Group #28479123 — subgroups

Subgroups — #28479123

Order 15104.

382 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 167 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	#21423841	C₂	1	3	yes	maximal
2	C₅₉⋊#1273	C₂	1	3	yes	maximal
3	C₅₉×#1273	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₅₉⋊C₈	C₂²	1	3	yes	—
5	C₄²×Dic₅₉	C₂²	1	1	yes	characteristic
6	#16837156	C₄	1	6	yes	—
7	#16837156	C₄	1	6	yes	—
8	C₂×C₄×C₈×C₅₉	C₂²	1	3	yes	—
9	C₂²×C₅₉⋊C₈	C₂×C₄	1	6	yes	—
10	C₂×C₄×Dic₅₉	C₂×C₄	1	3	yes	—
11	C₂×C₄×Dic₅₉	C₂×C₄	1	3	yes	—
12	C₄×C₅₉⋊C₈	—	2	3	no	—
13	C₄×C₅₉⋊C₈	—	2	3	no	—
14	C₂²×C₈×C₅₉	C₂×C₄	1	6	yes	—
15	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₅₉	—	2	6	no	—
17	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
18	C₂²×Dic₅₉	C₄²	1	1	yes	characteristic
19	C₂²×Dic₅₉	C₄²	1	3	yes	—
20	C₂×C₈×C₅₉	—	2	6	no	—
21	C₂×C₈×C₅₉	—	2	6	no	—
22	C₂×C₈×C₅₉	—	2	6	no	—
23	C₄²×C₅₉	#41	1	3	yes	—
24	C₄²×C₅₉	#41	1	1	yes	characteristic
25	C₄×Dic₅₉	—	2	6	no	—
26	C₄×Dic₅₉	—	2	6	no	—
27	C₂×C₅₉⋊C₈	—	2	6	no	—
28	C₂×C₅₉⋊C₈	—	2	6	no	—
29	C₂×C₅₉⋊C₈	—	2	6	no	—
30	C₂³×C₅₉	C₂×C₄²	1	1	yes	characteristic
31	C₂×C₄×C₅₉	#117	1	3	yes	—
32	C₂×C₄×C₅₉	#117	1	3	yes	—
33	C₂×C₄×C₅₉	#131	1	6	yes	—
34	C₂×C₄×C₅₉	#131	1	6	yes	—
35	C₂×Dic₅₉	—	2	3	no	—
36	C₂×Dic₅₉	—	2	3	no	—
37	C₂×Dic₅₉	—	2	6	no	—
38	C₈×C₅₉	—	4	6	no	—
39	C₅₉⋊C₈	—	4	6	no	—
40	#8135	—	59	1	no	maximal, Sylow
41	C₂²×C₅₉	#376	1	1	yes	characteristic
42	C₂²×C₅₉	#405	1	3	yes	—
43	C₂²×C₅₉	#405	1	3	yes	—
44	C₄×C₅₉	#433	1	6	yes	—
45	C₄×C₅₉	#433	1	6	yes	—
46	Dic₅₉	—	4	1	no	—
47	Dic₅₉	—	4	3	no	—
48	#1744	—	59	3	no	—
49	#1273	—	59	3	no	—
50	#1273	D₅₉	1	1	yes	characteristic
51	C₂×C₅₉	#1454	1	1	yes	characteristic
52	C₂×C₅₉	#1454	1	3	yes	—
53	C₂×C₅₉	#1746	1	3	yes	—
54	C₂×C₄×C₈	—	59	3	no	—
55	C₂×C₄×C₈	C₂×D₅₉	1	3	yes	—
56	C₄³	—	59	1	no	—
57	#336	—	59	6	no	—
58	#336	—	59	6	no	—
59	C₅₉	#8135	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₈	—	59	6	no	—
61	C₂²×C₈	C₄×D₅₉	1	6	yes	—
62	C₂×C₄²	—	59	3	no	—
63	C₂×C₄²	—	59	3	no	—
64	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
65	C₄×C₈	—	2	6	no	—
66	C₄×C₈	—	118	3	no	—
67	C₄×C₈	—	118	3	no	—
68	C₂²×C₄	—	59	1	no	—
69	C₂²×C₄	—	59	3	no	—
70	C₂²×C₄	C₂×C₄×D₅₉	1	3	yes	—
71	C₂×C₈	—	2	6	no	—
72	C₂×C₈	—	2	6	no	—
73	C₂×C₈	—	2	6	no	—
74	C₂×C₈	—	118	6	no	—
75	C₂×C₈	—	118	6	no	—
76	C₂×C₈	—	118	6	no	—
77	C₄²	—	118	6	no	—
78	C₄²	—	118	6	no	—
79	C₄²	#11745687	1	3	yes	—
80	C₄²	#11745689	1	1	yes	characteristic
81	C₂³	C₄²×D₅₉	1	1	yes	characteristic
82	C₂×C₄	—	118	3	no	—
83	C₂×C₄	—	118	3	no	—
84	C₂×C₄	—	118	6	no	—
85	C₂×C₄	#14273683	1	3	yes	—
86	C₂×C₄	#14273685	1	3	yes	—
87	C₂×C₄	#14273733	1	6	yes	—
88	C₂×C₄	#14273735	1	6	yes	—
89	C₈	—	4	6	no	—
90	C₈	—	236	6	no	—
91	C₂²	#16837247	1	1	yes	characteristic
92	C₂²	#16837356	1	3	yes	—
93	C₂²	#16837359	1	3	yes	—
94	C₄	—	236	1	no	—
95	C₄	—	236	3	no	—
96	C₄	#16837460	1	6	yes	—
97	C₄	#16837462	1	6	yes	—
98	C₂	#21422395	1	1	yes	characteristic
99	C₂	#21422397	1	3	yes	—
100	C₂	#21423846	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21423841

C₂

yes

maximal

C₅₉⋊#1273

C₂

yes

maximal

C₅₉×#1273

C₂

yes

maximal, characteristic

C₂×C₄×C₅₉⋊C₈

C₂²

yes

—

C₄²×Dic₅₉

C₂²

yes

characteristic

#16837156

C₄

yes

—

#16837156

C₄

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₂²×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₅₉

C₂×C₄

yes

—

C₂×C₄×Dic₅₉

C₂×C₄

yes

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

—

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

—

C₂²×Dic₅₉

C₄²

yes

characteristic

C₂²×Dic₅₉

C₄²

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

#41

yes

—

C₄²×C₅₉

#41

yes

characteristic

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

#131

yes

—

C₂×C₄×C₅₉

#131

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#405

yes

—

C₄×C₅₉

#433

yes

—

C₄×C₅₉

#433

yes

—

Dic₅₉

—

Dic₅₉

—

#1744

—

#1273

—

#1273

D₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#1746

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₉

yes

—

C₄³

—

#336

—

#336

—

C₅₉

#8135

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₅₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

C₄×C₈

—

C₄×C₈

—

118

—

C₄×C₈

—

118

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×D₅₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₄²

—

118

—

C₄²

—

118

—

C₄²

#11745687

yes

—

C₄²

#11745689

yes

characteristic

C₂³

C₄²×D₅₉

yes

characteristic

C₂×C₄

—

118

—

C₂×C₄

—

118

—

C₂×C₄

—

118

—

C₂×C₄

#14273683

yes

—

C₂×C₄

#14273685

yes

—

C₂×C₄

#14273733

yes

—

C₂×C₄

#14273735

yes

—

C₈

—

C₈

—

236

—

C₂²

#16837247

yes

characteristic

C₂²

#16837356

yes

—

C₂²

#16837359

yes

—

C₄

—

236

—

C₄

—

236

—

C₄

#16837460

yes

—

C₄

#16837462

yes

—

C₂

#21422395

yes

characteristic

C₂

#21422397

yes

—

100

C₂

#21423846

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₉
commutator, socle, prime radical	C₂³×C₅₉	C₂×C₄²
Fitting	C₅₉×#1273	C₂
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1273	D₅₉
hypocenter	C₅₉	#8135
LCS	G ▷ C₂³×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₅₉

commutator, socle, prime radical

C₂³×C₅₉

C₂×C₄²

Fitting

C₅₉×#1273

C₂

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂³×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
59. C₅₉	#8135	semidirect

H ≅ G/N

type

59. C₅₉

#8135

semidirect

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

subgroup	index
#21423841	2
C₅₉⋊#1273	2
C₅₉×#1273	2
#8135	59

subgroup

index

#21423841

C₅₉⋊#1273

C₅₉×#1273

#8135

Subgroups — #28479123

Conjugacy classes of subgroups

Complements

Maximal subgroups