Group #20275547 — subgroups

Subgroups — #20275547

Order 6784.

320 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 93 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	#16811173	C₂	1	1	yes	maximal, characteristic
2	#16810647	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13133145	C₂	1	4	yes	maximal, complemented
4	C₅₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11719945	C₂²	1	2	yes	—
6	C₂×#11719917	C₂²	1	2	yes	—
7	#13133217	C₄	1	1	yes	characteristic, complemented
8	#13133214	C₄	1	1	yes	characteristic, complemented
9	#13133205	C₄	1	2	yes	complemented
10	#13133191	—	2	2	no	—
11	C₄×D₂₁₂	—	2	2	no	—
12	C₄⋊#88322	—	2	2	no	—
13	#13133145	—	2	8	no	—
14	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	#95×C₅₃	—	2	2	no	—
17	C₂²×C₄×C₅₃	—	2	2	no	—
18	C₂²×C₄×C₅₃	D₄	1	2	yes	—
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂×C₂²⋊D₅₃	—	2	2	no	—
21	#11719945	—	2	4	no	—
22	#11719945	—	2	4	no	—
23	C₂×D₂₁₂	—	2	2	no	—
24	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₅₃	—	2	2	no	—
26	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
27	C₄²×C₅₃	—	2	2	no	—
28	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₅₃	D₄	1	2	yes	—
31	C₄²×C₅₃	D₄	1	4	yes	complemented
32	#11719921	—	2	4	no	—
33	C₄⋊Dic₅₃	—	2	2	no	—
34	#11719917	—	2	4	no	—
35	C₂×#88322	—	2	2	no	—
36	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×D₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	—	2	4	no	—
39	C₂×C₄×C₅₃	—	4	2	no	—
40	C₂×C₄×C₅₃	—	4	4	no	—
41	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#41	1	2	yes	—
47	C₂²⋊D₅₃	—	4	4	no	—
48	C₂×Dic₅₃	—	2	2	no	—
49	D₂₁₂	—	2	4	no	—
50	D₂₁₂	—	4	4	no	—
51	C₄×D₅₃	—	4	2	no	—
52	C₄×D₅₃	—	4	2	no	—
53	Dic₁₀₆	—	2	4	no	—
54	#88322	—	4	2	no	—
55	C₂²×C₅₃	—	2	2	no	—
56	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₅₃	#121	1	2	yes	—
58	C₂×D₅₃	—	4	2	no	—
59	C₂×D₅₃	—	4	4	no	—
60	C₄×C₅₃	—	2	2	no	—
61	C₄×C₅₃	—	2	2	no	—
62	C₄×C₅₃	—	4	4	no	—
63	C₄×C₅₃	C₄×D₄	1	2	yes	—
64	C₄×C₅₃	#124	1	1	yes	characteristic
65	C₄×C₅₃	#127	1	1	yes	characteristic
66	Dic₅₃	—	4	2	no	—
67	#1963	—	53	1	no	maximal, Sylow
68	C₂×C₅₃	—	2	2	no	—
69	C₂×C₅₃	#390	1	1	yes	characteristic
70	C₂×C₅₃	#421	1	2	yes	—
71	D₅₃	—	8	2	no	—
72	#532	—	53	1	no	—
73	#423	—	53	1	no	—
74	C₂×#104	—	53	4	no	—
75	#376	D₅₃	1	1	yes	characteristic, complemented
76	C₅₃	#1963	1	1	yes	characteristic, complemented, Sylow
77	C₂×#41	—	53	2	no	—
78	C₂×M₁₆	—	53	2	no	—
79	C₄⋊Q₈	—	53	1	no	—
80	#127	—	53	1	no	—
81	#124	—	53	2	no	—
82	#121	—	106	2	no	—
83	C₄×D₄	—	106	2	no	—
84	C₂×C₄²	C₂×D₅₃	1	2	yes	—
85	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
86	C₄⋊C₈	—	106	2	no	—
87	#104	—	106	8	no	—
88	#95	—	2	2	no	—
89	#41	—	106	4	no	—
90	#41	—	106	4	no	—
91	C₂×Q₈	—	53	2	no	—
92	C₂×D₄	—	53	2	no	—
93	C₂×D₄	—	106	2	no	—
94	C₂×D₄	—	106	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	106	2	no	—
97	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
98	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
99	M₁₆	—	106	4	no	—
100	C₂×C₈	—	106	2	no	—
101	C₄⋊C₄	—	106	2	no	—
102	C₂²⋊C₄	—	106	4	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₄×D₅₃	1	1	yes	characteristic
105	C₄²	C₄×D₅₃	1	1	yes	characteristic
106	C₄²	D₂₁₂	1	2	yes	—
107	C₄²	D₂₁₂	1	4	yes	complemented
108	C₂³	—	106	2	no	—
109	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
110	Q₈	—	106	4	no	—
111	D₄	—	106	4	no	—
112	D₄	—	212	4	no	—
113	D₄	—	212	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	106	2	no	—
118	C₂×C₄	—	212	2	no	—
119	C₂×C₄	—	212	2	no	—
120	C₂×C₄	#11719921	1	2	yes	—
121	C₂×C₄	#11719921	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
123	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
124	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
125	C₂×C₄	#11719945	1	2	yes	—
126	C₈	—	212	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	212	2	no	—
129	C₂²	—	212	4	no	—
130	C₂²	C₂×#11719921	1	1	yes	characteristic
131	C₂²	#13133191	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	4	no	—
135	C₄	—	212	2	no	—
136	C₄	C₄×D₂₁₂	1	2	yes	—
137	C₄	#13133205	1	1	yes	characteristic
138	C₄	#13133214	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	424	2	no	—
141	C₂	#16810525	1	1	yes	characteristic
142	C₂	#16810641	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811173

C₂

yes

maximal, characteristic

#16810647

C₂

yes

maximal, characteristic, complemented

C₂×#13133145

C₂

yes

maximal, complemented

C₅₃×#376

C₂

yes

maximal, characteristic, complemented

C₂×#11719945

C₂²

yes

—

C₂×#11719917

C₂²

yes

—

#13133217

C₄

yes

characteristic, complemented

#13133214

C₄

yes

characteristic, complemented

#13133205

C₄

yes

complemented

#13133191

—

C₄×D₂₁₂

—

C₄⋊#88322

—

#13133145

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

—

#11719945

—

#11719945

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

D₄

yes

complemented

#11719921

—

C₄⋊Dic₅₃

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

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

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#121

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1963

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#390

yes

characteristic

C₂×C₅₃

#421

yes

—

D₅₃

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#1963

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

106

—

C₄×D₄

—

106

—

C₂×C₄²

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

#104

—

106

—

#95

—

#41

—

106

—

#41

—

106

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

106

—

100

C₂×C₈

—

106

—

101

C₄⋊C₄

—

106

—

102

C₂²⋊C₄

—

106

—

103

C₄²

—

104

C₄²

C₄×D₅₃

yes

characteristic

105

C₄²

C₄×D₅₃

yes

characteristic

106

C₄²

D₂₁₂

yes

—

107

C₄²

D₂₁₂

yes

complemented

108

C₂³

—

106

—

109

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

110

Q₈

—

106

—

111

D₄

—

106

—

112

D₄

—

212

—

113

D₄

—

212

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

106

—

118

C₂×C₄

—

212

—

119

C₂×C₄

—

212

—

120

C₂×C₄

#11719921

yes

—

121

C₂×C₄

#11719921

yes

—

122

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

123

C₂×C₄

C₂×D₂₁₂

yes

—

124

C₂×C₄

C₂×D₂₁₂

yes

characteristic

125

C₂×C₄

#11719945

yes

—

126

C₈

—

212

—

127

C₂²

—

128

C₂²

—

212

—

129

C₂²

—

212

—

130

C₂²

C₂×#11719921

yes

characteristic

131

C₂²

#13133191

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

212

—

136

C₄

C₄×D₂₁₂

yes

—

137

C₄

#13133205

yes

characteristic

138

C₄

#13133214

yes

characteristic

139

C₂

—

140

C₂

—

424

—

141

C₂

#16810525

yes

characteristic

142

C₂

#16810641

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#376	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#376	D₅₃
hypocenter	C₅₃	#1963
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#376

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
2. #16810647	C₂	semidirect
3. C₂×#13133145	C₂	semidirect
4. C₅₃×#376	C₂	semidirect
7. #13133217	C₄	semidirect
8. #13133214	C₄	semidirect
9. #13133205	C₄	semidirect
31. C₄²×C₅₃	D₄	semidirect
75. #376	D₅₃	semidirect
76. C₅₃	#1963	semidirect
107. C₄²	D₂₁₂	semidirect

H ≅ G/N

type

2. #16810647

C₂

semidirect

3. C₂×#13133145

C₂

semidirect

4. C₅₃×#376

C₂

semidirect

7. #13133217

C₄

semidirect

8. #13133214

C₄

semidirect

9. #13133205

C₄

semidirect

31. C₄²×C₅₃

D₄

semidirect

75. #376

D₅₃

semidirect

76. C₅₃

#1963

semidirect

107. C₄²

D₂₁₂

semidirect

Subgroups — #20275547

Conjugacy classes of subgroups

Complements