Group #13116413 — subgroups

Subgroups — #13116413

Order 1664.

158 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 57 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	#11718785	C₂	1	1	yes	maximal, characteristic
2	#11718779	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#344	C₂	1	1	yes	maximal, characteristic
4	#11718151	C₂	1	1	yes	maximal, characteristic
5	#11718147	C₂	1	1	yes	maximal, characteristic
6	#11718124	C₂	1	1	yes	maximal, characteristic
7	#11718123	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₁₃×#117	C₂²	1	1	yes	characteristic
11	#88180	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×#6099	C₂²	1	1	yes	characteristic
14	C₂×#6099	C₂²	1	1	yes	characteristic
15	#88062	—	2	2	no	—
16	#88054	—	2	2	no	—
17	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₁₃	—	2	1	no	—
19	C₁₃×M₁₆	—	2	1	no	—
20	C₁₃×M₁₆	—	2	1	no	—
21	C₁₃×M₁₆	—	2	1	no	—
22	C₁₃×M₁₆	—	2	1	no	—
23	C₂×C₈×C₁₃	—	2	1	no	—
24	C₂×C₈×C₁₃	—	2	1	no	—
25	C₁₃×C₄⋊C₄	—	2	1	no	—
26	C₁₃×C₂²⋊C₄	—	2	1	no	—
27	C₄²×C₁₃	—	2	1	no	—
28	C₂²⋊Dic₁₃	—	2	1	no	—
29	C₄⋊Dic₁₃	—	2	1	no	—
30	C₄×Dic₁₃	—	2	1	no	—
31	C₄×Dic₁₃	—	2	1	no	—
32	C₄×Dic₁₃	Q₈	1	1	yes	characteristic
33	C₄×Dic₁₃	Q₈	1	1	yes	characteristic
34	#6099	—	2	1	no	—
35	#6099	—	2	1	no	—
36	#6099	—	2	1	no	—
37	#6099	—	2	1	no	—
38	C₂×#1055	—	2	1	no	—
39	C₂×#1055	D₄	1	2	yes	—
40	#2102	—	13	1	no	maximal, Sylow
41	C₂³×C₁₃	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₃	—	4	1	no	—
43	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
46	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
47	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
48	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
49	C₂×Dic₁₃	—	2	1	no	—
50	C₂×Dic₁₃	—	2	1	no	—
51	C₂×Dic₁₃	—	4	2	no	—
52	C₂×Dic₁₃	—	4	1	no	—
53	C₈×C₁₃	—	4	1	no	—
54	C₈×C₁₃	—	4	1	no	—
55	#1055	—	2	2	no	—
56	#1055	—	4	1	no	—
57	#430	—	13	1	no	—
58	#423	—	13	1	no	—
59	#344	—	13	1	no	—
60	#344	D₁₃	1	1	yes	characteristic
61	#339	—	13	1	no	—
62	#339	—	13	1	no	—
63	H(3,ℤ/4)	—	13	1	no	—
64	C₂²×C₁₃	—	4	1	no	—
65	C₂²×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
66	C₂²×C₁₃	#126	1	1	yes	characteristic
67	C₂²×C₁₃	#126	1	1	yes	characteristic
68	C₄×C₁₃	—	8	1	no	—
69	C₄×C₁₃	#123	1	1	yes	characteristic
70	C₄×C₁₃	#124	1	1	yes	characteristic
71	C₄×C₁₃	#125	1	1	yes	characteristic
72	C₄×C₁₃	#125	1	1	yes	characteristic
73	Dic₁₃	—	2	2	no	—
74	Dic₁₃	—	4	1	no	—
75	Dic₁₃	—	8	1	no	—
76	C₂×M₁₆	—	13	1	no	—
77	C₂×M₁₆	—	13	1	no	—
78	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
79	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
80	#117	—	13	1	no	—
81	#117	C₂×D₁₃	1	1	yes	characteristic
82	C₂×C₄²	—	13	1	no	—
83	#108	—	26	2	no	—
84	C₄⋊C₈	—	26	2	no	—
85	C₂×C₁₃	—	2	1	no	—
86	C₂×C₁₃	—	2	1	no	—
87	C₂×C₁₃	—	2	1	no	—
88	C₂×C₁₃	#400	1	1	yes	characteristic
89	C₂²×C₄	—	26	1	no	—
90	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
91	M₁₆	—	2	1	no	—
92	M₁₆	—	2	1	no	—
93	M₁₆	—	2	1	no	—
94	M₁₆	—	2	1	no	—
95	M₁₆	—	26	1	no	—
96	M₁₆	—	26	1	no	—
97	M₁₆	—	26	1	no	—
98	M₁₆	—	26	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	13	2	no	—
102	C₂×C₈	—	26	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	26	1	no	—
105	C₂²⋊C₄	—	2	1	no	—
106	C₂²⋊C₄	—	26	1	no	—
107	C₄²	—	2	1	no	—
108	C₄²	—	13	1	no	—
109	C₄²	—	13	1	no	—
110	C₄²	—	26	1	no	—
111	C₄²	—	26	1	no	—
112	C₁₃	#2102	1	1	yes	characteristic, complemented, Sylow
113	C₂³	#6127	1	1	yes	characteristic
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	26	1	no	—
116	C₂×C₄	—	26	1	no	—
117	C₂×C₄	—	52	2	no	—
118	C₂×C₄	—	52	1	no	—
119	C₂×C₄	#6127	1	1	yes	characteristic
120	C₂×C₄	#6127	1	1	yes	characteristic
121	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
122	C₂×C₄	#6129	1	1	yes	characteristic
123	C₂×C₄	#6129	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₁₃	1	1	yes	characteristic
125	C₈	—	4	1	no	—
126	C₈	—	4	1	no	—
127	C₈	—	26	2	no	—
128	C₈	—	52	1	no	—
129	C₂²	—	4	1	no	—
130	C₂²	#88130	1	1	yes	characteristic
131	C₂²	#88133	1	1	yes	characteristic
132	C₂²	#88150	1	1	yes	characteristic
133	C₄	—	8	1	no	—
134	C₄	—	26	2	no	—
135	C₄	—	52	1	no	—
136	C₄	—	104	1	no	—
137	C₄	#88137	1	1	yes	characteristic
138	C₄	#88139	1	1	yes	characteristic
139	C₄	#88143	1	1	yes	characteristic
140	C₄	#88143	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#11718319	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11718785

C₂

yes

maximal, characteristic

#11718779

C₂

yes

maximal, characteristic

C₁₃×#344

C₂

yes

maximal, characteristic

#11718151

C₂

yes

maximal, characteristic

#11718147

C₂

yes

maximal, characteristic

#11718124

C₂

yes

maximal, characteristic

#11718123

C₂

yes

maximal, characteristic

C₂×C₁₃×M₁₆

C₂²

yes

characteristic

C₂×C₁₃×M₁₆

C₂²

yes

characteristic

C₁₃×#117

C₂²

yes

characteristic

#88180

C₂²

yes

characteristic

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

C₂×#6099

C₂²

yes

characteristic

C₂×#6099

C₂²

yes

characteristic

#88062

—

#88054

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

Q₈

yes

characteristic

C₄×Dic₁₃

Q₈

yes

—

—

—

—

—

D₄

yes

—

—

C₂³×C₁₃

yes

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₈

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

—

—

—

—

—

D₁₃

yes

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂²⋊Q₈

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₃

yes

characteristic

C₂×M₁₆

C₂×D₁₃

yes

characteristic

#117

—

#117

C₂×D₁₃

yes

characteristic

C₂×C₄²

—

#108

—

C₄⋊C₈

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

#400

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₁₃

#2102

yes

characteristic, complemented, Sylow

113

C₂³

#6127

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

#6127

yes

characteristic

120

C₂×C₄

#6127

yes

characteristic

121

C₂×C₄

D₄×D₁₃

yes

characteristic

122

C₂×C₄

#6129

yes

characteristic

123

C₂×C₄

#6129

yes

characteristic

124

C₂×C₄

Q₈×D₁₃

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₂²

—

130

C₂²

#88130

yes

characteristic

131

C₂²

#88133

yes

characteristic

132

C₂²

#88150

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

104

—

137

C₄

#88137

yes

characteristic

138

C₄

#88139

yes

characteristic

139

C₄

#88143

yes

characteristic

140

C₄

#88143

yes

characteristic

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

#11718319

yes

characteristic

	subgroup	G/H
center	C₄	#88139
commutator, cosocle	C₂²×C₄×C₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#344	C₂
socle	C₂×C₁₃	#400
prime radical	C₂³×C₁₃	#41
hypercenter	#344	D₁₃
hypocenter	C₁₃	#2102
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₄

#88139

commutator, cosocle

C₂²×C₄×C₁₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#344

C₂

C₂×C₁₃

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
112. C₁₃	#2102	semidirect

H ≅ G/N

type

112. C₁₃

#2102

semidirect

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

subgroup	index
#11718785	2
#11718779	2
C₁₃×#344	2
#11718151	2
#11718147	2
#11718124	2
#11718123	2
#2102	13

subgroup

index

C₁₃×#344

Subgroups — #13116413

Conjugacy classes of subgroups

Complements

Maximal subgroups