Group #20283846 — subgroups

Subgroups — #20283846

Order 6784.

394 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 179 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	C₄×#11719949	C₂	1	1	yes	maximal, characteristic
2	Q₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810702	C₂	1	3	yes	maximal, complemented
4	#16810700	C₂	1	3	yes	maximal, complemented
5	#16810674	C₂	1	3	yes	maximal, complemented
6	#16810645	C₂	1	3	yes	maximal, complemented
7	C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719949	C₄	1	1	yes	characteristic
9	C₂×#11719912	C₂²	1	6	yes	—
10	Q₈×Dic₅₃	C₄	1	1	yes	characteristic
11	C₄×D₂₁₂	C₂²	1	3	yes	—
12	#13133179	C₄	1	3	yes	—
13	#13133175	C₄	1	3	yes	—
14	C₄²×D₅₃	C₂²	1	3	yes	—
15	C₄⋊#88322	C₂²	1	3	yes	—
16	#13133146	C₂²	1	3	yes	—
17	#13133146	C₂²	1	1	yes	characteristic
18	#13133129	C₂²	1	3	yes	—
19	#13133129	C₂²	1	3	yes	—
20	C₄×#88322	C₂²	1	3	yes	—
21	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₃	C₂²	1	3	yes	—
23	C₄×C₈×C₅₃	C₂²	1	3	yes	—
24	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	#11719949	—	2	4	no	—
26	C₂×D₂₁₂	C₂×C₄	1	3	yes	—
27	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	—
28	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	—
29	C₂×C₄×D₅₃	C₂³	1	3	yes	—
30	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₂³	1	3	yes	—
32	C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
33	C₄²×C₅₃	C₂³	1	3	yes	—
34	#11719921	C₂×C₄	1	6	yes	—
35	C₄⋊Dic₅₃	C₂×C₄	1	3	yes	—
36	C₄×Dic₅₃	C₂×C₄	1	3	yes	—
37	C₄×Dic₅₃	C₂×C₄	1	3	yes	—
38	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
39	C₂×#88322	C₂³	1	1	yes	characteristic
40	C₂×#88322	C₂³	1	3	yes	—
41	#11719912	—	4	6	no	—
42	C₂²×D₅₃	C₂²×C₄	1	3	yes	—
43	Q₈×C₅₃	M₁₆	1	4	yes	complemented
44	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₅₃	C₂²×C₄	1	3	yes	—
49	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
50	D₂₁₂	—	2	6	no	—
51	C₄×D₅₃	—	2	6	no	—
52	C₄×D₅₃	—	2	3	no	—
53	C₄×D₅₃	—	2	3	no	—
54	C₄×D₅₃	—	2	6	no	—
55	C₈×C₅₃	—	2	3	no	—
56	C₈×C₅₃	—	2	1	no	—
57	#88322	—	2	3	no	—
58	#88322	—	2	1	no	—
59	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
60	C₂×D₅₃	—	2	3	no	—
61	C₂×D₅₃	—	2	3	no	—
62	C₂×D₅₃	—	2	3	no	—
63	C₄×C₅₃	—	2	3	no	—
64	C₄×C₅₃	C₂×M₁₆	1	6	yes	—
65	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
66	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
67	Dic₅₃	—	2	3	no	—
68	Dic₅₃	—	2	1	no	—
69	Dic₅₃	#131	1	2	yes	—
70	#2991	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	#520	1	1	yes	characteristic
72	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
73	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
74	D₅₃	—	4	3	no	—
75	C₄×#41	—	53	1	no	—
76	C₈×Q₈	—	53	1	no	—
77	C₈×Q₈	D₅₃	1	1	yes	characteristic, complemented
78	#436	—	53	3	no	—
79	#436	—	53	3	no	—
80	#431	—	53	3	no	—
81	#423	—	53	3	no	—
82	C₅₃	#2991	1	1	yes	characteristic, complemented, Sylow
83	C₂×#41	—	53	1	no	—
84	C₂×M₁₆	—	53	6	no	—
85	C₄×Q₈	—	53	1	no	—
86	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
87	C₄×D₄	—	53	3	no	—
88	C₄×D₄	—	53	3	no	—
89	#117	—	53	3	no	—
90	C₂×C₄²	—	53	3	no	—
91	C₄⋊C₈	—	53	3	no	—
92	C₄⋊C₈	—	53	1	no	—
93	C₄⋊C₈	—	53	3	no	—
94	C₄⋊C₈	C₂×D₅₃	1	3	yes	—
95	C₂²⋊C₈	—	53	3	no	—
96	C₂²⋊C₈	—	53	3	no	—
97	C₄×C₈	—	53	3	no	—
98	C₄×C₈	C₂×D₅₃	1	3	yes	—
99	#41	—	106	4	no	—
100	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
101	C₂×D₄	—	53	3	no	—
102	C₂²×C₄	—	53	3	no	—
103	C₂²×C₄	—	53	3	no	—
104	C₂²×C₄	—	53	3	no	—
105	M₁₆	—	212	6	no	—
106	C₂×C₈	—	53	1	no	—
107	C₂×C₈	—	53	3	no	—
108	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₅₃	1	3	yes	—
110	C₄⋊C₄	—	53	3	no	—
111	C₄⋊C₄	C₄×D₅₃	1	3	yes	—
112	C₂²⋊C₄	—	53	6	no	—
113	C₄²	—	53	3	no	—
114	C₄²	—	53	3	no	—
115	C₄²	—	53	1	no	—
116	C₄²	C₂²×D₅₃	1	3	yes	—
117	C₂³	—	53	3	no	—
118	Q₈	#11719912	1	4	yes	complemented
119	D₄	—	106	6	no	—
120	C₂×C₄	—	53	1	no	—
121	C₂×C₄	—	53	1	no	—
122	C₂×C₄	—	53	3	no	—
123	C₂×C₄	—	106	6	no	—
124	C₂×C₄	—	106	6	no	—
125	C₂×C₄	—	106	3	no	—
126	C₂×C₄	—	106	3	no	—
127	C₂×C₄	C₂×C₄×D₅₃	1	3	yes	—
128	C₂×C₄	C₂×C₄×D₅₃	1	3	yes	—
129	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
130	C₈	—	2	3	no	—
131	C₈	—	2	1	no	—
132	C₈	—	106	3	no	—
133	C₈	—	106	1	no	—
134	C₂²	—	106	3	no	—
135	C₂²	—	106	3	no	—
136	C₂²	—	106	3	no	—
137	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	53	2	no	—
140	C₄	—	106	3	no	—
141	C₄	—	106	1	no	—
142	C₄	C₂×#11719912	1	6	yes	—
143	C₄	#13133263	1	1	yes	characteristic
144	C₄	#13133265	1	1	yes	characteristic
145	C₂	—	212	3	no	—
146	C₂	#16811105	1	1	yes	characteristic
147	C₂	C₂²×#11719912	1	1	yes	characteristic
148	C₂	#131×D₅₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#11719949

C₂

yes

maximal, characteristic

Q₈×#88322

C₂

yes

maximal, characteristic, complemented

#16810702

C₂

yes

maximal, complemented

#16810700

C₂

yes

maximal, complemented

#16810674

C₂

yes

maximal, complemented

#16810645

C₂

yes

maximal, complemented

C₈×Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#11719949

C₄

yes

characteristic

C₂×#11719912

C₂²

yes

—

Q₈×Dic₅₃

C₄

yes

characteristic

C₄×D₂₁₂

C₂²

yes

—

#13133179

C₄

yes

—

#13133175

C₄

yes

—

C₄²×D₅₃

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

#13133146

C₂²

yes

—

#13133146

C₂²

yes

characteristic

#13133129

C₂²

yes

—

#13133129

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

#11719949

—

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂³

yes

—

#11719921

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

—

#11719912

—

C₂²×D₅₃

C₂²×C₄

yes

—

Q₈×C₅₃

M₁₆

yes

complemented

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

characteristic

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

H(5,𝔽₂)

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#131

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂²×M₁₆

yes

characteristic

C₂×C₅₃

C₂×#131

yes

characteristic

D₅₃

—

C₄×#41

—

C₈×Q₈

—

C₈×Q₈

D₅₃

yes

characteristic, complemented

—

—

—

—

C₅₃

#2991

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

C₂×D₅₃

yes

characteristic

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

—

#41

—

106

—

100

C₂×Q₈

C₄×D₅₃

yes

characteristic

101

C₂×D₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

M₁₆

—

212

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²×D₅₃

yes

characteristic

109

C₂×C₈

C₂²×D₅₃

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₄×D₅₃

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

C₂²×D₅₃

yes

—

117

C₂³

—

118

Q₈

#11719912

yes

complemented

119

D₄

—

106

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

106

—

124

C₂×C₄

—

106

—

125

C₂×C₄

—

106

—

126

C₂×C₄

—

106

—

127

C₂×C₄

C₂×C₄×D₅₃

yes

—

128

C₂×C₄

C₂×C₄×D₅₃

yes

—

129

C₂×C₄

C₂³×D₅₃

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

106

—

133

C₈

—

106

—

134

C₂²

—

106

—

135

C₂²

—

106

—

136

C₂²

—

106

—

137

C₂²

C₂²×C₄×D₅₃

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

106

—

141

C₄

—

106

—

142

C₄

C₂×#11719912

yes

—

143

C₄

#13133263

yes

characteristic

144

C₄

#13133265

yes

characteristic

145

C₂

—

212

—

146

C₂

#16811105

yes

characteristic

147

C₂

C₂²×#11719912

yes

characteristic

148

C₂

#131×D₅₃

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₈×Q₈×C₅₃	C₂
hypercenter	C₈×Q₈	D₅₃
hypocenter	C₅₃	#2991
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₃

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

C₈×Q₈×C₅₃

C₂

hypercenter

C₈×Q₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₈×Q₈

N	H ≅ G/N	type
2. Q₈×#88322	C₂	semidirect
3. #16810702	C₂	semidirect
4. #16810700	C₂	semidirect
5. #16810674	C₂	semidirect
6. #16810645	C₂	semidirect
7. C₈×Q₈×C₅₃	C₂	semidirect
43. Q₈×C₅₃	M₁₆	semidirect
77. C₈×Q₈	D₅₃	semidirect
82. C₅₃	#2991	semidirect
118. Q₈	#11719912	semidirect

H ≅ G/N

type

2. Q₈×#88322

C₂

semidirect

3. #16810702

C₂

semidirect

4. #16810700

C₂

semidirect

5. #16810674

C₂

semidirect

6. #16810645

C₂

semidirect

7. C₈×Q₈×C₅₃

C₂

semidirect

43. Q₈×C₅₃

M₁₆

semidirect

77. C₈×Q₈

D₅₃

semidirect

82. C₅₃

#2991

semidirect

118. Q₈

#11719912

semidirect

Subgroups — #20283846

Conjugacy classes of subgroups

Complements