Group #28488538 — subgroups

Subgroups — #28488538

Order 15104.

542 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 195 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×#16837782	C₂	1	1	yes	maximal, characteristic
2	#21425057	C₂	1	4	yes	maximal
3	#21424253	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₉×#322	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₃₆	—	2	2	no	—
6	C₂×#14273727	C₂²	1	2	yes	—
7	#16837782	—	2	2	no	—
8	#16837782	C₄	1	4	yes	complemented
9	C₂×#14273659	—	2	2	no	—
10	#16837272	C₂²	1	2	yes	—
11	#16837165	—	2	8	no	—
12	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×#322	—	2	2	no	—
15	C₂²×Dic₁₁₈	—	2	2	no	—
16	C₂×Dic₂₃₆	—	2	8	no	—
17	C₂×Dic₂₃₆	—	4	4	no	—
18	C₂×Dic₂₃₆	—	4	4	no	—
19	#14273727	—	2	4	no	—
20	#14273727	—	2	2	no	—
21	#14273727	C₂×C₄	1	4	yes	—
22	C₂×C₄⋊Dic₅₉	—	2	4	no	—
23	C₂×C₄⋊Dic₅₉	—	2	4	no	—
24	C₂×C₄⋊Dic₅₉	—	2	2	no	—
25	#14273659	—	4	4	no	—
26	C₂²×C₈×C₅₉	D₄	1	4	yes	—
27	C₂²×C₈×C₅₉	D₄	1	2	yes	—
28	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₅₉	—	2	2	no	—
30	C₄×C₈×C₅₉	—	2	2	no	—
31	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₅₉	D₄	1	2	yes	—
33	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
35	C₂²×Dic₅₉	—	4	2	no	—
36	C₂²×Dic₅₉	—	4	2	no	—
37	C₂×Dic₁₁₈	—	2	4	no	—
38	C₂×Dic₁₁₈	—	4	4	no	—
39	C₂×Dic₁₁₈	—	4	4	no	—
40	C₂×Dic₁₁₈	—	4	2	no	—
41	C₂×C₈×C₅₉	—	2	4	no	—
42	C₂×C₈×C₅₉	—	2	8	no	—
43	C₂×C₈×C₅₉	C₂²⋊C₄	1	4	yes	—
44	C₂×C₈×C₅₉	D₈	1	4	yes	—
45	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
46	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₅₉	#41	1	2	yes	—
49	C₄⋊Dic₅₉	—	2	8	no	—
50	C₄⋊Dic₅₉	—	4	4	no	—
51	C₄⋊Dic₅₉	—	4	4	no	—
52	C₄⋊Dic₅₉	—	4	8	no	—
53	Dic₂₃₆	—	4	16	no	—
54	C₂³×C₅₉	#127	1	1	yes	characteristic
55	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
57	C₂×C₄×C₅₉	#121	1	2	yes	—
58	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
60	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
61	C₂×C₄×C₅₉	#137	1	8	yes	—
62	C₂×Dic₅₉	—	4	4	no	—
63	C₂×Dic₅₉	—	8	2	no	—
64	C₂×Dic₅₉	—	8	2	no	—
65	C₂×Dic₅₉	—	8	2	no	—
66	C₂×Dic₅₉	—	8	4	no	—
67	Dic₁₁₈	—	4	8	no	—
68	Dic₁₁₈	—	8	4	no	—
69	C₈×C₅₉	—	4	4	no	—
70	C₈×C₅₉	#102	1	8	yes	—
71	#9839	—	59	1	no	maximal, Sylow
72	C₂²×C₅₉	#390	1	1	yes	characteristic
73	C₂²×C₅₉	#492	1	1	yes	characteristic
74	C₂²×C₅₉	#493	1	1	yes	characteristic
75	C₂²×C₅₉	#497	1	4	yes	—
76	C₄×C₅₉	C₂×#102	1	2	yes	—
77	C₄×C₅₉	#441	1	4	yes	—
78	C₄×C₅₉	#470	1	4	yes	—
79	C₄×C₅₉	#486	1	2	yes	—
80	Dic₅₉	—	8	4	no	—
81	Dic₅₉	—	16	2	no	—
82	C₂×#494	—	59	1	no	—
83	#1990	—	59	4	no	—
84	#1844	—	59	1	no	—
85	C₂×#322	D₅₉	1	1	yes	characteristic
86	C₂×C₅₉	#1716	1	2	yes	—
87	C₂×C₅₉	#1722	1	2	yes	—
88	C₂×C₅₉	#1957	1	1	yes	characteristic
89	C₂×C₅₉	#1971	1	2	yes	—
90	C₂²×Q₁₆	—	118	2	no	—
91	C₂×C₄⋊Q₈	—	59	2	no	—
92	#494	—	59	4	no	—
93	#494	—	118	2	no	—
94	C₂×#106	—	118	2	no	—
95	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
96	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
97	#384	—	59	2	no	—
98	#340	—	118	8	no	—
99	#322	—	2	2	no	—
100	C₅₉	#9839	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₈	—	118	2	no	—
102	C₂×Q₁₆	—	118	8	no	—
103	C₂×Q₁₆	—	236	4	no	—
104	C₂×Q₁₆	—	236	4	no	—
105	C₂²×C₈	D₂₃₆	1	4	yes	—
106	C₂²×C₈	D₂₃₆	1	2	yes	—
107	C₄⋊Q₈	—	59	4	no	—
108	C₄⋊Q₈	—	118	4	no	—
109	C₄⋊Q₈	—	118	2	no	—
110	C₂×C₄⋊C₄	—	118	4	no	—
111	C₂×C₄⋊C₄	—	118	4	no	—
112	C₂×C₄⋊C₄	—	118	2	no	—
113	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
114	#106	—	236	4	no	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	C₄×C₈	C₄×D₅₉	1	2	yes	—
118	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
119	C₂×Q₈	—	118	4	no	—
120	C₂×Q₈	—	236	4	no	—
121	C₂×Q₈	—	236	4	no	—
122	C₂×Q₈	—	236	2	no	—
123	C₂²×C₄	—	236	2	no	—
124	C₂²×C₄	—	236	2	no	—
125	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
126	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
127	Q₁₆	—	236	16	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	8	no	—
130	C₂×C₈	#11745660	1	4	yes	—
131	C₂×C₈	D₄₇₂	1	4	yes	—
132	C₂×C₈	(C₂×D₅₉)⋊C₄	1	4	yes	—
133	C₄⋊C₄	—	118	8	no	—
134	C₄⋊C₄	—	236	4	no	—
135	C₄⋊C₄	—	236	8	no	—
136	C₄⋊C₄	—	236	4	no	—
137	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
138	C₄²	#11745685	1	2	yes	—
139	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
140	C₂³	#14273724	1	1	yes	characteristic
141	Q₈	—	472	4	no	—
142	Q₈	—	236	8	no	—
143	C₂×C₄	—	472	2	no	—
144	C₂×C₄	—	472	4	no	—
145	C₂×C₄	—	472	2	no	—
146	C₂×C₄	—	472	2	no	—
147	C₂×C₄	—	236	4	no	—
148	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
149	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
150	C₂×C₄	#14273701	1	2	yes	—
151	C₂×C₄	#14273715	1	1	yes	characteristic
152	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
153	C₂×C₄	C₂×#11745660	1	2	yes	—
154	C₂×C₄	#14273760	1	8	yes	—
155	C₈	—	4	4	no	—
156	C₈	#14273649	1	8	yes	—
157	C₂²	#16837304	1	1	yes	characteristic
158	C₂²	#16837775	1	1	yes	characteristic
159	C₂²	#16837779	1	1	yes	characteristic
160	C₂²	#16837796	1	4	yes	—
161	C₄	—	944	2	no	—
162	C₄	—	472	4	no	—
163	C₄	C₂×#14273649	1	2	yes	—
164	C₄	#16837510	1	4	yes	—
165	C₄	#16837660	1	4	yes	—
166	C₄	#16837749	1	2	yes	—
167	C₂	#21423729	1	2	yes	—
168	C₂	#21423767	1	2	yes	—
169	C₂	#21424902	1	1	yes	characteristic
170	C₂	#21424967	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837782

C₂

yes

maximal, characteristic

#21425057

C₂

yes

maximal

#21424253

C₂

yes

maximal, characteristic

C₂×C₅₉×#322

C₂

yes

maximal, characteristic

C₂²×Dic₂₃₆

—

C₂×#14273727

C₂²

yes

—

#16837782

—

#16837782

C₄

yes

complemented

C₂×#14273659

—

#16837272

C₂²

yes

—

#16837165

—

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#322

—

C₂²×Dic₁₁₈

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

#14273727

—

#14273727

—

#14273727

C₂×C₄

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

#14273659

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

#41

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₂₃₆

—

C₂³×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

—

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂×C₄×C₅₉

#137

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

#102

yes

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₄×C₅₉

C₂×#102

yes

—

C₄×C₅₉

#441

yes

—

C₄×C₅₉

#470

yes

—

C₄×C₅₉

#486

yes

—

Dic₅₉

—

Dic₅₉

—

—

—

—

D₅₉

yes

characteristic

C₂×C₅₉

#1716

yes

—

C₂×C₅₉

#1722

yes

—

C₂×C₅₉

#1957

yes

characteristic

C₂×C₅₉

#1971

yes

—

C₂²×Q₁₆

—

118

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

118

—

C₂×#106

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#384

—

#340

—

118

—

#322

—

100

C₅₉

#9839

yes

characteristic, complemented, Sylow

101

C₂²×Q₈

—

118

—

102

C₂×Q₁₆

—

118

—

103

C₂×Q₁₆

—

236

—

104

C₂×Q₁₆

—

236

—

105

C₂²×C₈

D₂₃₆

yes

—

106

C₂²×C₈

D₂₃₆

yes

—

107

C₄⋊Q₈

—

108

C₄⋊Q₈

—

118

—

109

C₄⋊Q₈

—

118

—

110

C₂×C₄⋊C₄

—

118

—

111

C₂×C₄⋊C₄

—

118

—

112

C₂×C₄⋊C₄

—

118

—

113

C₂×C₄²

C₂²×D₅₉

yes

characteristic

114

#106

—

236

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

C₄×D₅₉

yes

—

118

C₄×C₈

C₂²⋊D₅₉

yes

—

119

C₂×Q₈

—

118

—

120

C₂×Q₈

—

236

—

121

C₂×Q₈

—

236

—

122

C₂×Q₈

—

236

—

123

C₂²×C₄

—

236

—

124

C₂²×C₄

—

236

—

125

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

126

C₂²×C₄

C₂×D₂₃₆

yes

—

127

Q₁₆

—

236

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

#11745660

yes

—

131

C₂×C₈

D₄₇₂

yes

—

132

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

133

C₄⋊C₄

—

118

—

134

C₄⋊C₄

—

236

—

135

C₄⋊C₄

—

236

—

136

C₄⋊C₄

—

236

—

137

C₄²

C₂×C₄×D₅₉

yes

characteristic

138

C₄²

#11745685

yes

—

139

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

140

C₂³

#14273724

yes

characteristic

141

Q₈

—

472

—

142

Q₈

—

236

—

143

C₂×C₄

—

472

—

144

C₂×C₄

—

472

—

145

C₂×C₄

—

472

—

146

C₂×C₄

—

472

—

147

C₂×C₄

—

236

—

148

C₂×C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

149

C₂×C₄

C₄×D₂₃₆

yes

—

150

C₂×C₄

#14273701

yes

—

151

C₂×C₄

#14273715

yes

characteristic

152

C₂×C₄

C₂×D₄₇₂

yes

—

153

C₂×C₄

C₂×#11745660

yes

—

154

C₂×C₄

#14273760

yes

—

155

C₈

—

156

C₈

#14273649

yes

—

157

C₂²

#16837304

yes

characteristic

158

C₂²

#16837775

yes

characteristic

159

C₂²

#16837779

yes

characteristic

160

C₂²

#16837796

yes

—

161

C₄

—

944

—

162

C₄

—

472

—

163

C₄

C₂×#14273649

yes

—

164

C₄

#16837510

yes

—

165

C₄

#16837660

yes

—

166

C₄

#16837749

yes

—

167

C₂

#21423729

yes

—

168

C₂

#21423767

yes

—

169

C₂

#21424902

yes

characteristic

170

C₂

#21424967

yes

—

	subgroup	G/H
center	C₂³	#14273724
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#322	C₂
socle, prime radical	C₂³×C₅₉	#127
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#322	D₅₉
hypocenter	C₅₉	#9839
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂³

#14273724

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₂×C₅₉×#322

C₂

socle, prime radical

C₂³×C₅₉

#127

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
8. #16837782	C₄	semidirect
100. C₅₉	#9839	semidirect

H ≅ G/N

type

8. #16837782

C₄

semidirect

100. C₅₉

#9839

semidirect

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

subgroup	index
C₂×#16837782	2
#21425057	2
#21424253	2
C₂×C₅₉×#322	2
#9839	59

subgroup

index

C₂×C₅₉×#322

Subgroups — #28488538

Conjugacy classes of subgroups

Complements

Maximal subgroups