Group #15756730 — subgroups

Subgroups — #15756730

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	#13113592	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	#11718049	—	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	#8825	—	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	SD₁₆	1	8	yes	—
27	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
28	C₄²×C₁₃	C₂×D₄	1	2	yes	—
29	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
30	#6099	—	4	8	no	—
31	C₂×#1055	—	4	4	no	—
32	#1848	—	13	2	no	—
33	#1566	—	13	4	no	—
34	C₂×#322	D₁₃	1	1	yes	characteristic
35	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
36	C₂×C₄×C₁₃	—	2	4	no	—
37	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
40	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
41	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
42	C₈×C₁₃	—	2	8	no	—
43	C₈×C₁₃	#106	1	8	yes	—
44	#1055	—	8	4	no	—
45	C₂×#108	—	26	4	no	—
46	#423	—	13	4	no	—
47	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
49	#334	—	26	8	no	—
50	#322	Dic₁₃	1	4	yes	—
51	C₂²×C₁₃	—	2	2	no	—
52	C₂²×C₁₃	#384	1	1	yes	characteristic
53	C₂²×C₁₃	#498	1	2	yes	—
54	C₄×C₁₃	—	2	2	no	—
55	C₄×C₁₃	C₂×#106	1	2	yes	—
56	C₄×C₁₃	#430	1	4	yes	—
57	C₄×C₁₃	#492	1	2	yes	—
58	C₂×M₁₆	—	26	4	no	—
59	C₂²×C₈	Dic₂₆	1	4	yes	—
60	C₂²×C₈	D₅₂	1	2	yes	—
61	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
62	#108	—	52	8	no	—
63	C₄⋊C₈	—	26	8	no	—
64	C₄×C₈	Dic₂₆	1	2	yes	—
65	C₄×C₈	D₅₂	1	4	yes	—
66	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
67	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
68	C₂×C₁₃	—	2	2	no	—
69	C₂×C₁₃	#1844	1	1	yes	characteristic
70	C₂×C₁₃	#1849	1	2	yes	—
71	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
72	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
73	M₁₆	—	52	8	no	—
74	C₂×C₈	—	4	4	no	—
75	C₂×C₈	—	52	4	no	—
76	C₂×C₈	#6095	1	8	yes	—
77	C₂×C₈	C₄⋊Dic₁₃	1	8	yes	—
78	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
79	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
80	C₄²	C₂×D₅₂	1	2	yes	—
81	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
82	C₁₃	#8825	1	1	yes	characteristic, complemented, Sylow
83	C₂³	#88123	1	1	yes	characteristic
84	C₂×C₄	—	2	4	no	—
85	C₂×C₄	#88123	1	2	yes	—
86	C₂×C₄	#88128	1	1	yes	characteristic
87	C₂×C₄	C₂×#6095	1	4	yes	—
88	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
89	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
90	C₈	—	2	8	no	—
91	C₈	—	104	4	no	—
92	C₈	#88057	1	8	yes	—
93	C₂²	—	2	2	no	—
94	C₂²	#11718341	1	2	yes	—
95	C₂²	#11718596	1	1	yes	characteristic
96	C₄	—	2	2	no	—
97	C₄	#11718354	1	2	yes	—
98	C₄	C₂×#88057	1	2	yes	—
99	C₄	#11718786	1	4	yes	—
100	C₂	—	2	2	no	—
101	C₂	#13114993	1	1	yes	characteristic
102	C₂	#13115011	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13115008

C₂

yes

maximal

#13113592

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

—

#11718049

—

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

SD₁₆

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×SD₁₆

yes

—

C₈×C₁₃

—

C₈×C₁₃

#106

yes

—

#1055

—

C₂×#108

—

#423

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

C₂×C₄×C₈

C₂×D₁₃

yes

—

#334

—

#322

Dic₁₃

yes

—

C₂²×C₁₃

—

C₂²×C₁₃

#384

yes

characteristic

C₂²×C₁₃

#498

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₂×#106

yes

—

C₄×C₁₃

#430

yes

—

C₄×C₁₃

#492

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

#1844

yes

characteristic

C₂×C₁₃

#1849

yes

—

C₂²×C₄

C₂×Dic₂₆

yes

—

C₂²×C₄

C₂×D₅₂

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#6095

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

#8825

yes

characteristic, complemented, Sylow

C₂³

#88123

yes

characteristic

C₂×C₄

—

C₂×C₄

#88123

yes

—

C₂×C₄

#88128

yes

characteristic

C₂×C₄

C₂×#6095

yes

—

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

—

C₈

—

C₈

—

104

—

C₈

#88057

yes

—

C₂²

—

C₂²

#11718341

yes

—

C₂²

#11718596

yes

characteristic

C₄

—

C₄

#11718354

yes

—

C₄

C₂×#88057

yes

—

C₄

#11718786

yes

—

100

C₂

—

101

C₂

#13114993

yes

characteristic

102

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	#8825
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
82. C₁₃	#8825	semidirect

H ≅ G/N

type

82. C₁₃

#8825

semidirect

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

subgroup	index
#13115008	2
#13113592	2
C₂×C₁₃×#322	2
#8825	13

subgroup

index

#13115008

#13113592

C₂×C₁₃×#322

#8825

Subgroups — #15756730

Conjugacy classes of subgroups

Complements

Maximal subgroups