Group #13120603 — subgroups

Subgroups — #13120603

Order 1664.

528 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 157 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×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88200	C₂	1	2	yes	maximal
3	C₂×#88126	C₂	1	1	yes	maximal, characteristic
4	#11718891	C₂	1	2	yes	maximal, complemented
5	#11718654	C₂	1	4	yes	maximal
6	#11718641	C₂	1	4	yes	maximal, complemented
7	#11718597	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
9	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×C₄⋊Q₈	—	2	4	no	—
11	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	#88200	—	2	8	no	—
14	C₂×#6103	C₂²	1	2	yes	—
15	C₂×#6103	C₂²	1	4	yes	—
16	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
17	C₂×#6101	C₂²	1	2	yes	—
18	C₂×#6101	C₂²	1	4	yes	—
19	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
20	#88126	—	2	4	no	—
21	#88071	C₂²	1	4	yes	—
22	#88071	C₂²	1	8	yes	—
23	C₂³×D₁₃	C₂³	1	1	yes	characteristic
24	C₂×Q₈×C₁₃	—	2	4	no	—
25	C₂×Q₈×C₁₃	—	2	4	no	—
26	C₂×Q₈×C₁₃	—	2	4	no	—
27	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
28	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
29	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₁₃	C₂³	1	2	yes	—
31	C₂²×Dic₁₃	C₂³	1	4	yes	—
32	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₄×D₁₃	—	2	2	no	—
34	C₂×C₄×D₁₃	—	2	2	no	—
35	C₂×C₄×D₁₃	D₄	1	4	yes	—
36	C₁₃×C₄⋊C₄	—	2	8	no	—
37	C₄²×C₁₃	D₄	1	4	yes	—
38	#6103	—	2	8	no	—
39	#6103	—	2	4	no	—
40	C₄⋊Dic₁₃	—	2	4	no	—
41	#6101	—	2	4	no	—
42	#6101	—	2	8	no	—
43	C₄×Dic₁₃	—	2	2	no	—
44	#2621	—	13	1	no	maximal, Sylow
45	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
46	C₂²×D₁₃	—	2	1	no	—
47	C₂²×D₁₃	—	2	1	no	—
48	C₂²×D₁₃	—	2	4	no	—
49	C₂²×D₁₃	C₂×D₄	1	2	yes	—
50	Q₈×C₁₃	—	4	8	no	—
51	C₂×C₄×C₁₃	—	2	4	no	—
52	C₂×C₄×C₁₃	—	2	4	no	—
53	C₂×C₄×C₁₃	—	2	4	no	—
54	C₂×C₄×C₁₃	—	2	4	no	—
55	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
56	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₃	#41	1	2	yes	—
59	C₂×Dic₁₃	—	2	1	no	—
60	C₂×Dic₁₃	—	2	1	no	—
61	C₂×Dic₁₃	—	2	2	no	—
62	C₂×Dic₁₃	—	2	4	no	—
63	C₂×Dic₁₃	—	2	4	no	—
64	C₂×Dic₁₃	—	2	8	no	—
65	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
66	C₄×D₁₃	—	2	8	no	—
67	C₂×C₄⋊Q₈	D₁₃	1	1	yes	characteristic, complemented
68	C₂×C₂²⋊Q₈	—	13	2	no	—
69	C₂×#117	—	13	1	no	—
70	#398	—	13	4	no	—
71	#397	—	13	4	no	—
72	#391	—	13	2	no	—
73	#382	—	13	1	no	—
74	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
75	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
76	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
77	C₂²×C₁₃	2₋¹⁺⁴	1	4	yes	—
78	C₂×D₁₃	—	2	2	no	—
79	C₂×D₁₃	—	2	4	no	—
80	C₂×D₁₃	—	4	2	no	—
81	C₂×D₁₃	—	4	2	no	—
82	C₄×C₁₃	—	2	4	no	—
83	C₄×C₁₃	—	4	4	no	—
84	C₄×C₁₃	#121	1	4	yes	—
85	Dic₁₃	—	2	2	no	—
86	Dic₁₃	—	4	4	no	—
87	Dic₁₃	—	4	2	no	—
88	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
89	C₂³×C₄	—	13	1	no	—
90	C₄⋊Q₈	—	2	4	no	—
91	C₂²⋊Q₈	—	26	8	no	—
92	#117	—	26	4	no	—
93	C₂×C₄⋊C₄	—	13	2	no	—
94	C₂×C₄⋊C₄	—	13	4	no	—
95	C₂×C₄⋊C₄	—	13	2	no	—
96	C₂×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
97	C₂×C₂²⋊C₄	—	13	2	no	—
98	C₂×C₂²⋊C₄	—	13	4	no	—
99	C₂×C₄²	—	13	1	no	—
100	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
101	#95	—	13	4	no	—
102	#95	—	13	8	no	—
103	C₂×C₁₃	C₂×#121	1	1	yes	characteristic
104	C₂×C₁₃	#536	1	2	yes	—
105	C₂×C₁₃	#536	1	2	yes	—
106	C₂×C₁₃	#541	1	2	yes	—
107	D₁₃	—	4	2	no	—
108	C₂⁴	—	13	1	no	—
109	C₂×Q₈	—	2	4	no	—
110	C₂×Q₈	—	2	4	no	—
111	C₂×Q₈	—	2	4	no	—
112	C₂²×C₄	—	13	1	no	—
113	C₂²×C₄	—	13	4	no	—
114	C₂²×C₄	—	13	2	no	—
115	C₂²×C₄	—	13	4	no	—
116	C₂²×C₄	—	26	2	no	—
117	C₂²×C₄	—	26	2	no	—
118	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
119	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
120	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
121	C₄⋊C₄	—	2	8	no	—
122	C₄⋊C₄	—	26	8	no	—
123	C₄⋊C₄	—	26	4	no	—
124	C₄⋊C₄	—	26	4	no	—
125	C₂²⋊C₄	—	26	8	no	—
126	C₂²⋊C₄	—	26	4	no	—
127	C₄²	—	26	2	no	—
128	C₄²	C₂²⋊D₁₃	1	4	yes	—
129	C₁₃	#2621	1	1	yes	characteristic, complemented, Sylow
130	C₂³	—	13	2	no	—
131	C₂³	—	26	1	no	—
132	C₂³	—	26	4	no	—
133	C₂³	—	26	1	no	—
134	C₂³	C₂³×D₁₃	1	1	yes	characteristic
135	Q₈	—	4	8	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	13	2	no	—
141	C₂×C₄	—	26	1	no	—
142	C₂×C₄	—	26	1	no	—
143	C₂×C₄	—	26	8	no	—
144	C₂×C₄	—	26	4	no	—
145	C₂×C₄	—	26	8	no	—
146	C₂×C₄	—	26	4	no	—
147	C₂×C₄	—	26	2	no	—
148	C₂×C₄	D₄×D₁₃	1	2	yes	—
149	C₂×C₄	#6129	1	2	yes	—
150	C₂×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
151	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
152	C₂²	—	26	4	no	—
153	C₂²	—	26	2	no	—
154	C₂²	—	52	2	no	—
155	C₂²	—	52	2	no	—
156	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
157	C₂²	C₂×#6129	1	1	yes	characteristic
158	C₂²	#88254	1	4	yes	—
159	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
160	C₄	—	2	4	no	—
161	C₄	—	4	4	no	—
162	C₄	—	26	2	no	—
163	C₄	—	52	4	no	—
164	C₄	—	52	2	no	—
165	C₄	#88192	1	4	yes	—
166	C₂	—	52	2	no	—
167	C₂	#11719429	1	2	yes	—
168	C₂	#11719436	1	2	yes	—
169	C₂	C₂×#88192	1	1	yes	characteristic
170	C₂	#11719521	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

C₂×#88200

C₂

yes

maximal

C₂×#88126

C₂

yes

maximal, characteristic

#11718891

C₂

yes

maximal, complemented

#11718654

C₂

yes

maximal

#11718641

C₂

yes

maximal, complemented

#11718597

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₃

C₂²

yes

—

C₂²×C₄×D₁₃

C₂²

yes

characteristic

C₁₃×C₄⋊Q₈

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

#88200

—

C₂×#6103

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

#88126

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂³×D₁₃

C₂³

yes

characteristic

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

characteristic

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

D₄

yes

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

D₄

yes

—

#6103

—

#6103

—

C₄⋊Dic₁₃

—

#6101

—

#6101

—

C₄×Dic₁₃

—

#2621

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

C₂×D₄

yes

—

Q₈×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

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₄

yes

—

C₄×D₁₃

—

C₂×C₄⋊Q₈

D₁₃

yes

characteristic, complemented

C₂×C₂²⋊Q₈

—

—

—

—

—

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

characteristic

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#121

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

C₂×D₁₃

yes

—

C₂³×C₄

—

C₄⋊Q₈

—

C₂²⋊Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

100

C₂×C₄²

C₂×D₁₃

yes

characteristic

101

#95

—

102

#95

—

103

C₂×C₁₃

C₂×#121

yes

characteristic

104

C₂×C₁₃

#536

yes

—

105

C₂×C₁₃

#536

yes

—

106

C₂×C₁₃

#541

yes

—

107

D₁₃

—

108

C₂⁴

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

C₂²×D₁₃

yes

—

119

C₂²×C₄

C₂²×D₁₃

yes

—

120

C₂²×C₄

C₂²×D₁₃

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₂²⋊C₄

—

126

C₂²⋊C₄

—

127

C₄²

—

128

C₄²

C₂²⋊D₁₃

yes

—

129

C₁₃

#2621

yes

characteristic, complemented, Sylow

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

C₂³×D₁₃

yes

characteristic

135

Q₈

—

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

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

D₄×D₁₃

yes

—

149

C₂×C₄

#6129

yes

—

150

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

151

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

C₂×D₄×D₁₃

yes

characteristic

157

C₂²

C₂×#6129

yes

characteristic

158

C₂²

#88254

yes

—

159

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

#88192

yes

—

166

C₂

—

167

C₂

#11719429

yes

—

168

C₂

#11719436

yes

—

169

C₂

C₂×#88192

yes

characteristic

170

C₂

#11719521

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₁₃×C₄⋊Q₈	C₂
prime radical	C₂³×D₁₃	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#2621
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₂×C₁₃×C₄⋊Q₈

C₂

prime radical

C₂³×D₁₃

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×C₁₃×C₄⋊Q₈	C₂	semidirect
4. #11718891	C₂	semidirect
6. #11718641	C₂	semidirect
7. #11718597	C₂	semidirect
67. C₂×C₄⋊Q₈	D₁₃	semidirect
129. C₁₃	#2621	semidirect

H ≅ G/N

type

1. C₂×C₁₃×C₄⋊Q₈

C₂

semidirect

4. #11718891

C₂

semidirect

6. #11718641

C₂

semidirect

7. #11718597

C₂

semidirect

67. C₂×C₄⋊Q₈

D₁₃

semidirect

129. C₁₃

#2621

semidirect

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

subgroup	index
C₂×C₁₃×C₄⋊Q₈	2
C₂×#88200	2
C₂×#88126	2
#11718891	2
#11718654	2
#11718641	2
#11718597	2
#2621	13

subgroup

index

C₂×C₁₃×C₄⋊Q₈

Subgroups — #13120603

Conjugacy classes of subgroups

Complements

Maximal subgroups