Group #13115005 — subgroups

Subgroups — #13115005

Order 1664.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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×#382	C₂	1	1	yes	maximal, characteristic
2	C₂×#88048	C₂	1	2	yes	maximal
3	C₂×#88047	C₂	1	2	yes	maximal
4	#11718596	C₂	1	1	yes	maximal, characteristic
5	C₂×#88043	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₃	—	2	2	no	—
10	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂²×#1055	C₂²	1	2	yes	—
12	C₁₃×#95	—	2	2	no	—
13	#88048	—	2	4	no	—
14	#88047	—	2	4	no	—
15	#88043	C₄	1	4	yes	complemented
16	C₂²×C₄×C₁₃	—	2	1	no	—
17	C₂²×C₄×C₁₃	—	2	1	no	—
18	C₂²×C₄×C₁₃	D₄	1	2	yes	—
19	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₁₃	—	2	2	no	—
21	C₁₃×C₄⋊C₄	—	2	2	no	—
22	C₁₃×C₄⋊C₄	—	2	2	no	—
23	C₄²×C₁₃	C₂×C₄	1	2	yes	—
24	C₄²×C₁₃	Q₈	1	2	yes	—
25	C₄⋊Dic₁₃	—	2	2	no	—
26	C₄⋊Dic₁₃	—	2	2	no	—
27	C₄⋊Dic₁₃	—	4	4	no	—
28	C₂×#1055	C₂×C₄	1	4	yes	—
29	C₂×#1055	D₄	1	4	yes	—
30	C₂×#1055	Q₈	1	4	yes	—
31	#1847	—	13	1	no	maximal, Sylow
32	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₃	—	2	2	no	—
34	C₂×C₄×C₁₃	—	2	2	no	—
35	C₂×C₄×C₁₃	—	4	1	no	—
36	C₂×C₄×C₁₃	—	4	1	no	—
37	C₂×C₄×C₁₃	—	4	2	no	—
38	C₂×C₄×C₁₃	—	4	2	no	—
39	C₂×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
40	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×Dic₁₃	—	4	2	no	—
45	C₂×Dic₁₃	—	4	4	no	—
46	#1055	C₄⋊C₄	1	8	yes	complemented
47	C₂×#107	—	13	2	no	—
48	C₂×#106	—	13	2	no	—
49	C₂×#97	—	13	1	no	—
50	#384	—	13	1	no	—
51	#382	D₁₃	1	1	yes	characteristic
52	C₂²×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₁₃	#127	1	1	yes	characteristic
54	C₂²×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
55	C₂²×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
56	C₂²×C₁₃	#137	1	2	yes	—
57	C₄×C₁₃	—	2	4	no	—
58	C₄×C₁₃	—	8	1	no	—
59	C₄×C₁₃	—	8	1	no	—
60	C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
61	C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
62	Dic₁₃	—	8	2	no	—
63	C₂²×C₈	—	13	2	no	—
64	C₂×C₄⋊C₄	—	13	2	no	—
65	C₂×C₄⋊C₄	—	26	2	no	—
66	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
67	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
68	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
69	#107	—	26	4	no	—
70	#106	—	26	4	no	—
71	#97	—	13	4	no	—
72	#95	—	2	2	no	—
73	C₂×C₁₃	#384	1	1	yes	characteristic
74	C₂×C₁₃	#428	1	2	yes	—
75	C₂×C₁₃	#496	1	1	yes	characteristic
76	C₂×C₁₃	#497	1	1	yes	characteristic
77	C₂×C₁₃	#501	1	2	yes	—
78	C₂²×C₄	—	2	1	no	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	—	26	2	no	—
81	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
82	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
83	C₂×C₈	—	13	4	no	—
84	C₂×C₈	—	13	4	no	—
85	C₂×C₈	—	13	4	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	26	2	no	—
89	C₄⋊C₄	—	26	2	no	—
90	C₄⋊C₄	—	52	4	no	—
91	C₄²	Dic₂₆	1	2	yes	—
92	C₄²	C₄×D₁₃	1	2	yes	—
93	C₁₃	#1847	1	1	yes	characteristic, complemented, Sylow
94	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	52	2	no	—
102	C₂×C₄	—	52	4	no	—
103	C₂×C₄	#6101	1	4	yes	—
104	C₂×C₄	C₂×Dic₂₆	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
106	C₂×C₄	D₄×D₁₃	1	2	yes	—
107	C₂×C₄	Q₈×D₁₃	1	2	yes	—
108	C₈	—	13	8	no	—
109	C₂²	C₂×#6101	1	1	yes	characteristic
110	C₂²	#88194	1	1	yes	characteristic
111	C₂²	#88198	1	1	yes	characteristic
112	C₂²	#88203	1	2	yes	—
113	C₂²	#88205	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	1	no	—
116	C₄	—	8	1	no	—
117	C₄	—	104	2	no	—
118	C₄	#88142	1	2	yes	—
119	C₄	C₄⋊C₄×D₁₃	1	2	yes	—
120	C₂	#11718637	1	1	yes	characteristic
121	C₂	#11718656	1	2	yes	—
122	C₂	#11718719	1	1	yes	characteristic
123	C₂	#11718721	1	1	yes	characteristic
124	C₂	#11718726	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#382

C₂

yes

maximal, characteristic

C₂×#88048

C₂

yes

maximal

C₂×#88047

C₂

yes

maximal

#11718596

C₂

yes

maximal, characteristic

C₂×#88043

C₂

yes

maximal, characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂²×#1055

C₂²

yes

—

C₁₃×#95

—

#88048

—

#88047

—

#88043

C₄

yes

complemented

C₂²×C₄×C₁₃

—

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

Q₈

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

C₂×C₄

yes

—

C₂×#1055

D₄

yes

—

C₂×#1055

Q₈

yes

—

#1847

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

#1055

C₄⋊C₄

yes

complemented

C₂×#107

—

C₂×#106

—

C₂×#97

—

#384

—

#382

D₁₃

yes

characteristic

C₂²×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃

#127

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×C₁₃

C₂×C₄⋊C₄

yes

—

C₄×C₁₃

C₄⋊Q₈

yes

—

Dic₁₃

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

C₂×C₄²

C₂×D₁₃

yes

—

—

—

—

C₂×C₁₃

#384

yes

characteristic

C₂×C₁₃

#428

yes

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂²×C₄

—

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

Dic₂₆

yes

—

C₄²

C₄×D₁₃

yes

—

C₁₃

#1847

yes

characteristic, complemented, Sylow

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#6101

yes

—

104

C₂×C₄

C₂×Dic₂₆

yes

characteristic

105

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

106

C₂×C₄

D₄×D₁₃

yes

—

107

C₂×C₄

Q₈×D₁₃

yes

—

108

C₈

—

109

C₂²

C₂×#6101

yes

characteristic

110

C₂²

#88194

yes

characteristic

111

C₂²

#88198

yes

characteristic

112

C₂²

#88203

yes

—

113

C₂²

#88205

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

104

—

118

C₄

#88142

yes

—

119

C₄

C₄⋊C₄×D₁₃

yes

—

120

C₂

#11718637

yes

characteristic

121

C₂

#11718656

yes

—

122

C₂

#11718719

yes

characteristic

123

C₂

#11718721

yes

characteristic

124

C₂

#11718726

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#382	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	#382	D₁₃
hypocenter	C₁₃	#1847
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#382

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 ◁ #382

N	H ≅ G/N	type
15. #88043	C₄	semidirect
46. #1055	C₄⋊C₄	semidirect
93. C₁₃	#1847	semidirect

H ≅ G/N

type

15. #88043

C₄

semidirect

46. #1055

C₄⋊C₄

semidirect

93. C₁₃

#1847

semidirect

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

subgroup	index
C₁₃×#382	2
C₂×#88048	2
C₂×#88047	2
#11718596	2
C₂×#88043	2
#1847	13

subgroup

index

C₁₃×#382

Subgroups — #13115005

Conjugacy classes of subgroups

Complements

Maximal subgroups