Group #13113131 — subgroups

Subgroups — #13113131

Order 1664.

190 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 105 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×#431	C₂	1	1	yes	maximal, characteristic
2	#11718586	C₂	1	1	yes	maximal, characteristic
3	C₄×#6099	C₂	1	1	yes	maximal, characteristic
4	#11718047	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718046	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718046	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×#1055	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#6099	C₄	1	2	yes	—
10	C₁₃×C₄⋊C₈	C₄	1	2	yes	—
11	C₁₃×C₂²⋊C₈	C₄	1	2	yes	—
12	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²⋊#1055	C₄	1	2	yes	—
15	C₄⋊#1055	C₄	1	2	yes	—
16	#88043	C₄	1	2	yes	—
17	C₄×#1055	C₂²	1	1	yes	characteristic
18	C₄×#1055	C₂²	1	1	yes	characteristic
19	C₄×#1055	C₂²	1	1	yes	characteristic
20	C₄×#1055	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₃	C₂³	1	1	yes	characteristic
30	#6099	—	2	4	no	—
31	C₂×#1055	C₂×C₄	1	2	yes	—
32	C₂×#1055	C₂×C₄	1	2	yes	—
33	C₂×#1055	C₂×C₄	1	2	yes	—
34	C₂×#1055	C₂×C₄	1	2	yes	—
35	#1453	—	13	1	no	maximal, Sylow
36	C₂³×C₁₃	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₁₃	—	2	2	no	—
38	C₂×C₄×C₁₃	—	2	2	no	—
39	C₂×C₄×C₁₃	—	2	2	no	—
40	C₂×C₄×C₁₃	C₄²	1	1	yes	characteristic
41	C₂×C₄×C₁₃	C₄²	1	1	yes	characteristic
42	C₂×C₄×C₁₃	C₄²	1	1	yes	characteristic
43	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₁₃	—	2	2	no	—
47	C₈×C₁₃	—	2	2	no	—
48	#1055	—	2	2	no	—
49	#1055	—	2	2	no	—
50	#1055	—	2	2	no	—
51	#1055	M₁₆	1	4	yes	—
52	#431	—	13	1	no	—
53	#431	D₁₃	1	1	yes	characteristic
54	C₄×M₁₆	—	13	1	no	—
55	#322	—	13	1	no	—
56	#322	—	13	1	no	—
57	#322	—	13	1	no	—
58	C₈²	—	13	1	no	—
59	C₂²×C₁₃	—	2	1	no	—
60	C₂²×C₁₃	—	2	1	no	—
61	C₂²×C₁₃	—	2	1	no	—
62	C₂²×C₁₃	C₂×C₄²	1	1	yes	characteristic
63	C₄×C₁₃	—	2	1	no	—
64	C₄×C₁₃	—	2	1	no	—
65	C₄×C₁₃	—	2	1	no	—
66	C₄×C₁₃	C₂×M₁₆	1	2	yes	—
67	C₄×C₁₃	#131	1	2	yes	—
68	C₄×C₁₃	#131	1	2	yes	—
69	C₂×M₁₆	—	13	2	no	—
70	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
71	C₄⋊C₈	—	13	2	no	—
72	C₄⋊C₈	Dic₁₃	1	2	yes	—
73	C₂²⋊C₈	—	13	2	no	—
74	C₂²⋊C₈	Dic₁₃	1	2	yes	—
75	#97	—	13	2	no	—
76	C₄×C₈	—	13	1	no	—
77	C₄×C₈	—	13	1	no	—
78	C₄×C₈	—	13	1	no	—
79	C₄×C₈	—	13	1	no	—
80	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
81	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
82	C₂×C₁₃	—	4	1	no	—
83	C₂×C₁₃	C₄×M₁₆	1	1	yes	characteristic
84	C₂×C₁₃	#405	1	1	yes	characteristic
85	C₂×C₁₃	#405	1	1	yes	characteristic
86	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
88	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
89	M₁₆	—	26	4	no	—
90	C₂×C₈	—	13	2	no	—
91	C₂×C₈	—	13	2	no	—
92	C₂×C₈	—	13	2	no	—
93	C₂×C₈	—	13	2	no	—
94	C₂×C₈	C₂×Dic₁₃	1	2	yes	—
95	C₂×C₈	C₂×Dic₁₃	1	2	yes	—
96	C₄²	C₄×D₁₃	1	1	yes	characteristic
97	C₄²	C₄×D₁₃	1	1	yes	characteristic
98	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
99	C₄²	C₂²×D₁₃	1	1	yes	characteristic
100	C₁₃	#1453	1	1	yes	characteristic, complemented, Sylow
101	C₂³	C₄×Dic₁₃	1	1	yes	characteristic
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₄×Dic₁₃	1	1	yes	characteristic
106	C₂×C₄	C₄×Dic₁₃	1	1	yes	characteristic
107	C₂×C₄	C₄×Dic₁₃	1	1	yes	characteristic
108	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
109	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
110	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	2	2	no	—
113	C₈	—	13	4	no	—
114	C₈	—	26	2	no	—
115	C₈	—	26	2	no	—
116	C₈	—	26	2	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂×C₄×Dic₁₃	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	#88155	1	2	yes	—
125	C₄	M₁₆×D₁₃	1	2	yes	—
126	C₄	#88202	1	2	yes	—
127	C₂	—	4	1	no	—
128	C₂	#11718658	1	1	yes	characteristic
129	C₂	#11718762	1	1	yes	characteristic
130	C₂	M₁₆×Dic₁₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#431

C₂

yes

maximal, characteristic

#11718586

C₂

yes

maximal, characteristic

C₄×#6099

C₂

yes

maximal, characteristic

#11718047

C₂

yes

maximal, characteristic, complemented

#11718046

C₂

yes

maximal, characteristic, complemented

#11718046

C₂

yes

maximal, characteristic, complemented

C₈×#1055

C₂

yes

maximal, characteristic, complemented

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×#6099

C₄

yes

—

C₁₃×C₄⋊C₈

C₄

yes

—

C₁₃×C₂²⋊C₈

C₄

yes

—

C₄×C₈×C₁₃

C₂²

yes

characteristic

C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂²⋊#1055

C₄

yes

—

C₄⋊#1055

C₄

yes

—

#88043

C₄

yes

—

C₄×#1055

C₂²

yes

characteristic

C₄×#1055

C₂²

yes

characteristic

C₄×#1055

C₂²

yes

characteristic

C₄×#1055

C₂²

yes

characteristic

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₈×C₁₃

C₂×C₄

yes

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

C₂³

yes

characteristic

#6099

—

C₂×#1055

C₂×C₄

yes

—

C₂×#1055

C₂×C₄

yes

—

C₂×#1055

C₂×C₄

yes

—

C₂×#1055

C₂×C₄

yes

—

#1453

—

maximal, Sylow

C₂³×C₁₃

C₄²

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄²

yes

characteristic

C₂×C₄×C₁₃

C₄²

yes

characteristic

C₂×C₄×C₁₃

C₄²

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₈×C₁₃

—

C₈×C₁₃

—

—

—

—

M₁₆

yes

—

#431

—

#431

D₁₃

yes

characteristic

C₄×M₁₆

—

#322

—

#322

—

#322

—

C₈²

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₄²

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂×M₁₆

yes

—

C₄×C₁₃

#131

yes

—

C₄×C₁₃

#131

yes

—

C₂×M₁₆

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

Dic₁₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

Dic₁₃

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₁₃

yes

characteristic

C₄×C₈

C₂×D₁₃

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

C₄×M₁₆

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂²×C₄

C₄×D₁₃

yes

characteristic

C₂²×C₄

C₄×D₁₃

yes

characteristic

C₂²×C₄

C₂×Dic₁₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₃

yes

—

C₂×C₈

C₂×Dic₁₃

yes

—

C₄²

C₄×D₁₃

yes

characteristic

C₄²

C₄×D₁₃

yes

characteristic

C₄²

C₂×Dic₁₃

yes

characteristic

C₄²

C₂²×D₁₃

yes

characteristic

100

C₁₃

#1453

yes

characteristic, complemented, Sylow

101

C₂³

C₄×Dic₁₃

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₄×Dic₁₃

yes

characteristic

106

C₂×C₄

C₄×Dic₁₃

yes

characteristic

107

C₂×C₄

C₄×Dic₁₃

yes

characteristic

108

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

109

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

110

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×C₄×Dic₁₃

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#88155

yes

—

125

C₄

M₁₆×D₁₃

yes

—

126

C₄

#88202

yes

—

127

C₂

—

128

C₂

#11718658

yes

characteristic

129

C₂

#11718762

yes

characteristic

130

C₂

M₁₆×Dic₁₃

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₃
commutator, socle	C₂²×C₁₃	C₂×C₄²
Fitting	C₁₃×#431	C₂
prime radical	C₂³×C₁₃	C₄²
cosocle	C₄²×C₁₃	C₂³
hypercenter	#431	D₁₃
hypocenter	C₁₃	#1453
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ #431

subgroup

G/H

center, Frattini

C₄²

C₂²×D₁₃

commutator, socle

C₂²×C₁₃

C₂×C₄²

Fitting

C₁₃×#431

C₂

prime radical

C₂³×C₁₃

C₄²

cosocle

C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₄² ◁ #431

N	H ≅ G/N	type
4. #11718047	C₂	semidirect
5. #11718046	C₂	semidirect
6. #11718046	C₂	semidirect
7. C₈×#1055	C₂	semidirect
100. C₁₃	#1453	semidirect

H ≅ G/N

type

4. #11718047

C₂

semidirect

5. #11718046

C₂

semidirect

6. #11718046

C₂

semidirect

7. C₈×#1055

C₂

semidirect

100. C₁₃

#1453

semidirect

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

subgroup	index
C₁₃×#431	2
#11718586	2
C₄×#6099	2
#11718047	2
#11718046	2
C₈×#1055	2
#1453	13

subgroup

index

C₁₃×#431

Subgroups — #13113131

Conjugacy classes of subgroups

Complements

Maximal subgroups