Group #13115669 — subgroups

Subgroups — #13115669

Order 1664.

320 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₁₃×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×#88077	C₂	1	4	yes	maximal, complemented
3	#11718593	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718585	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₃×#41	C₂²	1	2	yes	—
6	C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
7	C₁₃×#127	C₄	1	1	yes	characteristic, complemented
8	C₁₃×#124	C₄	1	2	yes	complemented
9	C₁₃×#121	—	2	2	no	—
10	C₄×D₄×C₁₃	—	2	2	no	—
11	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
13	C₂×#6099	C₂²	1	2	yes	—
14	#88077	—	2	8	no	—
15	#88071	—	2	2	no	—
16	C₄⋊#1055	—	2	2	no	—
17	C₁₃×#41	—	2	4	no	—
18	C₁₃×#41	—	2	4	no	—
19	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
20	C₂×D₄×C₁₃	—	2	2	no	—
21	C₂×D₄×C₁₃	—	2	2	no	—
22	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₃	—	2	2	no	—
24	C₂²×C₄×C₁₃	D₄	1	2	yes	—
25	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₁₃	—	2	2	no	—
27	C₁₃×C₄⋊C₄	—	2	2	no	—
28	C₁₃×C₂²⋊C₄	—	2	4	no	—
29	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₁₃	D₄	1	2	yes	—
32	C₄×Dic₁₃	—	2	2	no	—
33	C₄×Dic₁₃	D₄	1	4	yes	complemented
34	#6099	—	2	4	no	—
35	C₂×#1055	—	2	2	no	—
36	#1963	—	13	1	no	maximal, Sylow
37	C₂³×C₁₃	—	2	2	no	—
38	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₁₃	—	2	4	no	—
40	D₄×C₁₃	—	2	4	no	—
41	D₄×C₁₃	—	4	4	no	—
42	D₄×C₁₃	—	4	4	no	—
43	C₂×C₄×C₁₃	—	2	2	no	—
44	C₂×C₄×C₁₃	—	4	2	no	—
45	C₂×C₄×C₁₃	—	4	2	no	—
46	C₂×C₄×C₁₃	—	4	2	no	—
47	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃	#41	1	2	yes	—
53	C₂×Dic₁₃	—	2	4	no	—
54	C₂×Dic₁₃	—	4	4	no	—
55	#1055	—	4	2	no	—
56	#532	D₁₃	1	1	yes	characteristic
57	#423	—	13	1	no	—
58	C₂×#104	—	13	4	no	—
59	#376	—	13	1	no	—
60	C₂²×C₁₃	—	2	2	no	—
61	C₂²×C₁₃	—	4	2	no	—
62	C₂²×C₁₃	—	4	4	no	—
63	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₁₃	#121	1	2	yes	—
65	C₄×C₁₃	—	2	2	no	—
66	C₄×C₁₃	—	2	2	no	—
67	C₄×C₁₃	—	4	2	no	—
68	C₄×C₁₃	C₄×D₄	1	2	yes	—
69	C₄×C₁₃	#124	1	1	yes	characteristic
70	C₄×C₁₃	#127	1	1	yes	characteristic
71	Dic₁₃	—	4	4	no	—
72	C₂×#41	C₂×D₁₃	1	2	yes	—
73	C₂×M₁₆	—	13	2	no	—
74	C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic, complemented
75	#127	Dic₁₃	1	1	yes	characteristic, complemented
76	#124	Dic₁₃	1	2	yes	complemented
77	#121	—	2	2	no	—
78	C₄×D₄	—	2	2	no	—
79	C₂×C₄²	—	13	2	no	—
80	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
81	C₄⋊C₈	—	26	2	no	—
82	#104	—	26	8	no	—
83	#95	—	26	2	no	—
84	C₂×C₁₃	—	2	2	no	—
85	C₂×C₁₃	—	8	2	no	—
86	C₂×C₁₃	#390	1	1	yes	characteristic
87	C₂×C₁₃	#421	1	2	yes	—
88	#41	—	2	4	no	—
89	#41	—	2	4	no	—
90	C₂×Q₈	C₂×Dic₁₃	1	2	yes	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	26	2	no	—
96	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
97	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
98	M₁₆	—	26	4	no	—
99	C₂×C₈	—	26	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₂²⋊C₄	—	2	4	no	—
102	C₄²	—	13	4	no	—
103	C₄²	—	26	2	no	—
104	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
105	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
106	C₄²	C₂²⋊D₁₃	1	2	yes	—
107	C₁₃	#1963	1	1	yes	characteristic, complemented, Sylow
108	C₂³	—	2	2	no	—
109	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
110	Q₈	—	2	4	no	—
111	D₄	—	2	4	no	—
112	D₄	—	4	4	no	—
113	D₄	—	4	4	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	26	4	no	—
119	C₂×C₄	—	52	4	no	—
120	C₂×C₄	C₂²⋊Dic₁₃	1	2	yes	—
121	C₂×C₄	C₂²⋊Dic₁₃	1	2	yes	—
122	C₂×C₄	D₄×D₁₃	1	2	yes	—
123	C₂×C₄	#6129	1	2	yes	—
124	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
126	C₈	—	52	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	4	2	no	—
129	C₂²	—	4	4	no	—
130	C₂²	#88192	1	2	yes	—
131	C₂²	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	2	no	—
135	C₄	—	52	4	no	—
136	C₄	D₄×Dic₁₃	1	2	yes	—
137	C₄	#88190	1	1	yes	characteristic
138	C₄	#88194	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	8	2	no	—
141	C₂	#11718882	1	1	yes	characteristic
142	C₂	#11718897	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#532

C₂

yes

maximal, characteristic

C₂×#88077

C₂

yes

maximal, complemented

#11718593

C₂

yes

maximal, characteristic, complemented

#11718585

C₂

yes

maximal, characteristic, complemented

C₂×C₁₃×#41

C₂²

yes

—

C₁₃×C₄⋊Q₈

C₄

yes

characteristic, complemented

C₁₃×#127

C₄

yes

characteristic, complemented

C₁₃×#124

C₄

yes

complemented

C₁₃×#121

—

C₄×D₄×C₁₃

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×#6099

C₂²

yes

—

#88077

—

#88071

—

C₄⋊#1055

—

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

D₄

yes

—

C₄×Dic₁₃

—

C₄×Dic₁₃

D₄

yes

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂×D₄

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

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

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

#1055

—

#532

D₁₃

yes

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃

#121

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×D₄

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

C₂×#41

C₂×D₁₃

yes

—

C₂×M₁₆

—

C₄⋊Q₈

Dic₁₃

yes

characteristic, complemented

#127

Dic₁₃

yes

characteristic, complemented

#124

Dic₁₃

yes

complemented

#121

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

#104

—

#95

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

#390

yes

characteristic

C₂×C₁₃

#421

yes

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₁₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₁₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

M₁₆

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

C₂×Dic₁₃

yes

characteristic

105

C₄²

C₂×Dic₁₃

yes

characteristic

106

C₄²

C₂²⋊D₁₃

yes

—

107

C₁₃

#1963

yes

characteristic, complemented, Sylow

108

C₂³

—

109

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

110

Q₈

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂²⋊Dic₁₃

yes

—

121

C₂×C₄

C₂²⋊Dic₁₃

yes

—

122

C₂×C₄

D₄×D₁₃

yes

—

123

C₂×C₄

#6129

yes

—

124

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

126

C₈

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

#88192

yes

—

131

C₂²

C₂×C₂²⋊Dic₁₃

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

D₄×Dic₁₃

yes

—

137

C₄

#88190

yes

characteristic

138

C₄

#88194

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

#11718882

yes

characteristic

142

C₂

#11718897

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting, prime radical	C₁₃×#532	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	#532	D₁₃
hypocenter	C₁₃	#1963
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting, prime radical

C₁₃×#532

C₂

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
2. C₂×#88077	C₂	semidirect
3. #11718593	C₂	semidirect
4. #11718585	C₂	semidirect
6. C₁₃×C₄⋊Q₈	C₄	semidirect
7. C₁₃×#127	C₄	semidirect
8. C₁₃×#124	C₄	semidirect
33. C₄×Dic₁₃	D₄	semidirect
74. C₄⋊Q₈	Dic₁₃	semidirect
75. #127	Dic₁₃	semidirect
76. #124	Dic₁₃	semidirect
107. C₁₃	#1963	semidirect

H ≅ G/N

type

2. C₂×#88077

C₂

semidirect

3. #11718593

C₂

semidirect

4. #11718585

C₂

semidirect

6. C₁₃×C₄⋊Q₈

C₄

semidirect

7. C₁₃×#127

C₄

semidirect

8. C₁₃×#124

C₄

semidirect

33. C₄×Dic₁₃

D₄

semidirect

74. C₄⋊Q₈

Dic₁₃

semidirect

75. #127

Dic₁₃

semidirect

76. #124

Dic₁₃

semidirect

107. C₁₃

#1963

semidirect

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

subgroup	index
C₁₃×#532	2
C₂×#88077	2
#11718593	2
#11718585	2
#1963	13

subgroup

index

C₁₃×#532

Subgroups — #13115669

Conjugacy classes of subgroups

Complements

Maximal subgroups