Group #13119322 — subgroups

Subgroups — #13119322

Order 1664.

512 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 165 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×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#11718891	C₂	1	2	yes	maximal, complemented
4	C₄×C₂²⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	#11718647	C₂	1	4	yes	maximal, complemented
6	#11718638	C₂	1	2	yes	maximal, complemented
7	#11718604	C₂	1	4	yes	maximal
8	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×Dic₁₃	C₂²	1	2	yes	—
11	C₁₃×C₂²⋊Q₈	—	2	4	no	—
12	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₁₃	—	2	4	no	—
17	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
19	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
20	#88071	C₂²	1	8	yes	—
21	#88071	C₂²	1	8	yes	—
22	#88071	C₂²	1	2	yes	—
23	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×Q₈×C₁₃	—	2	2	no	—
25	C₂×Q₈×C₁₃	—	2	4	no	—
26	C₂²×C₄×C₁₃	—	2	4	no	—
27	C₂²×C₄×C₁₃	D₄	1	4	yes	—
28	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
29	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×Dic₁₃	—	2	4	no	—
33	C₂²×Dic₁₃	—	2	8	no	—
34	C₂²×Dic₁₃	C₂³	1	4	yes	—
35	C₂²×Dic₁₃	C₂³	1	4	yes	—
36	C₁₃×C₄⋊C₄	—	2	2	no	—
37	C₁₃×C₄⋊C₄	—	2	4	no	—
38	C₁₃×C₂²⋊C₄	—	2	4	no	—
39	C₂²⋊Dic₁₃	—	2	4	no	—
40	C₄⋊Dic₁₃	—	2	4	no	—
41	C₄⋊Dic₁₃	—	2	4	no	—
42	C₄×Dic₁₃	—	2	4	no	—
43	#2503	—	13	1	no	maximal, Sylow
44	C₂³×C₁₃	—	2	2	no	—
45	C₂³×C₁₃	—	2	2	no	—
46	C₂³×C₁₃	C₂×D₄	1	2	yes	—
47	C₂³×C₁₃	#41	1	4	yes	—
48	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
49	Q₈×C₁₃	—	4	4	no	—
50	C₂×C₄×C₁₃	—	2	2	no	—
51	C₂×C₄×C₁₃	—	2	8	no	—
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	—	2	4	no	—
55	C₂×C₄×C₁₃	—	2	4	no	—
56	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₃	#41	1	4	yes	—
59	C₂×Dic₁₃	—	2	4	no	—
60	C₂×Dic₁₃	—	2	4	no	—
61	C₂×Dic₁₃	—	2	4	no	—
62	C₂×Dic₁₃	—	2	8	no	—
63	C₂×Dic₁₃	—	2	4	no	—
64	C₂×Dic₁₃	—	4	8	no	—
65	C₂×C₂²⋊Q₈	D₁₃	1	1	yes	characteristic
66	C₂²×C₄⋊C₄	—	13	1	no	—
67	#400	—	13	4	no	—
68	#391	—	13	2	no	—
69	#382	—	13	2	no	—
70	#381	—	13	4	no	—
71	C₄×C₂²⋊C₄	—	13	1	no	—
72	C₂²×C₁₃	—	2	2	no	—
73	C₂²×C₁₃	—	2	2	no	—
74	C₂²×C₁₃	—	2	8	no	—
75	C₂²×C₁₃	#124	1	4	yes	—
76	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
77	C₂²×C₁₃	C₂×#41	1	2	yes	—
78	C₂²×C₁₃	C₂×#41	1	2	yes	—
79	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
80	C₄×C₁₃	—	2	2	no	—
81	C₄×C₁₃	—	4	2	no	—
82	C₄×C₁₃	—	4	2	no	—
83	C₄×C₁₃	#123	1	4	yes	—
84	Dic₁₃	—	4	4	no	—
85	Dic₁₃	—	4	4	no	—
86	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
87	C₂³×C₄	—	13	2	no	—
88	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
89	C₂²⋊Q₈	—	2	4	no	—
90	C₂×C₄⋊C₄	—	13	2	no	—
91	C₂×C₄⋊C₄	—	13	2	no	—
92	C₂×C₄⋊C₄	—	26	4	no	—
93	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
94	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
95	C₂×C₂²⋊C₄	—	13	2	no	—
96	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
97	C₂×C₄²	—	13	2	no	—
98	#95	—	13	2	no	—
99	#95	—	13	8	no	—
100	#95	—	13	8	no	—
101	C₂×C₁₃	—	2	4	no	—
102	C₂×C₁₃	C₂×#123	1	1	yes	characteristic
103	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
104	C₂×C₁₃	#536	1	1	yes	characteristic
105	C₂×C₁₃	#552	1	4	yes	—
106	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
107	C₂×Q₈	—	2	4	no	—
108	C₂×Q₈	—	2	2	no	—
109	C₂²×C₄	—	2	4	no	—
110	C₂²×C₄	—	13	4	no	—
111	C₂²×C₄	—	13	4	no	—
112	C₂²×C₄	—	26	8	no	—
113	C₂²×C₄	—	26	4	no	—
114	C₂²×C₄	C₂²⋊D₁₃	1	4	yes	—
115	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
116	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
117	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
118	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	26	4	no	—
122	C₄⋊C₄	—	26	4	no	—
123	C₂²⋊C₄	—	2	4	no	—
124	C₂²⋊C₄	—	26	4	no	—
125	C₄²	—	26	4	no	—
126	C₁₃	#2503	1	1	yes	characteristic, complemented, Sylow
127	C₂³	—	2	2	no	—
128	C₂³	—	2	2	no	—
129	C₂³	#6131	1	4	yes	—
130	C₂³	C₂×C₂²⋊D₁₃	1	2	yes	—
131	C₂³	C₂³×D₁₃	1	1	yes	characteristic
132	Q₈	—	4	4	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	26	4	no	—
140	C₂×C₄	—	26	4	no	—
141	C₂×C₄	—	26	8	no	—
142	C₂×C₄	—	26	4	no	—
143	C₂×C₄	—	26	4	no	—
144	C₂×C₄	—	52	8	no	—
145	C₂×C₄	#6129	1	4	yes	—
146	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
148	C₂²	—	2	2	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	8	no	—
151	C₂²	#88201	1	4	yes	—
152	C₂²	C₂×#6129	1	2	yes	—
153	C₂²	C₂×#6131	1	2	yes	—
154	C₂²	#88257	1	2	yes	—
155	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
156	C₄	—	2	2	no	—
157	C₄	—	4	2	no	—
158	C₄	—	4	2	no	—
159	C₄	—	52	4	no	—
160	C₄	—	52	4	no	—
161	C₄	#88189	1	4	yes	—
162	C₂	—	2	4	no	—
163	C₂	#11719347	1	4	yes	—
164	C₂	C₂×#88189	1	1	yes	characteristic
165	C₂	C₂×#88201	1	1	yes	characteristic
166	C₂	#11719530	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×C₂²⋊Q₈

C₂

yes

maximal, characteristic

C₂²×C₄⋊Dic₁₃

C₂

yes

maximal, characteristic

#11718891

C₂

yes

maximal, complemented

C₄×C₂²⋊Dic₁₃

C₂

yes

maximal, characteristic

#11718647

C₂

yes

maximal, complemented

#11718638

C₂

yes

maximal, complemented

#11718604

C₂

yes

maximal

C₂²×Q₈×C₁₃

C₂²

yes

characteristic

C₂³×C₄×C₁₃

C₂²

yes

characteristic

C₂³×Dic₁₃

C₂²

yes

—

C₁₃×C₂²⋊Q₈

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂⁴×C₁₃

C₂³

yes

characteristic

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

#2503

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂×D₄

yes

—

C₂³×C₁₃

#41

yes

—

C₂³×C₁₃

C₂⁴

yes

characteristic

Q₈×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

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×C₂²⋊Q₈

D₁₃

yes

characteristic

C₂²×C₄⋊C₄

—

—

—

—

—

C₄×C₂²⋊C₄

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#124

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#123

yes

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

C₂×D₁₃

yes

characteristic

C₂³×C₄

—

C₂³×C₄

C₂×D₁₃

yes

characteristic

C₂²⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

#95

—

#95

—

100

#95

—

101

C₂×C₁₃

—

102

C₂×C₁₃

C₂×#123

yes

characteristic

103

C₂×C₁₃

C₂×#124

yes

characteristic

104

C₂×C₁₃

#536

yes

characteristic

105

C₂×C₁₃

#552

yes

—

106

C₂⁴

C₂²×D₁₃

yes

characteristic

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂²⋊D₁₃

yes

—

115

C₂²×C₄

C₂²×D₁₃

yes

—

116

C₂²×C₄

C₂²×D₁₃

yes

—

117

C₂²×C₄

C₂²×D₁₃

yes

characteristic

118

C₂²×C₄

C₂²×D₁₃

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₁₃

#2503

yes

characteristic, complemented, Sylow

127

C₂³

—

128

C₂³

—

129

C₂³

#6131

yes

—

130

C₂³

C₂×C₂²⋊D₁₃

yes

—

131

C₂³

C₂³×D₁₃

yes

characteristic

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

#6129

yes

—

146

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

147

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#88201

yes

—

152

C₂²

C₂×#6129

yes

—

153

C₂²

C₂×#6131

yes

—

154

C₂²

#88257

yes

—

155

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#88189

yes

—

162

C₂

—

163

C₂

#11719347

yes

—

164

C₂

C₂×#88189

yes

characteristic

165

C₂

C₂×#88201

yes

characteristic

166

C₂

#11719530

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₁₃×C₂²⋊Q₈	C₂
prime radical	C₂⁴×C₁₃	C₂³
hypercenter	C₂×C₂²⋊Q₈	D₁₃
hypocenter	C₁₃	#2503
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₂×C₁₃×C₂²⋊Q₈

C₂

prime radical

C₂⁴×C₁₃

C₂³

hypercenter

C₂×C₂²⋊Q₈

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

N	H ≅ G/N	type
3. #11718891	C₂	semidirect
5. #11718647	C₂	semidirect
6. #11718638	C₂	semidirect
126. C₁₃	#2503	semidirect

H ≅ G/N

type

3. #11718891

C₂

semidirect

5. #11718647

C₂

semidirect

6. #11718638

C₂

semidirect

126. C₁₃

#2503

semidirect

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

subgroup	index
C₂×C₁₃×C₂²⋊Q₈	2
C₂²×C₄⋊Dic₁₃	2
#11718891	2
C₄×C₂²⋊Dic₁₃	2
#11718647	2
#11718638	2
#11718604	2
#2503	13

subgroup

index

C₂×C₁₃×C₂²⋊Q₈

C₂²×C₄⋊Dic₁₃

C₄×C₂²⋊Dic₁₃

Subgroups — #13119322

Conjugacy classes of subgroups

Complements

Maximal subgroups