Group #13122824 — subgroups

Subgroups — #13122824

Order 1664.

368 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 141 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×#391	C₂	1	1	yes	maximal, characteristic
2	#11718891	C₂	1	2	yes	maximal
3	#11718647	C₂	1	2	yes	maximal
4	#11718589	C₂	1	2	yes	maximal
5	#11718322	C₂	1	4	yes	maximal
6	#11718316	C₂	1	4	yes	maximal
7	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
14	C₂×#6101	C₂²	1	4	yes	—
15	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
16	C₁₃×#95	C₂²	1	4	yes	—
17	#88071	C₂²	1	2	yes	—
18	#88071	C₂²	1	4	yes	—
19	#88071	C₂²	1	4	yes	—
20	#88071	C₂²	1	4	yes	—
21	C₂×Q₈×C₁₃	—	4	1	no	—
22	C₂×Q₈×C₁₃	—	4	1	no	—
23	C₂×Q₈×C₁₃	Q₈	1	4	yes	—
24	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
25	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
27	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
28	C₂²×Dic₁₃	C₂³	1	4	yes	—
29	C₂²×Dic₁₃	C₂³	1	4	yes	—
30	C₁₃×C₄⋊C₄	—	4	1	no	—
31	C₄²×C₁₃	—	4	1	no	—
32	C₄⋊Dic₁₃	—	2	4	no	—
33	C₄⋊Dic₁₃	—	4	2	no	—
34	C₄⋊Dic₁₃	—	4	2	no	—
35	C₄⋊Dic₁₃	—	4	2	no	—
36	#6101	—	2	8	no	—
37	C₄×Dic₁₃	—	4	2	no	—
38	#2809	—	13	1	no	maximal, Sylow
39	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
40	Q₈×C₁₃	—	4	4	no	—
41	C₂×C₄×C₁₃	—	2	2	no	—
42	C₂×C₄×C₁₃	—	2	2	no	—
43	C₂×C₄×C₁₃	—	2	2	no	—
44	C₂×C₄×C₁₃	—	2	1	no	—
45	C₂×C₄×C₁₃	—	2	1	no	—
46	C₂×C₄×C₁₃	—	2	2	no	—
47	C₂×C₄×C₁₃	—	2	2	no	—
48	C₂×C₄×C₁₃	—	2	4	no	—
49	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
51	C₂×C₄×C₁₃	#41	1	4	yes	—
52	C₂×Dic₁₃	—	2	4	no	—
53	C₂×Dic₁₃	—	2	4	no	—
54	C₂×Dic₁₃	—	2	4	no	—
55	C₂×Dic₁₃	—	2	4	no	—
56	C₂×Dic₁₃	—	2	4	no	—
57	C₂×Dic₁₃	—	2	4	no	—
58	#400	—	13	2	no	—
59	#398	—	13	4	no	—
60	#391	—	13	2	no	—
61	#391	D₁₃	1	1	yes	characteristic
62	#389	—	13	2	no	—
63	#387	—	13	4	no	—
64	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
65	C₂²×C₁₃	C₂×#41	1	2	yes	—
66	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
67	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
68	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
69	C₄×C₁₃	—	2	4	no	—
70	C₄×C₁₃	—	2	4	no	—
71	C₄×C₁₃	—	2	2	no	—
72	C₄×C₁₃	—	4	2	no	—
73	Dic₁₃	—	4	4	no	—
74	Dic₁₃	—	4	4	no	—
75	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
76	C₂×C₄⋊C₄	—	13	4	no	—
77	C₂×C₄⋊C₄	—	13	2	no	—
78	C₂×C₄⋊C₄	—	13	2	no	—
79	C₂×C₄⋊C₄	—	13	2	no	—
80	C₂×C₄⋊C₄	—	13	2	no	—
81	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
82	C₂×C₄²	—	13	2	no	—
83	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
84	#95	—	13	2	no	—
85	#95	—	13	4	no	—
86	#95	—	13	4	no	—
87	#95	—	13	4	no	—
88	#95	C₂×D₁₃	1	4	yes	—
89	C₂×C₁₃	#539	1	2	yes	—
90	C₂×C₁₃	#543	1	1	yes	characteristic
91	C₂×C₁₃	#555	1	1	yes	characteristic
92	C₂×C₁₃	#557	1	2	yes	—
93	C₂×C₁₃	#563	1	1	yes	characteristic
94	C₂×Q₈	—	4	1	no	—
95	C₂×Q₈	—	4	1	no	—
96	C₂×Q₈	Dic₂₆	1	4	yes	—
97	C₂²×C₄	—	13	4	no	—
98	C₂²×C₄	—	13	4	no	—
99	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
100	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
101	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
102	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
103	C₄⋊C₄	—	4	1	no	—
104	C₄⋊C₄	—	26	8	no	—
105	C₄⋊C₄	—	26	4	no	—
106	C₄⋊C₄	—	52	2	no	—
107	C₄⋊C₄	—	52	2	no	—
108	C₄⋊C₄	—	52	2	no	—
109	C₄²	—	4	1	no	—
110	C₄²	—	52	2	no	—
111	C₁₃	#2809	1	1	yes	characteristic, complemented, Sylow
112	C₂³	C₂³×D₁₃	1	1	yes	characteristic
113	Q₈	—	4	4	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	26	4	no	—
123	C₂×C₄	—	26	4	no	—
124	C₂×C₄	—	26	4	no	—
125	C₂×C₄	—	26	4	no	—
126	C₂×C₄	—	26	4	no	—
127	C₂×C₄	—	26	4	no	—
128	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
129	C₂×C₄	C₂×Dic₂₆	1	4	yes	—
130	C₂×C₄	#6131	1	4	yes	—
131	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
132	C₂²	#88251	1	1	yes	characteristic
133	C₂²	#88251	1	1	yes	characteristic
134	C₂²	C₂×#6131	1	2	yes	—
135	C₂²	#88257	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	4	no	—
138	C₄	—	2	4	no	—
139	C₄	—	4	2	no	—
140	C₄	—	52	4	no	—
141	C₄	—	52	4	no	—
142	C₂	#11719188	1	1	yes	characteristic
143	C₂	#11719275	1	2	yes	—
144	C₂	#11719350	1	2	yes	—
145	C₂	#11719377	1	1	yes	characteristic
146	C₂	#11719439	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#391

C₂

yes

maximal, characteristic

#11718891

C₂

yes

maximal

#11718647

C₂

yes

maximal

#11718589

C₂

yes

maximal

#11718322

C₂

yes

maximal

#11718316

C₂

yes

maximal

C₂²×Q₈×C₁₃

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₁₃×#95

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

Q₈

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

#2809

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

—

—

—

D₁₃

yes

characteristic

#389

—

#387

—

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

C₂×D₁₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

—

—

—

—

C₂×D₁₃

yes

—

C₂×C₁₃

#539

yes

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂×C₁₃

#563

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₂₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

100

C₂²×C₄

C₂²×D₁₃

yes

characteristic

101

C₂²×C₄

C₂²×D₁₃

yes

—

102

C₂²×C₄

C₂²×D₁₃

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₁₃

#2809

yes

characteristic, complemented, Sylow

112

C₂³

C₂³×D₁₃

yes

characteristic

113

Q₈

—

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

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×Dic₂₆

yes

—

129

C₂×C₄

C₂×Dic₂₆

yes

—

130

C₂×C₄

#6131

yes

—

131

C₂²

C₂²×Dic₂₆

yes

characteristic

132

C₂²

#88251

yes

characteristic

133

C₂²

#88251

yes

characteristic

134

C₂²

C₂×#6131

yes

—

135

C₂²

#88257

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₂

#11719188

yes

characteristic

143

C₂

#11719275

yes

—

144

C₂

#11719350

yes

—

145

C₂

#11719377

yes

characteristic

146

C₂

#11719439

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#391	C₂
hypercenter	#391	D₁₃
hypocenter	C₁₃	#2809
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #391

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#391

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #391

N	H ≅ G/N	type
111. C₁₃	#2809	semidirect

H ≅ G/N

type

111. C₁₃

#2809

semidirect

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

subgroup	index
C₁₃×#391	2
#11718891	2
#11718647	2
#11718589	2
#11718322	2
#11718316	2
#2809	13

subgroup

index

C₁₃×#391

Subgroups — #13122824

Conjugacy classes of subgroups

Complements

Maximal subgroups