Group #13115758 — subgroups

Subgroups — #13115758

Order 1664.

324 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 121 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×#88123	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#88056	C₂	1	4	yes	maximal
4	C₄×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₆	C₂²	1	2	yes	—
6	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₃	—	2	1	no	—
9	C₂×C₄⋊Dic₁₃	—	2	1	no	—
10	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₃	—	2	1	no	—
12	#88123	—	2	2	no	—
13	#88123	C₄	1	4	yes	complemented
14	C₁₃×C₄⋊C₈	—	2	2	no	—
15	#88071	—	2	1	no	—
16	#88056	—	2	8	no	—
17	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₁₃	—	2	2	no	—
21	C₂²×Dic₁₃	—	2	2	no	—
22	C₂×Dic₂₆	—	2	4	no	—
23	C₂×Dic₂₆	—	2	4	no	—
24	C₂×Dic₂₆	—	2	2	no	—
25	C₂×Dic₂₆	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₁₃	—	2	2	no	—
27	C₂×C₈×C₁₃	—	2	4	no	—
28	C₄²×C₁₃	C₂×C₄	1	2	yes	—
29	C₄²×C₁₃	D₄	1	2	yes	—
30	C₄⋊Dic₁₃	—	2	4	no	—
31	C₄⋊Dic₁₃	—	2	4	no	—
32	C₄⋊Dic₁₃	—	2	2	no	—
33	C₄⋊Dic₁₃	D₄	1	4	yes	—
34	C₄×Dic₁₃	—	4	2	no	—
35	#1981	—	13	1	no	maximal, Sylow
36	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃	—	2	2	no	—
38	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₃	SD₁₆	1	2	yes	—
41	C₂×C₄×C₁₃	Q₁₆	1	2	yes	—
42	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃	#41	1	2	yes	—
46	C₂×Dic₁₃	—	2	4	no	—
47	C₂×Dic₁₃	—	2	4	no	—
48	C₂×Dic₁₃	—	4	4	no	—
49	C₂×Dic₁₃	—	4	2	no	—
50	C₂×Dic₁₃	—	4	2	no	—
51	Dic₂₆	—	2	8	no	—
52	Dic₂₆	—	4	2	no	—
53	Dic₂₆	—	4	2	no	—
54	C₈×C₁₃	—	4	2	no	—
55	C₂×C₄⋊Q₈	—	13	1	no	—
56	C₂×C₄⋊C₈	D₁₃	1	1	yes	characteristic
57	C₂×#103	—	13	4	no	—
58	C₄×C₄⋊C₄	—	13	1	no	—
59	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₁₃	#121	1	2	yes	—
61	C₂²×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂²×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
63	C₂²×C₁₃	#137	1	2	yes	—
64	C₄×C₁₃	—	2	2	no	—
65	C₄×C₁₃	#103	1	4	yes	complemented
66	C₄×C₁₃	C₄×D₄	1	2	yes	—
67	C₄×C₁₃	#124	1	1	yes	characteristic
68	C₄×C₁₃	#127	1	1	yes	characteristic
69	Dic₁₃	—	4	4	no	—
70	Dic₁₃	—	4	4	no	—
71	C₂²×Q₈	—	13	2	no	—
72	C₂²×C₈	C₂×D₁₃	1	2	yes	—
73	C₄⋊Q₈	—	13	4	no	—
74	C₄⋊Q₈	—	26	2	no	—
75	C₂×C₄⋊C₄	—	13	2	no	—
76	C₂×C₄⋊C₄	—	26	1	no	—
77	C₂×C₄⋊C₄	—	26	1	no	—
78	C₂×C₄²	—	26	1	no	—
79	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
80	C₄⋊C₈	—	2	2	no	—
81	#103	—	26	8	no	—
82	#95	—	26	1	no	—
83	C₂×C₁₃	#390	1	1	yes	characteristic
84	C₂×C₁₃	C₂×#103	1	1	yes	characteristic
85	C₂×C₁₃	#419	1	1	yes	characteristic
86	C₂×C₁₃	#461	1	2	yes	—
87	C₂×C₁₃	#462	1	2	yes	—
88	C₂×Q₈	—	13	4	no	—
89	C₂×Q₈	—	26	2	no	—
90	C₂×Q₈	—	26	4	no	—
91	C₂×Q₈	—	26	4	no	—
92	C₂²×C₄	—	26	2	no	—
93	C₂²×C₄	—	26	2	no	—
94	C₂²×C₄	D₅₂	1	1	yes	characteristic
95	C₂²×C₄	D₅₂	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	—	2	2	no	—
99	C₄⋊C₄	—	13	4	no	—
100	C₄⋊C₄	—	26	4	no	—
101	C₄⋊C₄	—	26	4	no	—
102	C₄⋊C₄	—	26	2	no	—
103	C₄²	—	52	2	no	—
104	C₄²	C₄×D₁₃	1	2	yes	—
105	C₄²	C₂²⋊D₁₃	1	2	yes	—
106	C₁₃	#1981	1	1	yes	characteristic, complemented, Sylow
107	C₂³	C₂×D₅₂	1	1	yes	characteristic
108	Q₈	—	26	8	no	—
109	Q₈	—	52	2	no	—
110	Q₈	—	52	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	26	4	no	—
113	C₂×C₄	—	26	4	no	—
114	C₂×C₄	—	52	4	no	—
115	C₂×C₄	—	52	2	no	—
116	C₂×C₄	—	52	2	no	—
117	C₂×C₄	#6095	1	2	yes	—
118	C₂×C₄	Dic₅₂	1	2	yes	—
119	C₂×C₄	#6103	1	2	yes	—
120	C₂×C₄	#6103	1	2	yes	—
121	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
122	C₂×C₄	D₄×D₁₃	1	2	yes	—
123	C₂×C₄	#6131	1	2	yes	—
124	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
125	C₈	—	4	2	no	—
126	C₂²	#88149	1	2	yes	—
127	C₂²	C₂×#6095	1	1	yes	characteristic
128	C₂²	C₂×Dic₅₂	1	1	yes	characteristic
129	C₂²	#88163	1	2	yes	—
130	C₂²	C₂×#6103	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	52	4	no	—
133	C₄	—	52	4	no	—
134	C₄	#88056	1	4	yes	complemented
135	C₄	#88147	1	2	yes	—
136	C₄	#88194	1	1	yes	characteristic
137	C₄	#88201	1	1	yes	characteristic
138	C₂	#11718502	1	2	yes	—
139	C₂	#11718510	1	2	yes	—
140	C₂	#11718649	1	1	yes	characteristic
141	C₂	C₂×#88056	1	1	yes	characteristic
142	C₂	#11718781	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88123

C₂

yes

maximal, characteristic

C₂×C₁₃×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂×#88056

C₂

yes

maximal

C₄×C₄⋊Dic₁₃

C₂

yes

maximal, characteristic

C₂²×Dic₂₆

C₂²

yes

—

C₂²×C₈×C₁₃

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

—

#88123

—

#88123

C₄

yes

complemented

C₁₃×C₄⋊C₈

—

#88071

—

#88056

—

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

C₂×C₄

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

C₂×C₄

yes

—

C₄²×C₁₃

D₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

D₄

yes

—

C₄×Dic₁₃

—

#1981

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

SD₁₆

yes

—

C₂×C₄×C₁₃

Q₁₆

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

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

D₁₃

yes

characteristic

C₂×#103

—

C₄×C₄⋊C₄

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃

#121

yes

—

C₂²×C₁₃

C₂×SD₁₆

yes

characteristic

C₂²×C₁₃

C₂×Q₁₆

yes

characteristic

C₂²×C₁₃

#137

yes

—

C₄×C₁₃

—

C₄×C₁₃

#103

yes

complemented

C₄×C₁₃

C₄×D₄

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₁₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

#103

—

#95

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂×C₁₃

#462

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₅₂

yes

characteristic

C₂²×C₄

D₅₂

yes

characteristic

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄²

—

104

C₄²

C₄×D₁₃

yes

—

105

C₄²

C₂²⋊D₁₃

yes

—

106

C₁₃

#1981

yes

characteristic, complemented, Sylow

107

C₂³

C₂×D₅₂

yes

characteristic

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#6095

yes

—

118

C₂×C₄

Dic₅₂

yes

—

119

C₂×C₄

#6103

yes

—

120

C₂×C₄

#6103

yes

—

121

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

122

C₂×C₄

D₄×D₁₃

yes

—

123

C₂×C₄

#6131

yes

—

124

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

125

C₈

—

126

C₂²

#88149

yes

—

127

C₂²

C₂×#6095

yes

characteristic

128

C₂²

C₂×Dic₅₂

yes

characteristic

129

C₂²

#88163

yes

—

130

C₂²

C₂×#6103

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#88056

yes

complemented

135

C₄

#88147

yes

—

136

C₄

#88194

yes

characteristic

137

C₄

#88201

yes

characteristic

138

C₂

#11718502

yes

—

139

C₂

#11718510

yes

—

140

C₂

#11718649

yes

characteristic

141

C₂

C₂×#88056

yes

characteristic

142

C₂

#11718781

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₃
hypocenter	C₁₃	#1981
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₅₂

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₂×C₁₃×C₄⋊C₈

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₄

cosocle

C₂²×C₄×C₁₃

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₃

hypocenter

C₁₃

#1981

LCS

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
13. #88123	C₄	semidirect
65. C₄×C₁₃	#103	semidirect
106. C₁₃	#1981	semidirect
134. C₄	#88056	semidirect

H ≅ G/N

type

13. #88123

C₄

semidirect

65. C₄×C₁₃

#103

semidirect

106. C₁₃

#1981

semidirect

134. C₄

#88056

semidirect

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

subgroup	index
C₂×#88123	2
C₂×C₁₃×C₄⋊C₈	2
C₂×#88056	2
C₄×C₄⋊Dic₁₃	2
#1981	13

subgroup

index

C₂×#88123

C₂×C₁₃×C₄⋊C₈

C₂×#88056

C₄×C₄⋊Dic₁₃

#1981

Subgroups — #13115758

Conjugacy classes of subgroups

Complements

Maximal subgroups