Group #21425610 — subgroups

Subgroups — #21425610

Order 7552.

176 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 67 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	#16837444	C₂	1	1	yes	maximal, characteristic
2	#16837442	C₂	1	1	yes	maximal, characteristic
3	C₂×#14273658	C₂	1	1	yes	maximal, characteristic
4	#16837430	C₂	1	1	yes	maximal, characteristic
5	#16837170	C₂	1	1	yes	maximal, characteristic
6	#16837166	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#343	C₂	1	1	yes	maximal, characteristic
8	C₂×#11745664	C₂²	1	1	yes	characteristic
9	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
10	#14273686	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
12	#14273661	—	2	1	no	—
13	#14273660	—	2	1	no	—
14	#14273659	—	2	1	no	—
15	#14273658	—	2	1	no	—
16	#14273658	—	2	2	no	—
17	Dic₅₉⋊C₈	—	2	2	no	—
18	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	—	2	1	no	—
22	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
23	C₂²×Dic₅₉	—	2	1	no	—
24	M₁₆×C₅₉	—	2	1	no	—
25	M₁₆×C₅₉	—	2	1	no	—
26	M₁₆×C₅₉	—	2	1	no	—
27	M₁₆×C₅₉	Q₈	1	2	yes	—
28	C₂×C₈×C₅₉	—	2	1	no	—
29	C₂×C₈×C₅₉	—	2	1	no	—
30	C₄⋊C₄×C₅₉	—	2	1	no	—
31	C₄⋊C₄×C₅₉	—	2	1	no	—
32	C₂²⋊Dic₅₉	—	2	1	no	—
33	C₄⋊Dic₅₉	—	2	1	no	—
34	C₄⋊Dic₅₉	—	2	1	no	—
35	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
36	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
37	C₄×Dic₅₉	—	2	1	no	—
38	#11745664	—	2	1	no	—
39	#11745664	Q₈	1	2	yes	—
40	C₂×C₅₉⋊C₈	—	2	1	no	—
41	C₂×C₅₉⋊C₈	—	2	1	no	—
42	C₂×C₅₉⋊C₈	—	2	1	no	—
43	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
44	C₂³×C₅₉	#41	1	1	yes	characteristic
45	C₂×C₄×C₅₉	—	4	1	no	—
46	C₂×C₄×C₅₉	—	8	1	no	—
47	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
52	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
53	C₂×Dic₅₉	—	2	1	no	—
54	C₂×Dic₅₉	—	2	1	no	—
55	C₂×Dic₅₉	—	4	1	no	—
56	C₂×Dic₅₉	—	8	1	no	—
57	C₈×C₅₉	—	2	2	no	—
58	C₈×C₅₉	—	4	1	no	—
59	C₅₉⋊C₈	—	2	2	no	—
60	C₅₉⋊C₈	—	2	2	no	—
61	C₂²×C₅₉	—	2	1	no	—
62	C₂²×C₅₉	—	2	1	no	—
63	C₂²×C₅₉	#121	1	1	yes	characteristic
64	C₂²×C₅₉	#125	1	1	yes	characteristic
65	C₂²×C₅₉	#125	1	1	yes	characteristic
66	C₄×C₅₉	—	8	1	no	—
67	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
68	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
69	C₄×C₅₉	#123	1	1	yes	characteristic
70	C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
71	Dic₅₉	—	4	1	no	—
72	Dic₅₉	—	4	1	no	—
73	Dic₅₉	—	8	1	no	—
74	#2090	—	59	1	no	maximal, Sylow
75	C₂×C₅₉	—	2	1	no	—
76	C₂×C₅₉	—	2	1	no	—
77	C₂×C₅₉	#398	1	1	yes	characteristic
78	C₂×C₅₉	#471	1	1	yes	characteristic
79	C₂×C₅₉	#471	1	1	yes	characteristic
80	#428	—	59	1	no	—
81	#428	—	59	1	no	—
82	C₂×#106	—	59	1	no	—
83	#424	—	59	1	no	—
84	#343	—	59	1	no	—
85	#343	D₅₉	1	1	yes	characteristic
86	#341	—	59	1	no	—
87	C₅₉	#2090	1	1	yes	characteristic, complemented, Sylow
88	C₂×M₁₆	—	59	1	no	—
89	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
90	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
91	C₂²×C₈	—	59	1	no	—
92	#117	—	59	1	no	—
93	C₂×C₄⋊C₄	—	59	1	no	—
94	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
95	#107	—	118	1	no	—
96	#107	—	118	1	no	—
97	#106	—	118	1	no	—
98	#106	—	118	2	no	—
99	#106	—	118	1	no	—
100	C₄⋊C₈	—	118	2	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	118	1	no	—
103	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
104	M₁₆	—	2	1	no	—
105	M₁₆	—	2	1	no	—
106	M₁₆	—	2	1	no	—
107	M₁₆	—	59	2	no	—
108	M₁₆	—	118	1	no	—
109	M₁₆	Dic₁₁₈	1	2	yes	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	59	2	no	—
113	C₂×C₈	—	118	1	no	—
114	C₂×C₈	—	118	1	no	—
115	C₂×C₈	—	118	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	59	1	no	—
119	C₄⋊C₄	—	59	1	no	—
120	C₄⋊C₄	—	118	1	no	—
121	C₄⋊C₄	—	118	1	no	—
122	C₂²⋊C₄	—	118	1	no	—
123	C₄²	—	118	1	no	—
124	C₂³	#11745685	1	1	yes	characteristic
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	8	1	no	—
127	C₂×C₄	—	118	1	no	—
128	C₂×C₄	—	118	1	no	—
129	C₂×C₄	—	472	1	no	—
130	C₂×C₄	—	236	1	no	—
131	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
132	C₂×C₄	#11745685	1	1	yes	characteristic
133	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
134	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
135	C₂×C₄	#11745687	1	1	yes	characteristic
136	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	4	1	no	—
139	C₈	—	118	2	no	—
140	C₈	—	118	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#14273700	1	1	yes	characteristic
144	C₂²	#14273717	1	1	yes	characteristic
145	C₂²	#14273719	1	1	yes	characteristic
146	C₄	—	8	1	no	—
147	C₄	—	472	1	no	—
148	C₄	—	236	1	no	—
149	C₄	—	236	1	no	—
150	C₄	#14273704	1	1	yes	characteristic
151	C₄	#14273705	1	1	yes	characteristic
152	C₄	#14273709	1	1	yes	characteristic
153	C₄	#14273726	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#16837336	1	1	yes	characteristic
157	C₂	#16837664	1	1	yes	characteristic
158	C₂	#16837668	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837444

C₂

yes

maximal, characteristic

#16837442

C₂

yes

maximal, characteristic

C₂×#14273658

C₂

yes

maximal, characteristic

#16837430

C₂

yes

maximal, characteristic

#16837170

C₂

yes

maximal, characteristic

#16837166

C₂

yes

maximal, characteristic

C₅₉×#343

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

#14273686

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

—

—

—

—

—

Dic₅₉⋊C₈

—

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

Q₈

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

D₄

yes

characteristic

C₄×Dic₅₉

—

#11745664

—

#11745664

Q₈

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂³×C₅₉

#41

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

#123

yes

characteristic

C₄×C₅₉

C₄⋊Q₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#2090

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

—

—

—

D₅₉

yes

characteristic

#341

—

C₅₉

#2090

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂²×C₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

#107

—

118

—

#107

—

118

—

#106

—

118

—

#106

—

118

—

#106

—

118

—

100

C₄⋊C₈

—

118

—

101

C₂²×C₄

—

102

C₂²×C₄

—

118

—

103

C₂²×C₄

C₂²×D₅₉

yes

characteristic

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

118

—

109

M₁₆

Dic₁₁₈

yes

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

118

—

114

C₂×C₈

—

118

—

115

C₂×C₈

—

118

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

118

—

121

C₄⋊C₄

—

118

—

122

C₂²⋊C₄

—

118

—

123

C₄²

—

118

—

124

C₂³

#11745685

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

118

—

128

C₂×C₄

—

118

—

129

C₂×C₄

—

472

—

130

C₂×C₄

—

236

—

131

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

132

C₂×C₄

#11745685

yes

characteristic

133

C₂×C₄

D₄×D₅₉

yes

characteristic

134

C₂×C₄

D₄×D₅₉

yes

characteristic

135

C₂×C₄

#11745687

yes

characteristic

136

C₂×C₄

Q₈×D₅₉

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

118

—

140

C₈

—

118

—

141

C₂²

—

142

C₂²

—

143

C₂²

#14273700

yes

characteristic

144

C₂²

#14273717

yes

characteristic

145

C₂²

#14273719

yes

characteristic

146

C₄

—

147

C₄

—

472

—

148

C₄

—

236

—

149

C₄

—

236

—

150

C₄

#14273704

yes

characteristic

151

C₄

#14273705

yes

characteristic

152

C₄

#14273709

yes

characteristic

153

C₄

#14273726

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

#16837336

yes

characteristic

157

C₂

#16837664

yes

characteristic

158

C₂

#16837668

yes

characteristic

	subgroup	G/H
center	C₂²	#14273700
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#343	C₂
socle	C₂²×C₅₉	#121
hypercenter	#343	D₅₉
hypocenter	C₅₉	#2090
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

#14273700

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#343

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
87. C₅₉	#2090	semidirect

H ≅ G/N

type

87. C₅₉

#2090

semidirect

Subgroups — #21425610

Conjugacy classes of subgroups

Complements