Group #20291041 — subgroups

Subgroups — D₅₃⋊#362 #20291041

Order 6784.

216 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 64 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	D₅₃⋊M₃₂	C₂	1	1	yes	maximal, characteristic
2	D₅₃⋊#108	C₂	1	1	yes	maximal, characteristic
3	#16811542	C₂	1	4	yes	maximal, complemented
4	C₂×Q₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
5	D₅₃⋊M₁₆	—	2	1	no	—
6	#13133292	C₂²	1	2	yes	—
7	#13133291	—	2	1	no	—
8	D₅₃⋊C₁₆	—	2	1	no	—
9	#13133288	—	2	1	no	—
10	#13133287	C₂²	1	2	yes	—
11	C₂×Q₈×D₅₃	—	2	1	no	—
12	C₂×Dic₂₁₂	C₄	1	1	yes	characteristic
13	C₂×#11719925	—	2	1	no	—
14	Q₁₆×D₅₃	—	2	1	no	—
15	Q₁₆×D₅₃	—	2	1	no	—
16	Q₁₆×D₅₃	C₄	1	2	yes	—
17	Q₁₆×D₅₃	C₂²	1	2	yes	—
18	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
19	C₂×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
20	C₂×Q₈×C₅₃	—	2	1	no	—
21	Q₈×D₅₃	—	2	2	no	—
22	Q₈×D₅₃	—	2	2	no	—
23	Q₈×D₅₃	—	4	1	no	—
24	Q₈×D₅₃	—	4	1	no	—
25	C₂×C₄×D₅₃	—	4	1	no	—
26	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
27	C₂×Dic₁₀₆	—	2	1	no	—
28	C₂×Dic₁₀₆	—	4	1	no	—
29	#11719940	—	4	1	no	—
30	#11719936	—	2	2	no	—
31	D₅₃⋊C₈	—	4	1	no	—
32	Q₁₆×C₅₃	—	2	1	no	—
33	Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	#11719925	—	2	2	no	—
36	#11719925	—	2	2	no	—
37	C₂×#88322	D₄	1	1	yes	characteristic
38	Dic₂₁₂	—	2	1	no	—
39	Dic₂₁₂	C₂×C₄	1	2	yes	—
40	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
41	C₈×D₅₃	D₄	1	1	yes	characteristic
42	C₈×D₅₃	D₄	1	1	yes	characteristic
43	C₈×D₅₃	C₂³	1	1	yes	characteristic
44	#11719910	—	2	2	no	—
45	C₂²×D₅₃	SD₁₆	1	1	yes	characteristic
46	Q₈×C₅₃	—	2	2	no	—
47	Q₈×C₅₃	—	4	1	no	—
48	C₂×C₄×C₅₃	—	4	1	no	—
49	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×Dic₅₃	—	4	1	no	—
51	C₂×Dic₅₃	D₈	1	1	yes	characteristic
52	C₄×D₅₃	—	4	2	no	—
53	C₄×D₅₃	—	4	2	no	—
54	C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₄×D₅₃	D₈	1	1	yes	characteristic
56	C₄×D₅₃	SD₁₆	1	1	yes	characteristic
57	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
58	Dic₁₀₆	—	2	2	no	—
59	Dic₁₀₆	—	4	1	no	—
60	Dic₁₀₆	—	4	2	no	—
61	Dic₁₀₆	—	4	2	no	—
62	#88324	—	4	2	no	—
63	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	#88322	C₂²⋊C₄	1	1	yes	characteristic
66	#88322	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₅₃	#102	1	1	yes	characteristic
68	C₂×D₅₃	—	2	2	no	—
69	C₂×D₅₃	#102	1	1	yes	characteristic
70	C₂×D₅₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₄×C₅₃	—	4	2	no	—
72	C₄×C₅₃	#102	1	1	yes	characteristic
73	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	Dic₅₃	—	4	2	no	—
75	Dic₅₃	#102	1	1	yes	characteristic
76	Dic₅₃	C₂×D₈	1	1	yes	characteristic
77	C₂×#362	—	53	1	no	maximal, Sylow
78	C₂×C₅₃	—	2	1	no	—
79	C₂×C₅₃	C₂×#102	1	1	yes	characteristic
80	D₅₃	—	2	1	no	—
81	D₅₃	#362	1	2	yes	complemented
82	C₂²×Q₁₆	—	53	1	no	—
83	C₂×M₃₂	—	53	1	no	—
84	C₂×#108	—	53	1	no	—
85	#362	—	53	4	no	—
86	C₅₃	C₂×#362	1	1	yes	characteristic, complemented, Sylow
87	C₂²×Q₈	—	106	1	no	—
88	C₂×Q₁₆	—	53	2	no	—
89	C₂×Q₁₆	—	53	2	no	—
90	C₂×Q₁₆	—	53	1	no	—
91	C₂×Q₁₆	—	106	1	no	—
92	C₂×Q₁₆	—	106	1	no	—
93	C₂×Q₁₆	—	106	1	no	—
94	C₂×Q₁₆	#6157	1	1	yes	characteristic
95	C₂×M₁₆	—	106	1	no	—
96	C₂²×C₈	—	53	1	no	—
97	M₃₂	—	53	2	no	—
98	M₃₂	—	106	1	no	—
99	C₂×C₁₆	—	106	1	no	—
100	#108	—	53	2	no	—
101	#108	—	106	1	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	106	2	no	—
104	C₂×Q₈	—	106	2	no	—
105	C₂×Q₈	—	106	1	no	—
106	C₂×Q₈	—	212	1	no	—
107	C₂×Q₈	—	212	1	no	—
108	C₂×Q₈	—	212	1	no	—
109	C₂²×C₄	—	53	1	no	—
110	C₂²×C₄	—	212	1	no	—
111	Q₁₆	—	2	1	no	—
112	Q₁₆	—	53	2	no	—
113	Q₁₆	—	106	2	no	—
114	Q₁₆	—	106	2	no	—
115	Q₁₆	—	106	1	no	—
116	Q₁₆	C₂×#6157	1	2	yes	—
117	M₁₆	—	106	2	no	—
118	M₁₆	—	212	1	no	—
119	C₂×C₈	—	53	1	no	—
120	C₂×C₈	—	53	1	no	—
121	C₂×C₈	—	53	1	no	—
122	C₂×C₈	—	53	1	no	—
123	C₂×C₈	—	53	1	no	—
124	C₂×C₈	—	212	1	no	—
125	C₂×C₈	C₂×#6157	1	1	yes	characteristic
126	C₁₆	—	106	2	no	—
127	C₂³	—	53	1	no	—
128	Q₈	—	2	2	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	106	2	no	—
131	Q₈	—	212	2	no	—
132	Q₈	—	212	2	no	—
133	Q₈	—	212	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	53	1	no	—
136	C₂×C₄	—	53	1	no	—
137	C₂×C₄	—	53	1	no	—
138	C₂×C₄	—	53	1	no	—
139	C₂×C₄	—	53	1	no	—
140	C₂×C₄	—	212	2	no	—
141	C₂×C₄	—	212	2	no	—
142	C₂×C₄	—	212	1	no	—
143	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
144	C₈	—	53	1	no	—
145	C₈	—	53	1	no	—
146	C₈	—	212	2	no	—
147	C₈	C₂²⋊#6157	1	1	yes	characteristic
148	C₈	C₂²×#6157	1	1	yes	characteristic
149	C₂²	—	53	1	no	—
150	C₂²	—	53	1	no	—
151	C₂²	—	106	2	no	—
152	C₂²	#13133279	1	1	yes	characteristic
153	C₄	—	4	2	no	—
154	C₄	—	53	1	no	—
155	C₄	—	53	1	no	—
156	C₄	—	212	2	no	—
157	C₄	#13133279	1	1	yes	characteristic
158	C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
159	C₂	—	2	1	no	—
160	C₂	—	53	2	no	—
161	C₂	—	106	1	no	—
162	C₂	C₂×#13133279	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅₃⋊M₃₂

C₂

yes

maximal, characteristic

D₅₃⋊#108

C₂

yes

maximal, characteristic

#16811542

C₂

yes

maximal, complemented

C₂×Q₁₆×D₅₃

C₂

yes

maximal, characteristic

D₅₃⋊M₁₆

—

#13133292

C₂²

yes

—

#13133291

—

D₅₃⋊C₁₆

—

#13133288

—

#13133287

C₂²

yes

—

C₂×Q₈×D₅₃

—

C₂×Dic₂₁₂

C₄

yes

characteristic

C₂×#11719925

—

Q₁₆×D₅₃

—

Q₁₆×D₅₃

—

Q₁₆×D₅₃

C₄

yes

—

Q₁₆×D₅₃

C₂²

yes

—

C₂×C₈×D₅₃

C₂²

yes

characteristic

C₂×Q₁₆×C₅₃

C₄

yes

characteristic

C₂×Q₈×C₅₃

—

Q₈×D₅₃

—

Q₈×D₅₃

—

Q₈×D₅₃

—

Q₈×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

#11719940

—

#11719936

—

D₅₃⋊C₈

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

—

D₄

yes

characteristic

Dic₂₁₂

—

Dic₂₁₂

C₂×C₄

yes

—

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

D₄

yes

characteristic

C₈×D₅₃

D₄

yes

characteristic

C₈×D₅₃

C₂³

yes

characteristic

#11719910

—

C₂²×D₅₃

SD₁₆

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

D₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂²⋊C₄

yes

characteristic

C₄×D₅₃

D₈

yes

characteristic

C₄×D₅₃

SD₁₆

yes

characteristic

C₄×D₅₃

C₂×D₄

yes

characteristic

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

#88324

—

C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₈×C₅₃

C₂²×C₄

yes

characteristic

#88322

C₂²⋊C₄

yes

characteristic

#88322

C₂×D₄

yes

characteristic

C₂²×C₅₃

#102

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

#102

yes

characteristic

C₂×D₅₃

C₂×SD₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#102

yes

characteristic

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

Dic₅₃

—

Dic₅₃

#102

yes

characteristic

Dic₅₃

C₂×D₈

yes

characteristic

C₂×#362

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#102

yes

characteristic

D₅₃

—

D₅₃

#362

yes

complemented

C₂²×Q₁₆

—

C₂×M₃₂

—

C₂×#108

—

#362

—

C₅₃

C₂×#362

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

#6157

yes

characteristic

C₂×M₁₆

—

106

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

106

—

C₂×C₁₆

—

106

—

100

#108

—

101

#108

—

106

—

102

C₂×Q₈

—

103

C₂×Q₈

—

106

—

104

C₂×Q₈

—

106

—

105

C₂×Q₈

—

106

—

106

C₂×Q₈

—

212

—

107

C₂×Q₈

—

212

—

108

C₂×Q₈

—

212

—

109

C₂²×C₄

—

110

C₂²×C₄

—

212

—

111

Q₁₆

—

112

Q₁₆

—

113

Q₁₆

—

106

—

114

Q₁₆

—

106

—

115

Q₁₆

—

106

—

116

Q₁₆

C₂×#6157

yes

—

117

M₁₆

—

106

—

118

M₁₆

—

212

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

212

—

125

C₂×C₈

C₂×#6157

yes

characteristic

126

C₁₆

—

106

—

127

C₂³

—

128

Q₈

—

129

Q₈

—

130

Q₈

—

106

—

131

Q₈

—

212

—

132

Q₈

—

212

—

133

Q₈

—

212

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

212

—

141

C₂×C₄

—

212

—

142

C₂×C₄

—

212

—

143

C₂×C₄

C₂²⋊#6157

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

212

—

147

C₈

C₂²⋊#6157

yes

characteristic

148

C₈

C₂²×#6157

yes

characteristic

149

C₂²

—

150

C₂²

—

151

C₂²

—

106

—

152

C₂²

#13133279

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

212

—

157

C₄

#13133279

yes

characteristic

158

C₄

C₂×C₂²⋊#6157

yes

characteristic

159

C₂

—

160

C₂

—

161

C₂

—

106

—

162

C₂

C₂×#13133279

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#13133279
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₈	C₂²×#6157
Fitting	C₂×Q₁₆×C₅₃	C₄
socle	C₂×C₅₃	C₂×#102
hypercenter	C₂×Q₁₆	#6157
hypocenter	C₅₃	C₂×#362
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂

C₂×#13133279

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₈

C₂²×#6157

Fitting

C₂×Q₁₆×C₅₃

C₄

C₂×C₅₃

C₂×Q₁₆

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

N	H ≅ G/N	type
3. #16811542	C₂	semidirect
81. D₅₃	#362	semidirect
86. C₅₃	C₂×#362	semidirect

H ≅ G/N

type

3. #16811542

C₂

semidirect

81. D₅₃

#362

semidirect

86. C₅₃

C₂×#362

semidirect

Subgroups — D₅₃⋊#362 #20291041

Conjugacy classes of subgroups

Complements

Subgroups — D53⋊#362 #20291041

Conjugacy classes of subgroups

Complements

Subgroups — D₅₃⋊#362 #20291041