Group #13120051 — subgroups

Subgroups — #13120051

Order 1664.

468 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 173 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×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	#11718635	C₂	1	4	yes	maximal
5	#11718597	C₂	1	2	yes	maximal
6	#11718590	C₂	1	2	yes	maximal
7	#11718320	C₂	1	4	yes	maximal
8	C₂²×Dic₂₆	C₂²	1	2	yes	—
9	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
16	C₂×#6101	C₂²	1	2	yes	—
17	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
18	#88123	—	2	4	no	—
19	C₄×Dic₂₆	—	2	4	no	—
20	C₁₃×#95	C₂²	1	2	yes	—
21	#88071	C₂²	1	8	yes	—
22	#88071	C₂²	1	8	yes	—
23	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
24	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂²×Dic₁₃	C₂³	1	2	yes	—
28	C₂²×Dic₁₃	C₂³	1	4	yes	—
29	C₂²×Dic₁₃	C₂³	1	2	yes	—
30	C₂×Dic₂₆	—	2	2	no	—
31	C₂×Dic₂₆	—	2	2	no	—
32	C₂×Dic₂₆	—	2	4	no	—
33	C₂×Dic₂₆	—	2	8	no	—
34	C₂×Dic₂₆	Q₈	1	4	yes	complemented
35	C₁₃×C₄⋊C₄	—	2	4	no	—
36	C₄²×C₁₃	—	2	4	no	—
37	C₄²×C₁₃	D₄	1	4	yes	—
38	C₄⋊Dic₁₃	—	2	2	no	—
39	C₄⋊Dic₁₃	—	2	4	no	—
40	C₄⋊Dic₁₃	—	2	4	no	—
41	#6101	—	2	4	no	—
42	C₄×Dic₁₃	—	2	4	no	—
43	#2568	—	13	1	no	maximal, Sylow
44	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
45	C₂×C₄×C₁₃	—	2	2	no	—
46	C₂×C₄×C₁₃	—	2	4	no	—
47	C₂×C₄×C₁₃	—	2	4	no	—
48	C₂×C₄×C₁₃	—	2	4	no	—
49	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₁₃	#41	1	2	yes	—
54	C₂×C₄×C₁₃	#41	1	4	yes	—
55	C₂×Dic₁₃	—	2	2	no	—
56	C₂×Dic₁₃	—	2	2	no	—
57	C₂×Dic₁₃	—	2	2	no	—
58	C₂×Dic₁₃	—	2	4	no	—
59	C₂×Dic₁₃	—	2	8	no	—
60	C₂×Dic₁₃	—	2	4	no	—
61	C₂×Dic₁₃	C₂×Q₈	1	4	yes	—
62	Dic₂₆	—	2	8	no	—
63	Dic₂₆	—	4	8	no	—
64	C₂×C₄⋊Q₈	—	13	1	no	—
65	C₂×C₄×Q₈	—	13	1	no	—
66	#400	—	13	4	no	—
67	#391	—	13	4	no	—
68	#391	—	13	2	no	—
69	#382	—	13	2	no	—
70	C₄×C₄⋊C₄	D₁₃	1	1	yes	characteristic
71	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
72	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
73	C₂²×C₁₃	C₂×#41	1	2	yes	—
74	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
75	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
76	C₄×C₁₃	—	2	2	no	—
77	C₄×C₁₃	—	4	4	no	—
78	C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
79	C₄×C₁₃	#124	1	4	yes	—
80	Dic₁₃	—	2	4	no	—
81	Dic₁₃	—	4	4	no	—
82	Dic₁₃	—	4	2	no	—
83	C₂²×Q₈	—	13	1	no	—
84	C₂²×Q₈	—	13	2	no	—
85	C₄⋊Q₈	—	26	4	no	—
86	C₄×Q₈	—	26	4	no	—
87	C₂×C₄⋊C₄	—	13	2	no	—
88	C₂×C₄⋊C₄	—	13	1	no	—
89	C₂×C₄⋊C₄	—	13	2	no	—
90	C₂×C₄⋊C₄	—	13	2	no	—
91	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
92	C₂×C₄²	—	13	2	no	—
93	C₂×C₄²	C₂×D₁₃	1	2	yes	—
94	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
95	#95	—	13	8	no	—
96	#95	—	13	8	no	—
97	#95	C₂×D₁₃	1	2	yes	—
98	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
100	C₂×C₁₃	#536	1	1	yes	characteristic
101	C₂×C₁₃	#556	1	2	yes	—
102	C₂×C₁₃	#557	1	2	yes	—
103	C₂×Q₈	—	13	4	no	—
104	C₂×Q₈	—	26	2	no	—
105	C₂×Q₈	—	26	4	no	—
106	C₂×Q₈	—	26	8	no	—
107	C₂×Q₈	—	26	2	no	—
108	C₂²×C₄	—	13	2	no	—
109	C₂²×C₄	—	13	2	no	—
110	C₂²×C₄	—	13	4	no	—
111	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
112	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	26	4	no	—
117	C₄⋊C₄	—	26	2	no	—
118	C₄⋊C₄	—	26	4	no	—
119	C₄⋊C₄	—	26	4	no	—
120	C₄²	—	2	4	no	—
121	C₄²	—	26	4	no	—
122	C₄²	D₅₂	1	4	yes	—
123	C₁₃	#2568	1	1	yes	characteristic, complemented, Sylow
124	C₂³	C₂³×D₁₃	1	1	yes	characteristic
125	Q₈	—	26	8	no	—
126	Q₈	—	52	8	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	13	4	no	—
132	C₂×C₄	—	26	2	no	—
133	C₂×C₄	—	26	2	no	—
134	C₂×C₄	—	26	2	no	—
135	C₂×C₄	—	26	4	no	—
136	C₂×C₄	—	26	8	no	—
137	C₂×C₄	—	26	4	no	—
138	C₂×C₄	C₂×D₅₂	1	2	yes	—
139	C₂×C₄	C₂×D₅₂	1	2	yes	—
140	C₂×C₄	C₂×D₅₂	1	2	yes	—
141	C₂×C₄	#6127	1	4	yes	—
142	C₂×C₄	#6129	1	2	yes	—
143	C₂×C₄	Q₈×D₁₃	1	2	yes	—
144	C₂²	C₂²×D₅₂	1	1	yes	characteristic
145	C₂²	C₂×#6127	1	2	yes	—
146	C₂²	C₂×#6129	1	1	yes	characteristic
147	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
148	C₂²	#88257	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	4	4	no	—
151	C₄	—	26	4	no	—
152	C₄	—	52	4	no	—
153	C₄	—	52	2	no	—
154	C₄	#88129	1	4	yes	—
155	C₄	#88151	1	4	yes	—
156	C₂	C₂×#88129	1	1	yes	characteristic
157	C₂	C₂×#88151	1	1	yes	characteristic
158	C₂	#11719231	1	1	yes	characteristic
159	C₂	#11719247	1	2	yes	—
160	C₂	#11719272	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88123

C₂

yes

maximal, characteristic

C₂×C₄×Dic₂₆

C₂

yes

maximal, characteristic

C₄×C₁₃×C₄⋊C₄

C₂

yes

maximal, characteristic

#11718635

C₂

yes

maximal

#11718597

C₂

yes

maximal

#11718590

C₂

yes

maximal

#11718320

C₂

yes

maximal

C₂²×Dic₂₆

C₂²

yes

—

C₂²×Dic₂₆

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

#88123

—

C₄×Dic₂₆

—

C₁₃×#95

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

Q₈

yes

complemented

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

D₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

#2568

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

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×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×Q₈

yes

—

Dic₂₆

—

Dic₂₆

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

—

—

—

—

C₄×C₄⋊C₄

D₁₃

yes

characteristic

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

C₂×#41

yes

characteristic

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂²⋊Q₈

yes

—

C₄×C₁₃

#124

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

—

C₂×C₄²

C₂×D₁₃

yes

—

—

C₂×D₁₃

yes

—

C₂×C₁₃

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₁₃

C₂×#124

yes

characteristic

100

C₂×C₁₃

#536

yes

characteristic

101

C₂×C₁₃

#556

yes

—

102

C₂×C₁₃

#557

yes

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²×D₁₃

yes

—

112

C₂²×C₄

C₂²×D₁₃

yes

characteristic

113

C₂²×C₄

C₂²×D₁₃

yes

characteristic

114

C₂²×C₄

C₂²×D₁₃

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄²

—

121

C₄²

—

122

C₄²

D₅₂

yes

—

123

C₁₃

#2568

yes

characteristic, complemented, Sylow

124

C₂³

C₂³×D₁₃

yes

characteristic

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

—

138

C₂×C₄

C₂×D₅₂

yes

—

139

C₂×C₄

C₂×D₅₂

yes

—

140

C₂×C₄

C₂×D₅₂

yes

—

141

C₂×C₄

#6127

yes

—

142

C₂×C₄

#6129

yes

—

143

C₂×C₄

Q₈×D₁₃

yes

—

144

C₂²

C₂²×D₅₂

yes

characteristic

145

C₂²

C₂×#6127

yes

—

146

C₂²

C₂×#6129

yes

characteristic

147

C₂²

C₂×Q₈×D₁₃

yes

characteristic

148

C₂²

#88257

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#88129

yes

—

155

C₄

#88151

yes

—

156

C₂

C₂×#88129

yes

characteristic

157

C₂

C₂×#88151

yes

characteristic

158

C₂

#11719231

yes

characteristic

159

C₂

#11719247

yes

—

160

C₂

#11719272

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₄×C₁₃×C₄⋊C₄	C₂
hypercenter	C₄×C₄⋊C₄	D₁₃
hypocenter	C₁₃	#2568
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₄×C₁₃×C₄⋊C₄

C₂

hypercenter

C₄×C₄⋊C₄

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
34. C₂×Dic₂₆	Q₈	semidirect
123. C₁₃	#2568	semidirect

H ≅ G/N

type

34. C₂×Dic₂₆

Q₈

semidirect

123. C₁₃

#2568

semidirect

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

subgroup	index
C₂×#88123	2
C₂×C₄×Dic₂₆	2
C₄×C₁₃×C₄⋊C₄	2
#11718635	2
#11718597	2
#11718590	2
#11718320	2
#2568	13

subgroup

index

C₂×#88123

C₂×C₄×Dic₂₆

C₄×C₁₃×C₄⋊C₄

Subgroups — #13120051

Conjugacy classes of subgroups

Complements

Maximal subgroups