Group #28500323 — subgroups

Subgroups — #28500323

Order 15104.

232 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 91 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	#21425689	C₂	1	1	yes	maximal, characteristic
2	#21422850	C₂	1	2	yes	maximal
3	#21422162	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422161	C₂	1	2	yes	maximal, complemented
5	C₅₉×#1368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14273666	C₄	1	1	yes	characteristic
7	C₂×#14273666	C₄	1	1	yes	characteristic
8	#16837424	—	2	2	no	—
9	C₄×C₈×D₅₉	C₂²	1	1	yes	characteristic
10	#16837178	C₂²	1	1	yes	characteristic
11	C₁₆×Dic₅₉	C₄	1	1	yes	characteristic
12	C₁₆×Dic₅₉	C₄	1	1	yes	characteristic
13	#16837153	C₂²	1	2	yes	—
14	#16837152	—	2	2	no	—
15	C₅₉×#346	C₂²	1	1	yes	characteristic
16	C₅₉×#334	C₂²	1	2	yes	—
17	C₂×#11745659	—	4	2	no	—
18	C₂×C₈×D₅₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×D₅₉	Q₈	1	1	yes	characteristic
20	C₄²×D₅₉	D₄	1	1	yes	characteristic
21	#14273666	—	2	2	no	—
22	#14273666	—	2	2	no	—
23	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
26	Dic₅₉⋊C₈	—	4	2	no	—
27	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
28	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₅₉	Q₈	1	1	yes	characteristic
30	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₈×C₅₉	—	2	2	no	—
33	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
34	C₂×C₄×D₅₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×D₅₉	SD₁₆	1	2	yes	—
36	C₂×C₈×C₅₉	—	4	2	no	—
37	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
39	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₄×Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
41	C₄×Dic₅₉	SD₁₆	1	2	yes	—
42	C₂×C₅₉⋊C₈	—	4	2	no	—
43	C₂×C₅₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₅₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
45	#11745659	—	8	4	no	—
46	C₈×D₅₉	—	2	1	no	—
47	C₈×D₅₉	—	2	1	no	—
48	C₈×D₅₉	—	2	2	no	—
49	C₁₆×C₅₉	—	2	1	no	—
50	C₁₆×C₅₉	M₁₆	1	2	yes	complemented
51	C₅₉⋊C₁₆	—	2	1	no	—
52	C₅₉⋊C₁₆	M₁₆	1	2	yes	complemented
53	C₂²×D₅₉	#106	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
56	C₂×Dic₅₉	#106	1	1	yes	characteristic
57	C₂×Dic₅₉	#106	1	2	yes	—
58	C₄×D₅₉	—	2	1	no	—
59	C₄×D₅₉	—	2	1	no	—
60	C₄×D₅₉	—	2	4	no	—
61	C₈×C₅₉	—	2	1	no	—
62	C₈×C₅₉	—	8	2	no	—
63	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
64	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
65	C₅₉⋊C₈	—	2	1	no	—
66	C₅₉⋊C₈	—	8	2	no	—
67	C₅₉⋊C₈	#108	1	2	yes	—
68	#12020	—	59	1	no	maximal, Sylow
69	C₂²×C₅₉	C₂×#106	1	1	yes	characteristic
70	C₂×D₅₉	—	2	1	no	—
71	C₂×D₅₉	—	2	2	no	—
72	C₄×C₅₉	—	2	2	no	—
73	C₄×C₅₉	#423	1	1	yes	characteristic
74	C₄×C₅₉	C₂×#108	1	1	yes	characteristic
75	Dic₅₉	—	2	1	no	—
76	Dic₅₉	#365	1	4	yes	—
77	C₄×M₃₂	—	59	1	no	—
78	#1566	—	59	2	no	—
79	#1368	—	59	1	no	—
80	#1368	—	59	2	no	—
81	#1368	D₅₉	1	1	yes	characteristic, complemented
82	C₂×C₅₉	#1566	1	1	yes	characteristic
83	C₂×C₅₉	C₂×#365	1	2	yes	—
84	D₅₉	—	4	1	no	—
85	C₂×M₃₂	—	59	1	no	—
86	C₂×M₃₂	—	59	1	no	—
87	#423	—	118	2	no	—
88	C₂×C₄×C₈	—	59	1	no	—
89	#346	—	59	1	no	—
90	#346	C₂×D₅₉	1	1	yes	characteristic
91	C₄×C₁₆	—	59	1	no	—
92	C₄×C₁₆	—	59	1	no	—
93	#334	—	59	2	no	—
94	#334	—	118	2	no	—
95	#334	C₂×D₅₉	1	2	yes	—
96	C₅₉	#12020	1	1	yes	characteristic, complemented, Sylow
97	C₂×M₁₆	—	236	2	no	—
98	C₂²×C₈	—	59	1	no	—
99	C₂²×C₈	—	59	1	no	—
100	C₂×C₄²	—	59	1	no	—
101	M₃₂	—	118	2	no	—
102	M₃₂	—	118	2	no	—
103	C₂×C₁₆	—	59	1	no	—
104	C₂×C₁₆	—	59	1	no	—
105	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
106	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	118	2	no	—
109	C₄⋊C₈	—	236	2	no	—
110	C₄×C₈	—	59	1	no	—
111	C₄×C₈	—	59	1	no	—
112	C₄×C₈	—	59	1	no	—
113	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
114	C₂²×C₄	—	59	1	no	—
115	C₂²×C₄	—	59	2	no	—
116	M₁₆	—	472	4	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	59	1	no	—
119	C₂×C₈	—	59	1	no	—
120	C₂×C₈	—	118	2	no	—
121	C₂×C₈	—	118	1	no	—
122	C₂×C₈	—	118	1	no	—
123	C₂×C₈	—	236	2	no	—
124	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
125	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
126	C₄²	—	59	1	no	—
127	C₄²	—	59	2	no	—
128	C₄²	D₄×D₅₉	1	1	yes	characteristic
129	C₁₆	—	2	1	no	—
130	C₁₆	—	59	2	no	—
131	C₁₆	—	118	1	no	—
132	C₁₆	#11745659	1	2	yes	complemented
133	C₂³	—	59	1	no	—
134	C₂×C₄	—	59	2	no	—
135	C₂×C₄	—	59	1	no	—
136	C₂×C₄	—	118	4	no	—
137	C₂×C₄	—	118	1	no	—
138	C₂×C₄	—	118	1	no	—
139	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
140	C₂×C₄	SD₁₆×D₅₉	1	2	yes	—
141	C₈	—	2	1	no	—
142	C₈	—	8	2	no	—
143	C₈	—	59	2	no	—
144	C₈	—	118	1	no	—
145	C₈	—	472	2	no	—
146	C₈	M₁₆×D₅₉	1	1	yes	characteristic
147	C₈	C₂×#11745659	1	1	yes	characteristic
148	C₂²	—	118	1	no	—
149	C₂²	—	118	2	no	—
150	C₂²	#106×D₅₉	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	59	4	no	—
153	C₄	—	118	1	no	—
154	C₄	#16837424	1	1	yes	characteristic
155	C₄	#108×D₅₉	1	1	yes	characteristic
156	C₂	—	236	1	no	—
157	C₂	#21422850	1	1	yes	characteristic
158	C₂	#365×D₅₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21425689

C₂

yes

maximal, characteristic

#21422850

C₂

yes

maximal

#21422162

C₂

yes

maximal, characteristic, complemented

#21422161

C₂

yes

maximal, complemented

C₅₉×#1368

C₂

yes

maximal, characteristic, complemented

C₂×#14273666

C₄

yes

characteristic

C₂×#14273666

C₄

yes

characteristic

#16837424

—

C₄×C₈×D₅₉

C₂²

yes

characteristic

#16837178

C₂²

yes

characteristic

C₁₆×Dic₅₉

C₄

yes

characteristic

C₁₆×Dic₅₉

C₄

yes

characteristic

#16837153

C₂²

yes

—

#16837152

—

C₅₉×#346

C₂²

yes

characteristic

C₅₉×#334

C₂²

yes

—

C₂×#11745659

—

C₂×C₈×D₅₉

C₂×C₄

yes

characteristic

C₂×C₈×D₅₉

Q₈

yes

characteristic

C₄²×D₅₉

D₄

yes

characteristic

#14273666

—

#14273666

—

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

characteristic

C₄⋊(C₅₉⋊C₈)

—

Dic₅₉⋊C₈

—

C₄×C₅₉⋊C₈

D₄

yes

characteristic

C₈×Dic₅₉

C₂×C₄

yes

characteristic

C₈×Dic₅₉

Q₈

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄×D₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×D₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄×Dic₅₉

C₄⋊C₄

yes

characteristic

C₄×Dic₅₉

SD₁₆

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₄⋊C₄

yes

characteristic

#11745659

—

C₈×D₅₉

—

C₈×D₅₉

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₁₆×C₅₉

M₁₆

yes

complemented

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

M₁₆

yes

complemented

C₂²×D₅₉

#106

yes

characteristic

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂×Dic₅₉

#106

yes

characteristic

C₂×Dic₅₉

#106

yes

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×M₁₆

yes

characteristic

C₈×C₅₉

C₂×M₁₆

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

#108

yes

—

—

C₂²×C₅₉

yes

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

#365

yes

—

C₄×M₃₂

—

—

—

—

D₅₉

yes

characteristic, complemented

C₂×C₅₉

#1566

yes

characteristic

C₂×C₅₉

C₂×#365

yes

—

D₅₉

—

C₂×M₃₂

—

C₂×M₃₂

—

#423

—

118

—

C₂×C₄×C₈

—

#346

—

#346

C₂×D₅₉

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

#334

—

#334

—

118

—

#334

C₂×D₅₉

yes

—

C₅₉

#12020

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

236

—

C₂²×C₈

—

C₂²×C₈

—

100

C₂×C₄²

—

101

M₃₂

—

118

—

102

M₃₂

—

118

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

C₄×D₅₉

yes

characteristic

106

C₂×C₁₆

C₄×D₅₉

yes

characteristic

107

C₄⋊C₈

—

108

C₄⋊C₈

—

118

—

109

C₄⋊C₈

—

236

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

C₂²×D₅₉

yes

characteristic

114

C₂²×C₄

—

115

C₂²×C₄

—

116

M₁₆

—

472

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

118

—

121

C₂×C₈

—

118

—

122

C₂×C₈

—

118

—

123

C₂×C₈

—

236

—

124

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

125

C₂×C₈

Q₈×D₅₉

yes

characteristic

126

C₄²

—

127

C₄²

—

128

C₄²

D₄×D₅₉

yes

characteristic

129

C₁₆

—

130

C₁₆

—

131

C₁₆

—

118

—

132

C₁₆

#11745659

yes

complemented

133

C₂³

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

118

—

137

C₂×C₄

—

118

—

138

C₂×C₄

—

118

—

139

C₂×C₄

C₄⋊C₄×D₅₉

yes

characteristic

140

C₂×C₄

SD₁₆×D₅₉

yes

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

118

—

145

C₈

—

472

—

146

C₈

M₁₆×D₅₉

yes

characteristic

147

C₈

C₂×#11745659

yes

characteristic

148

C₂²

—

118

—

149

C₂²

—

118

—

150

C₂²

#106×D₅₉

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

118

—

154

C₄

#16837424

yes

characteristic

155

C₄

#108×D₅₉

yes

characteristic

156

C₂

—

236

—

157

C₂

#21422850

yes

characteristic

158

C₂

#365×D₅₉

yes

—

	subgroup	G/H
center	C₂²	#106×D₅₉
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1368	C₂
socle	C₂²×C₅₉	C₂×#106
prime radical	C₂²×D₅₉	#106
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	#1368	D₅₉
hypocenter	C₅₉	#12020
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

subgroup

G/H

center

C₂²

#106×D₅₉

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1368

C₂

C₂²×C₅₉

C₂²×D₅₉

C₄×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

N	H ≅ G/N	type
3. #21422162	C₂	semidirect
4. #21422161	C₂	semidirect
5. C₅₉×#1368	C₂	semidirect
50. C₁₆×C₅₉	M₁₆	semidirect
52. C₅₉⋊C₁₆	M₁₆	semidirect
81. #1368	D₅₉	semidirect
96. C₅₉	#12020	semidirect
132. C₁₆	#11745659	semidirect

H ≅ G/N

type

3. #21422162

C₂

semidirect

4. #21422161

C₂

semidirect

5. C₅₉×#1368

C₂

semidirect

50. C₁₆×C₅₉

M₁₆

semidirect

52. C₅₉⋊C₁₆

M₁₆

semidirect

81. #1368

D₅₉

semidirect

96. C₅₉

#12020

semidirect

132. C₁₆

#11745659

semidirect

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

subgroup	index
#21425689	2
#21422850	2
#21422162	2
#21422161	2
C₅₉×#1368	2
#12020	59

subgroup

index

C₅₉×#1368

Subgroups — #28500323

Conjugacy classes of subgroups

Complements

Maximal subgroups