Group #20275467 — subgroups

Subgroups — #20275467

Order 6784.

200 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 83 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×#13133152	C₂	1	1	yes	maximal, characteristic
2	#16810659	C₂	1	1	yes	maximal, characteristic
3	#16810651	C₂	1	1	yes	maximal, characteristic
4	#16810650	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133128	C₂	1	1	yes	maximal, characteristic
6	#16810383	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#339	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719917	C₂²	1	1	yes	characteristic
9	C₂²×#88322	C₂²	1	1	yes	characteristic
10	C₂²×#88322	C₂²	1	1	yes	characteristic
11	#13133176	C₂²	1	1	yes	characteristic
12	#13133152	—	2	2	no	—
13	#13133150	C₄	1	2	yes	complemented
14	#13133148	C₄	1	2	yes	complemented
15	C₄⋊#88322	—	2	2	no	—
16	#13133146	—	2	2	no	—
17	#13133128	—	2	2	no	—
18	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	#117×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	—	2	1	no	—
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	M₁₆×C₅₃	—	2	1	no	—
24	M₁₆×C₅₃	—	2	1	no	—
25	C₂×C₈×C₅₃	—	2	1	no	—
26	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
27	C₂²⋊C₄×C₅₃	—	2	1	no	—
28	C₄²×C₅₃	—	2	1	no	—
29	C₄²×C₅₃	—	2	1	no	—
30	C₄²×C₅₃	Q₈	1	2	yes	—
31	C₂²⋊Dic₅₃	—	2	1	no	—
32	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
33	C₄×Dic₅₃	—	2	1	no	—
34	#11719917	—	2	1	no	—
35	#11719917	—	2	1	no	—
36	C₂×#88322	—	2	1	no	—
37	C₂×#88322	—	2	1	no	—
38	C₂×#88322	—	2	1	no	—
39	C₂×#88322	C₂×C₄	1	1	yes	characteristic
40	C₂×#88322	C₂×C₄	1	1	yes	characteristic
41	C₂×#88322	D₄	1	2	yes	—
42	C₂×#88322	D₄	1	2	yes	—
43	C₂×#88322	Q₈	1	2	yes	—
44	C₂³×C₅₃	#41	1	1	yes	characteristic
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	4	2	no	—
48	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
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×Dic₅₃	—	2	2	no	—
55	C₈×C₅₃	—	4	1	no	—
56	#88322	—	2	2	no	—
57	#88322	—	4	1	no	—
58	#88322	C₄⋊C₄	1	2	yes	—
59	#88322	C₄⋊C₄	1	2	yes	—
60	C₂²×C₅₃	—	2	2	no	—
61	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
62	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₅₃	#125	1	1	yes	characteristic
64	C₄×C₅₃	—	4	2	no	—
65	C₄×C₅₃	—	4	2	no	—
66	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
68	C₄×C₅₃	#121	1	1	yes	characteristic
69	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
70	Dic₅₃	—	4	2	no	—
71	#1946	—	53	1	no	maximal, Sylow
72	C₂×C₅₃	—	2	1	no	—
73	C₂×C₅₃	—	2	1	no	—
74	C₂×C₅₃	#389	1	1	yes	characteristic
75	C₂×C₅₃	#444	1	2	yes	—
76	C₂×#108	—	53	1	no	—
77	#427	—	53	1	no	—
78	#424	—	53	1	no	—
79	#424	—	53	1	no	—
80	C₂×#97	—	53	1	no	—
81	#339	—	53	1	no	—
82	#339	D₅₃	1	1	yes	characteristic
83	C₅₃	#1946	1	1	yes	characteristic, complemented, Sylow
84	C₂×M₁₆	—	53	1	no	—
85	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
86	C₂²×C₈	—	53	1	no	—
87	C₂²×C₈	—	53	1	no	—
88	#117	—	53	1	no	—
89	#117	C₂×D₅₃	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
91	#108	—	106	2	no	—
92	#107	—	53	2	no	—
93	#106	—	53	2	no	—
94	C₄⋊C₈	—	106	2	no	—
95	C₄⋊C₈	—	106	2	no	—
96	#97	—	106	2	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	106	1	no	—
102	M₁₆	—	106	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	53	1	no	—
105	C₂×C₈	—	53	1	no	—
106	C₂×C₈	—	53	2	no	—
107	C₂×C₈	—	53	2	no	—
108	C₂×C₈	—	53	2	no	—
109	C₂×C₈	—	106	1	no	—
110	C₂×C₈	—	106	1	no	—
111	C₂×C₈	—	106	1	no	—
112	C₄⋊C₄	—	53	2	no	—
113	C₄⋊C₄	C₄×D₅₃	1	2	yes	—
114	C₂²⋊C₄	—	2	1	no	—
115	C₂²⋊C₄	—	106	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	106	1	no	—
119	C₄²	Dic₁₀₆	1	2	yes	—
120	C₂³	#11719947	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	106	2	no	—
125	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
126	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
127	C₂×C₄	#11719945	1	1	yes	characteristic
128	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
129	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
130	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
131	C₈	—	4	1	no	—
132	C₈	—	53	2	no	—
133	C₈	—	53	2	no	—
134	C₈	—	106	2	no	—
135	C₈	—	212	1	no	—
136	C₂²	—	2	2	no	—
137	C₂²	#13133177	1	1	yes	characteristic
138	C₂²	#13133194	1	1	yes	characteristic
139	C₂²	#13133207	1	1	yes	characteristic
140	C₄	—	4	2	no	—
141	C₄	—	4	2	no	—
142	C₄	—	212	2	no	—
143	C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
144	C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
145	C₄	#13133190	1	1	yes	characteristic
146	C₄	#13133216	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#16810520	1	1	yes	characteristic
150	C₂	#16810742	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133152

C₂

yes

maximal, characteristic

#16810659

C₂

yes

maximal, characteristic

#16810651

C₂

yes

maximal, characteristic

#16810650

C₂

yes

maximal, characteristic

C₂×#13133128

C₂

yes

maximal, characteristic

#16810383

C₂

yes

maximal, characteristic

C₅₃×#339

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

#13133176

C₂²

yes

characteristic

#13133152

—

#13133150

C₄

yes

complemented

#13133148

C₄

yes

—

—

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

Q₈

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

—

—

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

D₄

yes

—

C₂×#88322

D₄

yes

—

C₂×#88322

Q₈

yes

—

C₂³×C₅₃

#41

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

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

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

#88322

C₄⋊C₄

yes

—

#88322

C₄⋊C₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

#125

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

characteristic

C₄×C₅₃

#121

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

Dic₅₃

—

#1946

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#389

yes

characteristic

C₂×C₅₃

#444

yes

—

—

—

—

—

C₂×#97

—

#339

—

#339

D₅₃

yes

characteristic

C₅₃

#1946

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

characteristic

#108

—

106

—

#107

—

#106

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

#97

—

106

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

106

—

102

M₁₆

—

106

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

106

—

110

C₂×C₈

—

106

—

111

C₂×C₈

—

106

—

112

C₄⋊C₄

—

113

C₄⋊C₄

C₄×D₅₃

yes

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

106

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

106

—

119

C₄²

Dic₁₀₆

yes

—

120

C₂³

#11719947

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

106

—

125

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

126

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

127

C₂×C₄

#11719945

yes

characteristic

128

C₂×C₄

D₄×D₅₃

yes

characteristic

129

C₂×C₄

D₄×D₅₃

yes

characteristic

130

C₂×C₄

Q₈×D₅₃

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

106

—

135

C₈

—

212

—

136

C₂²

—

137

C₂²

#13133177

yes

characteristic

138

C₂²

#13133194

yes

characteristic

139

C₂²

#13133207

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

212

—

143

C₄

C₄⋊C₄×D₅₃

yes

characteristic

144

C₄

C₄×Dic₁₀₆

yes

characteristic

145

C₄

#13133190

yes

characteristic

146

C₄

#13133216

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#16810520

yes

characteristic

150

C₂

#16810742

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#339	C₂
socle	C₂²×C₅₃	C₄×D₄
hypercenter	#339	D₅₃
hypocenter	C₅₃	#1946
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

subgroup

G/H

center

C₂×C₄

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#339

C₂

socle

C₂²×C₅₃

C₄×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

N	H ≅ G/N	type
13. #13133150	C₄	semidirect
14. #13133148	C₄	semidirect
83. C₅₃	#1946	semidirect

H ≅ G/N

type

13. #13133150

C₄

semidirect

14. #13133148

C₄

semidirect

83. C₅₃

#1946

semidirect

Subgroups — #20275467

Conjugacy classes of subgroups

Complements