Group #28491803 — subgroups

Subgroups — #28491803

Order 15104.

364 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 89 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	#21432483	C₂	1	1	yes	maximal, characteristic
2	#21425073	C₂	1	1	yes	maximal, characteristic
3	#21422661	C₂	1	2	yes	maximal
4	#21421955	C₂	1	2	yes	maximal
5	C₅₉×#1276	C₂	1	1	yes	maximal, characteristic
6	#16837959	C₂²	1	1	yes	characteristic
7	#16837426	C₂²	1	1	yes	characteristic
8	#16837300	—	2	2	no	—
9	#16837286	—	2	4	no	—
10	#16837268	C₂²	1	2	yes	—
11	#16837162	—	2	2	no	—
12	#16837140	—	2	4	no	—
13	C₅₉×#431	C₂²	1	2	yes	—
14	C₄³×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×#11745664	—	2	1	no	—
16	#14273727	Q₈	1	2	yes	—
17	#14273719	—	2	1	no	—
18	#14273706	—	2	2	no	—
19	C₄×Dic₁₁₈	—	2	2	no	—
20	#14273686	—	2	1	no	—
21	C₂×C₄⋊Dic₅₉	—	2	2	no	—
22	C₂×Dic₅₉⋊C₄	—	2	4	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	C₅₉⋊#95	—	2	4	no	—
25	C₅₉⋊#95	—	2	2	no	—
26	C₅₉⋊#95	D₄	1	4	yes	—
27	C₂×C₄²×C₅₉	—	2	2	no	—
28	C₂×C₄²×C₅₉	D₄	1	2	yes	—
29	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₅₉	—	2	2	no	—
31	C₂²⋊C₈×C₅₉	—	2	2	no	—
32	C₄×C₈×C₅₉	—	2	2	no	—
33	C₂²×C₄×C₅₉	—	2	2	no	—
34	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
35	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₅₉	—	2	4	no	—
37	C₂×Dic₁₁₈	—	2	2	no	—
38	C₂×C₈×C₅₉	—	4	2	no	—
39	C₄²×C₅₉	—	2	2	no	—
40	C₄²×C₅₉	—	2	2	no	—
41	C₄²×C₅₉	—	4	4	no	—
42	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₅₉	#41	1	2	yes	—
44	C₄²×C₅₉	#41	1	1	yes	characteristic
45	C₂²⋊Dic₅₉	—	4	1	no	—
46	C₄⋊Dic₅₉	—	2	2	no	—
47	C₄⋊Dic₅₉	—	2	2	no	—
48	C₄⋊Dic₅₉	—	4	2	no	—
49	C₄⋊Dic₅₉	—	4	2	no	—
50	Dic₅₉⋊C₄	—	4	8	no	—
51	Dic₅₉⋊C₄	—	4	2	no	—
52	C₄×Dic₅₉	—	4	1	no	—
53	#11745664	—	4	2	no	—
54	C₂×C₅₉⋊C₈	—	4	1	no	—
55	C₂³×C₅₉	#120	1	1	yes	characteristic
56	C₂×C₄×C₅₉	—	2	4	no	—
57	C₂×C₄×C₅₉	—	4	2	no	—
58	C₂×C₄×C₅₉	—	4	2	no	—
59	C₂×C₄×C₅₉	—	4	2	no	—
60	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
61	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#124	1	2	yes	—
63	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
64	C₂×C₄×C₅₉	#135	1	2	yes	—
65	C₂×C₄×C₅₉	#137	1	2	yes	—
66	C₂×Dic₅₉	—	4	8	no	—
67	C₂×Dic₅₉	—	4	2	no	—
68	C₂×Dic₅₉	—	4	4	no	—
69	Dic₁₁₈	—	8	2	no	—
70	C₈×C₅₉	—	8	2	no	—
71	C₅₉⋊C₈	—	8	1	no	—
72	#10367	—	59	1	no	maximal, Sylow
73	C₂²×C₅₉	—	2	2	no	—
74	C₂²×C₅₉	#393	1	1	yes	characteristic
75	C₂²×C₅₉	#452	1	2	yes	—
76	C₄×C₅₉	—	2	2	no	—
77	C₄×C₅₉	—	2	2	no	—
78	C₄×C₅₉	—	4	4	no	—
79	C₄×C₅₉	#479	1	2	yes	—
80	C₄×C₅₉	#488	1	1	yes	characteristic
81	C₄×C₅₉	#491	1	1	yes	characteristic
82	Dic₅₉	—	8	4	no	—
83	Dic₅₉	—	8	2	no	—
84	#2864	—	59	1	no	—
85	#1994	—	59	1	no	—
86	#1518	—	59	2	no	—
87	#1286	—	59	2	no	—
88	#1276	D₅₉	1	1	yes	characteristic
89	C₂×C₅₉	—	2	2	no	—
90	C₂×C₅₉	#2026	1	1	yes	characteristic
91	C₂×C₅₉	#2031	1	2	yes	—
92	#533	—	59	1	no	—
93	#431	C₂×D₅₉	1	2	yes	—
94	#423	—	59	1	no	—
95	#389	—	118	2	no	—
96	#387	—	118	4	no	—
97	#382	—	59	2	no	—
98	C₄³	C₂×D₅₉	1	1	yes	characteristic
99	#339	—	118	2	no	—
100	#328	—	118	4	no	—
101	C₅₉	#10367	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	118	1	no	—
103	C₄⋊Q₈	—	59	2	no	—
104	#125	—	118	1	no	—
105	C₂²⋊Q₈	—	118	2	no	—
106	C₄×Q₈	—	118	2	no	—
107	#117	—	118	1	no	—
108	C₂×C₄⋊C₄	—	118	4	no	—
109	C₂×C₄⋊C₄	—	118	2	no	—
110	C₂×C₄²	—	2	2	no	—
111	C₂×C₄²	D₂₃₆	1	2	yes	—
112	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	118	2	no	—
115	C₂²⋊C₈	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	#95	—	59	4	no	—
118	#95	—	118	4	no	—
119	#95	—	118	2	no	—
120	C₂×Q₈	—	118	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	118	4	no	—
123	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
124	C₂²×C₄	D₄×D₅₉	1	2	yes	—
125	M₁₆	—	236	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	—	236	1	no	—
128	C₄⋊C₄	—	118	2	no	—
129	C₄⋊C₄	—	118	2	no	—
130	C₄⋊C₄	—	236	8	no	—
131	C₄⋊C₄	—	236	2	no	—
132	C₄⋊C₄	—	236	2	no	—
133	C₄⋊C₄	—	236	2	no	—
134	C₂²⋊C₄	—	236	1	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	4	4	no	—
138	C₄²	—	236	1	no	—
139	C₄²	#11745685	1	2	yes	—
140	C₄²	#11745687	1	1	yes	characteristic
141	C₄²	Q₈×D₅₉	1	1	yes	characteristic
142	C₂³	#14273697	1	1	yes	characteristic
143	Q₈	—	472	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	236	8	no	—
149	C₂×C₄	—	236	4	no	—
150	C₂×C₄	—	236	2	no	—
151	C₂×C₄	#14273705	1	2	yes	—
152	C₂×C₄	#14273707	1	1	yes	characteristic
153	C₂×C₄	#14273713	1	2	yes	—
154	C₂×C₄	#14273715	1	1	yes	characteristic
155	C₂×C₄	#14273749	1	2	yes	—
156	C₂×C₄	#14273760	1	2	yes	—
157	C₈	—	8	2	no	—
158	C₈	—	472	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	#16837315	1	1	yes	characteristic
161	C₂²	#16837566	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	4	4	no	—
165	C₄	—	472	4	no	—
166	C₄	—	472	2	no	—
167	C₄	#16837710	1	2	yes	—
168	C₄	#16837757	1	1	yes	characteristic
169	C₄	#16837771	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#21425248	1	1	yes	characteristic
172	C₂	#21425269	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21432483

C₂

yes

maximal, characteristic

#21425073

C₂

yes

maximal, characteristic

#21422661

C₂

yes

maximal

#21421955

C₂

yes

maximal

C₅₉×#1276

C₂

yes

maximal, characteristic

#16837959

C₂²

yes

characteristic

#16837426

C₂²

yes

—

—

C₂²

yes

—

#16837162

—

#16837140

—

C₅₉×#431

C₂²

yes

—

C₄³×C₅₉

C₂²

yes

characteristic

C₂×#11745664

—

#14273727

Q₈

yes

—

#14273719

—

#14273706

—

C₄×Dic₁₁₈

—

#14273686

—

C₂×C₄⋊Dic₅₉

—

C₂×Dic₅₉⋊C₄

—

C₄⋊(C₅₉⋊C₈)

—

C₅₉⋊#95

—

C₅₉⋊#95

—

C₅₉⋊#95

D₄

yes

—

C₂×C₄²×C₅₉

—

C₂×C₄²×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²⋊C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₄²×C₅₉

#41

yes

—

C₄²×C₅₉

#41

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#120

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

#124

yes

—

C₂×C₄×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

#135

yes

—

C₂×C₄×C₅₉

#137

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

#10367

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#393

yes

characteristic

C₂²×C₅₉

#452

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#479

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

—

—

—

—

D₅₉

yes

characteristic

C₂×C₅₉

—

C₂×C₅₉

#2026

yes

characteristic

C₂×C₅₉

#2031

yes

—

#533

—

#431

C₂×D₅₉

yes

—

#423

—

#389

—

118

—

#387

—

118

—

#382

—

C₄³

C₂×D₅₉

yes

characteristic

#339

—

118

—

100

#328

—

118

—

101

C₅₉

#10367

yes

characteristic, complemented, Sylow

102

C₂×M₁₆

—

118

—

103

C₄⋊Q₈

—

104

#125

—

118

—

105

C₂²⋊Q₈

—

118

—

106

C₄×Q₈

—

118

—

107

#117

—

118

—

108

C₂×C₄⋊C₄

—

118

—

109

C₂×C₄⋊C₄

—

118

—

110

C₂×C₄²

—

111

C₂×C₄²

D₂₃₆

yes

—

112

C₂×C₄²

C₂²×D₅₉

yes

characteristic

113

C₄⋊C₈

—

114

C₄⋊C₈

—

118

—

115

C₂²⋊C₈

—

116

C₄×C₈

—

117

#95

—

118

#95

—

118

—

119

#95

—

118

—

120

C₂×Q₈

—

118

—

121

C₂²×C₄

—

122

C₂²×C₄

—

118

—

123

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

124

C₂²×C₄

D₄×D₅₉

yes

—

125

M₁₆

—

236

—

126

C₂×C₈

—

127

C₂×C₈

—

236

—

128

C₄⋊C₄

—

118

—

129

C₄⋊C₄

—

118

—

130

C₄⋊C₄

—

236

—

131

C₄⋊C₄

—

236

—

132

C₄⋊C₄

—

236

—

133

C₄⋊C₄

—

236

—

134

C₂²⋊C₄

—

236

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

236

—

139

C₄²

#11745685

yes

—

140

C₄²

#11745687

yes

characteristic

141

C₄²

Q₈×D₅₉

yes

characteristic

142

C₂³

#14273697

yes

characteristic

143

Q₈

—

472

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

236

—

149

C₂×C₄

—

236

—

150

C₂×C₄

—

236

—

151

C₂×C₄

#14273705

yes

—

152

C₂×C₄

#14273707

yes

characteristic

153

C₂×C₄

#14273713

yes

—

154

C₂×C₄

#14273715

yes

characteristic

155

C₂×C₄

#14273749

yes

—

156

C₂×C₄

#14273760

yes

—

157

C₈

—

158

C₈

—

472

—

159

C₂²

—

160

C₂²

#16837315

yes

characteristic

161

C₂²

#16837566

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

472

—

166

C₄

—

472

—

167

C₄

#16837710

yes

—

168

C₄

#16837757

yes

characteristic

169

C₄

#16837771

yes

characteristic

170

C₂

—

171

C₂

#21425248

yes

characteristic

172

C₂

#21425269

yes

—

	subgroup	G/H
center	C₂²	#16837315
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1276	C₂
socle	C₂²×C₅₉	#393
prime radical	C₂³×C₅₉	#120
hypercenter	#1276	D₅₉
hypocenter	C₅₉	#10367
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₂²

#16837315

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1276

C₂

C₂²×C₅₉

C₂³×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
101. C₅₉	#10367	semidirect

H ≅ G/N

type

101. C₅₉

#10367

semidirect

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

subgroup	index
#21432483	2
#21425073	2
#21422661	2
#21421955	2
C₅₉×#1276	2
#10367	59

subgroup

index

C₅₉×#1276

Subgroups — #28491803

Conjugacy classes of subgroups

Complements

Maximal subgroups