Group #13115359 — subgroups

Subgroups — #13115359

Order 1664.

312 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 93 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×#536	C₂	1	1	yes	maximal, characteristic
2	#11719524	C₂	1	1	yes	maximal, characteristic
3	#11718901	C₂	1	1	yes	maximal, characteristic
4	#11718886	C₂	1	2	yes	maximal, complemented
5	#11718123	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
7	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₁₃×C₄⋊Q₈	C₄	1	2	yes	complemented
9	C₁₃×#124	C₄	1	2	yes	complemented
10	C₁₃×#123	—	2	2	no	—
11	C₁₃×C₂²⋊Q₈	—	2	2	no	—
12	C₁₃×#117	C₂²	1	1	yes	characteristic
13	#88202	—	2	4	no	—
14	#88180	C₂²	1	2	yes	—
15	C₂×#6099	—	2	2	no	—
16	C₂×#6099	C₂²	1	2	yes	—
17	#88075	—	2	4	no	—
18	#88074	—	2	2	no	—
19	C₁₃×#41	—	2	2	no	—
20	C₁₃×#41	—	2	2	no	—
21	C₂×Q₈×C₁₃	—	2	1	no	—
22	C₂×Q₈×C₁₃	—	2	2	no	—
23	C₂×Q₈×C₁₃	—	2	2	no	—
24	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×D₄×C₁₃	D₄	1	2	yes	—
28	C₂²×C₄×C₁₃	—	2	1	no	—
29	C₂²×C₄×C₁₃	D₄	1	2	yes	—
30	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₁₃×C₄⋊C₄	—	2	4	no	—
32	C₁₃×C₄⋊C₄	—	2	1	no	—
33	C₁₃×C₂²⋊C₄	—	2	4	no	—
34	C₁₃×C₂²⋊C₄	—	2	1	no	—
35	C₄²×C₁₃	C₂×C₄	1	2	yes	—
36	C₂²⋊Dic₁₃	—	2	2	no	—
37	C₄⋊Dic₁₃	—	2	2	no	—
38	C₄×Dic₁₃	—	2	2	no	—
39	#6099	—	2	4	no	—
40	#6099	—	2	2	no	—
41	#6099	—	2	2	no	—
42	C₂×#1055	—	2	4	no	—
43	C₂×#1055	D₄	1	4	yes	—
44	#1909	—	13	1	no	maximal, Sylow
45	C₂³×C₁₃	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
47	Q₈×C₁₃	—	2	2	no	—
48	Q₈×C₁₃	—	2	4	no	—
49	Q₈×C₁₃	—	4	1	no	—
50	Q₈×C₁₃	—	4	1	no	—
51	D₄×C₁₃	—	2	4	no	—
52	D₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	—	2	2	no	—
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	—	2	2	no	—
57	C₂×C₄×C₁₃	—	4	2	no	—
58	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₃	#41	1	2	yes	—
63	C₂×Dic₁₃	—	4	2	no	—
64	#1055	—	2	4	no	—
65	#568	—	13	1	no	—
66	#536	D₁₃	1	1	yes	characteristic
67	#419	—	13	2	no	—
68	#410	—	13	1	no	—
69	H(3,ℤ/4)	—	13	2	no	—
70	C₂²×C₁₃	—	2	2	no	—
71	C₂²×C₁₃	—	4	1	no	—
72	C₂²×C₁₃	—	4	2	no	—
73	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₁₃	#121	1	2	yes	—
75	C₄×C₁₃	—	2	2	no	—
76	C₄×C₁₃	—	4	2	no	—
77	C₄×C₁₃	—	4	2	no	—
78	C₄×C₁₃	C₄×D₄	1	2	yes	—
79	C₄×C₁₃	#120	1	1	yes	characteristic
80	C₄×C₁₃	#123	1	1	yes	characteristic
81	Dic₁₃	—	8	2	no	—
82	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
83	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
84	#131	—	26	4	no	—
85	C₂×M₁₆	—	13	2	no	—
86	C₂×M₁₆	—	26	2	no	—
87	C₄⋊Q₈	Dic₁₃	1	2	yes	complemented
88	#124	Dic₁₃	1	2	yes	complemented
89	#123	—	2	2	no	—
90	C₂²⋊Q₈	—	2	2	no	—
91	#117	—	13	2	no	—
92	#117	C₂×D₁₃	1	1	yes	characteristic
93	#103	—	26	4	no	—
94	#99	—	26	2	no	—
95	C₂×C₁₃	—	2	2	no	—
96	C₂×C₁₃	—	2	1	no	—
97	C₂×C₁₃	—	4	2	no	—
98	C₂×C₁₃	#386	1	1	yes	characteristic
99	#41	—	2	2	no	—
100	#41	—	2	2	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	C₂×Dic₁₃	1	2	yes	—
105	C₂×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
106	C₂×D₄	C₂×Dic₁₃	1	1	yes	characteristic
107	C₂×D₄	C₂²⋊D₁₃	1	2	yes	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
110	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
111	M₁₆	—	26	4	no	—
112	M₁₆	—	26	2	no	—
113	M₁₆	—	26	2	no	—
114	C₂×C₈	—	13	4	no	—
115	C₂×C₈	—	26	4	no	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	26	2	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₂²⋊C₄	—	2	1	no	—
121	C₂²⋊C₄	—	26	2	no	—
122	C₄²	—	26	2	no	—
123	C₄²	C₂×Dic₁₃	1	2	yes	—
124	C₁₃	#1909	1	1	yes	characteristic, complemented, Sylow
125	C₂³	C₂²⋊Dic₁₃	1	2	yes	—
126	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
127	Q₈	—	2	2	no	—
128	Q₈	—	2	4	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	1	no	—
131	D₄	—	2	4	no	—
132	D₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	52	2	no	—
139	C₂×C₄	C₂²⋊Dic₁₃	1	2	yes	—
140	C₂×C₄	D₄×D₁₃	1	2	yes	—
141	C₂×C₄	#6129	1	2	yes	—
142	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
144	C₈	—	26	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	2	no	—
148	C₂²	#88193	1	2	yes	—
149	C₂²	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	4	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	104	2	no	—
154	C₄	D₄×Dic₁₃	1	2	yes	—
155	C₄	#88189	1	1	yes	characteristic
156	C₄	#88191	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#11718881	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#536

C₂

yes

maximal, characteristic

#11719524

C₂

yes

maximal, characteristic

#11718901

C₂

yes

maximal, characteristic

#11718886

C₂

yes

maximal, complemented

#11718123

C₂

yes

maximal, complemented

C₂×C₁₃×#41

C₂²

yes

characteristic

C₂²×Q₈×C₁₃

C₂²

yes

characteristic

C₁₃×C₄⋊Q₈

C₄

yes

complemented

C₁₃×#124

C₄

yes

complemented

C₁₃×#123

—

C₁₃×C₂²⋊Q₈

—

C₁₃×#117

C₂²

yes

characteristic

#88202

—

#88180

C₂²

yes

—

C₂×#6099

—

C₂×#6099

C₂²

yes

—

#88075

—

#88074

—

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

C₂×Q₈×C₁₃

C₂×C₄

yes

characteristic

C₂×D₄×C₁₃

C₂×C₄

yes

characteristic

C₂×D₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

C₂×C₄

yes

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

—

—

—

—

D₄

yes

—

#1909

—

maximal, Sylow

C₂³×C₁₃

C₂²⋊C₄

yes

—

C₂³×C₁₃

C₂×D₄

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

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

—

#1055

—

#568

—

#536

D₁₃

yes

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃

#121

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×D₄

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

C₂×#41

C₂×D₁₃

yes

characteristic

C₂²×Q₈

C₂×D₁₃

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

Dic₁₃

yes

complemented

#124

Dic₁₃

yes

complemented

#123

—

C₂²⋊Q₈

—

#117

—

#117

C₂×D₁₃

yes

characteristic

#103

—

#99

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

#386

yes

characteristic

#41

—

100

#41

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

C₂×Dic₁₃

yes

—

105

C₂×Q₈

C₂×Dic₁₃

yes

characteristic

106

C₂×D₄

C₂×Dic₁₃

yes

characteristic

107

C₂×D₄

C₂²⋊D₁₃

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂²⋊D₁₃

yes

—

110

C₂²×C₄

C₂²×D₁₃

yes

characteristic

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

—

123

C₄²

C₂×Dic₁₃

yes

—

124

C₁₃

#1909

yes

characteristic, complemented, Sylow

125

C₂³

C₂²⋊Dic₁₃

yes

—

126

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

127

Q₈

—

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂²⋊Dic₁₃

yes

—

140

C₂×C₄

D₄×D₁₃

yes

—

141

C₂×C₄

#6129

yes

—

142

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

143

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

#88193

yes

—

149

C₂²

C₂×C₂²⋊Dic₁₃

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

104

—

154

C₄

D₄×Dic₁₃

yes

—

155

C₄

#88189

yes

characteristic

156

C₄

#88191

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#11718881

yes

characteristic

	subgroup	G/H
center	C₂	#11718881
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#536	C₂
socle	C₂×C₁₃	#386
prime radical	C₂×C₁₃×#41	C₂²
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	#536	D₁₃
hypocenter	C₁₃	#1909
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

subgroup

G/H

center

C₂

#11718881

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#536

C₂

socle

C₂×C₁₃

#386

prime radical

C₂×C₁₃×#41

C₂²

cosocle

C₂²×C₄×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #536

N	H ≅ G/N	type
4. #11718886	C₂	semidirect
5. #11718123	C₂	semidirect
8. C₁₃×C₄⋊Q₈	C₄	semidirect
9. C₁₃×#124	C₄	semidirect
87. C₄⋊Q₈	Dic₁₃	semidirect
88. #124	Dic₁₃	semidirect
124. C₁₃	#1909	semidirect

H ≅ G/N

type

4. #11718886

C₂

semidirect

5. #11718123

C₂

semidirect

8. C₁₃×C₄⋊Q₈

C₄

semidirect

9. C₁₃×#124

C₄

semidirect

87. C₄⋊Q₈

Dic₁₃

semidirect

88. #124

Dic₁₃

semidirect

124. C₁₃

#1909

semidirect

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

subgroup	index
C₁₃×#536	2
#11719524	2
#11718901	2
#11718886	2
#11718123	2
#1909	13

subgroup

index

C₁₃×#536

Subgroups — #13115359

Conjugacy classes of subgroups

Complements

Maximal subgroups