Group #20276298 — subgroups

Subgroups — #20276298

Order 6784.

148 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 59 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	#16810671	C₂	1	1	yes	maximal, characteristic
2	#16810650	C₂	1	1	yes	maximal, characteristic
3	#16810393	C₂	1	1	yes	maximal, characteristic
4	#16810387	C₂	1	1	yes	maximal, characteristic
5	#16810387	C₂	1	1	yes	maximal, characteristic
6	#16810381	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	#13133152	—	2	2	no	—
13	C₄⋊#88322	—	2	2	no	—
14	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	#117×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	M₁₆×C₅₃	—	2	1	no	—
19	M₁₆×C₅₃	—	2	1	no	—
20	M₁₆×C₅₃	—	2	1	no	—
21	M₁₆×C₅₃	—	2	1	no	—
22	C₂×C₈×C₅₃	—	2	1	no	—
23	C₂×C₈×C₅₃	—	2	1	no	—
24	C₄⋊C₄×C₅₃	—	2	1	no	—
25	C₂²⋊C₄×C₅₃	Q₈	1	2	yes	—
26	C₄²×C₅₃	—	2	1	no	—
27	#11719917	—	2	1	no	—
28	#11719917	—	2	1	no	—
29	#11719917	—	2	1	no	—
30	#11719917	—	2	1	no	—
31	#11719917	—	2	1	no	—
32	#11719917	—	2	1	no	—
33	C₂×#88322	—	2	1	no	—
34	C₂×#88322	—	2	1	no	—
35	C₂×#88322	—	2	1	no	—
36	C₂×#88322	—	2	1	no	—
37	C₂×#88322	—	2	1	no	—
38	C₂×#88322	D₄	1	2	yes	—
39	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
47	C₈×C₅₃	—	4	1	no	—
48	C₈×C₅₃	—	4	1	no	—
49	#88322	—	2	2	no	—
50	#88322	—	4	1	no	—
51	#88322	—	4	1	no	—
52	#88322	—	4	1	no	—
53	C₂²×C₅₃	—	4	1	no	—
54	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
55	C₂²×C₅₃	#125	1	1	yes	characteristic
56	C₂²×C₅₃	#125	1	1	yes	characteristic
57	C₄×C₅₃	—	4	2	no	—
58	C₄×C₅₃	#123	1	1	yes	characteristic
59	C₄×C₅₃	#124	1	1	yes	characteristic
60	C₄×C₅₃	#126	1	1	yes	characteristic
61	C₄×C₅₃	#126	1	1	yes	characteristic
62	#2103	—	53	1	no	maximal, Sylow
63	C₂×C₅₃	—	2	1	no	—
64	C₂×C₅₃	—	2	1	no	—
65	C₂×C₅₃	—	2	1	no	—
66	C₂×C₅₃	#400	1	1	yes	characteristic
67	#430	—	53	1	no	—
68	#424	—	53	1	no	—
69	#344	—	53	1	no	—
70	#344	D₅₃	1	1	yes	characteristic
71	#341	—	53	1	no	—
72	#341	—	53	1	no	—
73	#338	—	53	1	no	—
74	C₅₃	#2103	1	1	yes	characteristic, complemented, Sylow
75	C₂×M₁₆	—	53	1	no	—
76	C₂×M₁₆	—	53	1	no	—
77	C₂×M₁₆	—	53	1	no	—
78	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
79	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
80	C₂²×C₈	—	53	1	no	—
81	#117	C₂×D₅₃	1	1	yes	characteristic
82	#108	—	106	2	no	—
83	C₄⋊C₈	—	106	2	no	—
84	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
85	M₁₆	—	2	1	no	—
86	M₁₆	—	2	1	no	—
87	M₁₆	—	2	1	no	—
88	M₁₆	—	2	1	no	—
89	M₁₆	—	106	1	no	—
90	M₁₆	—	106	1	no	—
91	M₁₆	—	106	1	no	—
92	M₁₆	—	106	1	no	—
93	M₁₆	—	106	1	no	—
94	M₁₆	—	106	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	53	2	no	—
98	C₂×C₈	—	106	1	no	—
99	C₂×C₈	—	106	1	no	—
100	C₂×C₈	—	106	1	no	—
101	C₂×C₈	—	106	1	no	—
102	C₂×C₈	—	106	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₂²⋊C₄	Dic₁₀₆	1	2	yes	—
105	C₄²	—	2	1	no	—
106	C₂³	C₂×Dic₁₀₆	1	1	yes	characteristic
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	#11719945	1	1	yes	characteristic
109	C₂×C₄	#11719945	1	1	yes	characteristic
110	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
111	C₂×C₄	#11719947	1	1	yes	characteristic
112	C₂×C₄	#11719947	1	1	yes	characteristic
113	C₂×C₄	#11719949	1	1	yes	characteristic
114	C₈	—	4	1	no	—
115	C₈	—	4	1	no	—
116	C₈	—	106	2	no	—
117	C₈	—	212	1	no	—
118	C₈	—	212	1	no	—
119	C₈	—	212	1	no	—
120	C₂²	—	4	1	no	—
121	C₂²	#13133194	1	1	yes	characteristic
122	C₂²	#13133208	1	1	yes	characteristic
123	C₂²	#13133209	1	1	yes	characteristic
124	C₄	—	4	2	no	—
125	C₄	#13133202	1	1	yes	characteristic
126	C₄	#13133203	1	1	yes	characteristic
127	C₄	#13133210	1	1	yes	characteristic
128	C₄	#13133212	1	1	yes	characteristic
129	C₂	—	2	1	no	—
130	C₂	—	2	1	no	—
131	C₂	—	2	1	no	—
132	C₂	#16810565	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810671

C₂

yes

maximal, characteristic

#16810650

C₂

yes

maximal, characteristic

#16810393

C₂

yes

maximal, characteristic

#16810387

C₂

yes

maximal, characteristic

#16810387

C₂

yes

maximal, characteristic

#16810381

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

characteristic

#13133152

—

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×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

Q₈

yes

—

C₄²×C₅₃

—

—

—

—

—

—

—

—

—

—

—

—

D₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²⋊Q₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

—

—

—

D₅₃

yes

—

—

—

C₅₃

#2103

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

—

106

—

C₄⋊C₈

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

100

C₂×C₈

—

106

—

101

C₂×C₈

—

106

—

102

C₂×C₈

—

106

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

Dic₁₀₆

yes

—

105

C₄²

—

106

C₂³

C₂×Dic₁₀₆

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

#11719945

yes

characteristic

109

C₂×C₄

#11719945

yes

characteristic

110

C₂×C₄

D₄×D₅₃

yes

characteristic

111

C₂×C₄

#11719947

yes

characteristic

112

C₂×C₄

#11719947

yes

characteristic

113

C₂×C₄

#11719949

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

—

106

—

117

C₈

—

212

—

118

C₈

—

212

—

119

C₈

—

212

—

120

C₂²

—

121

C₂²

#13133194

yes

characteristic

122

C₂²

#13133208

yes

characteristic

123

C₂²

#13133209

yes

characteristic

124

C₄

—

125

C₄

#13133202

yes

characteristic

126

C₄

#13133203

yes

characteristic

127

C₄

#13133210

yes

characteristic

128

C₄

#13133212

yes

characteristic

129

C₂

—

130

C₂

—

131

C₂

—

132

C₂

#16810565

yes

characteristic

	subgroup	G/H
center	C₄	#13133203
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#344	C₂
socle	C₂×C₅₃	#400
hypercenter	#344	D₅₃
hypocenter	C₅₃	#2103
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₄

#13133203

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#344

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
74. C₅₃	#2103	semidirect

H ≅ G/N

type

74. C₅₃

#2103

semidirect

Subgroups — #20276298

Conjugacy classes of subgroups

Complements