Group #13131832 — subgroups

Subgroups — #13131832

Order 1664.

1424 conjugacy classes of subgroups; only the 750 normal subgroups are shown.

normal subgroups only

This group has 1424 conjugacy classes of subgroups — too many to draw. Only its 750 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	#11719652	C₂	1	1	yes	maximal, characteristic
2	#11719629	C₂	1	10	yes	maximal, complemented
3	D₁₃⋊#131	C₂	1	10	yes	maximal, complemented
4	#11719625	C₂	1	5	yes	maximal, complemented
5	C₂×#88240	C₂	1	5	yes	maximal, complemented
6	C₁₃×2₋¹⁺⁴	C₄	1	1	yes	characteristic
7	#88256	C₄	1	10	yes	—
8	#41×D₁₃	C₂²	1	10	yes	—
9	#88254	C₄	1	5	yes	—
10	C₂×#6131	C₂²	1	5	yes	—
11	#88240	C₂²	1	10	yes	complemented
12	#88240	C₂²	1	20	yes	complemented
13	#88240	C₂²	1	10	yes	complemented
14	#88238	C₂²	1	10	yes	complemented
15	#88238	C₂²	1	30	yes	complemented
16	D₁₃⋊M₁₆	C₂²	1	30	yes	complemented
17	C₂×#6118	C₂²	1	15	yes	complemented
18	C₂×D₁₃⋊C₈	C₂²	1	15	yes	complemented
19	C₁₃×#41	C₂×C₄	1	10	yes	—
20	C₂×Q₈×C₁₃	C₂×C₄	1	5	yes	—
21	#6131	C₂×C₄	1	20	yes	—
22	#6131	C₂³	1	10	yes	—
23	Q₈×D₁₃	C₂×C₄	1	10	yes	—
24	#6129	C₂³	1	10	yes	—
25	D₄×D₁₃	C₂×C₄	1	30	yes	—
26	#6127	C₂×C₄	1	30	yes	—
27	C₂×D₅₂	C₂×C₄	1	15	yes	—
28	C₂×C₄×D₁₃	C₂³	1	15	yes	—
29	#6122	C₂³	1	20	yes	—
30	C₂×#1057	C₂³	1	5	yes	—
31	C₂×#1057	C₂³	1	15	yes	—
32	#6118	C₂³	1	30	yes	—
33	#6118	C₂³	1	10	yes	—
34	D₁₃⋊C₈	C₂³	1	10	yes	—
35	D₁₃⋊C₈	C₂³	1	30	yes	—
36	#3573	—	13	1	no	maximal, Sylow
37	C₂²×D₁₃	C₂²×C₄	1	15	yes	—
38	Q₈×C₁₃	C₂²×C₄	1	10	yes	—
39	D₄×C₁₃	C₂²×C₄	1	10	yes	—
40	C₂×C₄×C₁₃	C₂²×C₄	1	15	yes	—
41	C₂²⋊D₁₃	C₂²×C₄	1	20	yes	—
42	C₂×Dic₁₃	C₂⁴	1	5	yes	—
43	D₅₂	C₂²×C₄	1	30	yes	—
44	C₄×D₁₃	C₂²×C₄	1	30	yes	—
45	C₄×D₁₃	C₂⁴	1	10	yes	—
46	Dic₂₆	C₂²×C₄	1	10	yes	—
47	#1057	C₂⁴	1	1	yes	characteristic
48	#1057	C₂⁴	1	5	yes	—
49	#1057	C₂⁴	1	10	yes	—
50	#585	—	13	1	no	—
51	#568	—	13	5	no	—
52	#568	—	13	10	no	—
53	C₂×#131	—	13	5	no	—
54	C₂×#131	—	13	10	no	—
55	C₂²×C₁₃	C₂³×C₄	1	5	yes	—
56	C₂×D₁₃	—	4	15	no	—
57	C₂×D₁₃	C₂³×C₄	1	10	yes	—
58	C₄×C₁₃	C₂³×C₄	1	10	yes	—
59	Dic₁₃	C₂³×C₄	1	5	yes	—
60	Dic₁₃	C₂⁵	1	1	yes	characteristic
61	2₋¹⁺⁴	—	13	5	no	—
62	2₋¹⁺⁴	#261	1	1	yes	characteristic
63	H(5,𝔽₂)	—	13	10	no	—
64	C₂×#41	—	13	10	no	—
65	C₂×#41	—	13	5	no	—
66	#131	—	13	10	no	—
67	#131	—	13	10	no	—
68	#131	—	13	30	no	—
69	#131	—	13	20	no	—
70	#131	—	13	10	no	—
71	C₂×M₁₆	—	13	15	no	—
72	C₂×M₁₆	—	13	30	no	—
73	C₂²×C₈	—	13	15	no	—
74	C₂×C₁₃	—	2	5	no	—
75	C₂×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
76	D₁₃	—	2	10	no	—
77	#41	—	13	10	no	—
78	#41	—	13	20	no	—
79	#41	—	13	10	no	—
80	#41	—	13	30	no	—
81	#41	C₂×#261	1	10	yes	—
82	C₂×Q₈	—	13	10	no	—
83	C₂×Q₈	C₂×#261	1	5	yes	—
84	C₂×D₄	—	13	30	no	—
85	C₂×D₄	—	13	15	no	—
86	C₂²×C₄	—	13	15	no	—
87	M₁₆	—	13	30	no	—
88	M₁₆	—	13	10	no	—
89	M₁₆	—	13	20	no	—
90	C₂×C₈	—	13	10	no	—
91	C₂×C₈	—	13	5	no	—
92	C₂×C₈	—	13	30	no	—
93	C₂×C₈	—	13	15	no	—
94	C₁₃	#3573	1	1	yes	characteristic, complemented, Sylow
95	C₂³	—	13	15	no	—
96	Q₈	—	13	10	no	—
97	Q₈	C₂²×#261	1	10	yes	—
98	D₄	—	13	20	no	—
99	D₄	—	13	30	no	—
100	D₄	C₂²×#261	1	10	yes	—
101	C₂×C₄	—	13	5	no	—
102	C₂×C₄	—	13	10	no	—
103	C₂×C₄	—	13	30	no	—
104	C₂×C₄	C₂²×#261	1	15	yes	—
105	C₈	—	13	1	no	—
106	C₈	—	13	10	no	—
107	C₈	—	13	5	no	—
108	C₂²	—	13	10	no	—
109	C₂²	—	52	15	no	—
110	C₂²	C₂³×#261	1	5	yes	—
111	C₄	—	13	5	no	—
112	C₄	—	13	1	no	—
113	C₄	C₂³×#261	1	10	yes	—
114	C₂	—	2	5	no	—
115	C₂	—	26	10	no	—
116	C₂	C₂⁴×#261	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11719652

C₂

yes

maximal, characteristic

#11719629

C₂

yes

maximal, complemented

D₁₃⋊#131

C₂

yes

maximal, complemented

#11719625

C₂

yes

maximal, complemented

C₂×#88240

C₂

yes

maximal, complemented

C₁₃×2₋¹⁺⁴

C₄

yes

characteristic

#88256

C₄

yes

—

#41×D₁₃

C₂²

yes

—

#88254

C₄

yes

—

C₂×#6131

C₂²

yes

—

#88240

C₂²

yes

complemented

#88240

C₂²

yes

complemented

#88240

C₂²

yes

complemented

#88238

C₂²

yes

complemented

#88238

C₂²

yes

complemented

D₁₃⋊M₁₆

C₂²

yes

complemented

C₂×#6118

C₂²

yes

complemented

C₂×D₁₃⋊C₈

C₂²

yes

complemented

C₁₃×#41

C₂×C₄

yes

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

#6131

C₂×C₄

yes

—

#6131

C₂³

yes

—

Q₈×D₁₃

C₂×C₄

yes

—

#6129

C₂³

yes

—

D₄×D₁₃

C₂×C₄

yes

—

#6127

C₂×C₄

yes

—

C₂×D₅₂

C₂×C₄

yes

—

C₂×C₄×D₁₃

C₂³

yes

—

#6122

C₂³

yes

—

C₂×#1057

C₂³

yes

—

C₂×#1057

C₂³

yes

—

#6118

C₂³

yes

—

#6118

C₂³

yes

—

D₁₃⋊C₈

C₂³

yes

—

D₁₃⋊C₈

C₂³

yes

—

#3573

—

maximal, Sylow

C₂²×D₁₃

C₂²×C₄

yes

—

Q₈×C₁₃

C₂²×C₄

yes

—

D₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂²⋊D₁₃

C₂²×C₄

yes

—

C₂×Dic₁₃

C₂⁴

yes

—

D₅₂

C₂²×C₄

yes

—

C₄×D₁₃

C₂²×C₄

yes

—

C₄×D₁₃

C₂⁴

yes

—

Dic₂₆

C₂²×C₄

yes

—

#1057

C₂⁴

yes

characteristic

#1057

C₂⁴

yes

—

#1057

C₂⁴

yes

—

—

—

—

—

—

C₂²×C₁₃

C₂³×C₄

yes

—

C₂×D₁₃

—

C₂×D₁₃

C₂³×C₄

yes

—

C₄×C₁₃

C₂³×C₄

yes

—

Dic₁₃

C₂³×C₄

yes

—

Dic₁₃

C₂⁵

yes

characteristic

2₋¹⁺⁴

—

2₋¹⁺⁴

#261

yes

characteristic

H(5,𝔽₂)

—

C₂×#41

—

C₂×#41

—

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₁₃

—

C₂×C₁₃

C₂⁴×C₄

yes

characteristic

D₁₃

—

—

—

—

—

yes

—

C₂×Q₈

—

C₂×Q₈

C₂×#261

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₁₃

#3573

yes

characteristic, complemented, Sylow

C₂³

—

Q₈

—

Q₈

C₂²×#261

yes

—

D₄

—

D₄

—

100

D₄

C₂²×#261

yes

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂²×#261

yes

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₂²

—

109

C₂²

—

110

C₂²

C₂³×#261

yes

—

111

C₄

—

112

C₄

—

113

C₄

C₂³×#261

yes

—

114

C₂

—

115

C₂

—

116

C₂

C₂⁴×#261

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×#261
commutator, socle	C₂×C₁₃	C₂⁴×C₄
Fitting	C₁₃×2₋¹⁺⁴	C₄
prime radical	#11719652	C₂
cosocle	Dic₁₃	C₂⁵
hypercenter	2₋¹⁺⁴	#261
hypocenter	C₁₃	#3573
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ 2₋¹⁺⁴

subgroup

G/H

center, Frattini

C₂

C₂⁴×#261

commutator, socle

C₂×C₁₃

C₂⁴×C₄

Fitting

C₁₃×2₋¹⁺⁴

C₄

prime radical

#11719652

C₂

cosocle

Dic₁₃

C₂⁵

2₋¹⁺⁴

C₁₃

G ▷ C₂×C₁₃ ▷ C₁₃

C₁ ◁ C₂ ◁ 2₋¹⁺⁴

N	H ≅ G/N	type
2. #11719629	C₂	semidirect
3. D₁₃⋊#131	C₂	semidirect
4. #11719625	C₂	semidirect
5. C₂×#88240	C₂	semidirect
11. #88240	C₂²	semidirect
12. #88240	C₂²	semidirect
13. #88240	C₂²	semidirect
14. #88238	C₂²	semidirect
15. #88238	C₂²	semidirect
16. D₁₃⋊M₁₆	C₂²	semidirect
17. C₂×#6118	C₂²	semidirect
18. C₂×D₁₃⋊C₈	C₂²	semidirect
94. C₁₃	#3573	semidirect

H ≅ G/N

type

2. #11719629

C₂

semidirect

3. D₁₃⋊#131

C₂

semidirect

4. #11719625

C₂

semidirect

5. C₂×#88240

C₂

semidirect

11. #88240

C₂²

semidirect

12. #88240

C₂²

semidirect

13. #88240

C₂²

semidirect

14. #88238

C₂²

semidirect

15. #88238

C₂²

semidirect

16. D₁₃⋊M₁₆

C₂²

semidirect

17. C₂×#6118

C₂²

semidirect

18. C₂×D₁₃⋊C₈

C₂²

semidirect

94. C₁₃

#3573

semidirect

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

subgroup	index
#11719652	2
#11719629	2
D₁₃⋊#131	2
#11719625	2
C₂×#88240	2
#3573	13

subgroup

index

D₁₃⋊#131

Subgroups — #13131832

Conjugacy classes of subgroups

Complements

Maximal subgroups