Group #13129649 — subgroups

Subgroups — #13129649

Order 1664.

5796 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.

normal subgroups only

This group has 5796 conjugacy classes of subgroups — too many to draw. Only its 3201 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×#585	C₂	1	1	yes	maximal, characteristic, complemented
2	D₁₃×2₋¹⁺⁴	C₂	1	12	yes	maximal, complemented
3	#11719650	C₂	1	20	yes	maximal, complemented
4	#11719648	C₂	1	15	yes	maximal, complemented
5	C₂×#88257	C₂	1	15	yes	maximal, complemented
6	C₁₃×2₋¹⁺⁴	C₂²	1	6	yes	complemented
7	C₁₃×H(5,𝔽₂)	C₂²	1	10	yes	complemented
8	C₂×C₁₃×#41	C₂²	1	15	yes	complemented
9	#88257	C₂²	1	60	yes	complemented
10	#88257	C₂²	1	120	yes	complemented
11	#88256	C₂²	1	20	yes	complemented
12	#41×D₁₃	C₂²	1	120	yes	complemented
13	#88254	C₂²	1	30	yes	complemented
14	C₂×Q₈×D₁₃	C₂²	1	30	yes	complemented
15	#88251	C₂²	1	90	yes	complemented
16	C₂×#6129	C₂²	1	90	yes	complemented
17	C₂×#6127	C₂²	1	45	yes	complemented
18	C₂²×Dic₂₆	C₂²	1	15	yes	complemented
19	C₁₃×#41	C₂³	1	60	yes	—
20	C₁₃×#41	C₂³	1	20	yes	—
21	C₂×Q₈×C₁₃	C₂³	1	15	yes	—
22	C₂×D₄×C₁₃	C₂³	1	45	yes	—
23	C₂²×C₄×C₁₃	C₂³	1	15	yes	—
24	C₂×C₂²⋊D₁₃	C₂³	1	90	yes	—
25	C₂²×Dic₁₃	C₂³	1	30	yes	—
26	#6131	C₂³	1	40	yes	—
27	Q₈×D₁₃	C₂³	1	120	yes	—
28	#6129	C₂³	1	360	yes	—
29	D₄×D₁₃	C₂³	1	120	yes	—
30	#6127	C₂³	1	60	yes	—
31	#6127	C₂³	1	180	yes	—
32	C₂×D₅₂	C₂³	1	15	yes	—
33	C₂×C₄×D₁₃	C₂³	1	90	yes	—
34	C₂×Dic₂₆	C₂³	1	45	yes	—
35	C₂×Dic₂₆	C₂³	1	90	yes	—
36	2₋¹⁺⁶	—	13	1	no	maximal, Sylow
37	C₂³×C₁₃	C₂⁴	1	15	yes	—
38	C₂²×D₁₃	C₂⁴	1	30	yes	—
39	Q₈×C₁₃	C₂⁴	1	20	yes	—
40	D₄×C₁₃	C₂⁴	1	60	yes	—
41	C₂×C₄×C₁₃	C₂⁴	1	15	yes	—
42	C₂×C₄×C₁₃	C₂⁴	1	45	yes	—
43	C₂²⋊D₁₃	C₂⁴	1	120	yes	—
44	C₂×Dic₁₃	C₂⁴	1	90	yes	—
45	D₅₂	C₂⁴	1	6	yes	—
46	D₅₂	C₂⁴	1	30	yes	—
47	C₄×D₁₃	C₂⁴	1	120	yes	—
48	Dic₂₆	C₂⁴	1	10	yes	—
49	Dic₂₆	C₂⁴	1	90	yes	—
50	#585	—	13	20	no	—
51	#585	—	13	15	no	—
52	#585	D₁₃	1	1	yes	characteristic, complemented
53	C₂×2₋¹⁺⁴	—	13	12	no	—
54	C₂×2₋¹⁺⁴	—	13	15	no	—
55	C₂²×C₁₃	—	4	15	no	—
56	C₂²×C₁₃	C₂⁵	1	15	yes	—
57	C₂×D₁₃	—	4	30	no	—
58	C₂×D₁₃	C₂⁵	1	12	yes	—
59	C₄×C₁₃	C₂⁵	1	1	yes	characteristic
60	C₄×C₁₃	C₂⁵	1	15	yes	—
61	Dic₁₃	C₂⁵	1	20	yes	—
62	2₋¹⁺⁴	—	13	120	no	—
63	2₋¹⁺⁴	—	13	60	no	—
64	2₋¹⁺⁴	—	13	30	no	—
65	2₋¹⁺⁴	C₂×D₁₃	1	6	yes	complemented
66	H(5,𝔽₂)	—	13	20	no	—
67	H(5,𝔽₂)	—	13	90	no	—
68	H(5,𝔽₂)	C₂×D₁₃	1	10	yes	complemented
69	C₂×#41	—	13	120	no	—
70	C₂×#41	—	13	90	no	—
71	C₂×#41	—	13	45	no	—
72	C₂×#41	C₂×D₁₃	1	15	yes	complemented
73	C₂²×Q₈	—	13	30	no	—
74	C₂²×Q₈	—	13	15	no	—
75	C₂×C₁₃	—	2	15	no	—
76	C₂×C₁₃	C₂⁶	1	1	yes	characteristic
77	D₁₃	—	2	12	no	—
78	#41	—	13	40	no	—
79	#41	—	13	360	no	—
80	#41	—	13	180	no	—
81	#41	—	13	60	no	—
82	#41	C₂²×D₁₃	1	60	yes	—
83	#41	C₂²×D₁₃	1	20	yes	—
84	C₂×Q₈	—	13	120	no	—
85	C₂×Q₈	—	13	90	no	—
86	C₂×Q₈	—	13	45	no	—
87	C₂×Q₈	C₂²×D₁₃	1	15	yes	—
88	C₂×D₄	—	13	90	no	—
89	C₂×D₄	—	13	120	no	—
90	C₂×D₄	—	13	15	no	—
91	C₂×D₄	C₂²×D₁₃	1	45	yes	—
92	C₂²×C₄	—	13	30	no	—
93	C₂²×C₄	—	13	90	no	—
94	C₂²×C₄	C₂²×D₁₃	1	15	yes	—
95	C₁₃	2₋¹⁺⁶	1	1	yes	characteristic, complemented, Sylow
96	C₂³	—	13	30	no	—
97	C₂³	C₂³×D₁₃	1	15	yes	—
98	Q₈	—	13	90	no	—
99	Q₈	—	13	10	no	—
100	Q₈	C₂³×D₁₃	1	20	yes	—
101	D₄	—	13	120	no	—
102	D₄	—	13	30	no	—
103	D₄	—	13	6	no	—
104	D₄	C₂³×D₁₃	1	60	yes	—
105	C₂×C₄	—	13	90	no	—
106	C₂×C₄	—	13	120	no	—
107	C₂×C₄	C₂³×D₁₃	1	45	yes	—
108	C₂×C₄	C₂³×D₁₃	1	15	yes	—
109	C₂²	—	4	15	no	—
110	C₂²	—	13	12	no	—
111	C₂²	—	52	30	no	—
112	C₂²	C₂⁴×D₁₃	1	15	yes	—
113	C₄	—	13	20	no	—
114	C₄	C₂⁴×D₁₃	1	15	yes	—
115	C₄	C₂⁴×D₁₃	1	1	yes	characteristic
116	C₂	—	2	15	no	—
117	C₂	—	26	12	no	—
118	C₂	C₂⁵×D₁₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#585

C₂

yes

maximal, characteristic, complemented

D₁₃×2₋¹⁺⁴

C₂

yes

maximal, complemented

#11719650

C₂

yes

maximal, complemented

#11719648

C₂

yes

maximal, complemented

C₂×#88257

C₂

yes

maximal, complemented

C₁₃×2₋¹⁺⁴

C₂²

yes

complemented

C₁₃×H(5,𝔽₂)

C₂²

yes

complemented

C₂×C₁₃×#41

C₂²

yes

complemented

#88257

C₂²

yes

complemented

#88257

C₂²

120

yes

complemented

#88256

C₂²

yes

complemented

#41×D₁₃

C₂²

120

yes

complemented

#88254

C₂²

yes

complemented

C₂×Q₈×D₁₃

C₂²

yes

complemented

#88251

C₂²

yes

complemented

C₂×#6129

C₂²

yes

complemented

C₂×#6127

C₂²

yes

complemented

C₂²×Dic₂₆

C₂²

yes

complemented

C₁₃×#41

C₂³

yes

—

C₁₃×#41

C₂³

yes

—

C₂×Q₈×C₁₃

C₂³

yes

—

C₂×D₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂×C₂²⋊D₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

#6131

C₂³

yes

—

Q₈×D₁₃

C₂³

120

yes

—

#6129

C₂³

360

yes

—

D₄×D₁₃

C₂³

120

yes

—

#6127

C₂³

yes

—

#6127

C₂³

180

yes

—

C₂×D₅₂

C₂³

yes

—

C₂×C₄×D₁₃

C₂³

yes

—

C₂×Dic₂₆

C₂³

yes

—

C₂×Dic₂₆

C₂³

yes

—

2₋¹⁺⁶

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

—

C₂²×D₁₃

C₂⁴

yes

—

Q₈×C₁₃

C₂⁴

yes

—

D₄×C₁₃

C₂⁴

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

—

C₂²⋊D₁₃

C₂⁴

120

yes

—

C₂×Dic₁₃

C₂⁴

yes

—

D₅₂

C₂⁴

yes

—

D₅₂

C₂⁴

yes

—

C₄×D₁₃

C₂⁴

120

yes

—

Dic₂₆

C₂⁴

yes

—

Dic₂₆

C₂⁴

yes

—

#585

—

#585

—

#585

D₁₃

yes

characteristic, complemented

C₂×2₋¹⁺⁴

—

C₂×2₋¹⁺⁴

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂⁵

yes

—

C₂×D₁₃

—

C₂×D₁₃

C₂⁵

yes

—

C₄×C₁₃

C₂⁵

yes

characteristic

C₄×C₁₃

C₂⁵

yes

—

Dic₁₃

C₂⁵

yes

—

2₋¹⁺⁴

—

120

—

2₋¹⁺⁴

—

2₋¹⁺⁴

—

2₋¹⁺⁴

C₂×D₁₃

yes

—

—

C₂×D₁₃

yes

complemented

C₂×#41

—

120

—

C₂×#41

—

C₂×#41

—

C₂×#41

C₂×D₁₃

yes

complemented

C₂²×Q₈

—

C₂²×Q₈

—

C₂×C₁₃

—

C₂×C₁₃

C₂⁶

yes

characteristic

D₁₃

—

#41

—

#41

—

360

—

#41

—

180

—

#41

—

#41

C₂²×D₁₃

yes

—

#41

C₂²×D₁₃

yes

—

C₂×Q₈

—

120

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₁₃

yes

—

C₂×D₄

—

C₂×D₄

—

120

—

C₂×D₄

—

C₂×D₄

C₂²×D₁₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₁₃

2₋¹⁺⁶

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

C₂³×D₁₃

yes

—

Q₈

—

Q₈

—

100

Q₈

C₂³×D₁₃

yes

—

101

D₄

—

120

—

102

D₄

—

103

D₄

—

104

D₄

C₂³×D₁₃

yes

—

105

C₂×C₄

—

106

C₂×C₄

—

120

—

107

C₂×C₄

C₂³×D₁₃

yes

—

108

C₂×C₄

C₂³×D₁₃

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

C₂⁴×D₁₃

yes

—

113

C₄

—

114

C₄

C₂⁴×D₁₃

yes

—

115

C₄

C₂⁴×D₁₃

yes

characteristic

116

C₂

—

117

C₂

—

118

C₂

C₂⁵×D₁₃

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁵×D₁₃
commutator, socle, cosocle	C₂×C₁₃	C₂⁶
Fitting	C₁₃×#585	C₂
hypercenter	#585	D₁₃
hypocenter	C₁₃	2₋¹⁺⁶
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ #585

subgroup

G/H

center, Frattini

C₂

C₂⁵×D₁₃

commutator, socle, cosocle

C₂×C₁₃

C₂⁶

Fitting

C₁₃×#585

C₂

hypercenter

#585

D₁₃

hypocenter

C₁₃

2₋¹⁺⁶

LCS

G ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂ ◁ #585

N	H ≅ G/N	type
1. C₁₃×#585	C₂	semidirect
2. D₁₃×2₋¹⁺⁴	C₂	semidirect
3. #11719650	C₂	semidirect
4. #11719648	C₂	semidirect
5. C₂×#88257	C₂	semidirect
6. C₁₃×2₋¹⁺⁴	C₂²	semidirect
7. C₁₃×H(5,𝔽₂)	C₂²	semidirect
8. C₂×C₁₃×#41	C₂²	semidirect
9. #88257	C₂²	semidirect
10. #88257	C₂²	semidirect
11. #88256	C₂²	semidirect
12. #41×D₁₃	C₂²	semidirect
13. #88254	C₂²	semidirect
14. C₂×Q₈×D₁₃	C₂²	semidirect
15. #88251	C₂²	semidirect
16. C₂×#6129	C₂²	semidirect
17. C₂×#6127	C₂²	semidirect
18. C₂²×Dic₂₆	C₂²	semidirect
52. #585	D₁₃	semidirect
65. 2₋¹⁺⁴	C₂×D₁₃	semidirect
68. H(5,𝔽₂)	C₂×D₁₃	semidirect
72. C₂×#41	C₂×D₁₃	semidirect
95. C₁₃	2₋¹⁺⁶	semidirect

H ≅ G/N

type

1. C₁₃×#585

C₂

semidirect

2. D₁₃×2₋¹⁺⁴

C₂

semidirect

3. #11719650

C₂

semidirect

4. #11719648

C₂

semidirect

5. C₂×#88257

C₂

semidirect

6. C₁₃×2₋¹⁺⁴

C₂²

semidirect

7. C₁₃×H(5,𝔽₂)

C₂²

semidirect

8. C₂×C₁₃×#41

C₂²

semidirect

9. #88257

C₂²

semidirect

10. #88257

C₂²

semidirect

11. #88256

C₂²

semidirect

12. #41×D₁₃

C₂²

semidirect

13. #88254

C₂²

semidirect

14. C₂×Q₈×D₁₃

C₂²

semidirect

15. #88251

C₂²

semidirect

16. C₂×#6129

C₂²

semidirect

17. C₂×#6127

C₂²

semidirect

18. C₂²×Dic₂₆

C₂²

semidirect

52. #585

D₁₃

semidirect

65. 2₋¹⁺⁴

C₂×D₁₃

semidirect

68. H(5,𝔽₂)

C₂×D₁₃

semidirect

72. C₂×#41

C₂×D₁₃

semidirect

95. C₁₃

2₋¹⁺⁶

semidirect

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

subgroup	index
C₁₃×#585	2
D₁₃×2₋¹⁺⁴	2
#11719650	2
#11719648	2
C₂×#88257	2
2₋¹⁺⁶	13

subgroup

index

C₁₃×#585

D₁₃×2₋¹⁺⁴

#11719650

#11719648

C₂×#88257

2₋¹⁺⁶

Subgroups — #13129649

Conjugacy classes of subgroups

Complements

Maximal subgroups