Group #20273119 — subgroups

Subgroups — #20273119

Order 6784.

216 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 89 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	#16811180	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#16810355	C₂	1	4	yes	maximal, complemented
4	C₅₃×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	—	2	1	no	—
6	#13133217	C₄	1	1	yes	characteristic
7	#13133217	C₄	1	1	yes	characteristic
8	#13133209	—	2	1	no	—
9	#13133196	—	2	1	no	—
10	#13133196	C₄	1	2	yes	—
11	C₄×Dic₁₀₆	—	2	1	no	—
12	C₄×Dic₁₀₆	C₂²	1	2	yes	—
13	#13133176	—	2	1	no	—
14	C₄⋊#88322	—	2	1	no	—
15	C₄⋊#88322	C₂²	1	2	yes	—
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₅₃	—	2	1	no	—
18	C₂²⋊C₈×C₅₃	—	2	1	no	—
19	C₄×C₈×C₅₃	C₂²	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂×Dic₁₀₆	—	2	1	no	—
24	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₅₃	—	2	2	no	—
26	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₅₃	D₄	1	1	yes	characteristic
28	C₄²×C₅₃	D₄	1	1	yes	characteristic
29	C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	C₂²⋊Dic₅₃	—	2	2	no	—
31	C₄⋊Dic₅₃	—	2	1	no	—
32	C₄⋊Dic₅₃	—	2	1	no	—
33	C₄⋊Dic₅₃	—	2	1	no	—
34	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
35	#11719919	—	2	2	no	—
36	#11719919	—	2	2	no	—
37	C₄×Dic₅₃	—	2	2	no	—
38	C₂×#88322	—	2	2	no	—
39	C₂×#88322	—	4	2	no	—
40	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	4	1	no	—
43	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
47	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
48	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₅₃	—	2	2	no	—
52	C₂×Dic₅₃	—	2	2	no	—
53	Dic₁₀₆	—	2	4	no	—
54	Dic₁₀₆	—	4	2	no	—
55	C₈×C₅₃	—	4	2	no	—
56	#88322	—	4	2	no	—
57	C₂²×C₅₃	—	2	1	no	—
58	C₂²×C₅₃	—	2	1	no	—
59	C₂²×C₅₃	#103	1	2	yes	—
60	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₅₃	—	2	1	no	—
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	—	2	1	no	—
64	C₄×C₅₃	#103	1	2	yes	—
65	C₄×C₅₃	#131	1	2	yes	—
66	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
68	Dic₅₃	—	4	2	no	—
69	Dic₅₃	—	4	2	no	—
70	#1481	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	—	2	2	no	—
72	C₂×C₅₃	#408	1	1	yes	characteristic
73	C₂×C₅₃	C₂×#103	1	1	yes	characteristic
74	C₂×C₅₃	#421	1	1	yes	characteristic
75	#533	—	53	1	no	—
76	#431	D₅₃	1	1	yes	characteristic
77	C₂×C₄⋊C₈	—	53	1	no	—
78	Q₈⋊C₈	—	53	4	no	—
79	C₅₃	#1481	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₈	—	106	1	no	—
81	C₄⋊Q₈	—	53	1	no	—
82	C₄⋊Q₈	—	53	1	no	—
83	#125	—	106	1	no	—
84	C₂²⋊Q₈	—	53	2	no	—
85	C₂²⋊Q₈	—	106	1	no	—
86	C₄×Q₈	—	53	2	no	—
87	C₄×Q₈	—	106	1	no	—
88	#117	—	106	1	no	—
89	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
90	C₄⋊C₈	—	2	1	no	—
91	C₄⋊C₈	—	53	2	no	—
92	C₄⋊C₈	—	106	1	no	—
93	C₂²⋊C₈	—	2	1	no	—
94	C₄×C₈	C₂×D₅₃	1	2	yes	—
95	C₂×Q₈	—	53	2	no	—
96	C₂×Q₈	—	106	1	no	—
97	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
98	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	106	2	no	—
102	C₂×C₈	—	212	2	no	—
103	C₄⋊C₄	—	53	2	no	—
104	C₄⋊C₄	—	106	2	no	—
105	C₄⋊C₄	—	106	1	no	—
106	C₄⋊C₄	—	106	1	no	—
107	C₄⋊C₄	—	106	2	no	—
108	C₄⋊C₄	—	106	1	no	—
109	C₂²⋊C₄	—	106	2	no	—
110	C₄²	—	106	2	no	—
111	C₄²	C₄×D₅₃	1	1	yes	characteristic
112	C₄²	D₂₁₂	1	1	yes	characteristic
113	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
114	C₄²	C₂²×D₅₃	1	1	yes	characteristic
115	C₂³	#11719921	1	1	yes	characteristic
116	Q₈	—	106	4	no	—
117	Q₈	—	212	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	106	2	no	—
121	C₂×C₄	—	106	2	no	—
122	C₂×C₄	#11719913	1	2	yes	—
123	C₂×C₄	Dic₂₁₂	1	2	yes	—
124	C₂×C₄	#11719921	1	1	yes	characteristic
125	C₂×C₄	#11719921	1	1	yes	characteristic
126	C₂×C₄	#11719921	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
128	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
129	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
130	C₈	—	4	2	no	—
131	C₈	—	212	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	#13133142	1	2	yes	—
135	C₂²	C₂×#11719921	1	1	yes	characteristic
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	212	2	no	—
140	C₄	—	212	2	no	—
141	C₄	#13133142	1	2	yes	—
142	C₄	#13133225	1	2	yes	—
143	C₄	C₂×#11719913	1	1	yes	characteristic
144	C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	#16810591	1	1	yes	characteristic
147	C₂	C₂×#13133142	1	1	yes	characteristic
148	C₂	#16810641	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic

#16810355

C₂

yes

maximal, complemented

C₅₃×#431

C₂

yes

maximal, characteristic

C₂²×#88322

—

#13133217

C₄

yes

characteristic

#13133217

C₄

yes

—

—

C₄

yes

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₂²

yes

—

#13133176

—

C₄⋊#88322

—

C₄⋊#88322

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#103

yes

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#103

yes

—

C₄×C₅₃

#131

yes

—

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#1481

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

characteristic

C₂×C₄⋊C₈

—

Q₈⋊C₈

—

C₅₃

#1481

yes

characteristic, complemented, Sylow

C₂²×C₈

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

106

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

C₄×Q₈

—

106

—

#117

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₅₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

106

—

102

C₂×C₈

—

212

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

106

—

105

C₄⋊C₄

—

106

—

106

C₄⋊C₄

—

106

—

107

C₄⋊C₄

—

106

—

108

C₄⋊C₄

—

106

—

109

C₂²⋊C₄

—

106

—

110

C₄²

—

106

—

111

C₄²

C₄×D₅₃

yes

characteristic

112

C₄²

D₂₁₂

yes

characteristic

113

C₄²

C₂²⋊D₅₃

yes

characteristic

114

C₄²

C₂²×D₅₃

yes

characteristic

115

C₂³

#11719921

yes

characteristic

116

Q₈

—

106

—

117

Q₈

—

212

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

106

—

121

C₂×C₄

—

106

—

122

C₂×C₄

#11719913

yes

—

123

C₂×C₄

Dic₂₁₂

yes

—

124

C₂×C₄

#11719921

yes

characteristic

125

C₂×C₄

#11719921

yes

characteristic

126

C₂×C₄

#11719921

yes

characteristic

127

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

128

C₂×C₄

C₂×D₂₁₂

yes

characteristic

129

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

130

C₈

—

131

C₈

—

212

—

132

C₂²

—

133

C₂²

—

134

C₂²

#13133142

yes

—

135

C₂²

C₂×#11719921

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

212

—

140

C₄

—

212

—

141

C₄

#13133142

yes

—

142

C₄

#13133225

yes

—

143

C₄

C₂×#11719913

yes

characteristic

144

C₄

C₂×Dic₂₁₂

yes

characteristic

145

C₂

—

146

C₂

#16810591

yes

characteristic

147

C₂

C₂×#13133142

yes

characteristic

148

C₂

#16810641

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#431	D₅₃
hypocenter	C₅₃	#1481
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#431

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #16810355	C₂	semidirect
79. C₅₃	#1481	semidirect

H ≅ G/N

type

3. #16810355

C₂

semidirect

79. C₅₃

#1481

semidirect

Subgroups — #20273119

Conjugacy classes of subgroups

Complements