Group #20276057 — subgroups

Subgroups — #20276057

Order 6784.

246 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 75 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	#16810666	C₂	1	2	yes	maximal
2	#16810492	C₂	1	1	yes	maximal, characteristic
3	#16810392	C₂	1	1	yes	maximal, characteristic
4	#16810382	C₂	1	2	yes	maximal
5	C₅₃×#544	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
8	C₂×#11719919	—	2	2	no	—
9	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
10	#13133150	—	2	2	no	—
11	#13133148	—	2	2	no	—
12	C₄⋊Q₈×C₅₃	—	2	2	no	—
13	#125×C₅₃	—	2	2	no	—
14	C₂²⋊Q₈×C₅₃	—	2	2	no	—
15	#117×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	2	2	no	—
18	C₂²×C₄×C₅₃	D₄	1	2	yes	—
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₅₃	—	2	1	no	—
21	C₂²×Dic₅₃	—	2	1	no	—
22	C₄⋊C₄×C₅₃	—	2	2	no	—
23	C₄⋊C₄×C₅₃	—	2	4	no	—
24	C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
26	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
27	C₂²⋊C₄×C₅₃	—	2	2	no	—
28	C₄²×C₅₃	—	2	2	no	—
29	C₄⋊Dic₅₃	—	2	2	no	—
30	#11719919	—	4	4	no	—
31	C₄×Dic₅₃	—	2	1	no	—
32	C₄×Dic₅₃	Q₈	1	2	yes	complemented
33	#11719917	—	2	2	no	—
34	#11719917	Q₈	1	4	yes	complemented
35	C₂×#88322	—	2	2	no	—
36	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₅₃	—	8	2	no	—
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	—	4	2	no	—
42	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
46	C₂×Dic₅₃	—	2	2	no	—
47	C₂×Dic₅₃	—	2	2	no	—
48	C₂×Dic₅₃	—	2	2	no	—
49	C₂×Dic₅₃	—	4	2	no	—
50	C₂×Dic₅₃	—	4	1	no	—
51	#88322	—	2	4	no	—
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	—	2	1	no	—
54	C₂²×C₅₃	#120	1	1	yes	characteristic
55	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
57	C₄×C₅₃	—	4	2	no	—
58	C₄×C₅₃	—	4	2	no	—
59	C₄×C₅₃	—	4	2	no	—
60	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
62	Dic₅₃	—	2	4	no	—
63	Dic₅₃	—	8	1	no	—
64	#2060	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	—	2	1	no	—
66	C₂×C₅₃	—	2	1	no	—
67	C₂×C₅₃	#395	1	1	yes	characteristic
68	C₂×C₅₃	#453	1	1	yes	characteristic
69	C₂×C₅₃	#456	1	1	yes	characteristic
70	#544	D₅₃	1	1	yes	characteristic
71	#428	—	53	2	no	—
72	#384	—	53	1	no	—
73	#343	—	53	1	no	—
74	#339	—	53	2	no	—
75	C₅₃	#2060	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	53	2	no	—
77	C₄⋊Q₈	—	2	2	no	—
78	#125	—	2	2	no	—
79	C₂²⋊Q₈	—	2	2	no	—
80	#117	C₂×D₅₃	1	2	yes	—
81	C₂×C₄⋊C₄	—	53	1	no	—
82	C₂×C₄⋊C₄	—	106	2	no	—
83	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
84	C₂×C₄²	—	53	1	no	—
85	#107	—	106	2	no	—
86	#106	—	106	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂²×C₄	—	106	1	no	—
89	C₂²×C₄	—	106	1	no	—
90	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
91	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
92	M₁₆	—	53	4	no	—
93	M₁₆	—	106	2	no	—
94	C₂×C₈	—	106	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	106	2	no	—
99	C₄⋊C₄	—	212	4	no	—
100	C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
101	C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
102	C₂²⋊C₄	—	2	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	53	2	no	—
105	C₄²	—	106	1	no	—
106	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
107	Q₈	—	8	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	106	2	no	—
113	C₂×C₄	—	106	2	no	—
114	C₂×C₄	—	106	2	no	—
115	C₂×C₄	—	212	2	no	—
116	C₂×C₄	—	212	1	no	—
117	C₂×C₄	Q₈×D₅₃	1	2	yes	—
118	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
119	C₂×C₄	#11719949	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
121	C₈	—	106	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	#13133186	1	1	yes	characteristic
125	C₂²	#13133193	1	1	yes	characteristic
126	C₂²	#13133215	1	1	yes	characteristic
127	C₄	—	4	2	no	—
128	C₄	—	4	2	no	—
129	C₄	—	4	2	no	—
130	C₄	—	106	4	no	—
131	C₄	—	424	1	no	—
132	C₄	#13133193	1	2	yes	—
133	C₄	#13133215	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#16810542	1	1	yes	characteristic
137	C₂	#16810791	1	1	yes	characteristic
138	C₂	#16810806	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810666

C₂

yes

maximal

#16810492

C₂

yes

maximal, characteristic

#16810392

C₂

yes

maximal, characteristic

#16810382

C₂

yes

maximal

C₅₃×#544

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×#11719919

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133150

—

#13133148

—

C₄⋊Q₈×C₅₃

—

#125×C₅₃

—

C₂²⋊Q₈×C₅₃

—

#117×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

Q₈

yes

complemented

#11719917

—

#11719917

Q₈

yes

complemented

C₂×#88322

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#120

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

C₄⋊Q₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

C₄⋊Q₈

yes

—

Dic₅₃

—

Dic₅₃

—

#2060

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

—

—

—

—

C₅₃

#2060

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₄²

—

#107

—

106

—

#106

—

106

—

C₂×Q₈

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

100

C₄⋊C₄

C₂²⋊D₅₃

yes

—

101

C₄⋊C₄

C₂²⋊D₅₃

yes

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

—

106

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

106

—

113

C₂×C₄

—

106

—

114

C₂×C₄

—

106

—

115

C₂×C₄

—

212

—

116

C₂×C₄

—

212

—

117

C₂×C₄

Q₈×D₅₃

yes

—

118

C₂×C₄

Q₈×D₅₃

yes

characteristic

119

C₂×C₄

#11719949

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

121

C₈

—

106

—

122

C₂²

—

123

C₂²

—

124

C₂²

#13133186

yes

characteristic

125

C₂²

#13133193

yes

characteristic

126

C₂²

#13133215

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

106

—

131

C₄

—

424

—

132

C₄

#13133193

yes

—

133

C₄

#13133215

yes

—

134

C₂

—

135

C₂

—

136

C₂

#16810542

yes

characteristic

137

C₂

#16810791

yes

characteristic

138

C₂

#16810806

yes

characteristic

	subgroup	G/H
center	C₂²	#13133186
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#544	C₂
socle	C₂²×C₅₃	#120
hypercenter	#544	D₅₃
hypocenter	C₅₃	#2060
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

subgroup

G/H

center

C₂²

#13133186

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#544

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

N	H ≅ G/N	type
32. C₄×Dic₅₃	Q₈	semidirect
34. #11719917	Q₈	semidirect
75. C₅₃	#2060	semidirect

H ≅ G/N

type

32. C₄×Dic₅₃

Q₈

semidirect

34. #11719917

Q₈

semidirect

75. C₅₃

#2060

semidirect

Subgroups — #20276057

Conjugacy classes of subgroups

Complements