Group #13120225 — subgroups

Subgroups — #13120225

Order 1664.

468 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 169 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×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic
3	#11718644	C₂	1	2	yes	maximal
4	#11718640	C₂	1	2	yes	maximal
5	#11718634	C₂	1	2	yes	maximal
6	#11718634	C₂	1	2	yes	maximal
7	#11718633	C₂	1	4	yes	maximal
8	C₄×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
10	C₁₃×#125	—	2	4	no	—
11	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
18	C₂×#6101	C₂²	1	2	yes	—
19	C₂×#6101	C₂²	1	4	yes	—
20	C₂×#6101	C₂²	1	4	yes	—
21	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
22	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
23	C₄×Dic₂₆	—	2	4	no	—
24	#88071	C₂²	1	2	yes	—
25	#88071	C₂²	1	4	yes	—
26	#88071	C₂²	1	4	yes	—
27	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
28	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
29	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×Dic₁₃	C₂³	1	2	yes	—
33	C₂²×Dic₁₃	C₂³	1	4	yes	—
34	C₂²×Dic₁₃	C₂³	1	2	yes	—
35	C₂×Dic₂₆	—	2	2	no	—
36	C₂×Dic₂₆	—	2	4	no	—
37	C₁₃×C₄⋊C₄	—	2	4	no	—
38	C₁₃×C₄⋊C₄	—	2	4	no	—
39	C₁₃×C₄⋊C₄	—	2	2	no	—
40	C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₄²×C₁₃	D₄	1	4	yes	—
42	C₄⋊Dic₁₃	—	2	2	no	—
43	C₄⋊Dic₁₃	Q₈	1	8	yes	complemented
44	#6101	—	2	4	no	—
45	#6101	—	2	8	no	—
46	#6101	—	4	4	no	—
47	C₄×Dic₁₃	—	2	4	no	—
48	C₄×Dic₁₃	—	4	2	no	—
49	#2584	—	13	1	no	maximal, Sylow
50	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
51	C₂×C₄×C₁₃	—	2	2	no	—
52	C₂×C₄×C₁₃	—	2	1	no	—
53	C₂×C₄×C₁₃	—	2	1	no	—
54	C₂×C₄×C₁₃	—	2	2	no	—
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	—	2	4	no	—
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
63	C₂×Dic₁₃	—	2	2	no	—
64	C₂×Dic₁₃	—	2	4	no	—
65	C₂×Dic₁₃	—	2	2	no	—
66	C₂×Dic₁₃	—	2	4	no	—
67	C₂×Dic₁₃	—	2	2	no	—
68	C₂×Dic₁₃	—	2	2	no	—
69	C₂×Dic₁₃	—	2	2	no	—
70	C₂×Dic₁₃	C₂×Q₈	1	8	yes	—
71	C₂×Dic₁₃	#41	1	4	yes	—
72	Dic₂₆	—	2	8	no	—
73	C₂×#125	D₁₃	1	1	yes	characteristic
74	C₂×C₄×Q₈	—	13	1	no	—
75	#398	—	13	2	no	—
76	#395	—	13	2	no	—
77	#389	—	13	4	no	—
78	#389	—	13	2	no	—
79	#389	—	13	2	no	—
80	C₄×C₄⋊C₄	—	13	1	no	—
81	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
82	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
83	C₂²×C₁₃	C₂×#41	1	2	yes	—
84	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
85	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
86	C₄×C₁₃	—	2	2	no	—
87	C₄×C₁₃	—	2	2	no	—
88	C₄×C₁₃	—	4	2	no	—
89	C₄×C₁₃	—	4	2	no	—
90	C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
91	Dic₁₃	—	2	4	no	—
92	Dic₁₃	—	2	8	no	—
93	Dic₁₃	—	4	2	no	—
94	C₂²×Q₈	—	13	1	no	—
95	#125	—	2	4	no	—
96	C₄×Q₈	—	26	4	no	—
97	C₂×C₄⋊C₄	—	13	2	no	—
98	C₂×C₄⋊C₄	—	13	2	no	—
99	C₂×C₄⋊C₄	—	13	1	no	—
100	C₂×C₄⋊C₄	—	13	4	no	—
101	C₂×C₄⋊C₄	—	13	4	no	—
102	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
103	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
104	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
105	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
106	C₂×C₄²	—	13	2	no	—
107	C₂×C₄²	—	13	2	no	—
108	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
109	#95	—	13	2	no	—
110	#95	—	13	4	no	—
111	#95	—	13	4	no	—
112	C₂×C₁₃	#525	1	1	yes	characteristic
113	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
114	C₂×C₁₃	#537	1	1	yes	characteristic
115	C₂×C₁₃	#554	1	2	yes	—
116	C₂×C₁₃	#557	1	2	yes	—
117	C₂×Q₈	—	26	4	no	—
118	C₂×Q₈	—	26	2	no	—
119	C₂²×C₄	—	13	4	no	—
120	C₂²×C₄	—	13	2	no	—
121	C₂²×C₄	—	13	2	no	—
122	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
123	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
124	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
127	C₄⋊C₄	—	2	4	no	—
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	13	8	no	—
132	C₄⋊C₄	—	26	8	no	—
133	C₄⋊C₄	—	26	4	no	—
134	C₄⋊C₄	—	26	2	no	—
135	C₄⋊C₄	—	52	4	no	—
136	C₄²	—	26	4	no	—
137	C₄²	—	52	2	no	—
138	C₄²	C₂²⋊D₁₃	1	4	yes	—
139	C₁₃	#2584	1	1	yes	characteristic, complemented, Sylow
140	C₂³	C₂³×D₁₃	1	1	yes	characteristic
141	Q₈	—	26	8	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	13	8	no	—
151	C₂×C₄	—	13	4	no	—
152	C₂×C₄	—	26	4	no	—
153	C₂×C₄	—	26	2	no	—
154	C₂×C₄	—	26	4	no	—
155	C₂×C₄	—	26	2	no	—
156	C₂×C₄	—	26	2	no	—
157	C₂×C₄	—	26	2	no	—
158	C₂×C₄	—	26	2	no	—
159	C₂×C₄	Q₈×D₁₃	1	4	yes	—
160	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
161	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
162	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
163	C₂²	C₂×Q₈×D₁₃	1	2	yes	—
164	C₂²	#41×D₁₃	1	2	yes	—
165	C₂²	#88256	1	1	yes	characteristic
166	C₂²	#88257	1	1	yes	characteristic
167	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	4	2	no	—
172	C₄	—	26	4	no	—
173	C₄	—	26	8	no	—
174	C₄	—	52	2	no	—
175	C₄	#88198	1	4	yes	—
176	C₂	#11719390	1	2	yes	—
177	C₂	#11719395	1	2	yes	—
178	C₂	C₂×#88198	1	1	yes	characteristic
179	C₂	#11719533	1	1	yes	characteristic
180	C₂	#11719538	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#125

C₂

yes

maximal, characteristic

C₂×C₄×Dic₂₆

C₂

yes

maximal, characteristic

#11718644

C₂

yes

maximal

#11718640

C₂

yes

maximal

#11718634

C₂

yes

maximal

#11718634

C₂

yes

maximal

#11718633

C₂

yes

maximal

C₄×C₄⋊Dic₁₃

C₂

yes

maximal, characteristic

C₂²×Dic₂₆

C₂²

yes

characteristic

C₁₃×#125

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₄×Dic₂₆

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

D₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

Q₈

yes

—

—

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

#2584

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

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×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

#41

yes

—

Dic₂₆

—

C₂×#125

D₁₃

yes

characteristic

C₂×C₄×Q₈

—

—

—

—

—

—

C₄×C₄⋊C₄

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

—

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₁₃

2₋¹⁺⁴

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄⋊Q₈

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

#125

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

103

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

104

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

105

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

106

C₂×C₄²

—

107

C₂×C₄²

—

108

C₂×C₄²

C₂×D₁₃

yes

characteristic

109

#95

—

110

#95

—

111

#95

—

112

C₂×C₁₃

#525

yes

characteristic

113

C₂×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

114

C₂×C₁₃

#537

yes

characteristic

115

C₂×C₁₃

#554

yes

—

116

C₂×C₁₃

#557

yes

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₂²×D₁₃

yes

—

123

C₂²×C₄

C₂²×D₁₃

yes

—

124

C₂²×C₄

C₂²×D₁₃

yes

characteristic

125

C₂²×C₄

C₂²×D₁₃

yes

characteristic

126

C₂²×C₄

C₂²×D₁₃

yes

characteristic

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

—

137

C₄²

—

138

C₄²

C₂²⋊D₁₃

yes

—

139

C₁₃

#2584

yes

characteristic, complemented, Sylow

140

C₂³

C₂³×D₁₃

yes

characteristic

141

Q₈

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

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

—

159

C₂×C₄

Q₈×D₁₃

yes

—

160

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

161

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

162

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

163

C₂²

C₂×Q₈×D₁₃

yes

—

164

C₂²

#41×D₁₃

yes

—

165

C₂²

#88256

yes

characteristic

166

C₂²

#88257

yes

characteristic

167

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

#88198

yes

—

176

C₂

#11719390

yes

—

177

C₂

#11719395

yes

—

178

C₂

C₂×#88198

yes

characteristic

179

C₂

#11719533

yes

characteristic

180

C₂

#11719538

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₁₃×#125	C₂
hypercenter	C₂×#125	D₁₃
hypocenter	C₁₃	#2584
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₂×C₁₃×#125

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
43. C₄⋊Dic₁₃	Q₈	semidirect
139. C₁₃	#2584	semidirect

H ≅ G/N

type

43. C₄⋊Dic₁₃

Q₈

semidirect

139. C₁₃

#2584

semidirect

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

subgroup	index
C₂×C₁₃×#125	2
C₂×C₄×Dic₂₆	2
#11718644	2
#11718640	2
#11718634	2
#11718633	2
C₄×C₄⋊Dic₁₃	2
#2584	13

subgroup

index

C₂×C₁₃×#125

C₂×C₄×Dic₂₆

C₄×C₄⋊Dic₁₃

Subgroups — #13120225

Conjugacy classes of subgroups

Complements

Maximal subgroups