Group #13115814 — subgroups

Subgroups — #13115814

Order 1664.

284 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 105 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×#533	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#11718585	C₂	1	1	yes	maximal, characteristic
4	#11718124	C₂	1	4	yes	maximal
5	C₁₃×C₄⋊Q₈	C₄	1	2	yes	complemented
6	C₁₃×#125	C₄	1	2	yes	complemented
7	C₁₃×C₂²⋊Q₈	—	2	2	no	—
8	C₄×Q₈×C₁₃	—	2	2	no	—
9	C₁₃×#117	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₃	—	2	2	no	—
12	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
13	C₂×#6099	C₂²	1	2	yes	—
14	C₄⋊#1055	—	2	2	no	—
15	C₂×Q₈×C₁₃	—	2	2	no	—
16	C₂²×C₄×C₁₃	D₄	1	2	yes	—
17	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₁₃	—	2	2	no	—
19	C₁₃×C₄⋊C₄	—	2	4	no	—
20	C₁₃×C₄⋊C₄	—	2	2	no	—
21	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
22	C₁₃×C₂²⋊C₄	—	2	2	no	—
23	C₄²×C₁₃	—	2	2	no	—
24	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₁₃	D₄	1	2	yes	—
27	C₄×Dic₁₃	—	2	8	no	—
28	C₄×Dic₁₃	—	2	2	no	—
29	C₄×Dic₁₃	Q₈	1	4	yes	complemented
30	#6099	—	2	4	no	—
31	C₂×#1055	—	2	2	no	—
32	#1994	—	13	1	no	maximal, Sylow
33	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₁₃	—	4	4	no	—
35	C₂×C₄×C₁₃	—	2	4	no	—
36	C₂×C₄×C₁₃	—	2	4	no	—
37	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₁₃	#41	1	2	yes	—
43	C₂×Dic₁₃	—	2	8	no	—
44	C₂×Dic₁₃	—	2	4	no	—
45	#1055	—	4	2	no	—
46	#533	D₁₃	1	1	yes	characteristic
47	#423	—	13	1	no	—
48	C₄³	—	13	1	no	—
49	#339	—	13	4	no	—
50	C₂²×C₁₃	—	2	2	no	—
51	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₃	C₂²⋊Q₈	1	2	yes	—
53	C₄×C₁₃	—	2	2	no	—
54	C₄×C₁₃	—	4	4	no	—
55	C₄×C₁₃	#104	1	4	yes	—
56	C₄×C₁₃	C₄×Q₈	1	2	yes	—
57	C₄×C₁₃	#124	1	1	yes	characteristic
58	C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
59	Dic₁₃	—	2	8	no	—
60	C₂×M₁₆	—	13	2	no	—
61	C₄⋊Q₈	Dic₁₃	1	2	yes	complemented
62	#125	Dic₁₃	1	2	yes	complemented
63	C₂²⋊Q₈	—	2	2	no	—
64	C₄×Q₈	—	2	2	no	—
65	#117	C₂×D₁₃	1	2	yes	—
66	C₂×C₄²	—	13	2	no	—
67	C₂×C₄²	—	26	2	no	—
68	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
69	C₄⋊C₈	—	26	2	no	—
70	C₂×C₁₃	—	2	2	no	—
71	C₂×C₁₃	#391	1	1	yes	characteristic
72	C₂×C₁₃	C₂×#104	1	2	yes	—
73	C₂×Q₈	—	2	2	no	—
74	C₂²×C₄	—	26	2	no	—
75	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
76	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
77	M₁₆	—	26	4	no	—
78	C₂×C₈	—	26	2	no	—
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₄²	—	2	2	no	—
84	C₄²	—	13	4	no	—
85	C₄²	—	26	8	no	—
86	C₄²	—	26	2	no	—
87	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
88	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
89	C₄²	C₂²⋊D₁₃	1	2	yes	—
90	C₁₃	#1994	1	1	yes	characteristic, complemented, Sylow
91	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
92	Q₈	—	4	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	26	8	no	—
96	C₂×C₄	—	26	4	no	—
97	C₂×C₄	C₂²⋊Dic₁₃	1	2	yes	—
98	C₂×C₄	C₂²⋊Dic₁₃	1	2	yes	—
99	C₂×C₄	Q₈×D₁₃	1	2	yes	—
100	C₂×C₄	#6131	1	2	yes	—
101	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
102	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
103	C₈	—	52	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	#88200	1	2	yes	—
106	C₂²	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
107	C₄	—	2	2	no	—
108	C₄	—	4	4	no	—
109	C₄	—	26	8	no	—
110	C₄	#88077	1	4	yes	—
111	C₄	#88198	1	1	yes	characteristic
112	C₄	Q₈×Dic₁₃	1	2	yes	—
113	C₄	#88201	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#11718891	1	1	yes	characteristic
116	C₂	C₂×#88077	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#533

C₂

yes

maximal, characteristic

C₄²×Dic₁₃

C₂

yes

maximal, characteristic

#11718585

C₂

yes

maximal, characteristic

#11718124

C₂

yes

maximal

C₁₃×C₄⋊Q₈

C₄

yes

complemented

C₁₃×#125

C₄

yes

complemented

C₁₃×C₂²⋊Q₈

—

C₄×Q₈×C₁₃

—

C₁₃×#117

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄×Dic₁₃

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×#6099

C₂²

yes

—

C₄⋊#1055

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

D₄

yes

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

Q₈

yes

—

—

—

C₂³×C₁₃

C₂×D₄

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

#1055

—

#533

D₁₃

yes

characteristic

#423

—

C₄³

—

#339

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃

C₂²⋊Q₈

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#104

yes

—

C₄×C₁₃

C₄×Q₈

yes

—

C₄×C₁₃

#124

yes

characteristic

C₄×C₁₃

C₄⋊Q₈

yes

characteristic

Dic₁₃

—

C₂×M₁₆

—

C₄⋊Q₈

Dic₁₃

yes

complemented

#125

Dic₁₃

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₁₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

C₂×C₁₃

—

C₂×C₁₃

#391

yes

characteristic

C₂×C₁₃

C₂×#104

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₁₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×Dic₁₃

yes

characteristic

C₄²

C₂×Dic₁₃

yes

characteristic

C₄²

C₂²⋊D₁₃

yes

—

C₁₃

#1994

yes

characteristic, complemented, Sylow

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₁₃

yes

—

C₂×C₄

C₂²⋊Dic₁₃

yes

—

C₂×C₄

Q₈×D₁₃

yes

—

100

C₂×C₄

#6131

yes

—

101

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

102

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

103

C₈

—

104

C₂²

—

105

C₂²

#88200

yes

—

106

C₂²

C₂×C₂²⋊Dic₁₃

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#88077

yes

—

111

C₄

#88198

yes

characteristic

112

C₄

Q₈×Dic₁₃

yes

—

113

C₄

#88201

yes

characteristic

114

C₂

—

115

C₂

#11718891

yes

characteristic

116

C₂

C₂×#88077

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#533	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	#533	D₁₃
hypocenter	C₁₃	#1994
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#533

C₂

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₃

C₂×D₄

cosocle

C₂²×C₄×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
5. C₁₃×C₄⋊Q₈	C₄	semidirect
6. C₁₃×#125	C₄	semidirect
29. C₄×Dic₁₃	Q₈	semidirect
61. C₄⋊Q₈	Dic₁₃	semidirect
62. #125	Dic₁₃	semidirect
90. C₁₃	#1994	semidirect

H ≅ G/N

type

5. C₁₃×C₄⋊Q₈

C₄

semidirect

6. C₁₃×#125

C₄

semidirect

29. C₄×Dic₁₃

Q₈

semidirect

61. C₄⋊Q₈

Dic₁₃

semidirect

62. #125

Dic₁₃

semidirect

90. C₁₃

#1994

semidirect

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

subgroup	index
C₁₃×#533	2
C₄²×Dic₁₃	2
#11718585	2
#11718124	2
#1994	13

subgroup

index

C₁₃×#533

C₄²×Dic₁₃

#11718585

#11718124

#1994

Subgroups — #13115814

Conjugacy classes of subgroups

Complements

Maximal subgroups