Group #13123919 — subgroups

Subgroups — #13123919

Order 1664.

356 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 177 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×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#431	C₂	1	1	yes	maximal, characteristic
3	#11718586	C₂	1	1	yes	maximal, characteristic
4	#11718492	C₂	1	4	yes	maximal, complemented
5	#11718380	C₂	1	4	yes	maximal
6	#11718340	C₂	1	4	yes	maximal, complemented
7	C₂²×Dic₂₆	C₄	1	1	yes	characteristic
8	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
10	C₂×#6101	C₄	1	2	yes	—
11	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
12	C₄×Dic₂₆	C₄	1	2	yes	—
13	C₄×Dic₂₆	C₄	1	2	yes	—
14	C₄×Dic₂₆	C₂²	1	2	yes	—
15	C₄×Dic₂₆	C₂²	1	2	yes	—
16	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
17	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₄×C₈×C₁₃	C₂²	1	2	yes	—
19	C₂²⋊#1055	C₂²	1	2	yes	—
20	#88055	C₂²	1	8	yes	—
21	#88054	C₂²	1	8	yes	—
22	C₄⋊#1055	C₂²	1	2	yes	—
23	C₄×#1055	C₂²	1	2	yes	—
24	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
28	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
29	C₂×Dic₂₆	—	2	4	no	—
30	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
31	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₁₃	C₂³	1	4	yes	—
33	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
39	#6101	C₂×C₄	1	4	yes	—
40	#6101	C₂×C₄	1	4	yes	—
41	C₄×Dic₁₃	C₂×C₄	1	4	yes	—
42	C₄×Dic₁₃	C₂³	1	4	yes	—
43	C₂×#1055	C₂³	1	4	yes	—
44	#2983	—	13	1	no	maximal, Sylow
45	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃	—	2	2	no	—
47	C₂×C₄×C₁₃	—	4	1	no	—
48	C₂×C₄×C₁₃	M₁₆	1	4	yes	—
49	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₁₃	—	2	4	no	—
56	C₂×Dic₁₃	—	4	2	no	—
57	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
58	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
59	Dic₂₆	—	2	4	no	—
60	Dic₂₆	—	2	4	no	—
61	C₈×C₁₃	—	2	4	no	—
62	#1055	—	2	4	no	—
63	C₂×C₄×Q₈	—	13	1	no	—
64	#446	—	13	4	no	—
65	#446	—	13	4	no	—
66	#432	—	13	4	no	—
67	#431	—	13	1	no	—
68	#431	D₁₃	1	1	yes	characteristic
69	C₂²×C₁₃	—	2	1	no	—
70	C₂²×C₁₃	—	2	1	no	—
71	C₂²×C₁₃	C₂×M₁₆	1	2	yes	—
72	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
73	C₄×C₁₃	—	2	1	no	—
74	C₄×C₁₃	—	2	1	no	—
75	C₄×C₁₃	—	2	1	no	—
76	C₄×C₁₃	C₂×M₁₆	1	2	yes	—
77	C₄×C₁₃	C₂×M₁₆	1	2	yes	—
78	C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
79	Dic₁₃	—	2	4	no	—
80	Dic₁₃	—	2	4	no	—
81	C₂²×Q₈	—	13	1	no	—
82	C₄×Q₈	—	13	2	no	—
83	C₄×Q₈	—	13	2	no	—
84	C₄×Q₈	—	13	2	no	—
85	C₄×Q₈	—	13	2	no	—
86	C₂×C₄⋊C₄	—	13	2	no	—
87	C₂×C₄⋊C₄	—	13	1	no	—
88	C₂×C₄²	—	13	2	no	—
89	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
90	C₄⋊C₈	—	13	8	no	—
91	C₄⋊C₈	—	13	2	no	—
92	C₄⋊C₈	C₂×D₁₃	1	2	yes	—
93	C₂²⋊C₈	—	13	2	no	—
94	C₂²⋊C₈	C₂×D₁₃	1	2	yes	—
95	#97	—	13	8	no	—
96	C₄×C₈	—	13	2	no	—
97	C₄×C₈	C₂×D₁₃	1	2	yes	—
98	C₂×C₁₃	—	2	2	no	—
99	C₂×C₁₃	#519	1	1	yes	characteristic
100	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
101	C₂×C₁₃	#568	1	1	yes	characteristic
102	C₂×Q₈	—	13	2	no	—
103	C₂×Q₈	—	13	2	no	—
104	C₂×Q₈	—	26	4	no	—
105	C₂²×C₄	—	13	2	no	—
106	C₂²×C₄	—	13	2	no	—
107	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
108	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
109	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
110	C₂×C₈	—	13	4	no	—
111	C₂×C₈	C₂²×D₁₃	1	4	yes	—
112	C₄⋊C₄	—	13	4	no	—
113	C₄⋊C₄	—	13	4	no	—
114	C₄⋊C₄	—	13	2	no	—
115	C₄⋊C₄	—	13	2	no	—
116	C₄²	—	13	4	no	—
117	C₄²	—	13	4	no	—
118	C₄²	C₄×D₁₃	1	1	yes	characteristic
119	C₄²	C₄×D₁₃	1	1	yes	characteristic
120	C₄²	C₂²×D₁₃	1	1	yes	characteristic
121	C₄²	C₂²×D₁₃	1	1	yes	characteristic
122	C₁₃	#2983	1	1	yes	characteristic, complemented, Sylow
123	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
124	Q₈	—	26	4	no	—
125	Q₈	—	26	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	13	4	no	—
129	C₂×C₄	—	13	4	no	—
130	C₂×C₄	—	26	4	no	—
131	C₂×C₄	—	52	2	no	—
132	C₂×C₄	#6094	1	4	yes	—
133	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	26	4	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂×#6094	1	2	yes	—
144	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	26	4	no	—
149	C₄	—	26	4	no	—
150	C₄	C₂×#6094	1	2	yes	—
151	C₄	C₂×#6094	1	2	yes	—
152	C₄	#88257	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	#11719221	1	1	yes	characteristic
155	C₂	C₂²×#6094	1	1	yes	characteristic
156	C₂	#11719450	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₂₆

C₂

yes

maximal, characteristic

C₁₃×#431

C₂

yes

maximal, characteristic

#11718586

C₂

yes

maximal, characteristic

#11718492

C₂

yes

maximal, complemented

#11718380

C₂

yes

maximal

#11718340

C₂

yes

maximal, complemented

C₂²×Dic₂₆

C₄

yes

characteristic

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₄

yes

characteristic

C₂×#6101

C₄

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₄×Dic₂₆

C₄

yes

—

C₄×Dic₂₆

C₄

yes

—

C₄×Dic₂₆

C₂²

yes

—

C₄×Dic₂₆

C₂²

yes

—

C₁₃×C₄⋊C₈

C₂²

yes

—

C₁₃×C₂²⋊C₈

C₂²

yes

—

C₄×C₈×C₁₃

C₂²

yes

—

C₂²⋊#1055

C₂²

yes

—

#88055

C₂²

yes

—

#88054

C₂²

yes

—

C₄⋊#1055

C₂²

yes

—

C₄×#1055

C₂²

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂×C₄

yes

—

C₂²×Dic₁₃

C₂×C₄

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

C₂×C₄

yes

—

C₂×Dic₂₆

C₂×C₄

yes

—

C₂×C₈×C₁₃

C₂³

yes

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

C₂³

yes

characteristic

C₄²×C₁₃

C₂³

yes

characteristic

C₄⋊Dic₁₃

C₂×C₄

yes

—

C₄⋊Dic₁₃

C₂×C₄

yes

—

#6101

C₂×C₄

yes

—

#6101

C₂×C₄

yes

—

C₄×Dic₁₃

C₂×C₄

yes

—

C₄×Dic₁₃

C₂³

yes

—

C₂×#1055

C₂³

yes

—

#2983

—

maximal, Sylow

C₂³×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

M₁₆

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂²×C₄

yes

—

C₂×Dic₁₃

C₂²×C₄

yes

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

#1055

—

C₂×C₄×Q₈

—

—

—

—

—

D₁₃

yes

characteristic

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×M₁₆

yes

—

C₂²×C₁₃

C₂³×C₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂×M₁₆

yes

—

C₄×C₁₃

C₂×M₁₆

yes

—

C₄×C₁₃

2₋¹⁺⁴

yes

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₁₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×D₁₃

yes

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₁₃

yes

—

C₂×C₁₃

—

C₂×C₁₃

#519

yes

characteristic

100

C₂×C₁₃

C₂²×M₁₆

yes

characteristic

101

C₂×C₁₃

#568

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

C₄×D₁₃

yes

characteristic

108

C₂²×C₄

C₄×D₁₃

yes

characteristic

109

C₂²×C₄

C₂²×D₁₃

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

C₂²×D₁₃

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄²

—

117

C₄²

—

118

C₄²

C₄×D₁₃

yes

characteristic

119

C₄²

C₄×D₁₃

yes

characteristic

120

C₄²

C₂²×D₁₃

yes

characteristic

121

C₄²

C₂²×D₁₃

yes

characteristic

122

C₁₃

#2983

yes

characteristic, complemented, Sylow

123

C₂³

C₂×C₄×D₁₃

yes

characteristic

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

#6094

yes

—

133

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

134

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

135

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

136

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

137

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

138

C₂×C₄

C₂³×D₁₃

yes

characteristic

139

C₈

—

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×#6094

yes

—

144

C₂²

C₂²×C₄×D₁₃

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

C₂×#6094

yes

—

151

C₄

C₂×#6094

yes

—

152

C₄

#88257

yes

—

153

C₂

—

154

C₂

#11719221

yes

characteristic

155

C₂

C₂²×#6094

yes

characteristic

156

C₂

#11719450

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#431	C₂
prime radical	C₂³×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#431	D₁₃
hypocenter	C₁₃	#2983
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₃

commutator, socle

C₂²×C₁₃

C₂³×C₄

Fitting

C₁₃×#431

C₂

prime radical

C₂³×C₁₃

C₂²×C₄

cosocle

C₂×C₄×C₁₃

C₂⁴

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
4. #11718492	C₂	semidirect
6. #11718340	C₂	semidirect
122. C₁₃	#2983	semidirect

H ≅ G/N

type

4. #11718492

C₂

semidirect

6. #11718340

C₂

semidirect

122. C₁₃

#2983

semidirect

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

subgroup	index
C₂×C₄×Dic₂₆	2
C₁₃×#431	2
#11718586	2
#11718492	2
#11718380	2
#11718340	2
#2983	13

subgroup

index

C₂×C₄×Dic₂₆

C₁₃×#431

Subgroups — #13123919

Conjugacy classes of subgroups

Complements

Maximal subgroups