Group #28475931 — subgroups

Subgroups — #28475931

Order 15104.

190 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 87 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	#21422859	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837153	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2163	C₂	1	1	yes	maximal, characteristic
4	#16837459	—	2	1	no	—
5	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
6	#16837155	—	2	1	no	—
7	#16837153	—	2	1	no	—
8	#16837153	C₄	1	2	yes	—
9	#16837153	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
11	C₅₉×C₄⋊C₁₆	—	2	1	no	—
12	C₅₉×C₂²⋊C₁₆	C₄	1	2	yes	—
13	C₅₉×#346	—	2	1	no	—
14	C₂²×C₅₉⋊C₈	—	4	1	no	—
15	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
16	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
17	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
18	C₂²⋊(C₅₉⋊C₈)	—	4	1	no	—
19	#14273638	—	4	1	no	—
20	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₅₉	—	2	2	no	—
24	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
27	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
30	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
31	C₂×C₈×C₅₉	—	2	2	no	—
32	C₂×C₈×C₅₉	—	4	1	no	—
33	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₄²×C₅₉	D₈	1	1	yes	characteristic
40	C₄²×C₅₉	Q₁₆	1	1	yes	characteristic
41	C₂×C₅₉⋊C₈	—	4	2	no	—
42	C₂×C₅₉⋊C₈	—	4	2	no	—
43	C₂×C₅₉⋊C₈	—	8	2	no	—
44	C₁₆×C₅₉	—	4	2	no	—
45	C₂³×C₅₉	#106	1	1	yes	characteristic
46	C₂×C₄×C₅₉	—	2	2	no	—
47	C₂×C₄×C₅₉	—	2	1	no	—
48	C₂×C₄×C₅₉	—	2	1	no	—
49	C₂×C₄×C₅₉	—	4	1	no	—
50	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
51	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
52	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
53	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
54	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
55	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
56	C₈×C₅₉	—	2	1	no	—
57	C₈×C₅₉	—	2	1	no	—
58	C₈×C₅₉	—	2	1	no	—
59	C₈×C₅₉	#108	1	2	yes	—
60	C₅₉⋊C₈	—	8	2	no	—
61	C₅₉⋊C₈	—	8	2	no	—
62	#7294	—	59	1	no	maximal, Sylow
63	C₂²×C₅₉	—	2	1	no	—
64	C₂²×C₅₉	—	2	1	no	—
65	C₂²×C₅₉	#340	1	1	yes	characteristic
66	C₂²×C₅₉	#365	1	2	yes	—
67	C₄×C₅₉	—	2	1	no	—
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	#341	1	1	yes	characteristic
73	C₄×C₅₉	#344	1	1	yes	characteristic
74	#2163	D₅₉	1	1	yes	characteristic
75	#1568	—	59	1	no	—
76	C₂×#334	—	59	1	no	—
77	C₂×C₅₉	—	2	2	no	—
78	C₂×C₅₉	#1301	1	1	yes	characteristic
79	C₂×C₅₉	#1383	1	1	yes	characteristic
80	C₂×C₅₉	#1389	1	1	yes	characteristic
81	#432	—	118	1	no	—
82	C₂×C₄⋊C₈	—	118	1	no	—
83	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
84	C₄⋊C₁₆	—	2	1	no	—
85	C₂²⋊C₁₆	Dic₅₉	1	2	yes	—
86	#346	—	2	1	no	—
87	#335	—	118	1	no	—
88	#334	—	59	2	no	—
89	#334	—	59	2	no	—
90	#334	—	118	1	no	—
91	C₅₉	#7294	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₈	—	236	1	no	—
93	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
94	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
95	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
96	C₂×C₁₆	—	2	2	no	—
97	C₄⋊C₈	—	118	2	no	—
98	C₄⋊C₈	—	118	2	no	—
99	C₄⋊C₈	—	236	1	no	—
100	C₂²⋊C₈	—	236	1	no	—
101	#97	—	236	1	no	—
102	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
103	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
104	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
106	C₂²×C₄	#11745660	1	1	yes	characteristic
107	C₂²×C₄	#11745660	1	1	yes	characteristic
108	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	4	1	no	—
111	C₂×C₈	—	472	2	no	—
112	C₂×C₈	—	236	2	no	—
113	C₂×C₈	—	236	2	no	—
114	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
115	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
116	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
117	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
118	C₄²	D₄₇₂	1	1	yes	characteristic
119	C₄²	Dic₂₃₆	1	1	yes	characteristic
120	C₄²	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
121	C₄²	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
122	C₁₆	—	4	2	no	—
123	C₂³	#14273659	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
129	C₂×C₄	#14273649	1	1	yes	characteristic
130	C₂×C₄	#14273649	1	1	yes	characteristic
131	C₂×C₄	#14273652	1	1	yes	characteristic
132	C₂×C₄	#14273652	1	1	yes	characteristic
133	C₂×C₄	#14273661	1	1	yes	characteristic
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	472	2	no	—
138	C₈	—	472	2	no	—
139	C₈	#14273662	1	2	yes	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	#16837165	1	1	yes	characteristic
143	C₂²	#16837225	1	2	yes	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	#16837166	1	1	yes	characteristic
150	C₄	#16837172	1	1	yes	characteristic
151	C₂	—	2	2	no	—
152	C₂	#21421992	1	1	yes	characteristic
153	C₂	#21422195	1	1	yes	characteristic
154	C₂	#21422211	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21422859

C₂

yes

maximal, characteristic

C₂×#16837153

C₂

yes

maximal, characteristic

C₅₉×#2163

C₂

yes

maximal, characteristic

#16837459

—

C₂×C₄⋊(C₅₉⋊C₈)

—

#16837155

—

#16837153

—

#16837153

C₄

yes

—

#16837153

C₄

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×C₄⋊C₁₆

—

C₅₉×C₂²⋊C₁₆

C₄

yes

—

C₅₉×#346

—

C₂²×C₅₉⋊C₈

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

#14273638

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

Q₈

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

D₄

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

C₄²

yes

characteristic

C₂×C₈×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₄²×C₅₉

C₂²⋊C₄

yes

characteristic

C₄²×C₅₉

C₂²⋊C₄

yes

characteristic

C₄²×C₅₉

D₈

yes

characteristic

C₄²×C₅₉

Q₁₆

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

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

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#108

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#7294

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#340

yes

characteristic

C₂²×C₅₉

#365

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

D₅₉

yes

characteristic

#1568

—

C₂×#334

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

118

—

C₂×C₄⋊C₈

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

Dic₅₉

yes

—

#346

—

#335

—

118

—

#334

—

#334

—

#334

—

118

—

C₅₉

#7294

yes

characteristic, complemented, Sylow

C₂²×C₈

—

236

—

C₂²×C₈

Dic₁₁₈

yes

characteristic

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

C₂×C₄²

D₂₃₆

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

118

—

C₄⋊C₈

—

118

—

C₄⋊C₈

—

236

—

100

C₂²⋊C₈

—

236

—

101

#97

—

236

—

102

C₄×C₈

C₄×D₅₉

yes

characteristic

103

C₄×C₈

C₄×D₅₉

yes

characteristic

104

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

105

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

106

C₂²×C₄

#11745660

yes

characteristic

107

C₂²×C₄

#11745660

yes

characteristic

108

C₂²×C₄

C₄⋊Dic₅₉

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

472

—

112

C₂×C₈

—

236

—

113

C₂×C₈

—

236

—

114

C₂×C₈

C₄×Dic₅₉

yes

characteristic

115

C₂×C₈

Dic₅₉⋊C₄

yes

characteristic

116

C₂×C₈

Dic₅₉⋊C₄

yes

characteristic

117

C₂×C₈

C₂²⋊Dic₅₉

yes

characteristic

118

C₄²

D₄₇₂

yes

characteristic

119

C₄²

Dic₂₃₆

yes

characteristic

120

C₄²

(C₂×D₅₉)⋊C₄

yes

characteristic

121

C₄²

(C₂×D₅₉)⋊C₄

yes

characteristic

122

C₁₆

—

123

C₂³

#14273659

yes

characteristic

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₅₉⋊#95

yes

characteristic

129

C₂×C₄

#14273649

yes

characteristic

130

C₂×C₄

#14273649

yes

characteristic

131

C₂×C₄

#14273652

yes

characteristic

132

C₂×C₄

#14273652

yes

characteristic

133

C₂×C₄

#14273661

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

472

—

138

C₈

—

472

—

139

C₈

#14273662

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

#16837165

yes

characteristic

143

C₂²

#16837225

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#16837166

yes

characteristic

150

C₄

#16837172

yes

characteristic

151

C₂

—

152

C₂

#21421992

yes

characteristic

153

C₂

#21422195

yes

characteristic

154

C₂

#21422211

yes

characteristic

	subgroup	G/H
center	C₂²	#16837165
commutator	C₂×C₈×C₅₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₉
Fitting	C₅₉×#2163	C₂
socle	C₂²×C₅₉	#340
prime radical	C₂³×C₅₉	#106
cosocle	C₂×C₄×C₈×C₅₉	C₂²
hypercenter	#2163	D₅₉
hypocenter	C₅₉	#7294
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2163

subgroup

G/H

center

C₂²

#16837165

commutator

C₂×C₈×C₅₉

C₄²

Frattini

C₂×C₄×C₈

C₂×D₅₉

Fitting

C₅₉×#2163

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄×C₈×C₅₉

C₂²

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2163

N	H ≅ G/N	type
91. C₅₉	#7294	semidirect

H ≅ G/N

type

91. C₅₉

#7294

semidirect

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

subgroup	index
#21422859	2
C₂×#16837153	2
C₅₉×#2163	2
#7294	59

subgroup

index

#21422859

C₂×#16837153

C₅₉×#2163

#7294

Subgroups — #28475931

Conjugacy classes of subgroups

Complements

Maximal subgroups