Group #13131831 — subgroups

Subgroups — #13131831

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	#11719650	C₂	1	1	yes	maximal, characteristic
2	#11719629	C₂	1	6	yes	maximal, complemented
3	D₁₃⋊#131	C₂	1	6	yes	maximal, complemented
4	#11719621	C₂	1	9	yes	maximal, complemented
5	C₂×#88238	C₂	1	9	yes	maximal, complemented
6	C₁₃×H(5,𝔽₂)	C₄	1	1	yes	characteristic
7	#88257	C₄	1	6	yes	—
8	#41×D₁₃	C₂²	1	6	yes	—
9	#88251	C₄	1	9	yes	—
10	C₂×#6129	C₂²	1	9	yes	—
11	C₂×#6122	C₂²	1	18	yes	complemented
12	C₂²×#1057	C₂²	1	6	yes	complemented
13	#88240	C₂²	1	2	yes	complemented
14	#88240	C₂²	1	6	yes	complemented
15	#88238	C₂²	1	18	yes	complemented
16	#88238	C₂²	1	36	yes	complemented
17	#88238	C₂²	1	18	yes	complemented
18	D₁₃⋊M₁₆	C₂²	1	18	yes	complemented
19	C₂×#6118	C₂²	1	9	yes	complemented
20	C₂×D₁₃⋊C₈	C₂²	1	9	yes	complemented
21	C₁₃×#41	C₂×C₄	1	6	yes	—
22	C₂×D₄×C₁₃	C₂×C₄	1	9	yes	—
23	C₂×C₂²⋊D₁₃	C₂×C₄	1	18	yes	—
24	C₂²×Dic₁₃	C₂³	1	6	yes	—
25	#6131	C₂³	1	2	yes	—
26	Q₈×D₁₃	C₂×C₄	1	6	yes	—
27	#6129	C₂×C₄	1	36	yes	—
28	#6129	C₂³	1	18	yes	—
29	D₄×D₁₃	C₂×C₄	1	18	yes	—
30	#6127	C₂×C₄	1	18	yes	—
31	C₂×C₄×D₁₃	C₂³	1	9	yes	—
32	C₂×Dic₂₆	C₂×C₄	1	9	yes	—
33	#6122	C₂³	1	36	yes	—
34	C₂×#1057	C₂³	1	9	yes	—
35	C₂×#1057	C₂³	1	9	yes	—
36	C₂×#1057	C₂³	1	18	yes	—
37	#6118	C₂³	1	18	yes	—
38	#6118	C₂³	1	6	yes	—
39	D₁₃⋊C₈	C₂³	1	6	yes	—
40	D₁₃⋊C₈	C₂³	1	18	yes	—
41	#3573	—	13	1	no	maximal, Sylow
42	C₂³×C₁₃	C₂²×C₄	1	6	yes	—
43	C₂²×D₁₃	C₂²×C₄	1	9	yes	—
44	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
45	D₄×C₁₃	C₂²×C₄	1	18	yes	—
46	C₂×C₄×C₁₃	C₂²×C₄	1	9	yes	—
47	C₂²⋊D₁₃	C₂²×C₄	1	36	yes	—
48	C₂×Dic₁₃	C₂²×C₄	1	18	yes	—
49	C₂×Dic₁₃	C₂⁴	1	9	yes	—
50	D₅₂	C₂²×C₄	1	6	yes	—
51	C₄×D₁₃	C₂²×C₄	1	18	yes	—
52	C₄×D₁₃	C₂⁴	1	6	yes	—
53	Dic₂₆	C₂²×C₄	1	18	yes	—
54	#1057	C₂⁴	1	1	yes	characteristic
55	#1057	C₂⁴	1	6	yes	—
56	#1057	C₂⁴	1	9	yes	—
57	#585	—	13	1	no	—
58	#568	—	13	9	no	—
59	#568	—	13	6	no	—
60	C₂×#131	—	13	9	no	—
61	C₂×#131	—	13	6	no	—
62	C₂²×C₁₃	—	4	6	no	—
63	C₂²×C₁₃	C₂³×C₄	1	9	yes	—
64	C₂×D₁₃	—	4	9	no	—
65	C₂×D₁₃	C₂³×C₄	1	6	yes	—
66	C₄×C₁₃	C₂³×C₄	1	6	yes	—
67	Dic₁₃	C₂³×C₄	1	9	yes	—
68	Dic₁₃	C₂⁵	1	1	yes	characteristic
69	2₋¹⁺⁴	—	13	6	no	—
70	H(5,𝔽₂)	—	13	9	no	—
71	H(5,𝔽₂)	#261	1	1	yes	characteristic
72	C₂×#41	—	13	6	no	—
73	C₂×#41	—	13	9	no	—
74	#131	—	13	18	no	—
75	#131	—	13	2	no	—
76	#131	—	13	36	no	—
77	#131	—	13	18	no	—
78	#131	—	13	6	no	—
79	C₂×M₁₆	—	13	9	no	—
80	C₂×M₁₆	—	13	18	no	—
81	C₂×M₁₆	—	13	18	no	—
82	C₂²×C₈	—	13	9	no	—
83	C₂²×C₈	—	13	6	no	—
84	C₂×C₁₃	—	2	9	no	—
85	C₂×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
86	D₁₃	—	2	6	no	—
87	#41	—	13	2	no	—
88	#41	—	13	18	no	—
89	#41	—	13	36	no	—
90	#41	—	13	18	no	—
91	#41	C₂×#261	1	6	yes	—
92	C₂×Q₈	—	13	6	no	—
93	C₂×Q₈	—	13	9	no	—
94	C₂×D₄	—	13	18	no	—
95	C₂×D₄	—	13	18	no	—
96	C₂×D₄	C₂×#261	1	9	yes	—
97	C₂²×C₄	—	13	6	no	—
98	C₂²×C₄	—	13	9	no	—
99	M₁₆	—	13	18	no	—
100	M₁₆	—	13	6	no	—
101	M₁₆	—	13	36	no	—
102	C₂×C₈	—	13	6	no	—
103	C₂×C₈	—	13	9	no	—
104	C₂×C₈	—	13	18	no	—
105	C₂×C₈	—	13	9	no	—
106	C₂×C₈	—	13	18	no	—
107	C₁₃	#3573	1	1	yes	characteristic, complemented, Sylow
108	C₂³	—	13	9	no	—
109	C₂³	C₂²×#261	1	6	yes	—
110	Q₈	—	13	18	no	—
111	Q₈	C₂²×#261	1	2	yes	—
112	D₄	—	13	36	no	—
113	D₄	—	13	6	no	—
114	D₄	C₂²×#261	1	18	yes	—
115	C₂×C₄	—	13	9	no	—
116	C₂×C₄	—	13	18	no	—
117	C₂×C₄	—	13	6	no	—
118	C₂×C₄	—	13	18	no	—
119	C₂×C₄	C₂²×#261	1	9	yes	—
120	C₈	—	13	1	no	—
121	C₈	—	13	6	no	—
122	C₈	—	13	9	no	—
123	C₂²	—	4	6	no	—
124	C₂²	—	13	6	no	—
125	C₂²	—	52	9	no	—
126	C₂²	C₂³×#261	1	9	yes	—
127	C₄	—	13	9	no	—
128	C₄	—	13	1	no	—
129	C₄	C₂³×#261	1	6	yes	—
130	C₂	—	2	9	no	—
131	C₂	—	26	6	no	—
132	C₂	C₂⁴×#261	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11719650

C₂

yes

maximal, characteristic

#11719629

C₂

yes

maximal, complemented

D₁₃⋊#131

C₂

yes

maximal, complemented

#11719621

C₂

yes

maximal, complemented

C₂×#88238

C₂

yes

maximal, complemented

C₁₃×H(5,𝔽₂)

C₄

yes

characteristic

#88257

C₄

yes

—

#41×D₁₃

C₂²

yes

—

#88251

C₄

yes

—

C₂×#6129

C₂²

yes

—

C₂×#6122

C₂²

yes

complemented

C₂²×#1057

C₂²

yes

complemented

#88240

C₂²

yes

complemented

#88240

C₂²

yes

complemented

#88238

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×D₄×C₁₃

C₂×C₄

yes

—

C₂×C₂²⋊D₁₃

C₂×C₄

yes

—

C₂²×Dic₁₃

C₂³

yes

—

#6131

C₂³

yes

—

Q₈×D₁₃

C₂×C₄

yes

—

#6129

C₂×C₄

yes

—

#6129

C₂³

yes

—

D₄×D₁₃

C₂×C₄

yes

—

#6127

C₂×C₄

yes

—

C₂×C₄×D₁₃

C₂³

yes

—

C₂×Dic₂₆

C₂×C₄

yes

—

#6122

C₂³

yes

—

C₂×#1057

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

C₂³×C₁₃

C₂²×C₄

yes

—

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

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

—

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

—

—

yes

C₂×#41

—

C₂×#41

—

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₁₃

—

C₂×C₁₃

C₂⁴×C₄

yes

characteristic

D₁₃

—

—

—

—

—

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×#261

yes

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₁₃

#3573

yes

characteristic, complemented, Sylow

108

C₂³

—

109

C₂³

C₂²×#261

yes

—

110

Q₈

—

111

Q₈

C₂²×#261

yes

—

112

D₄

—

113

D₄

—

114

D₄

C₂²×#261

yes

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²×#261

yes

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

C₂³×#261

yes

—

127

C₄

—

128

C₄

—

129

C₄

C₂³×#261

yes

—

130

C₂

—

131

C₂

—

132

C₂

C₂⁴×#261

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×#261
commutator, socle	C₂×C₁₃	C₂⁴×C₄
Fitting	C₁₃×H(5,𝔽₂)	C₄
prime radical	#11719650	C₂
cosocle	Dic₁₃	C₂⁵
hypercenter	H(5,𝔽₂)	#261
hypocenter	C₁₃	#3573
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

subgroup

G/H

center, Frattini

C₂

C₂⁴×#261

commutator, socle

C₂×C₁₃

C₂⁴×C₄

Fitting

C₁₃×H(5,𝔽₂)

C₄

prime radical

#11719650

C₂

cosocle

Dic₁₃

C₂⁵

C₁₃

G ▷ C₂×C₁₃ ▷ C₁₃

C₁ ◁ C₂ ◁ H(5,𝔽₂)

N	H ≅ G/N	type
2. #11719629	C₂	semidirect
3. D₁₃⋊#131	C₂	semidirect
4. #11719621	C₂	semidirect
5. C₂×#88238	C₂	semidirect
11. C₂×#6122	C₂²	semidirect
12. C₂²×#1057	C₂²	semidirect
13. #88240	C₂²	semidirect
14. #88240	C₂²	semidirect
15. #88238	C₂²	semidirect
16. #88238	C₂²	semidirect
17. #88238	C₂²	semidirect
18. D₁₃⋊M₁₆	C₂²	semidirect
19. C₂×#6118	C₂²	semidirect
20. C₂×D₁₃⋊C₈	C₂²	semidirect
107. C₁₃	#3573	semidirect

H ≅ G/N

type

2. #11719629

C₂

semidirect

3. D₁₃⋊#131

C₂

semidirect

4. #11719621

C₂

semidirect

5. C₂×#88238

C₂

semidirect

11. C₂×#6122

C₂²

semidirect

12. C₂²×#1057

C₂²

semidirect

13. #88240

C₂²

semidirect

14. #88240

C₂²

semidirect

15. #88238

C₂²

semidirect

16. #88238

C₂²

semidirect

17. #88238

C₂²

semidirect

18. D₁₃⋊M₁₆

C₂²

semidirect

19. C₂×#6118

C₂²

semidirect

20. C₂×D₁₃⋊C₈

C₂²

semidirect

107. C₁₃

#3573

semidirect

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

subgroup	index
#11719650	2
#11719629	2
D₁₃⋊#131	2
#11719621	2
C₂×#88238	2
#3573	13

subgroup

index

D₁₃⋊#131

Subgroups — #13131831

Conjugacy classes of subgroups

Complements

Maximal subgroups