Group #13116159 — subgroups

Subgroups — #13116159

Order 1664.

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

normal subgroups only

This group has 288 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	C₂²×Dic₅₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#88056	C₂	1	2	yes	maximal
4	#11718590	C₂	1	2	yes	maximal
5	#11718143	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆	C₂²	1	2	yes	—
7	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂×Dic₅₂	—	2	4	no	—
12	C₂×Dic₅₂	—	2	2	no	—
13	C₁₃×C₄⋊C₈	—	2	2	no	—
14	#88071	—	2	2	no	—
15	#88071	—	2	2	no	—
16	#88056	—	2	4	no	—
17	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₁₃	—	2	2	no	—
21	C₂²×Dic₁₃	—	2	2	no	—
22	C₂×Dic₂₆	—	2	2	no	—
23	C₂×Dic₂₆	—	2	2	no	—
24	C₂×Dic₂₆	—	2	2	no	—
25	C₂×Dic₂₆	—	2	2	no	—
26	C₂×Dic₂₆	—	2	2	no	—
27	C₂×Dic₂₆	D₄	1	4	yes	—
28	C₂×C₈×C₁₃	—	2	2	no	—
29	C₂×C₈×C₁₃	—	2	1	no	—
30	C₂×C₈×C₁₃	—	2	2	no	—
31	C₂×C₈×C₁₃	D₄	1	2	yes	—
32	C₄²×C₁₃	—	2	1	no	—
33	C₄²×C₁₃	—	2	1	no	—
34	C₄⋊Dic₁₃	—	2	2	no	—
35	C₄⋊Dic₁₃	—	2	2	no	—
36	Dic₅₂	—	2	8	no	—
37	#2056	—	13	1	no	maximal, Sylow
38	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃	—	2	1	no	—
40	C₂×C₄×C₁₃	—	2	1	no	—
41	C₂×C₄×C₁₃	—	2	1	no	—
42	C₂×C₄×C₁₃	—	2	2	no	—
43	C₂×C₄×C₁₃	Q₁₆	1	2	yes	complemented
44	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
47	C₂×C₄×C₁₃	#41	1	2	yes	—
48	C₂×Dic₁₃	—	2	4	no	—
49	C₂×Dic₁₃	—	4	2	no	—
50	C₂×Dic₁₃	—	4	2	no	—
51	C₂×Dic₁₃	—	4	2	no	—
52	C₂×Dic₁₃	—	4	2	no	—
53	C₂×Dic₁₃	—	4	2	no	—
54	Dic₂₆	—	2	4	no	—
55	Dic₂₆	—	2	4	no	—
56	Dic₂₆	—	4	2	no	—
57	Dic₂₆	—	4	2	no	—
58	C₈×C₁₃	—	2	2	no	—
59	C₈×C₁₃	—	4	1	no	—
60	C₂²×Q₁₆	—	13	1	no	—
61	C₂×C₄⋊C₈	D₁₃	1	1	yes	characteristic
62	C₂×#103	—	13	2	no	—
63	#391	—	13	2	no	—
64	#340	—	13	1	no	—
65	C₂²×C₁₃	#121	1	2	yes	—
66	C₂²×C₁₃	#121	1	1	yes	characteristic
67	C₂²×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
68	C₂²×C₁₃	#135	1	1	yes	characteristic
69	C₂²×C₁₃	#137	1	2	yes	—
70	C₄×C₁₃	—	2	2	no	—
71	C₄×C₁₃	—	4	1	no	—
72	C₄×C₁₃	#120	1	1	yes	characteristic
73	C₄×C₁₃	#123	1	2	yes	—
74	C₄×C₁₃	#124	1	1	yes	characteristic
75	Dic₁₃	—	4	4	no	—
76	Dic₁₃	—	8	2	no	—
77	C₂²×Q₈	—	13	2	no	—
78	C₂×Q₁₆	—	26	4	no	—
79	C₂×Q₁₆	—	26	2	no	—
80	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
81	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
82	C₂×C₄⋊C₄	—	13	2	no	—
83	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
84	C₄⋊C₈	—	2	2	no	—
85	#103	—	26	4	no	—
86	#95	—	26	2	no	—
87	#95	—	26	2	no	—
88	C₂×C₁₃	#394	1	1	yes	characteristic
89	C₂×C₁₃	#462	1	2	yes	—
90	C₂×C₁₃	#464	1	2	yes	—
91	C₂×C₁₃	#467	1	1	yes	characteristic
92	C₂×C₁₃	#470	1	1	yes	characteristic
93	C₂×Q₈	—	13	4	no	—
94	C₂×Q₈	—	26	2	no	—
95	C₂×Q₈	—	26	2	no	—
96	C₂×Q₈	—	26	2	no	—
97	C₂×Q₈	—	26	2	no	—
98	C₂×Q₈	—	26	2	no	—
99	C₂²×C₄	—	26	2	no	—
100	C₂²×C₄	—	26	2	no	—
101	C₂²×C₄	D₅₂	1	1	yes	characteristic
102	C₂²×C₄	D₅₂	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
104	Q₁₆	—	26	8	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	C₂²⋊D₁₃	1	2	yes	—
109	C₄⋊C₄	—	26	2	no	—
110	C₄⋊C₄	—	26	2	no	—
111	C₄²	—	2	1	no	—
112	C₄²	—	2	1	no	—
113	C₁₃	#2056	1	1	yes	characteristic, complemented, Sylow
114	C₂³	C₂×D₅₂	1	1	yes	characteristic
115	Q₈	—	26	4	no	—
116	Q₈	—	26	4	no	—
117	Q₈	—	52	2	no	—
118	Q₈	—	52	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	26	4	no	—
124	C₂×C₄	—	52	2	no	—
125	C₂×C₄	—	52	2	no	—
126	C₂×C₄	—	52	2	no	—
127	C₂×C₄	—	52	2	no	—
128	C₂×C₄	—	52	2	no	—
129	C₂×C₄	Dic₅₂	1	2	yes	complemented
130	C₂×C₄	#6127	1	1	yes	characteristic
131	C₂×C₄	D₄×D₁₃	1	2	yes	—
132	C₂×C₄	#6131	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	—	4	1	no	—
136	C₂²	#88149	1	2	yes	—
137	C₂²	#88158	1	1	yes	characteristic
138	C₂²	C₂×Dic₅₂	1	1	yes	characteristic
139	C₂²	#88163	1	2	yes	—
140	C₂²	#88184	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	4	1	no	—
143	C₄	—	52	4	no	—
144	C₄	—	104	2	no	—
145	C₄	#88148	1	2	yes	—
146	C₄	#88191	1	1	yes	characteristic
147	C₄	#88201	1	1	yes	characteristic
148	C₂	#11718506	1	2	yes	—
149	C₂	#11718510	1	2	yes	—
150	C₂	#11718652	1	1	yes	characteristic
151	C₂	#11718777	1	1	yes	characteristic
152	C₂	#11718798	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₂

C₂

yes

maximal, characteristic

C₂×C₁₃×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂×#88056

C₂

yes

maximal

#11718590

C₂

yes

maximal

#11718143

C₂

yes

maximal, characteristic

C₂²×Dic₂₆

C₂²

yes

—

C₂²×C₈×C₁₃

C₂²

yes

characteristic

C₂²×C₈×C₁₃

C₂²

yes

characteristic

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₁₃×C₄⋊C₈

—

#88071

—

#88071

—

#88056

—

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

D₄

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

D₄

yes

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

Dic₅₂

—

#2056

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

Q₁₆

yes

complemented

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₂²×Q₁₆

—

C₂×C₄⋊C₈

D₁₃

yes

—

—

—

C₂²×C₁₃

#121

yes

—

C₂²×C₁₃

#121

yes

characteristic

C₂²×C₁₃

C₂×Q₁₆

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#120

yes

characteristic

C₄×C₁₃

#123

yes

—

C₄×C₁₃

#124

yes

characteristic

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂×D₁₃

yes

characteristic

C₂²×C₈

C₂×D₁₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

#103

—

#95

—

#95

—

C₂×C₁₃

#394

yes

characteristic

C₂×C₁₃

#462

yes

—

C₂×C₁₃

#464

yes

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

D₅₂

yes

characteristic

102

C₂²×C₄

D₅₂

yes

characteristic

103

C₂²×C₄

C₂²×D₁₃

yes

characteristic

104

Q₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²⋊D₁₃

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄²

—

112

C₄²

—

113

C₁₃

#2056

yes

characteristic, complemented, Sylow

114

C₂³

C₂×D₅₂

yes

characteristic

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

Dic₅₂

yes

complemented

130

C₂×C₄

#6127

yes

characteristic

131

C₂×C₄

D₄×D₁₃

yes

—

132

C₂×C₄

#6131

yes

—

133

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

134

C₈

—

135

C₈

—

136

C₂²

#88149

yes

—

137

C₂²

#88158

yes

characteristic

138

C₂²

C₂×Dic₅₂

yes

characteristic

139

C₂²

#88163

yes

—

140

C₂²

#88184

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

104

—

145

C₄

#88148

yes

—

146

C₄

#88191

yes

characteristic

147

C₄

#88201

yes

characteristic

148

C₂

#11718506

yes

—

149

C₂

#11718510

yes

—

150

C₂

#11718652

yes

characteristic

151

C₂

#11718777

yes

characteristic

152

C₂

#11718798

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₂
commutator, cosocle	C₂²×C₄×C₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₁₃
hypocenter	C₁₃	#2056
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₅₂

commutator, cosocle

C₂²×C₄×C₁₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₂×C₁₃×C₄⋊C₈

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₁₃

hypocenter

C₁₃

#2056

LCS

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
43. C₂×C₄×C₁₃	Q₁₆	semidirect
113. C₁₃	#2056	semidirect
129. C₂×C₄	Dic₅₂	semidirect

H ≅ G/N

type

43. C₂×C₄×C₁₃

Q₁₆

semidirect

113. C₁₃

#2056

semidirect

129. C₂×C₄

Dic₅₂

semidirect

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

subgroup	index
C₂²×Dic₅₂	2
C₂×C₁₃×C₄⋊C₈	2
C₂×#88056	2
#11718590	2
#11718143	2
#2056	13

subgroup

index

C₂²×Dic₅₂

C₂×C₁₃×C₄⋊C₈

Subgroups — #13116159

Conjugacy classes of subgroups

Complements

Maximal subgroups