Group #13123108 — subgroups

Subgroups — #13123108

Order 1664.

356 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 133 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×#382	C₂	1	1	yes	maximal, characteristic
2	#11718647	C₂	1	2	yes	maximal
3	#11718640	C₂	1	2	yes	maximal
4	#11718597	C₂	1	2	yes	maximal
5	#11718321	C₂	1	4	yes	maximal
6	#11718317	C₂	1	4	yes	maximal
7	C₂²×Dic₂₆	C₂²	1	2	yes	—
8	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
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	C₁₃×#95	C₂²	1	4	yes	—
16	#88071	C₂²	1	4	yes	—
17	#88071	C₂²	1	4	yes	—
18	#88071	C₂²	1	2	yes	—
19	#88071	C₂²	1	2	yes	—
20	#88071	C₂²	1	4	yes	—
21	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
22	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
23	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
25	C₂²×Dic₁₃	C₂³	1	4	yes	—
26	C₂²×Dic₁₃	C₂³	1	4	yes	—
27	C₂×Dic₂₆	—	4	4	no	—
28	C₂×Dic₂₆	—	4	2	no	—
29	C₁₃×C₄⋊C₄	—	4	1	no	—
30	C₁₃×C₄⋊C₄	—	4	1	no	—
31	C₄²×C₁₃	Q₈	1	4	yes	complemented
32	C₄⋊Dic₁₃	—	4	2	no	—
33	C₄⋊Dic₁₃	—	4	2	no	—
34	C₄⋊Dic₁₃	—	4	2	no	—
35	#6101	—	4	4	no	—
36	#2844	—	13	1	no	maximal, Sylow
37	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
38	C₂×C₄×C₁₃	—	2	2	no	—
39	C₂×C₄×C₁₃	—	2	1	no	—
40	C₂×C₄×C₁₃	—	2	1	no	—
41	C₂×C₄×C₁₃	—	2	2	no	—
42	C₂×C₄×C₁₃	—	2	4	no	—
43	C₂×C₄×C₁₃	—	2	2	no	—
44	C₂×C₄×C₁₃	—	2	2	no	—
45	C₂×C₄×C₁₃	—	2	4	no	—
46	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
48	C₂×Dic₁₃	—	2	4	no	—
49	C₂×Dic₁₃	—	2	4	no	—
50	C₂×Dic₁₃	—	2	4	no	—
51	C₂×Dic₁₃	—	2	4	no	—
52	C₂×Dic₁₃	—	2	4	no	—
53	C₂×Dic₁₃	—	2	4	no	—
54	Dic₂₆	—	4	8	no	—
55	#400	—	13	2	no	—
56	#400	—	13	4	no	—
57	#395	—	13	2	no	—
58	#395	—	13	4	no	—
59	#382	—	13	2	no	—
60	#382	D₁₃	1	1	yes	characteristic
61	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
62	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
63	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
64	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
65	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
66	C₄×C₁₃	—	2	2	no	—
67	C₄×C₁₃	—	2	4	no	—
68	C₄×C₁₃	—	4	2	no	—
69	C₄×C₁₃	—	4	2	no	—
70	Dic₁₃	—	4	4	no	—
71	Dic₁₃	—	4	4	no	—
72	C₂²×Q₈	—	13	2	no	—
73	C₂×C₄⋊C₄	—	13	4	no	—
74	C₂×C₄⋊C₄	—	13	2	no	—
75	C₂×C₄⋊C₄	—	13	2	no	—
76	C₂×C₄⋊C₄	—	13	2	no	—
77	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
78	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
79	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
80	#95	—	13	4	no	—
81	#95	—	13	4	no	—
82	#95	—	13	2	no	—
83	#95	—	13	4	no	—
84	#95	—	13	2	no	—
85	#95	C₂×D₁₃	1	4	yes	—
86	C₂×C₁₃	#544	1	1	yes	characteristic
87	C₂×C₁₃	#544	1	2	yes	—
88	C₂×C₁₃	H(3,𝔽₄)	1	1	yes	characteristic
89	C₂×C₁₃	#563	1	1	yes	characteristic
90	C₂×C₁₃	#563	1	2	yes	—
91	C₂×Q₈	—	52	2	no	—
92	C₂×Q₈	—	52	4	no	—
93	C₂²×C₄	—	13	4	no	—
94	C₂²×C₄	—	13	4	no	—
95	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
97	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
99	C₄⋊C₄	—	4	1	no	—
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	52	4	no	—
102	C₄⋊C₄	—	52	2	no	—
103	C₄⋊C₄	—	52	2	no	—
104	C₄⋊C₄	—	52	2	no	—
105	C₄²	Dic₂₆	1	4	yes	complemented
106	C₁₃	#2844	1	1	yes	characteristic, complemented, Sylow
107	C₂³	C₂³×D₁₃	1	1	yes	characteristic
108	Q₈	—	52	8	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	26	4	no	—
118	C₂×C₄	—	26	4	no	—
119	C₂×C₄	—	26	4	no	—
120	C₂×C₄	—	26	4	no	—
121	C₂×C₄	—	26	4	no	—
122	C₂×C₄	—	26	4	no	—
123	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
124	C₂×C₄	C₂×Dic₂₆	1	4	yes	—
125	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
126	C₂²	#88251	1	1	yes	characteristic
127	C₂²	#88254	1	1	yes	characteristic
128	C₂²	#88256	1	2	yes	—
129	C₂²	#88257	1	2	yes	—
130	C₄	—	2	2	no	—
131	C₄	—	2	4	no	—
132	C₄	—	4	2	no	—
133	C₄	—	4	2	no	—
134	C₄	—	52	4	no	—
135	C₄	—	52	4	no	—
136	C₂	#11719205	1	1	yes	characteristic
137	C₂	#11719224	1	2	yes	—
138	C₂	#11719377	1	1	yes	characteristic
139	C₂	#11719388	1	1	yes	characteristic
140	C₂	#11719404	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#382

C₂

yes

maximal, characteristic

#11718647

C₂

yes

maximal

#11718640

C₂

yes

maximal

#11718597

C₂

yes

maximal

#11718321

C₂

yes

maximal

#11718317

C₂

yes

maximal

C₂²×Dic₂₆

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

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×#6101

C₂²

yes

—

C₁₃×#95

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

Q₈

yes

complemented

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

#2844

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

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×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

—

—

—

—

—

D₁₃

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

2₋¹⁺⁴

yes

characteristic

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

C₂×C₄²

C₂×D₁₃

yes

—

—

—

—

—

C₂×D₁₃

yes

—

C₂×C₁₃

#544

yes

characteristic

C₂×C₁₃

#544

yes

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₃

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄²

Dic₂₆

yes

complemented

106

C₁₃

#2844

yes

characteristic, complemented, Sylow

107

C₂³

C₂³×D₁₃

yes

characteristic

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

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

C₂×Dic₂₆

yes

—

124

C₂×C₄

C₂×Dic₂₆

yes

—

125

C₂²

C₂²×Dic₂₆

yes

characteristic

126

C₂²

#88251

yes

characteristic

127

C₂²

#88254

yes

characteristic

128

C₂²

#88256

yes

—

129

C₂²

#88257

yes

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₂

#11719205

yes

characteristic

137

C₂

#11719224

yes

—

138

C₂

#11719377

yes

characteristic

139

C₂

#11719388

yes

characteristic

140

C₂

#11719404

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#382	C₂
hypercenter	#382	D₁₃
hypocenter	C₁₃	#2844
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#382

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
31. C₄²×C₁₃	Q₈	semidirect
105. C₄²	Dic₂₆	semidirect
106. C₁₃	#2844	semidirect

H ≅ G/N

type

31. C₄²×C₁₃

Q₈

semidirect

105. C₄²

Dic₂₆

semidirect

106. C₁₃

#2844

semidirect

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

subgroup	index
C₁₃×#382	2
#11718647	2
#11718640	2
#11718597	2
#11718321	2
#11718317	2
#2844	13

subgroup

index

C₁₃×#382

Subgroups — #13123108

Conjugacy classes of subgroups

Complements

Maximal subgroups