Group #20275574 — subgroups

Subgroups — #20275574

Order 6784.

312 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂²×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133140	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810528	C₂	1	2	yes	maximal
5	C₂×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
7	C₂×Dic₂₁₂	—	2	2	no	—
8	C₂×Dic₂₁₂	—	2	2	no	—
9	C₂×Dic₂₁₂	C₄	1	4	yes	complemented
10	C₂²×#88322	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
12	#13133140	—	2	4	no	—
13	C₈×Dic₅₃	—	2	2	no	—
14	#13133124	—	2	4	no	—
15	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	#107×C₅₃	—	2	2	no	—
18	C₂²×C₄×C₅₃	—	2	2	no	—
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₅₃	—	2	2	no	—
21	C₂²×Dic₅₃	D₄	1	2	yes	—
22	C₂×Dic₁₀₆	—	2	2	no	—
23	C₂×Dic₁₀₆	—	2	2	no	—
24	C₂×Dic₁₀₆	—	2	2	no	—
25	C₂×Dic₁₀₆	—	2	2	no	—
26	C₂×Dic₁₀₆	—	2	2	no	—
27	C₂×Dic₁₀₆	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₅₃	D₄	1	2	yes	—
30	C₂×C₈×C₅₃	D₄	1	2	yes	—
31	C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₄×Dic₅₃	—	2	1	no	—
34	C₄×Dic₅₃	—	2	1	no	—
35	C₂×#88322	—	2	1	no	—
36	C₂×#88322	—	2	1	no	—
37	C₂×#88322	D₄	1	2	yes	—
38	Dic₂₁₂	—	2	8	no	—
39	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	—	4	2	no	—
41	C₂×C₄×C₅₃	—	4	4	no	—
42	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
48	C₂×Dic₅₃	—	2	2	no	—
49	C₂×Dic₅₃	—	2	2	no	—
50	C₂×Dic₅₃	—	2	4	no	—
51	C₂×Dic₅₃	—	4	2	no	—
52	C₂×Dic₅₃	—	4	2	no	—
53	C₂×Dic₅₃	Q₁₆	1	4	yes	complemented
54	Dic₁₀₆	—	2	4	no	—
55	Dic₁₀₆	—	2	4	no	—
56	Dic₁₀₆	—	4	2	no	—
57	Dic₁₀₆	—	4	2	no	—
58	C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
59	#88322	—	2	2	no	—
60	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₅₃	#121	1	1	yes	characteristic
62	C₂²×C₅₃	#127	1	1	yes	characteristic
63	C₂²×C₅₃	C₂×Q₁₆	1	2	yes	—
64	C₂²×C₅₃	#135	1	2	yes	—
65	C₄×C₅₃	—	8	2	no	—
66	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
68	C₄×C₅₃	#121	1	1	yes	characteristic
69	C₄×C₅₃	#124	1	1	yes	characteristic
70	Dic₅₃	—	2	4	no	—
71	Dic₅₃	—	4	4	no	—
72	#1969	—	53	1	no	maximal, Sylow
73	C₂×C₅₃	#390	1	1	yes	characteristic
74	C₂×C₅₃	C₄×Q₁₆	1	2	yes	—
75	C₂×C₅₃	#467	1	2	yes	—
76	C₂×C₅₃	#494	1	1	yes	characteristic
77	C₂×C₅₃	#495	1	1	yes	characteristic
78	C₂²×Q₁₆	—	53	1	no	—
79	C₂×#107	D₅₃	1	1	yes	characteristic
80	C₂×#103	—	53	2	no	—
81	C₂×C₄×C₈	—	53	1	no	—
82	#391	—	53	2	no	—
83	C₅₃	#1969	1	1	yes	characteristic, complemented, Sylow
84	C₂²×Q₈	—	53	2	no	—
85	C₂×Q₁₆	—	53	4	no	—
86	C₂×Q₁₆	—	106	2	no	—
87	C₂×Q₁₆	—	106	2	no	—
88	C₂²×C₈	—	53	1	no	—
89	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
90	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
91	C₂×C₄²	—	53	1	no	—
92	#107	—	2	2	no	—
93	#103	—	106	4	no	—
94	C₄×C₈	—	106	2	no	—
95	#95	—	106	4	no	—
96	C₂×Q₈	—	53	4	no	—
97	C₂×Q₈	—	106	2	no	—
98	C₂×Q₈	—	106	2	no	—
99	C₂×Q₈	—	106	2	no	—
100	C₂×Q₈	—	106	2	no	—
101	C₂×Q₈	—	106	2	no	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	53	2	no	—
104	C₂²×C₄	—	106	2	no	—
105	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
106	Q₁₆	—	106	8	no	—
107	C₂×C₈	—	53	2	no	—
108	C₂×C₈	—	106	1	no	—
109	C₂×C₈	—	106	1	no	—
110	C₂×C₈	C₄×D₅₃	1	2	yes	—
111	C₂×C₈	D₂₁₂	1	2	yes	—
112	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄²	—	106	1	no	—
116	C₄²	—	106	1	no	—
117	C₂³	D₄×D₅₃	1	1	yes	characteristic
118	Q₈	—	106	4	no	—
119	Q₈	—	106	4	no	—
120	Q₈	—	212	2	no	—
121	Q₈	—	212	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	4	no	—
124	C₂×C₄	—	53	4	no	—
125	C₂×C₄	—	106	2	no	—
126	C₂×C₄	—	106	2	no	—
127	C₂×C₄	—	106	4	no	—
128	C₂×C₄	—	212	2	no	—
129	C₂×C₄	—	212	2	no	—
130	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
131	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
132	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
133	C₂×C₄	#11719949	1	1	yes	characteristic
134	C₂×C₄	#11719949	1	1	yes	characteristic
135	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
136	C₈	—	106	2	no	—
137	C₈	#11719921	1	4	yes	—
138	C₂²	#13133179	1	1	yes	characteristic
139	C₂²	#13133192	1	1	yes	characteristic
140	C₂²	#13133213	1	1	yes	characteristic
141	C₂²	Q₁₆×D₅₃	1	2	yes	—
142	C₂²	#13133236	1	2	yes	—
143	C₄	—	8	2	no	—
144	C₄	—	106	4	no	—
145	C₄	—	212	4	no	—
146	C₄	C₂×#11719921	1	1	yes	characteristic
147	C₄	#13133179	1	1	yes	characteristic
148	C₄	#13133192	1	1	yes	characteristic
149	C₄	#13133206	1	1	yes	characteristic
150	C₂	#16810524	1	1	yes	characteristic
151	C₂	#16810717	1	2	yes	—
152	C₂	#16810865	1	2	yes	—
153	C₂	#16811001	1	1	yes	characteristic
154	C₂	#16811006	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₁₂

C₂

yes

maximal, characteristic

C₂×#13133140

C₂

yes

maximal

C₂×C₈×Dic₅₃

C₂

yes

maximal, characteristic

#16810528

C₂

yes

maximal

C₂×#107×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

C₄

yes

complemented

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133140

—

C₈×Dic₅₃

—

#13133124

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

#107×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

D₄

yes

—

Dic₂₁₂

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

Q₁₆

yes

complemented

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

C₂²⋊C₄

yes

—

#88322

—

C₂²×C₅₃

C₄×D₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×Q₁₆

yes

—

C₂²×C₅₃

#135

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₄×Q₁₆

yes

—

C₂×C₅₃

#467

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂²×Q₁₆

—

C₂×#107

D₅₃

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

—

#391

—

C₅₃

#1969

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

#107

—

#103

—

106

—

C₄×C₈

—

106

—

#95

—

106

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

100

C₂×Q₈

—

106

—

101

C₂×Q₈

—

106

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

106

—

105

C₂²×C₄

C₂²×D₅₃

yes

characteristic

106

Q₁₆

—

106

—

107

C₂×C₈

—

108

C₂×C₈

—

106

—

109

C₂×C₈

—

106

—

110

C₂×C₈

C₄×D₅₃

yes

—

111

C₂×C₈

D₂₁₂

yes

—

112

C₂×C₈

C₂²⋊D₅₃

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

—

106

—

116

C₄²

—

106

—

117

C₂³

D₄×D₅₃

yes

characteristic

118

Q₈

—

106

—

119

Q₈

—

106

—

120

Q₈

—

212

—

121

Q₈

—

212

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

106

—

126

C₂×C₄

—

106

—

127

C₂×C₄

—

106

—

128

C₂×C₄

—

212

—

129

C₂×C₄

—

212

—

130

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

131

C₂×C₄

C₂×D₂₁₂

yes

characteristic

132

C₂×C₄

D₄×D₅₃

yes

characteristic

133

C₂×C₄

#11719949

yes

characteristic

134

C₂×C₄

#11719949

yes

characteristic

135

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

136

C₈

—

106

—

137

C₈

#11719921

yes

—

138

C₂²

#13133179

yes

characteristic

139

C₂²

#13133192

yes

characteristic

140

C₂²

#13133213

yes

characteristic

141

C₂²

Q₁₆×D₅₃

yes

—

142

C₂²

#13133236

yes

—

143

C₄

—

144

C₄

—

106

—

145

C₄

—

212

—

146

C₄

C₂×#11719921

yes

characteristic

147

C₄

#13133179

yes

characteristic

148

C₄

#13133192

yes

characteristic

149

C₄

#13133206

yes

characteristic

150

C₂

#16810524

yes

characteristic

151

C₂

#16810717

yes

—

152

C₂

#16810865

yes

—

153

C₂

#16811001

yes

characteristic

154

C₂

#16811006

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#107×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#107	D₅₃
hypocenter	C₅₃	#1969
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#107×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
9. C₂×Dic₂₁₂	C₄	semidirect
53. C₂×Dic₅₃	Q₁₆	semidirect
83. C₅₃	#1969	semidirect

H ≅ G/N

type

9. C₂×Dic₂₁₂

C₄

semidirect

53. C₂×Dic₅₃

Q₁₆

semidirect

83. C₅₃

#1969

semidirect

Subgroups — #20275574

Conjugacy classes of subgroups

Complements