Group #28425629 — subgroups

Subgroups — C₅₉×#13520 #28425629

Order 15104.

238 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 106 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	C₅₉×#2161	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#2151	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1953	C₂	1	2	yes	maximal
4	C₅₉×#1385	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1381	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
8	C₅₉×#430	—	2	2	no	—
9	C₂×#108×C₅₉	C₂²	1	2	yes	—
10	C₂×#108×C₅₉	C₂²	1	2	yes	—
11	C₅₉×#405	C₂²	1	1	yes	characteristic
12	C₅₉×#368	—	2	2	no	—
13	C₅₉×C₄⋊C₁₆	—	2	2	no	—
14	C₅₉×#344	—	2	2	no	—
15	C₂×M₁₆×C₅₉	—	2	2	no	—
16	C₂×M₁₆×C₅₉	—	2	2	no	—
17	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
19	#117×C₅₉	D₄	1	1	yes	characteristic
20	M₃₂×C₅₉	—	2	2	no	—
21	C₂×C₁₆×C₅₉	—	2	1	no	—
22	C₂×C₁₆×C₅₉	—	2	2	no	—
23	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
24	#108×C₅₉	—	2	2	no	—
25	#108×C₅₉	—	2	2	no	—
26	#108×C₅₉	—	2	2	no	—
27	#108×C₅₉	—	4	4	no	—
28	#108×C₅₉	Q₈	1	4	yes	—
29	#97×C₅₉	—	2	1	no	—
30	C₄×C₈×C₅₉	—	2	1	no	—
31	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
32	M₁₆×C₅₉	—	2	4	no	—
33	M₁₆×C₅₉	—	2	2	no	—
34	M₁₆×C₅₉	—	4	2	no	—
35	M₁₆×C₅₉	—	4	2	no	—
36	M₁₆×C₅₉	—	4	2	no	—
37	C₂×C₈×C₅₉	—	2	1	no	—
38	C₂×C₈×C₅₉	—	4	2	no	—
39	C₂×C₈×C₅₉	—	4	2	no	—
40	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
42	C₂×C₈×C₅₉	#41	1	2	yes	—
43	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
44	C₄⋊C₄×C₅₉	—	2	1	no	—
45	C₂²⋊C₄×C₅₉	Q₁₆	1	2	yes	—
46	C₄²×C₅₉	—	2	1	no	—
47	C₁₆×C₅₉	—	2	2	no	—
48	C₁₆×C₅₉	—	4	1	no	—
49	C₂³×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
50	C₂×C₄×C₅₉	—	2	2	no	—
51	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
53	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
54	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
55	C₈×C₅₉	—	4	1	no	—
56	C₈×C₅₉	—	4	4	no	—
57	C₈×C₅₉	—	8	2	no	—
58	C₈×C₅₉	#123	1	1	yes	characteristic
59	C₈×C₅₉	#125	1	2	yes	—
60	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
61	#13520	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂²×C₅₉	—	4	1	no	—
63	C₂²×C₅₉	#467	1	1	yes	characteristic
64	C₂²×C₅₉	#477	1	2	yes	—
65	C₄×C₅₉	—	4	2	no	—
66	C₄×C₅₉	#398	1	1	yes	characteristic
67	C₄×C₅₉	#469	1	1	yes	characteristic
68	C₄×C₅₉	#478	1	2	yes	—
69	#2161	C₂×C₅₉	1	1	yes	characteristic
70	#2151	C₂×C₅₉	1	1	yes	characteristic
71	#1953	C₂×C₅₉	1	2	yes	—
72	#1385	C₂×C₅₉	1	1	yes	characteristic
73	#1381	C₂×C₅₉	1	2	yes	—
74	C₂×C₅₉	—	2	1	no	—
75	C₂×C₅₉	—	2	2	no	—
76	C₂×C₅₉	#2087	1	1	yes	characteristic
77	C₂×M₃₂	C₂²×C₅₉	1	1	yes	characteristic
78	C₂²×C₁₆	C₂²×C₅₉	1	1	yes	characteristic
79	#430	—	2	2	no	—
80	C₂×#108	C₂²×C₅₉	1	2	yes	—
81	C₂×#108	C₂²×C₅₉	1	2	yes	—
82	#405	C₂²×C₅₉	1	1	yes	characteristic
83	#368	—	2	2	no	—
84	C₄⋊C₁₆	—	2	2	no	—
85	#344	—	2	2	no	—
86	C₅₉	#13520	1	1	yes	characteristic, direct product factor, Sylow
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	—	2	2	no	—
89	C₂×M₁₆	D₄×C₅₉	1	1	yes	characteristic
90	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
91	#117	D₄×C₅₉	1	1	yes	characteristic
92	M₃₂	—	2	2	no	—
93	C₂×C₁₆	—	2	1	no	—
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	D₄×C₅₉	1	2	yes	—
96	#108	—	2	2	no	—
97	#108	—	2	2	no	—
98	#108	—	2	2	no	—
99	#108	—	4	4	no	—
100	#108	Q₈×C₅₉	1	4	yes	—
101	#97	—	2	1	no	—
102	C₄×C₈	—	2	1	no	—
103	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
104	M₁₆	—	2	4	no	—
105	M₁₆	—	2	2	no	—
106	M₁₆	—	4	2	no	—
107	M₁₆	—	4	2	no	—
108	M₁₆	—	4	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
113	C₂×C₈	C₂×Q₈×C₅₉	1	2	yes	—
114	C₂×C₈	#41×C₅₉	1	2	yes	—
115	C₂×C₈	#41×C₅₉	1	1	yes	characteristic
116	C₄⋊C₄	—	2	1	no	—
117	C₂²⋊C₄	Q₁₆×C₅₉	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₁₆	—	2	2	no	—
120	C₁₆	—	4	1	no	—
121	C₂³	C₂×Q₁₆×C₅₉	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
124	C₂×C₄	C₂²⋊Q₈×C₅₉	1	2	yes	—
125	C₂×C₄	#135×C₅₉	1	1	yes	characteristic
126	C₂×C₄	AGL(1,ℤ/8)×C₅₉	1	2	yes	—
127	C₈	—	4	1	no	—
128	C₈	—	4	4	no	—
129	C₈	—	8	2	no	—
130	C₈	#123×C₅₉	1	1	yes	characteristic
131	C₈	#125×C₅₉	1	2	yes	—
132	C₈	C₄⋊Q₈×C₅₉	1	1	yes	characteristic
133	C₂²	—	4	1	no	—
134	C₂²	C₅₉×#467	1	1	yes	characteristic
135	C₂²	C₅₉×#477	1	2	yes	—
136	C₄	—	4	2	no	—
137	C₄	C₅₉×#398	1	1	yes	characteristic
138	C₄	C₅₉×#469	1	1	yes	characteristic
139	C₄	C₅₉×#478	1	2	yes	—
140	C₂	—	2	1	no	—
141	C₂	—	2	2	no	—
142	C₂	C₅₉×#2087	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₉×#2161

C₂

yes

maximal, characteristic

C₅₉×#2151

C₂

yes

maximal, characteristic

C₅₉×#1953

C₂

yes

maximal

C₅₉×#1385

C₂

yes

maximal, characteristic

C₅₉×#1381

C₂

yes

maximal

C₂×M₃₂×C₅₉

C₂²

yes

characteristic

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₅₉×#430

—

C₂×#108×C₅₉

C₂²

yes

—

C₂×#108×C₅₉

C₂²

yes

—

C₅₉×#405

C₂²

yes

characteristic

C₅₉×#368

—

C₅₉×C₄⋊C₁₆

—

C₅₉×#344

—

C₂×M₁₆×C₅₉

—

C₂×M₁₆×C₅₉

—

C₂×M₁₆×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

#117×C₅₉

D₄

yes

characteristic

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

D₄

yes

—

#108×C₅₉

—

#108×C₅₉

—

#108×C₅₉

—

#108×C₅₉

—

#108×C₅₉

Q₈

yes

—

#97×C₅₉

—

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

—

C₂×C₈×C₅₉

#41

yes

—

C₂×C₈×C₅₉

#41

yes

characteristic

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

Q₁₆

yes

—

C₄²×C₅₉

—

C₁₆×C₅₉

—

C₁₆×C₅₉

—

C₂³×C₅₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₉

#135

yes

characteristic

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#123

yes

characteristic

C₈×C₅₉

#125

yes

—

C₈×C₅₉

C₄⋊Q₈

yes

characteristic

#13520

C₅₉

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#467

yes

characteristic

C₂²×C₅₉

#477

yes

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

#2161

C₂×C₅₉

yes

characteristic

#2151

C₂×C₅₉

yes

characteristic

#1953

C₂×C₅₉

yes

—

#1385

C₂×C₅₉

yes

characteristic

#1381

C₂×C₅₉

yes

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#2087

yes

characteristic

C₂×M₃₂

C₂²×C₅₉

yes

characteristic

C₂²×C₁₆

C₂²×C₅₉

yes

characteristic

#430

—

C₂×#108

C₂²×C₅₉

yes

—

C₂×#108

C₂²×C₅₉

yes

—

#405

C₂²×C₅₉

yes

characteristic

#368

—

C₄⋊C₁₆

—

#344

—

C₅₉

#13520

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₄×C₅₉

yes

characteristic

C₂²×C₈

C₂³×C₅₉

yes

characteristic

#117

D₄×C₅₉

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄×C₅₉

yes

—

—

—

—

—

100

#108

Q₈×C₅₉

yes

—

101

#97

—

102

C₄×C₈

—

103

C₂²×C₄

C₂×D₄×C₅₉

yes

characteristic

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₂×D₄×C₅₉

yes

characteristic

113

C₂×C₈

C₂×Q₈×C₅₉

yes

—

114

C₂×C₈

#41×C₅₉

yes

—

115

C₂×C₈

#41×C₅₉

yes

characteristic

116

C₄⋊C₄

—

117

C₂²⋊C₄

Q₁₆×C₅₉

yes

—

118

C₄²

—

119

C₁₆

—

120

C₁₆

—

121

C₂³

C₂×Q₁₆×C₅₉

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

#121×C₅₉

yes

characteristic

124

C₂×C₄

C₂²⋊Q₈×C₅₉

yes

—

125

C₂×C₄

#135×C₅₉

yes

characteristic

126

C₂×C₄

AGL(1,ℤ/8)×C₅₉

yes

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

#123×C₅₉

yes

characteristic

131

C₈

#125×C₅₉

yes

—

132

C₈

C₄⋊Q₈×C₅₉

yes

characteristic

133

C₂²

—

134

C₂²

C₅₉×#467

yes

characteristic

135

C₂²

C₅₉×#477

yes

—

136

C₄

—

137

C₄

C₅₉×#398

yes

characteristic

138

C₄

C₅₉×#469

yes

characteristic

139

C₄

C₅₉×#478

yes

—

140

C₂

—

141

C₂

—

142

C₂

C₅₉×#2087

yes

characteristic

	subgroup	G/H
center	C₄×C₅₉	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₉
socle	C₂×C₅₉	#2087
prime radical	C₂³×C₅₉	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₉ ◁ C₂²×C₄×C₅₉ ◁ C₅₉×#405 ◁ G

subgroup

G/H

center

C₄×C₅₉

#469

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₅₉

socle

C₂×C₅₉

#2087

prime radical

C₂³×C₅₉

C₂×Q₁₆

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₅₉ ◁ C₂²×C₄×C₅₉ ◁ C₅₉×#405 ◁ G

N	H ≅ G/N	type
61. #13520	C₅₉	direct (N × H)
86. C₅₉	#13520	direct (N × H)

H ≅ G/N

type

61. #13520

C₅₉

direct (N × H)

86. C₅₉

#13520

direct (N × H)

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

subgroup	index
C₅₉×#2161	2
C₅₉×#2151	2
C₅₉×#1953	2
C₅₉×#1385	2
C₅₉×#1381	2
#13520	59

subgroup

index

C₅₉×#2161

C₅₉×#2151

C₅₉×#1953

C₅₉×#1385

C₅₉×#1381

#13520

Subgroups — C₅₉×#13520 #28425629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C59×#13520 #28425629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#13520 #28425629