Group #28509225 — subgroups

Subgroups — #28509225

Order 15104.

170 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 75 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	M₃₂×Dic₅₉	C₂	1	1	yes	maximal, characteristic
2	#21422293	C₂	1	2	yes	maximal, complemented
3	#21422236	C₂	1	1	yes	maximal, characteristic
4	#21422231	C₂	1	2	yes	maximal, complemented
5	C₅₉×#1400	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₅₉	C₄	1	1	yes	characteristic
8	#16837236	C₂²	1	2	yes	—
9	#16837177	C₄	1	1	yes	characteristic
10	#16837177	C₂²	1	1	yes	characteristic
11	C₁₆×Dic₅₉	C₄	1	1	yes	characteristic
12	C₁₆×Dic₅₉	C₂²	1	1	yes	characteristic
13	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×M₆₄	C₂²	1	2	yes	—
15	C₂²×C₅₉⋊C₈	C₈	1	1	yes	characteristic
16	C₂×C₄×Dic₅₉	C₈	1	1	yes	characteristic
17	#14273667	—	2	2	no	—
18	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₈×Dic₅₉	C₈	1	1	yes	characteristic
21	C₈×Dic₅₉	C₈	1	1	yes	characteristic
22	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
23	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
24	C₅₉⋊C₃₂	—	2	2	no	—
25	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₅₉	—	2	2	no	—
27	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
29	C₃₂×C₅₉	—	2	2	no	—
30	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
31	C₂²×Dic₅₉	—	2	1	no	—
32	C₂×C₈×C₅₉	—	2	1	no	—
33	C₂×C₈×C₅₉	—	2	1	no	—
34	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
36	C₄×Dic₅₉	—	2	1	no	—
37	C₄×Dic₅₉	C₂×C₈	1	1	yes	characteristic
38	C₄×Dic₅₉	C₂×C₈	1	1	yes	characteristic
39	C₂×C₅₉⋊C₈	—	2	2	no	—
40	C₂×C₅₉⋊C₈	C₂×C₈	1	1	yes	characteristic
41	C₂×C₅₉⋊C₈	C₂×C₈	1	1	yes	characteristic
42	C₁₆×C₅₉	—	2	1	no	—
43	C₁₆×C₅₉	#41	1	2	yes	—
44	C₅₉⋊C₁₆	—	2	1	no	—
45	C₅₉⋊C₁₆	M₁₆	1	2	yes	—
46	C₂³×C₅₉	M₃₂	1	1	yes	characteristic
47	C₂×C₄×C₅₉	—	2	1	no	—
48	C₂×C₄×C₅₉	—	2	1	no	—
49	C₂×C₄×C₅₉	M₃₂	1	1	yes	characteristic
50	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
51	C₂×Dic₅₉	—	2	1	no	—
52	C₂×Dic₅₉	—	2	1	no	—
53	C₂×Dic₅₉	—	4	2	no	—
54	C₈×C₅₉	—	2	1	no	—
55	C₈×C₅₉	#117	1	1	yes	characteristic
56	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
57	C₅₉⋊C₈	—	2	1	no	—
58	C₅₉⋊C₈	M₃₂	1	2	yes	—
59	#13642	—	59	1	no	maximal, Sylow
60	C₂²×C₅₉	—	2	1	no	—
61	C₂²×C₅₉	—	4	1	no	—
62	C₂²×C₅₉	C₂×M₃₂	1	1	yes	characteristic
63	C₄×C₅₉	—	2	1	no	—
64	C₄×C₅₉	#431	1	1	yes	characteristic
65	C₄×C₅₉	C₂×M₃₂	1	1	yes	characteristic
66	Dic₅₉	—	2	2	no	—
67	Dic₅₉	—	4	1	no	—
68	C₄×M₃₂	—	59	1	no	—
69	#1422	—	59	2	no	—
70	#1400	—	59	1	no	—
71	#1400	D₅₉	1	1	yes	characteristic
72	#1398	—	59	2	no	—
73	C₂×C₅₉	—	2	1	no	—
74	C₂×C₅₉	—	4	1	no	—
75	C₂×C₅₉	#2163	1	1	yes	characteristic
76	C₂×M₃₂	—	59	1	no	—
77	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
78	C₂×C₄×C₈	—	59	1	no	—
79	M₆₄	—	59	2	no	—
80	M₆₄	C₂×D₅₉	1	2	yes	—
81	#346	—	59	1	no	—
82	#346	—	59	1	no	—
83	C₄×C₁₆	—	59	1	no	—
84	C₄×C₁₆	—	59	1	no	—
85	C₅₉	#13642	1	1	yes	characteristic, complemented, Sylow
86	C₂²×C₈	—	59	1	no	—
87	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
88	C₂×C₄²	—	59	1	no	—
89	M₃₂	—	2	2	no	—
90	M₃₂	—	118	2	no	—
91	C₂×C₁₆	—	59	1	no	—
92	C₂×C₁₆	—	59	1	no	—
93	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
94	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
95	C₄×C₈	—	59	1	no	—
96	C₄×C₈	—	59	1	no	—
97	C₄×C₈	—	59	1	no	—
98	C₄×C₈	—	59	1	no	—
99	C₃₂	—	2	2	no	—
100	C₃₂	—	118	2	no	—
101	C₂²×C₄	—	118	1	no	—
102	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	59	1	no	—
106	C₂×C₈	—	59	1	no	—
107	C₂×C₈	—	118	2	no	—
108	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
109	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
110	C₄²	—	59	1	no	—
111	C₄²	—	59	1	no	—
112	C₄²	—	118	1	no	—
113	C₁₆	—	2	1	no	—
114	C₁₆	—	59	2	no	—
115	C₁₆	—	118	1	no	—
116	C₁₆	#11745687	1	2	yes	—
117	C₂³	#14273666	1	1	yes	characteristic
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	118	1	no	—
121	C₂×C₄	—	118	1	no	—
122	C₂×C₄	—	236	2	no	—
123	C₂×C₄	#14273666	1	1	yes	characteristic
124	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	59	2	no	—
127	C₈	—	118	1	no	—
128	C₈	#14273683	1	1	yes	characteristic
129	C₈	M₁₆×D₅₉	1	1	yes	characteristic
130	C₂²	—	2	1	no	—
131	C₂²	—	4	1	no	—
132	C₂²	C₂×#14273666	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	118	2	no	—
135	C₄	—	236	1	no	—
136	C₄	#16837452	1	1	yes	characteristic
137	C₄	M₃₂×D₅₉	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	4	1	no	—
140	C₂	#21425896	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₃₂×Dic₅₉

C₂

yes

maximal, characteristic

#21422293

C₂

yes

maximal, complemented

#21422236

C₂

yes

maximal, characteristic

#21422231

C₂

yes

maximal, complemented

C₅₉×#1400

C₂

yes

maximal, characteristic

C₂×#14273667

C₄

yes

characteristic

C₂×C₈×Dic₅₉

C₄

yes

characteristic

#16837236

C₂²

yes

—

#16837177

C₄

yes

characteristic

#16837177

C₂²

yes

characteristic

C₁₆×Dic₅₉

C₄

yes

characteristic

C₁₆×Dic₅₉

C₂²

yes

characteristic

C₂×M₃₂×C₅₉

C₂²

yes

characteristic

C₅₉×M₆₄

C₂²

yes

—

C₂²×C₅₉⋊C₈

C₈

yes

characteristic

C₂×C₄×Dic₅₉

C₈

yes

characteristic

#14273667

—

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

characteristic

C₈×Dic₅₉

C₈

yes

characteristic

C₈×Dic₅₉

C₈

yes

characteristic

C₈×Dic₅₉

C₂×C₄

yes

characteristic

C₈×Dic₅₉

C₂×C₄

yes

characteristic

C₅₉⋊C₃₂

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂³

yes

characteristic

C₃₂×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₈

yes

characteristic

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₈

yes

characteristic

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₄×Dic₅₉

—

C₄×Dic₅₉

C₂×C₈

yes

characteristic

C₄×Dic₅₉

C₂×C₈

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₈

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₈

yes

characteristic

C₁₆×C₅₉

—

C₁₆×C₅₉

#41

yes

—

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

M₁₆

yes

—

C₂³×C₅₉

M₃₂

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

M₃₂

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#117

yes

characteristic

C₈×C₅₉

C₂×M₁₆

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

M₃₂

yes

—

#13642

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×M₃₂

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

#431

yes

characteristic

C₄×C₅₉

C₂×M₃₂

yes

characteristic

Dic₅₉

—

Dic₅₉

—

C₄×M₃₂

—

#1422

—

#1400

—

#1400

D₅₉

yes

characteristic

#1398

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#2163

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

C₂×D₅₉

yes

characteristic

C₂×C₄×C₈

—

M₆₄

—

M₆₄

C₂×D₅₉

yes

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

C₅₉

#13642

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₅₉

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

118

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₅₉

yes

characteristic

C₂×C₁₆

C₂²×D₅₉

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₃₂

—

100

C₃₂

—

118

—

101

C₂²×C₄

—

118

—

102

C₂²×C₄

C₈×D₅₉

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

118

—

108

C₂×C₈

C₈×D₅₉

yes

characteristic

109

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

110

C₄²

—

111

C₄²

—

112

C₄²

—

118

—

113

C₁₆

—

114

C₁₆

—

115

C₁₆

—

118

—

116

C₁₆

#11745687

yes

—

117

C₂³

#14273666

yes

characteristic

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

118

—

121

C₂×C₄

—

118

—

122

C₂×C₄

—

236

—

123

C₂×C₄

#14273666

yes

characteristic

124

C₂×C₄

C₂×C₈×D₅₉

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

118

—

128

C₈

#14273683

yes

characteristic

129

C₈

M₁₆×D₅₉

yes

characteristic

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×#14273666

yes

characteristic

133

C₄

—

134

C₄

—

118

—

135

C₄

—

236

—

136

C₄

#16837452

yes

characteristic

137

C₄

M₃₂×D₅₉

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

#21425896

yes

characteristic

	subgroup	G/H
center	C₈	#14273683
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂×C₁₆	C₂²×D₅₉
Fitting	C₅₉×#1400	C₂
socle	C₂×C₅₉	#2163
prime radical	C₂³×C₅₉	M₃₂
cosocle	C₂×C₁₆×C₅₉	C₂³
hypercenter	#1400	D₅₉
hypocenter	C₅₉	#13642
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #1400

subgroup

G/H

center

C₈

#14273683

commutator

C₂×C₄×C₅₉

C₂²×C₈

Frattini

C₂×C₁₆

C₂²×D₅₉

Fitting

C₅₉×#1400

C₂

socle

C₂×C₅₉

#2163

prime radical

C₂³×C₅₉

M₃₂

cosocle

C₂×C₁₆×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #1400

N	H ≅ G/N	type
2. #21422293	C₂	semidirect
4. #21422231	C₂	semidirect
85. C₅₉	#13642	semidirect

H ≅ G/N

type

2. #21422293

C₂

semidirect

4. #21422231

C₂

semidirect

85. C₅₉

#13642

semidirect

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

subgroup	index
M₃₂×Dic₅₉	2
#21422293	2
#21422236	2
#21422231	2
C₅₉×#1400	2
#13642	59

subgroup

index

M₃₂×Dic₅₉

C₅₉×#1400

Subgroups — #28509225

Conjugacy classes of subgroups

Complements

Maximal subgroups