Group #28482366 — subgroups

Subgroups — #28482366

Order 15104.

342 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 155 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	#21424248	C₂	1	1	yes	maximal, characteristic
2	#21424213	C₂	1	1	yes	maximal, characteristic
3	#21421931	C₂	1	4	yes	maximal
4	C₅₉×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
8	#16837157	—	2	2	no	—
9	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
10	C₄³×C₅₉	C₂²	1	1	yes	characteristic
11	C₅₉×#336	C₄	1	2	yes	—
12	C₅₉×#336	C₄	1	2	yes	—
13	C₂²×C₅₉⋊C₈	—	2	2	no	—
14	C₂²×C₅₉⋊C₈	—	2	2	no	—
15	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
16	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
17	C₄⋊(C₅₉⋊C₈)	—	4	4	no	—
18	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₅₉	D₄	1	2	yes	—
22	C₂×C₄²×C₅₉	Q₈	1	2	yes	—
23	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₅₉	—	2	2	no	—
25	C₄×C₈×C₅₉	—	2	2	no	—
26	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
27	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₈×C₅₉	—	2	4	no	—
32	C₂×C₈×C₅₉	—	2	4	no	—
33	C₂×C₈×C₅₉	—	2	4	no	—
34	C₄²×C₅₉	—	2	4	no	—
35	C₄²×C₅₉	—	2	2	no	—
36	C₄²×C₅₉	—	2	2	no	—
37	C₄²×C₅₉	—	2	2	no	—
38	C₄²×C₅₉	—	2	2	no	—
39	C₄²×C₅₉	#41	1	2	yes	—
40	C₄²×C₅₉	#41	1	2	yes	—
41	C₂×C₅₉⋊C₈	—	4	2	no	—
42	C₂×C₅₉⋊C₈	—	4	2	no	—
43	C₂×C₅₉⋊C₈	—	4	4	no	—
44	C₂×C₅₉⋊C₈	—	4	2	no	—
45	C₂×C₅₉⋊C₈	—	4	2	no	—
46	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	—	2	2	no	—
48	C₂×C₄×C₅₉	—	2	2	no	—
49	C₂×C₄×C₅₉	—	2	2	no	—
50	C₂×C₄×C₅₉	—	2	2	no	—
51	C₂×C₄×C₅₉	—	2	2	no	—
52	C₂×C₄×C₅₉	—	2	2	no	—
53	C₂×C₄×C₅₉	#108	1	4	yes	—
54	C₂×C₄×C₅₉	#117	1	2	yes	—
55	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
56	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
57	C₂×C₄×C₅₉	#125	1	2	yes	—
58	C₂×C₄×C₅₉	#131	1	2	yes	—
59	C₂×C₄×C₅₉	#131	1	2	yes	—
60	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
61	C₂×C₄×C₅₉	#137	1	2	yes	—
62	C₈×C₅₉	—	4	4	no	—
63	C₅₉⋊C₈	—	8	2	no	—
64	C₅₉⋊C₈	—	8	2	no	—
65	#8789	—	59	1	no	maximal, Sylow
66	C₂²×C₅₉	#382	1	1	yes	characteristic
67	C₂²×C₅₉	#424	1	1	yes	characteristic
68	C₂²×C₅₉	#424	1	1	yes	characteristic
69	C₂²×C₅₉	#428	1	2	yes	—
70	C₂²×C₅₉	C₂×#108	1	2	yes	—
71	C₄×C₅₉	—	2	4	no	—
72	C₄×C₅₉	—	2	2	no	—
73	C₄×C₅₉	—	4	2	no	—
74	C₄×C₅₉	#433	1	4	yes	—
75	C₄×C₅₉	#489	1	2	yes	—
76	C₄×C₅₉	#490	1	1	yes	characteristic
77	C₄×C₅₉	#491	1	1	yes	characteristic
78	#1842	—	59	1	no	—
79	#1831	—	59	1	no	—
80	#1744	D₅₉	1	1	yes	characteristic
81	#1277	—	59	4	no	—
82	C₂×C₅₉	#1573	1	2	yes	—
83	C₂×C₅₉	#1574	1	2	yes	—
84	C₂×C₅₉	#1832	1	1	yes	characteristic
85	C₂×C₅₉	#1836	1	1	yes	characteristic
86	C₂×C₅₉	#1837	1	1	yes	characteristic
87	C₂×C₄⋊C₈	—	59	2	no	—
88	C₂×C₄⋊C₈	—	59	2	no	—
89	C₂×C₄⋊C₈	—	118	2	no	—
90	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
91	C₄³	C₂×D₅₉	1	1	yes	characteristic
92	#336	—	118	2	no	—
93	#336	Dic₅₉	1	2	yes	—
94	#336	Dic₅₉	1	2	yes	—
95	C₅₉	#8789	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	118	2	no	—
97	C₂²×C₈	—	118	2	no	—
98	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
99	C₂×C₄²	Dic₁₁₈	1	2	yes	—
100	C₂×C₄²	D₂₃₆	1	2	yes	—
101	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
102	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
103	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
104	C₄⋊C₈	—	118	4	no	—
105	C₄⋊C₈	—	118	4	no	—
106	C₄⋊C₈	—	236	4	no	—
107	C₄×C₈	—	2	2	no	—
108	C₄×C₈	—	2	2	no	—
109	C₂²×C₄	C₄⋊Dic₅₉	1	2	yes	—
110	C₂²×C₄	C₄⋊Dic₅₉	1	2	yes	—
111	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
112	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
113	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	—	236	2	no	—
118	C₂×C₈	—	236	2	no	—
119	C₂×C₈	—	236	4	no	—
120	C₂×C₈	—	236	2	no	—
121	C₂×C₈	—	236	2	no	—
122	C₄²	—	2	4	no	—
123	C₄²	—	2	2	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	#11745685	1	2	yes	—
128	C₄²	#11745685	1	2	yes	—
129	C₂³	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	#14273663	1	4	yes	—
137	C₂×C₄	#14273686	1	2	yes	—
138	C₂×C₄	#14273715	1	1	yes	characteristic
139	C₂×C₄	#14273715	1	1	yes	characteristic
140	C₂×C₄	#14273719	1	2	yes	—
141	C₂×C₄	#14273734	1	2	yes	—
142	C₂×C₄	#14273734	1	2	yes	—
143	C₂×C₄	#14273755	1	2	yes	—
144	C₂×C₄	#14273760	1	2	yes	—
145	C₈	—	4	4	no	—
146	C₈	—	472	2	no	—
147	C₈	—	472	2	no	—
148	C₂²	#16837268	1	1	yes	characteristic
149	C₂²	#16837429	1	1	yes	characteristic
150	C₂²	#16837429	1	1	yes	characteristic
151	C₂²	#16837444	1	2	yes	—
152	C₂²	C₂×#14273663	1	2	yes	—
153	C₄	—	2	4	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	2	no	—
156	C₄	#16837463	1	4	yes	—
157	C₄	#16837762	1	2	yes	—
158	C₄	#16837767	1	1	yes	characteristic
159	C₄	#16837771	1	1	yes	characteristic
160	C₂	#21422881	1	2	yes	—
161	C₂	#21422885	1	2	yes	—
162	C₂	#21424216	1	1	yes	characteristic
163	C₂	#21424230	1	1	yes	characteristic
164	C₂	#21424234	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21424248

C₂

yes

maximal, characteristic

#21424213

C₂

yes

maximal, characteristic

#21421931

C₂

yes

maximal

C₅₉×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₅₉⋊C₈)

—

C₂×C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#16837157

—

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₄³×C₅₉

C₂²

yes

characteristic

C₅₉×#336

C₄

yes

—

C₅₉×#336

C₄

yes

—

C₂²×C₅₉⋊C₈

—

C₂²×C₅₉⋊C₈

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

Q₈

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

C₄⋊C₄

yes

—

C₂²×C₄×C₅₉

C₄⋊C₄

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

#41

yes

—

C₄²×C₅₉

#41

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#108

yes

—

C₂×C₄×C₅₉

#117

yes

—

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₂×C₄×C₅₉

#131

yes

—

C₂×C₄×C₅₉

#131

yes

—

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₉

#137

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

C₂×#108

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#433

yes

—

C₄×C₅₉

#489

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

—

—

D₅₉

yes

characteristic

#1277

—

C₂×C₅₉

#1573

yes

—

C₂×C₅₉

#1574

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

—

C₄³

C₂×D₅₉

yes

characteristic

#336

—

118

—

#336

Dic₅₉

yes

—

#336

Dic₅₉

yes

—

C₅₉

#8789

yes

characteristic, complemented, Sylow

C₂²×C₈

—

118

—

C₂²×C₈

—

118

—

C₂²×C₈

C₂×Dic₅₉

yes

—

C₂×C₄²

Dic₁₁₈

yes

—

100

C₂×C₄²

D₂₃₆

yes

—

101

C₂×C₄²

C₂×Dic₅₉

yes

characteristic

102

C₂×C₄²

C₂×Dic₅₉

yes

characteristic

103

C₂×C₄²

C₂²×D₅₉

yes

characteristic

104

C₄⋊C₈

—

118

—

105

C₄⋊C₈

—

118

—

106

C₄⋊C₈

—

236

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₂²×C₄

C₄⋊Dic₅₉

yes

—

110

C₂²×C₄

C₄⋊Dic₅₉

yes

—

111

C₂²×C₄

C₂×Dic₁₁₈

yes

characteristic

112

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

113

C₂²×C₄

C₂²×Dic₅₉

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

236

—

118

C₂×C₈

—

236

—

119

C₂×C₈

—

236

—

120

C₂×C₈

—

236

—

121

C₂×C₈

—

236

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

#11745685

yes

—

128

C₄²

#11745685

yes

—

129

C₂³

C₂×C₄⋊Dic₅₉

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#14273663

yes

—

137

C₂×C₄

#14273686

yes

—

138

C₂×C₄

#14273715

yes

characteristic

139

C₂×C₄

#14273715

yes

characteristic

140

C₂×C₄

#14273719

yes

—

141

C₂×C₄

#14273734

yes

—

142

C₂×C₄

#14273734

yes

—

143

C₂×C₄

#14273755

yes

—

144

C₂×C₄

#14273760

yes

—

145

C₈

—

146

C₈

—

472

—

147

C₈

—

472

—

148

C₂²

#16837268

yes

characteristic

149

C₂²

#16837429

yes

characteristic

150

C₂²

#16837429

yes

characteristic

151

C₂²

#16837444

yes

—

152

C₂²

C₂×#14273663

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#16837463

yes

—

157

C₄

#16837762

yes

—

158

C₄

#16837767

yes

characteristic

159

C₄

#16837771

yes

characteristic

160

C₂

#21422881

yes

—

161

C₂

#21422885

yes

—

162

C₂

#21424216

yes

characteristic

163

C₂

#21424230

yes

characteristic

164

C₂

#21424234

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₃₆
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1744	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1744	D₅₉
hypocenter	C₅₉	#8789
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₂₃₆

commutator

C₂²×C₄×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1744

C₂

socle, prime radical

C₂³×C₅₉

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
95. C₅₉	#8789	semidirect

H ≅ G/N

type

95. C₅₉

#8789

semidirect

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

subgroup	index
#21424248	2
#21424213	2
#21421931	2
C₅₉×#1744	2
#8789	59

subgroup

index

C₅₉×#1744

Subgroups — #28482366

Conjugacy classes of subgroups

Complements

Maximal subgroups