Group #15756728 — subgroups

Subgroups — #15756728

Order 3328.

338 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 163 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	#13113591	C₂	1	4	yes	maximal
3	C₂×#11718047	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#108	—	2	2	no	—
6	C₁₃×#423	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88059	—	2	2	no	—
9	#11718585	C₂²	1	2	yes	—
10	C₂×C₄×#1055	C₂²	1	2	yes	—
11	#11718072	—	2	8	no	—
12	#11718047	C₄	1	4	yes	—
13	C₂×C₁₃×M₁₆	—	2	2	no	—
14	C₂²×C₈×C₁₃	D₄	1	2	yes	—
15	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
16	C₂×#6099	—	2	2	no	—
17	C₂²×#1055	Q₈	1	4	yes	—
18	C₁₃×#108	—	4	4	no	—
19	C₁₃×C₄⋊C₈	—	2	4	no	—
20	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₁₃	Q₈	1	2	yes	—
22	#88059	—	4	4	no	—
23	C₄⋊#1055	—	2	4	no	—
24	C₄×#1055	C₂×C₄	1	4	yes	—
25	C₄×#1055	D₄	1	4	yes	—
26	#8825	—	13	1	no	maximal, Sylow
27	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
29	C₁₃×M₁₆	—	4	4	no	—
30	C₂×C₈×C₁₃	—	4	2	no	—
31	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
32	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
33	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
34	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
35	C₄²×C₁₃	C₂×D₄	1	2	yes	—
36	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
37	#6099	—	4	4	no	—
38	C₂×#1055	—	4	4	no	—
39	C₂×#1055	—	4	2	no	—
40	C₂×#1055	C₄⋊C₄	1	8	yes	—
41	#1848	—	13	1	no	—
42	#1848	D₁₃	1	1	yes	characteristic
43	#1566	—	13	4	no	—
44	C₂×#322	—	13	1	no	—
45	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₃	—	2	4	no	—
47	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
50	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
51	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
52	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
53	C₈×C₁₃	—	8	2	no	—
54	C₈×C₁₃	#106	1	8	yes	—
55	#1055	—	2	8	no	—
56	#1055	—	8	2	no	—
57	C₂×#108	—	2	2	no	—
58	C₂×#108	—	26	2	no	—
59	#423	—	13	2	no	—
60	#423	C₂×D₁₃	1	2	yes	—
61	C₂×C₄×C₈	—	13	2	no	—
62	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
63	#334	—	26	8	no	—
64	#322	—	13	4	no	—
65	C₂²×C₁₃	—	2	2	no	—
66	C₂²×C₁₃	#384	1	1	yes	characteristic
67	C₂²×C₁₃	#498	1	2	yes	—
68	C₄×C₁₃	—	2	2	no	—
69	C₄×C₁₃	C₂×#106	1	2	yes	—
70	C₄×C₁₃	#430	1	4	yes	—
71	C₄×C₁₃	#492	1	1	yes	characteristic
72	C₄×C₁₃	#492	1	1	yes	characteristic
73	C₂×M₁₆	—	2	2	no	—
74	C₂×M₁₆	—	26	2	no	—
75	C₂²×C₈	—	13	4	no	—
76	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
77	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
78	#108	—	4	4	no	—
79	#108	—	52	4	no	—
80	C₄⋊C₈	—	2	4	no	—
81	C₄⋊C₈	—	26	4	no	—
82	C₄×C₈	—	13	4	no	—
83	C₄×C₈	—	13	4	no	—
84	C₄×C₈	Dic₂₆	1	2	yes	—
85	C₄×C₈	C₄×D₁₃	1	2	yes	—
86	C₂×C₁₃	—	2	2	no	—
87	C₂×C₁₃	#1844	1	1	yes	characteristic
88	C₂×C₁₃	#1849	1	2	yes	—
89	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
90	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
91	M₁₆	—	4	4	no	—
92	M₁₆	—	52	4	no	—
93	C₂×C₈	—	4	2	no	—
94	C₂×C₈	—	13	8	no	—
95	C₂×C₈	—	52	4	no	—
96	C₂×C₈	—	52	2	no	—
97	C₂×C₈	#6101	1	4	yes	—
98	C₂×C₈	#6105	1	4	yes	—
99	C₂×C₈	#6106	1	4	yes	—
100	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
101	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
102	C₄²	D₄×D₁₃	1	2	yes	—
103	C₁₃	#8825	1	1	yes	characteristic, complemented, Sylow
104	C₂³	#88198	1	1	yes	characteristic
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	#88142	1	2	yes	—
107	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
108	C₂×C₄	C₂×#6101	1	1	yes	characteristic
109	C₂×C₄	C₂×#6105	1	2	yes	—
110	C₂×C₄	#88194	1	1	yes	characteristic
111	C₂×C₄	C₂×#6106	1	2	yes	—
112	C₈	—	8	2	no	—
113	C₈	—	26	8	no	—
114	C₈	—	104	2	no	—
115	C₈	#88048	1	8	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	#11718637	1	1	yes	characteristic
118	C₂²	#11718740	1	2	yes	—
119	C₄	—	2	2	no	—
120	C₄	#11718554	1	4	yes	—
121	C₄	C₂×#88048	1	2	yes	—
122	C₄	#11718737	1	1	yes	characteristic
123	C₄	#11718746	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#13114992	1	1	yes	characteristic
126	C₂	#13115009	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13115008

C₂

yes

maximal, characteristic

#13113591

C₂

yes

maximal

C₂×#11718047

C₂

yes

maximal, characteristic

C₁₃×#1848

C₂

yes

maximal, characteristic

C₂×C₁₃×#108

—

C₁₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88059

—

#11718585

C₂²

yes

—

C₂×C₄×#1055

C₂²

yes

—

#11718072

—

#11718047

C₄

yes

—

C₂×C₁₃×M₁₆

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×#6099

—

C₂²×#1055

Q₈

yes

—

C₁₃×#108

—

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

#88059

—

C₄⋊#1055

—

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

D₄

yes

—

#8825

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×Q₈

yes

—

—

—

C₄⋊C₄

yes

—

#1848

—

#1848

D₁₃

yes

characteristic

#1566

—

C₂×#322

—

C₂³×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₈×C₁₃

—

C₈×C₁₃

#106

yes

—

—

—

—

—

—

C₂×D₁₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

#334

—

#322

—

C₂²×C₁₃

—

C₂²×C₁₃

#384

yes

characteristic

C₂²×C₁₃

#498

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₂×#106

yes

—

C₄×C₁₃

#430

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₁₃

yes

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₂₆

yes

—

C₄×C₈

C₄×D₁₃

yes

—

C₂×C₁₃

—

C₂×C₁₃

#1844

yes

characteristic

C₂×C₁₃

#1849

yes

—

C₂²×C₄

Q₈×D₁₃

yes

—

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#6101

yes

—

C₂×C₈

#6105

yes

—

C₂×C₈

#6106

yes

—

100

C₄²

C₂×Dic₂₆

yes

characteristic

101

C₄²

C₂×C₄×D₁₃

yes

characteristic

102

C₄²

D₄×D₁₃

yes

—

103

C₁₃

#8825

yes

characteristic, complemented, Sylow

104

C₂³

#88198

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

#88142

yes

—

107

C₂×C₄

C₄⋊C₄×D₁₃

yes

—

108

C₂×C₄

C₂×#6101

yes

characteristic

109

C₂×C₄

C₂×#6105

yes

—

110

C₂×C₄

#88194

yes

characteristic

111

C₂×C₄

C₂×#6106

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

104

—

115

C₈

#88048

yes

—

116

C₂²

—

117

C₂²

#11718637

yes

characteristic

118

C₂²

#11718740

yes

—

119

C₄

—

120

C₄

#11718554

yes

—

121

C₄

C₂×#88048

yes

—

122

C₄

#11718737

yes

characteristic

123

C₄

#11718746

yes

characteristic

124

C₂

—

125

C₂

#13114992

yes

characteristic

126

C₂

#13115009

yes

—

	subgroup	G/H
center	C₂×C₄	#88194
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1848	C₂
socle	C₂²×C₁₃	#384
prime radical	C₂³×C₁₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1848	D₁₃
hypocenter	C₁₃	#8825
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂×C₄

#88194

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1848

C₂

socle

C₂²×C₁₃

#384

prime radical

C₂³×C₁₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
103. C₁₃	#8825	semidirect

H ≅ G/N

type

103. C₁₃

#8825

semidirect

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

subgroup	index
#13115008	2
#13113591	2
C₂×#11718047	2
C₁₃×#1848	2
#8825	13

subgroup

index

C₁₃×#1848

Subgroups — #15756728

Conjugacy classes of subgroups

Complements

Maximal subgroups