Group #13123703 — subgroups

Subgroups — #13123703

Order 1664.

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

normal subgroups only

This group has 456 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	C₂²×Dic₅₂	C₂	1	1	yes	maximal, characteristic
2	#11719221	C₂	1	2	yes	maximal
3	C₁₃×#427	C₂	1	1	yes	maximal, characteristic
4	#11718762	C₂	1	1	yes	maximal, characteristic
5	#11718665	C₂	1	2	yes	maximal
6	#11718534	C₂	1	4	yes	maximal, complemented
7	#11718514	C₂	1	4	yes	maximal, complemented
8	C₂²×Dic₂₆	C₂²	1	2	yes	—
9	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×#117	C₂²	1	2	yes	—
11	#88180	C₂²	1	1	yes	characteristic
12	C₂×#6099	C₂²	1	1	yes	characteristic
13	C₂×Dic₅₂	C₄	1	8	yes	complemented
14	C₂×Dic₅₂	C₂²	1	2	yes	—
15	C₂×Dic₅₂	C₂²	1	2	yes	—
16	#88141	C₂²	1	8	yes	—
17	#88131	—	2	4	no	—
18	C₄×Dic₂₆	—	2	4	no	—
19	C₁₃×#107	C₂²	1	2	yes	—
20	C₁₃×#106	C₂²	1	2	yes	—
21	#88055	C₂²	1	2	yes	—
22	C₈×Dic₁₃	C₂²	1	2	yes	—
23	#88050	C₂²	1	8	yes	—
24	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₁₃	—	2	2	no	—
26	C₂×Dic₂₆	—	2	2	no	—
27	C₂×Dic₂₆	—	2	4	no	—
28	C₂×Dic₂₆	C₂×C₄	1	8	yes	—
29	C₂×Dic₂₆	C₂³	1	4	yes	—
30	C₂×C₈×C₁₃	C₂×C₄	1	4	yes	—
31	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
34	C₁₃×C₂²⋊C₄	—	2	2	no	—
35	C₄²×C₁₃	—	2	2	no	—
36	C₂²⋊Dic₁₃	D₄	1	2	yes	—
37	C₄⋊Dic₁₃	D₄	1	2	yes	—
38	#6101	—	2	8	no	—
39	C₄×Dic₁₃	—	2	8	no	—
40	C₄×Dic₁₃	C₂³	1	2	yes	—
41	#6099	—	2	2	no	—
42	C₂×#1055	C₂³	1	2	yes	—
43	Dic₅₂	C₂×C₄	1	8	yes	—
44	Dic₅₂	C₂×C₄	1	8	yes	—
45	#2950	—	13	1	no	maximal, Sylow
46	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃	—	2	4	no	—
48	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃	#41	1	2	yes	—
51	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₁₃	—	2	8	no	—
53	C₂×Dic₁₃	—	2	4	no	—
54	C₂×Dic₁₃	C₂×D₄	1	4	yes	—
55	Dic₂₆	—	2	8	no	—
56	Dic₂₆	—	2	4	no	—
57	Dic₂₆	C₂²×C₄	1	8	yes	—
58	C₈×C₁₃	C₂²×C₄	1	2	yes	—
59	C₈×C₁₃	C₂²×C₄	1	2	yes	—
60	#1055	—	2	2	no	—
61	C₂²×Q₁₆	—	13	1	no	—
62	#519	—	13	2	no	—
63	#441	—	13	4	no	—
64	C₄×Q₁₆	—	13	4	no	—
65	#427	D₁₃	1	1	yes	characteristic
66	#419	—	13	2	no	—
67	#405	—	13	1	no	—
68	C₂²×C₁₃	—	2	2	no	—
69	C₂²×C₁₃	C₄×D₄	1	2	yes	—
70	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
71	C₄×C₁₃	—	4	4	no	—
72	C₄×C₁₃	C₄×D₄	1	2	yes	—
73	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
74	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
75	Dic₁₃	—	2	8	no	—
76	Dic₁₃	—	2	4	no	—
77	C₂²×Q₈	—	13	2	no	—
78	C₂×Q₁₆	—	13	8	no	—
79	C₂×Q₁₆	—	13	2	no	—
80	C₂×Q₁₆	—	13	2	no	—
81	C₂×M₁₆	—	13	1	no	—
82	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
83	C₄×Q₈	—	13	8	no	—
84	C₄×Q₈	—	26	4	no	—
85	#117	—	13	1	no	—
86	#117	—	26	4	no	—
87	#117	C₂×D₁₃	1	2	yes	—
88	#107	C₂×D₁₃	1	2	yes	—
89	#106	C₂×D₁₃	1	2	yes	—
90	#103	—	13	8	no	—
91	#97	—	13	2	no	—
92	C₄×C₈	—	13	2	no	—
93	C₂×C₁₃	—	2	2	no	—
94	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
95	C₂×C₁₃	#578	1	2	yes	—
96	C₂×Q₈	—	13	8	no	—
97	C₂×Q₈	—	13	4	no	—
98	C₂×Q₈	—	26	2	no	—
99	C₂×Q₈	—	26	4	no	—
100	C₂²×C₄	—	26	2	no	—
101	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
102	Q₁₆	—	13	8	no	—
103	Q₁₆	—	13	8	no	—
104	M₁₆	—	26	2	no	—
105	C₂×C₈	—	13	2	no	—
106	C₂×C₈	C₄×D₁₃	1	4	yes	—
107	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
108	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
109	C₄⋊C₄	—	13	2	no	—
110	C₄⋊C₄	—	26	8	no	—
111	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₂²⋊C₄	—	13	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	—	13	2	no	—
116	C₄²	—	26	8	no	—
117	C₁₃	#2950	1	1	yes	characteristic, complemented, Sylow
118	C₂³	D₄×D₁₃	1	1	yes	characteristic
119	Q₈	—	13	8	no	—
120	Q₈	—	26	8	no	—
121	Q₈	—	26	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	13	4	no	—
124	C₂×C₄	—	26	8	no	—
125	C₂×C₄	—	26	4	no	—
126	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
127	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
128	C₂×C₄	#6131	1	2	yes	—
129	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
130	C₈	—	26	2	no	—
131	C₈	C₂×C₄×D₁₃	1	2	yes	—
132	C₈	C₂×C₄×D₁₃	1	2	yes	—
133	C₂²	—	2	2	no	—
134	C₂²	#88147	1	2	yes	—
135	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
136	C₄	—	4	4	no	—
137	C₄	—	26	8	no	—
138	C₄	—	26	4	no	—
139	C₄	#88147	1	2	yes	—
140	C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
141	C₄	C₂×#6131	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×#88147	1	1	yes	characteristic
144	C₂	#11719476	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₂

C₂

yes

maximal, characteristic

#11719221

C₂

yes

maximal

C₁₃×#427

C₂

yes

maximal, characteristic

#11718762

C₂

yes

maximal, characteristic

#11718665

C₂

yes

maximal

#11718534

C₂

yes

maximal, complemented

#11718514

C₂

yes

maximal, complemented

C₂²×Dic₂₆

C₂²

yes

—

C₂²×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×#117

C₂²

yes

—

#88180

C₂²

yes

characteristic

C₂×#6099

C₂²

yes

characteristic

C₂×Dic₅₂

C₄

yes

complemented

C₂×Dic₅₂

C₂²

yes

—

C₂×Dic₅₂

C₂²

yes

—

#88141

C₂²

yes

—

#88131

—

C₄×Dic₂₆

—

C₁₃×#107

C₂²

yes

—

C₁₃×#106

C₂²

yes

—

#88055

C₂²

yes

—

C₈×Dic₁₃

C₂²

yes

—

#88050

C₂²

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

C₂×C₄

yes

—

C₂×Dic₂₆

C₂³

yes

—

C₂×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₈×C₁₃

C₂³

yes

characteristic

C₂×C₈×C₁₃

C₂³

yes

characteristic

C₁₃×C₄⋊C₄

C₂³

yes

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₂²⋊Dic₁₃

D₄

yes

—

C₄⋊Dic₁₃

D₄

yes

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

C₂³

yes

—

#6099

—

C₂×#1055

C₂³

yes

—

Dic₅₂

C₂×C₄

yes

—

Dic₅₂

C₂×C₄

yes

—

#2950

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₄

yes

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

C₂²×C₄

yes

—

C₈×C₁₃

C₂²×C₄

yes

—

C₈×C₁₃

C₂²×C₄

yes

—

#1055

—

C₂²×Q₁₆

—

#519

—

#441

—

C₄×Q₁₆

—

#427

D₁₃

yes

characteristic

#419

—

#405

—

C₂²×C₁₃

—

C₂²×C₁₃

C₄×D₄

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

C₄×D₄

yes

—

C₄×C₁₃

C₂³×C₄

yes

characteristic

C₄×C₁₃

C₂×#41

yes

characteristic

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₁₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

—

#117

C₂×D₁₃

yes

—

#107

C₂×D₁₃

yes

—

#106

C₂×D₁₃

yes

—

#103

—

#97

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

C₂×C₄×D₄

yes

characteristic

C₂×C₁₃

#578

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂²×D₁₃

yes

characteristic

102

Q₁₆

—

103

Q₁₆

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

C₄×D₁₃

yes

—

107

C₂×C₈

C₂²×D₁₃

yes

characteristic

108

C₂×C₈

C₂²×D₁₃

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂²×D₁₃

yes

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

117

C₁₃

#2950

yes

characteristic, complemented, Sylow

118

C₂³

D₄×D₁₃

yes

characteristic

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₂×C₄×D₁₃

yes

—

127

C₂×C₄

D₄×D₁₃

yes

characteristic

128

C₂×C₄

#6131

yes

—

129

C₂×C₄

C₂³×D₁₃

yes

characteristic

130

C₈

—

131

C₈

C₂×C₄×D₁₃

yes

—

132

C₈

C₂×C₄×D₁₃

yes

—

133

C₂²

—

134

C₂²

#88147

yes

—

135

C₂²

C₂×D₄×D₁₃

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#88147

yes

—

140

C₄

C₂²×C₄×D₁₃

yes

characteristic

141

C₄

C₂×#6131

yes

characteristic

142

C₂

—

143

C₂

C₂×#88147

yes

characteristic

144

C₂

#11719476

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#427	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂³×C₁₃	C₂×D₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#427	D₁₃
hypocenter	C₁₃	#2950
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₁₃

commutator

C₄×C₁₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₃

Fitting

C₁₃×#427

C₂

socle

C₂²×C₁₃

C₂²×D₄

prime radical

C₂³×C₁₃

C₂×D₄

cosocle

C₂×C₄×C₁₃

C₂⁴

D₁₃

C₁₃

G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
6. #11718534	C₂	semidirect
7. #11718514	C₂	semidirect
13. C₂×Dic₅₂	C₄	semidirect
117. C₁₃	#2950	semidirect

H ≅ G/N

type

6. #11718534

C₂

semidirect

7. #11718514

C₂

semidirect

13. C₂×Dic₅₂

C₄

semidirect

117. C₁₃

#2950

semidirect

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

subgroup	index
C₂²×Dic₅₂	2
#11719221	2
C₁₃×#427	2
#11718762	2
#11718665	2
#11718534	2
#11718514	2
#2950	13

subgroup

index

C₂²×Dic₅₂

C₁₃×#427

Subgroups — #13123703

Conjugacy classes of subgroups

Complements

Maximal subgroups