Group #13115793 — subgroups

Subgroups — #13115793

Order 1664.

260 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 109 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×#88124	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
4	#11718121	C₂	1	2	yes	maximal
5	#11718121	C₂	1	2	yes	maximal
6	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₃	—	2	1	no	—
8	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂×#6101	—	2	1	no	—
12	C₂×#6101	—	2	1	no	—
13	C₂²×#1055	C₂²	1	2	yes	—
14	#88124	—	2	2	no	—
15	#88124	C₄	1	4	yes	complemented
16	C₁₃×#95	—	2	1	no	—
17	C₄⋊#1055	—	2	2	no	—
18	C₂²×C₄×C₁₃	—	2	2	no	—
19	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₁₃	—	2	1	no	—
23	C₂²×Dic₁₃	—	2	1	no	—
24	C₁₃×C₄⋊C₄	—	2	2	no	—
25	C₁₃×C₄⋊C₄	Q₈	1	4	yes	—
26	C₄²×C₁₃	—	4	2	no	—
27	C₄²×C₁₃	C₂×C₄	1	2	yes	—
28	C₄²×C₁₃	D₄	1	2	yes	—
29	C₄⋊Dic₁₃	—	2	1	no	—
30	C₄⋊Dic₁₃	—	2	1	no	—
31	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
32	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
33	#6101	—	2	4	no	—
34	#6101	—	2	4	no	—
35	C₂×#1055	—	2	2	no	—
36	C₂×#1055	—	2	4	no	—
37	#1989	—	13	1	no	maximal, Sylow
38	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃	—	2	2	no	—
40	C₂×C₄×C₁₃	—	2	1	no	—
41	C₂×C₄×C₁₃	—	2	1	no	—
42	C₂×C₄×C₁₃	—	2	4	no	—
43	C₂×C₄×C₁₃	—	4	2	no	—
44	C₂×C₄×C₁₃	—	4	2	no	—
45	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₁₃	#41	1	2	yes	—
51	C₂×Dic₁₃	—	2	2	no	—
52	C₂×Dic₁₃	—	2	2	no	—
53	C₂×Dic₁₃	—	4	2	no	—
54	C₂×Dic₁₃	—	4	1	no	—
55	C₂×Dic₁₃	—	4	1	no	—
56	#1055	—	4	2	no	—
57	C₂×#125	—	13	1	no	—
58	C₂×C₄⋊C₈	—	13	1	no	—
59	C₄×C₄⋊C₄	D₁₃	1	1	yes	characteristic
60	#340	—	13	2	no	—
61	#340	—	13	2	no	—
62	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₁₃	C₂²⋊Q₈	1	2	yes	—
64	C₂²×C₁₃	#135	1	2	yes	—
65	C₂²×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₁₃	#137	1	1	yes	characteristic
67	C₄×C₁₃	—	2	2	no	—
68	C₄×C₁₃	—	2	2	no	—
69	C₄×C₁₃	—	4	4	no	—
70	C₄×C₁₃	C₄×Q₈	1	2	yes	—
71	C₄×C₁₃	#124	1	1	yes	characteristic
72	C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
73	Dic₁₃	—	4	2	no	—
74	Dic₁₃	—	4	2	no	—
75	C₂²×C₈	—	13	2	no	—
76	#125	—	13	4	no	—
77	#125	—	26	2	no	—
78	C₂×C₄⋊C₄	—	13	1	no	—
79	C₂×C₄⋊C₄	—	13	1	no	—
80	C₂×C₄⋊C₄	—	26	1	no	—
81	C₂×C₄⋊C₄	—	26	1	no	—
82	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
83	C₂×C₄²	—	2	1	no	—
84	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
85	C₄⋊C₈	—	26	2	no	—
86	#95	—	2	1	no	—
87	C₂×C₁₃	#391	1	1	yes	characteristic
88	C₂×C₁₃	#416	1	1	yes	characteristic
89	C₂×C₁₃	#417	1	1	yes	characteristic
90	C₂×C₁₃	#478	1	2	yes	—
91	C₂×C₁₃	#479	1	2	yes	—
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	—	26	1	no	—
94	C₂²×C₄	—	26	1	no	—
95	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
98	C₂×C₈	—	26	2	no	—
99	C₂×C₈	—	26	4	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	13	2	no	—
102	C₄⋊C₄	—	13	2	no	—
103	C₄⋊C₄	—	26	4	no	—
104	C₄⋊C₄	—	26	4	no	—
105	C₄⋊C₄	—	26	1	no	—
106	C₄⋊C₄	—	26	1	no	—
107	C₄⋊C₄	Dic₂₆	1	4	yes	—
108	C₄²	—	4	2	no	—
109	C₄²	C₄×D₁₃	1	2	yes	—
110	C₄²	D₅₂	1	2	yes	—
111	C₁₃	#1989	1	1	yes	characteristic, complemented, Sylow
112	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	26	2	no	—
120	C₂×C₄	—	26	2	no	—
121	C₂×C₄	—	52	1	no	—
122	C₂×C₄	—	52	1	no	—
123	C₂×C₄	—	52	2	no	—
124	C₂×C₄	#6103	1	2	yes	—
125	C₂×C₄	#6103	1	2	yes	—
126	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
127	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
128	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
129	C₂×C₄	#6127	1	2	yes	—
130	C₈	—	52	2	no	—
131	C₂²	#88178	1	2	yes	—
132	C₂²	C₂×#6103	1	1	yes	characteristic
133	C₂²	#88186	1	1	yes	characteristic
134	C₂²	#88196	1	1	yes	characteristic
135	C₂²	#88204	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	4	4	no	—
139	C₄	—	52	2	no	—
140	C₄	—	52	2	no	—
141	C₄	C₄×Dic₂₆	1	2	yes	—
142	C₄	#88123	1	1	yes	characteristic
143	C₄	#88129	1	1	yes	characteristic
144	C₂	#11718590	1	1	yes	characteristic
145	C₂	#11718627	1	1	yes	characteristic
146	C₂	#11718663	1	1	yes	characteristic
147	C₂	#11718670	1	2	yes	—
148	C₂	#11718683	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88124

C₂

yes

maximal, characteristic

C₄×C₁₃×C₄⋊C₄

C₂

yes

maximal, characteristic

C₂×C₄⋊#1055

C₂

yes

maximal, characteristic

#11718121

C₂

yes

maximal

#11718121

C₂

yes

maximal

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×#6101

—

C₂×#6101

—

C₂²×#1055

C₂²

yes

—

#88124

—

#88124

C₄

yes

complemented

C₁₃×#95

—

C₄⋊#1055

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

Q₈

yes

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×C₄

yes

—

C₄²×C₁₃

D₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

C₂×C₄

yes

—

C₄⋊Dic₁₃

C₂×C₄

yes

—

—

—

—

—

—

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×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

characteristic

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

#1055

—

C₂×#125

—

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

D₁₃

yes

characteristic

#340

—

#340

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃

C₂²⋊Q₈

yes

—

C₂²×C₁₃

#135

yes

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×Q₈

yes

—

C₄×C₁₃

#124

yes

characteristic

C₄×C₁₃

C₄⋊Q₈

yes

characteristic

Dic₁₃

—

Dic₁₃

—

C₂²×C₈

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

#95

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂×C₁₃

#479

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

Dic₂₆

yes

—

108

C₄²

—

109

C₄²

C₄×D₁₃

yes

—

110

C₄²

D₅₂

yes

—

111

C₁₃

#1989

yes

characteristic, complemented, Sylow

112

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#6103

yes

—

125

C₂×C₄

#6103

yes

—

126

C₂×C₄

C₂×Dic₂₆

yes

—

127

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

128

C₂×C₄

C₂×D₅₂

yes

characteristic

129

C₂×C₄

#6127

yes

—

130

C₈

—

131

C₂²

#88178

yes

—

132

C₂²

C₂×#6103

yes

characteristic

133

C₂²

#88186

yes

characteristic

134

C₂²

#88196

yes

characteristic

135

C₂²

#88204

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

C₄×Dic₂₆

yes

—

142

C₄

#88123

yes

characteristic

143

C₄

#88129

yes

characteristic

144

C₂

#11718590

yes

characteristic

145

C₂

#11718627

yes

characteristic

146

C₂

#11718663

yes

characteristic

147

C₂

#11718670

yes

—

148

C₂

#11718683

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₄×C₁₃×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₃
hypocenter	C₁₃	#1989
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₄×C₁₃×C₄⋊C₄

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₄

cosocle

C₂²×C₄×C₁₃

C₂³

hypercenter

C₄×C₄⋊C₄

D₁₃

hypocenter

C₁₃

#1989

LCS

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
15. #88124	C₄	semidirect
111. C₁₃	#1989	semidirect

H ≅ G/N

type

15. #88124

C₄

semidirect

111. C₁₃

#1989

semidirect

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

subgroup	index
C₂×#88124	2
C₄×C₁₃×C₄⋊C₄	2
C₂×C₄⋊#1055	2
#11718121	2
#1989	13

subgroup

index

C₂×#88124

C₄×C₁₃×C₄⋊C₄

C₂×C₄⋊#1055

#11718121

#1989

Subgroups — #13115793

Conjugacy classes of subgroups

Complements

Maximal subgroups