Group #28491799 — subgroups

Subgroups — #28491799

Order 15104.

364 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 93 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	#21425073	C₂	1	1	yes	maximal, characteristic
2	#21422659	C₂	1	2	yes	maximal
3	#21421954	C₂	1	2	yes	maximal
4	#21421929	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#2864	C₂	1	1	yes	maximal, characteristic
6	#16837959	C₂²	1	1	yes	characteristic
7	#16837455	C₂²	1	2	yes	—
8	#16837426	C₂²	1	1	yes	characteristic
9	#16837162	—	2	2	no	—
10	#16837138	—	2	4	no	—
11	C₅₉×#389	—	2	2	no	—
12	C₅₉×#387	—	2	4	no	—
13	C₅₉×#382	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	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
22	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
23	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
24	C₄×C₅₉⋊C₈	—	2	2	no	—
25	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
26	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
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	#95×C₅₉	—	2	4	no	—
31	#95×C₅₉	—	2	2	no	—
32	#95×C₅₉	D₄	1	4	yes	—
33	C₂²×C₄×C₅₉	—	2	2	no	—
34	C₂²×C₄×C₅₉	—	2	4	no	—
35	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
36	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₁₁₈	—	2	2	no	—
38	C₄⋊C₄×C₅₉	—	4	8	no	—
39	C₄⋊C₄×C₅₉	—	4	2	no	—
40	C₄⋊C₄×C₅₉	—	4	2	no	—
41	C₄²×C₅₉	—	2	2	no	—
42	C₄²×C₅₉	—	2	2	no	—
43	C₄²×C₅₉	—	4	4	no	—
44	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₅₉	#41	1	1	yes	characteristic
46	C₄²×C₅₉	#41	1	2	yes	—
47	C₂²⋊Dic₅₉	—	4	1	no	—
48	C₄⋊Dic₅₉	—	2	2	no	—
49	C₄⋊Dic₅₉	—	2	2	no	—
50	Dic₅₉⋊C₄	—	4	2	no	—
51	C₄×Dic₅₉	—	4	1	no	—
52	#11745664	—	4	2	no	—
53	C₂×C₅₉⋊C₈	—	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	—	4	8	no	—
61	C₂×C₄×C₅₉	—	4	4	no	—
62	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
63	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₅₉	#124	1	2	yes	—
65	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#135	1	2	yes	—
67	C₂×C₄×C₅₉	#137	1	2	yes	—
68	C₂×Dic₅₉	—	4	2	no	—
69	Dic₁₁₈	—	8	2	no	—
70	C₅₉⋊C₈	—	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	—	8	4	no	—
80	C₄×C₅₉	#479	1	2	yes	—
81	C₄×C₅₉	#488	1	1	yes	characteristic
82	C₄×C₅₉	#491	1	1	yes	characteristic
83	Dic₅₉	—	8	2	no	—
84	#2864	D₅₉	1	1	yes	characteristic
85	#1994	—	59	1	no	—
86	#1518	—	59	2	no	—
87	#1286	—	59	2	no	—
88	#1276	—	59	1	no	—
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	—	59	2	no	—
94	#423	—	59	1	no	—
95	#389	—	2	2	no	—
96	#387	—	2	4	no	—
97	#382	C₂×D₅₉	1	2	yes	—
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	—	2	4	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄²	—	2	2	no	—
111	C₂×C₄²	C₂²⋊D₅₉	1	2	yes	—
112	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
113	C₄⋊C₈	—	118	2	no	—
114	C₄⋊C₈	—	118	2	no	—
115	C₂²⋊C₈	—	118	2	no	—
116	C₄×C₈	—	118	2	no	—
117	#95	—	2	4	no	—
118	#95	—	2	2	no	—
119	#95	C₂²⋊D₅₉	1	4	yes	—
120	C₂×Q₈	—	118	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
124	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
125	M₁₆	—	236	2	no	—
126	C₂×C₈	—	236	2	no	—
127	C₂×C₈	—	236	1	no	—
128	C₄⋊C₄	—	4	8	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	118	2	no	—
132	C₄⋊C₄	—	118	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	Q₈×D₅₉	1	1	yes	characteristic
140	C₄²	#11745689	1	1	yes	characteristic
141	C₄²	#11745689	1	2	yes	—
142	C₂³	#14273696	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	—	4	8	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	236	2	no	—
151	C₂×C₄	#14273703	1	2	yes	—
152	C₂×C₄	#14273703	1	1	yes	characteristic
153	C₂×C₄	#14273716	1	2	yes	—
154	C₂×C₄	#14273716	1	1	yes	characteristic
155	C₂×C₄	#14273747	1	2	yes	—
156	C₂×C₄	#14273757	1	2	yes	—
157	C₈	—	472	2	no	—
158	C₈	—	472	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	#16837313	1	1	yes	characteristic
161	C₂²	#16837564	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	4	4	no	—
165	C₄	—	8	4	no	—
166	C₄	—	472	2	no	—
167	C₄	#16837706	1	2	yes	—
168	C₄	#16837759	1	1	yes	characteristic
169	C₄	#16837772	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#21425244	1	1	yes	characteristic
172	C₂	#21425265	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21425073

C₂

yes

maximal, characteristic

#21422659

C₂

yes

maximal

#21421954

C₂

yes

maximal

#21421929

C₂

yes

maximal, characteristic

C₅₉×#2864

C₂

yes

maximal, characteristic

#16837959

C₂²

yes

characteristic

#16837455

C₂²

yes

—

#16837426

C₂²

yes

characteristic

#16837162

—

#16837138

—

C₅₉×#389

—

C₅₉×#387

—

C₅₉×#382

C₂²

yes

—

C₄³×C₅₉

C₂²

yes

characteristic

C₂×#11745664

—

#14273727

Q₈

yes

—

#14273719

—

#14273706

—

C₄×Dic₁₁₈

—

#14273686

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

—

C₂×C₄²×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#95×C₅₉

—

#95×C₅₉

—

#95×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂×Q₈

yes

C₄²×C₅₉

yes

C₄²×C₅₉

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#120

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

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

—

Dic₁₁₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#10367

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#393

yes

characteristic

C₂²×C₅₉

#452

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#479

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#2864

D₅₉

yes

—

—

—

—

C₂×C₅₉

—

C₂×C₅₉

#2026

yes

characteristic

C₂×C₅₉

#2031

yes

—

—

—

—

—

—

C₂×D₅₉

yes

—

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

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄²

—

111

C₂×C₄²

C₂²⋊D₅₉

yes

—

112

C₂×C₄²

C₂²×D₅₉

yes

characteristic

113

C₄⋊C₈

—

118

—

114

C₄⋊C₈

—

118

—

115

C₂²⋊C₈

—

118

—

116

C₄×C₈

—

118

—

117

#95

—

118

#95

—

119

#95

C₂²⋊D₅₉

yes

—

120

C₂×Q₈

—

118

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

—

124

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

125

M₁₆

—

236

—

126

C₂×C₈

—

236

—

127

C₂×C₈

—

236

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

118

—

132

C₄⋊C₄

—

118

—

133

C₄⋊C₄

—

236

—

134

C₂²⋊C₄

—

236

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

236

—

139

C₄²

Q₈×D₅₉

yes

characteristic

140

C₄²

#11745689

yes

characteristic

141

C₄²

#11745689

yes

—

142

C₂³

#14273696

yes

characteristic

143

Q₈

—

472

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

236

—

151

C₂×C₄

#14273703

yes

—

152

C₂×C₄

#14273703

yes

characteristic

153

C₂×C₄

#14273716

yes

—

154

C₂×C₄

#14273716

yes

characteristic

155

C₂×C₄

#14273747

yes

—

156

C₂×C₄

#14273757

yes

—

157

C₈

—

472

—

158

C₈

—

472

—

159

C₂²

—

160

C₂²

#16837313

yes

characteristic

161

C₂²

#16837564

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

472

—

167

C₄

#16837706

yes

—

168

C₄

#16837759

yes

characteristic

169

C₄

#16837772

yes

characteristic

170

C₂

—

171

C₂

#21425244

yes

characteristic

172

C₂

#21425265

yes

—

	subgroup	G/H
center	C₂²	#16837313
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#2864	C₂
socle	C₂²×C₅₉	#393
prime radical	C₂³×C₅₉	#120
hypercenter	#2864	D₅₉
hypocenter	C₅₉	#10367
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2864

subgroup

G/H

center

C₂²

#16837313

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#2864

C₂

C₂²×C₅₉

C₂³×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2864

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
#21425073	2
#21422659	2
#21421954	2
#21421929	2
C₅₉×#2864	2
#10367	59

subgroup

index

C₅₉×#2864

Subgroups — #28491799

Conjugacy classes of subgroups

Complements

Maximal subgroups