Group #20285224 — subgroups

Subgroups — #20285224

Order 6784.

356 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 145 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	#16811178	C₂	1	2	yes	maximal
2	#16811026	C₂	1	2	yes	maximal, complemented
3	#16811023	C₂	1	4	yes	maximal, complemented
4	#16811021	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810658	C₂	1	4	yes	maximal
6	C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₅₃	C₂²	1	2	yes	—
9	#13133217	C₂²	1	2	yes	—
10	#13133208	C₂²	1	4	yes	—
11	#13133197	—	2	2	no	—
12	#13133193	—	2	2	no	—
13	Q₈×Dic₅₃	—	2	2	no	—
14	#13133182	—	2	2	no	—
15	#13133175	C₂²	1	4	yes	—
16	C₄²×D₅₃	C₂²	1	1	yes	characteristic
17	#13133150	C₂²	1	8	yes	—
18	#13133149	C₂²	1	4	yes	—
19	C₄×#88322	C₂²	1	1	yes	characteristic
20	C₈×Dic₅₃	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
22	#106×C₅₃	C₂²	1	4	yes	—
23	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₅₃	—	2	2	no	—
25	C₂×C₄×D₅₃	D₄	1	2	yes	—
26	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₀₆	—	2	2	no	—
28	C₂×C₈×C₅₃	C₂³	1	2	yes	—
29	C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
31	C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	#11719921	—	2	4	no	—
33	C₄⋊Dic₅₃	—	2	2	no	—
34	C₄⋊Dic₅₃	—	2	4	no	—
35	C₄⋊Dic₅₃	C₂³	1	4	yes	—
36	#11719919	—	2	4	no	—
37	C₄×Dic₅₃	—	2	4	no	—
38	C₄×Dic₅₃	D₄	1	2	yes	—
39	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
40	C₂×#88322	C₂³	1	2	yes	—
41	C₈×D₅₃	Q₈	1	8	yes	complemented
42	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₅₃	—	4	4	no	—
44	C₂×C₄×C₅₃	—	2	4	no	—
45	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₅₃	—	2	4	no	—
48	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
49	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
50	C₄×D₅₃	—	2	4	no	—
51	C₄×D₅₃	C₂×Q₈	1	4	yes	—
52	Dic₁₀₆	—	4	4	no	—
53	C₈×C₅₃	C₂×Q₈	1	4	yes	—
54	#88322	C₂×Q₈	1	4	yes	—
55	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
56	C₂×D₅₃	—	2	2	no	—
57	C₂×D₅₃	C₄⋊Q₈	1	2	yes	—
58	C₄×C₅₃	—	4	4	no	—
59	C₄×C₅₃	#135	1	4	yes	—
60	C₄×C₅₃	C₂²×Q₈	1	2	yes	—
61	Dic₅₃	—	2	2	no	—
62	Dic₅₃	—	4	4	no	—
63	Dic₅₃	C₄⋊Q₈	1	2	yes	—
64	#3160	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₅₃	C₂×#135	1	2	yes	—
67	D₅₃	—	2	2	no	—
68	#533	—	53	2	no	—
69	#500	—	53	2	no	—
70	#499	—	53	4	no	—
71	#498	—	53	1	no	—
72	#498	D₅₃	1	1	yes	characteristic, complemented
73	#427	—	53	4	no	—
74	C₂×C₄×C₈	—	53	1	no	—
75	C₅₃	#3160	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₈	—	53	2	no	—
77	C₄⋊Q₈	—	53	2	no	—
78	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
79	#125	—	53	4	no	—
80	C₂²⋊Q₈	—	106	2	no	—
81	C₂²⋊Q₈	—	106	2	no	—
82	C₄×Q₈	—	106	2	no	—
83	C₄×Q₈	—	106	2	no	—
84	#117	—	53	4	no	—
85	C₂×C₄²	—	53	1	no	—
86	#107	—	53	8	no	—
87	#106	—	53	4	no	—
88	#106	C₂×D₅₃	1	4	yes	—
89	C₄×C₈	—	53	2	no	—
90	C₄×C₈	—	53	1	no	—
91	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	106	2	no	—
94	C₂²×C₄	—	53	1	no	—
95	C₂²×C₄	—	53	2	no	—
96	C₂×C₈	—	53	2	no	—
97	C₂×C₈	—	53	8	no	—
98	C₂×C₈	C₂²×D₅₃	1	2	yes	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	53	4	no	—
101	C₄⋊C₄	—	106	4	no	—
102	C₄⋊C₄	—	106	4	no	—
103	C₄⋊C₄	—	106	2	no	—
104	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
105	C₂²⋊C₄	—	106	4	no	—
106	C₄²	—	53	2	no	—
107	C₄²	—	53	1	no	—
108	C₄²	—	106	4	no	—
109	C₄²	C₂²×D₅₃	1	1	yes	characteristic
110	C₂³	—	53	1	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	212	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	53	1	no	—
115	C₂×C₄	—	53	4	no	—
116	C₂×C₄	—	53	2	no	—
117	C₂×C₄	—	106	4	no	—
118	C₂×C₄	—	106	4	no	—
119	C₂×C₄	D₄×D₅₃	1	2	yes	—
120	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
121	C₈	—	53	4	no	—
122	C₈	Q₈×D₅₃	1	4	yes	—
123	C₂²	—	53	2	no	—
124	C₂²	—	106	2	no	—
125	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
126	C₄	—	4	4	no	—
127	C₄	—	53	2	no	—
128	C₄	—	106	2	no	—
129	C₄	—	212	4	no	—
130	C₄	#13133238	1	4	yes	—
131	C₄	C₂×Q₈×D₅₃	1	2	yes	—
132	C₂	—	106	2	no	—
133	C₂	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
134	C₂	C₂×#13133238	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811178

C₂

yes

maximal

#16811026

C₂

yes

maximal, complemented

#16811023

C₂

yes

maximal, complemented

#16811021

C₂

yes

maximal, characteristic, complemented

#16810658

C₂

yes

maximal

C₄×C₈×D₅₃

C₂

yes

maximal, characteristic

C₅₃×#498

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₅₃

C₂²

yes

—

#13133217

C₂²

yes

—

#13133208

C₂²

yes

—

#13133197

—

#13133193

—

Q₈×Dic₅₃

—

#13133182

—

#13133175

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

#13133150

C₂²

yes

—

#13133149

C₂²

yes

—

C₄×#88322

C₂²

yes

characteristic

C₈×Dic₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

#106×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

—

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

—

C₈×D₅₃

Q₈

yes

complemented

C₂²×D₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

C₂×Q₈

yes

—

Dic₁₀₆

—

C₈×C₅₃

C₂×Q₈

yes

—

#88322

C₂×Q₈

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

C₄⋊Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

#135

yes

—

C₄×C₅₃

C₂²×Q₈

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₄⋊Q₈

yes

—

#3160

—

maximal, Sylow

C₂×C₅₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₅₃

C₂×#135

yes

—

D₅₃

—

—

—

—

—

D₅₃

yes

characteristic, complemented

#427

—

C₂×C₄×C₈

—

C₅₃

#3160

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

—

#125

—

C₂²⋊Q₈

—

106

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

106

—

C₄×Q₈

—

106

—

#117

—

C₂×C₄²

—

#107

—

#106

—

#106

C₂×D₅₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×D₅₃

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

106

—

102

C₄⋊C₄

—

106

—

103

C₄⋊C₄

—

106

—

104

C₄⋊C₄

C₂²×D₅₃

yes

—

105

C₂²⋊C₄

—

106

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

106

—

109

C₄²

C₂²×D₅₃

yes

characteristic

110

C₂³

—

111

Q₈

—

112

Q₈

—

212

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

106

—

118

C₂×C₄

—

106

—

119

C₂×C₄

D₄×D₅₃

yes

—

120

C₂×C₄

C₂³×D₅₃

yes

characteristic

121

C₈

—

122

C₈

Q₈×D₅₃

yes

—

123

C₂²

—

124

C₂²

—

106

—

125

C₂²

C₂×D₄×D₅₃

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

106

—

129

C₄

—

212

—

130

C₄

#13133238

yes

—

131

C₄

C₂×Q₈×D₅₃

yes

—

132

C₂

—

106

—

133

C₂

C₄⋊Q₈×D₅₃

yes

characteristic

134

C₂

C₂×#13133238

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#498	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#498	D₅₃
hypocenter	C₅₃	#3160
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#498

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #16811026	C₂	semidirect
3. #16811023	C₂	semidirect
4. #16811021	C₂	semidirect
7. C₅₃×#498	C₂	semidirect
41. C₈×D₅₃	Q₈	semidirect
72. #498	D₅₃	semidirect
75. C₅₃	#3160	semidirect

H ≅ G/N

type

2. #16811026

C₂

semidirect

3. #16811023

C₂

semidirect

4. #16811021

C₂

semidirect

7. C₅₃×#498

C₂

semidirect

41. C₈×D₅₃

Q₈

semidirect

72. #498

D₅₃

semidirect

75. C₅₃

#3160

semidirect

Subgroups — #20285224

Conjugacy classes of subgroups

Complements