Group #28499886 — subgroups

Subgroups — #28499886

Order 15104.

274 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 79 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	#21425908	C₂	1	2	yes	maximal
2	#21422131	C₂	1	4	yes	maximal, complemented
3	C₅₉×#3139	C₂	1	1	yes	maximal, characteristic
4	#16837223	—	2	2	no	—
5	#16837184	—	2	2	no	—
6	#16837180	C₂²	1	4	yes	—
7	C₅₉×#533	C₄	1	1	yes	characteristic
8	C₂×C₄⋊Q₈×C₅₉	C₄	1	1	yes	characteristic
9	C₅₉×#491	C₄	1	2	yes	—
10	C₅₉×#491	C₂²	1	2	yes	—
11	C₅₉×#487	—	2	2	no	—
12	C₅₉×#419	—	2	2	no	—
13	C₄×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	#14273667	—	2	4	no	—
15	C₂²×Q₈×C₅₉	—	2	1	no	—
16	C₂×M₁₆×C₅₉	D₄	1	2	yes	—
17	C₄⋊Q₈×C₅₉	—	2	1	no	—
18	C₄⋊Q₈×C₅₉	—	2	1	no	—
19	C₄⋊Q₈×C₅₉	—	4	1	no	—
20	C₄⋊Q₈×C₅₉	C₂×C₄	1	2	yes	—
21	#125×C₅₉	C₂×C₄	1	2	yes	—
22	C₂²⋊Q₈×C₅₉	—	4	1	no	—
23	C₄×Q₈×C₅₉	—	4	1	no	—
24	#117×C₅₉	—	2	1	no	—
25	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
26	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
27	#103×C₅₉	—	2	4	no	—
28	#103×C₅₉	—	2	4	no	—
29	#97×C₅₉	C₂×C₄	1	1	yes	characteristic
30	#97×C₅₉	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₅₉	D₄	1	2	yes	—
32	C₂×Q₈×C₅₉	—	2	2	no	—
33	C₂×Q₈×C₅₉	—	4	1	no	—
34	C₂×Q₈×C₅₉	—	4	4	no	—
35	C₂×Q₈×C₅₉	—	4	1	no	—
36	C₂²×C₄×C₅₉	—	2	1	no	—
37	C₂²×C₄×C₅₉	—	4	1	no	—
38	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
39	M₁₆×C₅₉	—	4	2	no	—
40	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
41	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
42	C₄⋊C₄×C₅₉	—	2	2	no	—
43	C₄⋊C₄×C₅₉	—	4	2	no	—
44	C₄⋊C₄×C₅₉	—	4	1	no	—
45	C₄⋊C₄×C₅₉	—	4	2	no	—
46	C₄⋊C₄×C₅₉	—	4	2	no	—
47	C₂²⋊C₄×C₅₉	—	4	1	no	—
48	C₄²×C₅₉	—	2	1	no	—
49	C₄²×C₅₉	—	4	1	no	—
50	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
52	C₅₉⋊C₁₆	—	4	4	no	—
53	C₂³×C₅₉	#100	1	1	yes	characteristic
54	Q₈×C₅₉	—	4	4	no	—
55	Q₈×C₅₉	—	8	2	no	—
56	Q₈×C₅₉	—	8	2	no	—
57	C₂×C₄×C₅₉	—	2	1	no	—
58	C₂×C₄×C₅₉	—	2	2	no	—
59	C₂×C₄×C₅₉	—	2	1	no	—
60	C₂×C₄×C₅₉	—	4	2	no	—
61	C₂×C₄×C₅₉	—	4	2	no	—
62	C₂×C₄×C₅₉	—	4	2	no	—
63	C₂×C₄×C₅₉	—	8	2	no	—
64	C₂×C₄×C₅₉	#100	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₈×C₅₉	—	2	2	no	—
67	C₈×C₅₉	—	2	2	no	—
68	#11916	—	59	1	no	maximal, Sylow
69	C₂²×C₅₉	—	2	1	no	—
70	C₂²×C₅₉	—	4	1	no	—
71	C₂²×C₅₉	C₂×#100	1	1	yes	characteristic
72	C₄×C₅₉	—	2	2	no	—
73	C₄×C₅₉	—	2	1	no	—
74	C₄×C₅₉	—	4	1	no	—
75	C₄×C₅₉	—	8	2	no	—
76	C₄×C₅₉	—	8	2	no	—
77	C₄×C₅₉	#421	1	2	yes	—
78	#3139	D₅₉	1	1	yes	characteristic
79	#2165	—	59	2	no	—
80	#1356	—	59	4	no	—
81	C₂×C₅₉	—	2	1	no	—
82	C₂×C₅₉	—	4	1	no	—
83	C₂×C₅₉	#1533	1	1	yes	characteristic
84	#533	Dic₅₉	1	1	yes	characteristic
85	C₂×C₄⋊Q₈	Dic₅₉	1	1	yes	characteristic
86	#491	Dic₅₉	1	2	yes	—
87	#491	C₂×D₅₉	1	2	yes	—
88	#487	—	2	2	no	—
89	#419	—	2	2	no	—
90	C₄×M₁₆	C₂×D₅₉	1	1	yes	characteristic
91	#364	—	118	2	no	—
92	#349	—	118	2	no	—
93	#347	—	59	4	no	—
94	C₅₉	#11916	1	1	yes	characteristic, complemented, Sylow
95	C₂²×Q₈	—	2	1	no	—
96	C₂×M₁₆	C₂²⋊D₅₉	1	2	yes	—
97	C₄⋊Q₈	—	2	1	no	—
98	C₄⋊Q₈	—	2	1	no	—
99	C₄⋊Q₈	—	4	1	no	—
100	C₄⋊Q₈	C₂×Dic₅₉	1	2	yes	—
101	#125	C₂×Dic₅₉	1	2	yes	—
102	C₂²⋊Q₈	—	4	1	no	—
103	C₄×Q₈	—	4	1	no	—
104	#117	—	2	1	no	—
105	C₂×C₄⋊C₄	—	4	1	no	—
106	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
107	M₃₂	—	118	4	no	—
108	#103	—	2	4	no	—
109	#103	—	2	4	no	—
110	#97	C₂×Dic₅₉	1	1	yes	characteristic
111	#97	C₂²×D₅₉	1	1	yes	characteristic
112	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
113	C₂×Q₈	—	2	2	no	—
114	C₂×Q₈	—	4	1	no	—
115	C₂×Q₈	—	4	4	no	—
116	C₂×Q₈	—	4	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	4	1	no	—
119	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
120	M₁₆	—	4	2	no	—
121	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
122	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	1	no	—
128	C₂²⋊C₄	—	4	1	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	4	1	no	—
131	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
132	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
133	C₁₆	—	236	4	no	—
134	C₂³	#14273644	1	1	yes	characteristic
135	Q₈	—	4	4	no	—
136	Q₈	—	8	2	no	—
137	Q₈	—	8	2	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	#14273644	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	2	2	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	C₂×#14273644	1	1	yes	characteristic
152	C₄	—	2	2	no	—
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	8	2	no	—
156	C₄	—	8	2	no	—
157	C₄	#16837419	1	2	yes	—
158	C₂	—	2	1	no	—
159	C₂	—	4	1	no	—
160	C₂	#21422720	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21425908

C₂

yes

maximal

#21422131

C₂

yes

maximal, complemented

C₅₉×#3139

C₂

yes

maximal, characteristic

#16837223

—

#16837184

—

#16837180

C₂²

yes

—

C₅₉×#533

C₄

yes

characteristic

C₂×C₄⋊Q₈×C₅₉

C₄

yes

characteristic

C₅₉×#491

C₄

yes

—

C₅₉×#491

C₂²

yes

—

C₅₉×#487

—

C₅₉×#419

—

C₄×M₁₆×C₅₉

C₂²

yes

characteristic

#14273667

—

C₂²×Q₈×C₅₉

—

C₂×M₁₆×C₅₉

D₄

yes

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

C₂×C₄

yes

—

#125×C₅₉

C₂×C₄

yes

—

C₂²⋊Q₈×C₅₉

—

C₄×Q₈×C₅₉

—

#117×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₂×C₄

yes

characteristic

#103×C₅₉

—

#103×C₅₉

—

#97×C₅₉

C₂×C₄

yes

characteristic

#97×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

M₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂²⋊C₄

yes

characteristic

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₅₉⋊C₁₆

—

C₂³×C₅₉

#100

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#100

yes

characteristic

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

—

#11916

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×#100

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#421

yes

—

#3139

D₅₉

yes

characteristic

#2165

—

#1356

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#1533

yes

characteristic

#533

Dic₅₉

yes

characteristic

C₂×C₄⋊Q₈

Dic₅₉

yes

characteristic

#491

Dic₅₉

yes

—

#491

C₂×D₅₉

yes

—

#487

—

#419

—

C₄×M₁₆

C₂×D₅₉

yes

characteristic

#364

—

118

—

#349

—

118

—

#347

—

C₅₉

#11916

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

C₂²⋊D₅₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

100

C₄⋊Q₈

C₂×Dic₅₉

yes

—

101

#125

C₂×Dic₅₉

yes

—

102

C₂²⋊Q₈

—

103

C₄×Q₈

—

104

#117

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄²

C₂×Dic₅₉

yes

characteristic

107

M₃₂

—

118

—

108

#103

—

109

#103

—

110

#97

C₂×Dic₅₉

yes

characteristic

111

#97

C₂²×D₅₉

yes

characteristic

112

C₄×C₈

C₂²⋊D₅₉

yes

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂²⋊Dic₅₉

yes

characteristic

120

M₁₆

—

121

C₂×C₈

C₂²⋊Dic₅₉

yes

—

122

C₂×C₈

C₂×C₂²⋊D₅₉

yes

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²⋊Dic₅₉

yes

characteristic

132

C₄²

C₂²×Dic₅₉

yes

characteristic

133

C₁₆

—

236

—

134

C₂³

#14273644

yes

characteristic

135

Q₈

—

136

Q₈

—

137

Q₈

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

#14273644

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

147

C₈

—

148

C₈

—

149

C₂²

—

150

C₂²

—

151

C₂²

C₂×#14273644

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#16837419

yes

—

158

C₂

—

159

C₂

—

160

C₂

#21422720

yes

characteristic

	subgroup	G/H
center	C₂	#21422720
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	#97	C₂²×D₅₉
Fitting	C₅₉×#3139	C₂
socle	C₂×C₅₉	#1533
prime radical	C₂³×C₅₉	#100
cosocle	#97×C₅₉	C₂³
hypercenter	#3139	D₅₉
hypocenter	C₅₉	#11916
LCS	G ▷ C₄²×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #3139

subgroup

G/H

center

C₂

#21422720

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

#97

C₂²×D₅₉

Fitting

C₅₉×#3139

C₂

C₂×C₅₉

C₂³×C₅₉

#97×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #3139

N	H ≅ G/N	type
2. #21422131	C₂	semidirect
94. C₅₉	#11916	semidirect

H ≅ G/N

type

2. #21422131

C₂

semidirect

94. C₅₉

#11916

semidirect

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

subgroup	index
#21425908	2
#21422131	2
C₅₉×#3139	2
#11916	59

subgroup

index

#21425908

#21422131

C₅₉×#3139

#11916

Subgroups — #28499886

Conjugacy classes of subgroups

Complements

Maximal subgroups