Group #20285153 — subgroups

Subgroups — C₂×#16811015 #20285153

Order 6784.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#11719925	C₂	1	2	yes	maximal, complemented
2	C₂²×#11719924	C₂	1	2	yes	maximal
3	C₂×#13133205	C₂	1	1	yes	maximal, characteristic
4	#16811015	C₂	1	8	yes	maximal, direct product factor
5	C₂×#13133128	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
8	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
9	C₂×#11719925	—	2	8	no	—
10	C₂×#11719925	C₂²	1	8	yes	complemented
11	C₂×#11719924	—	2	8	no	—
12	C₂×#11719924	C₂²	1	8	yes	—
13	C₂²×#88322	C₂²	1	2	yes	—
14	#13133205	—	2	2	no	—
15	#13133205	C₂²	1	4	yes	—
16	C₂×#11719921	—	2	2	no	—
17	#13133128	C₂²	1	4	yes	complemented
18	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₅₃	—	2	2	no	—
20	C₄⋊Q₈×C₅₃	C₂²	1	4	yes	complemented
21	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂³×D₅₃	—	2	1	no	—
24	C₂×Q₈×C₅₃	—	2	2	no	—
25	C₂×Q₈×C₅₃	—	2	4	no	—
26	C₂×Q₈×C₅₃	—	2	4	no	—
27	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
28	C₂²×C₄×C₅₃	—	2	2	no	—
29	C₂²×C₄×C₅₃	D₄	1	2	yes	—
30	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₅₃	—	2	1	no	—
32	C₂×D₂₁₂	—	2	4	no	—
33	C₂×D₂₁₂	—	2	1	no	—
34	C₂×D₂₁₂	C₂³	1	2	yes	—
35	C₂×Dic₁₀₆	—	2	4	no	—
36	C₂×Dic₁₀₆	—	2	1	no	—
37	C₂×Dic₁₀₆	C₂³	1	2	yes	—
38	C₄⋊C₄×C₅₃	—	2	8	no	—
39	C₄²×C₅₃	D₄	1	2	yes	—
40	C₄²×C₅₃	C₂³	1	2	yes	—
41	#11719925	—	2	16	no	—
42	#11719924	—	2	16	no	—
43	#11719921	—	2	8	no	—
44	C₂×#88322	D₄	1	8	yes	—
45	C₂×#88322	C₂³	1	4	yes	—
46	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×D₅₃	—	2	2	no	—
48	C₂²×D₅₃	—	4	2	no	—
49	C₂²×D₅₃	—	4	4	no	—
50	Q₈×C₅₃	—	2	4	no	—
51	Q₈×C₅₃	—	2	4	no	—
52	Q₈×C₅₃	—	4	4	no	—
53	C₂×C₄×C₅₃	—	2	4	no	—
54	C₂×C₄×C₅₃	—	2	4	no	—
55	C₂×C₄×C₅₃	—	4	4	no	—
56	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
57	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
58	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₅₃	—	2	2	no	—
61	C₂×Dic₅₃	—	4	2	no	—
62	D₂₁₂	—	2	4	no	—
63	D₂₁₂	—	4	2	no	—
64	Dic₁₀₆	—	2	4	no	—
65	Dic₁₀₆	—	4	2	no	—
66	#88322	C₂×D₄	1	8	yes	—
67	C₂²×C₅₃	#127	1	2	yes	—
68	C₂²×C₅₃	#137	1	4	yes	—
69	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
70	C₂×D₅₃	—	4	2	no	—
71	C₂×D₅₃	—	4	4	no	—
72	C₂×D₅₃	—	8	4	no	—
73	C₄×C₅₃	—	2	4	no	—
74	C₄×C₅₃	—	4	4	no	—
75	C₄×C₅₃	#127	1	2	yes	—
76	C₄×C₅₃	C₂²×D₄	1	2	yes	—
77	Dic₅₃	—	4	2	no	—
78	C₂×#497	—	53	1	no	maximal, Sylow
79	C₂×C₅₃	#497	1	4	yes	complemented
80	C₂×C₅₃	C₂×#127	1	1	yes	characteristic
81	C₂×C₅₃	C₂×#137	1	2	yes	—
82	D₅₃	—	8	2	no	—
83	C₂²×Q₁₆	—	53	2	no	—
84	C₂²×SD₁₆	—	53	2	no	—
85	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic, complemented
86	C₂×#124	—	53	1	no	—
87	#497	—	53	8	no	—
88	C₂×#97	—	53	1	no	—
89	C₅₃	C₂×#497	1	1	yes	characteristic, complemented, Sylow
90	C₂²×Q₈	—	53	1	no	—
91	C₂²×Q₈	C₂×D₅₃	1	2	yes	—
92	C₂²×D₄	—	53	1	no	—
93	C₂×Q₁₆	—	53	8	no	—
94	C₂×Q₁₆	—	106	8	no	—
95	C₂×SD₁₆	—	53	8	no	—
96	C₂×SD₁₆	—	106	8	no	—
97	C₂²×C₈	—	53	2	no	—
98	C₄⋊Q₈	—	2	2	no	—
99	C₄⋊Q₈	C₂×D₅₃	1	4	yes	complemented
100	#124	—	53	4	no	—
101	#124	—	106	2	no	—
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₂²⋊C₄	—	106	2	no	—
104	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
105	#97	—	53	4	no	—
106	C₂⁴	—	106	1	no	—
107	C₂×Q₈	—	2	2	no	—
108	C₂×Q₈	—	2	4	no	—
109	C₂×Q₈	—	2	4	no	—
110	C₂×Q₈	—	53	2	no	—
111	C₂×Q₈	—	106	1	no	—
112	C₂×Q₈	—	106	4	no	—
113	C₂×Q₈	C₂²×D₅₃	1	4	yes	—
114	C₂×D₄	—	53	2	no	—
115	C₂×D₄	—	106	4	no	—
116	C₂×D₄	—	106	1	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	106	1	no	—
119	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
120	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
121	Q₁₆	—	106	16	no	—
122	SD₁₆	—	106	16	no	—
123	C₂×C₈	—	53	8	no	—
124	C₂×C₈	—	53	4	no	—
125	C₄⋊C₄	—	2	8	no	—
126	C₂²⋊C₄	—	106	8	no	—
127	C₄²	C₂²⋊D₅₃	1	2	yes	—
128	C₄²	C₂²×D₅₃	1	2	yes	—
129	C₂³	—	106	2	no	—
130	C₂³	—	212	2	no	—
131	C₂³	—	212	4	no	—
132	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
133	Q₈	—	2	4	no	—
134	Q₈	—	2	4	no	—
135	Q₈	—	4	4	no	—
136	Q₈	—	106	4	no	—
137	Q₈	—	212	2	no	—
138	D₄	—	106	4	no	—
139	D₄	—	212	2	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	—	106	2	no	—
144	C₂×C₄	—	212	2	no	—
145	C₂×C₄	D₄×D₅₃	1	4	yes	—
146	C₂×C₄	C₂×C₂²⋊D₅₃	1	4	yes	—
147	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
148	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
149	C₈	—	53	8	no	—
150	C₂²	—	424	4	no	—
151	C₂²	—	212	4	no	—
152	C₂²	—	212	2	no	—
153	C₂²	#13133213	1	2	yes	—
154	C₂²	#13133248	1	4	yes	—
155	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
156	C₄	—	2	4	no	—
157	C₄	—	4	4	no	—
158	C₄	—	212	2	no	—
159	C₄	#13133213	1	2	yes	—
160	C₄	C₂×D₄×D₅₃	1	2	yes	—
161	C₂	—	424	2	no	—
162	C₂	#16811015	1	4	yes	direct product factor
163	C₂	C₂×#13133213	1	1	yes	characteristic
164	C₂	C₂×#13133248	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719925

C₂

yes

maximal, complemented

C₂²×#11719924

C₂

yes

maximal

C₂×#13133205

C₂

yes

maximal, characteristic

#16811015

C₂

yes

maximal, direct product factor

C₂×#13133128

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₀₆

C₂²

yes

characteristic

C₂²×D₂₁₂

C₂²

yes

characteristic

C₂×#11719925

—

C₂×#11719925

C₂²

yes

complemented

C₂×#11719924

—

C₂×#11719924

C₂²

yes

—

C₂²×#88322

C₂²

yes

—

#13133205

—

#13133205

C₂²

yes

—

C₂×#11719921

—

#13133128

C₂²

yes

complemented

C₂²×Q₈×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

complemented

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂³

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

—

—

—

D₄

yes

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

Dic₁₀₆

—

Dic₁₀₆

—

#88322

C₂×D₄

yes

—

C₂²×C₅₃

#127

yes

—

C₂²×C₅₃

#137

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#127

yes

—

C₄×C₅₃

C₂²×D₄

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

C₂²×Q₁₆

—

C₂²×SD₁₆

—

C₂×C₄⋊Q₈

D₅₃

yes

characteristic, complemented

C₂×#124

—

#497

—

C₂×#97

—

C₅₃

C₂×#497

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅₃

yes

—

C₂²×D₄

—

C₂×Q₁₆

—

C₂×Q₁₆

—

106

—

C₂×SD₁₆

—

C₂×SD₁₆

—

106

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

complemented

100

#124

—

101

#124

—

106

—

102

C₂×C₄⋊C₄

—

103

C₂×C₂²⋊C₄

—

106

—

104

C₂×C₄²

C₂×D₅₃

yes

characteristic

105

#97

—

106

C₂⁴

—

106

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

106

—

112

C₂×Q₈

—

106

—

113

C₂×Q₈

C₂²×D₅₃

yes

—

114

C₂×D₄

—

115

C₂×D₄

—

106

—

116

C₂×D₄

—

106

—

117

C₂²×C₄

—

118

C₂²×C₄

—

106

—

119

C₂²×C₄

C₂²⋊D₅₃

yes

—

120

C₂²×C₄

C₂²×D₅₃

yes

characteristic

121

Q₁₆

—

106

—

122

SD₁₆

—

106

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₄⋊C₄

—

126

C₂²⋊C₄

—

106

—

127

C₄²

C₂²⋊D₅₃

yes

—

128

C₄²

C₂²×D₅₃

yes

—

129

C₂³

—

106

—

130

C₂³

—

212

—

131

C₂³

—

212

—

132

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

106

—

137

Q₈

—

212

—

138

D₄

—

106

—

139

D₄

—

212

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

106

—

144

C₂×C₄

—

212

—

145

C₂×C₄

D₄×D₅₃

yes

—

146

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

147

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

148

C₂×C₄

C₂³×D₅₃

yes

characteristic

149

C₈

—

150

C₂²

—

424

—

151

C₂²

—

212

—

152

C₂²

—

212

—

153

C₂²

#13133213

yes

—

154

C₂²

#13133248

yes

—

155

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

212

—

159

C₄

#13133213

yes

—

160

C₄

C₂×D₄×D₅₃

yes

—

161

C₂

—

424

—

162

C₂

#16811015

yes

direct product factor

163

C₂

C₂×#13133213

yes

characteristic

164

C₂

C₂×#13133248

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	C₂×#497
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₅₃

hypocenter

C₅₃

C₂×#497

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
1. C₂²×#11719925	C₂	semidirect
4. #16811015	C₂	direct (N × H)
5. C₂×#13133128	C₂	semidirect
6. C₂×C₄⋊Q₈×C₅₃	C₂	semidirect
10. C₂×#11719925	C₂²	semidirect
17. #13133128	C₂²	semidirect
20. C₄⋊Q₈×C₅₃	C₂²	semidirect
79. C₂×C₅₃	#497	semidirect
85. C₂×C₄⋊Q₈	D₅₃	semidirect
89. C₅₃	C₂×#497	semidirect
99. C₄⋊Q₈	C₂×D₅₃	semidirect
162. C₂	#16811015	direct (N × H)

H ≅ G/N

type

1. C₂²×#11719925

C₂

semidirect

4. #16811015

C₂

direct (N × H)

5. C₂×#13133128

C₂

semidirect

6. C₂×C₄⋊Q₈×C₅₃

C₂

semidirect

10. C₂×#11719925

C₂²

semidirect

17. #13133128

C₂²

semidirect

20. C₄⋊Q₈×C₅₃

C₂²

semidirect

79. C₂×C₅₃

#497

semidirect

85. C₂×C₄⋊Q₈

D₅₃

semidirect

89. C₅₃

C₂×#497

semidirect

99. C₄⋊Q₈

C₂×D₅₃

semidirect

162. C₂

#16811015

direct (N × H)

Subgroups — C₂×#16811015 #20285153

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16811015 #20285153

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811015 #20285153