Group #15756616 — subgroups

Subgroups — #15756616

Order 3328.

270 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 107 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	#13115008	C₂	1	1	yes	maximal, characteristic
2	#13113128	C₂	1	1	yes	maximal, characteristic
3	#13112694	C₂	1	4	yes	maximal
4	C₁₃×#1835	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₁₃×#382	C₂²	1	2	yes	—
7	C₂×#88059	—	2	2	no	—
8	#11718586	C₂²	1	2	yes	—
9	#11718585	C₂²	1	2	yes	—
10	C₁₃×#340	—	2	2	no	—
11	#11718046	C₄	1	4	yes	complemented
12	C₂²×C₈×C₁₃	D₄	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
14	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
15	C₂×#6099	—	2	2	no	—
16	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
17	C₄×C₈×C₁₃	Q₈	1	2	yes	—
18	C₁₃×#95	—	2	4	no	—
19	C₂²⋊#1055	—	2	2	no	—
20	#88059	—	4	4	no	—
21	C₄⋊#1055	—	2	2	no	—
22	C₄⋊#1055	—	2	4	no	—
23	C₄×#1055	C₂×C₄	1	4	yes	—
24	#8794	—	13	1	no	maximal, Sylow
25	C₂²×C₄×C₁₃	—	4	2	no	—
26	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₃	#41	1	2	yes	—
28	C₂×C₈×C₁₃	—	2	2	no	—
29	C₂×C₈×C₁₃	—	2	2	no	—
30	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
31	C₁₃×C₄⋊C₄	—	4	2	no	—
32	C₁₃×C₄⋊C₄	—	4	2	no	—
33	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
35	C₄²×C₁₃	#41	1	2	yes	—
36	#6099	—	4	4	no	—
37	C₂×#1055	—	2	4	no	—
38	C₂×#1055	—	4	2	no	—
39	#1848	—	13	1	no	—
40	#1835	D₁₃	1	1	yes	characteristic
41	#1452	—	13	1	no	—
42	#1286	—	13	4	no	—
43	C₂³×C₁₃	#124	1	1	yes	characteristic
44	C₂×C₄×C₁₃	—	4	2	no	—
45	C₂×C₄×C₁₃	—	8	4	no	—
46	C₂×C₄×C₁₃	—	8	2	no	—
47	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₃	#117	1	2	yes	—
49	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
50	C₂×C₄×C₁₃	#125	1	2	yes	—
51	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
52	C₂×C₄×C₁₃	#137	1	2	yes	—
53	C₈×C₁₃	—	2	4	no	—
54	#1055	—	4	4	no	—
55	#1055	—	8	2	no	—
56	#431	—	13	2	no	—
57	C₂×#108	—	26	2	no	—
58	#423	—	13	2	no	—
59	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
60	#382	C₂×D₁₃	1	2	yes	—
61	#340	—	2	2	no	—
62	#322	—	13	4	no	—
63	C₂²×C₁₃	—	2	2	no	—
64	C₂²×C₁₃	#382	1	1	yes	characteristic
65	C₂²×C₁₃	#489	1	2	yes	—
66	C₄×C₁₃	—	2	2	no	—
67	C₄×C₁₃	—	2	2	no	—
68	C₄×C₁₃	—	16	2	no	—
69	C₄×C₁₃	#428	1	2	yes	—
70	C₄×C₁₃	#490	1	1	yes	characteristic
71	C₄×C₁₃	#491	1	1	yes	characteristic
72	C₂×M₁₆	—	26	2	no	—
73	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
74	C₂×C₄⋊C₄	—	2	2	no	—
75	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
76	#108	—	52	4	no	—
77	C₄⋊C₈	—	26	2	no	—
78	C₄⋊C₈	—	26	4	no	—
79	C₂²⋊C₈	—	26	2	no	—
80	C₄×C₈	—	13	4	no	—
81	C₄×C₈	Dic₂₆	1	2	yes	—
82	C₄×C₈	C₄×D₁₃	1	2	yes	—
83	#95	—	2	4	no	—
84	C₂×C₁₃	—	2	2	no	—
85	C₂×C₁₃	#1833	1	2	yes	—
86	C₂×C₁₃	#1836	1	1	yes	characteristic
87	C₂²×C₄	—	4	2	no	—
88	C₂²×C₄	#6129	1	2	yes	—
89	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
90	M₁₆	—	52	4	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	26	4	no	—
94	C₂×C₈	—	52	2	no	—
95	C₂×C₈	#6101	1	4	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
99	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
100	C₄²	#6131	1	2	yes	—
101	C₁₃	#8794	1	1	yes	characteristic, complemented, Sylow
102	C₂³	#88190	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	8	4	no	—
105	C₂×C₄	—	8	2	no	—
106	C₂×C₄	#88144	1	2	yes	—
107	C₂×C₄	#88146	1	2	yes	—
108	C₂×C₄	C₂×#6101	1	1	yes	characteristic
109	C₂×C₄	#88201	1	1	yes	characteristic
110	C₂×C₄	#88203	1	2	yes	—
111	C₂×C₄	#88205	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	—	52	4	no	—
114	C₈	—	104	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	#11718638	1	1	yes	characteristic
117	C₂²	#11718716	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	16	2	no	—
121	C₄	#11718656	1	2	yes	—
122	C₄	#11718728	1	1	yes	characteristic
123	C₄	#11718730	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#13114959	1	2	yes	—
126	C₂	#13114969	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13115008

C₂

yes

maximal, characteristic

#13113128

C₂

yes

maximal, characteristic

#13112694

C₂

yes

maximal

C₁₃×#1835

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×#382

C₂²

yes

—

C₂×#88059

—

#11718586

C₂²

yes

—

#11718585

C₂²

yes

—

C₁₃×#340

—

#11718046

C₄

yes

complemented

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×#6099

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

C₁₃×#95

—

C₂²⋊#1055

—

—

—

—

C₂×C₄

yes

—

#8794

—

maximal, Sylow

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

characteristic

C₄²×C₁₃

#41

yes

—

—

—

—

—

D₁₃

yes

characteristic

#1452

—

#1286

—

C₂³×C₁₃

#124

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

#124

yes

characteristic

C₂×C₄×C₁₃

#125

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₈×C₁₃

—

—

—

—

—

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

#382

C₂×D₁₃

yes

—

#340

—

#322

—

C₂²×C₁₃

—

C₂²×C₁₃

#382

yes

characteristic

C₂²×C₁₃

#489

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#428

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₁₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

Dic₂₆

yes

—

C₄×C₈

C₄×D₁₃

yes

—

#95

—

C₂×C₁₃

—

C₂×C₁₃

#1833

yes

—

C₂×C₁₃

#1836

yes

characteristic

C₂²×C₄

—

C₂²×C₄

#6129

yes

—

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#6101

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×Dic₂₆

yes

characteristic

C₄²

C₂×C₄×D₁₃

yes

characteristic

100

C₄²

#6131

yes

—

101

C₁₃

#8794

yes

characteristic, complemented, Sylow

102

C₂³

#88190

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

#88144

yes

—

107

C₂×C₄

#88146

yes

—

108

C₂×C₄

C₂×#6101

yes

characteristic

109

C₂×C₄

#88201

yes

characteristic

110

C₂×C₄

#88203

yes

—

111

C₂×C₄

#88205

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

104

—

115

C₂²

—

116

C₂²

#11718638

yes

characteristic

117

C₂²

#11718716

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#11718656

yes

—

122

C₄

#11718728

yes

characteristic

123

C₄

#11718730

yes

characteristic

124

C₂

—

125

C₂

#13114959

yes

—

126

C₂

#13114969

yes

characteristic

	subgroup	G/H
center	C₂²	#11718638
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1835	C₂
socle	C₂²×C₁₃	#382
prime radical	C₂³×C₁₃	#124
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1835	D₁₃
hypocenter	C₁₃	#8794
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

subgroup

G/H

center

C₂²

#11718638

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1835

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

N	H ≅ G/N	type
11. #11718046	C₄	semidirect
101. C₁₃	#8794	semidirect

H ≅ G/N

type

11. #11718046

C₄

semidirect

101. C₁₃

#8794

semidirect

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

subgroup	index
#13115008	2
#13113128	2
#13112694	2
C₁₃×#1835	2
#8794	13

subgroup

index

C₁₃×#1835

Subgroups — #15756616

Conjugacy classes of subgroups

Complements

Maximal subgroups