Group #13122910 — subgroups

Subgroups — #13122910

Order 1664.

456 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 133 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×#88130	C₂	1	2	yes	maximal, complemented
2	C₁₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718655	C₂	1	6	yes	maximal, complemented
4	#11718652	C₂	1	3	yes	maximal
5	#11718644	C₂	1	3	yes	maximal, complemented
6	C₂²×D₅₂	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
8	C₂×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#6103	C₂²	1	6	yes	—
11	C₂×#6103	C₂²	1	6	yes	—
12	C₂×C₄⋊Dic₁₃	C₂²	1	3	yes	—
13	C₂×#6101	C₂²	1	6	yes	—
14	#88130	—	2	8	no	—
15	#88071	C₂²	1	3	yes	—
16	#88071	C₂²	1	3	yes	—
17	C₂³×D₁₃	C₂³	1	2	yes	—
18	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
20	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
21	C₂²×Dic₁₃	C₂³	1	6	yes	—
22	C₂×D₅₂	—	4	3	no	—
23	C₁₃×C₄⋊C₄	—	4	3	no	—
24	C₁₃×C₄⋊C₄	—	4	3	no	—
25	C₄²×C₁₃	D₄	1	4	yes	complemented
26	#6103	—	2	12	no	—
27	#6103	—	4	6	no	—
28	C₄⋊Dic₁₃	—	4	3	no	—
29	#6101	—	2	12	no	—
30	#2823	—	13	1	no	maximal, Sylow
31	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
32	C₂²×D₁₃	—	2	2	no	—
33	C₂²×D₁₃	—	2	6	no	—
34	C₂²×D₁₃	—	2	6	no	—
35	C₂×C₄×C₁₃	—	2	3	no	—
36	C₂×C₄×C₁₃	—	2	3	no	—
37	C₂×C₄×C₁₃	—	2	3	no	—
38	C₂×C₄×C₁₃	—	2	6	no	—
39	C₂×C₄×C₁₃	—	2	3	no	—
40	C₂×C₄×C₁₃	C₂×D₄	1	6	yes	—
41	C₂×Dic₁₃	—	2	6	no	—
42	C₂×Dic₁₃	—	2	6	no	—
43	C₂×Dic₁₃	—	2	6	no	—
44	D₅₂	—	4	4	no	—
45	C₂×#126	—	13	2	no	—
46	#398	—	13	3	no	—
47	#396	—	13	6	no	—
48	#394	—	13	3	no	—
49	#384	D₁₃	1	1	yes	characteristic, complemented
50	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
51	C₂²×C₁₃	H(5,𝔽₂)	1	3	yes	—
52	C₂²×C₁₃	2₋¹⁺⁴	1	3	yes	—
53	C₂×D₁₃	—	4	6	no	—
54	C₂×D₁₃	—	4	2	no	—
55	C₂×D₁₃	—	4	6	no	—
56	C₄×C₁₃	—	2	6	no	—
57	C₄×C₁₃	—	4	1	no	—
58	C₄×C₁₃	—	4	3	no	—
59	Dic₁₃	—	4	6	no	—
60	C₂²×D₄	—	13	1	no	—
61	#126	—	26	8	no	—
62	C₂×C₄⋊C₄	—	13	6	no	—
63	C₂×C₄⋊C₄	—	13	3	no	—
64	C₂×C₄⋊C₄	C₂×D₁₃	1	3	yes	—
65	C₂×C₄⋊C₄	C₂×D₁₃	1	3	yes	—
66	C₂×C₂²⋊C₄	—	13	6	no	—
67	C₂×C₂²⋊C₄	—	13	6	no	—
68	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
69	#95	—	13	3	no	—
70	#95	—	13	3	no	—
71	C₂×C₁₃	#537	1	3	yes	—
72	C₂×C₁₃	#560	1	1	yes	characteristic
73	C₂×C₁₃	#563	1	3	yes	—
74	D₁₃	—	8	2	no	—
75	C₂⁴	—	13	2	no	—
76	C₂×D₄	—	52	3	no	—
77	C₂²×C₄	—	13	6	no	—
78	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
80	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
81	C₄⋊C₄	—	4	3	no	—
82	C₄⋊C₄	—	4	3	no	—
83	C₄⋊C₄	—	26	12	no	—
84	C₄⋊C₄	—	52	3	no	—
85	C₂²⋊C₄	—	26	12	no	—
86	C₂²⋊C₄	—	52	6	no	—
87	C₄²	D₅₂	1	4	yes	complemented
88	C₁₃	#2823	1	1	yes	characteristic, complemented, Sylow
89	C₂³	—	26	2	no	—
90	C₂³	—	26	6	no	—
91	C₂³	—	26	6	no	—
92	C₂³	C₂³×D₁₃	1	1	yes	characteristic
93	D₄	—	52	4	no	—
94	C₂×C₄	—	2	3	no	—
95	C₂×C₄	—	2	3	no	—
96	C₂×C₄	—	2	6	no	—
97	C₂×C₄	—	2	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	26	6	no	—
100	C₂×C₄	—	26	6	no	—
101	C₂×C₄	—	26	6	no	—
102	C₂×C₄	C₂×D₅₂	1	6	yes	—
103	C₂²	—	52	2	no	—
104	C₂²	—	52	6	no	—
105	C₂²	—	52	6	no	—
106	C₂²	C₂²×D₅₂	1	1	yes	characteristic
107	C₂²	#88251	1	3	yes	—
108	C₂²	#88254	1	3	yes	—
109	C₄	—	2	6	no	—
110	C₄	—	4	3	no	—
111	C₄	—	4	1	no	—
112	C₄	—	52	6	no	—
113	C₂	—	104	2	no	—
114	C₂	#11719215	1	3	yes	—
115	C₂	#11719425	1	1	yes	characteristic
116	C₂	#11719440	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88130

C₂

yes

maximal, complemented

C₁₃×#384

C₂

yes

maximal, characteristic, complemented

#11718655

C₂

yes

maximal, complemented

#11718652

C₂

yes

maximal

#11718644

C₂

yes

maximal, complemented

C₂²×D₅₂

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×#6103

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

#88130

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂³×D₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×D₅₂

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

D₄

yes

complemented

#6103

—

#6103

—

C₄⋊Dic₁₃

—

#6101

—

#2823

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

C₂²×D₄

—

#126

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#95

—

#95

—

C₂×C₁₃

#537

yes

—

C₂×C₁₃

#560

yes

characteristic

C₂×C₁₃

#563

yes

—

D₁₃

—

C₂⁴

—

C₂×D₄

—

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

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

D₅₂

yes

complemented

C₁₃

#2823

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₁₃

yes

characteristic

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₂×D₅₂

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₂²×D₅₂

yes

characteristic

107

C₂²

#88251

yes

—

108

C₂²

#88254

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₂

—

104

—

114

C₂

#11719215

yes

—

115

C₂

#11719425

yes

characteristic

116

C₂

#11719440

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#384	C₂
prime radical	C₂²×D₅₂	C₂²
hypercenter	#384	D₁₃
hypocenter	C₁₃	#2823
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#384

C₂

prime radical

C₂²×D₅₂

C₂²

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
1. C₂×#88130	C₂	semidirect
2. C₁₃×#384	C₂	semidirect
3. #11718655	C₂	semidirect
5. #11718644	C₂	semidirect
25. C₄²×C₁₃	D₄	semidirect
49. #384	D₁₃	semidirect
87. C₄²	D₅₂	semidirect
88. C₁₃	#2823	semidirect

H ≅ G/N

type

1. C₂×#88130

C₂

semidirect

2. C₁₃×#384

C₂

semidirect

3. #11718655

C₂

semidirect

5. #11718644

C₂

semidirect

25. C₄²×C₁₃

D₄

semidirect

49. #384

D₁₃

semidirect

87. C₄²

D₅₂

semidirect

88. C₁₃

#2823

semidirect

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

subgroup	index
C₂×#88130	2
C₁₃×#384	2
#11718655	2
#11718652	2
#11718644	2
#2823	13

subgroup

index

C₁₃×#384

Subgroups — #13122910

Conjugacy classes of subgroups

Complements

Maximal subgroups