Group #13119821 — subgroups

Subgroups — #13119821

Order 1664.

540 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 197 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×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#88198	C₂	1	2	yes	maximal
3	C₂×#88123	C₂	1	1	yes	maximal, characteristic
4	#11718644	C₂	1	4	yes	maximal
5	#11718635	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
7	#11718589	C₂	1	4	yes	maximal
8	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×Dic₂₆	C₂²	1	2	yes	—
10	C₄×Q₈×C₁₃	—	2	4	no	—
11	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	#88198	—	2	8	no	—
16	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
19	C₂×#6101	C₂²	1	8	yes	—
20	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
21	#88123	—	2	4	no	—
22	#88071	C₂²	1	8	yes	—
23	#88071	C₂²	1	2	yes	—
24	C₂×Q₈×C₁₃	—	2	2	no	—
25	C₂×Q₈×C₁₃	—	2	2	no	—
26	C₂×Q₈×C₁₃	D₄	1	4	yes	—
27	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
28	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×Dic₁₃	C₂³	1	4	yes	—
33	C₂²×Dic₁₃	C₂³	1	4	yes	—
34	C₂×Dic₂₆	—	2	4	no	—
35	C₂×Dic₂₆	—	2	4	no	—
36	C₂×Dic₂₆	—	2	4	no	—
37	C₁₃×C₄⋊C₄	—	2	4	no	—
38	C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₄²×C₁₃	—	2	4	no	—
40	C₄²×C₁₃	D₄	1	4	yes	—
41	C₄⋊Dic₁₃	—	2	4	no	—
42	C₄⋊Dic₁₃	—	2	4	no	—
43	C₄⋊Dic₁₃	Q₈	1	8	yes	complemented
44	#6101	—	2	16	no	—
45	C₄×Dic₁₃	—	2	4	no	—
46	#2546	—	13	1	no	maximal, Sylow
47	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
48	Q₈×C₁₃	—	2	8	no	—
49	C₂×C₄×C₁₃	—	2	2	no	—
50	C₂×C₄×C₁₃	—	2	2	no	—
51	C₂×C₄×C₁₃	—	2	2	no	—
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	—	2	4	no	—
54	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
55	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
60	C₂×C₄×C₁₃	#41	1	2	yes	—
61	C₂×Dic₁₃	—	2	4	no	—
62	C₂×Dic₁₃	—	2	4	no	—
63	C₂×Dic₁₃	—	2	4	no	—
64	C₂×Dic₁₃	—	2	4	no	—
65	C₂×Dic₁₃	—	2	4	no	—
66	C₂×Dic₁₃	C₂×Q₈	1	8	yes	—
67	Dic₂₆	—	4	8	no	—
68	C₂×C₄⋊Q₈	—	13	2	no	—
69	C₂×C₄⋊Q₈	—	13	1	no	—
70	C₂×C₄×Q₈	D₁₃	1	1	yes	characteristic
71	#398	—	13	4	no	—
72	#391	—	13	2	no	—
73	#389	—	13	4	no	—
74	C₄×C₄⋊C₄	—	13	1	no	—
75	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
76	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
77	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
78	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
79	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
80	C₄×C₁₃	—	2	4	no	—
81	C₄×C₁₃	—	2	2	no	—
82	C₄×C₁₃	—	4	2	no	—
83	C₄×C₁₃	#121	1	4	yes	—
84	C₄×C₁₃	C₄⋊Q₈	1	4	yes	complemented
85	Dic₁₃	—	2	8	no	—
86	Dic₁₃	—	4	4	no	—
87	C₂²×Q₈	—	13	2	no	—
88	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
89	C₄⋊Q₈	—	26	8	no	—
90	C₄⋊Q₈	—	26	4	no	—
91	C₄×Q₈	—	2	4	no	—
92	C₂×C₄⋊C₄	—	13	2	no	—
93	C₂×C₄⋊C₄	—	13	8	no	—
94	C₂×C₄⋊C₄	—	13	2	no	—
95	C₂×C₄⋊C₄	—	13	2	no	—
96	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
97	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
98	C₂×C₄²	—	13	2	no	—
99	C₂×C₄²	C₂×D₁₃	1	2	yes	—
100	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
101	#95	—	13	2	no	—
102	#95	—	13	8	no	—
103	C₂×C₁₃	C₂×#121	1	1	yes	characteristic
104	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₁₃	#536	1	1	yes	characteristic
106	C₂×C₁₃	D₄×Q₈	1	2	yes	—
107	C₂×C₁₃	#557	1	2	yes	—
108	C₂×Q₈	—	2	2	no	—
109	C₂×Q₈	—	2	2	no	—
110	C₂×Q₈	—	26	4	no	—
111	C₂×Q₈	—	26	4	no	—
112	C₂×Q₈	—	26	4	no	—
113	C₂×Q₈	D₅₂	1	4	yes	—
114	C₂²×C₄	—	13	4	no	—
115	C₂²×C₄	—	13	4	no	—
116	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
117	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
118	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
119	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
120	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	13	8	no	—
124	C₄⋊C₄	—	26	16	no	—
125	C₄⋊C₄	—	26	4	no	—
126	C₄⋊C₄	—	26	4	no	—
127	C₄²	—	2	4	no	—
128	C₄²	—	26	4	no	—
129	C₄²	C₂²⋊D₁₃	1	4	yes	—
130	C₁₃	#2546	1	1	yes	characteristic, complemented, Sylow
131	C₂³	C₂³×D₁₃	1	1	yes	characteristic
132	Q₈	—	2	8	no	—
133	Q₈	—	52	8	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	13	8	no	—
140	C₂×C₄	—	26	4	no	—
141	C₂×C₄	—	26	4	no	—
142	C₂×C₄	—	26	4	no	—
143	C₂×C₄	—	26	4	no	—
144	C₂×C₄	—	26	4	no	—
145	C₂×C₄	C₂×D₅₂	1	4	yes	—
146	C₂×C₄	C₂×D₅₂	1	2	yes	—
147	C₂×C₄	#6127	1	2	yes	—
148	C₂×C₄	Q₈×D₁₃	1	4	yes	—
149	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
150	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
151	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
152	C₂²	C₂²×D₅₂	1	1	yes	characteristic
153	C₂²	C₂×#6127	1	1	yes	characteristic
154	C₂²	C₂×Q₈×D₁₃	1	2	yes	—
155	C₂²	#88257	1	2	yes	—
156	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	2	2	no	—
159	C₄	—	4	2	no	—
160	C₄	—	26	8	no	—
161	C₄	—	52	4	no	—
162	C₄	#88184	1	4	yes	—
163	C₄	#88198	1	4	yes	complemented
164	C₂	#11719272	1	2	yes	—
165	C₂	Q₈×D₅₂	1	2	yes	—
166	C₂	C₂×#88184	1	1	yes	characteristic
167	C₂	C₂×#88198	1	1	yes	characteristic
168	C₂	#11719530	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Q₈×C₁₃

C₂

yes

maximal, characteristic

C₂×#88198

C₂

yes

maximal

C₂×#88123

C₂

yes

maximal, characteristic

#11718644

C₂

yes

maximal

#11718635

C₂

yes

maximal

C₄×C₄⋊Dic₁₃

C₂

yes

maximal, characteristic

#11718589

C₂

yes

maximal

C₂²×Q₈×C₁₃

C₂²

yes

characteristic

C₂²×Dic₂₆

C₂²

yes

—

C₄×Q₈×C₁₃

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

#88198

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

#88123

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

D₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

Q₈

yes

complemented

#6101

—

C₄×Dic₁₃

—

#2546

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×Q₈

yes

—

Dic₂₆

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

D₁₃

yes

—

—

—

C₄×C₄⋊C₄

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

—

C₂²×C₁₃

C₂×#41

yes

characteristic

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#121

yes

—

C₄×C₁₃

C₄⋊Q₈

yes

complemented

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₁₃

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

—

100

C₂×C₄²

C₂×D₁₃

yes

characteristic

101

#95

—

102

#95

—

103

C₂×C₁₃

C₂×#121

yes

characteristic

104

C₂×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

105

C₂×C₁₃

#536

yes

characteristic

106

C₂×C₁₃

D₄×Q₈

yes

—

107

C₂×C₁₃

#557

yes

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

D₅₂

yes

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂²×D₁₃

yes

—

117

C₂²×C₄

C₂²×D₁₃

yes

—

118

C₂²×C₄

C₂²×D₁₃

yes

characteristic

119

C₂²×C₄

C₂²×D₁₃

yes

characteristic

120

C₂²×C₄

C₂²×D₁₃

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄²

—

128

C₄²

—

129

C₄²

C₂²⋊D₁₃

yes

—

130

C₁₃

#2546

yes

characteristic, complemented, Sylow

131

C₂³

C₂³×D₁₃

yes

characteristic

132

Q₈

—

133

Q₈

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

C₂×D₅₂

yes

—

146

C₂×C₄

C₂×D₅₂

yes

—

147

C₂×C₄

#6127

yes

—

148

C₂×C₄

Q₈×D₁₃

yes

—

149

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

150

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

151

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

152

C₂²

C₂²×D₅₂

yes

characteristic

153

C₂²

C₂×#6127

yes

characteristic

154

C₂²

C₂×Q₈×D₁₃

yes

—

155

C₂²

#88257

yes

—

156

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

#88184

yes

—

163

C₄

#88198

yes

complemented

164

C₂

#11719272

yes

—

165

C₂

Q₈×D₅₂

yes

—

166

C₂

C₂×#88184

yes

characteristic

167

C₂

C₂×#88198

yes

characteristic

168

C₂

#11719530

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₄×Q₈×C₁₃	C₂
hypercenter	C₂×C₄×Q₈	D₁₃
hypocenter	C₁₃	#2546
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₂×C₄×Q₈×C₁₃

C₂

hypercenter

C₂×C₄×Q₈

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
43. C₄⋊Dic₁₃	Q₈	semidirect
84. C₄×C₁₃	C₄⋊Q₈	semidirect
130. C₁₃	#2546	semidirect
163. C₄	#88198	semidirect

H ≅ G/N

type

43. C₄⋊Dic₁₃

Q₈

semidirect

84. C₄×C₁₃

C₄⋊Q₈

semidirect

130. C₁₃

#2546

semidirect

163. C₄

#88198

semidirect

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

subgroup	index
C₂×C₄×Q₈×C₁₃	2
C₂×#88198	2
C₂×#88123	2
#11718644	2
#11718635	2
C₄×C₄⋊Dic₁₃	2
#11718589	2
#2546	13

subgroup

index

C₂×C₄×Q₈×C₁₃

C₄×C₄⋊Dic₁₃

Subgroups — #13119821

Conjugacy classes of subgroups

Complements

Maximal subgroups