Group #15756733 — subgroups

Subgroups — #15756733

Order 3328.

338 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 187 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	2	yes	maximal
2	#13113595	C₂	1	4	yes	maximal
3	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
5	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×#88059	—	2	4	no	—
7	#11718585	C₂²	1	4	yes	—
8	C₁₃×#322	C₄	1	4	yes	—
9	#11718050	—	2	8	no	—
10	C₂²×C₈×C₁₃	D₄	1	2	yes	—
11	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
12	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
13	C₂×#6099	—	2	4	no	—
14	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
15	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
16	C₄×C₈×C₁₃	D₄	1	4	yes	—
17	C₄×C₈×C₁₃	Q₈	1	2	yes	—
18	#88059	—	4	8	no	—
19	C₄⋊#1055	—	2	8	no	—
20	#8826	—	13	1	no	maximal, Sylow
21	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
23	C₂×C₈×C₁₃	—	4	4	no	—
24	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
25	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
26	C₂×C₈×C₁₃	D₈	1	4	yes	—
27	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
28	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
29	C₄²×C₁₃	C₂×D₄	1	2	yes	—
30	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
31	#6099	—	4	8	no	—
32	C₂×#1055	—	4	4	no	—
33	#1848	—	13	2	no	—
34	#1567	—	13	4	no	—
35	C₂×#322	D₁₃	1	1	yes	characteristic
36	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
37	C₂×C₄×C₁₃	—	2	4	no	—
38	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
41	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
42	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
43	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
44	C₈×C₁₃	—	2	8	no	—
45	C₈×C₁₃	#107	1	8	yes	—
46	#1055	—	8	4	no	—
47	C₂×#108	—	26	4	no	—
48	#423	—	13	4	no	—
49	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
51	#335	—	26	8	no	—
52	#322	Dic₁₃	1	4	yes	—
53	C₂²×C₁₃	—	2	2	no	—
54	C₂²×C₁₃	#384	1	1	yes	characteristic
55	C₂²×C₁₃	#500	1	2	yes	—
56	C₄×C₁₃	—	2	2	no	—
57	C₄×C₁₃	C₂×#107	1	2	yes	—
58	C₄×C₁₃	#430	1	4	yes	—
59	C₄×C₁₃	#493	1	1	yes	characteristic
60	C₄×C₁₃	#494	1	1	yes	characteristic
61	C₂×M₁₆	—	26	4	no	—
62	C₂²×C₈	Dic₂₆	1	4	yes	—
63	C₂²×C₈	D₅₂	1	2	yes	—
64	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
65	#108	—	52	8	no	—
66	C₄⋊C₈	—	26	8	no	—
67	C₄×C₈	Dic₂₆	1	2	yes	—
68	C₄×C₈	D₅₂	1	4	yes	—
69	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
70	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
71	C₂×C₁₃	—	2	2	no	—
72	C₂×C₁₃	#1845	1	1	yes	characteristic
73	C₂×C₁₃	#1849	1	2	yes	—
74	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
75	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
76	M₁₆	—	52	8	no	—
77	C₂×C₈	—	4	4	no	—
78	C₂×C₈	—	52	4	no	—
79	C₂×C₈	D₁₀₄	1	4	yes	—
80	C₂×C₈	Dic₅₂	1	4	yes	—
81	C₂×C₈	C₄⋊Dic₁₃	1	8	yes	—
82	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
83	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
84	C₄²	C₂×D₅₂	1	2	yes	—
85	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
86	C₁₃	#8826	1	1	yes	characteristic, complemented, Sylow
87	C₂³	#88123	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	#88123	1	2	yes	—
90	C₂×C₄	#88128	1	1	yes	characteristic
91	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
92	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
93	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
94	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
95	C₈	—	2	8	no	—
96	C₈	—	104	4	no	—
97	C₈	#88058	1	8	yes	—
98	C₂²	—	2	2	no	—
99	C₂²	#11718343	1	2	yes	—
100	C₂²	#11718596	1	1	yes	characteristic
101	C₄	—	2	2	no	—
102	C₄	#11718356	1	1	yes	characteristic
103	C₄	#11718360	1	1	yes	characteristic
104	C₄	C₂×#88058	1	2	yes	—
105	C₄	#11718786	1	4	yes	—
106	C₂	—	2	2	no	—
107	C₂	#13114996	1	1	yes	characteristic
108	C₂	#13115011	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13115008

C₂

yes

maximal

#13113595

C₂

yes

maximal

C₂×C₁₃×#322

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88059

—

#11718585

C₂²

yes

—

C₁₃×#322

C₄

yes

—

#11718050

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂²×C₈×C₁₃

Q₈

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×#6099

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

—

—

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

Q₁₆

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×Q₈

yes

—

—

—

—

D₁₃

yes

characteristic

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×D₈

yes

—

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₈×C₁₃

—

C₈×C₁₃

#107

yes

—

#1055

—

C₂×#108

—

#423

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

C₂×C₄×C₈

C₂×D₁₃

yes

—

#335

—

#322

Dic₁₃

yes

—

C₂²×C₁₃

—

C₂²×C₁₃

#384

yes

characteristic

C₂²×C₁₃

#500

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₂×#107

yes

—

C₄×C₁₃

#430

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₂×M₁₆

—

C₂²×C₈

Dic₂₆

yes

—

C₂²×C₈

D₅₂

yes

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

#108

—

C₄⋊C₈

—

C₄×C₈

Dic₂₆

yes

—

C₄×C₈

D₅₂

yes

—

C₄×C₈

C₂×Dic₁₃

yes

—

C₄×C₈

C₂×Dic₁₃

yes

—

C₂×C₁₃

—

C₂×C₁₃

#1845

yes

characteristic

C₂×C₁₃

#1849

yes

—

C₂²×C₄

C₂×Dic₂₆

yes

—

C₂²×C₄

C₂×D₅₂

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₀₄

yes

—

C₂×C₈

Dic₅₂

yes

—

C₂×C₈

C₄⋊Dic₁₃

yes

—

C₂×C₈

C₄⋊Dic₁₃

yes

—

C₄²

C₂×Dic₂₆

yes

characteristic

C₄²

C₂×D₅₂

yes

—

C₄²

C₂²×Dic₁₃

yes

characteristic

C₁₃

#8826

yes

characteristic, complemented, Sylow

C₂³

#88123

yes

characteristic

C₂×C₄

—

C₂×C₄

#88123

yes

—

C₂×C₄

#88128

yes

characteristic

C₂×C₄

C₂×D₁₀₄

yes

—

C₂×C₄

C₂×Dic₅₂

yes

—

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

—

C₈

—

C₈

—

104

—

C₈

#88058

yes

—

C₂²

—

C₂²

#11718343

yes

—

100

C₂²

#11718596

yes

characteristic

101

C₄

—

102

C₄

#11718356

yes

characteristic

103

C₄

#11718360

yes

characteristic

104

C₄

C₂×#88058

yes

—

105

C₄

#11718786

yes

—

106

C₂

—

107

C₂

#13114996

yes

characteristic

108

C₂

#13115011

yes

—

	subgroup	G/H
center	C₂×C₄	#88128
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#322	C₂
socle	C₂²×C₁₃	#384
prime radical	C₂³×C₁₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#322	D₁₃
hypocenter	C₁₃	#8826
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#88128

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₂×C₁₃×#322

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 ◁ C₂×#322

N	H ≅ G/N	type
86. C₁₃	#8826	semidirect

H ≅ G/N

type

86. C₁₃

#8826

semidirect

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

subgroup	index
#13115008	2
#13113595	2
C₂×C₁₃×#322	2
#8826	13

subgroup

index

#13115008

#13113595

C₂×C₁₃×#322

#8826

Subgroups — #15756733

Conjugacy classes of subgroups

Complements

Maximal subgroups