Group #13115683 — subgroups

Subgroups — #13115683

Order 1664.

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

normal subgroups only

This group has 312 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	C₂×#88129	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88064	C₂	1	2	yes	maximal, complemented
4	C₂×#88063	C₂	1	2	yes	maximal
5	C₄×#6099	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₅₂	C₂²	1	1	yes	characteristic
7	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#6103	C₄	1	4	yes	—
11	C₂×#6099	C₂²	1	2	yes	—
12	#88129	—	2	1	no	—
13	#88129	—	2	1	no	—
14	#88129	—	2	2	no	—
15	C₁₃×C₄⋊C₈	—	2	2	no	—
16	#88064	—	2	4	no	—
17	#88063	—	2	4	no	—
18	#88043	—	2	1	no	—
19	C₄×#1055	—	2	1	no	—
20	C₂³×D₁₃	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
24	C₂×D₅₂	—	2	1	no	—
25	C₂×D₅₂	—	2	1	no	—
26	C₂×D₅₂	—	2	2	no	—
27	C₂×D₅₂	—	2	2	no	—
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	2	no	—
33	C₁₃×M₁₆	—	2	2	no	—
34	C₂×C₈×C₁₃	—	2	2	no	—
35	C₄²×C₁₃	D₄	1	1	yes	characteristic
36	C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₄²×C₁₃	D₄	1	2	yes	—
38	#6103	—	2	4	no	—
39	#6103	—	2	4	no	—
40	#6099	—	2	2	no	—
41	#6099	D₄	1	4	yes	—
42	C₂×#1055	—	2	2	no	—
43	#1967	—	13	1	no	maximal, Sylow
44	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×D₁₃	—	2	2	no	—
46	C₂²×D₁₃	—	2	4	no	—
47	C₂²×D₁₃	—	4	4	no	—
48	C₂×C₄×C₁₃	—	4	2	no	—
49	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃	#41	1	2	yes	—
55	C₂×Dic₁₃	—	2	2	no	—
56	C₂×Dic₁₃	—	2	4	no	—
57	D₅₂	—	4	2	no	—
58	D₅₂	—	4	1	no	—
59	D₅₂	—	4	1	no	—
60	Dic₂₆	—	4	2	no	—
61	Dic₂₆	—	4	1	no	—
62	Dic₂₆	—	4	1	no	—
63	C₈×C₁₃	—	4	2	no	—
64	#1055	—	2	4	no	—
65	C₂×#124	—	13	1	no	—
66	#423	D₁₃	1	1	yes	characteristic, complemented
67	C₂×#101	—	13	2	no	—
68	C₂×#100	—	13	2	no	—
69	C₄×M₁₆	—	13	1	no	—
70	C₂²×C₁₃	—	2	2	no	—
71	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₁₃	C₄×D₄	1	2	yes	—
73	C₂×D₁₃	—	4	2	no	—
74	C₂×D₁₃	—	4	4	no	—
75	C₂×D₁₃	—	8	4	no	—
76	C₄×C₁₃	—	2	2	no	—
77	C₄×C₁₃	—	2	2	no	—
78	C₄×C₁₃	#121	1	2	yes	—
79	C₄×C₁₃	#124	1	1	yes	characteristic
80	C₄×C₁₃	#127	1	1	yes	characteristic
81	Dic₁₃	—	4	2	no	—
82	C₂²×Q₈	—	13	1	no	—
83	C₂²×D₄	—	13	1	no	—
84	C₂×M₁₆	—	13	2	no	—
85	C₂×M₁₆	C₂×D₁₃	1	2	yes	—
86	#124	—	26	1	no	—
87	#124	—	26	1	no	—
88	#124	—	26	2	no	—
89	C₂×C₂²⋊C₄	—	13	4	no	—
90	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
91	C₄⋊C₈	—	2	2	no	—
92	#101	—	26	4	no	—
93	#100	—	26	4	no	—
94	#97	—	26	1	no	—
95	C₄×C₈	—	26	1	no	—
96	C₂×C₁₃	—	2	2	no	—
97	C₂×C₁₃	#390	1	1	yes	characteristic
98	C₂×C₁₃	#413	1	2	yes	—
99	D₁₃	—	8	2	no	—
100	C₂⁴	—	13	2	no	—
101	C₂×Q₈	—	26	1	no	—
102	C₂×Q₈	—	26	2	no	—
103	C₂×Q₈	—	26	2	no	—
104	C₂×Q₈	—	26	1	no	—
105	C₂×D₄	—	26	1	no	—
106	C₂×D₄	—	26	2	no	—
107	C₂×D₄	—	26	2	no	—
108	C₂×D₄	—	26	1	no	—
109	C₂²×C₄	—	13	2	no	—
110	C₂²×C₄	C₄×D₁₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
112	M₁₆	—	2	2	no	—
113	M₁₆	—	2	2	no	—
114	M₁₆	—	13	4	no	—
115	M₁₆	—	26	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	26	2	no	—
118	C₂²⋊C₄	—	26	4	no	—
119	C₂²⋊C₄	—	26	4	no	—
120	C₄²	D₅₂	1	1	yes	characteristic
121	C₄²	D₅₂	1	1	yes	characteristic
122	C₄²	C₂²⋊D₁₃	1	2	yes	—
123	C₁₃	#1967	1	1	yes	characteristic, complemented, Sylow
124	C₂³	—	26	2	no	—
125	C₂³	—	26	4	no	—
126	C₂³	—	52	4	no	—
127	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
128	Q₈	—	52	2	no	—
129	Q₈	—	52	1	no	—
130	Q₈	—	52	1	no	—
131	D₄	—	52	2	no	—
132	D₄	—	52	1	no	—
133	D₄	—	52	1	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	26	4	no	—
136	C₂×C₄	—	26	2	no	—
137	C₂×C₄	#6103	1	2	yes	—
138	C₂×C₄	#6103	1	2	yes	—
139	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
140	C₂×C₄	D₄×D₁₃	1	2	yes	—
141	C₂×C₄	#6131	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
143	C₈	—	4	2	no	—
144	C₈	—	26	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	52	4	no	—
147	C₂²	—	52	2	no	—
148	C₂²	—	104	4	no	—
149	C₂²	#88147	1	2	yes	—
150	C₂²	C₂×#6103	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	52	2	no	—
154	C₄	#88149	1	2	yes	—
155	C₄	#88194	1	1	yes	characteristic
156	C₄	#88201	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	—	104	2	no	—
159	C₂	#11718649	1	1	yes	characteristic
160	C₂	#11718793	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88129

C₂

yes

maximal, characteristic

C₁₃×#423

C₂

yes

maximal, characteristic, complemented

C₂×#88064

C₂

yes

maximal, complemented

C₂×#88063

C₂

yes

maximal

C₄×#6099

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₂

C₂²

yes

characteristic

C₂²×Dic₂₆

C₂²

yes

characteristic

C₂×C₁₃×M₁₆

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×#6103

C₄

yes

—

C₂×#6099

C₂²

yes

—

#88129

—

#88129

—

#88129

—

C₁₃×C₄⋊C₈

—

—

—

—

—

C₂³×D₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂×C₄

yes

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

D₄

yes

characteristic

C₄²×C₁₃

D₄

yes

characteristic

C₄²×C₁₃

D₄

yes

—

—

—

—

D₄

yes

—

C₂×#1055

—

#1967

—

maximal, Sylow

C₂³×C₁₃

C₂²×C₄

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

D₅₂

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

#1055

—

C₂×#124

—

#423

D₁₃

yes

characteristic, complemented

C₂×#101

—

C₂×#100

—

C₄×M₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃

C₄×D₄

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#121

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

C₂²×Q₈

—

C₂²×D₄

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₃

yes

—

#124

—

#124

—

#124

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

#101

—

#100

—

#97

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

#390

yes

characteristic

C₂×C₁₃

#413

yes

—

D₁₃

—

100

C₂⁴

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄×D₁₃

yes

—

111

C₂²×C₄

C₂²×D₁₃

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

D₅₂

yes

characteristic

121

C₄²

D₅₂

yes

characteristic

122

C₄²

C₂²⋊D₁₃

yes

—

123

C₁₃

#1967

yes

characteristic, complemented, Sylow

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₂×C₄×D₁₃

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#6103

yes

—

138

C₂×C₄

#6103

yes

—

139

C₂×C₄

C₂×D₅₂

yes

characteristic

140

C₂×C₄

D₄×D₁₃

yes

—

141

C₂×C₄

#6131

yes

—

142

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

143

C₈

—

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

104

—

149

C₂²

#88147

yes

—

150

C₂²

C₂×#6103

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#88149

yes

—

155

C₄

#88194

yes

characteristic

156

C₄

#88201

yes

characteristic

157

C₂

—

158

C₂

—

104

—

159

C₂

#11718649

yes

characteristic

160

C₂

#11718793

yes

—

	subgroup	G/H
center	C₂²	C₂×#6103
commutator	C₂³×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#423	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₅₂	C₂²
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	#423	D₁₃
hypocenter	C₁₃	#1967
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×#6103

commutator

C₂³×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#423

C₂

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

prime radical

C₂²×D₅₂

C₂²

cosocle

C₂²×C₄×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. C₁₃×#423	C₂	semidirect
3. C₂×#88064	C₂	semidirect
5. C₄×#6099	C₂	semidirect
66. #423	D₁₃	semidirect
123. C₁₃	#1967	semidirect

H ≅ G/N

type

2. C₁₃×#423

C₂

semidirect

3. C₂×#88064

C₂

semidirect

5. C₄×#6099

C₂

semidirect

66. #423

D₁₃

semidirect

123. C₁₃

#1967

semidirect

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

subgroup	index
C₂×#88129	2
C₁₃×#423	2
C₂×#88064	2
C₂×#88063	2
C₄×#6099	2
#1967	13

subgroup

index

C₁₃×#423

Subgroups — #13115683

Conjugacy classes of subgroups

Complements

Maximal subgroups