Group #20272980 — subgroups

Subgroups — #20272980

Order 6784.

608 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 229 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×C₄⋊Dic₅₃	C₂	1	3	yes	maximal
2	C₂×#13133124	C₂	1	3	yes	maximal
3	C₂×#95×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₅₃	C₂²	1	3	yes	—
5	C₂³×Dic₅₃	C₂²	1	1	yes	characteristic
6	C₂×C₄⋊Dic₅₃	—	2	6	no	—
7	C₂×C₄⋊Dic₅₃	—	2	12	no	—
8	#13133124	—	2	6	no	—
9	C₂³×C₄×C₅₃	C₂²	1	3	yes	—
10	#95×C₅₃	C₄	1	4	yes	complemented
11	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	—	2	6	no	—
13	C₂²×C₄×C₅₃	—	2	6	no	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	6	yes	—
15	C₂²×C₄×C₅₃	D₄	1	6	yes	—
16	C₂²×C₄×C₅₃	Q₈	1	6	yes	—
17	C₂²×Dic₅₃	—	2	3	no	—
18	C₂²×Dic₅₃	—	2	3	no	—
19	C₂²×Dic₅₃	—	2	6	no	—
20	C₂²×Dic₅₃	—	2	6	no	—
21	C₂²×Dic₅₃	—	2	3	no	—
22	C₂²×Dic₅₃	—	2	3	no	—
23	C₂²×Dic₅₃	D₄	1	6	yes	—
24	C₂²×Dic₅₃	Q₈	1	2	yes	—
25	C₄⋊Dic₅₃	—	2	24	no	—
26	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
27	C₂³×C₅₃	C₂×D₄	1	3	yes	—
28	C₂³×C₅₃	C₂×D₄	1	3	yes	—
29	C₂³×C₅₃	C₂×Q₈	1	3	yes	—
30	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
31	C₂³×C₅₃	#41	1	3	yes	—
32	C₂³×C₅₃	#41	1	1	yes	characteristic
33	C₂×C₄×C₅₃	—	2	12	no	—
34	C₂×C₄×C₅₃	—	2	6	no	—
35	C₂×C₄×C₅₃	—	2	12	no	—
36	C₂×C₄×C₅₃	—	2	6	no	—
37	C₂×C₄×C₅₃	C₄⋊C₄	1	12	yes	complemented
38	C₂×Dic₅₃	—	2	12	no	—
39	C₂×Dic₅₃	—	2	6	no	—
40	C₂×Dic₅₃	—	2	6	no	—
41	C₂×Dic₅₃	—	4	3	no	—
42	C₂×Dic₅₃	—	4	6	no	—
43	C₂×Dic₅₃	—	4	3	no	—
44	C₂×Dic₅₃	—	4	3	no	—
45	C₂×Dic₅₃	—	4	1	no	—
46	C₂²×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
47	C₂²×C₅₃	C₄×D₄	1	3	yes	—
48	C₂²×C₅₃	C₄×Q₈	1	1	yes	characteristic
49	C₂²×C₅₃	#120	1	3	yes	—
50	C₂²×C₅₃	#121	1	3	yes	—
51	C₂²×C₅₃	C₂²⋊Q₈	1	6	yes	—
52	C₂²×C₅₃	C₂²⋊Q₈	1	3	yes	—
53	C₂²×C₅₃	#123	1	3	yes	—
54	C₂²×C₅₃	#125	1	3	yes	—
55	C₂²×C₅₃	#127	1	1	yes	characteristic
56	C₂²×C₅₃	C₄⋊Q₈	1	3	yes	—
57	C₂²×C₅₃	C₄⋊Q₈	1	3	yes	—
58	C₄×C₅₃	—	2	12	no	—
59	Dic₅₃	—	4	6	no	—
60	Dic₅₃	—	4	2	no	—
61	#1444	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	#384	1	1	yes	characteristic
63	C₂×C₅₃	#385	1	3	yes	—
64	C₂×C₅₃	#389	1	3	yes	—
65	C₂×C₅₃	#392	1	1	yes	characteristic
66	C₂×C₅₃	#395	1	3	yes	—
67	C₂×C₅₃	#398	1	3	yes	—
68	C₂×C₅₃	#399	1	1	yes	characteristic
69	C₂²×C₄⋊C₄	—	53	3	no	—
70	C₂×#95	—	53	3	no	—
71	C₂×#95	D₅₃	1	1	yes	characteristic
72	C₅₃	#1444	1	1	yes	characteristic, complemented, Sylow
73	C₂³×C₄	—	53	3	no	—
74	C₂³×C₄	—	53	1	no	—
75	C₂³×C₄	C₂×D₅₃	1	3	yes	—
76	C₂×C₄⋊C₄	—	106	12	no	—
77	C₂×C₄⋊C₄	—	106	6	no	—
78	#95	—	106	6	no	—
79	#95	Dic₅₃	1	4	yes	complemented
80	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
81	C₂²×C₄	—	2	6	no	—
82	C₂²×C₄	—	2	6	no	—
83	C₂²×C₄	—	53	6	no	—
84	C₂²×C₄	—	53	2	no	—
85	C₂²×C₄	—	106	3	no	—
86	C₂²×C₄	—	106	3	no	—
87	C₂²×C₄	—	106	6	no	—
88	C₂²×C₄	—	106	6	no	—
89	C₂²×C₄	—	106	3	no	—
90	C₂²×C₄	—	106	3	no	—
91	C₂²×C₄	Dic₁₀₆	1	6	yes	—
92	C₂²×C₄	D₂₁₂	1	6	yes	—
93	C₂²×C₄	C₂×Dic₅₃	1	6	yes	—
94	C₄⋊C₄	—	106	24	no	—
95	C₂³	C₂×Dic₁₀₆	1	3	yes	—
96	C₂³	C₂×D₂₁₂	1	3	yes	—
97	C₂³	D₄×D₅₃	1	3	yes	—
98	C₂³	#11719947	1	3	yes	—
99	C₂³	Q₈×D₅₃	1	1	yes	characteristic
100	C₂³	#11719949	1	1	yes	characteristic
101	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
102	C₂×C₄	—	2	12	no	—
103	C₂×C₄	—	2	6	no	—
104	C₂×C₄	—	2	12	no	—
105	C₂×C₄	—	2	6	no	—
106	C₂×C₄	—	106	12	no	—
107	C₂×C₄	—	106	6	no	—
108	C₂×C₄	—	106	6	no	—
109	C₂×C₄	—	212	3	no	—
110	C₂×C₄	—	212	6	no	—
111	C₂×C₄	—	212	3	no	—
112	C₂×C₄	—	212	3	no	—
113	C₂×C₄	—	212	1	no	—
114	C₂×C₄	C₄⋊Dic₅₃	1	12	yes	complemented
115	C₂²	C₂×C₄⋊Dic₅₃	1	3	yes	—
116	C₂²	D₄×Dic₅₃	1	3	yes	—
117	C₂²	Q₈×Dic₅₃	1	1	yes	characteristic
118	C₂²	#13133187	1	3	yes	—
119	C₂²	#13133192	1	3	yes	—
120	C₂²	#13133194	1	6	yes	—
121	C₂²	#13133197	1	3	yes	—
122	C₂²	#13133201	1	3	yes	—
123	C₂²	#13133208	1	3	yes	—
124	C₂²	#13133214	1	1	yes	characteristic
125	C₂²	#13133216	1	3	yes	—
126	C₂²	#13133217	1	3	yes	—
127	C₄	—	2	12	no	—
128	C₄	—	212	6	no	—
129	C₄	—	212	2	no	—
130	C₂	#16810493	1	1	yes	characteristic
131	C₂	#16810498	1	3	yes	—
132	C₂	#16810522	1	3	yes	—
133	C₂	#16810533	1	1	yes	characteristic
134	C₂	#16810544	1	3	yes	—
135	C₂	#16810555	1	3	yes	—
136	C₂	#16810560	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₅₃

C₂

yes

maximal

C₂×#13133124

C₂

yes

maximal

C₂×#95×C₅₃

C₂

yes

maximal, characteristic

C₂³×Dic₅₃

C₂²

yes

—

C₂³×Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

#13133124

—

C₂³×C₄×C₅₃

C₂²

yes

—

#95×C₅₃

C₄

yes

complemented

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

Q₈

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

—

C₂²×Dic₅₃

Q₈

yes

—

C₄⋊Dic₅₃

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂³×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

C₂³×C₅₃

#41

yes

—

C₂³×C₅₃

#41

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

complemented

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

—

C₂²×C₅₃

C₄×D₄

yes

—

C₂²×C₅₃

C₄×Q₈

yes

characteristic

C₂²×C₅₃

#120

yes

—

C₂²×C₅₃

#121

yes

—

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₂²×C₅₃

#123

yes

—

C₂²×C₅₃

#125

yes

—

C₂²×C₅₃

#127

yes

characteristic

C₂²×C₅₃

C₄⋊Q₈

yes

—

C₂²×C₅₃

C₄⋊Q₈

yes

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#389

yes

—

C₂×C₅₃

#392

yes

characteristic

C₂×C₅₃

#395

yes

—

C₂×C₅₃

#398

yes

—

C₂×C₅₃

#399

yes

characteristic

C₂²×C₄⋊C₄

—

C₂×#95

—

C₂×#95

D₅₃

yes

characteristic

C₅₃

#1444

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

—

106

—

#95

—

106

—

#95

Dic₅₃

yes

complemented

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

Dic₁₀₆

yes

—

C₂²×C₄

D₂₁₂

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₄⋊C₄

—

106

—

C₂³

C₂×Dic₁₀₆

yes

—

C₂³

C₂×D₂₁₂

yes

—

C₂³

D₄×D₅₃

yes

—

C₂³

#11719947

yes

—

C₂³

Q₈×D₅₃

yes

characteristic

100

C₂³

#11719949

yes

characteristic

101

C₂³

C₂²×Dic₅₃

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

106

—

107

C₂×C₄

—

106

—

108

C₂×C₄

—

106

—

109

C₂×C₄

—

212

—

110

C₂×C₄

—

212

—

111

C₂×C₄

—

212

—

112

C₂×C₄

—

212

—

113

C₂×C₄

—

212

—

114

C₂×C₄

C₄⋊Dic₅₃

yes

complemented

115

C₂²

C₂×C₄⋊Dic₅₃

yes

—

116

C₂²

D₄×Dic₅₃

yes

—

117

C₂²

Q₈×Dic₅₃

yes

characteristic

118

C₂²

#13133187

yes

—

119

C₂²

#13133192

yes

—

120

C₂²

#13133194

yes

—

121

C₂²

#13133197

yes

—

122

C₂²

#13133201

yes

—

123

C₂²

#13133208

yes

—

124

C₂²

#13133214

yes

characteristic

125

C₂²

#13133216

yes

—

126

C₂²

#13133217

yes

—

127

C₄

—

128

C₄

—

212

—

129

C₄

—

212

—

130

C₂

#16810493

yes

characteristic

131

C₂

#16810498

yes

—

132

C₂

#16810522

yes

—

133

C₂

#16810533

yes

characteristic

134

C₂

#16810544

yes

—

135

C₂

#16810555

yes

—

136

C₂

#16810560

yes

characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₅₃
commutator	C₂³×C₅₃	C₂²×C₄
Fitting	C₂×#95×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂³
hypercenter	C₂×#95	D₅₃
hypocenter	C₅₃	#1444
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#95

subgroup

G/H

center, Frattini

C₂⁴

C₂²×D₅₃

commutator

C₂³×C₅₃

C₂²×C₄

Fitting

C₂×#95×C₅₃

C₂

socle

C₂⁴×C₅₃

C₂³

hypercenter

C₂×#95

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂⁴ ◁ C₂×#95

N	H ≅ G/N	type
10. #95×C₅₃	C₄	semidirect
37. C₂×C₄×C₅₃	C₄⋊C₄	semidirect
72. C₅₃	#1444	semidirect
79. #95	Dic₅₃	semidirect
114. C₂×C₄	C₄⋊Dic₅₃	semidirect

H ≅ G/N

type

10. #95×C₅₃

C₄

semidirect

37. C₂×C₄×C₅₃

C₄⋊C₄

semidirect

72. C₅₃

#1444

semidirect

79. #95

Dic₅₃

semidirect

114. C₂×C₄

C₄⋊Dic₅₃

semidirect

Subgroups — #20272980

Conjugacy classes of subgroups

Complements