Group #28475944 — subgroups

Subgroups — #28475944

Order 15104.

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

normal subgroups only

This group has 216 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	#21425895	C₂	1	1	yes	maximal, characteristic
2	#21422855	C₂	1	1	yes	maximal, characteristic
3	C₄×#108×C₅₉	C₂	1	1	yes	maximal, characteristic
4	#16837426	—	2	1	no	—
5	C₄⋊(C₅₉⋊C₁₆)	C₄	1	2	yes	—
6	C₂²⋊(C₅₉⋊C₁₆)	—	2	1	no	—
7	C₄×C₅₉⋊C₁₆	—	2	1	no	—
8	#16837155	—	2	1	no	—
9	#16837155	C₄	1	2	yes	—
10	C₂×#108×C₅₉	—	2	1	no	—
11	C₄×M₁₆×C₅₉	—	2	1	no	—
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×#341	C₄	1	2	yes	—
14	C₂×#11745664	—	4	1	no	—
15	C₂×C₅₉⋊C₁₆	—	2	2	no	—
16	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
17	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
18	C₂×M₁₆×C₅₉	—	2	2	no	—
19	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
22	#108×C₅₉	—	2	4	no	—
23	#97×C₅₉	—	4	1	no	—
24	C₄×C₈×C₅₉	—	4	1	no	—
25	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
28	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
29	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₅₉	Q₁₆	1	1	yes	characteristic
32	M₁₆×C₅₉	—	2	4	no	—
33	M₁₆×C₅₉	—	4	2	no	—
34	C₂×C₈×C₅₉	—	2	2	no	—
35	C₂×C₈×C₅₉	—	4	1	no	—
36	C₂×C₈×C₅₉	—	4	2	no	—
37	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₅₉	D₈	1	1	yes	characteristic
44	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
45	#11745664	—	8	2	no	—
46	C₂×C₅₉⋊C₈	—	4	2	no	—
47	C₅₉⋊C₁₆	—	4	2	no	—
48	C₂³×C₅₉	#103	1	1	yes	characteristic
49	C₂×C₄×C₅₉	—	2	1	no	—
50	C₂×C₄×C₅₉	—	2	1	no	—
51	C₂×C₄×C₅₉	—	2	1	no	—
52	C₂×C₄×C₅₉	—	2	1	no	—
53	C₂×C₄×C₅₉	—	2	2	no	—
54	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
55	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
56	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
57	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
58	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
59	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
60	C₈×C₅₉	—	2	1	no	—
61	C₈×C₅₉	—	2	1	no	—
62	C₈×C₅₉	—	2	1	no	—
63	C₈×C₅₉	—	4	4	no	—
64	C₈×C₅₉	#104	1	2	yes	—
65	C₅₉⋊C₈	—	8	2	no	—
66	#7298	—	59	1	no	maximal, Sylow
67	C₂²×C₅₉	—	2	1	no	—
68	C₂²×C₅₉	—	2	1	no	—
69	C₂²×C₅₉	—	2	1	no	—
70	C₂²×C₅₉	#340	1	1	yes	characteristic
71	C₄×C₅₉	—	2	1	no	—
72	C₄×C₅₉	—	2	1	no	—
73	C₄×C₅₉	—	2	1	no	—
74	C₄×C₅₉	#339	1	1	yes	characteristic
75	C₄×C₅₉	#344	1	1	yes	characteristic
76	C₄×C₅₉	#361	1	1	yes	characteristic
77	C₄×C₅₉	#362	1	1	yes	characteristic
78	C₄×C₅₉	#368	1	2	yes	—
79	#2162	—	59	1	no	—
80	C₄×#108	D₅₉	1	1	yes	characteristic
81	#1567	—	59	1	no	—
82	C₂×C₅₉	—	4	1	no	—
83	C₂×C₅₉	#1303	1	1	yes	characteristic
84	C₂×C₅₉	#1381	1	1	yes	characteristic
85	C₂×C₅₉	#1387	1	1	yes	characteristic
86	C₂×#108	—	2	1	no	—
87	#423	—	118	1	no	—
88	C₄×M₁₆	—	2	1	no	—
89	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
90	C₄⋊C₁₆	—	59	2	no	—
91	C₂²⋊C₁₆	—	118	1	no	—
92	C₄×C₁₆	—	118	1	no	—
93	#341	Dic₅₉	1	2	yes	—
94	#335	—	59	2	no	—
95	#335	—	118	1	no	—
96	C₅₉	#7298	1	1	yes	characteristic, complemented, Sylow
97	C₂×M₁₆	—	2	2	no	—
98	C₂×M₁₆	—	236	1	no	—
99	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
100	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
101	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
102	C₂×C₁₆	—	118	2	no	—
103	#108	—	2	4	no	—
104	C₄⋊C₈	—	118	2	no	—
105	C₄⋊C₈	—	236	1	no	—
106	#97	—	4	1	no	—
107	C₄×C₈	—	4	1	no	—
108	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
109	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
110	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
111	C₄×C₈	D₂₃₆	1	1	yes	characteristic
112	C₂²×C₄	#11745671	1	1	yes	characteristic
113	C₂²×C₄	#11745672	1	1	yes	characteristic
114	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
115	M₁₆	—	2	4	no	—
116	M₁₆	—	4	2	no	—
117	M₁₆	—	472	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	—	236	2	no	—
122	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
123	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
124	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
125	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
126	C₄²	Dic₅₉⋊C₄	1	1	yes	characteristic
127	C₄²	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
128	C₄²	#11745669	1	1	yes	characteristic
129	C₄²	#11745670	1	1	yes	characteristic
130	C₁₆	—	236	2	no	—
131	C₂³	#14273651	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
138	C₂×C₄	#14273647	1	1	yes	characteristic
139	C₂×C₄	#14273648	1	1	yes	characteristic
140	C₂×C₄	#14273650	1	1	yes	characteristic
141	C₂×C₄	#14273658	1	1	yes	characteristic
142	C₂×C₄	#14273660	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	4	4	no	—
147	C₈	—	472	2	no	—
148	C₈	#14273655	1	2	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#16837164	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	#16837162	1	1	yes	characteristic
157	C₄	#16837172	1	1	yes	characteristic
158	C₄	#16837214	1	1	yes	characteristic
159	C₄	#16837217	1	1	yes	characteristic
160	C₄	#16837231	1	2	yes	—
161	C₂	—	4	1	no	—
162	C₂	#21421999	1	1	yes	characteristic
163	C₂	#21422189	1	1	yes	characteristic
164	C₂	#21422206	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21425895

C₂

yes

maximal, characteristic

#21422855

C₂

yes

maximal, characteristic

C₄×#108×C₅₉

C₂

yes

maximal, characteristic

#16837426

—

C₄⋊(C₅₉⋊C₁₆)

C₄

yes

—

C₂²⋊(C₅₉⋊C₁₆)

—

C₄×C₅₉⋊C₁₆

—

#16837155

—

#16837155

C₄

yes

—

C₂×#108×C₅₉

—

C₄×M₁₆×C₅₉

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#341

C₄

yes

—

C₂×#11745664

—

C₂×C₅₉⋊C₁₆

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

#108×C₅₉

—

#97×C₅₉

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

Q₈

yes

characteristic

C₂²×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₉

SD₁₆

yes

characteristic

C₂²×C₄×C₅₉

Q₁₆

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄²

yes

characteristic

C₂×C₈×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₄²×C₅₉

C₂²⋊C₄

yes

characteristic

C₄²×C₅₉

C₄⋊C₄

yes

characteristic

C₄²×C₅₉

D₈

yes

characteristic

C₄²×C₅₉

SD₁₆

yes

characteristic

#11745664

—

C₂×C₅₉⋊C₈

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

#103

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#104

yes

—

C₅₉⋊C₈

—

#7298

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#340

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

#2162

—

C₄×#108

D₅₉

yes

characteristic

#1567

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

118

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

118

—

C₄×C₁₆

—

118

—

#341

Dic₅₉

yes

—

#335

—

#335

—

118

—

C₅₉

#7298

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

236

—

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

100

C₂²×C₈

C₂²⋊D₅₉

yes

characteristic

101

C₂×C₄²

C₂²⋊D₅₉

yes

characteristic

102

C₂×C₁₆

—

118

—

103

#108

—

104

C₄⋊C₈

—

118

—

105

C₄⋊C₈

—

236

—

106

#97

—

107

C₄×C₈

—

108

C₄×C₈

Dic₁₁₈

yes

characteristic

109

C₄×C₈

C₄×D₅₉

yes

characteristic

110

C₄×C₈

C₄×D₅₉

yes

characteristic

111

C₄×C₈

D₂₃₆

yes

characteristic

112

C₂²×C₄

#11745671

yes

characteristic

113

C₂²×C₄

#11745672

yes

characteristic

114

C₂²×C₄

C₂²⋊Dic₅₉

yes

characteristic

115

M₁₆

—

116

M₁₆

—

117

M₁₆

—

472

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

236

—

122

C₂×C₈

C₄×Dic₅₉

yes

characteristic

123

C₂×C₈

Dic₅₉⋊C₄

yes

characteristic

124

C₂×C₈

C₄⋊Dic₅₉

yes

characteristic

125

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

characteristic

126

C₄²

Dic₅₉⋊C₄

yes

characteristic

127

C₄²

(C₂×D₅₉)⋊C₄

yes

characteristic

128

C₄²

#11745669

yes

characteristic

129

C₄²

#11745670

yes

characteristic

130

C₁₆

—

236

—

131

C₂³

#14273651

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₅₉⋊#95

yes

characteristic

138

C₂×C₄

#14273647

yes

characteristic

139

C₂×C₄

#14273648

yes

characteristic

140

C₂×C₄

#14273650

yes

characteristic

141

C₂×C₄

#14273658

yes

characteristic

142

C₂×C₄

#14273660

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

472

—

148

C₈

#14273655

yes

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

#16837164

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#16837162

yes

characteristic

157

C₄

#16837172

yes

characteristic

158

C₄

#16837214

yes

characteristic

159

C₄

#16837217

yes

characteristic

160

C₄

#16837231

yes

—

161

C₂

—

162

C₂

#21421999

yes

characteristic

163

C₂

#21422189

yes

characteristic

164

C₂

#21422206

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14273647
commutator	C₂×C₈×C₅₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₉
Fitting	C₄×#108×C₅₉	C₂
socle	C₂²×C₅₉	#340
prime radical	C₂³×C₅₉	#103
cosocle	C₂×C₄×C₈×C₅₉	C₂²
hypercenter	C₄×#108	D₅₉
hypocenter	C₅₉	#7298
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

subgroup

G/H

center

C₂×C₄

#14273647

commutator

C₂×C₈×C₅₉

C₄²

Frattini

C₂×C₄×C₈

C₂×D₅₉

Fitting

C₄×#108×C₅₉

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄×C₈×C₅₉

C₂²

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

N	H ≅ G/N	type
96. C₅₉	#7298	semidirect

H ≅ G/N

type

96. C₅₉

#7298

semidirect

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

subgroup	index
#21425895	2
#21422855	2
C₄×#108×C₅₉	2
#7298	59

subgroup

index

#21425895

#21422855

C₄×#108×C₅₉

#7298

Subgroups — #28475944

Conjugacy classes of subgroups

Complements

Maximal subgroups