₄×C₁₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting, prime radical

C₂×C₁₃×#127

C₂

socle

C₂³×C₁₃

C₂×D₄

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
2. C₂²×#6105	C₂	semidirect
3. C₂×#88072	C₂	semidirect
4. #11718590	C₂	semidirect
5. #11718116	C₂	semidirect
35. C₂×Dic₂₆	D₄	semidirect
41. C₂×#1055	D₄	semidirect
56. C₂×C₄×C₁₃	SD₁₆	semidirect
131. C₁₃	#2014	semidirect
152. C₂×C₄	#6105	semidirect

H ≅ G/N

type

2. C₂²×#6105

C₂

semidirect

3. C₂×#88072

C₂

semidirect

4. #11718590

C₂

semidirect

5. #11718116

C₂

semidirect

35. C₂×Dic₂₆

D₄

semidirect

41. C₂×#1055

D₄

semidirect

56. C₂×C₄×C₁₃

SD₁₆

semidirect

131. C₁₃

#2014

semidirect

152. C₂×C₄

#6105

semidirect

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

subgroup	index
C₂×C₁₃×#127	2
C₂²×#6105	2
C₂×#88072	2
#11718590	2
#11718116	2
#2014	13

subgroup

index

C₂×C₁₃×#127

C₂²×#6105

Subgroups — #13115924

Conjugacy classes of subgroups

Complements

Maximal subgroups