Group #13115977 — subgroups

Subgroups — #13115977

Order 1664.

290 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 93 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×#88123	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#391	C₂	1	1	yes	maximal, characteristic
3	C₂×#88075	C₂	1	2	yes	maximal
4	#11718121	C₂	1	2	yes	maximal
5	#11718116	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×Dic₂₆	—	2	1	no	—
8	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₃	—	2	1	no	—
11	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
12	C₂²×#1055	C₂²	1	2	yes	—
13	#88123	—	2	4	no	—
14	C₁₃×#95	—	2	1	no	—
15	C₁₃×#95	—	2	1	no	—
16	#88075	—	2	4	no	—
17	C₂×Q₈×C₁₃	—	2	1	no	—
18	C₂×Q₈×C₁₃	—	2	2	no	—
19	C₂×Q₈×C₁₃	—	2	2	no	—
20	C₂×Q₈×C₁₃	Q₈	1	2	yes	complemented
21	C₂²×C₄×C₁₃	—	2	1	no	—
22	C₂²×C₄×C₁₃	—	2	1	no	—
23	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₁₃	—	2	2	no	—
27	C₂×Dic₂₆	—	2	4	no	—
28	C₂×Dic₂₆	—	4	2	no	—
29	C₂×Dic₂₆	—	4	2	no	—
30	C₁₃×C₄⋊C₄	—	2	2	no	—
31	C₄²×C₁₃	—	2	2	no	—
32	C₄⋊Dic₁₃	—	2	4	no	—
33	C₄⋊Dic₁₃	—	2	2	no	—
34	C₄⋊Dic₁₃	D₄	1	4	yes	—
35	C₂×#1055	—	2	2	no	—
36	C₂×#1055	—	2	2	no	—
37	C₂×#1055	—	2	2	no	—
38	#2024	—	13	1	no	maximal, Sylow
39	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₁₃	—	2	4	no	—
41	Q₈×C₁₃	—	4	2	no	—
42	C₂×C₄×C₁₃	—	2	1	no	—
43	C₂×C₄×C₁₃	—	2	1	no	—
44	C₂×C₄×C₁₃	—	2	1	no	—
45	C₂×C₄×C₁₃	—	2	1	no	—
46	C₂×C₄×C₁₃	—	2	2	no	—
47	C₂×C₄×C₁₃	—	4	1	no	—
48	C₂×C₄×C₁₃	—	4	1	no	—
49	C₂×C₄×C₁₃	—	4	1	no	—
50	C₂×C₄×C₁₃	—	4	1	no	—
51	C₂×C₄×C₁₃	—	4	1	no	—
52	C₂×C₄×C₁₃	SD₁₆	1	2	yes	—
53	C₂×C₄×C₁₃	Q₁₆	1	2	yes	—
54	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃	#41	1	2	yes	—
57	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
58	C₂×Dic₁₃	—	2	4	no	—
59	C₂×Dic₁₃	—	4	2	no	—
60	C₂×Dic₁₃	—	4	2	no	—
61	Dic₂₆	—	4	8	no	—
62	#1055	—	4	2	no	—
63	C₂×C₄⋊Q₈	—	13	1	no	—
64	C₂×#103	—	13	2	no	—
65	#391	D₁₃	1	1	yes	characteristic
66	#340	—	13	2	no	—
67	#336	—	13	1	no	—
68	C₂²×C₁₃	#120	1	1	yes	characteristic
69	C₂²×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂²×C₁₃	#124	1	1	yes	characteristic
71	C₂²×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₂²×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
73	C₂²×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₂²×C₁₃	#137	1	1	yes	characteristic
75	C₄×C₁₃	—	2	2	no	—
76	C₄×C₁₃	—	2	2	no	—
77	C₄×C₁₃	—	4	2	no	—
78	C₄×C₁₃	—	8	1	no	—
79	C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
80	C₄×C₁₃	#124	1	2	yes	—
81	Dic₁₃	—	4	4	no	—
82	C₂²×Q₈	—	26	1	no	—
83	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
84	C₂²×C₈	—	13	2	no	—
85	C₄⋊Q₈	—	26	4	no	—
86	C₂×C₄⋊C₄	—	13	2	no	—
87	C₂×C₄⋊C₄	—	26	1	no	—
88	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
89	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
90	#103	—	26	4	no	—
91	#95	—	2	1	no	—
92	#95	—	2	1	no	—
93	C₂×C₁₃	#393	1	1	yes	characteristic
94	C₂×C₁₃	#450	1	1	yes	characteristic
95	C₂×C₁₃	#451	1	1	yes	characteristic
96	C₂×C₁₃	#475	1	1	yes	characteristic
97	C₂×C₁₃	#477	1	1	yes	characteristic
98	C₂×C₁₃	#487	1	1	yes	characteristic
99	C₂×C₁₃	#489	1	1	yes	characteristic
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	26	4	no	—
104	C₂×Q₈	—	52	2	no	—
105	C₂×Q₈	—	52	2	no	—
106	C₂×Q₈	Dic₂₆	1	2	yes	complemented
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	26	2	no	—
110	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
111	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
112	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
113	C₂×C₈	—	26	2	no	—
114	C₂×C₈	—	26	2	no	—
115	C₂×C₈	—	26	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	13	4	no	—
118	C₄⋊C₄	—	26	4	no	—
119	C₄⋊C₄	—	26	2	no	—
120	C₄²	—	2	2	no	—
121	C₁₃	#2024	1	1	yes	characteristic, complemented, Sylow
122	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
123	Q₈	—	2	4	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	52	8	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	26	4	no	—
137	C₂×C₄	—	52	2	no	—
138	C₂×C₄	—	52	2	no	—
139	C₂×C₄	#6106	1	2	yes	—
140	C₂×C₄	#6107	1	2	yes	—
141	C₂×C₄	C₂×Dic₂₆	1	1	yes	characteristic
142	C₂×C₄	#6127	1	1	yes	characteristic
143	C₂×C₄	D₄×D₁₃	1	2	yes	—
144	C₂×C₄	#6129	1	2	yes	—
145	C₈	—	52	2	no	—
146	C₂²	#88178	1	1	yes	characteristic
147	C₂²	#88190	1	1	yes	characteristic
148	C₂²	#88191	1	1	yes	characteristic
149	C₂²	C₂×#6106	1	1	yes	characteristic
150	C₂²	C₂×#6107	1	1	yes	characteristic
151	C₂²	#88203	1	1	yes	characteristic
152	C₂²	#88205	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	2	no	—
156	C₄	—	8	1	no	—
157	C₄	—	52	4	no	—
158	C₄	#88132	1	2	yes	—
159	C₄	#88139	1	2	yes	—
160	C₂	#11718608	1	1	yes	characteristic
161	C₂	#11718681	1	1	yes	characteristic
162	C₂	#11718682	1	1	yes	characteristic
163	C₂	#11718707	1	1	yes	characteristic
164	C₂	#11718711	1	1	yes	characteristic
165	C₂	#11718716	1	1	yes	characteristic
166	C₂	#11718742	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88123

C₂

yes

maximal, characteristic

C₁₃×#391

C₂

yes

maximal, characteristic

C₂×#88075

C₂

yes

maximal

#11718121

C₂

yes

maximal

#11718116

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₃

C₂²

yes

characteristic

C₂²×Dic₂₆

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂²×#1055

C₂²

yes

—

#88123

—

C₁₃×#95

—

C₁₃×#95

—

#88075

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

Q₈

yes

complemented

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

D₄

yes

—

—

—

—

—

C₂³×C₁₃

C₂×D₄

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

SD₁₆

yes

—

C₂×C₄×C₁₃

Q₁₆

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

#41

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

#1055

—

C₂×C₄⋊Q₈

—

C₂×#103

—

#391

D₁₃

yes

characteristic

#340

—

#336

—

C₂²×C₁₃

#120

yes

characteristic

C₂²×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₃

#124

yes

characteristic

C₂²×C₁₃

C₂×SD₁₆

yes

characteristic

C₂²×C₁₃

C₂×Q₁₆

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂²⋊Q₈

yes

—

C₄×C₁₃

#124

yes

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₁₃

yes

characteristic

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

C₂×C₄²

C₂×D₁₃

yes

—

—

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

Dic₂₆

yes

complemented

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²⋊D₁₃

yes

characteristic

111

C₂²×C₄

C₂²⋊D₁₃

yes

characteristic

112

C₂²×C₄

C₂²×D₁₃

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄²

—

121

C₁₃

#2024

yes

characteristic, complemented, Sylow

122

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

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

—

138

C₂×C₄

—

139

C₂×C₄

#6106

yes

—

140

C₂×C₄

#6107

yes

—

141

C₂×C₄

C₂×Dic₂₆

yes

characteristic

142

C₂×C₄

#6127

yes

characteristic

143

C₂×C₄

D₄×D₁₃

yes

—

144

C₂×C₄

#6129

yes

—

145

C₈

—

146

C₂²

#88178

yes

characteristic

147

C₂²

#88190

yes

characteristic

148

C₂²

#88191

yes

characteristic

149

C₂²

C₂×#6106

yes

characteristic

150

C₂²

C₂×#6107

yes

characteristic

151

C₂²

#88203

yes

characteristic

152

C₂²

#88205

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

#88132

yes

—

159

C₄

#88139

yes

—

160

C₂

#11718608

yes

characteristic

161

C₂

#11718681

yes

characteristic

162

C₂

#11718682

yes

characteristic

163

C₂

#11718707

yes

characteristic

164

C₂

#11718711

yes

characteristic

165

C₂

#11718716

yes

characteristic

166

C₂

#11718742

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃
commutator, cosocle	C₂²×C₄×C₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#391	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
hypercenter	#391	D₁₃
hypocenter	C₁₃	#2024
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃

commutator, cosocle

C₂²×C₄×C₁₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#391

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₄

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
20. C₂×Q₈×C₁₃	Q₈	semidirect
106. C₂×Q₈	Dic₂₆	semidirect
121. C₁₃	#2024	semidirect

H ≅ G/N

type

20. C₂×Q₈×C₁₃

Q₈

semidirect

106. C₂×Q₈

Dic₂₆

semidirect

121. C₁₃

#2024

semidirect

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

subgroup	index
C₂×#88123	2
C₁₃×#391	2
C₂×#88075	2
#11718121	2
#11718116	2
#2024	13

subgroup

index

C₁₃×#391

Subgroups — #13115977

Conjugacy classes of subgroups

Complements

Maximal subgroups