Group #13118022 — subgroups

Subgroups — #13118022

Order 1664.

408 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 213 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×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
2	#11718646	C₂	1	4	yes	maximal
3	#11718642	C₂	1	4	yes	maximal
4	#11718642	C₂	1	4	yes	maximal
5	#11718597	C₂	1	2	yes	maximal
6	C₁₃×#125	C₄	1	8	yes	complemented
7	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
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	4	yes	—
13	C₂×C₄×Dic₁₃	C₂²	1	4	yes	—
14	#88071	C₂²	1	4	yes	—
15	#88071	C₂²	1	4	yes	—
16	#88071	C₂²	1	4	yes	—
17	#88071	C₂²	1	4	yes	—
18	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
20	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
21	C₂²×Dic₁₃	C₂³	1	4	yes	—
22	C₂²×Dic₁₃	C₂³	1	4	yes	—
23	C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
24	C₁₃×C₄⋊C₄	C₂×C₄	1	16	yes	—
25	C₄²×C₁₃	C₂×C₄	1	4	yes	—
26	C₄⋊Dic₁₃	—	4	2	no	—
27	C₄⋊Dic₁₃	—	4	4	no	—
28	C₄⋊Dic₁₃	—	4	2	no	—
29	C₄×Dic₁₃	—	4	4	no	—
30	#2382	—	13	1	no	maximal, Sylow
31	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
32	C₂×C₄×C₁₃	—	2	4	no	—
33	C₂×C₄×C₁₃	—	2	2	no	—
34	C₂×C₄×C₁₃	—	2	1	no	—
35	C₂×C₄×C₁₃	—	2	1	no	—
36	C₂×C₄×C₁₃	—	2	4	no	—
37	C₂×C₄×C₁₃	—	2	2	no	—
38	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
40	C₂×C₄×C₁₃	C₂²×C₄	1	8	yes	—
41	C₂×Dic₁₃	—	2	4	no	—
42	C₂×Dic₁₃	—	2	4	no	—
43	C₂×Dic₁₃	—	2	4	no	—
44	C₂×Dic₁₃	—	2	4	no	—
45	C₂×Dic₁₃	—	2	4	no	—
46	C₂×Dic₁₃	—	2	4	no	—
47	C₂×#125	D₁₃	1	1	yes	characteristic
48	#389	—	13	4	no	—
49	#387	—	13	4	no	—
50	#387	—	13	4	no	—
51	#382	—	13	2	no	—
52	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
53	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
54	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
55	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
56	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
57	C₄×C₁₃	—	2	2	no	—
58	C₄×C₁₃	—	2	8	no	—
59	C₄×C₁₃	—	2	4	no	—
60	Dic₁₃	—	4	4	no	—
61	Dic₁₃	—	4	4	no	—
62	#125	Dic₁₃	1	8	yes	complemented
63	C₂×C₄⋊C₄	—	13	2	no	—
64	C₂×C₄⋊C₄	—	13	2	no	—
65	C₂×C₄⋊C₄	—	13	4	no	—
66	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
67	C₂×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
68	C₂×C₄²	—	13	4	no	—
69	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
70	#95	—	13	4	no	—
71	#95	—	13	4	no	—
72	#95	—	13	4	no	—
73	#95	—	13	4	no	—
74	C₂×C₁₃	#519	1	1	yes	characteristic
75	C₂×C₁₃	#520	1	2	yes	—
76	C₂×C₁₃	#562	1	1	yes	characteristic
77	C₂×C₁₃	#563	1	2	yes	—
78	C₂×C₁₃	#564	1	1	yes	characteristic
79	C₂²×C₄	—	13	4	no	—
80	C₂²×C₄	—	13	4	no	—
81	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
82	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
83	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
84	C₄⋊C₄	—	52	4	no	—
85	C₄⋊C₄	—	52	2	no	—
86	C₄⋊C₄	—	52	2	no	—
87	C₄⋊C₄	C₂×Dic₁₃	1	8	yes	—
88	C₄⋊C₄	C₂×Dic₁₃	1	16	yes	—
89	C₄²	—	52	4	no	—
90	C₄²	C₂×Dic₁₃	1	4	yes	—
91	C₁₃	#2382	1	1	yes	characteristic, complemented, Sylow
92	C₂³	C₂³×D₁₃	1	1	yes	characteristic
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	26	4	no	—
100	C₂×C₄	—	26	4	no	—
101	C₂×C₄	—	26	4	no	—
102	C₂×C₄	—	26	4	no	—
103	C₂×C₄	—	26	4	no	—
104	C₂×C₄	—	26	4	no	—
105	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
106	C₂×C₄	C₂²×Dic₁₃	1	4	yes	—
107	C₂×C₄	C₂²×Dic₁₃	1	8	yes	—
108	C₂²	#88254	1	1	yes	characteristic
109	C₂²	#88254	1	1	yes	characteristic
110	C₂²	#88256	1	2	yes	—
111	C₂²	#88257	1	2	yes	—
112	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	2	8	no	—
115	C₄	—	2	4	no	—
116	C₄	—	52	4	no	—
117	C₄	—	52	4	no	—
118	C₂	C₁₃⋊#564	1	1	yes	characteristic
119	C₂	#11719404	1	2	yes	—
120	C₂	#11719405	1	1	yes	characteristic
121	C₂	#11719517	1	1	yes	characteristic
122	C₂	#11719532	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#125

C₂

yes

maximal, characteristic

#11718646

C₂

yes

maximal

#11718642

C₂

yes

maximal

#11718642

C₂

yes

maximal

#11718597

C₂

yes

maximal

C₁₃×#125

C₄

yes

complemented

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

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

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

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

C₂×C₄

yes

—

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₁₃

C₂×C₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

#2382

—

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

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×#125

D₁₃

yes

—

—

—

—

C₂²×C₁₃

C₂³×C₄

yes

characteristic

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₂²×C₁₃

2₋¹⁺⁴

yes

characteristic

C₂²×C₁₃

2₋¹⁺⁴

yes

characteristic

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

#125

Dic₁₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

—

—

—

—

C₂×C₁₃

#519

yes

characteristic

C₂×C₁₃

#520

yes

—

C₂×C₁₃

#562

yes

characteristic

C₂×C₁₃

#563

yes

—

C₂×C₁₃

#564

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₁₃

yes

—

C₄⋊C₄

C₂×Dic₁₃

yes

—

C₄²

—

C₄²

C₂×Dic₁₃

yes

—

C₁₃

#2382

yes

characteristic, complemented, Sylow

C₂³

C₂³×D₁₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²×Dic₁₃

yes

—

106

C₂×C₄

C₂²×Dic₁₃

yes

—

107

C₂×C₄

C₂²×Dic₁₃

yes

—

108

C₂²

#88254

yes

characteristic

109

C₂²

#88254

yes

characteristic

110

C₂²

#88256

yes

—

111

C₂²

#88257

yes

—

112

C₂²

C₂³×Dic₁₃

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₂

C₁₃⋊#564

yes

characteristic

119

C₂

#11719404

yes

—

120

C₂

#11719405

yes

characteristic

121

C₂

#11719517

yes

characteristic

122

C₂

#11719532

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₄
Fitting	C₂×C₁₃×#125	C₂
socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
hypercenter	C₂×#125	D₁₃
hypocenter	C₁₃	#2382
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator

C₂²×C₁₃

C₂³×C₄

Fitting

C₂×C₁₃×#125

C₂

socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
6. C₁₃×#125	C₄	semidirect
62. #125	Dic₁₃	semidirect
91. C₁₃	#2382	semidirect

H ≅ G/N

type

6. C₁₃×#125

C₄

semidirect

62. #125

Dic₁₃

semidirect

91. C₁₃

#2382

semidirect

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

subgroup	index
C₂×C₁₃×#125	2
#11718646	2
#11718642	2
#11718597	2
#2382	13

subgroup

index

C₂×C₁₃×#125

Subgroups — #13118022

Conjugacy classes of subgroups

Complements

Maximal subgroups