Group #20273423 — subgroups

Subgroups — #20273423

Order 6784.

198 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 73 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	#16811180	C₂	1	1	yes	maximal, characteristic
2	#16810680	C₂	1	1	yes	maximal, characteristic
3	#16810371	C₂	1	2	yes	maximal, complemented
4	#16810368	C₂	1	2	yes	maximal, complemented
5	C₅₃×#431	C₂	1	1	yes	maximal, characteristic
6	#13133217	C₂²	1	1	yes	characteristic
7	#13133217	C₂²	1	1	yes	characteristic
8	#13133209	—	2	1	no	—
9	#13133196	—	2	1	no	—
10	#13133196	C₄	1	2	yes	—
11	C₄×Dic₁₀₆	—	2	1	no	—
12	C₄×Dic₁₀₆	C₄	1	2	yes	—
13	#13133176	—	2	1	no	—
14	C₄⋊#88322	C₂²	1	2	yes	—
15	C₂²⋊#88322	—	2	1	no	—
16	#13133128	—	2	1	no	—
17	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂²⋊C₈×C₅₃	—	2	1	no	—
20	C₄×C₈×C₅₃	—	2	1	no	—
21	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
24	C₂×Dic₁₀₆	—	2	1	no	—
25	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₅₃	—	2	2	no	—
27	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₅₃	D₄	1	1	yes	characteristic
29	C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₂²⋊Dic₅₃	—	2	2	no	—
32	C₄⋊Dic₅₃	—	2	1	no	—
33	C₄⋊Dic₅₃	—	2	1	no	—
34	C₄⋊Dic₅₃	—	2	1	no	—
35	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
36	#11719919	—	2	2	no	—
37	#11719919	—	2	2	no	—
38	C₄×Dic₅₃	—	2	2	no	—
39	C₂×#88322	—	2	2	no	—
40	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	4	1	no	—
44	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₅₃	—	2	2	no	—
51	C₂×Dic₅₃	—	2	2	no	—
52	Dic₁₀₆	—	4	2	no	—
53	Dic₁₀₆	—	4	2	no	—
54	C₈×C₅₃	—	4	2	no	—
55	#88322	—	4	2	no	—
56	C₂²×C₅₃	—	2	1	no	—
57	C₂²×C₅₃	—	2	1	no	—
58	C₂²×C₅₃	—	2	1	no	—
59	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₅₃	—	2	1	no	—
61	C₄×C₅₃	—	2	1	no	—
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	—	2	1	no	—
64	C₄×C₅₃	#135	1	2	yes	—
65	C₄×C₅₃	#137	1	2	yes	—
66	Dic₅₃	—	4	2	no	—
67	Dic₅₃	—	4	2	no	—
68	#1554	—	53	1	no	maximal, Sylow
69	C₂×C₅₃	—	4	1	no	—
70	C₂×C₅₃	#413	1	1	yes	characteristic
71	C₂×C₅₃	#416	1	1	yes	characteristic
72	C₂×C₅₃	#419	1	1	yes	characteristic
73	#533	—	53	1	no	—
74	#432	—	53	1	no	—
75	#431	D₅₃	1	1	yes	characteristic
76	#333	—	53	2	no	—
77	#332	—	53	2	no	—
78	C₅₃	#1554	1	1	yes	characteristic, complemented, Sylow
79	C₄⋊Q₈	—	53	1	no	—
80	C₄⋊Q₈	—	53	1	no	—
81	#125	—	106	1	no	—
82	C₂²⋊Q₈	—	53	2	no	—
83	C₂²⋊Q₈	—	106	1	no	—
84	C₄×Q₈	—	53	2	no	—
85	C₄×Q₈	—	106	1	no	—
86	#117	—	106	1	no	—
87	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
88	C₄⋊C₈	—	53	2	no	—
89	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
90	C₂²⋊C₈	—	2	1	no	—
91	C₂²⋊C₈	—	106	1	no	—
92	#97	—	106	1	no	—
93	C₄×C₈	—	2	1	no	—
94	C₂×Q₈	—	53	2	no	—
95	C₂×Q₈	—	106	1	no	—
96	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
97	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	106	2	no	—
101	C₄⋊C₄	—	53	2	no	—
102	C₄⋊C₄	—	106	1	no	—
103	C₄⋊C₄	—	106	1	no	—
104	C₄⋊C₄	—	106	1	no	—
105	C₄⋊C₄	—	106	2	no	—
106	C₄⋊C₄	—	106	2	no	—
107	C₂²⋊C₄	—	106	2	no	—
108	C₄²	—	106	2	no	—
109	C₄²	C₄×D₅₃	1	1	yes	characteristic
110	C₄²	D₂₁₂	1	1	yes	characteristic
111	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
112	C₄²	C₂²×D₅₃	1	1	yes	characteristic
113	C₂³	#11719921	1	1	yes	characteristic
114	Q₈	—	212	2	no	—
115	Q₈	—	212	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	106	2	no	—
120	C₂×C₄	—	106	2	no	—
121	C₂×C₄	#11719921	1	1	yes	characteristic
122	C₂×C₄	#11719921	1	1	yes	characteristic
123	C₂×C₄	#11719921	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
125	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₈	—	212	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂×#11719921	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	212	2	no	—
138	C₄	—	212	2	no	—
139	C₄	#13133239	1	2	yes	—
140	C₄	#13133247	1	2	yes	—
141	C₂	—	4	1	no	—
142	C₂	#16810605	1	1	yes	characteristic
143	C₂	#16810620	1	1	yes	characteristic
144	C₂	#16810631	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal, characteristic

#16810680

C₂

yes

maximal, characteristic

#16810371

C₂

yes

maximal, complemented

#16810368

C₂

yes

maximal, complemented

C₅₃×#431

C₂

yes

maximal, characteristic

#13133217

C₂²

yes

characteristic

#13133217

C₂²

yes

—

—

C₄

yes

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₄

yes

—

#13133176

—

C₄⋊#88322

C₂²

yes

—

C₂²⋊#88322

—

#13133128

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#135

yes

—

C₄×C₅₃

#137

yes

—

Dic₅₃

—

Dic₅₃

—

#1554

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

#333

—

#332

—

C₅₃

#1554

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

106

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

C₄×Q₈

—

106

—

#117

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

106

—

#97

—

106

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

106

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

106

—

103

C₄⋊C₄

—

106

—

104

C₄⋊C₄

—

106

—

105

C₄⋊C₄

—

106

—

106

C₄⋊C₄

—

106

—

107

C₂²⋊C₄

—

106

—

108

C₄²

—

106

—

109

C₄²

C₄×D₅₃

yes

characteristic

110

C₄²

D₂₁₂

yes

characteristic

111

C₄²

C₂²⋊D₅₃

yes

characteristic

112

C₄²

C₂²×D₅₃

yes

characteristic

113

C₂³

#11719921

yes

characteristic

114

Q₈

—

212

—

115

Q₈

—

212

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

106

—

120

C₂×C₄

—

106

—

121

C₂×C₄

#11719921

yes

characteristic

122

C₂×C₄

#11719921

yes

characteristic

123

C₂×C₄

#11719921

yes

characteristic

124

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

125

C₂×C₄

C₂×D₂₁₂

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

127

C₈

—

128

C₈

—

212

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×#11719921

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

212

—

138

C₄

—

212

—

139

C₄

#13133239

yes

—

140

C₄

#13133247

yes

—

141

C₂

—

142

C₂

#16810605

yes

characteristic

143

C₂

#16810620

yes

characteristic

144

C₂

#16810631

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11719921
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#431	D₅₃
hypocenter	C₅₃	#1554
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂×#11719921

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#431

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #16810371	C₂	semidirect
4. #16810368	C₂	semidirect
78. C₅₃	#1554	semidirect

H ≅ G/N

type

3. #16810371

C₂

semidirect

4. #16810368

C₂

semidirect

78. C₅₃

#1554

semidirect

Subgroups — #20273423

Conjugacy classes of subgroups

Complements