Group #20279924 — subgroups

Subgroups — #20279924

Order 6784.

500 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 189 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×C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	#16810542	C₂	1	4	yes	maximal
3	#16810528	C₂	1	2	yes	maximal
4	#16810519	C₂	1	4	yes	maximal
5	#16810492	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
9	C₄×Dic₁₀₆	—	2	4	no	—
10	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	C₂×#11719919	C₂²	1	2	yes	—
13	C₂×#11719919	C₂²	1	8	yes	—
14	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
15	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
16	#13133124	C₂²	1	8	yes	—
17	#13133124	C₂²	1	2	yes	—
18	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₅₃	—	2	4	no	—
20	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	—	2	4	no	—
24	C₂×Q₈×C₅₃	—	4	4	no	—
25	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
26	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₅₃	C₂³	1	2	yes	—
30	C₂²×Dic₅₃	C₂³	1	2	yes	—
31	C₂²×Dic₅₃	C₂³	1	4	yes	—
32	C₂×Dic₁₀₆	—	2	2	no	—
33	C₂×Dic₁₀₆	—	2	2	no	—
34	C₂×Dic₁₀₆	Q₈	1	4	yes	complemented
35	C₄⋊C₄×C₅₃	—	2	4	no	—
36	C₄⋊C₄×C₅₃	—	2	4	no	—
37	C₄²×C₅₃	D₄	1	4	yes	—
38	C₄⋊Dic₅₃	—	2	2	no	—
39	C₄⋊Dic₅₃	Q₈	1	8	yes	complemented
40	#11719919	—	2	4	no	—
41	#11719919	—	4	8	no	—
42	C₄×Dic₅₃	—	2	4	no	—
43	C₄×Dic₅₃	—	2	4	no	—
44	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
45	Q₈×C₅₃	—	4	8	no	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	4	no	—
48	C₂×C₄×C₅₃	—	2	8	no	—
49	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
54	C₂×C₄×C₅₃	#41	1	2	yes	—
55	C₂×Dic₅₃	—	2	2	no	—
56	C₂×Dic₅₃	—	2	2	no	—
57	C₂×Dic₅₃	—	2	4	no	—
58	C₂×Dic₅₃	—	2	4	no	—
59	C₂×Dic₅₃	—	2	2	no	—
60	C₂×Dic₅₃	—	2	4	no	—
61	C₂×Dic₅₃	C₂×Q₈	1	4	yes	—
62	C₂×Dic₅₃	C₂×Q₈	1	8	yes	—
63	Dic₁₀₆	—	2	8	no	—
64	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
65	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
66	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
67	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
68	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
69	C₄×C₅₃	—	2	2	no	—
70	C₄×C₅₃	—	4	4	no	—
71	C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
72	C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
73	Dic₅₃	—	2	4	no	—
74	Dic₅₃	—	2	8	no	—
75	Dic₅₃	—	4	2	no	—
76	#2567	—	53	1	no	maximal, Sylow
77	C₂×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₅₃	#535	1	1	yes	characteristic
80	C₂×C₅₃	#554	1	2	yes	—
81	C₂×C₅₃	Q₈²	1	2	yes	—
82	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
83	C₂×C₄×Q₈	—	53	1	no	—
84	#395	—	53	4	no	—
85	#391	—	53	2	no	—
86	#389	—	53	4	no	—
87	#384	—	53	2	no	—
88	C₄×C₄⋊C₄	—	53	1	no	—
89	C₅₃	#2567	1	1	yes	characteristic, complemented, Sylow
90	C₂²×Q₈	—	53	1	no	—
91	C₂²×Q₈	C₂×D₅₃	1	2	yes	—
92	C₄⋊Q₈	—	2	4	no	—
93	C₄×Q₈	—	106	4	no	—
94	C₂×C₄⋊C₄	—	53	2	no	—
95	C₂×C₄⋊C₄	—	53	2	no	—
96	C₂×C₄⋊C₄	—	53	1	no	—
97	C₂×C₄⋊C₄	—	53	8	no	—
98	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
99	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
100	C₂×C₄²	—	53	2	no	—
101	C₂×C₄²	—	53	2	no	—
102	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
103	#95	—	53	2	no	—
104	#95	—	53	8	no	—
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	—	4	4	no	—
107	C₂×Q₈	—	53	4	no	—
108	C₂×Q₈	—	106	2	no	—
109	C₂×Q₈	—	106	2	no	—
110	C₂²×C₄	—	53	2	no	—
111	C₂²×C₄	—	53	4	no	—
112	C₂²×C₄	—	53	2	no	—
113	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
114	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	53	8	no	—
120	C₄⋊C₄	—	106	4	no	—
121	C₄⋊C₄	—	106	2	no	—
122	C₄⋊C₄	—	212	8	no	—
123	C₄²	—	106	4	no	—
124	C₄²	—	106	4	no	—
125	C₄²	C₂²⋊D₅₃	1	4	yes	—
126	C₂³	C₂³×D₅₃	1	1	yes	characteristic
127	Q₈	—	4	8	no	—
128	Q₈	—	106	8	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	8	no	—
132	C₂×C₄	—	53	8	no	—
133	C₂×C₄	—	53	4	no	—
134	C₂×C₄	—	106	2	no	—
135	C₂×C₄	—	106	4	no	—
136	C₂×C₄	—	106	2	no	—
137	C₂×C₄	—	106	4	no	—
138	C₂×C₄	—	106	2	no	—
139	C₂×C₄	—	106	4	no	—
140	C₂×C₄	Q₈×D₅₃	1	2	yes	—
141	C₂×C₄	Q₈×D₅₃	1	4	yes	—
142	C₂×C₄	#11719949	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
144	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
145	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
146	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
147	C₂²	C₂×Q₈×D₅₃	1	2	yes	—
148	C₂²	C₂×Q₈×D₅₃	1	1	yes	characteristic
149	C₂²	C₂×#11719949	1	1	yes	characteristic
150	C₂²	#13133262	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	4	4	no	—
153	C₄	—	106	8	no	—
154	C₄	—	106	4	no	—
155	C₄	—	212	2	no	—
156	C₄	#13133193	1	4	yes	—
157	C₄	#13133215	1	4	yes	—
158	C₂	C₂×#13133193	1	1	yes	characteristic
159	C₂	C₂×#13133215	1	1	yes	characteristic
160	C₂	#16811190	1	1	yes	characteristic
161	C₂	#16811334	1	2	yes	—
162	C₂	#16811353	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₀₆

C₂

yes

maximal, characteristic

#16810542

C₂

yes

maximal

#16810528

C₂

yes

maximal

#16810519

C₂

yes

maximal

#16810492

C₂

yes

maximal

C₄×C₄⋊Dic₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

characteristic

C₄×Dic₁₀₆

—

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

Q₈

yes

complemented

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

Q₈

yes

complemented

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂³×C₅₃

C₂⁴

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×Q₈

yes

—

C₂×Dic₅₃

C₂×Q₈

yes

—

Dic₁₀₆

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₅₃

C₂²×Q₈

yes

—

C₂²×C₅₃

C₂×#41

yes

characteristic

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

C₄⋊Q₈

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#2567

—

maximal, Sylow

C₂×C₅₃

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₅₃

C₂×C₄⋊Q₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

Q₈²

yes

—

C₂×C₄⋊Q₈

D₅₃

yes

characteristic

C₂×C₄×Q₈

—

—

—

—

—

C₄×C₄⋊C₄

—

C₅₃

#2567

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅₃

yes

—

C₄⋊Q₈

—

C₄×Q₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

100

C₂×C₄²

—

101

C₂×C₄²

—

102

C₂×C₄²

C₂×D₅₃

yes

characteristic

103

#95

—

104

#95

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

106

—

109

C₂×Q₈

—

106

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂²×D₅₃

yes

—

114

C₂²×C₄

C₂²×D₅₃

yes

characteristic

115

C₂²×C₄

C₂²×D₅₃

yes

characteristic

116

C₂²×C₄

C₂²×D₅₃

yes

characteristic

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

106

—

121

C₄⋊C₄

—

106

—

122

C₄⋊C₄

—

212

—

123

C₄²

—

106

—

124

C₄²

—

106

—

125

C₄²

C₂²⋊D₅₃

yes

—

126

C₂³

C₂³×D₅₃

yes

characteristic

127

Q₈

—

128

Q₈

—

106

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

106

—

135

C₂×C₄

—

106

—

136

C₂×C₄

—

106

—

137

C₂×C₄

—

106

—

138

C₂×C₄

—

106

—

139

C₂×C₄

—

106

—

140

C₂×C₄

Q₈×D₅₃

yes

—

141

C₂×C₄

Q₈×D₅₃

yes

—

142

C₂×C₄

#11719949

yes

—

143

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

144

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

145

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

146

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

147

C₂²

C₂×Q₈×D₅₃

yes

—

148

C₂²

C₂×Q₈×D₅₃

yes

characteristic

149

C₂²

C₂×#11719949

yes

characteristic

150

C₂²

#13133262

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

106

—

154

C₄

—

106

—

155

C₄

—

212

—

156

C₄

#13133193

yes

—

157

C₄

#13133215

yes

—

158

C₂

C₂×#13133193

yes

characteristic

159

C₂

C₂×#13133215

yes

characteristic

160

C₂

#16811190

yes

characteristic

161

C₂

#16811334

yes

—

162

C₂

#16811353

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#2567
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

hypercenter

C₂×C₄⋊Q₈

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
34. C₂×Dic₁₀₆	Q₈	semidirect
39. C₄⋊Dic₅₃	Q₈	semidirect
89. C₅₃	#2567	semidirect

H ≅ G/N

type

34. C₂×Dic₁₀₆

Q₈

semidirect

39. C₄⋊Dic₅₃

Q₈

semidirect

89. C₅₃

#2567

semidirect

Subgroups — #20279924

Conjugacy classes of subgroups

Complements