Group #20274707 — subgroups

Subgroups — #20274707

Order 6784.

336 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 113 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	#16810647	C₂	1	2	yes	maximal, complemented
2	C₂×#13133134	C₂	1	2	yes	maximal
3	#16810392	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	C₂²	1	4	yes	—
6	C₄⋊#88322	—	2	2	no	—
7	C₄⋊#88322	—	2	2	no	—
8	#13133134	—	2	4	no	—
9	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₅₃	C₄	1	2	yes	—
11	C₄×D₄×C₅₃	—	2	2	no	—
12	C₄×D₄×C₅₃	—	2	2	no	—
13	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂⁴×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×D₄×C₅₃	—	2	2	no	—
18	C₂×D₄×C₅₃	—	2	2	no	—
19	C₂×D₄×C₅₃	D₄	1	2	yes	—
20	C₂×D₄×C₅₃	Q₈	1	2	yes	—
21	C₂²×C₄×C₅₃	—	2	2	no	—
22	C₂²×C₄×C₅₃	—	2	2	no	—
23	C₂²×C₄×C₅₃	—	4	4	no	—
24	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂²⋊C₄×C₅₃	—	2	4	no	—
30	C₄²×C₅₃	D₄	1	2	yes	—
31	C₄²×C₅₃	D₄	1	2	yes	—
32	#11719917	—	2	4	no	—
33	#11719917	—	2	4	no	—
34	C₂×#88322	—	2	4	no	—
35	C₂³×C₅₃	—	2	2	no	—
36	C₂³×C₅₃	—	2	2	no	—
37	C₂³×C₅₃	—	4	2	no	—
38	C₂³×C₅₃	—	4	2	no	—
39	C₂³×C₅₃	C₄⋊C₄	1	4	yes	—
40	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	D₄×C₅₃	—	2	4	no	—
42	D₄×C₅₃	—	2	4	no	—
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	4	1	no	—
46	C₂×C₄×C₅₃	—	4	4	no	—
47	C₂×C₄×C₅₃	—	4	4	no	—
48	C₂×C₄×C₅₃	—	4	2	no	—
49	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#41	1	2	yes	—
56	#88322	—	4	4	no	—
57	C₂²×C₅₃	—	2	1	no	—
58	C₂²×C₅₃	—	2	1	no	—
59	C₂²×C₅₃	—	2	4	no	—
60	C₂²×C₅₃	—	4	4	no	—
61	C₂²×C₅₃	—	4	4	no	—
62	C₂²×C₅₃	—	4	2	no	—
63	C₂²×C₅₃	—	4	2	no	—
64	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₅₃	#117	1	1	yes	characteristic
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	—	2	2	no	—
69	C₄×C₅₃	—	4	2	no	—
70	C₄×C₅₃	#100	1	2	yes	complemented
71	C₄×C₅₃	#121	1	2	yes	—
72	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
73	#1800	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	—	2	1	no	—
75	C₂×C₅₃	—	2	1	no	—
76	C₂×C₅₃	—	4	4	no	—
77	C₂×C₅₃	#380	1	1	yes	characteristic
78	C₂×C₅₃	C₂×#100	1	1	yes	characteristic
79	C₂×C₅₃	#413	1	1	yes	characteristic
80	C₂×C₄×D₄	D₅₃	1	1	yes	characteristic
81	#423	—	53	2	no	—
82	C₂×#100	—	53	2	no	—
83	#343	—	53	2	no	—
84	C₅₃	#1800	1	1	yes	characteristic, complemented, Sylow
85	C₂²×D₄	C₂×D₅₃	1	1	yes	characteristic
86	C₂³×C₄	Dic₅₃	1	2	yes	—
87	C₂×M₁₆	—	53	4	no	—
88	C₄×D₄	—	2	2	no	—
89	C₄×D₄	—	2	2	no	—
90	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
91	C₂×C₂²⋊C₄	Dic₅₃	1	2	yes	—
92	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
93	C₄⋊C₈	—	106	2	no	—
94	C₄⋊C₈	—	106	2	no	—
95	#100	—	106	4	no	—
96	C₂⁴	C₂×Dic₅₃	1	2	yes	—
97	C₂×D₄	—	2	2	no	—
98	C₂×D₄	—	2	2	no	—
99	C₂×D₄	Dic₁₀₆	1	2	yes	—
100	C₂×D₄	D₂₁₂	1	2	yes	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	4	4	no	—
104	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
105	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
106	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
107	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
108	M₁₆	—	106	4	no	—
109	M₁₆	—	106	4	no	—
110	C₂×C₈	—	106	4	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₂²⋊C₄	—	2	4	no	—
113	C₄²	C₂²⋊D₅₃	1	2	yes	—
114	C₄²	C₂²⋊D₅₃	1	2	yes	—
115	C₂³	—	2	2	no	—
116	C₂³	—	2	2	no	—
117	C₂³	—	4	2	no	—
118	C₂³	—	4	2	no	—
119	C₂³	C₄⋊Dic₅₃	1	4	yes	—
120	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
121	D₄	—	2	4	no	—
122	D₄	—	2	4	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	4	no	—
127	C₂×C₄	—	4	4	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
130	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
131	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
132	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
133	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
134	C₂×C₄	#11719945	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
136	C₈	—	212	4	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	4	no	—
140	C₂²	—	4	4	no	—
141	C₂²	—	4	4	no	—
142	C₂²	—	4	2	no	—
143	C₂²	—	4	2	no	—
144	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
145	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
146	C₂²	#13133176	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	2	no	—
150	C₄	#13133134	1	2	yes	complemented
151	C₄	#13133191	1	2	yes	—
152	C₄	#13133196	1	2	yes	—
153	C₂	—	2	1	no	—
154	C₂	—	2	1	no	—
155	C₂	—	4	4	no	—
156	C₂	#16810483	1	1	yes	characteristic
157	C₂	C₂×#13133134	1	1	yes	characteristic
158	C₂	#16810607	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810647

C₂

yes

maximal, complemented

C₂×#13133134

C₂

yes

maximal

#16810392

C₂

yes

maximal, complemented

C₂×C₄×D₄×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₄⋊#88322

—

C₄⋊#88322

—

#13133134

—

C₂²×D₄×C₅₃

C₂²

yes

characteristic

C₂³×C₄×C₅₃

C₄

yes

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

D₄

yes

—

C₂×D₄×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

D₄

yes

—

#11719917

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₄⋊C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#117

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#100

yes

complemented

C₄×C₅₃

#121

yes

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

#1800

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×D₄

D₅₃

yes

—

—

—

C₅₃

#1800

yes

characteristic, complemented, Sylow

C₂²×D₄

C₂×D₅₃

yes

characteristic

C₂³×C₄

Dic₅₃

yes

—

C₂×M₁₆

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₂²⋊C₄

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

#100

—

106

—

C₂⁴

C₂×Dic₅₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₁₀₆

yes

—

100

C₂×D₄

D₂₁₂

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

105

C₂²×C₄

C₂×Dic₅₃

yes

—

106

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

107

C₂²×C₄

C₂²×D₅₃

yes

characteristic

108

M₁₆

—

106

—

109

M₁₆

—

106

—

110

C₂×C₈

—

106

—

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

C₂²⋊D₅₃

yes

—

114

C₄²

C₂²⋊D₅₃

yes

—

115

C₂³

—

116

C₂³

—

117

C₂³

—

118

C₂³

—

119

C₂³

C₄⋊Dic₅₃

yes

—

120

C₂³

C₂²×Dic₅₃

yes

characteristic

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂²⋊Dic₅₃

yes

—

130

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

131

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

132

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

133

C₂×C₄

C₂×D₂₁₂

yes

characteristic

134

C₂×C₄

#11719945

yes

—

135

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

136

C₈

—

212

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

145

C₂²

C₂×C₄⋊Dic₅₃

yes

characteristic

146

C₂²

#13133176

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#13133134

yes

complemented

151

C₄

#13133191

yes

—

152

C₄

#13133196

yes

—

153

C₂

—

154

C₂

—

155

C₂

—

156

C₂

#16810483

yes

characteristic

157

C₂

C₂×#13133134

yes

characteristic

158

C₂

#16810607

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₃
commutator	C₂³×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄×D₄×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₂×C₄×D₄	D₅₃
hypocenter	C₅₃	#1800
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×D₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₃

commutator

C₂³×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄×D₄×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

hypercenter

C₂×C₄×D₄

D₅₃

hypocenter

C₅₃

#1800

LCS

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×D₄

N	H ≅ G/N	type
1. #16810647	C₂	semidirect
3. #16810392	C₂	semidirect
70. C₄×C₅₃	#100	semidirect
84. C₅₃	#1800	semidirect
150. C₄	#13133134	semidirect

H ≅ G/N

type

1. #16810647

C₂

semidirect

3. #16810392

C₂

semidirect

70. C₄×C₅₃

#100

semidirect

84. C₅₃

#1800

semidirect

150. C₄

#13133134

semidirect

Subgroups — #20274707

Conjugacy classes of subgroups

Complements