Group #13120048 — subgroups

Subgroups — #13120048

Order 1664.

468 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 181 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×#88123	C₂	1	1	yes	maximal, characteristic
3	#11718891	C₂	1	2	yes	maximal
4	#11718647	C₂	1	4	yes	maximal
5	#11718638	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
7	#11718590	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	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
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	#88123	—	2	4	no	—
20	#88071	C₂²	1	8	yes	—
21	#88071	C₂²	1	8	yes	—
22	#88071	C₂²	1	2	yes	—
23	C₂×Q₈×C₁₃	—	2	2	no	—
24	C₂×Q₈×C₁₃	—	2	2	no	—
25	C₂×Q₈×C₁₃	Q₈	1	4	yes	complemented
26	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
27	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
29	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₃	C₂³	1	4	yes	—
32	C₂²×Dic₁₃	C₂³	1	4	yes	—
33	C₂×Dic₂₆	—	2	4	no	—
34	C₂×Dic₂₆	—	2	8	no	—
35	C₁₃×C₄⋊C₄	—	2	4	no	—
36	C₁₃×C₄⋊C₄	—	2	2	no	—
37	C₄²×C₁₃	—	2	4	no	—
38	C₄²×C₁₃	D₄	1	4	yes	—
39	C₄⋊Dic₁₃	—	2	4	no	—
40	C₄⋊Dic₁₃	—	2	4	no	—
41	C₄⋊Dic₁₃	—	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	Q₈×C₁₃	—	2	8	no	—
46	C₂×C₄×C₁₃	—	2	2	no	—
47	C₂×C₄×C₁₃	—	2	2	no	—
48	C₂×C₄×C₁₃	—	2	2	no	—
49	C₂×C₄×C₁₃	—	2	2	no	—
50	C₂×C₄×C₁₃	—	2	4	no	—
51	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
55	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₁₃	#41	1	2	yes	—
57	C₂×C₄×C₁₃	#41	1	4	yes	—
58	C₂×Dic₁₃	—	2	4	no	—
59	C₂×Dic₁₃	—	2	4	no	—
60	C₂×Dic₁₃	—	2	4	no	—
61	C₂×Dic₁₃	—	2	8	no	—
62	C₂×Dic₁₃	—	2	4	no	—
63	Dic₂₆	—	4	8	no	—
64	C₂×C₄⋊Q₈	—	13	1	no	—
65	C₂×C₄×Q₈	D₁₃	1	1	yes	characteristic
66	#400	—	13	4	no	—
67	#391	—	13	2	no	—
68	#391	—	13	4	no	—
69	#382	—	13	2	no	—
70	C₄×C₄⋊C₄	—	13	1	no	—
71	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
72	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
73	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
74	C₂²×C₁₃	C₂×#41	1	2	yes	—
75	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
76	C₄×C₁₃	—	2	4	no	—
77	C₄×C₁₃	—	2	2	no	—
78	C₄×C₁₃	—	4	2	no	—
79	C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
80	C₄×C₁₃	#124	1	4	yes	—
81	Dic₁₃	—	4	4	no	—
82	Dic₁₃	—	4	4	no	—
83	C₂²×Q₈	—	13	2	no	—
84	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
85	C₄⋊Q₈	—	26	4	no	—
86	C₄×Q₈	—	2	4	no	—
87	C₂×C₄⋊C₄	—	13	2	no	—
88	C₂×C₄⋊C₄	—	13	2	no	—
89	C₂×C₄⋊C₄	—	13	2	no	—
90	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
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	2	no	—
96	#95	—	13	8	no	—
97	#95	—	13	8	no	—
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	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	26	8	no	—
106	C₂×Q₈	—	26	4	no	—
107	C₂×Q₈	Dic₂₆	1	4	yes	complemented
108	C₂²×C₄	—	13	4	no	—
109	C₂²×C₄	—	13	4	no	—
110	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₃	1	2	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	—	2	2	no	—
117	C₄⋊C₄	—	26	4	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	C₂²⋊D₁₃	1	4	yes	—
123	C₁₃	#2568	1	1	yes	characteristic, complemented, Sylow
124	C₂³	C₂³×D₁₃	1	1	yes	characteristic
125	Q₈	—	2	8	no	—
126	Q₈	—	52	8	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	26	4	no	—
133	C₂×C₄	—	26	4	no	—
134	C₂×C₄	—	26	8	no	—
135	C₂×C₄	—	26	4	no	—
136	C₂×C₄	—	26	4	no	—
137	C₂×C₄	C₂×Dic₂₆	1	4	yes	—
138	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
139	C₂×C₄	#6127	1	2	yes	—
140	C₂×C₄	#6131	1	4	yes	—
141	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
144	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
145	C₂²	C₂×#6127	1	1	yes	characteristic
146	C₂²	C₂×#6131	1	2	yes	—
147	C₂²	#88257	1	2	yes	—
148	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	2	no	—
152	C₄	—	52	4	no	—
153	C₄	—	52	4	no	—
154	C₄	#88178	1	4	yes	—
155	C₄	#88201	1	4	yes	—
156	C₂	#11719275	1	2	yes	—
157	C₂	#11719289	1	2	yes	—
158	C₂	C₂×#88178	1	1	yes	characteristic
159	C₂	C₂×#88201	1	1	yes	characteristic
160	C₂	#11719530	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Q₈×C₁₃

C₂

yes

maximal, characteristic

C₂×#88123

C₂

yes

maximal, characteristic

#11718891

C₂

yes

maximal

#11718647

C₂

yes

maximal

#11718638

C₂

yes

maximal

C₄×C₄⋊Dic₁₃

C₂

yes

maximal, characteristic

#11718590

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

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

#88123

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

Q₈

yes

complemented

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

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

D₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

#2568

—

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×Q₈

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

—

Dic₂₆

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

D₁₃

yes

—

—

—

—

C₄×C₄⋊C₄

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂²⋊Q₈

yes

—

C₄×C₁₃

#124

yes

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₁₃

yes

characteristic

C₄⋊Q₈

—

C₄×Q₈

—

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

—

C₂×C₄²

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

Dic₂₆

yes

complemented

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²×D₁₃

yes

—

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

C₂²⋊D₁₃

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

C₂×Dic₂₆

yes

—

138

C₂×C₄

C₂×Dic₂₆

yes

—

139

C₂×C₄

#6127

yes

—

140

C₂×C₄

#6131

yes

—

141

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

142

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

143

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

144

C₂²

C₂²×Dic₂₆

yes

characteristic

145

C₂²

C₂×#6127

yes

characteristic

146

C₂²

C₂×#6131

yes

—

147

C₂²

#88257

yes

—

148

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#88178

yes

—

155

C₄

#88201

yes

—

156

C₂

#11719275

yes

—

157

C₂

#11719289

yes

—

158

C₂

C₂×#88178

yes

characteristic

159

C₂

C₂×#88201

yes

characteristic

160

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	#2568
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
25. C₂×Q₈×C₁₃	Q₈	semidirect
107. C₂×Q₈	Dic₂₆	semidirect
123. C₁₃	#2568	semidirect

H ≅ G/N

type

25. C₂×Q₈×C₁₃

Q₈

semidirect

107. C₂×Q₈

Dic₂₆

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×C₄×Q₈×C₁₃	2
C₂×#88123	2
#11718891	2
#11718647	2
#11718638	2
C₄×C₄⋊Dic₁₃	2
#11718590	2
#2568	13

subgroup

index

C₂×C₄×Q₈×C₁₃

C₄×C₄⋊Dic₁₃

Subgroups — #13120048

Conjugacy classes of subgroups

Complements

Maximal subgroups