Group #13115813 — subgroups

Subgroups — #13115813

Order 1664.

284 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 101 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	#11719175	C₂	1	1	yes	maximal, characteristic
2	C₄³×C₁₃	C₂	1	1	yes	maximal, characteristic
3	#11718585	C₂	1	1	yes	maximal, characteristic
4	#11718115	C₂	1	4	yes	maximal
5	C₂×C₄²×C₁₃	—	2	2	no	—
6	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	#88180	C₂²	1	2	yes	—
9	#88178	—	2	2	no	—
10	C₂×#6099	C₂²	1	2	yes	—
11	#88124	C₄	1	2	yes	complemented
12	#88123	C₄	1	2	yes	complemented
13	C₄×Dic₂₆	—	2	2	no	—
14	C₄⋊#1055	—	2	2	no	—
15	C₂²×C₄×C₁₃	—	2	2	no	—
16	C₂²×C₄×C₁₃	D₄	1	2	yes	—
17	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×Dic₂₆	—	2	2	no	—
19	C₄²×C₁₃	—	2	8	no	—
20	C₄²×C₁₃	—	2	2	no	—
21	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₁₃	D₄	1	2	yes	—
24	C₄²×C₁₃	Q₈	1	4	yes	complemented
25	C₂²⋊Dic₁₃	—	2	2	no	—
26	C₄⋊Dic₁₃	—	2	2	no	—
27	C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
28	#6101	—	2	4	no	—
29	C₄×Dic₁₃	—	2	2	no	—
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	C₂×C₄×C₁₃	—	2	8	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	4	no	—
44	Dic₂₆	—	4	4	no	—
45	#1055	—	4	2	no	—
46	#533	—	13	1	no	—
47	#423	—	13	1	no	—
48	C₄³	D₁₃	1	1	yes	characteristic
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	8	no	—
54	C₄×C₁₃	—	2	2	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	—	4	4	no	—
60	C₂×M₁₆	—	13	2	no	—
61	C₄⋊Q₈	—	13	2	no	—
62	#125	—	13	2	no	—
63	C₂²⋊Q₈	—	26	2	no	—
64	C₄×Q₈	—	26	2	no	—
65	#117	—	13	2	no	—
66	C₂×C₄²	—	2	2	no	—
67	C₂×C₄²	C₂×D₁₃	1	2	yes	—
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	—	26	2	no	—
74	C₂²×C₄	—	2	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	—	13	4	no	—
80	C₄⋊C₄	—	26	4	no	—
81	C₄⋊C₄	—	26	2	no	—
82	C₂²⋊C₄	—	26	2	no	—
83	C₄²	—	2	8	no	—
84	C₄²	—	2	2	no	—
85	C₄²	—	26	2	no	—
86	C₄²	Dic₂₆	1	4	yes	complemented
87	C₄²	C₄×D₁₃	1	1	yes	characteristic
88	C₄²	C₄×D₁₃	1	1	yes	characteristic
89	C₄²	D₅₂	1	2	yes	—
90	C₁₃	#1994	1	1	yes	characteristic, complemented, Sylow
91	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
92	Q₈	—	52	4	no	—
93	C₂×C₄	—	2	8	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	26	4	no	—
97	C₂×C₄	#6103	1	2	yes	—
98	C₂×C₄	#6103	1	2	yes	—
99	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
100	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
101	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
102	C₂×C₄	#6127	1	2	yes	—
103	C₈	—	52	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	#88178	1	2	yes	—
106	C₂²	C₂×#6103	1	1	yes	characteristic
107	C₄	—	2	8	no	—
108	C₄	—	2	2	no	—
109	C₄	—	52	4	no	—
110	C₄	#88045	1	4	yes	—
111	C₄	C₄×Dic₂₆	1	2	yes	—
112	C₄	#88123	1	1	yes	characteristic
113	C₄	#88129	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	C₂×#88045	1	2	yes	—
116	C₂	#11718590	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11719175

C₂

yes

maximal, characteristic

C₄³×C₁₃

C₂

yes

maximal, characteristic

#11718585

C₂

yes

maximal, characteristic

#11718115

C₂

yes

maximal

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

#88180

C₂²

yes

—

#88178

—

C₂×#6099

C₂²

yes

—

#88124

C₄

yes

complemented

#88123

C₄

yes

complemented

C₄×Dic₂₆

—

C₄⋊#1055

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

D₄

yes

—

C₄²×C₁₃

Q₈

yes

complemented

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

C₂×C₄

yes

—

#6101

—

C₄×Dic₁₃

—

—

—

—

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×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

—

Dic₂₆

—

#1055

—

#533

—

#423

—

C₄³

D₁₃

yes

characteristic

#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

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

—

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

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Dic₂₆

yes

complemented

C₄²

C₄×D₁₃

yes

characteristic

C₄²

C₄×D₁₃

yes

characteristic

C₄²

D₅₂

yes

—

C₁₃

#1994

yes

characteristic, complemented, Sylow

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#6103

yes

—

C₂×C₄

#6103

yes

—

C₂×C₄

C₂×Dic₂₆

yes

—

100

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

101

C₂×C₄

C₂×D₅₂

yes

characteristic

102

C₂×C₄

#6127

yes

—

103

C₈

—

104

C₂²

—

105

C₂²

#88178

yes

—

106

C₂²

C₂×#6103

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#88045

yes

—

111

C₄

C₄×Dic₂₆

yes

—

112

C₄

#88123

yes

characteristic

113

C₄

#88129

yes

characteristic

114

C₂

—

115

C₂

C₂×#88045

yes

—

116

C₂

#11718590

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₄³×C₁₃	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	C₄³	D₁₃
hypocenter	C₁₃	#1994
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

subgroup

G/H

center

C₂×C₄

C₂×D₅₂

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₄³×C₁₃

C₂

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₃

C₂×D₄

cosocle

C₂²×C₄×C₁₃

C₂³

hypercenter

C₄³

D₁₃

hypocenter

C₁₃

#1994

LCS

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

N	H ≅ G/N	type
11. #88124	C₄	semidirect
12. #88123	C₄	semidirect
24. C₄²×C₁₃	Q₈	semidirect
86. C₄²	Dic₂₆	semidirect
90. C₁₃	#1994	semidirect

H ≅ G/N

type

11. #88124

C₄

semidirect

12. #88123

C₄

semidirect

24. C₄²×C₁₃

Q₈

semidirect

86. C₄²

Dic₂₆

semidirect

90. C₁₃

#1994

semidirect

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

subgroup	index
#11719175	2
C₄³×C₁₃	2
#11718585	2
#11718115	2
#1994	13

subgroup

index

C₄³×C₁₃

Subgroups — #13115813

Conjugacy classes of subgroups

Complements

Maximal subgroups