Group #13114918 — subgroups

Subgroups — #13114918

Order 1664.

272 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 77 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×#88146	C₂	1	1	yes	maximal, characteristic
2	#11718787	C₂	1	2	yes	maximal
3	C₁₃×#343	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718146	C₂	1	2	yes	maximal, complemented
5	#11718122	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
8	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
9	C₂×#6103	C₄	1	2	yes	—
10	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₃	C₄	1	2	yes	—
12	C₂×#6099	C₂²	1	2	yes	—
13	#88146	—	2	4	no	—
14	#88060	—	2	4	no	—
15	C₂³×D₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
20	C₂×C₄×D₁₃	—	2	2	no	—
21	C₂×C₄×D₁₃	—	2	1	no	—
22	C₂×C₄×D₁₃	—	2	1	no	—
23	C₂×C₄×D₁₃	D₄	1	4	yes	—
24	C₁₃×M₁₆	—	2	2	no	—
25	C₁₃×M₁₆	—	2	2	no	—
26	C₂×C₈×C₁₃	—	2	2	no	—
27	C₁₃×C₄⋊C₄	—	2	2	no	—
28	#6103	—	2	4	no	—
29	C₄⋊Dic₁₃	—	2	2	no	—
30	C₄×Dic₁₃	—	2	4	no	—
31	#6099	—	2	2	no	—
32	#6099	—	2	2	no	—
33	C₂×#1055	—	2	2	no	—
34	#1822	—	13	1	no	maximal, Sylow
35	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×D₁₃	—	2	1	no	—
37	C₂²×D₁₃	—	2	1	no	—
38	C₂²×D₁₃	—	4	1	no	—
39	C₂²×D₁₃	—	4	1	no	—
40	C₂²×D₁₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₃	—	2	2	no	—
42	C₂×C₄×C₁₃	—	4	2	no	—
43	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₃	#41	1	2	yes	—
45	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
46	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
47	C₂×Dic₁₃	—	2	1	no	—
48	C₂×Dic₁₃	—	2	1	no	—
49	C₂×Dic₁₃	—	2	2	no	—
50	C₂×Dic₁₃	—	4	2	no	—
51	C₂×Dic₁₃	C₂²⋊C₄	1	2	yes	—
52	C₄×D₁₃	—	2	4	no	—
53	C₄×D₁₃	—	2	4	no	—
54	C₈×C₁₃	—	4	2	no	—
55	#1055	—	4	2	no	—
56	C₂×#117	—	13	1	no	—
57	#407	—	13	2	no	—
58	#343	—	13	2	no	—
59	#343	—	13	1	no	—
60	#343	D₁₃	1	1	yes	characteristic, complemented
61	C₂²×C₁₃	—	2	1	no	—
62	C₂²×C₁₃	—	2	1	no	—
63	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₁₃	#117	1	1	yes	characteristic
65	C₂²×C₁₃	#117	1	1	yes	characteristic
66	C₂×D₁₃	—	2	2	no	—
67	C₂×D₁₃	—	2	2	no	—
68	C₂×D₁₃	—	2	2	no	—
69	C₂×D₁₃	—	4	2	no	—
70	C₂×D₁₃	—	4	2	no	—
71	C₄×C₁₃	—	4	2	no	—
72	C₄×C₁₃	#123	1	2	yes	—
73	C₄×C₁₃	#123	1	2	yes	—
74	Dic₁₃	—	2	2	no	—
75	Dic₁₃	—	4	2	no	—
76	C₂³×C₄	—	13	1	no	—
77	C₂×M₁₆	—	13	2	no	—
78	C₂×M₁₆	C₂×D₁₃	1	2	yes	—
79	#117	—	26	4	no	—
80	C₂×C₄⋊C₄	—	13	1	no	—
81	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
82	C₂×C₂²⋊C₄	—	13	2	no	—
83	C₂×C₄²	—	13	2	no	—
84	C₂²⋊C₈	—	26	4	no	—
85	C₂×C₁₃	—	2	1	no	—
86	C₂×C₁₃	—	2	1	no	—
87	C₂×C₁₃	#381	1	1	yes	characteristic
88	C₂×C₁₃	#413	1	1	yes	characteristic
89	C₂×C₁₃	#413	1	1	yes	characteristic
90	D₁₃	—	4	2	no	—
91	C₂⁴	—	13	1	no	—
92	C₂²×C₄	—	13	1	no	—
93	C₂²×C₄	—	13	4	no	—
94	C₂²×C₄	—	13	2	no	—
95	C₂²×C₄	—	26	2	no	—
96	C₂²×C₄	—	26	1	no	—
97	C₂²×C₄	—	26	1	no	—
98	C₂²×C₄	C₄×D₁₃	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
100	M₁₆	—	2	2	no	—
101	M₁₆	—	2	2	no	—
102	M₁₆	—	26	2	no	—
103	M₁₆	—	26	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	26	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	26	2	no	—
108	C₂²⋊C₄	—	26	4	no	—
109	C₄²	—	26	4	no	—
110	C₁₃	#1822	1	1	yes	characteristic, complemented, Sylow
111	C₂³	—	13	2	no	—
112	C₂³	—	26	1	no	—
113	C₂³	—	26	1	no	—
114	C₂³	—	52	1	no	—
115	C₂³	—	52	1	no	—
116	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	13	2	no	—
120	C₂×C₄	—	26	1	no	—
121	C₂×C₄	—	26	1	no	—
122	C₂×C₄	—	26	4	no	—
123	C₂×C₄	—	26	4	no	—
124	C₂×C₄	—	26	2	no	—
125	C₂×C₄	—	52	2	no	—
126	C₂×C₄	#6127	1	2	yes	—
127	C₂×C₄	D₄×D₁₃	1	2	yes	—
128	C₂×C₄	#6129	1	1	yes	characteristic
129	C₂×C₄	#6131	1	1	yes	characteristic
130	C₈	—	4	2	no	—
131	C₈	—	52	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	26	2	no	—
135	C₂²	—	26	2	no	—
136	C₂²	—	26	2	no	—
137	C₂²	—	52	2	no	—
138	C₂²	—	52	2	no	—
139	C₂²	#88126	1	1	yes	characteristic
140	C₂²	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
141	C₂²	#88146	1	1	yes	characteristic
142	C₄	—	4	2	no	—
143	C₄	—	26	2	no	—
144	C₄	—	52	2	no	—
145	C₄	#88137	1	2	yes	—
146	C₄	#88148	1	2	yes	—
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	—	52	2	no	—
150	C₂	#11718325	1	1	yes	characteristic
151	C₂	#11718406	1	1	yes	characteristic
152	C₂	#11718412	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88146

C₂

yes

maximal, characteristic

#11718787

C₂

yes

maximal

C₁₃×#343

C₂

yes

maximal, characteristic, complemented

#11718146

C₂

yes

maximal, complemented

#11718122

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₃

C₂²

yes

characteristic

C₂×C₁₃×M₁₆

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×#6103

C₄

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄×Dic₁₃

C₄

yes

—

C₂×#6099

C₂²

yes

—

#88146

—

#88060

—

C₂³×D₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂×C₄

yes

characteristic

C₂²×Dic₁₃

C₂×C₄

yes

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

D₄

yes

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

#6103

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

—

—

—

—

C₂³×C₁₃

C₂²×C₄

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂²⋊C₄

yes

—

C₄×D₁₃

—

C₄×D₁₃

—

C₈×C₁₃

—

—

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#123

yes

—

C₄×C₁₃

#123

yes

—

Dic₁₃

—

Dic₁₃

—

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₃

yes

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

D₁₃

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₁₃

#1822

yes

characteristic, complemented, Sylow

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

—

115

C₂³

—

116

C₂³

C₂×C₄×D₁₃

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

#6127

yes

—

127

C₂×C₄

D₄×D₁₃

yes

—

128

C₂×C₄

#6129

yes

characteristic

129

C₂×C₄

#6131

yes

characteristic

130

C₈

—

131

C₈

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

#88126

yes

characteristic

140

C₂²

C₂²⋊C₄×D₁₃

yes

characteristic

141

C₂²

#88146

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#88137

yes

—

146

C₄

#88148

yes

—

147

C₂

—

148

C₂

—

149

C₂

—

150

C₂

#11718325

yes

characteristic

151

C₂

#11718406

yes

characteristic

152

C₂

#11718412

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₃
commutator	C₂³×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#343	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂³×D₁₃	C₂×C₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	#343	D₁₃
hypocenter	C₁₃	#1822
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₃

commutator

C₂³×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#343

C₂

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

prime radical

C₂³×D₁₃

C₂×C₄

cosocle

C₂²×C₄×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
3. C₁₃×#343	C₂	semidirect
4. #11718146	C₂	semidirect
5. #11718122	C₂	semidirect
60. #343	D₁₃	semidirect
110. C₁₃	#1822	semidirect

H ≅ G/N

type

3. C₁₃×#343

C₂

semidirect

4. #11718146

C₂

semidirect

5. #11718122

C₂

semidirect

60. #343

D₁₃

semidirect

110. C₁₃

#1822

semidirect

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

subgroup	index
C₂×#88146	2
#11718787	2
C₁₃×#343	2
#11718146	2
#11718122	2
#1822	13

subgroup

index

C₁₃×#343

Subgroups — #13114918

Conjugacy classes of subgroups

Complements

Maximal subgroups