Group #20275504 — subgroups

Subgroups — #20275504

Order 6784.

184 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 79 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	#16810672	C₂	1	1	yes	maximal, characteristic
2	#16810671	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133152	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133152	C₂	1	1	yes	maximal, characteristic
5	#16810579	C₂	1	1	yes	maximal, characteristic
6	#16810393	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#344	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719917	C₂²	1	1	yes	characteristic
9	C₂×#11719917	C₂²	1	1	yes	characteristic
10	C₂×#11719917	C₂²	1	1	yes	characteristic
11	C₂²×#88322	C₂²	1	1	yes	characteristic
12	#13133153	—	2	2	no	—
13	#13133152	—	2	2	no	—
14	#13133152	—	2	1	no	—
15	#13133152	—	2	1	no	—
16	#13133152	C₄	1	4	yes	complemented
17	#13133128	—	2	1	no	—
18	C₄×#88322	—	2	1	no	—
19	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	#117×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	M₁₆×C₅₃	—	2	1	no	—
24	M₁₆×C₅₃	—	2	1	no	—
25	M₁₆×C₅₃	—	2	1	no	—
26	M₁₆×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₂×C₈×C₅₃	—	2	1	no	—
29	C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₂²⋊C₄×C₅₃	Q₈	1	2	yes	—
31	C₄²×C₅₃	—	2	1	no	—
32	#11719917	—	2	1	no	—
33	#11719917	—	2	1	no	—
34	#11719917	—	2	1	no	—
35	#11719917	—	2	1	no	—
36	#11719917	C₂×C₄	1	2	yes	—
37	#11719917	D₄	1	2	yes	—
38	C₂×#88322	—	2	1	no	—
39	C₂×#88322	—	2	1	no	—
40	C₂×#88322	—	2	1	no	—
41	C₂×#88322	C₂×C₄	1	2	yes	—
42	C₂×#88322	D₄	1	1	yes	characteristic
43	C₂×#88322	D₄	1	1	yes	characteristic
44	C₂×#88322	Q₈	1	2	yes	—
45	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
53	C₈×C₅₃	—	2	2	no	—
54	C₈×C₅₃	—	4	1	no	—
55	#88322	—	2	2	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	—	4	1	no	—
61	C₂²×C₅₃	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₄×C₅₃	—	4	2	no	—
65	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
67	C₄×C₅₃	#121	1	1	yes	characteristic
68	C₄×C₅₃	#125	1	1	yes	characteristic
69	#1953	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	—	2	1	no	—
71	C₂×C₅₃	—	2	1	no	—
72	C₂×C₅₃	—	2	1	no	—
73	C₂×C₅₃	#389	1	1	yes	characteristic
74	#430	—	53	1	no	—
75	#430	—	53	1	no	—
76	C₂×#108	—	53	1	no	—
77	C₂×#108	—	53	1	no	—
78	#405	—	53	1	no	—
79	#344	—	53	1	no	—
80	#344	D₅₃	1	1	yes	characteristic
81	C₅₃	#1953	1	1	yes	characteristic, complemented, Sylow
82	C₂×M₁₆	—	53	1	no	—
83	C₂×M₁₆	—	53	1	no	—
84	C₂×M₁₆	—	53	1	no	—
85	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
86	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
87	C₂²×C₈	—	53	1	no	—
88	#117	C₂×D₅₃	1	1	yes	characteristic
89	#108	—	53	4	no	—
90	#108	—	106	2	no	—
91	#108	—	106	1	no	—
92	#108	—	106	1	no	—
93	#108	—	106	2	no	—
94	#97	—	106	1	no	—
95	C₄×C₈	—	106	1	no	—
96	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
97	M₁₆	—	2	1	no	—
98	M₁₆	—	2	1	no	—
99	M₁₆	—	2	1	no	—
100	M₁₆	—	53	2	no	—
101	M₁₆	—	53	2	no	—
102	M₁₆	—	106	1	no	—
103	M₁₆	—	106	1	no	—
104	M₁₆	—	106	1	no	—
105	M₁₆	—	106	1	no	—
106	M₁₆	C₄×D₅₃	1	2	yes	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	53	2	no	—
110	C₂×C₈	—	53	1	no	—
111	C₂×C₈	—	53	1	no	—
112	C₂×C₈	—	53	2	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	C₂²⋊C₄	Dic₁₀₆	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₂³	C₂×Dic₁₀₆	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
122	C₂×C₄	#11719945	1	1	yes	characteristic
123	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
124	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
125	C₂×C₄	#11719947	1	1	yes	characteristic
126	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	—	4	1	no	—
129	C₈	—	53	2	no	—
130	C₈	—	53	2	no	—
131	C₈	—	106	2	no	—
132	C₈	—	106	2	no	—
133	C₈	—	212	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	C₄×Dic₁₀₆	1	1	yes	characteristic
136	C₂²	#13133194	1	1	yes	characteristic
137	C₂²	#13133216	1	1	yes	characteristic
138	C₄	—	4	2	no	—
139	C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
140	C₄	#13133177	1	1	yes	characteristic
141	C₄	#13133190	1	1	yes	characteristic
142	C₄	#13133207	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#16810520	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810672

C₂

yes

maximal, characteristic

#16810671

C₂

yes

maximal, characteristic

C₂×#13133152

C₂

yes

maximal, characteristic

C₂×#13133152

C₂

yes

maximal, characteristic

#16810579

C₂

yes

maximal, characteristic

#16810393

C₂

yes

maximal, characteristic

C₅₃×#344

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

—

—

—

—

C₄

yes

complemented

#13133128

—

C₄×#88322

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

Q₈

yes

—

C₄²×C₅₃

—

—

—

—

—

C₂×C₄

yes

—

#11719917

D₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#88322

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

C₂×#88322

Q₈

yes

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

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

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

C₄⋊C₄

yes

—

#88322

C₄⋊C₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×Q₈

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

C₄⋊Q₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

—

—

—

—

—

—

D₅₃

yes

characteristic

C₅₃

#1953

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

#117

C₂×D₅₃

yes

characteristic

#108

—

#108

—

106

—

#108

—

106

—

#108

—

106

—

#108

—

106

—

#97

—

106

—

C₄×C₈

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

106

—

103

M₁₆

—

106

—

104

M₁₆

—

106

—

105

M₁₆

—

106

—

106

M₁₆

C₄×D₅₃

yes

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

106

—

114

C₂×C₈

—

106

—

115

C₂×C₈

—

106

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

Dic₁₀₆

yes

—

118

C₄²

—

119

C₂³

C₂×Dic₁₀₆

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

122

C₂×C₄

#11719945

yes

characteristic

123

C₂×C₄

D₄×D₅₃

yes

characteristic

124

C₂×C₄

D₄×D₅₃

yes

characteristic

125

C₂×C₄

#11719947

yes

characteristic

126

C₂×C₄

Q₈×D₅₃

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

106

—

132

C₈

—

106

—

133

C₈

—

212

—

134

C₂²

—

135

C₂²

C₄×Dic₁₀₆

yes

characteristic

136

C₂²

#13133194

yes

characteristic

137

C₂²

#13133216

yes

characteristic

138

C₄

—

139

C₄

C₄⋊C₄×D₅₃

yes

characteristic

140

C₄

#13133177

yes

characteristic

141

C₄

#13133190

yes

characteristic

142

C₄

#13133207

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

#16810520

yes

characteristic

	subgroup	G/H
center	C₄	#13133190
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#344	C₂
socle	C₂×C₅₃	#389
hypercenter	#344	D₅₃
hypocenter	C₅₃	#1953
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₄

#13133190

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#344

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
16. #13133152	C₄	semidirect
81. C₅₃	#1953	semidirect

H ≅ G/N

type

16. #13133152

C₄

semidirect

81. C₅₃

#1953

semidirect

Subgroups — #20275504

Conjugacy classes of subgroups

Complements