Group #13123555 — subgroups

Subgroups — #13123555

Order 1664.

484 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 189 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×#88155	C₂	1	2	yes	maximal, complemented
2	#11719182	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718771	C₂	1	2	yes	maximal, complemented
5	#11718585	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718367	C₂	1	8	yes	maximal, complemented
7	C₂×#6127	C₂²	1	2	yes	—
8	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	#88184	C₄	1	4	yes	—
12	C₂×#6099	C₂²	1	2	yes	—
13	#88155	—	2	8	no	—
14	C₂×#6094	C₂²	1	4	yes	—
15	C₂×C₈×D₁₃	C₂²	1	4	yes	—
16	#88129	C₄	1	2	yes	—
17	#88128	C₄	1	1	yes	characteristic
18	C₄×D₅₂	C₂²	1	4	yes	—
19	#88123	C₄	1	1	yes	characteristic
20	C₁₃×#97	C₂²	1	4	yes	—
21	#88060	C₂²	1	8	yes	—
22	C₄⋊#1055	C₂²	1	4	yes	—
23	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂×C₂²⋊D₁₃	C₂×C₄	1	4	yes	—
26	#6127	—	2	4	no	—
27	#6127	—	2	4	no	—
28	C₂×D₅₂	C₂×C₄	1	2	yes	—
29	C₂×D₅₂	C₂×C₄	1	4	yes	—
30	C₂×C₄×D₁₃	C₂³	1	4	yes	—
31	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₁₃	—	2	2	no	—
33	C₂×C₈×C₁₃	D₄	1	4	yes	complemented
34	C₂×C₈×C₁₃	C₂³	1	2	yes	—
35	C₂×C₈×C₁₃	C₂³	1	2	yes	—
36	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₁₃	C₂³	1	2	yes	—
39	#6103	C₂×C₄	1	8	yes	—
40	C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
41	#6099	—	2	4	no	—
42	C₂×#1055	C₂³	1	4	yes	—
43	#6094	—	2	8	no	—
44	C₈×D₁₃	—	2	8	no	—
45	#2932	—	13	1	no	maximal, Sylow
46	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×D₁₃	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₁₃	—	4	2	no	—
49	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₃	#41	1	2	yes	—
54	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
55	C₂²⋊D₁₃	—	2	8	no	—
56	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
57	D₅₂	—	2	4	no	—
58	D₅₂	—	4	4	no	—
59	C₄×D₁₃	—	2	4	no	—
60	C₄×D₁₃	—	2	4	no	—
61	Dic₂₆	—	2	4	no	—
62	C₈×C₁₃	—	2	2	no	—
63	C₈×C₁₃	C₂×D₄	1	4	yes	—
64	#1055	—	2	4	no	—
65	C₂×#131	—	13	2	no	—
66	#532	—	13	1	no	—
67	#435	—	13	8	no	—
68	#423	—	13	1	no	—
69	#408	—	13	2	no	—
70	C₂×#97	D₁₃	1	1	yes	characteristic, complemented
71	C₂²×C₁₃	—	2	2	no	—
72	C₂²×C₁₃	C₄×D₄	1	2	yes	—
73	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
74	C₂×D₁₃	—	2	4	no	—
75	C₂×D₁₃	—	2	8	no	—
76	C₄×C₁₃	—	2	2	no	—
77	C₄×C₁₃	—	2	2	no	—
78	C₄×C₁₃	C₄×D₄	1	2	yes	—
79	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
80	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
81	Dic₁₃	—	2	4	no	—
82	C₂×#41	—	13	2	no	—
83	#131	—	26	8	no	—
84	C₂×M₁₆	—	13	2	no	—
85	C₂×M₁₆	—	13	4	no	—
86	C₂²×C₈	—	13	4	no	—
87	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
88	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
89	C₄⋊Q₈	—	13	1	no	—
90	#127	—	13	1	no	—
91	#124	—	13	2	no	—
92	#121	—	13	4	no	—
93	C₄×D₄	—	13	4	no	—
94	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
95	C₄⋊C₈	—	13	4	no	—
96	C₂²⋊C₈	—	13	8	no	—
97	#97	C₂×D₁₃	1	4	yes	—
98	C₂×C₁₃	—	2	2	no	—
99	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
100	C₂×C₁₃	#568	1	2	yes	—
101	D₁₃	—	4	4	no	—
102	#41	—	26	4	no	—
103	#41	—	26	4	no	—
104	C₂×Q₈	—	13	2	no	—
105	C₂×D₄	—	13	4	no	—
106	C₂×D₄	—	13	2	no	—
107	C₂×D₄	—	13	4	no	—
108	C₂²×C₄	—	13	4	no	—
109	C₂²×C₄	C₄×D₁₃	1	2	yes	—
110	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
111	M₁₆	—	26	4	no	—
112	M₁₆	—	26	8	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	13	4	no	—
115	C₂×C₈	—	26	8	no	—
116	C₂×C₈	D₅₂	1	4	yes	complemented
117	C₂×C₈	C₂²×D₁₃	1	2	yes	—
118	C₂×C₈	C₂²×D₁₃	1	2	yes	—
119	C₄⋊C₄	—	13	4	no	—
120	C₂²⋊C₄	—	13	8	no	—
121	C₄²	C₄×D₁₃	1	1	yes	characteristic
122	C₄²	C₄×D₁₃	1	1	yes	characteristic
123	C₄²	C₂²×D₁₃	1	2	yes	—
124	C₁₃	#2932	1	1	yes	characteristic, complemented, Sylow
125	C₂³	—	13	4	no	—
126	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
127	Q₈	—	26	4	no	—
128	D₄	—	26	8	no	—
129	D₄	—	26	4	no	—
130	D₄	—	52	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	13	4	no	—
133	C₂×C₄	—	26	4	no	—
134	C₂×C₄	—	26	4	no	—
135	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
138	C₂×C₄	C₂×D₅₂	1	2	yes	—
139	C₂×C₄	#6127	1	2	yes	—
140	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	26	4	no	—
143	C₈	C₂×D₅₂	1	4	yes	—
144	C₂²	—	2	2	no	—
145	C₂²	—	26	4	no	—
146	C₂²	—	26	8	no	—
147	C₂²	C₄×D₅₂	1	2	yes	—
148	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	26	4	no	—
152	C₄	C₄×D₅₂	1	2	yes	—
153	C₄	C₂²×D₅₂	1	1	yes	characteristic
154	C₄	C₂×#6127	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	52	4	no	—
157	C₂	C₂×C₄×D₅₂	1	1	yes	characteristic
158	C₂	#11719450	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88155

C₂

yes

maximal, complemented

#11719182

C₂

yes

maximal, characteristic

C₂×C₁₃×#97

C₂

yes

maximal, characteristic, complemented

#11718771

C₂

yes

maximal, complemented

#11718585

C₂

yes

maximal, characteristic, complemented

#11718367

C₂

yes

maximal, complemented

C₂×#6127

C₂²

yes

—

C₂²×C₈×C₁₃

C₂²

yes

characteristic

C₂²×C₈×C₁₃

C₂²

yes

characteristic

C₂×C₄²×C₁₃

C₂²

yes

characteristic

#88184

C₄

yes

—

C₂×#6099

C₂²

yes

—

#88155

—

C₂×#6094

C₂²

yes

—

C₂×C₈×D₁₃

C₂²

yes

—

#88129

C₄

yes

—

#88128

C₄

yes

characteristic

C₄×D₅₂

C₂²

yes

—

#88123

C₄

yes

characteristic

C₁₃×#97

C₂²

yes

—

#88060

C₂²

yes

—

C₄⋊#1055

C₂²

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂×C₂²⋊D₁₃

C₂×C₄

yes

—

#6127

—

#6127

—

C₂×D₅₂

C₂×C₄

yes

—

C₂×D₅₂

C₂×C₄

yes

—

C₂×C₄×D₁₃

C₂³

yes

—

C₂×Dic₂₆

C₂×C₄

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

D₄

yes

complemented

C₂×C₈×C₁₃

C₂³

yes

—

C₂×C₈×C₁₃

C₂³

yes

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

C₂³

yes

—

#6103

C₂×C₄

yes

—

C₄⋊Dic₁₃

C₂×C₄

yes

—

#6099

—

C₂×#1055

C₂³

yes

—

#6094

—

C₈×D₁₃

—

#2932

—

maximal, Sylow

C₂³×C₁₃

C₂²×C₄

yes

characteristic

C₂²×D₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

C₂²⋊D₁₃

—

C₂×Dic₁₃

C₂²×C₄

yes

—

D₅₂

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

C₂×D₄

yes

—

—

—

—

—

—

—

C₂×#97

D₁₃

yes

characteristic, complemented

C₂²×C₁₃

—

C₂²×C₁₃

C₄×D₄

yes

—

C₂²×C₁₃

C₂³×C₄

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×D₄

yes

—

C₄×C₁₃

C₂²×D₄

yes

characteristic

C₄×C₁₃

C₂×#41

yes

characteristic

Dic₁₃

—

C₂×#41

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₁₃

yes

characteristic

C₂²×C₈

C₂×D₁₃

yes

characteristic

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

C₂×D₁₃

yes

—

C₂×C₁₃

—

C₂×C₁₃

C₂×C₄×D₄

yes

characteristic

100

C₂×C₁₃

#568

yes

—

101

D₁₃

—

102

#41

—

103

#41

—

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₄×D₁₃

yes

—

110

C₂²×C₄

C₂²×D₁₃

yes

characteristic

111

M₁₆

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

D₅₂

yes

complemented

117

C₂×C₈

C₂²×D₁₃

yes

—

118

C₂×C₈

C₂²×D₁₃

yes

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

C₄×D₁₃

yes

characteristic

122

C₄²

C₄×D₁₃

yes

characteristic

123

C₄²

C₂²×D₁₃

yes

—

124

C₁₃

#2932

yes

characteristic, complemented, Sylow

125

C₂³

—

126

C₂³

C₂×C₄×D₁₃

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×C₄×D₁₃

yes

—

136

C₂×C₄

C₂×C₄×D₁₃

yes

—

137

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

138

C₂×C₄

C₂×D₅₂

yes

—

139

C₂×C₄

#6127

yes

—

140

C₂×C₄

C₂³×D₁₃

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

C₂×D₅₂

yes

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

C₄×D₅₂

yes

—

148

C₂²

C₂²×C₄×D₁₃

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

C₄×D₅₂

yes

—

153

C₄

C₂²×D₅₂

yes

characteristic

154

C₄

C₂×#6127

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

C₂×C₄×D₅₂

yes

characteristic

158

C₂

#11719450

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₂×C₁₃×#97	C₂
prime radical	#11719182	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₂×#97	D₁₃
hypocenter	C₁₃	#2932
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₃

commutator, socle

C₂²×C₁₃

C₂³×C₄

Fitting

C₂×C₁₃×#97

C₂

prime radical

#11719182

C₂

cosocle

C₂×C₄×C₁₃

C₂⁴

hypercenter

C₂×#97

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₄ ◁ C₂×#97

N	H ≅ G/N	type
1. C₂×#88155	C₂	semidirect
3. C₂×C₁₃×#97	C₂	semidirect
4. #11718771	C₂	semidirect
5. #11718585	C₂	semidirect
6. #11718367	C₂	semidirect
33. C₂×C₈×C₁₃	D₄	semidirect
70. C₂×#97	D₁₃	semidirect
116. C₂×C₈	D₅₂	semidirect
124. C₁₃	#2932	semidirect

H ≅ G/N

type

1. C₂×#88155

C₂

semidirect

3. C₂×C₁₃×#97

C₂

semidirect

4. #11718771

C₂

semidirect

5. #11718585

C₂

semidirect

6. #11718367

C₂

semidirect

33. C₂×C₈×C₁₃

D₄

semidirect

70. C₂×#97

D₁₃

semidirect

116. C₂×C₈

D₅₂

semidirect

124. C₁₃

#2932

semidirect

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

subgroup	index
C₂×#88155	2
#11719182	2
C₂×C₁₃×#97	2
#11718771	2
#11718585	2
#11718367	2
#2932	13

subgroup

index

C₂×C₁₃×#97

Subgroups — #13123555

Conjugacy classes of subgroups

Complements

Maximal subgroups