Group #13122904 — subgroups

Subgroups — #13122904

Order 1664.

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

normal subgroups only

This group has 380 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×#88124	C₂	1	2	yes	maximal
2	C₁₃×#382	C₂	1	1	yes	maximal, characteristic
3	#11718644	C₂	1	4	yes	maximal
4	#11718322	C₂	1	4	yes	maximal
5	#11718320	C₂	1	4	yes	maximal
6	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₃	C₂²	1	4	yes	—
11	C₂×#6101	C₂²	1	4	yes	—
12	C₂×#6101	C₂²	1	4	yes	—
13	#88124	—	2	8	no	—
14	C₁₃×#95	C₂²	1	4	yes	—
15	#88071	C₂²	1	4	yes	—
16	#88071	C₂²	1	8	yes	—
17	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
18	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
19	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₁₃	C₂³	1	4	yes	—
21	C₂²×Dic₁₃	C₂³	1	4	yes	—
22	C₁₃×C₄⋊C₄	—	4	2	no	—
23	C₄²×C₁₃	D₄	1	4	yes	—
24	C₄⋊Dic₁₃	—	2	4	no	—
25	C₄⋊Dic₁₃	—	2	4	no	—
26	C₄⋊Dic₁₃	—	4	4	no	—
27	#6101	—	2	8	no	—
28	#6101	—	2	8	no	—
29	#2822	—	13	1	no	maximal, Sylow
30	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
31	C₂×C₄×C₁₃	—	2	1	no	—
32	C₂×C₄×C₁₃	—	2	4	no	—
33	C₂×C₄×C₁₃	—	2	1	no	—
34	C₂×C₄×C₁₃	—	2	4	no	—
35	C₂×C₄×C₁₃	—	2	8	no	—
36	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
38	C₂×Dic₁₃	—	2	4	no	—
39	C₂×Dic₁₃	—	2	4	no	—
40	C₂×Dic₁₃	—	2	4	no	—
41	C₂×Dic₁₃	—	2	8	no	—
42	C₂×Dic₁₃	—	2	4	no	—
43	C₂×#125	—	13	2	no	—
44	#400	—	13	4	no	—
45	#398	—	13	4	no	—
46	#398	—	13	4	no	—
47	#382	D₁₃	1	1	yes	characteristic
48	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
49	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
50	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
51	C₂²×C₁₃	2₋¹⁺⁴	1	4	yes	—
52	C₄×C₁₃	—	2	4	no	—
53	C₄×C₁₃	—	2	2	no	—
54	C₄×C₁₃	—	4	4	no	—
55	Dic₁₃	—	4	4	no	—
56	Dic₁₃	—	4	4	no	—
57	#125	—	26	8	no	—
58	C₂×C₄⋊C₄	—	13	4	no	—
59	C₂×C₄⋊C₄	—	13	4	no	—
60	C₂×C₄⋊C₄	—	13	2	no	—
61	C₂×C₄⋊C₄	—	13	2	no	—
62	C₂×C₄⋊C₄	—	13	4	no	—
63	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
64	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
65	#95	—	13	8	no	—
66	#95	—	13	4	no	—
67	#95	C₂×D₁₃	1	4	yes	—
68	C₂×C₁₃	#536	1	2	yes	—
69	C₂×C₁₃	#537	1	1	yes	characteristic
70	C₂×C₁₃	#563	1	2	yes	—
71	C₂×C₁₃	#564	1	2	yes	—
72	C₂²×C₄	—	13	4	no	—
73	C₂²×C₄	—	13	4	no	—
74	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
75	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
76	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
77	C₄⋊C₄	—	4	2	no	—
78	C₄⋊C₄	—	26	8	no	—
79	C₄⋊C₄	—	26	8	no	—
80	C₄⋊C₄	—	26	4	no	—
81	C₄⋊C₄	—	26	4	no	—
82	C₄⋊C₄	—	52	4	no	—
83	C₄²	D₅₂	1	4	yes	—
84	C₁₃	#2822	1	1	yes	characteristic, complemented, Sylow
85	C₂³	C₂³×D₁₃	1	1	yes	characteristic
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	2	8	no	—
91	C₂×C₄	—	26	4	no	—
92	C₂×C₄	—	26	4	no	—
93	C₂×C₄	—	26	4	no	—
94	C₂×C₄	—	26	4	no	—
95	C₂×C₄	—	26	8	no	—
96	C₂×C₄	C₂×D₅₂	1	2	yes	—
97	C₂×C₄	C₂×D₅₂	1	4	yes	—
98	C₂²	C₂²×D₅₂	1	1	yes	characteristic
99	C₂²	#88251	1	1	yes	characteristic
100	C₂²	#88254	1	1	yes	characteristic
101	C₂²	#88257	1	4	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	4	no	—
104	C₄	—	4	4	no	—
105	C₄	—	52	4	no	—
106	C₄	—	52	4	no	—
107	C₂	#11719215	1	1	yes	characteristic
108	C₂	#11719231	1	2	yes	—
109	C₂	#11719377	1	2	yes	—
110	C₂	C₁₃⋊#564	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88124

C₂

yes

maximal

C₁₃×#382

C₂

yes

maximal, characteristic

#11718644

C₂

yes

maximal

#11718322

C₂

yes

maximal

#11718320

C₂

yes

maximal

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×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

#88124

—

C₁₃×#95

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

D₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

—

—

—

—

D₁₃

yes

characteristic

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₁₃

2₋¹⁺⁴

yes

characteristic

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

C₂×D₁₃

yes

—

—

C₂×D₁₃

yes

—

C₂×C₁₃

#536

yes

—

C₂×C₁₃

#537

yes

characteristic

C₂×C₁₃

#563

yes

—

C₂×C₁₃

#564

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₅₂

yes

—

C₁₃

#2822

yes

characteristic, complemented, Sylow

C₂³

C₂³×D₁₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₅₂

yes

—

C₂×C₄

C₂×D₅₂

yes

—

C₂²

C₂²×D₅₂

yes

characteristic

C₂²

#88251

yes

characteristic

100

C₂²

#88254

yes

characteristic

101

C₂²

#88257

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₂

#11719215

yes

characteristic

108

C₂

#11719231

yes

—

109

C₂

#11719377

yes

—

110

C₂

C₁₃⋊#564

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#382	C₂
hypercenter	#382	D₁₃
hypocenter	C₁₃	#2822
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#382

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
84. C₁₃	#2822	semidirect

H ≅ G/N

type

84. C₁₃

#2822

semidirect

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

subgroup	index
C₂×#88124	2
C₁₃×#382	2
#11718644	2
#11718322	2
#11718320	2
#2822	13

subgroup

index

C₁₃×#382

Subgroups — #13122904

Conjugacy classes of subgroups

Complements

Maximal subgroups