Group #20276228 — subgroups

Subgroups — #20276228

Order 6784.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#16810666	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133148	C₂	1	1	yes	maximal, characteristic
3	#16810650	C₂	1	1	yes	maximal, characteristic
4	#16810392	C₂	1	1	yes	maximal, characteristic
5	#16810391	C₂	1	1	yes	maximal, characteristic
6	#16810387	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#428	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719917	C₂²	1	1	yes	characteristic
9	C₂×#11719917	C₂²	1	1	yes	characteristic
10	C₂²×#88322	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
12	#13133150	—	2	1	no	—
13	#13133148	—	2	1	no	—
14	#13133148	—	2	2	no	—
15	C₄⋊#88322	—	2	2	no	—
16	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	#117×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	#107×C₅₃	—	2	1	no	—
20	#106×C₅₃	—	2	1	no	—
21	C₂²×C₄×C₅₃	—	2	1	no	—
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₅₃	—	2	1	no	—
24	M₁₆×C₅₃	—	2	1	no	—
25	M₁₆×C₅₃	Q₈	1	2	yes	—
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₄⋊C₄×C₅₃	—	2	1	no	—
28	C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
30	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
31	C₂²⋊C₄×C₅₃	—	2	1	no	—
32	C₄²×C₅₃	—	2	1	no	—
33	C₄⋊Dic₅₃	—	2	1	no	—
34	C₄⋊Dic₅₃	—	2	1	no	—
35	#11719917	—	2	1	no	—
36	#11719917	—	2	1	no	—
37	#11719917	—	2	1	no	—
38	#11719917	Q₈	1	2	yes	—
39	C₂×#88322	—	2	1	no	—
40	C₂×#88322	—	2	1	no	—
41	C₂×#88322	—	2	1	no	—
42	C₂×#88322	—	2	1	no	—
43	C₂×#88322	D₄	1	2	yes	—
44	C₂³×C₅₃	#41	1	1	yes	characteristic
45	C₂×C₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	—	4	1	no	—
48	C₂×C₄×C₅₃	—	8	1	no	—
49	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
55	C₂×Dic₅₃	—	4	1	no	—
56	C₂×Dic₅₃	—	8	1	no	—
57	C₈×C₅₃	—	2	2	no	—
58	#88322	—	2	2	no	—
59	#88322	—	2	2	no	—
60	#88322	—	4	1	no	—
61	C₂²×C₅₃	—	2	1	no	—
62	C₂²×C₅₃	—	2	1	no	—
63	C₂²×C₅₃	#121	1	1	yes	characteristic
64	C₂²×C₅₃	#125	1	1	yes	characteristic
65	C₂²×C₅₃	#125	1	1	yes	characteristic
66	C₄×C₅₃	—	4	1	no	—
67	C₄×C₅₃	—	4	1	no	—
68	C₄×C₅₃	—	8	1	no	—
69	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
70	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₄×C₅₃	#123	1	1	yes	characteristic
72	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
73	Dic₅₃	—	8	1	no	—
74	#2090	—	53	1	no	maximal, Sylow
75	C₂×C₅₃	—	2	1	no	—
76	C₂×C₅₃	—	2	1	no	—
77	C₂×C₅₃	#398	1	1	yes	characteristic
78	C₂×C₅₃	#471	1	1	yes	characteristic
79	C₂×C₅₃	#471	1	1	yes	characteristic
80	#428	—	53	1	no	—
81	#428	D₅₃	1	1	yes	characteristic
82	C₂×#106	—	53	1	no	—
83	#424	—	53	1	no	—
84	#343	—	53	1	no	—
85	#343	—	53	1	no	—
86	#341	—	53	1	no	—
87	C₅₃	#2090	1	1	yes	characteristic, complemented, Sylow
88	C₂×M₁₆	—	53	1	no	—
89	C₂×M₁₆	—	53	1	no	—
90	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
91	C₂²×C₈	—	53	1	no	—
92	#117	C₂×D₅₃	1	1	yes	characteristic
93	C₂×C₄⋊C₄	—	53	1	no	—
94	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
95	#107	—	2	1	no	—
96	#107	—	106	1	no	—
97	#106	—	2	1	no	—
98	#106	—	106	2	no	—
99	#106	—	106	1	no	—
100	C₄⋊C₈	—	106	2	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	106	1	no	—
103	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
104	M₁₆	—	2	1	no	—
105	M₁₆	—	53	2	no	—
106	M₁₆	—	106	1	no	—
107	M₁₆	—	106	1	no	—
108	M₁₆	—	106	1	no	—
109	M₁₆	Dic₁₀₆	1	2	yes	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	53	2	no	—
112	C₂×C₈	—	106	1	no	—
113	C₂×C₈	—	106	1	no	—
114	C₂×C₈	—	106	1	no	—
115	C₂×C₈	—	106	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	106	1	no	—
119	C₄⋊C₄	—	106	1	no	—
120	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
121	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
122	C₂²⋊C₄	—	2	1	no	—
123	C₄²	—	2	1	no	—
124	C₂³	#11719947	1	1	yes	characteristic
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	424	1	no	—
130	C₂×C₄	—	212	1	no	—
131	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
132	C₂×C₄	#11719945	1	1	yes	characteristic
133	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
134	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
135	C₂×C₄	#11719949	1	1	yes	characteristic
136	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	106	2	no	—
139	C₈	—	106	2	no	—
140	C₈	—	212	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#13133189	1	1	yes	characteristic
144	C₂²	#13133207	1	1	yes	characteristic
145	C₂²	#13133208	1	1	yes	characteristic
146	C₄	—	4	1	no	—
147	C₄	—	4	1	no	—
148	C₄	—	8	1	no	—
149	C₄	—	424	1	no	—
150	C₄	#13133193	1	1	yes	characteristic
151	C₄	#13133196	1	1	yes	characteristic
152	C₄	#13133202	1	1	yes	characteristic
153	C₄	#13133216	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#16810556	1	1	yes	characteristic
157	C₂	#16810884	1	1	yes	characteristic
158	C₂	#16810888	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810666

C₂

yes

maximal, characteristic

C₂×#13133148

C₂

yes

maximal, characteristic

#16810650

C₂

yes

maximal, characteristic

#16810392

C₂

yes

maximal, characteristic

#16810391

C₂

yes

maximal, characteristic

#16810387

C₂

yes

maximal, characteristic

C₅₃×#428

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

—

—

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

#107×C₅₃

—

#106×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

Q₈

yes

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

—

—

—

Q₈

yes

—

—

—

—

—

D₄

yes

—

C₂³×C₅₃

#41

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

#123

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

Dic₅₃

—

#2090

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

—

—

—

—

—

C₅₃

#2090

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

#117

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

#107

—

#107

—

106

—

#106

—

#106

—

106

—

#106

—

106

—

100

C₄⋊C₈

—

106

—

101

C₂²×C₄

—

102

C₂²×C₄

—

106

—

103

C₂²×C₄

C₂²×D₅₃

yes

characteristic

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

106

—

107

M₁₆

—

106

—

108

M₁₆

—

106

—

109

M₁₆

Dic₁₀₆

yes

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

106

—

113

C₂×C₈

—

106

—

114

C₂×C₈

—

106

—

115

C₂×C₈

—

106

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

106

—

119

C₄⋊C₄

—

106

—

120

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

121

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

122

C₂²⋊C₄

—

123

C₄²

—

124

C₂³

#11719947

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

424

—

130

C₂×C₄

—

212

—

131

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

132

C₂×C₄

#11719945

yes

characteristic

133

C₂×C₄

D₄×D₅₃

yes

characteristic

134

C₂×C₄

Q₈×D₅₃

yes

characteristic

135

C₂×C₄

#11719949

yes

characteristic

136

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

137

C₈

—

138

C₈

—

106

—

139

C₈

—

106

—

140

C₈

—

212

—

141

C₂²

—

142

C₂²

—

143

C₂²

#13133189

yes

characteristic

144

C₂²

#13133207

yes

characteristic

145

C₂²

#13133208

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

424

—

150

C₄

#13133193

yes

characteristic

151

C₄

#13133196

yes

characteristic

152

C₄

#13133202

yes

characteristic

153

C₄

#13133216

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

#16810556

yes

characteristic

157

C₂

#16810884

yes

characteristic

158

C₂

#16810888

yes

characteristic

	subgroup	G/H
center	C₂²	#13133189
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#428	C₂
socle	C₂²×C₅₃	#121
hypercenter	#428	D₅₃
hypocenter	C₅₃	#2090
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

subgroup

G/H

center

C₂²

#13133189

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#428

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

N	H ≅ G/N	type
87. C₅₃	#2090	semidirect

H ≅ G/N

type

87. C₅₃

#2090

semidirect

Subgroups — #20276228

Conjugacy classes of subgroups

Complements