Group #15756593 — subgroups

Subgroups — #15756593

Order 3328.

322 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 147 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	#13114953	C₂	1	2	yes	maximal
2	#13112671	C₂	1	4	yes	maximal
3	C₁₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
5	C₄³×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×C₄⋊#1055	C₂²	1	4	yes	—
7	C₁₃×#336	C₄	1	4	yes	—
8	#11718116	—	2	4	no	—
9	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
10	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
11	C₂×C₄²×C₁₃	D₄	1	2	yes	—
12	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
13	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
14	C₂²×#1055	—	2	4	no	—
15	C₄×C₈×C₁₃	—	2	4	no	—
16	C₄⋊#1055	—	2	8	no	—
17	#8788	—	13	1	no	maximal, Sylow
18	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
19	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
22	C₂×C₈×C₁₃	—	2	4	no	—
23	C₂×C₈×C₁₃	—	2	8	no	—
24	C₄²×C₁₃	—	2	4	no	—
25	C₄²×C₁₃	—	2	4	no	—
26	C₄²×C₁₃	—	4	2	no	—
27	C₄²×C₁₃	#41	1	4	yes	—
28	C₂×#1055	—	4	4	no	—
29	C₂×#1055	—	4	8	no	—
30	#1831	—	13	2	no	—
31	#1744	D₁₃	1	1	yes	characteristic
32	#1277	—	13	4	no	—
33	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₃	—	2	2	no	—
35	C₂×C₄×C₁₃	—	2	4	no	—
36	C₂×C₄×C₁₃	—	2	4	no	—
37	C₂×C₄×C₁₃	—	2	4	no	—
38	C₂×C₄×C₁₃	#117	1	2	yes	—
39	C₂×C₄×C₁₃	#124	1	2	yes	—
40	C₂×C₄×C₁₃	#125	1	2	yes	—
41	C₂×C₄×C₁₃	#131	1	4	yes	—
42	C₂×C₄×C₁₃	#135	1	4	yes	—
43	C₈×C₁₃	—	4	4	no	—
44	#1055	—	8	4	no	—
45	C₂×C₄⋊C₈	—	13	4	no	—
46	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
47	C₄³	C₂×D₁₃	1	1	yes	characteristic
48	#336	—	26	4	no	—
49	#336	Dic₁₃	1	4	yes	—
50	C₂²×C₁₃	#382	1	1	yes	characteristic
51	C₂²×C₁₃	#424	1	2	yes	—
52	C₂²×C₁₃	#427	1	2	yes	—
53	C₂²×C₁₃	#430	1	2	yes	—
54	C₄×C₁₃	—	2	2	no	—
55	C₄×C₁₃	—	4	4	no	—
56	C₄×C₁₃	#433	1	4	yes	—
57	C₄×C₁₃	#488	1	4	yes	—
58	C₂²×C₈	—	26	4	no	—
59	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
60	C₂×C₄²	Dic₂₆	1	2	yes	—
61	C₂×C₄²	D₅₂	1	2	yes	—
62	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
63	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
64	C₄⋊C₈	—	26	8	no	—
65	C₄×C₈	—	2	4	no	—
66	C₂×C₁₃	#1572	1	4	yes	—
67	C₂×C₁₃	#1832	1	1	yes	characteristic
68	C₂×C₁₃	#1835	1	1	yes	characteristic
69	C₂×C₁₃	#1838	1	1	yes	characteristic
70	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
71	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
72	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
73	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
74	C₂×C₈	—	2	4	no	—
75	C₂×C₈	—	2	8	no	—
76	C₂×C₈	—	52	4	no	—
77	C₂×C₈	—	52	8	no	—
78	C₄²	—	2	4	no	—
79	C₄²	—	2	4	no	—
80	C₄²	—	4	2	no	—
81	C₄²	#6127	1	4	yes	—
82	C₁₃	#8788	1	1	yes	characteristic, complemented, Sylow
83	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	#88124	1	2	yes	—
89	C₂×C₄	#88129	1	2	yes	—
90	C₂×C₄	#88158	1	4	yes	—
91	C₂×C₄	#88180	1	2	yes	—
92	C₂×C₄	#88202	1	4	yes	—
93	C₈	—	4	4	no	—
94	C₈	—	104	4	no	—
95	C₂²	#11718597	1	1	yes	characteristic
96	C₂²	#11718660	1	2	yes	—
97	C₂²	#11718767	1	2	yes	—
98	C₂²	#11718786	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	4	4	no	—
101	C₄	#11718358	1	4	yes	—
102	C₄	#11718661	1	4	yes	—
103	C₂	#13113616	1	4	yes	—
104	C₂	#13114956	1	1	yes	characteristic
105	C₂	#13114967	1	1	yes	characteristic
106	C₂	#13114977	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114953

C₂

yes

maximal

#13112671

C₂

yes

maximal

C₁₃×#1744

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

—

C₄³×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊#1055

C₂²

yes

—

C₁₃×#336

C₄

yes

—

#11718116

—

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

Q₈

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂²×#1055

—

C₄×C₈×C₁₃

—

C₄⋊#1055

—

#8788

—

maximal, Sylow

C₂²×C₄×C₁₃

C₄⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

#41

yes

—

—

—

—

D₁₃

yes

characteristic

#1277

—

C₂³×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

#124

yes

—

C₂×C₄×C₁₃

#125

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₂×C₄×C₁₃

#135

yes

—

C₈×C₁₃

—

#1055

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

C₄³

C₂×D₁₃

yes

characteristic

#336

—

#336

Dic₁₃

yes

—

C₂²×C₁₃

#382

yes

characteristic

C₂²×C₁₃

#424

yes

—

C₂²×C₁₃

#427

yes

—

C₂²×C₁₃

#430

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#433

yes

—

C₄×C₁₃

#488

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₁₃

yes

—

C₂×C₄²

Dic₂₆

yes

—

C₂×C₄²

D₅₂

yes

—

C₂×C₄²

C₂×Dic₁₃

yes

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄×C₈

—

C₂×C₁₃

#1572

yes

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂²×C₄

C₄⋊Dic₁₃

yes

—

C₂²×C₄

C₂×Dic₂₆

yes

characteristic

C₂²×C₄

C₂×D₅₂

yes

characteristic

C₂²×C₄

C₂²×Dic₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#6127

yes

—

C₁₃

#8788

yes

characteristic, complemented, Sylow

C₂³

C₂×C₄⋊Dic₁₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#88124

yes

—

C₂×C₄

#88129

yes

—

C₂×C₄

#88158

yes

—

C₂×C₄

#88180

yes

—

C₂×C₄

#88202

yes

—

C₈

—

C₈

—

104

—

C₂²

#11718597

yes

characteristic

C₂²

#11718660

yes

—

C₂²

#11718767

yes

—

C₂²

#11718786

yes

—

C₄

—

100

C₄

—

101

C₄

#11718358

yes

—

102

C₄

#11718661

yes

—

103

C₂

#13113616

yes

—

104

C₂

#13114956

yes

characteristic

105

C₂

#13114967

yes

characteristic

106

C₂

#13114977

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1744	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1744	D₁₃
hypocenter	C₁₃	#8788
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₅₂

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1744

C₂

socle, prime radical

C₂³×C₁₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
82. C₁₃	#8788	semidirect

H ≅ G/N

type

82. C₁₃

#8788

semidirect

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

subgroup	index
#13114953	2
#13112671	2
C₁₃×#1744	2
#8788	13

subgroup

index

#13114953

#13112671

C₁₃×#1744

#8788

Subgroups — #15756593

Conjugacy classes of subgroups

Complements

Maximal subgroups