Group #20291376 — subgroups

Subgroups — #20291376

Order 6784.

440 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 198 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	#16811702	C₂	1	2	yes	maximal
2	#16811638	C₂	1	6	yes	maximal, complemented
3	D₅₃⋊#108	C₂	1	3	yes	maximal, complemented
4	C₂×#13133287	C₂	1	3	yes	maximal, complemented
5	#131×D₅₃	C₂	1	1	yes	maximal, characteristic
6	#13133305	—	2	2	no	—
7	#13133304	—	2	6	no	—
8	C₂×#11719936	C₂²	1	6	yes	—
9	D₅₃⋊M₁₆	C₂²	1	6	yes	—
10	#13133288	C₂²	1	6	yes	—
11	#13133287	C₂²	1	3	yes	—
12	#13133287	C₂²	1	6	yes	—
13	#13133287	C₂²	1	1	yes	characteristic
14	#41×D₅₃	C₂²	1	1	yes	characteristic
15	#13133225	C₄	1	3	yes	—
16	#13133224	C₄	1	1	yes	characteristic
17	#13133223	C₄	1	3	yes	—
18	M₁₆×D₅₃	C₂²	1	3	yes	—
19	C₂×C₈×D₅₃	C₂²	1	3	yes	—
20	#131×C₅₃	C₄	1	1	yes	characteristic
21	#41×C₅₃	C₂×C₄	1	1	yes	characteristic
22	#11719949	D₄	1	1	yes	characteristic
23	Q₈×D₅₃	Q₈	1	1	yes	characteristic
24	#11719947	D₄	1	3	yes	—
25	D₄×D₅₃	Q₈	1	3	yes	—
26	#11719945	C₂×C₄	1	3	yes	—
27	C₂×C₄×D₅₃	C₂³	1	3	yes	—
28	#11719940	—	2	6	no	—
29	C₂×#88324	—	2	6	no	—
30	#11719936	—	2	6	no	—
31	#11719936	C₂³	1	6	yes	—
32	#11719936	C₂³	1	2	yes	—
33	D₅₃⋊C₈	—	2	6	no	—
34	M₁₆×C₅₃	C₂×C₄	1	3	yes	—
35	C₂×C₈×C₅₃	C₂×C₄	1	3	yes	—
36	#11719917	C₂×C₄	1	3	yes	—
37	C₂×#88322	C₂×C₄	1	3	yes	—
38	#11719912	C₂×C₄	1	6	yes	—
39	C₈×D₅₃	C₂×C₄	1	3	yes	—
40	C₈×D₅₃	C₂×C₄	1	3	yes	—
41	C₈×D₅₃	C₂³	1	1	yes	characteristic
42	C₈×D₅₃	C₂³	1	3	yes	—
43	C₂²×D₅₃	C₂×Q₈	1	3	yes	—
44	Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	D₄×C₅₃	C₄⋊C₄	1	3	yes	—
46	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
47	C₂²⋊D₅₃	C₄⋊C₄	1	6	yes	—
48	C₂×Dic₅₃	C₂×D₄	1	3	yes	—
49	D₂₁₂	C₄⋊C₄	1	3	yes	—
50	C₄×D₅₃	C₂²×C₄	1	3	yes	—
51	C₄×D₅₃	C₂×D₄	1	3	yes	—
52	C₄×D₅₃	C₂×Q₈	1	3	yes	—
53	C₄×D₅₃	C₂⁴	1	1	yes	characteristic
54	Dic₁₀₆	C₄⋊C₄	1	3	yes	—
55	#88324	—	2	6	no	—
56	#88324	—	2	2	no	—
57	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₈×C₅₃	C₂²×C₄	1	3	yes	—
59	#88322	C₂²×C₄	1	1	yes	characteristic
60	#88322	C₂²×C₄	1	3	yes	—
61	C₂²×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
62	C₂×D₅₃	—	4	3	no	—
63	C₂×D₅₃	C₂×C₄⋊C₄	1	3	yes	—
64	C₂×D₅₃	C₂²×Q₈	1	1	yes	characteristic
65	C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
66	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
67	Dic₅₃	C₂×C₄⋊C₄	1	3	yes	—
68	Dic₅₃	C₂²×D₄	1	1	yes	characteristic
69	#2916	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	—	2	3	no	—
71	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
72	D₅₃	—	2	1	no	—
73	D₅₃	—	2	3	no	—
74	#568	—	53	2	no	—
75	C₂×#131	—	53	1	no	—
76	#430	—	53	6	no	—
77	C₂×#108	—	53	3	no	—
78	C₂×#108	—	53	3	no	—
79	C₅₃	#2916	1	1	yes	characteristic, complemented, Sylow
80	C₂×#41	—	53	1	no	—
81	#131	—	53	1	no	—
82	#131	—	53	3	no	—
83	#131	—	53	3	no	—
84	#131	—	106	6	no	—
85	#131	—	106	2	no	—
86	#131	#6157	1	1	yes	characteristic
87	C₂×M₁₆	—	53	3	no	—
88	C₂×M₁₆	—	53	6	no	—
89	C₂×M₁₆	—	53	6	no	—
90	C₂²×C₈	—	53	3	no	—
91	#108	—	53	3	no	—
92	#108	—	53	1	no	—
93	#108	—	53	6	no	—
94	#108	—	53	6	no	—
95	#41	—	53	1	no	—
96	#41	—	53	3	no	—
97	#41	—	53	3	no	—
98	#41	C₂×#6157	1	1	yes	characteristic
99	C₂×Q₈	—	53	1	no	—
100	C₂×D₄	—	53	3	no	—
101	C₂²×C₄	—	53	3	no	—
102	M₁₆	—	53	3	no	—
103	M₁₆	—	53	6	no	—
104	M₁₆	—	53	6	no	—
105	M₁₆	—	53	2	no	—
106	M₁₆	—	106	6	no	—
107	M₁₆	—	106	6	no	—
108	M₁₆	C₂×#6157	1	3	yes	—
109	C₂×C₈	—	53	3	no	—
110	C₂×C₈	—	53	1	no	—
111	C₂×C₈	—	53	3	no	—
112	C₂×C₈	—	53	3	no	—
113	C₂×C₈	—	53	3	no	—
114	C₂×C₈	—	106	6	no	—
115	C₂×C₈	—	106	6	no	—
116	C₂×C₈	C₂×#6157	1	3	yes	—
117	C₂³	—	53	3	no	—
118	Q₈	—	53	3	no	—
119	Q₈	C₄⋊#6157	1	1	yes	characteristic
120	D₄	—	53	6	no	—
121	D₄	—	53	3	no	—
122	D₄	C₄⋊#6157	1	3	yes	—
123	C₂×C₄	—	53	3	no	—
124	C₂×C₄	—	53	3	no	—
125	C₂×C₄	—	53	3	no	—
126	C₂×C₄	—	53	3	no	—
127	C₂×C₄	—	53	1	no	—
128	C₂×C₄	C₂²×#6157	1	3	yes	—
129	C₈	—	53	1	no	—
130	C₈	—	53	3	no	—
131	C₈	—	106	6	no	—
132	C₈	—	106	2	no	—
133	C₈	C₂²×#6157	1	1	yes	characteristic
134	C₈	C₂²×#6157	1	3	yes	—
135	C₂²	—	53	1	no	—
136	C₂²	—	53	3	no	—
137	C₂²	—	212	3	no	—
138	C₂²	C₂×C₄⋊#6157	1	3	yes	—
139	C₄	—	53	1	no	—
140	C₄	—	53	3	no	—
141	C₄	C₂×C₄⋊#6157	1	3	yes	—
142	C₄	C₂³×#6157	1	1	yes	characteristic
143	C₂	—	2	3	no	—
144	C₂	—	106	1	no	—
145	C₂	—	106	3	no	—
146	C₂	C₂²×C₄⋊#6157	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811702

C₂

yes

maximal

#16811638

C₂

yes

maximal, complemented

D₅₃⋊#108

C₂

yes

maximal, complemented

C₂×#13133287

C₂

yes

maximal, complemented

#131×D₅₃

C₂

yes

maximal, characteristic

#13133305

—

#13133304

—

C₂×#11719936

C₂²

yes

—

D₅₃⋊M₁₆

C₂²

yes

—

#13133288

C₂²

yes

—

#13133287

C₂²

yes

—

#13133287

C₂²

yes

—

#13133287

C₂²

yes

characteristic

#41×D₅₃

C₂²

yes

characteristic

#13133225

C₄

yes

—

#13133224

C₄

yes

characteristic

#13133223

C₄

yes

—

M₁₆×D₅₃

C₂²

yes

—

C₂×C₈×D₅₃

C₂²

yes

—

#131×C₅₃

C₄

yes

characteristic

#41×C₅₃

C₂×C₄

yes

characteristic

#11719949

D₄

yes

characteristic

Q₈×D₅₃

Q₈

yes

characteristic

#11719947

D₄

yes

—

D₄×D₅₃

Q₈

yes

—

#11719945

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

—

—

—

C₂³

yes

—

#11719936

C₂³

yes

—

D₅₃⋊C₈

—

M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

#11719917

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

#11719912

C₂×C₄

yes

—

C₈×D₅₃

C₂×C₄

yes

—

C₈×D₅₃

C₂×C₄

yes

—

C₈×D₅₃

C₂³

yes

characteristic

C₈×D₅₃

C₂³

yes

—

C₂²×D₅₃

C₂×Q₈

yes

—

Q₈×C₅₃

C₄⋊C₄

yes

characteristic

D₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂²⋊D₅₃

C₄⋊C₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

—

D₂₁₂

C₄⋊C₄

yes

—

C₄×D₅₃

C₂²×C₄

yes

—

C₄×D₅₃

C₂×D₄

yes

—

C₄×D₅₃

C₂×Q₈

yes

—

C₄×D₅₃

C₂⁴

yes

characteristic

Dic₁₀₆

C₄⋊C₄

yes

—

#88324

—

#88324

—

C₈×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

C₂²×C₄

yes

—

#88322

C₂²×C₄

yes

characteristic

#88322

C₂²×C₄

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×D₅₃

—

C₂×D₅₃

C₂×C₄⋊C₄

yes

—

C₂×D₅₃

C₂²×Q₈

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₄×C₅₃

C₂³×C₄

yes

characteristic

Dic₅₃

C₂×C₄⋊C₄

yes

—

Dic₅₃

C₂²×D₄

yes

characteristic

#2916

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂²×C₄⋊C₄

yes

characteristic

D₅₃

—

D₅₃

—

—

—

—

—

—

C₅₃

#2916

yes

characteristic, complemented, Sylow

C₂×#41

—

—

—

—

—

106

—

#131

—

106

—

#131

#6157

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

—

—

—

—

—

—

—

yes

C₂×Q₈

—

100

C₂×D₄

—

101

C₂²×C₄

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

106

—

107

M₁₆

—

106

—

108

M₁₆

C₂×#6157

yes

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

106

—

115

C₂×C₈

—

106

—

116

C₂×C₈

C₂×#6157

yes

—

117

C₂³

—

118

Q₈

—

119

Q₈

C₄⋊#6157

yes

characteristic

120

D₄

—

121

D₄

—

122

D₄

C₄⋊#6157

yes

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂²×#6157

yes

—

129

C₈

—

130

C₈

—

131

C₈

—

106

—

132

C₈

—

106

—

133

C₈

C₂²×#6157

yes

characteristic

134

C₈

C₂²×#6157

yes

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

212

—

138

C₂²

C₂×C₄⋊#6157

yes

—

139

C₄

—

140

C₄

—

141

C₄

C₂×C₄⋊#6157

yes

—

142

C₄

C₂³×#6157

yes

characteristic

143

C₂

—

144

C₂

—

106

—

145

C₂

—

106

—

146

C₂

C₂²×C₄⋊#6157

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₄⋊#6157
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₄	C₂³×#6157
Fitting	#131×C₅₃	C₄
socle	C₂×C₅₃	C₂²×C₄⋊C₄
hypercenter	#131	#6157
hypocenter	C₅₃	#2916
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

subgroup

G/H

center

C₂

C₂²×C₄⋊#6157

commutator

C₄×C₅₃

C₂³×C₄

Frattini

C₄

C₂³×#6157

Fitting

#131×C₅₃

C₄

socle

C₂×C₅₃

C₂²×C₄⋊C₄

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131

N	H ≅ G/N	type
2. #16811638	C₂	semidirect
3. D₅₃⋊#108	C₂	semidirect
4. C₂×#13133287	C₂	semidirect
79. C₅₃	#2916	semidirect

H ≅ G/N

type

2. #16811638

C₂

semidirect

3. D₅₃⋊#108

C₂

semidirect

4. C₂×#13133287

C₂

semidirect

79. C₅₃

#2916

semidirect

Subgroups — #20291376

Conjugacy classes of subgroups

Complements