Group #20275876 — subgroups

Subgroups — #20275876

Order 6784.

246 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 69 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	#16811254	C₂	1	1	yes	maximal, characteristic
2	#16810632	C₂	1	2	yes	maximal
3	#16810392	C₂	1	1	yes	maximal, characteristic
4	#16810383	C₂	1	2	yes	maximal
5	C₅₃×#391	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	#13133216	—	2	2	no	—
8	#13133208	—	2	2	no	—
9	#13133196	—	2	2	no	—
10	#13133176	C₂²	1	2	yes	—
11	#13133141	—	2	2	no	—
12	#13133141	—	2	2	no	—
13	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	#95×C₅₃	—	2	2	no	—
17	C₂×Q₈×C₅₃	—	2	1	no	—
18	C₂×Q₈×C₅₃	—	2	2	no	—
19	C₂×Q₈×C₅₃	—	2	2	no	—
20	C₂×Q₈×C₅₃	Q₈	1	2	yes	complemented
21	C₂²×C₄×C₅₃	—	2	1	no	—
22	C₂²×C₄×C₅₃	—	2	1	no	—
23	C₂²×C₄×C₅₃	D₄	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂×Dic₁₀₆	—	2	2	no	—
26	C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₄²×C₅₃	—	2	2	no	—
28	C₂²⋊Dic₅₃	—	2	2	no	—
29	C₄⋊Dic₅₃	—	2	2	no	—
30	C₄⋊Dic₅₃	D₄	1	2	yes	—
31	C₄⋊Dic₅₃	D₄	1	2	yes	—
32	#11719919	—	2	4	no	—
33	C₄×Dic₅₃	—	2	2	no	—
34	#11719917	—	2	2	no	—
35	#11719917	—	2	2	no	—
36	C₂×#88322	—	2	2	no	—
37	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₅₃	—	2	4	no	—
39	Q₈×C₅₃	—	4	2	no	—
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	4	1	no	—
44	C₂×C₄×C₅₃	—	4	1	no	—
45	C₂×C₄×C₅₃	—	4	1	no	—
46	C₂×C₄×C₅₃	—	4	1	no	—
47	C₂×C₄×C₅₃	—	4	2	no	—
48	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#41	1	2	yes	—
52	C₂×Dic₅₃	—	2	2	no	—
53	C₂×Dic₅₃	—	2	2	no	—
54	Dic₁₀₆	—	8	2	no	—
55	#88322	—	4	2	no	—
56	C₂²×C₅₃	—	2	1	no	—
57	C₂²×C₅₃	—	2	1	no	—
58	C₂²×C₅₃	#120	1	1	yes	characteristic
59	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂²×C₅₃	#124	1	1	yes	characteristic
61	C₄×C₅₃	—	4	2	no	—
62	C₄×C₅₃	—	4	2	no	—
63	C₄×C₅₃	—	4	2	no	—
64	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₅₃	#124	1	2	yes	—
66	Dic₅₃	—	4	2	no	—
67	Dic₅₃	—	4	2	no	—
68	#2028	—	53	1	no	maximal, Sylow
69	C₂×C₅₃	—	2	1	no	—
70	C₂×C₅₃	—	2	1	no	—
71	C₂×C₅₃	#393	1	1	yes	characteristic
72	C₂×C₅₃	#455	1	1	yes	characteristic
73	C₂×C₅₃	#456	1	1	yes	characteristic
74	#544	—	53	1	no	—
75	#419	—	53	2	no	—
76	#391	D₅₃	1	1	yes	characteristic
77	#343	—	53	1	no	—
78	#339	—	53	2	no	—
79	C₅₃	#2028	1	1	yes	characteristic, complemented, Sylow
80	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
81	C₂×M₁₆	—	53	2	no	—
82	C₄⋊Q₈	—	106	2	no	—
83	#125	—	106	2	no	—
84	C₂²⋊Q₈	—	106	2	no	—
85	#117	—	53	2	no	—
86	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
87	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
88	#103	—	106	2	no	—
89	#103	—	106	2	no	—
90	#95	—	2	2	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	106	2	no	—
95	C₂×Q₈	Dic₁₀₆	1	2	yes	complemented
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
99	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
100	M₁₆	—	106	2	no	—
101	M₁₆	—	106	2	no	—
102	C₂×C₈	—	106	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	53	2	no	—
105	C₄⋊C₄	—	53	2	no	—
106	C₄⋊C₄	—	106	2	no	—
107	C₄⋊C₄	—	106	4	no	—
108	C₂²⋊C₄	—	106	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	106	2	no	—
111	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
112	Q₈	—	2	4	no	—
113	Q₈	—	4	2	no	—
114	Q₈	—	424	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	106	2	no	—
124	C₂×C₄	—	106	2	no	—
125	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
126	C₂×C₄	#11719945	1	1	yes	characteristic
127	C₂×C₄	D₄×D₅₃	1	2	yes	—
128	C₂×C₄	#11719947	1	2	yes	—
129	C₈	—	212	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	#13133185	1	1	yes	characteristic
133	C₂²	#13133196	1	1	yes	characteristic
134	C₂²	#13133204	1	1	yes	characteristic
135	C₄	—	4	2	no	—
136	C₄	—	4	2	no	—
137	C₄	—	4	2	no	—
138	C₄	—	212	2	no	—
139	C₄	—	212	2	no	—
140	C₄	#13133194	1	2	yes	—
141	C₄	#13133203	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#16810535	1	1	yes	characteristic
145	C₂	#16810800	1	1	yes	characteristic
146	C₂	#16810805	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811254

C₂

yes

maximal, characteristic

#16810632

C₂

yes

maximal

#16810392

C₂

yes

maximal, characteristic

#16810383

C₂

yes

maximal

C₅₃×#391

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

—

—

—

C₂²

yes

—

#13133141

—

#13133141

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

Q₈

yes

complemented

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

—

C₄⋊Dic₅₃

D₄

yes

—

#11719919

—

C₄×Dic₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#120

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

#124

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

#124

yes

—

Dic₅₃

—

Dic₅₃

—

#2028

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

#343

—

#339

—

C₅₃

#2028

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

C₄⋊Q₈

—

106

—

#125

—

106

—

C₂²⋊Q₈

—

106

—

#117

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

characteristic

#103

—

106

—

#103

—

106

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

Dic₁₀₆

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

100

M₁₆

—

106

—

101

M₁₆

—

106

—

102

C₂×C₈

—

106

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

106

—

107

C₄⋊C₄

—

106

—

108

C₂²⋊C₄

—

106

—

109

C₄²

—

110

C₄²

—

106

—

111

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

424

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

106

—

124

C₂×C₄

—

106

—

125

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

126

C₂×C₄

#11719945

yes

characteristic

127

C₂×C₄

D₄×D₅₃

yes

—

128

C₂×C₄

#11719947

yes

—

129

C₈

—

212

—

130

C₂²

—

131

C₂²

—

132

C₂²

#13133185

yes

characteristic

133

C₂²

#13133196

yes

characteristic

134

C₂²

#13133204

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

212

—

139

C₄

—

212

—

140

C₄

#13133194

yes

—

141

C₄

#13133203

yes

—

142

C₂

—

143

C₂

—

144

C₂

#16810535

yes

characteristic

145

C₂

#16810800

yes

characteristic

146

C₂

#16810805

yes

characteristic

	subgroup	G/H
center	C₂²	#13133185
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#391	C₂
socle	C₂²×C₅₃	#120
hypercenter	#391	D₅₃
hypocenter	C₅₃	#2028
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

subgroup

G/H

center

C₂²

#13133185

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#391

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

N	H ≅ G/N	type
20. C₂×Q₈×C₅₃	Q₈	semidirect
79. C₅₃	#2028	semidirect
95. C₂×Q₈	Dic₁₀₆	semidirect

H ≅ G/N

type

20. C₂×Q₈×C₅₃

Q₈

semidirect

79. C₅₃

#2028

semidirect

95. C₂×Q₈

Dic₁₀₆

semidirect

Subgroups — #20275876

Conjugacy classes of subgroups

Complements