Group #13116012 — subgroups

Subgroups — #13116012

Order 1664.

278 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 67 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×#543	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88046	C₂	1	3	yes	maximal
3	#11718119	C₂	1	3	yes	maximal, complemented
4	C₂²×C₂²⋊D₁₃	C₂²	1	1	yes	characteristic
5	C₁₃×C₂²⋊Q₈	—	2	6	no	—
6	C₂×C₁₃×C₂²⋊C₄	C₂²	1	3	yes	—
7	C₂×C₂²⋊Dic₁₃	C₂²	1	3	yes	—
8	#88046	—	2	6	no	—
9	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
10	C₂³×D₁₃	Q₈	1	1	yes	characteristic, complemented
11	C₂×Q₈×C₁₃	—	2	2	no	—
12	C₂²×C₄×C₁₃	—	2	3	no	—
13	C₂×C₂²⋊D₁₃	—	2	3	no	—
14	C₂×C₂²⋊D₁₃	—	2	3	no	—
15	C₂²×Dic₁₃	—	2	3	no	—
16	C₂²×Dic₁₃	Q₈	1	1	yes	characteristic, complemented
17	C₁₃×C₄⋊C₄	—	2	6	no	—
18	C₁₃×C₂²⋊C₄	—	2	3	no	—
19	C₁₃×C₂²⋊C₄	D₄	1	6	yes	—
20	C₂²⋊Dic₁₃	—	2	3	no	—
21	C₂²⋊Dic₁₃	—	2	3	no	—
22	#2032	—	13	1	no	maximal, Sylow
23	C₂³×C₁₃	—	8	1	no	—
24	C₂³×C₁₃	C₂×D₄	1	3	yes	—
25	C₂³×C₁₃	C₂×Q₈	1	1	yes	characteristic
26	C₂³×C₁₃	#41	1	3	yes	—
27	C₂²×D₁₃	—	2	3	no	—
28	C₂²×D₁₃	—	8	1	no	—
29	Q₈×C₁₃	—	8	2	no	—
30	C₂×C₄×C₁₃	—	2	6	no	—
31	C₂×C₄×C₁₃	—	8	3	no	—
32	C₂²⋊D₁₃	—	4	1	no	—
33	C₂²⋊D₁₃	—	4	3	no	—
34	C₂×Dic₁₃	—	2	3	no	—
35	C₂×Dic₁₃	—	4	3	no	—
36	C₂×Dic₁₃	—	8	3	no	—
37	#543	D₁₃	1	1	yes	characteristic, complemented
38	C₂×#99	—	13	3	no	—
39	#342	—	13	3	no	—
40	C₂²×C₁₃	—	4	3	no	—
41	C₂²×C₁₃	—	4	3	no	—
42	C₂²×C₁₃	—	4	1	no	—
43	C₂²×C₁₃	#120	1	1	yes	characteristic
44	C₂²×C₁₃	C₂²⋊Q₈	1	3	yes	—
45	C₂²×C₁₃	#124	1	3	yes	—
46	C₂×D₁₃	—	4	1	no	—
47	C₂×D₁₃	—	8	3	no	—
48	C₄×C₁₃	—	4	6	no	—
49	Dic₁₃	—	4	1	no	—
50	Dic₁₃	—	8	3	no	—
51	C₂²×D₄	—	13	1	no	—
52	C₂²⋊Q₈	—	2	6	no	—
53	C₂×C₂²⋊C₄	—	13	3	no	—
54	C₂×C₂²⋊C₄	C₂×D₁₃	1	3	yes	—
55	#99	—	26	6	no	—
56	C₂×C₁₃	—	2	3	no	—
57	C₂×C₁₃	—	2	1	no	—
58	C₂×C₁₃	—	2	3	no	—
59	C₂×C₁₃	#393	1	1	yes	characteristic
60	D₁₃	—	8	1	no	—
61	C₂⁴	—	13	1	no	—
62	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
63	C₂×Q₈	—	2	2	no	—
64	C₂×D₄	—	26	3	no	—
65	C₂×D₄	—	26	3	no	—
66	C₂²×C₄	—	2	3	no	—
67	C₂²×C₄	—	13	1	no	—
68	C₂²×C₄	—	26	3	no	—
69	C₄⋊C₄	—	2	6	no	—
70	C₂²⋊C₄	—	2	3	no	—
71	C₂²⋊C₄	—	26	3	no	—
72	C₂²⋊C₄	—	26	3	no	—
73	C₂²⋊C₄	C₂²⋊D₁₃	1	6	yes	—
74	C₁₃	#2032	1	1	yes	characteristic, complemented, Sylow
75	C₂³	—	8	1	no	—
76	C₂³	—	26	3	no	—
77	C₂³	—	104	1	no	—
78	C₂³	Q₈×D₁₃	1	1	yes	characteristic
79	C₂³	#6131	1	3	yes	—
80	C₂³	C₂×C₂²⋊D₁₃	1	3	yes	—
81	Q₈	—	8	2	no	—
82	D₄	—	52	3	no	—
83	D₄	—	52	1	no	—
84	C₂×C₄	—	2	6	no	—
85	C₂×C₄	—	8	3	no	—
86	C₂×C₄	—	26	3	no	—
87	C₂×C₄	—	52	3	no	—
88	C₂×C₄	—	104	3	no	—
89	C₂²	—	4	3	no	—
90	C₂²	—	4	3	no	—
91	C₂²	—	4	1	no	—
92	C₂²	—	52	1	no	—
93	C₂²	—	104	3	no	—
94	C₂²	#88200	1	3	yes	—
95	C₂²	#88201	1	3	yes	—
96	C₂²	#88207	1	1	yes	characteristic
97	C₄	—	4	6	no	—
98	C₄	—	52	1	no	—
99	C₄	—	104	3	no	—
100	C₂	—	2	1	no	—
101	C₂	—	2	3	no	—
102	C₂	—	2	3	no	—
103	C₂	—	104	1	no	—
104	C₂	#11718892	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#543

C₂

yes

maximal, characteristic, complemented

C₂×#88046

C₂

yes

maximal

#11718119

C₂

yes

maximal, complemented

C₂²×C₂²⋊D₁₃

C₂²

yes

characteristic

C₁₃×C₂²⋊Q₈

—

C₂×C₁₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

#88046

—

C₂⁴×C₁₃

C₂³

yes

characteristic

C₂³×D₁₃

Q₈

yes

characteristic, complemented

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

—

C₂×C₂²⋊D₁₃

—

C₂×C₂²⋊D₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

Q₈

yes

characteristic, complemented

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₁₃×C₂²⋊C₄

D₄

yes

—

C₂²⋊Dic₁₃

—

C₂²⋊Dic₁₃

—

#2032

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

C₂×D₄

yes

—

C₂³×C₁₃

C₂×Q₈

yes

characteristic

C₂³×C₁₃

#41

yes

—

C₂²×D₁₃

—

C₂²×D₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂²⋊D₁₃

—

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

#543

D₁₃

yes

characteristic, complemented

C₂×#99

—

#342

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#120

yes

characteristic

C₂²×C₁₃

C₂²⋊Q₈

yes

—

C₂²×C₁₃

#124

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×D₄

—

C₂²⋊Q₈

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₁₃

yes

—

#99

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

#393

yes

characteristic

D₁₃

—

C₂⁴

—

C₂⁴

C₂²×D₁₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂²⋊D₁₃

yes

—

C₁₃

#2032

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

C₂³

—

104

—

C₂³

Q₈×D₁₃

yes

characteristic

C₂³

#6131

yes

—

C₂³

C₂×C₂²⋊D₁₃

yes

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

104

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

104

—

C₂²

#88200

yes

—

C₂²

#88201

yes

—

C₂²

#88207

yes

characteristic

C₄

—

C₄

—

C₄

—

104

—

100

C₂

—

101

C₂

—

102

C₂

—

103

C₂

—

104

—

104

C₂

#11718892

yes

characteristic

	subgroup	G/H
center	C₂	#11718892
commutator, cosocle	C₂⁴×C₁₃	C₂³
Frattini	C₂⁴	C₂²×D₁₃
Fitting	C₁₃×#543	C₂
socle	C₂×C₁₃	#393
prime radical	C₂²×C₂²⋊D₁₃	C₂²
hypercenter	#543	D₁₃
hypocenter	C₁₃	#2032
LCS	G ▷ C₂⁴×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #543

subgroup

G/H

center

C₂

#11718892

commutator, cosocle

C₂⁴×C₁₃

C₂³

Frattini

C₂⁴

C₂²×D₁₃

Fitting

C₁₃×#543

C₂

socle

C₂×C₁₃

#393

prime radical

C₂²×C₂²⋊D₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂⁴×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂ ◁ C₂⁴ ◁ #543

N	H ≅ G/N	type
1. C₁₃×#543	C₂	semidirect
3. #11718119	C₂	semidirect
10. C₂³×D₁₃	Q₈	semidirect
16. C₂²×Dic₁₃	Q₈	semidirect
37. #543	D₁₃	semidirect
74. C₁₃	#2032	semidirect

H ≅ G/N

type

1. C₁₃×#543

C₂

semidirect

3. #11718119

C₂

semidirect

10. C₂³×D₁₃

Q₈

semidirect

16. C₂²×Dic₁₃

Q₈

semidirect

37. #543

D₁₃

semidirect

74. C₁₃

#2032

semidirect

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

subgroup	index
C₁₃×#543	2
C₂×#88046	2
#11718119	2
#2032	13

subgroup

index

C₁₃×#543

C₂×#88046

#11718119

#2032

Subgroups — #13116012

Conjugacy classes of subgroups

Complements

Maximal subgroups