Group #28500355 — subgroups

Subgroups — #28500355

Order 15104.

210 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 99 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	#21425899	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837153	C₂	1	1	yes	maximal, characteristic
3	#21422161	C₂	1	4	yes	maximal, complemented
4	C₅₉×#1568	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
6	C₄⋊(C₅₉⋊C₁₆)	C₄	1	2	yes	—
7	C₂²⋊(C₅₉⋊C₁₆)	C₄	1	2	yes	—
8	#16837178	C₂²	1	2	yes	—
9	#16837153	—	2	1	no	—
10	#16837153	C₂²	1	2	yes	—
11	C₅₉×#432	—	2	1	no	—
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×#335	—	2	1	no	—
14	C₅₉×#334	C₂²	1	2	yes	—
15	C₂²×C₅₉⋊C₈	—	4	1	no	—
16	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	—
17	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
18	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
19	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
22	C₄⋊C₈×C₅₉	—	2	2	no	—
23	C₂²⋊C₈×C₅₉	—	4	1	no	—
24	#97×C₅₉	—	4	1	no	—
25	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
27	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
28	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
32	C₂×C₈×C₅₉	—	2	2	no	—
33	C₂×C₈×C₅₉	—	4	1	no	—
34	C₂×C₈×C₅₉	—	4	2	no	—
35	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
39	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
41	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
42	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₅₉⋊C₈	—	4	2	no	—
44	C₂×C₅₉⋊C₈	—	8	2	no	—
45	C₅₉⋊C₁₆	—	2	4	no	—
46	C₂³×C₅₉	#106	1	1	yes	characteristic
47	C₂×C₄×C₅₉	—	2	2	no	—
48	C₂×C₄×C₅₉	—	2	1	no	—
49	C₂×C₄×C₅₉	—	2	1	no	—
50	C₂×C₄×C₅₉	—	4	1	no	—
51	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
52	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
53	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
57	C₈×C₅₉	—	2	1	no	—
58	C₈×C₅₉	—	2	1	no	—
59	C₈×C₅₉	—	2	1	no	—
60	C₈×C₅₉	—	8	2	no	—
61	C₈×C₅₉	#131	1	2	yes	—
62	C₅₉⋊C₈	—	8	2	no	—
63	#12028	—	59	1	no	maximal, Sylow
64	C₂²×C₅₉	—	2	1	no	—
65	C₂²×C₅₉	—	2	1	no	—
66	C₂²×C₅₉	#365	1	2	yes	—
67	C₂²×C₅₉	C₂×#106	1	1	yes	characteristic
68	C₄×C₅₉	—	2	1	no	—
69	C₄×C₅₉	—	2	1	no	—
70	C₄×C₅₉	—	2	1	no	—
71	C₄×C₅₉	—	2	1	no	—
72	C₄×C₅₉	—	2	1	no	—
73	C₄×C₅₉	#424	1	1	yes	characteristic
74	C₄×C₅₉	#430	1	1	yes	characteristic
75	#2163	—	59	1	no	—
76	#1568	D₅₉	1	1	yes	characteristic
77	C₂×#334	—	59	1	no	—
78	#1368	—	59	4	no	—
79	C₂×C₅₉	—	2	2	no	—
80	C₂×C₅₉	#1570	1	1	yes	characteristic
81	C₂×C₅₉	C₂×#365	1	1	yes	characteristic
82	C₂×C₅₉	#2155	1	1	yes	characteristic
83	#432	—	2	1	no	—
84	C₂×C₄⋊C₈	—	118	1	no	—
85	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
86	C₄⋊C₁₆	—	59	2	no	—
87	C₂²⋊C₁₆	—	59	2	no	—
88	#346	—	59	2	no	—
89	#335	—	2	1	no	—
90	#334	—	59	2	no	—
91	#334	—	118	1	no	—
92	#334	C₂×D₅₉	1	2	yes	—
93	C₅₉	#12028	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	—	236	1	no	—
95	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
96	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
97	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
98	C₂×C₁₆	—	59	4	no	—
99	C₄⋊C₈	—	2	2	no	—
100	C₄⋊C₈	—	118	2	no	—
101	C₄⋊C₈	—	236	1	no	—
102	C₂²⋊C₈	—	4	1	no	—
103	#97	—	4	1	no	—
104	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
105	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
106	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
107	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
108	C₂²×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
109	C₂²×C₄	#11745670	1	1	yes	characteristic
110	C₂²×C₄	#11745671	1	1	yes	characteristic
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	—	472	2	no	—
115	C₂×C₈	—	236	2	no	—
116	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
117	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
118	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
119	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
120	C₄²	Dic₅₉⋊C₄	1	1	yes	characteristic
121	C₄²	#11745670	1	1	yes	characteristic
122	C₄²	#11745671	1	1	yes	characteristic
123	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
124	C₁₆	—	118	4	no	—
125	C₂³	#14273658	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	#14273658	1	1	yes	characteristic
131	C₂×C₄	#14273658	1	1	yes	characteristic
132	C₂×C₄	#14273658	1	1	yes	characteristic
133	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
134	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
135	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
136	C₈	—	2	1	no	—
137	C₈	—	2	1	no	—
138	C₈	—	2	1	no	—
139	C₈	—	8	2	no	—
140	C₈	—	472	2	no	—
141	C₈	#14273735	1	2	yes	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	#16837226	1	2	yes	—
145	C₂²	C₂×#14273658	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	#16837430	1	1	yes	characteristic
152	C₄	#16837450	1	1	yes	characteristic
153	C₂	—	2	2	no	—
154	C₂	#21422868	1	1	yes	characteristic
155	C₂	C₂×#16837226	1	1	yes	characteristic
156	C₂	#21425868	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21425899

C₂

yes

maximal, characteristic

C₂×#16837153

C₂

yes

maximal, characteristic

#21422161

C₂

yes

maximal, complemented

C₅₉×#1568

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₁₆)

C₄

yes

—

C₂²⋊(C₅₉⋊C₁₆)

C₄

yes

—

#16837178

C₂²

yes

—

#16837153

—

#16837153

C₂²

yes

—

C₅₉×#432

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#335

—

C₅₉×#334

C₂²

yes

—

C₂²×C₅₉⋊C₈

—

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

Q₈

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²⋊C₈×C₅₉

—

#97×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

Q₈

yes

characteristic

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂²×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₅₉

SD₁₆

yes

characteristic

C₂²×C₄×C₅₉

SD₁₆

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₄²×C₅₉

C₄⋊C₄

yes

characteristic

C₄²×C₅₉

SD₁₆

yes

characteristic

C₄²×C₅₉

SD₁₆

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

#106

yes

characteristic

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

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#131

yes

—

C₅₉⋊C₈

—

#12028

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#365

yes

—

C₂²×C₅₉

C₂×#106

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

—

D₅₉

yes

characteristic

C₂×#334

—

#1368

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₄⋊C₈

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

—

—

—

—

118

—

#334

C₂×D₅₉

yes

—

C₅₉

#12028

yes

characteristic, complemented, Sylow

C₂²×C₈

—

236

—

C₂²×C₈

Dic₁₁₈

yes

characteristic

C₂²×C₈

C₄×D₅₉

yes

characteristic

C₂×C₄²

C₂²⋊D₅₉

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

100

C₄⋊C₈

—

118

—

101

C₄⋊C₈

—

236

—

102

C₂²⋊C₈

—

103

#97

—

104

C₄×C₈

Dic₁₁₈

yes

characteristic

105

C₄×C₈

C₄×D₅₉

yes

characteristic

106

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

107

C₄×C₈

C₂²×D₅₉

yes

characteristic

108

C₂²×C₄

Dic₅₉⋊C₄

yes

characteristic

109

C₂²×C₄

#11745670

yes

characteristic

110

C₂²×C₄

#11745671

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

472

—

115

C₂×C₈

—

236

—

116

C₂×C₈

Dic₅₉⋊C₄

yes

characteristic

117

C₂×C₈

Dic₅₉⋊C₄

yes

characteristic

118

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

119

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

120

C₄²

Dic₅₉⋊C₄

yes

characteristic

121

C₄²

#11745670

yes

characteristic

122

C₄²

#11745671

yes

characteristic

123

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

124

C₁₆

—

118

—

125

C₂³

#14273658

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#14273658

yes

characteristic

131

C₂×C₄

#14273658

yes

characteristic

132

C₂×C₄

#14273658

yes

characteristic

133

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

134

C₂×C₄

C₂×#11745671

yes

characteristic

135

C₂×C₄

C₂×#11745670

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

472

—

141

C₈

#14273735

yes

—

142

C₂²

—

143

C₂²

—

144

C₂²

#16837226

yes

—

145

C₂²

C₂×#14273658

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#16837430

yes

characteristic

152

C₄

#16837450

yes

characteristic

153

C₂

—

154

C₂

#21422868

yes

characteristic

155

C₂

C₂×#16837226

yes

characteristic

156

C₂

#21425868

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14273658
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1568	C₂
socle	C₂²×C₅₉	C₂×#106
prime radical	C₂³×C₅₉	#106
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	#1568	D₅₉
hypocenter	C₅₉	#12028
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1568

subgroup

G/H

center

C₂²

C₂×#14273658

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1568

C₂

C₂²×C₅₉

C₂³×C₅₉

C₄×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1568

N	H ≅ G/N	type
3. #21422161	C₂	semidirect
93. C₅₉	#12028	semidirect

H ≅ G/N

type

3. #21422161

C₂

semidirect

93. C₅₉

#12028

semidirect

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

subgroup	index
#21425899	2
C₂×#16837153	2
#21422161	2
C₅₉×#1568	2
#12028	59

subgroup

index

C₅₉×#1568

Subgroups — #28500355

Conjugacy classes of subgroups

Complements

Maximal subgroups