Group #13115978 — subgroups

Subgroups — #13115978

Order 1664.

290 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 91 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	C₂×#88056	C₂	1	2	yes	maximal
3	#11718590	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#336	C₂	1	1	yes	maximal, characteristic
5	#11718143	C₂	1	2	yes	maximal
6	C₂²×Dic₂₆	—	2	1	no	—
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	1	no	—
11	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
13	#88123	—	2	4	no	—
14	#88071	—	2	1	no	—
15	#88071	—	2	1	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	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	4	no	—
26	C₂×Dic₂₆	—	2	1	no	—
27	C₂×Dic₂₆	—	4	2	no	—
28	C₂×Dic₂₆	—	4	2	no	—
29	C₂×Dic₂₆	Q₈	1	2	yes	complemented
30	C₂×C₈×C₁₃	—	2	2	no	—
31	C₂×C₈×C₁₃	—	2	2	no	—
32	C₂×C₈×C₁₃	—	2	2	no	—
33	C₄²×C₁₃	—	2	2	no	—
34	C₄⋊Dic₁₃	—	2	4	no	—
35	C₄⋊Dic₁₃	—	2	2	no	—
36	C₄⋊Dic₁₃	—	2	2	no	—
37	C₄⋊Dic₁₃	D₄	1	4	yes	—
38	#2024	—	13	1	no	maximal, Sylow
39	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃	—	2	1	no	—
41	C₂×C₄×C₁₃	—	2	1	no	—
42	C₂×C₄×C₁₃	—	2	1	no	—
43	C₂×C₄×C₁₃	—	2	1	no	—
44	C₂×C₄×C₁₃	SD₁₆	1	2	yes	—
45	C₂×C₄×C₁₃	Q₁₆	1	2	yes	—
46	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₃	#41	1	2	yes	—
49	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
50	C₂×Dic₁₃	—	2	4	no	—
51	C₂×Dic₁₃	—	2	2	no	—
52	C₂×Dic₁₃	—	4	1	no	—
53	C₂×Dic₁₃	—	4	1	no	—
54	C₂×Dic₁₃	—	4	2	no	—
55	C₂×Dic₁₃	—	4	2	no	—
56	C₂×Dic₁₃	—	4	1	no	—
57	C₂×Dic₁₃	—	4	1	no	—
58	C₂×Dic₁₃	—	4	1	no	—
59	Dic₂₆	—	2	4	no	—
60	Dic₂₆	—	4	8	no	—
61	Dic₂₆	—	4	2	no	—
62	C₈×C₁₃	—	4	2	no	—
63	C₂×C₄⋊Q₈	—	13	1	no	—
64	C₂×#103	—	13	2	no	—
65	#391	—	13	1	no	—
66	#340	—	13	2	no	—
67	#336	D₁₃	1	1	yes	characteristic
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	C₂²⋊Q₈	1	2	yes	—
78	C₄×C₁₃	#124	1	2	yes	—
79	Dic₁₃	—	4	2	no	—
80	Dic₁₃	—	4	4	no	—
81	Dic₁₃	—	8	1	no	—
82	C₂²×Q₈	—	13	1	no	—
83	C₂²×Q₈	—	26	1	no	—
84	C₂²×C₈	C₂×D₁₃	1	2	yes	—
85	C₄⋊Q₈	—	26	4	no	—
86	C₂×C₄⋊C₄	—	13	2	no	—
87	C₂×C₄⋊C₄	—	13	1	no	—
88	C₂×C₄⋊C₄	—	26	1	no	—
89	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
90	#103	—	26	4	no	—
91	#95	—	26	1	no	—
92	#95	—	26	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	—	13	2	no	—
101	C₂×Q₈	—	26	4	no	—
102	C₂×Q₈	—	26	1	no	—
103	C₂×Q₈	—	26	2	no	—
104	C₂×Q₈	—	26	2	no	—
105	C₂×Q₈	—	52	2	no	—
106	C₂×Q₈	—	52	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	1	yes	characteristic
111	C₂²×C₄	D₅₂	1	1	yes	characteristic
112	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₄⋊C₄	—	13	4	no	—
117	C₄⋊C₄	—	26	4	no	—
118	C₄⋊C₄	—	26	2	no	—
119	C₄⋊C₄	—	26	2	no	—
120	C₄²	—	2	2	no	—
121	C₁₃	#2024	1	1	yes	characteristic, complemented, Sylow
122	C₂³	C₂×D₅₂	1	1	yes	characteristic
123	Q₈	—	26	4	no	—
124	Q₈	—	52	8	no	—
125	Q₈	—	52	2	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	—	26	4	no	—
131	C₂×C₄	—	26	2	no	—
132	C₂×C₄	—	52	1	no	—
133	C₂×C₄	—	52	1	no	—
134	C₂×C₄	—	52	2	no	—
135	C₂×C₄	—	52	2	no	—
136	C₂×C₄	—	52	1	no	—
137	C₂×C₄	—	52	1	no	—
138	C₂×C₄	—	52	1	no	—
139	C₂×C₄	#6095	1	2	yes	—
140	C₂×C₄	Dic₅₂	1	2	yes	—
141	C₂×C₄	#6127	1	2	yes	—
142	C₂×C₄	D₄×D₁₃	1	2	yes	—
143	C₂×C₄	#6129	1	1	yes	characteristic
144	C₂×C₄	Q₈×D₁₃	1	1	yes	characteristic
145	C₈	—	4	2	no	—
146	C₂²	#88129	1	1	yes	characteristic
147	C₂²	#88136	1	1	yes	characteristic
148	C₂²	#88151	1	1	yes	characteristic
149	C₂²	C₂×#6095	1	1	yes	characteristic
150	C₂²	C₂×Dic₅₂	1	1	yes	characteristic
151	C₂²	#88162	1	1	yes	characteristic
152	C₂²	#88163	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	52	2	no	—
156	C₄	—	52	4	no	—
157	C₄	—	104	1	no	—
158	C₄	#88139	1	2	yes	—
159	C₄	#88150	1	2	yes	—
160	C₂	#11718334	1	1	yes	characteristic
161	C₂	#11718342	1	1	yes	characteristic
162	C₂	#11718375	1	1	yes	characteristic
163	C₂	#11718392	1	1	yes	characteristic
164	C₂	#11718400	1	1	yes	characteristic
165	C₂	#11718511	1	1	yes	characteristic
166	C₂	#11718512	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88123

C₂

yes

maximal, characteristic

C₂×#88056

C₂

yes

maximal

#11718590

C₂

yes

maximal, characteristic

C₁₃×#336

C₂

yes

maximal, characteristic

#11718143

C₂

yes

maximal

C₂²×Dic₂₆

—

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×C₄⋊Dic₁₃

C₂²

yes

—

—

—

—

—

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

characteristic

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

Q₈

yes

complemented

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

D₄

yes

—

#2024

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

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

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₂×C₄⋊Q₈

—

—

—

—

D₁₃

yes

characteristic

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

C₂²⋊Q₈

yes

—

C₄×C₁₃

#124

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₁₃

yes

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

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

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

D₅₂

yes

characteristic

111

C₂²×C₄

D₅₂

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×D₅₂

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

#6095

yes

—

140

C₂×C₄

Dic₅₂

yes

—

141

C₂×C₄

#6127

yes

—

142

C₂×C₄

D₄×D₁₃

yes

—

143

C₂×C₄

#6129

yes

characteristic

144

C₂×C₄

Q₈×D₁₃

yes

characteristic

145

C₈

—

146

C₂²

#88129

yes

characteristic

147

C₂²

#88136

yes

characteristic

148

C₂²

#88151

yes

characteristic

149

C₂²

C₂×#6095

yes

characteristic

150

C₂²

C₂×Dic₅₂

yes

characteristic

151

C₂²

#88162

yes

characteristic

152

C₂²

#88163

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

104

—

158

C₄

#88139

yes

—

159

C₄

#88150

yes

—

160

C₂

#11718334

yes

characteristic

161

C₂

#11718342

yes

characteristic

162

C₂

#11718375

yes

characteristic

163

C₂

#11718392

yes

characteristic

164

C₂

#11718400

yes

characteristic

165

C₂

#11718511

yes

characteristic

166

C₂

#11718512

yes

characteristic

	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	#2024
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
29. C₂×Dic₂₆	Q₈	semidirect
121. C₁₃	#2024	semidirect

H ≅ G/N

type

29. C₂×Dic₂₆

Q₈

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
C₂×#88056	2
#11718590	2
C₁₃×#336	2
#11718143	2
#2024	13

subgroup

index

C₁₃×#336

Subgroups — #13115978

Conjugacy classes of subgroups

Complements

Maximal subgroups