Group #13113239 — subgroups

Subgroups — #13113239

Order 1664.

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

normal subgroups only

This group has 216 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×#533	C₂	1	1	yes	maximal, characteristic
2	#11718586	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#1055	C₂	1	4	yes	maximal, complemented
5	C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₁₃×#125	—	2	1	no	—
8	C₁₃×C₂²⋊Q₈	—	2	1	no	—
9	C₁₃×C₂²⋊Q₈	C₄	1	2	yes	—
10	C₄×Q₈×C₁₃	—	2	1	no	—
11	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
12	C₁₃×#117	—	2	1	no	—
13	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²×#1055	—	2	1	no	—
15	C₂²⋊#1055	—	2	1	no	—
16	C₄⋊#1055	—	2	1	no	—
17	C₄⋊#1055	—	2	1	no	—
18	C₄⋊#1055	C₂²	1	2	yes	—
19	C₄×#1055	C₂²	1	2	yes	—
20	C₂×Q₈×C₁₃	—	2	1	no	—
21	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
25	C₁₃×C₄⋊C₄	—	2	1	no	—
26	C₁₃×C₄⋊C₄	—	2	1	no	—
27	C₁₃×C₄⋊C₄	—	2	1	no	—
28	C₁₃×C₄⋊C₄	—	2	2	no	—
29	C₁₃×C₄⋊C₄	—	2	2	no	—
30	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₁₃×C₂²⋊C₄	—	2	2	no	—
32	C₄²×C₁₃	—	2	2	no	—
33	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₄²×C₁₃	D₄	1	1	yes	characteristic
36	C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×#1055	—	2	2	no	—
38	C₂×#1055	—	2	2	no	—
39	C₂×#1055	—	4	2	no	—
40	#1481	—	13	1	no	maximal, Sylow
41	C₂³×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
42	Q₈×C₁₃	—	2	4	no	—
43	Q₈×C₁₃	—	4	2	no	—
44	C₂×C₄×C₁₃	—	2	2	no	—
45	C₂×C₄×C₁₃	—	2	2	no	—
46	C₂×C₄×C₁₃	—	2	2	no	—
47	C₂×C₄×C₁₃	—	4	1	no	—
48	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃	SD₁₆	1	2	yes	—
52	C₂×C₄×C₁₃	Q₁₆	1	2	yes	—
53	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
56	#1055	—	4	2	no	—
57	#1055	—	4	2	no	—
58	#533	D₁₃	1	1	yes	characteristic
59	#431	—	13	1	no	—
60	C₂×C₄⋊C₈	—	13	1	no	—
61	Q₈⋊C₈	—	13	4	no	—
62	C₂²×C₁₃	—	2	1	no	—
63	C₂²×C₁₃	—	2	1	no	—
64	C₂²×C₁₃	#103	1	2	yes	—
65	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₄×C₁₃	—	2	1	no	—
67	C₄×C₁₃	—	2	1	no	—
68	C₄×C₁₃	—	2	1	no	—
69	C₄×C₁₃	—	4	2	no	—
70	C₄×C₁₃	—	4	2	no	—
71	C₄×C₁₃	#103	1	2	yes	—
72	C₄×C₁₃	#131	1	2	yes	—
73	C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
74	C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
75	C₂²×C₈	—	26	1	no	—
76	C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic
77	C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic
78	#125	—	2	1	no	—
79	C₂²⋊Q₈	—	2	1	no	—
80	C₂²⋊Q₈	Dic₁₃	1	2	yes	—
81	C₄×Q₈	—	2	1	no	—
82	C₄×Q₈	C₂×D₁₃	1	2	yes	—
83	#117	—	2	1	no	—
84	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
85	C₄⋊C₈	—	13	2	no	—
86	C₄⋊C₈	—	26	1	no	—
87	C₄⋊C₈	—	26	1	no	—
88	C₂²⋊C₈	—	26	1	no	—
89	C₄×C₈	—	13	2	no	—
90	C₂×C₁₃	—	2	2	no	—
91	C₂×C₁₃	#408	1	1	yes	characteristic
92	C₂×C₁₃	C₂×#103	1	1	yes	characteristic
93	C₂×C₁₃	#421	1	1	yes	characteristic
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	C₂×Dic₁₃	1	2	yes	—
96	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
99	C₂×C₈	—	26	2	no	—
100	C₂×C₈	—	26	2	no	—
101	C₂×C₈	—	52	2	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
111	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
112	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
113	C₄²	C₂²×D₁₃	1	1	yes	characteristic
114	C₁₃	#1481	1	1	yes	characteristic, complemented, Sylow
115	C₂³	C₂²⋊Dic₁₃	1	1	yes	characteristic
116	Q₈	—	2	4	no	—
117	Q₈	—	4	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	#6106	1	2	yes	—
123	C₂×C₄	#6107	1	2	yes	—
124	C₂×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
125	C₂×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
126	C₂×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
127	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
129	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
130	C₈	—	52	2	no	—
131	C₈	—	52	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	#88075	1	2	yes	—
135	C₂²	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	4	2	no	—
140	C₄	—	4	2	no	—
141	C₄	#88075	1	2	yes	—
142	C₄	C₂×#6106	1	1	yes	characteristic
143	C₄	C₂×#6107	1	1	yes	characteristic
144	C₄	#88202	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	C₂×#88075	1	1	yes	characteristic
147	C₂	#11718895	1	1	yes	characteristic
148	C₂	#11718897	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#533

C₂

yes

maximal, characteristic

#11718586

C₂

yes

maximal, characteristic

C₂×C₄⋊#1055

C₂

yes

maximal, characteristic

Q₈⋊#1055

C₂

yes

maximal, complemented

C₁₃×C₄⋊Q₈

C₄

yes

characteristic

C₁₃×C₄⋊Q₈

C₄

yes

characteristic

C₁₃×#125

—

C₁₃×C₂²⋊Q₈

—

C₁₃×C₂²⋊Q₈

C₄

yes

—

C₄×Q₈×C₁₃

—

C₄×Q₈×C₁₃

C₂²

yes

—

C₁₃×#117

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂²×#1055

—

C₂²⋊#1055

—

C₄⋊#1055

—

C₄⋊#1055

—

C₄⋊#1055

C₂²

yes

—

C₄×#1055

C₂²

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

D₄

yes

characteristic

C₄²×C₁₃

D₄

yes

characteristic

C₄²×C₁₃

C₂³

yes

—

—

—

—

C₂³×C₁₃

C₂²⋊C₄

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

SD₁₆

yes

—

C₂×C₄×C₁₃

Q₁₆

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

—

D₁₃

yes

characteristic

#431

—

C₂×C₄⋊C₈

—

Q₈⋊C₈

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#103

yes

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#103

yes

—

C₄×C₁₃

#131

yes

—

C₄×C₁₃

C₂×SD₁₆

yes

characteristic

C₄×C₁₃

C₂×Q₁₆

yes

characteristic

C₂²×C₈

—

C₄⋊Q₈

Dic₁₃

yes

characteristic

C₄⋊Q₈

Dic₁₃

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

Dic₁₃

yes

—

C₄×Q₈

—

C₄×Q₈

C₂×D₁₃

yes

—

#117

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₃

yes

—

C₂²×C₄

C₂×Dic₁₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂×Dic₁₃

yes

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₄²

C₂×Dic₁₃

yes

characteristic

111

C₄²

C₂²⋊D₁₃

yes

characteristic

112

C₄²

C₂²⋊D₁₃

yes

characteristic

113

C₄²

C₂²×D₁₃

yes

characteristic

114

C₁₃

#1481

yes

characteristic, complemented, Sylow

115

C₂³

C₂²⋊Dic₁₃

yes

characteristic

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#6106

yes

—

123

C₂×C₄

#6107

yes

—

124

C₂×C₄

C₂²⋊Dic₁₃

yes

characteristic

125

C₂×C₄

C₂²⋊Dic₁₃

yes

characteristic

126

C₂×C₄

C₂²⋊Dic₁₃

yes

characteristic

127

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

129

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

130

C₈

—

131

C₈

—

132

C₂²

—

133

C₂²

—

134

C₂²

#88075

yes

—

135

C₂²

C₂×C₂²⋊Dic₁₃

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#88075

yes

—

142

C₄

C₂×#6106

yes

characteristic

143

C₄

C₂×#6107

yes

characteristic

144

C₄

#88202

yes

—

145

C₂

—

146

C₂

C₂×#88075

yes

characteristic

147

C₂

#11718895

yes

characteristic

148

C₂

#11718897

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₄²	C₂²×D₁₃
Fitting	C₁₃×#533	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₃	C₂²⋊C₄
cosocle	C₄²×C₁₃	C₂³
hypercenter	#533	D₁₃
hypocenter	C₁₃	#1481
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₄²

C₂²×D₁₃

Fitting

C₁₃×#533

C₂

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₃

C₂²⋊C₄

cosocle

C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
4. Q₈⋊#1055	C₂	semidirect
114. C₁₃	#1481	semidirect

H ≅ G/N

type

4. Q₈⋊#1055

C₂

semidirect

114. C₁₃

#1481

semidirect

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

subgroup	index
C₁₃×#533	2
#11718586	2
C₂×C₄⋊#1055	2
Q₈⋊#1055	2
#1481	13

subgroup

index

C₁₃×#533

#11718586

C₂×C₄⋊#1055

Q₈⋊#1055

#1481

Subgroups — #13113239

Conjugacy classes of subgroups

Complements

Maximal subgroups