Group #20276376 — subgroups

Subgroups — C₂×#16810418 #20276376

Order 6784.

284 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₂×#13133132	C₂	1	2	yes	maximal
2	#16810418	C₂	1	4	yes	maximal, direct product factor
3	C₂×#124×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
5	#13133132	—	2	2	no	—
6	#13133132	C₂²	1	4	yes	—
7	C₂²×Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
8	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
9	#124×C₅₃	—	2	2	no	—
10	#124×C₅₃	C₄	1	2	yes	complemented
11	#124×C₅₃	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
13	C₂×C₄²×C₅₃	C₄	1	1	yes	characteristic, complemented
14	C₂⁴×C₅₃	D₄	1	2	yes	—
15	C₂×Q₈×C₅₃	—	2	1	no	—
16	C₂×Q₈×C₅₃	—	4	1	no	—
17	C₂×Q₈×C₅₃	—	4	1	no	—
18	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	complemented
19	C₂×D₄×C₅₃	—	4	1	no	—
20	C₂×D₄×C₅₃	—	4	1	no	—
21	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×D₄×C₅₃	D₄	1	2	yes	—
23	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	—	2	1	no	—
25	C₂²×C₄×C₅₃	—	2	1	no	—
26	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×Dic₅₃	—	4	2	no	—
28	C₂²⋊C₄×C₅₃	—	2	4	no	—
29	C₂²⋊C₄×C₅₃	—	2	4	no	—
30	C₄²×C₅₃	—	2	1	no	—
31	C₄²×C₅₃	C₂×C₄	1	2	yes	complemented
32	C₂²⋊Dic₅₃	—	2	4	no	—
33	C₂²⋊Dic₅₃	—	4	2	no	—
34	C₂³×C₅₃	—	2	2	no	—
35	C₂³×C₅₃	—	2	2	no	—
36	C₂³×C₅₃	—	4	4	no	—
37	C₂³×C₅₃	C₂²⋊C₄	1	2	yes	—
38	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂³×C₅₃	C₂×D₄	1	2	yes	—
40	Q₈×C₅₃	—	4	2	no	—
41	Q₈×C₅₃	—	4	2	no	—
42	D₄×C₅₃	—	4	1	no	—
43	D₄×C₅₃	—	4	2	no	—
44	D₄×C₅₃	—	4	1	no	—
45	C₂×C₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	4	2	no	—
50	C₂×C₄×C₅₃	—	4	1	no	—
51	C₂×C₄×C₅₃	—	4	1	no	—
52	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×Dic₅₃	—	4	4	no	—
55	C₂×Dic₅₃	—	8	4	no	—
56	C₂²×C₅₃	—	2	2	no	—
57	C₂²×C₅₃	—	2	2	no	—
58	C₂²×C₅₃	—	2	2	no	—
59	C₂²×C₅₃	—	4	2	no	—
60	C₂²×C₅₃	—	4	2	no	—
61	C₂²×C₅₃	—	4	4	no	—
62	C₂²×C₅₃	—	8	2	no	—
63	C₂²×C₅₃	#99	1	1	yes	characteristic
64	C₂²×C₅₃	#99	1	1	yes	characteristic
65	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₄×C₅₃	—	2	1	no	—
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	—	4	2	no	—
69	C₄×C₅₃	—	4	2	no	—
70	Dic₅₃	—	8	4	no	—
71	C₂×#354	—	53	1	no	maximal, Sylow
72	C₂×C₅₃	—	2	1	no	—
73	C₂×C₅₃	—	2	1	no	—
74	C₂×C₅₃	—	4	2	no	—
75	C₂×C₅₃	—	4	2	no	—
76	C₂×C₅₃	#354	1	2	yes	complemented
77	C₂×C₅₃	C₂×#99	1	1	yes	characteristic
78	C₂×#124	D₅₃	1	1	yes	characteristic
79	C₂×#99	—	53	2	no	—
80	#354	—	53	4	no	—
81	C₅₃	C₂×#354	1	1	yes	characteristic, complemented, Sylow
82	C₂²×Q₈	Dic₅₃	1	1	yes	characteristic, complemented
83	C₂²×D₄	C₂×D₅₃	1	1	yes	characteristic
84	#124	—	2	2	no	—
85	#124	Dic₅₃	1	2	yes	complemented
86	#124	C₂×D₅₃	1	2	yes	—
87	C₂×C₂²⋊C₄	—	2	2	no	—
88	C₂×C₂²⋊C₄	—	106	2	no	—
89	C₂×C₄²	Dic₅₃	1	1	yes	characteristic, complemented
90	#99	—	53	4	no	—
91	#99	—	106	2	no	—
92	C₂⁴	C₂²⋊D₅₃	1	2	yes	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	4	1	no	—
95	C₂×Q₈	—	4	1	no	—
96	C₂×Q₈	C₂×Dic₅₃	1	2	yes	complemented
97	C₂×D₄	—	4	1	no	—
98	C₂×D₄	—	4	1	no	—
99	C₂×D₄	C₂×Dic₅₃	1	1	yes	characteristic
100	C₂×D₄	C₂²⋊D₅₃	1	2	yes	—
101	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	2	1	no	—
104	C₂²×C₄	—	212	2	no	—
105	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
106	C₂²⋊C₄	—	2	4	no	—
107	C₂²⋊C₄	—	2	4	no	—
108	C₂²⋊C₄	—	106	4	no	—
109	C₂²⋊C₄	—	212	2	no	—
110	C₄²	—	2	1	no	—
111	C₄²	C₂×Dic₅₃	1	2	yes	complemented
112	C₂³	—	2	2	no	—
113	C₂³	—	2	2	no	—
114	C₂³	—	4	4	no	—
115	C₂³	C₂²⋊Dic₅₃	1	2	yes	—
116	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
117	C₂³	C₂×C₂²⋊D₅₃	1	2	yes	—
118	Q₈	—	4	2	no	—
119	Q₈	—	4	2	no	—
120	D₄	—	4	1	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	424	4	no	—
131	C₂×C₄	—	212	4	no	—
132	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
133	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
134	C₂²	—	2	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	—	4	2	no	—
138	C₂²	—	4	2	no	—
139	C₂²	—	4	4	no	—
140	C₂²	—	8	2	no	—
141	C₂²	#13133132	1	1	yes	characteristic
142	C₂²	#13133132	1	1	yes	characteristic
143	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	4	2	no	—
147	C₄	—	4	2	no	—
148	C₄	—	424	4	no	—
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	—	4	2	no	—
152	C₂	—	4	2	no	—
153	C₂	#16810418	1	2	yes	direct product factor
154	C₂	C₂×#13133132	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133132

C₂

yes

maximal

#16810418

C₂

yes

maximal, direct product factor

C₂×#124×C₅₃

C₂

yes

maximal, characteristic

C₂×C₂²⋊Dic₅₃

—

#13133132

—

#13133132

C₂²

yes

—

C₂²×Q₈×C₅₃

C₄

yes

characteristic, complemented

C₂²×D₄×C₅₃

C₂²

yes

characteristic

#124×C₅₃

—

#124×C₅₃

C₄

yes

complemented

#124×C₅₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₄

yes

characteristic, complemented

C₂⁴×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

complemented

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

D₄

yes

—

C₂×D₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

complemented

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂³×C₅₃

C₂×D₄

yes

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

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

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

C₂×#354

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#354

yes

complemented

C₂×C₅₃

C₂×#99

yes

characteristic

C₂×#124

D₅₃

yes

characteristic

C₂×#99

—

#354

—

C₅₃

C₂×#354

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₅₃

yes

characteristic, complemented

C₂²×D₄

C₂×D₅₃

yes

characteristic

#124

—

#124

Dic₅₃

yes

complemented

#124

C₂×D₅₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

106

—

C₂×C₄²

Dic₅₃

yes

characteristic, complemented

#99

—

#99

—

106

—

C₂⁴

C₂²⋊D₅₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₃

yes

complemented

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₅₃

yes

characteristic

100

C₂×D₄

C₂²⋊D₅₃

yes

—

101

C₂×D₄

C₂²×D₅₃

yes

characteristic

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

212

—

105

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

106

C₂²⋊C₄

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

106

—

109

C₂²⋊C₄

—

212

—

110

C₄²

—

111

C₄²

C₂×Dic₅₃

yes

complemented

112

C₂³

—

113

C₂³

—

114

C₂³

—

115

C₂³

C₂²⋊Dic₅₃

yes

—

116

C₂³

C₂²⋊Dic₅₃

yes

characteristic

117

C₂³

C₂×C₂²⋊D₅₃

yes

—

118

Q₈

—

119

Q₈

—

120

D₄

—

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

—

130

C₂×C₄

—

424

—

131

C₂×C₄

—

212

—

132

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

133

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

#13133132

yes

characteristic

142

C₂²

#13133132

yes

characteristic

143

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

424

—

149

C₂

—

150

C₂

—

151

C₂

—

152

C₂

—

153

C₂

#16810418

yes

direct product factor

154

C₂

C₂×#13133132

yes

characteristic

	subgroup	G/H
center	C₂²	#13133132
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×D₄	C₂²×D₅₃
Fitting	C₂×#124×C₅₃	C₂
socle	C₂²×C₅₃	#99
hypercenter	C₂×#124	D₅₃
hypocenter	C₅₃	C₂×#354
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×#124

subgroup

G/H

center

C₂²

#13133132

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×D₄

C₂²×D₅₃

Fitting

C₂×#124×C₅₃

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×#124

N	H ≅ G/N	type
2. #16810418	C₂	direct (N × H)
7. C₂²×Q₈×C₅₃	C₄	semidirect
10. #124×C₅₃	C₄	semidirect
13. C₂×C₄²×C₅₃	C₄	semidirect
18. C₂×Q₈×C₅₃	C₂×C₄	semidirect
31. C₄²×C₅₃	C₂×C₄	semidirect
76. C₂×C₅₃	#354	semidirect
81. C₅₃	C₂×#354	semidirect
82. C₂²×Q₈	Dic₅₃	semidirect
85. #124	Dic₅₃	semidirect
89. C₂×C₄²	Dic₅₃	semidirect
96. C₂×Q₈	C₂×Dic₅₃	semidirect
111. C₄²	C₂×Dic₅₃	semidirect
153. C₂	#16810418	direct (N × H)

H ≅ G/N

type

2. #16810418

C₂

direct (N × H)

7. C₂²×Q₈×C₅₃

C₄

semidirect

10. #124×C₅₃

C₄

semidirect

13. C₂×C₄²×C₅₃

C₄

semidirect

18. C₂×Q₈×C₅₃

C₂×C₄

semidirect

31. C₄²×C₅₃

C₂×C₄

semidirect

76. C₂×C₅₃

#354

semidirect

81. C₅₃

C₂×#354

semidirect

82. C₂²×Q₈

Dic₅₃

semidirect

85. #124

Dic₅₃

semidirect

89. C₂×C₄²

Dic₅₃

semidirect

96. C₂×Q₈

C₂×Dic₅₃

semidirect

111. C₄²

C₂×Dic₅₃

semidirect

153. C₂

#16810418

direct (N × H)

Subgroups — C₂×#16810418 #20276376

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810418 #20276376

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810418 #20276376