Group #28479124 — subgroups

Subgroups — #28479124

Order 15104.

382 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 179 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	#21423842	C₂	1	1	yes	maximal, characteristic
2	#21423842	C₂	1	1	yes	maximal, characteristic
3	C₅₉⋊#1273	C₂	1	4	yes	maximal
4	C₅₉×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
7	#16837157	C₄	1	4	yes	—
8	#16837157	C₄	1	4	yes	—
9	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
10	C₄³×C₅₉	C₂²	1	1	yes	characteristic
11	C₅₉×#336	C₄	1	4	yes	—
12	C₂²×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
13	C₂²×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
14	C₄×C₅₉⋊C₈	—	2	4	no	—
15	C₄×C₅₉⋊C₈	—	2	4	no	—
16	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
17	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
21	C₄×C₈×C₅₉	—	2	4	no	—
22	C₂²×C₄×C₅₉	C₄²	1	4	yes	—
23	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₈×C₅₉	—	2	4	no	—
27	C₂×C₈×C₅₉	—	2	8	no	—
28	C₄²×C₅₉	—	2	4	no	—
29	C₄²×C₅₉	—	2	4	no	—
30	C₄²×C₅₉	—	2	4	no	—
31	C₄²×C₅₉	#41	1	4	yes	—
32	C₂×C₅₉⋊C₈	—	2	4	no	—
33	C₂×C₅₉⋊C₈	—	2	4	no	—
34	C₂×C₅₉⋊C₈	—	2	8	no	—
35	C₂×C₅₉⋊C₈	—	2	8	no	—
36	C₂³×C₅₉	C₂×C₄²	1	1	yes	characteristic
37	C₂×C₄×C₅₉	—	2	4	no	—
38	C₂×C₄×C₅₉	—	2	4	no	—
39	C₂×C₄×C₅₉	—	2	4	no	—
40	C₂×C₄×C₅₉	#117	1	2	yes	—
41	C₂×C₄×C₅₉	#117	1	2	yes	—
42	C₂×C₄×C₅₉	#117	1	2	yes	—
43	C₂×C₄×C₅₉	#131	1	4	yes	—
44	C₂×C₄×C₅₉	#131	1	4	yes	—
45	C₂×C₄×C₅₉	#131	1	4	yes	—
46	C₈×C₅₉	—	4	4	no	—
47	C₅₉⋊C₈	—	4	4	no	—
48	C₅₉⋊C₈	—	4	4	no	—
49	#8135	—	59	1	no	maximal, Sylow
50	C₂²×C₅₉	#376	1	1	yes	characteristic
51	C₂²×C₅₉	#405	1	2	yes	—
52	C₂²×C₅₉	#405	1	2	yes	—
53	C₂²×C₅₉	#405	1	2	yes	—
54	C₄×C₅₉	—	4	4	no	—
55	C₄×C₅₉	#433	1	4	yes	—
56	C₄×C₅₉	#433	1	4	yes	—
57	C₄×C₅₉	#433	1	4	yes	—
58	#1744	—	59	1	no	—
59	#1744	—	59	1	no	—
60	#1744	D₅₉	1	1	yes	characteristic
61	#1273	—	59	4	no	—
62	C₂×C₅₉	#1454	1	4	yes	—
63	C₂×C₅₉	#1746	1	1	yes	characteristic
64	C₂×C₅₉	#1746	1	1	yes	characteristic
65	C₂×C₅₉	#1746	1	1	yes	characteristic
66	C₂×C₄×C₈	—	59	2	no	—
67	C₂×C₄×C₈	—	59	2	no	—
68	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
69	C₄³	C₂×D₅₉	1	1	yes	characteristic
70	#336	—	59	4	no	—
71	#336	—	59	4	no	—
72	#336	Dic₅₉	1	4	yes	—
73	C₅₉	#8135	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₈	—	59	4	no	—
75	C₂²×C₈	—	59	4	no	—
76	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
77	C₂×C₄²	C₄×D₅₉	1	2	yes	—
78	C₂×C₄²	C₄×D₅₉	1	2	yes	—
79	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
80	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
81	C₄×C₈	—	2	4	no	—
82	C₄×C₈	—	118	4	no	—
83	C₄×C₈	—	118	4	no	—
84	C₂²×C₄	C₄×Dic₅₉	1	4	yes	—
85	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
86	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
87	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	2	8	no	—
90	C₂×C₈	—	118	4	no	—
91	C₂×C₈	—	118	4	no	—
92	C₂×C₈	—	118	8	no	—
93	C₂×C₈	—	118	8	no	—
94	C₄²	—	2	4	no	—
95	C₄²	—	2	4	no	—
96	C₄²	—	2	4	no	—
97	C₄²	#11745685	1	4	yes	—
98	C₂³	C₂×C₄×Dic₅₉	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	#14273684	1	2	yes	—
103	C₂×C₄	#14273684	1	2	yes	—
104	C₂×C₄	#14273686	1	2	yes	—
105	C₂×C₄	#14273733	1	4	yes	—
106	C₂×C₄	#14273734	1	4	yes	—
107	C₂×C₄	#14273735	1	4	yes	—
108	C₈	—	4	4	no	—
109	C₈	—	236	4	no	—
110	C₈	—	236	4	no	—
111	C₂²	#16837248	1	1	yes	characteristic
112	C₂²	#16837357	1	2	yes	—
113	C₂²	#16837358	1	2	yes	—
114	C₂²	#16837360	1	2	yes	—
115	C₄	—	4	4	no	—
116	C₄	#16837461	1	4	yes	—
117	C₄	#16837463	1	4	yes	—
118	C₄	#16837464	1	4	yes	—
119	C₂	#21422396	1	4	yes	—
120	C₂	#21423847	1	1	yes	characteristic
121	C₂	#21423848	1	1	yes	characteristic
122	C₂	#21423849	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21423842

C₂

yes

maximal, characteristic

#21423842

C₂

yes

maximal, characteristic

C₅₉⋊#1273

C₂

yes

maximal

C₅₉×#1744

C₂

yes

maximal, characteristic

C₂×C₄×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

—

#16837157

C₄

yes

—

#16837157

C₄

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₄³×C₅₉

C₂²

yes

characteristic

C₅₉×#336

C₄

yes

—

C₂²×C₅₉⋊C₈

C₂×C₄

yes

—

C₂²×C₅₉⋊C₈

C₂×C₄

yes

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

—

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

C₄²

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

#41

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

#131

yes

—

C₂×C₄×C₅₉

#131

yes

—

C₂×C₄×C₅₉

#131

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#405

yes

—

C₂²×C₅₉

#405

yes

—

C₄×C₅₉

—

C₄×C₅₉

#433

yes

—

C₄×C₅₉

#433

yes

—

C₄×C₅₉

#433

yes

—

#1744

—

#1744

—

#1744

D₅₉

yes

characteristic

#1273

—

C₂×C₅₉

#1454

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₉

yes

—

C₄³

C₂×D₅₉

yes

—

—

Dic₅₉

yes

—

C₅₉

#8135

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₅₉

yes

—

C₂×C₄²

C₄×D₅₉

yes

—

C₂×C₄²

C₄×D₅₉

yes

—

C₂×C₄²

C₂×Dic₅₉

yes

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

C₄×C₈

—

C₄×C₈

—

118

—

C₄×C₈

—

118

—

C₂²×C₄

C₄×Dic₅₉

yes

—

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

C₂²×C₄

C₂²×Dic₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#11745685

yes

—

C₂³

C₂×C₄×Dic₅₉

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

#14273684

yes

—

103

C₂×C₄

#14273684

yes

—

104

C₂×C₄

#14273686

yes

—

105

C₂×C₄

#14273733

yes

—

106

C₂×C₄

#14273734

yes

—

107

C₂×C₄

#14273735

yes

—

108

C₈

—

109

C₈

—

236

—

110

C₈

—

236

—

111

C₂²

#16837248

yes

characteristic

112

C₂²

#16837357

yes

—

113

C₂²

#16837358

yes

—

114

C₂²

#16837360

yes

—

115

C₄

—

116

C₄

#16837461

yes

—

117

C₄

#16837463

yes

—

118

C₄

#16837464

yes

—

119

C₂

#21422396

yes

—

120

C₂

#21423847

yes

characteristic

121

C₂

#21423848

yes

characteristic

122

C₂

#21423849

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₉
commutator, socle, prime radical	C₂³×C₅₉	C₂×C₄²
Fitting	C₅₉×#1744	C₂
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1744	D₅₉
hypocenter	C₅₉	#8135
LCS	G ▷ C₂³×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄² ◁ #1744

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₅₉

commutator, socle, prime radical

C₂³×C₅₉

C₂×C₄²

Fitting

C₅₉×#1744

C₂

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂³×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₄² ◁ #1744

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

H ≅ G/N

type

73. C₅₉

#8135

semidirect

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

subgroup	index
#21423842	2
C₅₉⋊#1273	2
C₅₉×#1744	2
#8135	59

subgroup

index

#21423842

C₅₉⋊#1273

C₅₉×#1744

#8135

Subgroups — #28479124

Conjugacy classes of subgroups

Complements

Maximal subgroups