Group #28479121 — subgroups

Subgroups — #28479121

Order 15104.

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

normal subgroups only

This group has 414 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	#21423842	C₂	1	1	yes	maximal, characteristic
2	C₄×#14273638	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×#14273638	C₄	1	4	yes	—
6	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	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	#14273638	—	2	8	no	—
15	C₄×C₅₉⋊C₈	—	2	4	no	—
16	C₄×C₅₉⋊C₈	—	2	4	no	—
17	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
18	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₅₉	—	2	4	no	—
23	C₂²×C₄×C₅₉	C₄²	1	4	yes	—
24	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈×C₅₉	—	2	4	no	—
28	C₂×C₈×C₅₉	—	2	8	no	—
29	C₄²×C₅₉	—	2	4	no	—
30	C₄²×C₅₉	—	2	4	no	—
31	C₄²×C₅₉	—	2	4	no	—
32	C₄²×C₅₉	#41	1	4	yes	—
33	C₂×C₅₉⋊C₈	—	2	4	no	—
34	C₂×C₅₉⋊C₈	—	2	4	no	—
35	C₂×C₅₉⋊C₈	—	2	4	no	—
36	C₂×C₅₉⋊C₈	—	2	8	no	—
37	C₂×C₅₉⋊C₈	M₁₆	1	8	yes	—
38	C₂³×C₅₉	C₂×C₄²	1	1	yes	characteristic
39	C₂×C₄×C₅₉	—	2	4	no	—
40	C₂×C₄×C₅₉	—	2	4	no	—
41	C₂×C₄×C₅₉	—	2	4	no	—
42	C₂×C₄×C₅₉	#117	1	2	yes	—
43	C₂×C₄×C₅₉	#117	1	2	yes	—
44	C₂×C₄×C₅₉	#117	1	2	yes	—
45	C₂×C₄×C₅₉	C₂×M₁₆	1	4	yes	—
46	C₂×C₄×C₅₉	#131	1	4	yes	—
47	C₂×C₄×C₅₉	#131	1	4	yes	—
48	C₈×C₅₉	—	4	4	no	—
49	C₅₉⋊C₈	—	2	8	no	—
50	C₅₉⋊C₈	—	4	4	no	—
51	#8134	—	59	1	no	maximal, Sylow
52	C₂²×C₅₉	#376	1	1	yes	characteristic
53	C₂²×C₅₉	C₄×M₁₆	1	2	yes	—
54	C₂²×C₅₉	#405	1	2	yes	—
55	C₂²×C₅₉	#405	1	2	yes	—
56	C₄×C₅₉	—	4	4	no	—
57	C₄×C₅₉	#432	1	4	yes	—
58	C₄×C₅₉	#433	1	4	yes	—
59	C₄×C₅₉	#433	1	4	yes	—
60	#1744	—	59	1	no	—
61	#1744	D₅₉	1	1	yes	characteristic
62	C₄×#97	—	59	1	no	—
63	#1273	—	59	4	no	—
64	C₂×C₅₉	#1453	1	4	yes	—
65	C₂×C₅₉	#1745	1	1	yes	characteristic
66	C₂×C₅₉	#1746	1	1	yes	characteristic
67	C₂×C₅₉	#1746	1	1	yes	characteristic
68	C₂×#97	—	59	4	no	—
69	C₂×C₄×C₈	—	59	2	no	—
70	C₂×C₄×C₈	—	59	2	no	—
71	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
72	C₄³	C₂×D₅₉	1	1	yes	characteristic
73	#336	—	59	4	no	—
74	#336	Dic₅₉	1	4	yes	—
75	C₅₉	#8134	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₈	—	59	4	no	—
77	C₂²×C₈	—	59	4	no	—
78	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
79	C₂×C₄²	C₄×D₅₉	1	2	yes	—
80	C₂×C₄²	C₄×D₅₉	1	2	yes	—
81	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
82	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
83	#97	—	118	8	no	—
84	C₄×C₈	—	2	4	no	—
85	C₄×C₈	—	118	4	no	—
86	C₄×C₈	—	118	4	no	—
87	C₂²×C₄	C₄×Dic₅₉	1	4	yes	—
88	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
89	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
90	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	2	8	no	—
93	C₂×C₈	—	59	8	no	—
94	C₂×C₈	—	118	4	no	—
95	C₂×C₈	—	118	4	no	—
96	C₂×C₈	—	118	4	no	—
97	C₂×C₈	—	118	8	no	—
98	C₄²	—	2	4	no	—
99	C₄²	—	2	4	no	—
100	C₄²	—	2	4	no	—
101	C₄²	#11745685	1	4	yes	—
102	C₂³	C₂×C₄×Dic₅₉	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	#14273684	1	2	yes	—
107	C₂×C₄	#14273684	1	2	yes	—
108	C₂×C₄	#14273686	1	2	yes	—
109	C₂×C₄	M₁₆×D₅₉	1	4	yes	—
110	C₂×C₄	#14273733	1	4	yes	—
111	C₂×C₄	#14273734	1	4	yes	—
112	C₈	—	4	4	no	—
113	C₈	—	118	8	no	—
114	C₈	—	236	4	no	—
115	C₂²	#16837248	1	1	yes	characteristic
116	C₂²	M₁₆×Dic₅₉	1	2	yes	—
117	C₂²	#16837357	1	2	yes	—
118	C₂²	#16837358	1	2	yes	—
119	C₄	—	4	4	no	—
120	C₄	#16837457	1	4	yes	—
121	C₄	#16837461	1	4	yes	—
122	C₄	#16837463	1	4	yes	—
123	C₂	#21422391	1	4	yes	—
124	C₂	#21423844	1	1	yes	characteristic
125	C₂	#21423847	1	1	yes	characteristic
126	C₂	#21423848	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21423842

C₂

yes

maximal, characteristic

C₄×#14273638

C₂

yes

maximal, characteristic

C₅₉⋊#1273

C₂

yes

maximal

C₅₉×#1744

C₂

yes

maximal, characteristic

C₂×#14273638

C₄

yes

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄×C₅₉⋊C₈

C₂²

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

—

#14273638

—

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

—

C₂×C₅₉⋊C₈

M₁₆

yes

—

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

C₂×M₁₆

yes

—

C₂×C₄×C₅₉

#131

yes

—

C₂×C₄×C₅₉

#131

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

C₄×M₁₆

yes

—

C₂²×C₅₉

#405

yes

—

C₂²×C₅₉

#405

yes

—

C₄×C₅₉

—

C₄×C₅₉

#432

yes

—

C₄×C₅₉

#433

yes

—

C₄×C₅₉

#433

yes

—

#1744

—

#1744

D₅₉

yes

characteristic

C₄×#97

—

#1273

—

C₂×C₅₉

#1453

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₉

yes

—

C₄³

C₂×D₅₉

yes

characteristic

#336

—

#336

Dic₅₉

yes

—

C₅₉

#8134

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

#97

—

118

—

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

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₄²

—

C₄²

—

100

C₄²

—

101

C₄²

#11745685

yes

—

102

C₂³

C₂×C₄×Dic₅₉

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

#14273684

yes

—

107

C₂×C₄

#14273684

yes

—

108

C₂×C₄

#14273686

yes

—

109

C₂×C₄

M₁₆×D₅₉

yes

—

110

C₂×C₄

#14273733

yes

—

111

C₂×C₄

#14273734

yes

—

112

C₈

—

113

C₈

—

118

—

114

C₈

—

236

—

115

C₂²

#16837248

yes

characteristic

116

C₂²

M₁₆×Dic₅₉

yes

—

117

C₂²

#16837357

yes

—

118

C₂²

#16837358

yes

—

119

C₄

—

120

C₄

#16837457

yes

—

121

C₄

#16837461

yes

—

122

C₄

#16837463

yes

—

123

C₂

#21422391

yes

—

124

C₂

#21423844

yes

characteristic

125

C₂

#21423847

yes

characteristic

126

C₂

#21423848

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	#8134
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
75. C₅₉	#8134	semidirect

H ≅ G/N

type

75. C₅₉

#8134

semidirect

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

subgroup	index
#21423842	2
C₄×#14273638	2
C₅₉⋊#1273	2
C₅₉×#1744	2
#8134	59

subgroup

index

#21423842

C₄×#14273638

C₅₉⋊#1273

C₅₉×#1744

#8134

Subgroups — #28479121

Conjugacy classes of subgroups

Complements

Maximal subgroups