Group #13115938 — subgroups

Subgroups — #13115938

Order 1664.

370 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 99 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	2	yes	maximal
2	C₂×#88123	C₂	1	1	yes	maximal, characteristic
3	C₂×#88056	C₂	1	2	yes	maximal
4	#11718590	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆	C₂²	1	2	yes	—
7	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₃	—	2	2	no	—
11	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₅₂	—	2	4	no	—
13	C₂×Dic₅₂	—	2	4	no	—
14	C₂×Dic₅₂	—	2	4	no	—
15	#88123	—	2	4	no	—
16	#88071	—	2	2	no	—
17	#88056	—	2	4	no	—
18	C₂²×C₄×C₁₃	D₄	1	2	yes	—
19	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₁₃	—	2	1	no	—
21	C₂²×Dic₁₃	—	2	2	no	—
22	C₂²×Dic₁₃	—	2	1	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	—	2	2	no	—
28	C₂×Dic₂₆	—	2	2	no	—
29	C₂×Dic₂₆	—	2	2	no	—
30	C₂×Dic₂₆	—	2	1	no	—
31	C₂×Dic₂₆	D₄	1	4	yes	—
32	C₂×Dic₂₆	D₄	1	2	yes	—
33	C₂×C₈×C₁₃	—	2	2	no	—
34	C₂×C₈×C₁₃	—	2	2	no	—
35	C₂×C₈×C₁₃	D₄	1	4	yes	—
36	C₄²×C₁₃	—	2	2	no	—
37	C₄⋊Dic₁₃	—	2	8	no	—
38	C₄⋊Dic₁₃	—	2	2	no	—
39	Dic₅₂	—	2	16	no	—
40	#2016	—	13	1	no	maximal, Sylow
41	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃	—	2	2	no	—
43	C₂×C₄×C₁₃	—	2	2	no	—
44	C₂×C₄×C₁₃	Q₁₆	1	4	yes	complemented
45	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
49	C₂×Dic₁₃	—	2	4	no	—
50	C₂×Dic₁₃	—	2	2	no	—
51	C₂×Dic₁₃	—	4	1	no	—
52	C₂×Dic₁₃	—	4	2	no	—
53	C₂×Dic₁₃	—	4	1	no	—
54	C₂×Dic₁₃	—	4	2	no	—
55	C₂×Dic₁₃	—	4	1	no	—
56	C₂×Dic₁₃	—	4	2	no	—
57	Dic₂₆	—	2	4	no	—
58	Dic₂₆	—	2	4	no	—
59	Dic₂₆	—	2	4	no	—
60	Dic₂₆	—	4	2	no	—
61	Dic₂₆	—	4	2	no	—
62	Dic₂₆	—	4	2	no	—
63	C₈×C₁₃	—	2	4	no	—
64	C₂²×Q₁₆	—	13	2	no	—
65	C₂×C₄⋊Q₈	—	13	1	no	—
66	C₂×#103	—	13	2	no	—
67	#391	—	13	1	no	—
68	#336	D₁₃	1	1	yes	characteristic
69	C₂²×C₁₃	#120	1	1	yes	characteristic
70	C₂²×C₁₃	#121	1	1	yes	characteristic
71	C₂²×C₁₃	#127	1	1	yes	characteristic
72	C₂²×C₁₃	C₂×Q₁₆	1	2	yes	—
73	C₂²×C₁₃	#137	1	2	yes	—
74	C₄×C₁₃	—	2	4	no	—
75	C₄×C₁₃	#120	1	2	yes	—
76	C₄×C₁₃	#121	1	2	yes	—
77	Dic₁₃	—	4	2	no	—
78	Dic₁₃	—	4	4	no	—
79	Dic₁₃	—	8	1	no	—
80	C₂²×Q₈	—	13	2	no	—
81	C₂²×Q₈	—	13	1	no	—
82	C₂×Q₁₆	—	26	4	no	—
83	C₂×Q₁₆	—	26	4	no	—
84	C₂×Q₁₆	—	26	4	no	—
85	C₂²×C₈	C₂×D₁₃	1	2	yes	—
86	C₄⋊Q₈	—	26	4	no	—
87	C₂×C₄⋊C₄	—	13	1	no	—
88	C₂×C₄⋊C₄	—	26	2	no	—
89	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
90	#103	—	26	4	no	—
91	#95	—	26	2	no	—
92	C₂×C₁₃	#392	1	1	yes	characteristic
93	C₂×C₁₃	#451	1	2	yes	—
94	C₂×C₁₃	#462	1	2	yes	—
95	C₂×C₁₃	#494	1	1	yes	characteristic
96	C₂×C₁₃	#497	1	1	yes	characteristic
97	C₂×Q₈	—	13	4	no	—
98	C₂×Q₈	—	13	2	no	—
99	C₂×Q₈	—	26	2	no	—
100	C₂×Q₈	—	26	1	no	—
101	C₂×Q₈	—	26	2	no	—
102	C₂×Q₈	—	26	2	no	—
103	C₂×Q₈	—	26	2	no	—
104	C₂×Q₈	—	26	2	no	—
105	C₂×Q₈	—	26	2	no	—
106	C₂×Q₈	—	26	2	no	—
107	C₂²×C₄	—	26	1	no	—
108	C₂²×C₄	—	26	2	no	—
109	C₂²×C₄	—	26	1	no	—
110	C₂²×C₄	D₅₂	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
112	Q₁₆	—	26	16	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	D₅₂	1	4	yes	—
116	C₄⋊C₄	—	26	8	no	—
117	C₄⋊C₄	—	26	2	no	—
118	C₄²	—	2	2	no	—
119	C₁₃	#2016	1	1	yes	characteristic, complemented, Sylow
120	C₂³	C₂×D₅₂	1	1	yes	characteristic
121	Q₈	—	26	4	no	—
122	Q₈	—	26	4	no	—
123	Q₈	—	26	4	no	—
124	Q₈	—	52	2	no	—
125	Q₈	—	52	2	no	—
126	Q₈	—	52	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	26	4	no	—
130	C₂×C₄	—	26	2	no	—
131	C₂×C₄	—	52	1	no	—
132	C₂×C₄	—	52	2	no	—
133	C₂×C₄	—	52	1	no	—
134	C₂×C₄	—	52	2	no	—
135	C₂×C₄	—	52	1	no	—
136	C₂×C₄	—	52	2	no	—
137	C₂×C₄	Dic₅₂	1	4	yes	complemented
138	C₂×C₄	C₂×D₅₂	1	2	yes	—
139	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
140	C₂×C₄	D₄×D₁₃	1	2	yes	—
141	C₂×C₄	#6131	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₂²	#88128	1	1	yes	characteristic
144	C₂²	#88136	1	1	yes	characteristic
145	C₂²	#88149	1	1	yes	characteristic
146	C₂²	C₂×Dic₅₂	1	2	yes	—
147	C₂²	#88163	1	2	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	52	4	no	—
150	C₄	—	52	2	no	—
151	C₄	—	104	1	no	—
152	C₄	#88136	1	2	yes	—
153	C₄	#88149	1	2	yes	—
154	C₂	#11718332	1	1	yes	characteristic
155	C₂	#11718360	1	1	yes	characteristic
156	C₂	#11718376	1	1	yes	characteristic
157	C₂	#11718400	1	2	yes	—
158	C₂	#11718510	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₂

C₂

yes

maximal

C₂×#88123

C₂

yes

maximal, characteristic

C₂×#88056

C₂

yes

maximal

#11718590

C₂

yes

maximal, characteristic

C₁₃×#336

C₂

yes

maximal, characteristic

C₂²×Dic₂₆

C₂²

yes

—

C₂²×Dic₂₆

C₂²

yes

characteristic

C₂²×C₈×C₁₃

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

#88123

—

#88071

—

#88056

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

D₄

yes

—

C₂×Dic₂₆

D₄

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

D₄

yes

—

C₄²×C₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

Dic₅₂

—

#2016

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

Q₁₆

yes

complemented

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

#41

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×#103

—

#391

—

#336

D₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

C₂×Q₁₆

yes

—

C₂²×C₁₃

#137

yes

—

C₄×C₁₃

—

C₄×C₁₃

#120

yes

—

C₄×C₁₃

#121

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂×D₁₃

yes

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#103

—

#95

—

C₂×C₁₃

#392

yes

characteristic

C₂×C₁₃

#451

yes

—

C₂×C₁₃

#462

yes

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

D₅₂

yes

—

111

C₂²×C₄

C₂²×D₁₃

yes

characteristic

112

Q₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

D₅₂

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄²

—

119

C₁₃

#2016

yes

characteristic, complemented, Sylow

120

C₂³

C₂×D₅₂

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

Dic₅₂

yes

complemented

138

C₂×C₄

C₂×D₅₂

yes

—

139

C₂×C₄

D₄×D₁₃

yes

characteristic

140

C₂×C₄

D₄×D₁₃

yes

—

141

C₂×C₄

#6131

yes

characteristic

142

C₈

—

143

C₂²

#88128

yes

characteristic

144

C₂²

#88136

yes

characteristic

145

C₂²

#88149

yes

characteristic

146

C₂²

C₂×Dic₅₂

yes

—

147

C₂²

#88163

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

104

—

152

C₄

#88136

yes

—

153

C₄

#88149

yes

—

154

C₂

#11718332

yes

characteristic

155

C₂

#11718360

yes

characteristic

156

C₂

#11718376

yes

characteristic

157

C₂

#11718400

yes

—

158

C₂

#11718510

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₂
commutator, cosocle	C₂²×C₄×C₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#336	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
hypercenter	#336	D₁₃
hypocenter	C₁₃	#2016
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₅₂

commutator, cosocle

C₂²×C₄×C₁₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#336

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₄

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
44. C₂×C₄×C₁₃	Q₁₆	semidirect
119. C₁₃	#2016	semidirect
137. C₂×C₄	Dic₅₂	semidirect

H ≅ G/N

type

44. C₂×C₄×C₁₃

Q₁₆

semidirect

119. C₁₃

#2016

semidirect

137. 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×#88123	2
C₂×#88056	2
#11718590	2
C₁₃×#336	2
#2016	13

subgroup

index

C₂²×Dic₅₂

C₁₃×#336

Subgroups — #13115938

Conjugacy classes of subgroups

Complements

Maximal subgroups