Group #20277057 — subgroups

Subgroups — C₄×#13133174 #20277057

Order 6784.

752 conjugacy classes of subgroups; only the 357 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 357 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×#13133174	C₂	1	2	yes	maximal
2	C₂×C₄²×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄×#11719919	C₂	1	4	yes	maximal, complemented
4	C₄×#11719921	C₂	1	4	yes	maximal
5	#16810469	C₂	1	2	yes	maximal, complemented
6	C₄²×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₅₃	C₂²	1	2	yes	—
9	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
10	#13133174	C₄	1	16	yes	direct product factor
11	C₂×#11719919	C₂²	1	4	yes	—
12	C₂×#11719921	C₂²	1	4	yes	—
13	C₂×C₄×Dic₅₃	C₂²	1	4	yes	—
14	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
16	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
17	C₄²×D₅₃	C₄	1	8	yes	complemented
18	#13133124	C₂²	1	8	yes	—
19	C₂×C₄²×C₅₃	C₂²	1	4	yes	—
20	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₂³×D₅₃	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
26	C₂²×Dic₅₃	C₂³	1	2	yes	—
27	C₂²×Dic₅₃	C₂³	1	4	yes	—
28	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
29	C₂×C₄×D₅₃	—	2	8	no	—
30	C₂×C₄×D₅₃	C₂×C₄	1	8	yes	—
31	C₂×C₄×D₅₃	C₂×C₄	1	4	yes	—
32	C₂×C₄×D₅₃	C₂×C₄	1	8	yes	—
33	C₄²×C₅₃	—	2	8	no	—
34	C₄²×C₅₃	C₂×C₄	1	4	yes	—
35	C₄²×C₅₃	C₂×C₄	1	8	yes	complemented
36	#11719921	C₂×C₄	1	16	yes	—
37	#11719919	C₂×C₄	1	16	yes	complemented
38	C₄×Dic₅₃	—	2	8	no	—
39	C₄×Dic₅₃	C₂×C₄	1	4	yes	—
40	C₄×Dic₅₃	C₂×C₄	1	8	yes	—
41	C₄×Dic₅₃	C₂×C₄	1	8	yes	complemented
42	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
43	C₂²×D₅₃	—	2	4	no	—
44	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
45	C₂²×D₅₃	C₂²×C₄	1	4	yes	—
46	C₂×C₄×C₅₃	—	2	8	no	—
47	C₂×C₄×C₅₃	C₂²×C₄	1	8	yes	—
48	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
50	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
51	C₂×C₄×C₅₃	#41	1	8	yes	—
52	C₂×Dic₅₃	—	2	4	no	—
53	C₂×Dic₅₃	—	2	8	no	—
54	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
55	C₂×Dic₅₃	C₂²×C₄	1	8	yes	—
56	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
57	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
58	C₄×D₅₃	—	2	16	no	—
59	C₄×D₅₃	C₄²	1	16	yes	complemented
60	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
61	C₂²×C₅₃	C₂³×C₄	1	2	yes	—
62	C₂²×C₅₃	C₂×#41	1	4	yes	—
63	C₂×D₅₃	—	2	4	no	—
64	C₂×D₅₃	—	2	8	no	—
65	C₂×D₅₃	C₂×C₄²	1	4	yes	—
66	C₄×C₅₃	—	2	8	no	—
67	C₄×C₅₃	C₂×C₄²	1	4	yes	—
68	C₄×C₅₃	#117	1	8	yes	complemented
69	Dic₅₃	—	2	4	no	—
70	Dic₅₃	—	2	8	no	—
71	Dic₅₃	C₂×C₄²	1	4	yes	—
72	C₄×#117	—	53	1	no	maximal, Sylow
73	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
74	C₂×C₅₃	C₂×#117	1	2	yes	—
75	C₂×C₅₃	C₄×#41	1	4	yes	—
76	D₅₃	—	2	4	no	—
77	C₂×#117	—	53	2	no	—
78	C₂²×C₄²	—	53	1	no	—
79	C₄×C₄⋊C₄	—	53	4	no	—
80	C₄×C₂²⋊C₄	—	53	4	no	—
81	#376	—	53	2	no	—
82	C₄³	—	53	1	no	—
83	C₄³	D₅₃	1	1	yes	characteristic, complemented
84	C₅₃	C₄×#117	1	1	yes	characteristic, complemented, Sylow
85	C₂³×C₄	—	53	1	no	—
86	C₂³×C₄	—	53	2	no	—
87	#117	—	53	16	no	—
88	C₂×C₄⋊C₄	—	53	4	no	—
89	C₂×C₂²⋊C₄	—	53	4	no	—
90	C₂×C₄²	—	53	2	no	—
91	C₂×C₄²	—	53	1	no	—
92	C₂×C₄²	—	53	2	no	—
93	C₂×C₄²	—	53	8	no	—
94	C₂×C₄²	—	53	4	no	—
95	C₂×C₄²	C₂×D₅₃	1	4	yes	—
96	C₂×C₄²	C₂×D₅₃	1	2	yes	—
97	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
98	#95	—	53	8	no	—
99	C₂⁴	—	53	1	no	—
100	C₂²×C₄	—	53	1	no	—
101	C₂²×C₄	—	53	4	no	—
102	C₂²×C₄	—	53	8	no	—
103	C₂²×C₄	—	53	2	no	—
104	C₂²×C₄	—	53	4	no	—
105	C₂²×C₄	—	53	8	no	—
106	C₂²×C₄	—	106	8	no	—
107	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
108	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
109	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
110	C₄⋊C₄	—	53	16	no	—
111	C₂²⋊C₄	—	53	16	no	—
112	C₄²	—	2	8	no	—
113	C₄²	—	53	8	no	—
114	C₄²	—	53	4	no	—
115	C₄²	—	53	8	no	—
116	C₄²	—	106	8	no	—
117	C₄²	C₄×D₅₃	1	4	yes	—
118	C₄²	C₄×D₅₃	1	8	yes	complemented
119	C₂³	—	53	4	no	—
120	C₂³	—	53	2	no	—
121	C₂³	—	106	4	no	—
122	C₂³	C₂³×D₅₃	1	1	yes	characteristic
123	C₂×C₄	—	2	8	no	—
124	C₂×C₄	—	53	2	no	—
125	C₂×C₄	—	53	4	no	—
126	C₂×C₄	—	53	16	no	—
127	C₂×C₄	—	53	4	no	—
128	C₂×C₄	—	53	8	no	—
129	C₂×C₄	—	106	8	no	—
130	C₂×C₄	—	106	4	no	—
131	C₂×C₄	—	106	16	no	—
132	C₂×C₄	C₂×C₄×D₅₃	1	8	yes	—
133	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
134	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
135	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
136	C₂×C₄	#11719945	1	8	yes	—
137	C₂²	—	53	4	no	—
138	C₂²	—	106	4	no	—
139	C₂²	—	106	8	no	—
140	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
141	C₂²	C₂²×C₄×D₅₃	1	2	yes	—
142	C₂²	C₂×#11719945	1	4	yes	—
143	C₄	—	2	8	no	—
144	C₄	—	53	4	no	—
145	C₄	—	106	4	no	—
146	C₄	—	106	8	no	—
147	C₄	C₄²×D₅₃	1	4	yes	—
148	C₄	#13133174	1	8	yes	direct product factor
149	C₂	—	106	4	no	—
150	C₂	C₂×C₄²×D₅₃	1	1	yes	characteristic
151	C₂	C₂×#13133174	1	2	yes	—
152	C₂	C₄×#11719945	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133174

C₂

yes

maximal

C₂×C₄²×D₅₃

C₂

yes

maximal, characteristic

C₄×#11719919

C₂

yes

maximal, complemented

C₄×#11719921

C₂

yes

maximal

#16810469

C₂

yes

maximal, complemented

C₄²×Dic₅₃

C₂

yes

maximal, characteristic, complemented

C₄³×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₂²

yes

—

C₂²×C₄×D₅₃

C₂²

yes

characteristic

#13133174

C₄

yes

direct product factor

C₂×#11719919

C₂²

yes

—

C₂×#11719921

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₄²×D₅₃

C₄

yes

complemented

#13133124

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

complemented

#11719921

C₂×C₄

yes

—

#11719919

C₂×C₄

yes

complemented

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

complemented

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

C₂²×C₄

yes

—

C₂²×D₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₄×D₅₃

—

C₄×D₅₃

C₄²

yes

complemented

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂²×C₅₃

C₂³×C₄

yes

—

C₂²×C₅₃

C₂×#41

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂×C₄²

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

—

C₄×C₅₃

#117

yes

complemented

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×C₄²

yes

—

C₄×#117

—

maximal, Sylow

C₂×C₅₃

C₂²×C₄²

yes

characteristic

C₂×C₅₃

C₂×#117

yes

—

C₂×C₅₃

C₄×#41

yes

—

D₅₃

—

C₂×#117

—

C₂²×C₄²

—

C₄×C₄⋊C₄

—

C₄×C₂²⋊C₄

—

#376

—

C₄³

—

C₄³

D₅₃

yes

characteristic, complemented

C₅₃

C₄×#117

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#95

—

C₂⁴

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

106

—

107

C₂²×C₄

C₂²×D₅₃

yes

—

108

C₂²×C₄

C₂²×D₅₃

yes

characteristic

109

C₂²×C₄

C₂²×D₅₃

yes

—

110

C₄⋊C₄

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

106

—

117

C₄²

C₄×D₅₃

yes

—

118

C₄²

C₄×D₅₃

yes

complemented

119

C₂³

—

120

C₂³

—

121

C₂³

—

106

—

122

C₂³

C₂³×D₅₃

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

106

—

130

C₂×C₄

—

106

—

131

C₂×C₄

—

106

—

132

C₂×C₄

C₂×C₄×D₅₃

yes

—

133

C₂×C₄

C₂×C₄×D₅₃

yes

—

134

C₂×C₄

C₂×C₄×D₅₃

yes

—

135

C₂×C₄

C₂×C₄×D₅₃

yes

—

136

C₂×C₄

#11719945

yes

—

137

C₂²

—

138

C₂²

—

106

—

139

C₂²

—

106

—

140

C₂²

C₂²×C₄×D₅₃

yes

characteristic

141

C₂²

C₂²×C₄×D₅₃

yes

—

142

C₂²

C₂×#11719945

yes

—

143

C₄

—

144

C₄

—

145

C₄

—

106

—

146

C₄

—

106

—

147

C₄

C₄²×D₅₃

yes

—

148

C₄

#13133174

yes

direct product factor

149

C₂

—

106

—

150

C₂

C₂×C₄²×D₅₃

yes

characteristic

151

C₂

C₂×#13133174

yes

—

152

C₂

C₄×#11719945

yes

—

	subgroup	G/H
center	C₂×C₄²	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₄²
Frattini	C₂³	C₂³×D₅₃
Fitting	C₄³×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₄³	D₅₃
hypocenter	C₅₃	C₄×#117
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄³

subgroup

G/H

center

C₂×C₄²

C₂×D₅₃

commutator

C₂×C₅₃

C₂²×C₄²

Frattini

C₂³

C₂³×D₅₃

Fitting

C₄³×C₅₃

C₂

socle

C₂³×C₅₃

C₂⁴

hypercenter

C₄³

D₅₃

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄² ◁ C₄³

N	H ≅ G/N	type
3. C₄×#11719919	C₂	semidirect
5. #16810469	C₂	semidirect
6. C₄²×Dic₅₃	C₂	semidirect
7. C₄³×C₅₃	C₂	semidirect
10. #13133174	C₄	direct (N × H)
17. C₄²×D₅₃	C₄	semidirect
35. C₄²×C₅₃	C₂×C₄	semidirect
37. #11719919	C₂×C₄	semidirect
41. C₄×Dic₅₃	C₂×C₄	semidirect
59. C₄×D₅₃	C₄²	semidirect
68. C₄×C₅₃	#117	semidirect
83. C₄³	D₅₃	semidirect
84. C₅₃	C₄×#117	semidirect
118. C₄²	C₄×D₅₃	semidirect
148. C₄	#13133174	direct (N × H)

H ≅ G/N

type

3. C₄×#11719919

C₂

semidirect

5. #16810469

C₂

semidirect

6. C₄²×Dic₅₃

C₂

semidirect

7. C₄³×C₅₃

C₂

semidirect

10. #13133174

C₄

direct (N × H)

17. C₄²×D₅₃

C₄

semidirect

35. C₄²×C₅₃

C₂×C₄

semidirect

37. #11719919

C₂×C₄

semidirect

41. C₄×Dic₅₃

C₂×C₄

semidirect

59. C₄×D₅₃

C₄²

semidirect

68. C₄×C₅₃

#117

semidirect

83. C₄³

D₅₃

semidirect

84. C₅₃

C₄×#117

semidirect

118. C₄²

C₄×D₅₃

semidirect

148. C₄

#13133174

direct (N × H)

Subgroups — C₄×#13133174 #20277057

Conjugacy classes of subgroups

Complements

Subgroups — C4×#13133174 #20277057

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#13133174 #20277057