Group #13115388 — subgroups

Subgroups — #13115388

Order 1664.

308 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₁₃×#568	C₂	1	1	yes	maximal, characteristic
2	#11719530	C₂	1	1	yes	maximal, characteristic
3	#11718898	C₂	1	1	yes	maximal, characteristic
4	C₂×#88064	C₂	1	2	yes	maximal, complemented
5	#11718151	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
7	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
8	C₁₃×#131	—	2	4	no	—
9	C₂×C₁₃×M₁₆	—	2	2	no	—
10	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
11	#88198	—	2	1	no	—
12	#88190	—	2	1	no	—
13	#88189	C₄	1	4	yes	—
14	#88180	C₂²	1	1	yes	characteristic
15	#88178	—	2	2	no	—
16	C₂×#6099	C₂²	1	2	yes	—
17	#88076	—	2	1	no	—
18	#88073	—	2	1	no	—
19	#88064	—	2	4	no	—
20	C₁₃×#41	—	2	2	no	—
21	C₁₃×#41	—	2	2	no	—
22	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
23	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
24	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₃	D₄	1	2	yes	—
26	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
28	C₂×Dic₂₆	—	2	2	no	—
29	C₂×Dic₂₆	—	2	1	no	—
30	C₂×Dic₂₆	—	2	1	no	—
31	C₂×Dic₂₆	—	2	2	no	—
32	C₁₃×M₁₆	—	2	4	no	—
33	C₁₃×M₁₆	—	2	2	no	—
34	C₁₃×M₁₆	D₄	1	4	yes	—
35	C₂×C₈×C₁₃	—	2	4	no	—
36	C₂×C₈×C₁₃	—	2	2	no	—
37	C₂²⋊Dic₁₃	—	2	4	no	—
38	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
39	C₄⋊Dic₁₃	—	2	1	no	—
40	#6101	—	2	4	no	—
41	C₄×Dic₁₃	—	2	1	no	—
42	#6099	—	2	2	no	—
43	#6099	—	2	2	no	—
44	C₂×#1055	—	2	2	no	—
45	#1915	—	13	1	no	maximal, Sylow
46	C₂³×C₁₃	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	Q₈×C₁₃	—	2	2	no	—
49	D₄×C₁₃	—	2	4	no	—
50	D₄×C₁₃	—	2	2	no	—
51	C₂×C₄×C₁₃	—	2	2	no	—
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₃	#41	1	2	yes	—
58	C₂×Dic₁₃	—	2	2	no	—
59	C₂×Dic₁₃	—	2	2	no	—
60	C₂×Dic₁₃	—	2	4	no	—
61	Dic₂₆	—	4	1	no	—
62	Dic₂₆	—	4	2	no	—
63	Dic₂₆	—	4	1	no	—
64	C₈×C₁₃	—	2	4	no	—
65	#1055	—	4	2	no	—
66	#568	D₁₃	1	1	yes	characteristic
67	#536	—	13	1	no	—
68	#413	—	13	1	no	—
69	C₂×#101	—	13	2	no	—
70	#344	—	13	2	no	—
71	C₂²×C₁₃	—	2	2	no	—
72	C₂²×C₁₃	—	4	1	no	—
73	C₂²×C₁₃	—	4	2	no	—
74	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₁₃	C₄×D₄	1	2	yes	—
76	C₄×C₁₃	—	2	2	no	—
77	C₄×C₁₃	#120	1	1	yes	characteristic
78	C₄×C₁₃	#121	1	2	yes	—
79	C₄×C₁₃	#123	1	1	yes	characteristic
80	Dic₁₃	—	4	2	no	—
81	Dic₁₃	—	4	2	no	—
82	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
83	C₂²×Q₈	—	13	1	no	—
84	#131	—	2	4	no	—
85	C₂×M₁₆	—	2	2	no	—
86	C₂×M₁₆	—	13	2	no	—
87	C₂×M₁₆	C₂×D₁₃	1	2	yes	—
88	C₄⋊Q₈	—	26	1	no	—
89	#124	—	26	1	no	—
90	#123	—	13	4	no	—
91	C₂²⋊Q₈	—	26	2	no	—
92	#117	—	13	1	no	—
93	#101	—	26	1	no	—
94	#101	—	26	4	no	—
95	#100	—	26	1	no	—
96	C₂×C₁₃	—	2	1	no	—
97	C₂×C₁₃	—	2	2	no	—
98	C₂×C₁₃	—	4	2	no	—
99	C₂×C₁₃	#386	1	1	yes	characteristic
100	#41	—	2	2	no	—
101	#41	—	2	2	no	—
102	C₂×Q₈	—	26	1	no	—
103	C₂×Q₈	—	26	2	no	—
104	C₂×Q₈	—	26	2	no	—
105	C₂×Q₈	—	26	1	no	—
106	C₂×Q₈	D₅₂	1	1	yes	characteristic
107	C₂×D₄	C₄×D₁₃	1	2	yes	—
108	C₂×D₄	D₅₂	1	1	yes	characteristic
109	C₂²×C₄	—	13	2	no	—
110	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
112	M₁₆	—	2	4	no	—
113	M₁₆	—	2	2	no	—
114	M₁₆	—	26	2	no	—
115	M₁₆	—	26	2	no	—
116	M₁₆	C₂²⋊D₁₃	1	4	yes	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	26	2	no	—
120	C₄⋊C₄	—	26	4	no	—
121	C₄⋊C₄	—	26	1	no	—
122	C₂²⋊C₄	—	13	2	no	—
123	C₂²⋊C₄	—	26	4	no	—
124	C₄²	—	26	1	no	—
125	C₁₃	#1915	1	1	yes	characteristic, complemented, Sylow
126	C₂³	#6103	1	2	yes	—
127	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
128	Q₈	—	2	2	no	—
129	Q₈	—	52	1	no	—
130	Q₈	—	52	2	no	—
131	Q₈	—	52	1	no	—
132	D₄	—	2	4	no	—
133	D₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	26	2	no	—
137	C₂×C₄	—	26	2	no	—
138	C₂×C₄	—	26	4	no	—
139	C₂×C₄	#6103	1	2	yes	—
140	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
141	C₂×C₄	#6127	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
144	C₈	—	2	4	no	—
145	C₈	—	52	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	2	no	—
149	C₂²	C₂×#6103	1	1	yes	characteristic
150	C₂²	C₄×C₂²⋊D₁₃	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	52	2	no	—
153	C₄	—	52	2	no	—
154	C₄	#88183	1	1	yes	characteristic
155	C₄	#88184	1	2	yes	—
156	C₄	#88207	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#11718874	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#568

C₂

yes

maximal, characteristic

#11719530

C₂

yes

maximal, characteristic

#11718898

C₂

yes

maximal, characteristic

C₂×#88064

C₂

yes

maximal, complemented

#11718151

C₂

yes

maximal, complemented

C₂×C₁₃×#41

C₂²

yes

characteristic

C₂²×Dic₂₆

C₂²

yes

characteristic

C₁₃×#131

—

C₂×C₁₃×M₁₆

—

C₂×C₁₃×M₁₆

C₂²

yes

—

#88198

—

#88190

—

#88189

C₄

yes

—

#88180

C₂²

yes

characteristic

#88178

—

C₂×#6099

C₂²

yes

—

#88076

—

#88073

—

#88064

—

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

D₄

yes

characteristic

C₂×D₄×C₁₃

C₂×C₄

yes

—

C₂×D₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂×C₄

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

D₄

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂²⋊Dic₁₃

—

C₂²⋊Dic₁₃

C₂×C₄

yes

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

—

—

—

—

C₂³×C₁₃

C₂²⋊C₄

yes

—

C₂³×C₁₃

C₂²×C₄

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

#1055

—

#568

D₁₃

yes

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃

C₄×D₄

yes

—

C₄×C₁₃

—

C₄×C₁₃

#120

yes

characteristic

C₄×C₁₃

#121

yes

—

C₄×C₁₃

#123

yes

characteristic

Dic₁₃

—

Dic₁₃

—

C₂×#41

C₂×D₁₃

yes

characteristic

C₂²×Q₈

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₃

yes

—

C₄⋊Q₈

—

#124

—

#123

—

C₂²⋊Q₈

—

—

—

—

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

#386

yes

characteristic

100

#41

—

101

#41

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

D₅₂

yes

characteristic

107

C₂×D₄

C₄×D₁₃

yes

—

108

C₂×D₄

D₅₂

yes

characteristic

109

C₂²×C₄

—

110

C₂²×C₄

C₂²⋊D₁₃

yes

—

111

C₂²×C₄

C₂²×D₁₃

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

C₂²⋊D₁₃

yes

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₂²⋊C₄

—

123

C₂²⋊C₄

—

124

C₄²

—

125

C₁₃

#1915

yes

characteristic, complemented, Sylow

126

C₂³

#6103

yes

—

127

C₂³

C₂×C₄×D₁₃

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

#6103

yes

—

140

C₂×C₄

C₂×D₅₂

yes

characteristic

141

C₂×C₄

#6127

yes

—

142

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

143

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

144

C₈

—

145

C₈

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×#6103

yes

characteristic

150

C₂²

C₄×C₂²⋊D₁₃

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#88183

yes

characteristic

155

C₄

#88184

yes

—

156

C₄

#88207

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#11718874

yes

characteristic

	subgroup	G/H
center	C₂	#11718874
commutator	C₂³×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#568	C₂
socle	C₂×C₁₃	#386
prime radical	C₂×C₁₃×#41	C₂²
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	#568	D₁₃
hypocenter	C₁₃	#1915
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₂

#11718874

commutator

C₂³×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#568

C₂

socle

C₂×C₁₃

#386

prime radical

C₂×C₁₃×#41

C₂²

cosocle

C₂²×C₄×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
4. C₂×#88064	C₂	semidirect
5. #11718151	C₂	semidirect
125. C₁₃	#1915	semidirect

H ≅ G/N

type

4. C₂×#88064

C₂

semidirect

5. #11718151

C₂

semidirect

125. C₁₃

#1915

semidirect

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

subgroup	index
C₁₃×#568	2
#11719530	2
#11718898	2
C₂×#88064	2
#11718151	2
#1915	13

subgroup

index

C₁₃×#568

Subgroups — #13115388

Conjugacy classes of subgroups

Complements

Maximal subgroups