Group #13115804 — subgroups

Subgroups — #13115804

Order 1664.

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

normal subgroups only

This group has 260 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	C₂×#88124	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#384	C₂	1	1	yes	maximal, characteristic
3	C₂×#88043	C₂	1	1	yes	maximal, characteristic
4	#11718121	C₂	1	4	yes	maximal
5	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
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×#6101	—	2	2	no	—
10	C₂²×#1055	C₂²	1	2	yes	—
11	#88124	—	2	2	no	—
12	#88124	C₄	1	4	yes	complemented
13	#88043	—	2	2	no	—
14	C₂²×C₄×C₁₃	—	2	2	no	—
15	C₂²×C₄×C₁₃	D₄	1	2	yes	—
16	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₁₃	—	2	2	no	—
18	C₁₃×C₄⋊C₄	—	2	4	no	—
19	C₁₃×C₄⋊C₄	—	4	4	no	—
20	C₄²×C₁₃	C₂×C₄	1	2	yes	—
21	C₄²×C₁₃	D₄	1	2	yes	—
22	C₄⋊Dic₁₃	—	2	2	no	—
23	C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
24	#6101	—	2	8	no	—
25	C₂×#1055	—	2	2	no	—
26	C₂×#1055	—	2	2	no	—
27	C₂×#1055	Q₈	1	4	yes	—
28	#1992	—	13	1	no	maximal, Sylow
29	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₃	—	2	2	no	—
31	C₂×C₄×C₁₃	—	2	2	no	—
32	C₂×C₄×C₁₃	—	4	2	no	—
33	C₂×C₄×C₁₃	—	4	4	no	—
34	C₂×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₁₃	#41	1	2	yes	—
39	C₂×Dic₁₃	—	2	4	no	—
40	C₂×Dic₁₃	—	4	2	no	—
41	C₂×Dic₁₃	—	4	2	no	—
42	#1055	—	2	4	no	—
43	C₂×#125	—	13	1	no	—
44	C₂×#97	—	13	1	no	—
45	#384	D₁₃	1	1	yes	characteristic
46	#340	—	13	4	no	—
47	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₃	#124	1	1	yes	characteristic
49	C₂²×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
51	C₂²×C₁₃	#137	1	2	yes	—
52	C₄×C₁₃	—	2	4	no	—
53	C₄×C₁₃	—	8	2	no	—
54	C₄×C₁₃	C₄×Q₈	1	2	yes	—
55	C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
56	Dic₁₃	—	4	4	no	—
57	C₂²×C₈	—	13	2	no	—
58	#125	—	13	4	no	—
59	#125	—	26	2	no	—
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄⋊C₄	—	13	2	no	—
62	C₂×C₄⋊C₄	—	26	2	no	—
63	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
64	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
65	#97	—	26	2	no	—
66	C₂×C₁₃	#391	1	1	yes	characteristic
67	C₂×C₁₃	#417	1	2	yes	—
68	C₂×C₁₃	#490	1	1	yes	characteristic
69	C₂×C₁₃	#491	1	1	yes	characteristic
70	C₂×C₁₃	#501	1	2	yes	—
71	C₂²×C₄	—	2	2	no	—
72	C₂²×C₄	—	26	2	no	—
73	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
75	C₂×C₈	—	13	4	no	—
76	C₂×C₈	—	26	2	no	—
77	C₂×C₈	—	26	2	no	—
78	C₄⋊C₄	—	2	4	no	—
79	C₄⋊C₄	—	4	4	no	—
80	C₄⋊C₄	—	13	4	no	—
81	C₄⋊C₄	—	26	8	no	—
82	C₄⋊C₄	—	26	2	no	—
83	C₄²	C₄×D₁₃	1	2	yes	—
84	C₄²	D₅₂	1	2	yes	—
85	C₁₃	#1992	1	1	yes	characteristic, complemented, Sylow
86	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	4	4	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	26	4	no	—
92	C₂×C₄	—	52	2	no	—
93	C₂×C₄	—	52	2	no	—
94	C₂×C₄	#6103	1	4	yes	—
95	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
96	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
97	C₂×C₄	#6129	1	2	yes	—
98	C₂×C₄	Q₈×D₁₃	1	2	yes	—
99	C₈	—	26	4	no	—
100	C₂²	C₂×#6103	1	1	yes	characteristic
101	C₂²	#88186	1	2	yes	—
102	C₂²	#88190	1	1	yes	characteristic
103	C₂²	#88196	1	2	yes	—
104	C₂²	#88198	1	1	yes	characteristic
105	C₄	—	2	4	no	—
106	C₄	—	8	2	no	—
107	C₄	—	52	4	no	—
108	C₄	#88141	1	2	yes	—
109	C₄	#88151	1	2	yes	—
110	C₂	#11718627	1	2	yes	—
111	C₂	#11718635	1	1	yes	characteristic
112	C₂	#11718732	1	1	yes	characteristic
113	C₂	#11718741	1	2	yes	—
114	C₂	#11718743	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88124

C₂

yes

maximal, characteristic

C₁₃×#384

C₂

yes

maximal, characteristic

C₂×#88043

C₂

yes

maximal, characteristic

#11718121

C₂

yes

maximal

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

—

C₂²×#1055

C₂²

yes

—

#88124

—

#88124

C₄

yes

complemented

#88043

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×C₄

yes

—

C₄²×C₁₃

D₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

C₂×C₄

yes

—

—

—

—

Q₈

yes

—

#1992

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

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

—

C₂×Dic₁₃

—

#1055

—

C₂×#125

—

C₂×#97

—

#384

D₁₃

yes

characteristic

#340

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃

#124

yes

characteristic

C₂²×C₁₃

C₄⋊Q₈

yes

characteristic

C₂²×C₁₃

AGL(1,ℤ/8)

yes

—

C₂²×C₁₃

#137

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×Q₈

yes

—

C₄×C₁₃

C₂²⋊Q₈

yes

—

Dic₁₃

—

C₂²×C₈

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#97

—

C₂×C₁₃

#391

yes

characteristic

C₂×C₁₃

#417

yes

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₁₃

yes

—

C₄²

D₅₂

yes

—

C₁₃

#1992

yes

characteristic, complemented, Sylow

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#6103

yes

—

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

C₂×C₄

C₂×D₅₂

yes

characteristic

C₂×C₄

#6129

yes

—

C₂×C₄

Q₈×D₁₃

yes

—

C₈

—

100

C₂²

C₂×#6103

yes

characteristic

101

C₂²

#88186

yes

—

102

C₂²

#88190

yes

characteristic

103

C₂²

#88196

yes

—

104

C₂²

#88198

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

#88141

yes

—

109

C₄

#88151

yes

—

110

C₂

#11718627

yes

—

111

C₂

#11718635

yes

characteristic

112

C₂

#11718732

yes

characteristic

113

C₂

#11718741

yes

—

114

C₂

#11718743

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#384	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	#384	D₁₃
hypocenter	C₁₃	#1992
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#384

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₄

cosocle

C₂²×C₄×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
12. #88124	C₄	semidirect
85. C₁₃	#1992	semidirect

H ≅ G/N

type

12. #88124

C₄

semidirect

85. C₁₃

#1992

semidirect

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

subgroup	index
C₂×#88124	2
C₁₃×#384	2
C₂×#88043	2
#11718121	2
#1992	13

subgroup

index

C₁₃×#384

Subgroups — #13115804

Conjugacy classes of subgroups

Complements

Maximal subgroups