Group #20275544 — subgroups

Subgroups — #20275544

Order 6784.

360 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 105 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	#16810647	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13133144	C₂	1	4	yes	maximal, complemented
3	C₄²×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#532	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	C₂²	1	2	yes	—
6	C₂×C₄×Dic₅₃	—	2	2	no	—
7	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
8	C₄⋊#88322	—	2	2	no	—
9	#13133144	—	2	8	no	—
10	C₂×#41×C₅₃	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
12	#127×C₅₃	C₄	1	1	yes	characteristic, complemented
13	#124×C₅₃	C₄	1	2	yes	complemented
14	#121×C₅₃	—	2	2	no	—
15	C₄×D₄×C₅₃	—	2	2	no	—
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	#41×C₅₃	—	2	4	no	—
18	#41×C₅₃	—	2	4	no	—
19	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₂×D₄×C₅₃	—	2	2	no	—
21	C₂×D₄×C₅₃	—	2	2	no	—
22	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₅₃	—	2	2	no	—
24	C₂²×C₄×C₅₃	D₄	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₅₃	—	2	2	no	—
27	C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₂²⋊C₄×C₅₃	—	2	4	no	—
29	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₅₃	D₄	1	2	yes	—
32	C₄×Dic₅₃	—	2	8	no	—
33	C₄×Dic₅₃	—	2	2	no	—
34	C₄×Dic₅₃	D₄	1	4	yes	complemented
35	#11719917	—	2	4	no	—
36	C₂×#88322	—	2	2	no	—
37	C₂³×C₅₃	—	2	2	no	—
38	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₅₃	—	2	4	no	—
40	D₄×C₅₃	—	2	4	no	—
41	D₄×C₅₃	—	4	4	no	—
42	D₄×C₅₃	—	4	4	no	—
43	C₂×C₄×C₅₃	—	2	4	no	—
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	4	2	no	—
46	C₂×C₄×C₅₃	—	4	2	no	—
47	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#41	1	2	yes	—
53	C₂×Dic₅₃	—	2	8	no	—
54	C₂×Dic₅₃	—	2	4	no	—
55	#88322	—	4	2	no	—
56	C₂²×C₅₃	—	2	2	no	—
57	C₂²×C₅₃	—	4	2	no	—
58	C₂²×C₅₃	—	4	4	no	—
59	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₅₃	#121	1	2	yes	—
61	C₄×C₅₃	—	2	2	no	—
62	C₄×C₅₃	—	4	2	no	—
63	C₄×C₅₃	#104	1	4	yes	complemented
64	C₄×C₅₃	C₄×D₄	1	2	yes	—
65	C₄×C₅₃	#124	1	1	yes	characteristic
66	C₄×C₅₃	#127	1	1	yes	characteristic
67	Dic₅₃	—	2	8	no	—
68	#1962	—	53	1	no	maximal, Sylow
69	C₂×C₅₃	—	2	2	no	—
70	C₂×C₅₃	—	8	2	no	—
71	C₂×C₅₃	#390	1	1	yes	characteristic
72	C₂×C₅₃	C₂×#104	1	2	yes	—
73	#532	D₅₃	1	1	yes	characteristic
74	#423	—	53	1	no	—
75	C₂×#104	—	53	4	no	—
76	C₄³	—	53	1	no	—
77	C₅₃	#1962	1	1	yes	characteristic, complemented, Sylow
78	C₂×#41	C₂×D₅₃	1	2	yes	—
79	C₂×M₁₆	—	53	2	no	—
80	C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic, complemented
81	#127	Dic₅₃	1	1	yes	characteristic, complemented
82	#124	Dic₅₃	1	2	yes	complemented
83	#121	—	2	2	no	—
84	C₄×D₄	—	2	2	no	—
85	C₂×C₄²	—	53	2	no	—
86	C₂×C₄²	—	106	2	no	—
87	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
88	C₄⋊C₈	—	106	2	no	—
89	#104	—	106	8	no	—
90	#41	—	2	4	no	—
91	#41	—	2	4	no	—
92	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	2	no	—
95	C₂×D₄	C₂×Dic₅₃	1	2	yes	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	106	2	no	—
98	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
99	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
100	M₁₆	—	106	4	no	—
101	C₂×C₈	—	106	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₂²⋊C₄	—	2	4	no	—
104	C₄²	—	53	4	no	—
105	C₄²	—	106	8	no	—
106	C₄²	—	106	2	no	—
107	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
108	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
109	C₄²	C₂²⋊D₅₃	1	2	yes	—
110	C₂³	—	2	2	no	—
111	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
112	Q₈	—	2	4	no	—
113	D₄	—	2	4	no	—
114	D₄	—	4	4	no	—
115	D₄	—	4	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	106	8	no	—
121	C₂×C₄	—	106	4	no	—
122	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
123	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
124	C₂×C₄	D₄×D₅₃	1	2	yes	—
125	C₂×C₄	#11719947	1	2	yes	—
126	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
128	C₈	—	212	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	4	2	no	—
131	C₂²	—	4	4	no	—
132	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
133	C₂²	#13133188	1	2	yes	—
134	C₄	—	2	2	no	—
135	C₄	—	4	2	no	—
136	C₄	—	106	8	no	—
137	C₄	#13133144	1	4	yes	complemented
138	C₄	D₄×Dic₅₃	1	2	yes	—
139	C₄	#13133204	1	1	yes	characteristic
140	C₄	#13133213	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	—	8	2	no	—
143	C₂	#16810526	1	1	yes	characteristic
144	C₂	C₂×#13133144	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810647

C₂

yes

maximal, characteristic, complemented

C₂×#13133144

C₂

yes

maximal, complemented

C₄²×Dic₅₃

C₂

yes

maximal, characteristic, complemented

C₅₃×#532

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₄⋊#88322

—

#13133144

—

C₂×#41×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₄

yes

characteristic, complemented

#127×C₅₃

C₄

yes

characteristic, complemented

#124×C₅₃

C₄

yes

complemented

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

complemented

#11719917

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#121

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#104

yes

complemented

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1962

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#390

yes

characteristic

C₂×C₅₃

C₂×#104

yes

—

#532

D₅₃

yes

characteristic

#423

—

C₂×#104

—

C₄³

—

C₅₃

#1962

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

—

C₂×M₁₆

—

C₄⋊Q₈

Dic₅₃

yes

characteristic, complemented

#127

Dic₅₃

yes

characteristic, complemented

#124

Dic₅₃

yes

complemented

#121

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

#104

—

106

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

100

M₁₆

—

106

—

101

C₂×C₈

—

106

—

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₄²

—

106

—

106

C₄²

—

106

—

107

C₄²

C₂×Dic₅₃

yes

characteristic

108

C₄²

C₂×Dic₅₃

yes

characteristic

109

C₄²

C₂²⋊D₅₃

yes

—

110

C₂³

—

111

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

112

Q₈

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

106

—

121

C₂×C₄

—

106

—

122

C₂×C₄

C₂²⋊Dic₅₃

yes

—

123

C₂×C₄

C₂²⋊Dic₅₃

yes

—

124

C₂×C₄

D₄×D₅₃

yes

—

125

C₂×C₄

#11719947

yes

—

126

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

127

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

128

C₈

—

212

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

133

C₂²

#13133188

yes

—

134

C₄

—

135

C₄

—

136

C₄

—

106

—

137

C₄

#13133144

yes

complemented

138

C₄

D₄×Dic₅₃

yes

—

139

C₄

#13133204

yes

characteristic

140

C₄

#13133213

yes

characteristic

141

C₂

—

142

C₂

—

143

C₂

#16810526

yes

characteristic

144

C₂

C₂×#13133144

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#532	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#532	D₅₃
hypocenter	C₅₃	#1962
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#532

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #16810647	C₂	semidirect
2. C₂×#13133144	C₂	semidirect
3. C₄²×Dic₅₃	C₂	semidirect
11. C₄⋊Q₈×C₅₃	C₄	semidirect
12. #127×C₅₃	C₄	semidirect
13. #124×C₅₃	C₄	semidirect
34. C₄×Dic₅₃	D₄	semidirect
63. C₄×C₅₃	#104	semidirect
77. C₅₃	#1962	semidirect
80. C₄⋊Q₈	Dic₅₃	semidirect
81. #127	Dic₅₃	semidirect
82. #124	Dic₅₃	semidirect
137. C₄	#13133144	semidirect

H ≅ G/N

type

1. #16810647

C₂

semidirect

2. C₂×#13133144

C₂

semidirect

3. C₄²×Dic₅₃

C₂

semidirect

11. C₄⋊Q₈×C₅₃

C₄

semidirect

12. #127×C₅₃

C₄

semidirect

13. #124×C₅₃

C₄

semidirect

34. C₄×Dic₅₃

D₄

semidirect

63. C₄×C₅₃

#104

semidirect

77. C₅₃

#1962

semidirect

80. C₄⋊Q₈

Dic₅₃

semidirect

81. #127

Dic₅₃

semidirect

82. #124

Dic₅₃

semidirect

137. C₄

#13133144

semidirect

Subgroups — #20275544

Conjugacy classes of subgroups

Complements