Group #20275490 — subgroups

Subgroups — #20275490

Order 6784.

200 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 97 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	2	yes	maximal
2	#16810647	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
4	#16810387	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	C₂²×#88322	C₂²	1	2	yes	—
8	#13133152	—	2	4	no	—
9	C₄⋊#88322	—	2	1	no	—
10	C₄⋊#88322	—	2	1	no	—
11	C₄⋊#88322	C₄	1	2	yes	—
12	C₄⋊#88322	C₄	1	2	yes	—
13	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	#97×C₅₃	—	2	1	no	—
17	C₄×C₈×C₅₃	—	2	1	no	—
18	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	M₁₆×C₅₃	—	2	1	no	—
22	M₁₆×C₅₃	—	2	1	no	—
23	M₁₆×C₅₃	D₄	1	2	yes	—
24	M₁₆×C₅₃	Q₈	1	2	yes	—
25	C₂×C₈×C₅₃	—	2	1	no	—
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₄²×C₅₃	D₄	1	1	yes	characteristic
31	#11719917	—	2	2	no	—
32	#11719917	—	2	2	no	—
33	C₂×#88322	—	2	2	no	—
34	C₂×#88322	—	2	2	no	—
35	C₂×#88322	—	2	2	no	—
36	C₂×#88322	C₂×C₄	1	4	yes	—
37	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	4	1	no	—
40	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
50	C₈×C₅₃	—	2	2	no	—
51	C₈×C₅₃	—	2	2	no	—
52	#88322	—	2	4	no	—
53	#88322	—	4	2	no	—
54	C₂²×C₅₃	—	2	1	no	—
55	C₂²×C₅₃	—	2	1	no	—
56	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₅₃	#125	1	1	yes	characteristic
58	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	—	2	1	no	—
61	C₄×C₅₃	—	2	1	no	—
62	C₄×C₅₃	#108	1	2	yes	complemented
63	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
64	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
65	C₄×C₅₃	#121	1	1	yes	characteristic
66	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
67	#1950	—	53	1	no	maximal, Sylow
68	C₂×C₅₃	—	2	1	no	—
69	C₂×C₅₃	—	2	1	no	—
70	C₂×C₅₃	#389	1	1	yes	characteristic
71	C₂×C₅₃	C₂×#108	1	1	yes	characteristic
72	C₂×C₅₃	#430	1	1	yes	characteristic
73	C₂×#108	—	53	2	no	—
74	#423	—	53	1	no	—
75	C₂×C₄⋊C₈	—	53	1	no	—
76	C₄×M₁₆	D₅₃	1	1	yes	characteristic
77	#341	—	53	2	no	—
78	C₅₃	#1950	1	1	yes	characteristic, complemented, Sylow
79	C₂×M₁₆	—	53	2	no	—
80	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
81	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
82	C₂²×C₈	—	53	2	no	—
83	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
84	#108	—	106	4	no	—
85	C₄⋊C₈	—	53	2	no	—
86	C₄⋊C₈	—	53	2	no	—
87	C₄⋊C₈	—	106	1	no	—
88	C₄⋊C₈	—	106	1	no	—
89	#97	—	2	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
92	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
94	M₁₆	—	2	1	no	—
95	M₁₆	—	2	1	no	—
96	M₁₆	—	106	2	no	—
97	M₁₆	—	106	2	no	—
98	M₁₆	Dic₁₀₆	1	2	yes	—
99	M₁₆	D₂₁₂	1	2	yes	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	53	4	no	—
103	C₂×C₈	—	106	2	no	—
104	C₂×C₈	—	106	2	no	—
105	C₂×C₈	—	106	2	no	—
106	C₄²	C₄×D₅₃	1	1	yes	characteristic
107	C₄²	C₄×D₅₃	1	1	yes	characteristic
108	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
109	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
110	C₂³	C₂×Dic₁₀₆	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	#11719919	1	1	yes	characteristic
114	C₂×C₄	#11719919	1	1	yes	characteristic
115	C₂×C₄	#11719919	1	1	yes	characteristic
116	C₂×C₄	#11719919	1	1	yes	characteristic
117	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
118	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
119	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
120	C₂×C₄	#11719945	1	1	yes	characteristic
121	C₂×C₄	#11719945	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	—	2	2	no	—
125	C₈	—	106	4	no	—
126	C₈	—	212	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	C₂×#11719919	1	1	yes	characteristic
130	C₂²	#13133209	1	1	yes	characteristic
131	C₂²	#13133217	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	#13133152	1	2	yes	complemented
136	C₄	C₄×D₂₁₂	1	1	yes	characteristic
137	C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
138	C₄	#13133191	1	1	yes	characteristic
139	C₄	#13133196	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	2	1	no	—
142	C₂	#16810521	1	1	yes	characteristic
143	C₂	C₂×#13133152	1	1	yes	characteristic
144	C₂	#16810671	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133152

C₂

yes

maximal

#16810647

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic

#16810387

C₂

yes

maximal

C₄×M₁₆×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂²×#88322

C₂²

yes

—

—

—

—

C₄

yes

—

C₄⋊#88322

C₄

yes

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

D₄

yes

—

M₁₆×C₅₃

Q₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

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

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#125

yes

characteristic

C₂²×C₅₃

C₄⋊Q₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#108

yes

complemented

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

characteristic

C₄×C₅₃

#121

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

#1950

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

C₂×C₄⋊C₈

—

C₄×M₁₆

D₅₃

yes

characteristic

#341

—

C₅₃

#1950

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#108

—

106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

#97

—

C₄×C₈

—

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

Dic₁₀₆

yes

—

M₁₆

D₂₁₂

yes

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

106

—

104

C₂×C₈

—

106

—

105

C₂×C₈

—

106

—

106

C₄²

C₄×D₅₃

yes

characteristic

107

C₄²

C₄×D₅₃

yes

characteristic

108

C₄²

C₂²⋊D₅₃

yes

characteristic

109

C₄²

C₂²⋊D₅₃

yes

characteristic

110

C₂³

C₂×Dic₁₀₆

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

#11719919

yes

characteristic

114

C₂×C₄

#11719919

yes

characteristic

115

C₂×C₄

#11719919

yes

characteristic

116

C₂×C₄

#11719919

yes

characteristic

117

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

118

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

119

C₂×C₄

C₂×D₂₁₂

yes

characteristic

120

C₂×C₄

#11719945

yes

characteristic

121

C₂×C₄

#11719945

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

106

—

126

C₈

—

212

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×#11719919

yes

characteristic

130

C₂²

#13133209

yes

characteristic

131

C₂²

#13133217

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#13133152

yes

complemented

136

C₄

C₄×D₂₁₂

yes

characteristic

137

C₄

C₄×Dic₁₀₆

yes

characteristic

138

C₄

#13133191

yes

characteristic

139

C₄

#13133196

yes

characteristic

140

C₂

—

141

C₂

—

142

C₂

#16810521

yes

characteristic

143

C₂

C₂×#13133152

yes

characteristic

144

C₂

#16810671

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₄×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₄×M₁₆	D₅₃
hypocenter	C₅₃	#1950
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₄×M₁₆×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₄

hypercenter

C₄×M₁₆

D₅₃

hypocenter

C₅₃

#1950

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
62. C₄×C₅₃	#108	semidirect
78. C₅₃	#1950	semidirect
135. C₄	#13133152	semidirect

H ≅ G/N

type

62. C₄×C₅₃

#108

semidirect

78. C₅₃

#1950

semidirect

135. C₄

#13133152

semidirect

Subgroups — #20275490

Conjugacy classes of subgroups

Complements