Group #20274242 — subgroups

Subgroups — #20274242

Order 6784.

140 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16811026	C₂	1	1	yes	maximal, characteristic
2	#16810919	C₂	1	1	yes	maximal, characteristic
3	#16810915	C₂	1	1	yes	maximal, characteristic
4	#16810750	C₂	1	1	yes	maximal, characteristic
5	#16810373	C₂	1	1	yes	maximal, characteristic
6	#16810371	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#333	C₂	1	1	yes	maximal, characteristic
8	#13133217	C₂²	1	1	yes	characteristic
9	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
10	#13133151	—	2	1	no	—
11	#13133150	—	2	1	no	—
12	#13133150	—	2	1	no	—
13	#13133150	—	2	1	no	—
14	C₄⋊#88322	C₂²	1	1	yes	characteristic
15	#13133146	—	2	1	no	—
16	#13133142	—	2	1	no	—
17	#13133140	—	2	1	no	—
18	#13133127	—	2	1	no	—
19	C₄×#88322	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	—	2	1	no	—
24	C₂×Dic₁₀₆	—	2	1	no	—
25	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₅₃	—	2	1	no	—
32	C₄⋊Dic₅₃	—	2	1	no	—
33	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
34	#11719919	—	2	1	no	—
35	C₄×Dic₅₃	—	2	1	no	—
36	C₂×#88322	—	2	1	no	—
37	C₂×#88322	D₄	1	1	yes	characteristic
38	C₂×#88322	D₄	1	1	yes	characteristic
39	Q₈×C₅₃	—	8	1	no	—
40	C₂×C₄×C₅₃	—	4	1	no	—
41	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
44	C₂×Dic₅₃	—	2	1	no	—
45	C₂×Dic₅₃	—	2	1	no	—
46	C₂×Dic₅₃	—	4	1	no	—
47	Dic₁₀₆	—	2	1	no	—
48	Dic₁₀₆	—	2	1	no	—
49	Dic₁₀₆	—	8	1	no	—
50	C₈×C₅₃	—	4	1	no	—
51	C₈×C₅₃	—	4	1	no	—
52	#88322	—	2	1	no	—
53	#88322	—	4	1	no	—
54	#88322	Q₁₆	1	2	yes	—
55	C₂²×C₅₃	#121	1	1	yes	characteristic
56	C₄×C₅₃	—	2	1	no	—
57	C₄×C₅₃	—	8	1	no	—
58	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
59	C₄×C₅₃	#137	1	1	yes	characteristic
60	C₄×C₅₃	#137	1	1	yes	characteristic
61	C₄×C₅₃	#137	1	1	yes	characteristic
62	Dic₅₃	—	4	1	no	—
63	Dic₅₃	—	4	1	no	—
64	Dic₅₃	—	8	1	no	—
65	#1696	—	53	1	no	maximal, Sylow
66	C₂×C₅₃	#462	1	1	yes	characteristic
67	C₂×C₅₃	#470	1	1	yes	characteristic
68	C₂×C₅₃	#472	1	1	yes	characteristic
69	#500	—	53	1	no	—
70	#477	—	53	1	no	—
71	#477	—	53	1	no	—
72	#446	—	53	1	no	—
73	#334	—	53	1	no	—
74	#333	—	53	1	no	—
75	#333	D₅₃	1	1	yes	characteristic
76	C₅₃	#1696	1	1	yes	characteristic, complemented, Sylow
77	C₄⋊Q₈	—	53	1	no	—
78	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
79	C₄×Q₈	—	53	1	no	—
80	#107	—	106	1	no	—
81	#107	—	106	1	no	—
82	#107	—	106	1	no	—
83	#107	—	106	1	no	—
84	C₄⋊C₈	—	53	1	no	—
85	C₄⋊C₈	—	106	1	no	—
86	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
87	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
88	#103	—	106	1	no	—
89	#103	—	106	1	no	—
90	#97	—	106	1	no	—
91	C₄×C₈	—	53	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	53	1	no	—
94	C₂×Q₈	—	106	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	53	1	no	—
98	C₂×C₈	—	53	1	no	—
99	C₂×C₈	—	106	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	53	1	no	—
103	C₄⋊C₄	—	106	1	no	—
104	C₄⋊C₄	—	106	1	no	—
105	C₄⋊C₄	—	106	1	no	—
106	C₄²	—	106	1	no	—
107	C₄²	C₂²×D₅₃	1	1	yes	characteristic
108	Q₈	—	8	1	no	—
109	Q₈	—	106	1	no	—
110	Q₈	—	106	1	no	—
111	Q₈	—	424	1	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	106	1	no	—
114	C₂×C₄	—	106	1	no	—
115	C₂×C₄	—	212	1	no	—
116	C₂×C₄	#11719945	1	1	yes	characteristic
117	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
118	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
119	C₈	—	4	1	no	—
120	C₈	—	4	1	no	—
121	C₈	—	53	2	no	—
122	C₈	—	106	1	no	—
123	C₈	—	212	1	no	—
124	C₂²	#13133190	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	8	1	no	—
127	C₄	—	424	1	no	—
128	C₄	—	212	1	no	—
129	C₄	—	212	1	no	—
130	C₄	Q₁₆×D₅₃	1	1	yes	characteristic
131	C₄	#13133246	1	1	yes	characteristic
132	C₄	#13133246	1	1	yes	characteristic
133	C₄	#13133249	1	1	yes	characteristic
134	C₂	#16810832	1	1	yes	characteristic
135	C₂	#16810880	1	1	yes	characteristic
136	C₂	#16810892	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811026

C₂

yes

maximal, characteristic

#16810919

C₂

yes

maximal, characteristic

#16810915

C₂

yes

maximal, characteristic

#16810750

C₂

yes

maximal, characteristic

#16810373

C₂

yes

maximal, characteristic

#16810371

C₂

yes

maximal, characteristic

C₅₃×#333

C₂

yes

maximal, characteristic

#13133217

C₂²

yes

characteristic

C₄×Dic₁₀₆

C₂²

yes

—

—

—

—

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

#88322

Q₁₆

yes

—

C₂²×C₅₃

#121

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

—

—

—

D₅₃

yes

characteristic

C₅₃

#1696

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

C₄×Q₈

—

#107

—

106

—

#107

—

106

—

#107

—

106

—

#107

—

106

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₈

C₂×D₅₃

yes

characteristic

#103

—

106

—

#103

—

106

—

#97

—

106

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

106

—

104

C₄⋊C₄

—

106

—

105

C₄⋊C₄

—

106

—

106

C₄²

—

106

—

107

C₄²

C₂²×D₅₃

yes

characteristic

108

Q₈

—

109

Q₈

—

106

—

110

Q₈

—

106

—

111

Q₈

—

424

—

112

C₂×C₄

—

113

C₂×C₄

—

106

—

114

C₂×C₄

—

106

—

115

C₂×C₄

—

212

—

116

C₂×C₄

#11719945

yes

characteristic

117

C₂×C₄

D₄×D₅₃

yes

characteristic

118

C₂×C₄

D₄×D₅₃

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

106

—

123

C₈

—

212

—

124

C₂²

#13133190

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

424

—

128

C₄

—

212

—

129

C₄

—

212

—

130

C₄

Q₁₆×D₅₃

yes

characteristic

131

C₄

#13133246

yes

characteristic

132

C₄

#13133246

yes

characteristic

133

C₄

#13133249

yes

characteristic

134

C₂

#16810832

yes

characteristic

135

C₂

#16810880

yes

characteristic

136

C₂

#16810892

yes

characteristic

	subgroup	G/H
center	C₂²	#13133190
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#333	C₂
socle	C₂²×C₅₃	#121
hypercenter	#333	D₅₃
hypocenter	C₅₃	#1696
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#13133190

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#333

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
76. C₅₃	#1696	semidirect

H ≅ G/N

type

76. C₅₃

#1696

semidirect

Subgroups — #20274242

Conjugacy classes of subgroups

Complements