Group #28504107 — subgroups

Subgroups — #28504107

Order 15104.

306 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 147 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	C₂×#16837232	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837177	C₂	1	1	yes	maximal, characteristic
3	#108×Dic₅₉	C₂	1	2	yes	maximal
4	#21422189	C₂	1	2	yes	maximal
5	C₂×C₅₉×#368	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14273663	C₂²	1	2	yes	—
8	M₁₆×Dic₅₉	—	2	2	no	—
9	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
10	#16837232	C₄	1	4	yes	complemented
11	#16837177	C₄	1	4	yes	—
12	#16837166	—	2	2	no	—
13	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×#108×C₅₉	C₂²	1	2	yes	—
15	C₅₉×#368	C₄	1	4	yes	complemented
16	C₂×#11745664	—	2	2	no	—
17	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
18	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
19	C₂×C₅₉⋊C₁₆	—	2	1	no	—
20	C₂×C₅₉⋊C₁₆	—	2	1	no	—
21	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
22	#14273663	—	2	2	no	—
23	#14273663	C₂×C₄	1	4	yes	—
24	#14273637	—	4	2	no	—
25	C₈×Dic₅₉	—	4	2	no	—
26	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
27	C₈×Dic₅₉	Q₈	1	2	yes	—
28	C₂×M₁₆×C₅₉	—	2	2	no	—
29	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
33	#108×C₅₉	—	2	2	no	—
34	#108×C₅₉	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₅₉	Q₁₆	1	2	yes	—
37	M₁₆×C₅₉	—	2	4	no	—
38	M₁₆×C₅₉	—	4	2	no	—
39	C₂×C₈×C₅₉	—	4	2	no	—
40	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
45	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
46	C₄×Dic₅₉	C₄⋊C₄	1	2	yes	—
47	C₄×Dic₅₉	D₈	1	2	yes	—
48	#11745664	—	2	4	no	—
49	#11745664	—	4	2	no	—
50	C₂×C₅₉⋊C₈	—	2	1	no	—
51	C₂×C₅₉⋊C₈	—	2	1	no	—
52	C₂×C₅₉⋊C₈	—	4	2	no	—
53	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
54	C₁₆×C₅₉	C₄²	1	4	yes	—
55	C₅₉⋊C₁₆	—	2	2	no	—
56	C₂³×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
63	C₂×Dic₅₉	—	4	2	no	—
64	C₂×Dic₅₉	#107	1	4	yes	—
65	C₈×C₅₉	—	4	4	no	—
66	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
67	C₈×C₅₉	#117	1	1	yes	characteristic
68	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
69	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
70	C₅₉⋊C₈	—	2	2	no	—
71	C₅₉⋊C₈	—	4	4	no	—
72	#12676	—	59	1	no	maximal, Sylow
73	C₂²×C₅₉	—	2	2	no	—
74	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
75	C₂²×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
76	C₂²×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
77	C₄×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
78	C₄×C₅₉	#427	1	1	yes	characteristic
79	C₄×C₅₉	C₄×D₈	1	1	yes	characteristic
80	C₄×C₅₉	C₄×D₈	1	1	yes	characteristic
81	Dic₅₉	—	2	4	no	—
82	C₂×#368	—	59	1	no	—
83	C₂×#368	D₅₉	1	1	yes	characteristic
84	C₂×#346	—	59	1	no	—
85	C₄×#108	—	59	2	no	—
86	#1381	—	59	2	no	—
87	C₂×C₅₉	—	2	1	no	—
88	C₂×C₅₉	—	2	1	no	—
89	C₂×C₅₉	C₄×#107	1	1	yes	characteristic
90	C₂×C₅₉	#2161	1	2	yes	—
91	C₂²×C₁₆	—	59	1	no	—
92	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
93	C₂×#108	—	59	2	no	—
94	C₂×#108	C₂×D₅₉	1	2	yes	—
95	C₄×M₁₆	—	118	2	no	—
96	C₂×C₄×C₈	—	59	1	no	—
97	#368	—	59	4	no	—
98	#368	Dic₅₉	1	4	yes	complemented
99	#346	—	59	4	no	—
100	#341	—	118	2	no	—
101	C₅₉	#12676	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	2	2	no	—
103	C₂×M₁₆	—	118	2	no	—
104	C₂²×C₈	—	59	1	no	—
105	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
106	C₂×C₄²	—	59	1	no	—
107	C₂×C₁₆	—	59	2	no	—
108	C₂×C₁₆	—	118	1	no	—
109	C₂×C₁₆	—	118	1	no	—
110	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
111	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
112	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
113	#108	—	2	2	no	—
114	#108	—	59	4	no	—
115	#108	—	118	2	no	—
116	#108	C₂×Dic₅₉	1	4	yes	—
117	#97	—	236	2	no	—
118	C₄×C₈	—	59	2	no	—
119	C₄×C₈	—	59	2	no	—
120	C₄×C₈	—	236	2	no	—
121	C₂²×C₄	—	59	2	no	—
122	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
123	M₁₆	—	2	4	no	—
124	M₁₆	—	4	2	no	—
125	M₁₆	—	118	4	no	—
126	M₁₆	—	236	2	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	—	59	2	no	—
129	C₂×C₈	—	118	1	no	—
130	C₂×C₈	—	118	1	no	—
131	C₂×C₈	—	236	2	no	—
132	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
133	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
134	C₂×C₈	#11745687	1	1	yes	characteristic
135	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
136	C₂×C₈	#11745689	1	1	yes	characteristic
137	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
138	C₄²	—	59	2	no	—
139	C₄²	—	59	2	no	—
140	C₁₆	—	118	2	no	—
141	C₁₆	C₄×Dic₅₉	1	4	yes	—
142	C₂³	Q₁₆×D₅₉	1	1	yes	characteristic
143	C₂×C₄	—	59	4	no	—
144	C₂×C₄	—	236	2	no	—
145	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
146	C₂×C₄	#14273689	1	1	yes	characteristic
147	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
148	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
149	C₂×C₄	#14273746	1	1	yes	characteristic
150	C₂×C₄	#14273750	1	1	yes	characteristic
151	C₈	—	4	4	no	—
152	C₈	—	118	2	no	—
153	C₈	—	236	4	no	—
154	C₈	C₂×C₄×Dic₅₉	1	1	yes	characteristic
155	C₈	#14273685	1	1	yes	characteristic
156	C₈	#14273692	1	1	yes	characteristic
157	C₈	Q₈×Dic₅₉	1	1	yes	characteristic
158	C₂²	—	2	2	no	—
159	C₂²	#107×D₅₉	1	1	yes	characteristic
160	C₂²	#16837496	1	1	yes	characteristic
161	C₂²	Q₁₆×Dic₅₉	1	1	yes	characteristic
162	C₄	—	118	4	no	—
163	C₄	C₄⋊C₄×Dic₅₉	1	1	yes	characteristic
164	C₄	#16837439	1	1	yes	characteristic
165	C₄	#16837484	1	1	yes	characteristic
166	C₄	D₈×Dic₅₉	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#107×Dic₅₉	1	1	yes	characteristic
170	C₂	#21425889	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837232

C₂

yes

maximal, characteristic

C₂×#16837177

C₂

yes

maximal, characteristic

#108×Dic₅₉

C₂

yes

maximal

#21422189

C₂

yes

maximal

C₂×C₅₉×#368

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₂×#14273663

C₂²

yes

—

M₁₆×Dic₅₉

—

C₂×C₈×Dic₅₉

C₂²

yes

characteristic

#16837232

C₄

yes

complemented

#16837177

C₄

yes

—

#16837166

—

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×#108×C₅₉

C₂²

yes

—

C₅₉×#368

C₄

yes

complemented

C₂×#11745664

—

C₂²×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

—

#14273663

—

#14273663

C₂×C₄

yes

—

#14273637

—

C₈×Dic₅₉

—

C₈×Dic₅₉

C₂×C₄

yes

—

C₈×Dic₅₉

Q₈

yes

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

#108×C₅₉

—

#108×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

Q₁₆

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

C₂×C₈×C₅₉

yes

C₂×C₈×C₅₉

yes

C₄×Dic₅₉

C₄⋊C₄

yes

—

C₄×Dic₅₉

D₈

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₄⋊C₄

yes

—

C₁₆×C₅₉

C₄²

yes

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

#107

yes

—

C₈×C₅₉

—

C₈×C₅₉

C₂×C₄²

yes

characteristic

C₈×C₅₉

#117

yes

characteristic

C₈×C₅₉

C₄×Q₈

yes

characteristic

C₈×C₅₉

C₄×Q₈

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#12676

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₂×#107

yes

characteristic

C₂²×C₅₉

C₄×Q₁₆

yes

characteristic

C₂²×C₅₉

C₄×Q₁₆

yes

characteristic

C₄×C₅₉

C₄×C₄⋊C₄

yes

characteristic

C₄×C₅₉

#427

yes

characteristic

C₄×C₅₉

C₄×D₈

yes

characteristic

C₄×C₅₉

C₄×D₈

yes

characteristic

Dic₅₉

—

C₂×#368

—

C₂×#368

D₅₉

yes

—

—

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

C₄×#107

yes

characteristic

C₂×C₅₉

#2161

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

C₂×#108

—

C₂×#108

C₂×D₅₉

yes

—

C₄×M₁₆

—

118

—

C₂×C₄×C₈

—

#368

—

#368

Dic₅₉

yes

complemented

#346

—

100

#341

—

118

—

101

C₅₉

#12676

yes

characteristic, complemented, Sylow

102

C₂×M₁₆

—

103

C₂×M₁₆

—

118

—

104

C₂²×C₈

—

105

C₂²×C₈

C₂²×D₅₉

yes

characteristic

106

C₂×C₄²

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

118

—

109

C₂×C₁₆

—

118

—

110

C₂×C₁₆

C₄×D₅₉

yes

—

111

C₂×C₁₆

C₄×D₅₉

yes

—

112

C₂×C₁₆

C₂×Dic₅₉

yes

—

113

#108

—

114

#108

—

115

#108

—

118

—

116

#108

C₂×Dic₅₉

yes

—

117

#97

—

236

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

236

—

121

C₂²×C₄

—

122

C₂²×C₄

D₄×D₅₉

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

M₁₆

—

118

—

126

M₁₆

—

236

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

118

—

130

C₂×C₈

—

118

—

131

C₂×C₈

—

236

—

132

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

133

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

134

C₂×C₈

#11745687

yes

characteristic

135

C₂×C₈

Q₈×D₅₉

yes

characteristic

136

C₂×C₈

#11745689

yes

characteristic

137

C₂×C₈

C₂²×Dic₅₉

yes

characteristic

138

C₄²

—

139

C₄²

—

140

C₁₆

—

118

—

141

C₁₆

C₄×Dic₅₉

yes

—

142

C₂³

Q₁₆×D₅₉

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

236

—

145

C₂×C₄

C₄⋊C₄×D₅₉

yes

characteristic

146

C₂×C₄

#14273689

yes

characteristic

147

C₂×C₄

D₄×Dic₅₉

yes

characteristic

148

C₂×C₄

D₈×D₅₉

yes

characteristic

149

C₂×C₄

#14273746

yes

characteristic

150

C₂×C₄

#14273750

yes

characteristic

151

C₈

—

152

C₈

—

118

—

153

C₈

—

236

—

154

C₈

C₂×C₄×Dic₅₉

yes

characteristic

155

C₈

#14273685

yes

characteristic

156

C₈

#14273692

yes

characteristic

157

C₈

Q₈×Dic₅₉

yes

characteristic

158

C₂²

—

159

C₂²

#107×D₅₉

yes

characteristic

160

C₂²

#16837496

yes

characteristic

161

C₂²

Q₁₆×Dic₅₉

yes

characteristic

162

C₄

—

118

—

163

C₄

C₄⋊C₄×Dic₅₉

yes

characteristic

164

C₄

#16837439

yes

characteristic

165

C₄

#16837484

yes

characteristic

166

C₄

D₈×Dic₅₉

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#107×Dic₅₉

yes

characteristic

170

C₂

#21425889

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₉
commutator	C₈×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#368	C₂
socle	C₂²×C₅₉	C₂×#107
prime radical	C₂³×C₅₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#368	D₅₉
hypocenter	C₅₉	#12676
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

subgroup

G/H

center

C₂×C₄

D₈×D₅₉

commutator

C₈×C₅₉

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₂×C₅₉×#368

C₂

socle

C₂²×C₅₉

C₂×#107

prime radical

C₂³×C₅₉

C₂×Q₁₆

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

N	H ≅ G/N	type
10. #16837232	C₄	semidirect
15. C₅₉×#368	C₄	semidirect
98. #368	Dic₅₉	semidirect
101. C₅₉	#12676	semidirect

H ≅ G/N

type

10. #16837232

C₄

semidirect

15. C₅₉×#368

C₄

semidirect

98. #368

Dic₅₉

semidirect

101. C₅₉

#12676

semidirect

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

subgroup	index
C₂×#16837232	2
C₂×#16837177	2
#108×Dic₅₉	2
#21422189	2
C₂×C₅₉×#368	2
#12676	59

subgroup

index

#108×Dic₅₉

C₂×C₅₉×#368

Subgroups — #28504107

Conjugacy classes of subgroups

Complements

Maximal subgroups