Group #20276066 — subgroups

Subgroups — #20276066

Order 6784.

230 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 75 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	#16810665	C₂	1	2	yes	maximal
3	#16810488	C₂	1	1	yes	maximal, characteristic
4	#16810384	C₂	1	2	yes	maximal
5	C₅₃×#343	C₂	1	1	yes	maximal, characteristic
6	#13133216	—	2	2	no	—
7	#13133208	—	2	2	no	—
8	#13133196	—	2	2	no	—
9	#13133176	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
12	#13133151	—	2	2	no	—
13	#13133149	—	2	2	no	—
14	#13133124	—	2	2	no	—
15	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	D₄	1	2	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₅₃	—	2	1	no	—
20	C₂²×Dic₅₃	—	2	1	no	—
21	C₂×Dic₁₀₆	—	2	2	no	—
22	M₁₆×C₅₃	—	2	2	no	—
23	M₁₆×C₅₃	Q₈	1	4	yes	complemented
24	C₂×C₈×C₅₃	—	2	2	no	—
25	C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₂²⋊Dic₅₃	—	2	2	no	—
27	C₄⋊Dic₅₃	—	2	2	no	—
28	C₄⋊Dic₅₃	—	2	2	no	—
29	C₄⋊Dic₅₃	D₄	1	2	yes	—
30	C₄⋊Dic₅₃	D₄	1	2	yes	—
31	#11719919	—	2	4	no	—
32	C₄×Dic₅₃	—	2	2	no	—
33	C₄×Dic₅₃	—	2	1	no	—
34	C₄×Dic₅₃	Q₈	1	2	yes	complemented
35	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	—	2	2	no	—
37	C₂×C₄×C₅₃	—	4	2	no	—
38	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
42	C₂×Dic₅₃	—	2	2	no	—
43	C₂×Dic₅₃	—	2	2	no	—
44	C₂×Dic₅₃	—	2	2	no	—
45	C₂×Dic₅₃	—	4	1	no	—
46	C₂×Dic₅₃	—	4	1	no	—
47	C₂×Dic₅₃	—	4	2	no	—
48	C₂×Dic₅₃	—	4	1	no	—
49	Dic₁₀₆	—	8	2	no	—
50	C₈×C₅₃	—	2	4	no	—
51	C₂²×C₅₃	—	2	1	no	—
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	#120	1	1	yes	characteristic
54	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
55	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
56	C₄×C₅₃	—	4	2	no	—
57	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
59	Dic₅₃	—	4	2	no	—
60	Dic₅₃	—	4	2	no	—
61	Dic₅₃	—	4	2	no	—
62	Dic₅₃	—	8	1	no	—
63	#2061	—	53	1	no	maximal, Sylow
64	C₂×C₅₃	—	2	1	no	—
65	C₂×C₅₃	—	2	1	no	—
66	C₂×C₅₃	#395	1	1	yes	characteristic
67	C₂×C₅₃	#454	1	1	yes	characteristic
68	C₂×C₅₃	#455	1	1	yes	characteristic
69	#544	—	53	1	no	—
70	#428	—	53	2	no	—
71	#382	—	53	1	no	—
72	#343	D₅₃	1	1	yes	characteristic
73	#339	—	53	2	no	—
74	C₅₃	#2061	1	1	yes	characteristic, complemented, Sylow
75	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
76	C₄⋊Q₈	—	106	2	no	—
77	#125	—	106	2	no	—
78	C₂²⋊Q₈	—	106	2	no	—
79	#117	—	53	2	no	—
80	C₂×C₄⋊C₄	—	53	1	no	—
81	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
82	C₂×C₄²	—	53	1	no	—
83	#107	—	106	2	no	—
84	#106	—	106	2	no	—
85	#95	—	106	2	no	—
86	C₂×Q₈	—	106	2	no	—
87	C₂²×C₄	—	106	1	no	—
88	C₂²×C₄	—	106	1	no	—
89	C₂²×C₄	D₂₁₂	1	2	yes	—
90	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
91	M₁₆	—	2	2	no	—
92	M₁₆	Dic₁₀₆	1	4	yes	complemented
93	C₂×C₈	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	53	2	no	—
96	C₄⋊C₄	—	53	2	no	—
97	C₄⋊C₄	—	106	2	no	—
98	C₄⋊C₄	—	106	4	no	—
99	C₄⋊C₄	—	106	2	no	—
100	C₂²⋊C₄	—	106	2	no	—
101	C₄²	—	53	2	no	—
102	C₄²	—	106	2	no	—
103	C₄²	—	106	1	no	—
104	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
105	Q₈	—	424	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	106	2	no	—
109	C₂×C₄	—	106	2	no	—
110	C₂×C₄	—	106	2	no	—
111	C₂×C₄	—	212	2	no	—
112	C₂×C₄	—	212	1	no	—
113	C₂×C₄	—	212	1	no	—
114	C₂×C₄	—	212	1	no	—
115	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
116	C₂×C₄	D₄×D₅₃	1	2	yes	—
117	C₂×C₄	#11719947	1	1	yes	characteristic
118	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	#13133187	1	1	yes	characteristic
123	C₂²	#13133197	1	1	yes	characteristic
124	C₂²	#13133217	1	1	yes	characteristic
125	C₄	—	4	2	no	—
126	C₄	—	424	1	no	—
127	C₄	—	212	2	no	—
128	C₄	—	212	2	no	—
129	C₄	—	212	2	no	—
130	C₄	#13133194	1	2	yes	—
131	C₄	#13133216	1	2	yes	—
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#16810544	1	1	yes	characteristic
135	C₂	#16810798	1	1	yes	characteristic
136	C₂	#16810803	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811254

C₂

yes

maximal, characteristic

#16810665

C₂

yes

maximal

#16810488

C₂

yes

maximal, characteristic

#16810384

C₂

yes

maximal

C₅₃×#343

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

—

—

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

M₁₆×C₅₃

—

M₁₆×C₅₃

Q₈

yes

complemented

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

—

C₄⋊Dic₅₃

D₄

yes

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

Q₈

yes

complemented

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#120

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

C₄⋊Q₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

C₄⋊Q₈

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#2061

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

D₅₃

yes

characteristic

#339

—

C₅₃

#2061

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₂×D₅₃

yes

—

C₄⋊Q₈

—

106

—

#125

—

106

—

C₂²⋊Q₈

—

106

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₄²

—

#107

—

106

—

#106

—

106

—

#95

—

106

—

C₂×Q₈

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

D₂₁₂

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

Dic₁₀₆

yes

complemented

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

100

C₂²⋊C₄

—

106

—

101

C₄²

—

102

C₄²

—

106

—

103

C₄²

—

106

—

104

C₂³

C₂×D₂₁₂

yes

characteristic

105

Q₈

—

424

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

106

—

109

C₂×C₄

—

106

—

110

C₂×C₄

—

106

—

111

C₂×C₄

—

212

—

112

C₂×C₄

—

212

—

113

C₂×C₄

—

212

—

114

C₂×C₄

—

212

—

115

C₂×C₄

C₂×Dic₁₀₆

yes

—

116

C₂×C₄

D₄×D₅₃

yes

—

117

C₂×C₄

#11719947

yes

characteristic

118

C₂×C₄

Q₈×D₅₃

yes

characteristic

119

C₈

—

120

C₂²

—

121

C₂²

—

122

C₂²

#13133187

yes

characteristic

123

C₂²

#13133197

yes

characteristic

124

C₂²

#13133217

yes

characteristic

125

C₄

—

126

C₄

—

424

—

127

C₄

—

212

—

128

C₄

—

212

—

129

C₄

—

212

—

130

C₄

#13133194

yes

—

131

C₄

#13133216

yes

—

132

C₂

—

133

C₂

—

134

C₂

#16810544

yes

characteristic

135

C₂

#16810798

yes

characteristic

136

C₂

#16810803

yes

characteristic

	subgroup	G/H
center	C₂²	#13133187
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#343	C₂
socle	C₂²×C₅₃	#120
hypercenter	#343	D₅₃
hypocenter	C₅₃	#2061
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

#13133187

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#343

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
23. M₁₆×C₅₃	Q₈	semidirect
34. C₄×Dic₅₃	Q₈	semidirect
74. C₅₃	#2061	semidirect
92. M₁₆	Dic₁₀₆	semidirect

H ≅ G/N

type

23. M₁₆×C₅₃

Q₈

semidirect

34. C₄×Dic₅₃

Q₈

semidirect

74. C₅₃

#2061

semidirect

92. M₁₆

Dic₁₀₆

semidirect

Subgroups — #20276066

Conjugacy classes of subgroups

Complements