Group #20274161 — subgroups

Subgroups — #20274161

Order 6784.

166 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16810971	C₂	1	1	yes	maximal, characteristic
2	#16810823	C₂	1	1	yes	maximal, characteristic
3	Q₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810375	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810369	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810366	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719924	—	2	1	no	—
9	#13133214	C₂²	1	1	yes	characteristic
10	C₄⋊#88322	—	2	1	no	—
11	C₄⋊#88322	C₂²	1	1	yes	characteristic
12	C₄⋊#88322	C₂²	1	1	yes	characteristic
13	#13133137	—	2	1	no	—
14	#13133137	—	2	1	no	—
15	#13133137	—	2	1	no	—
16	C₄×#88322	—	2	1	no	—
17	C₄×#88322	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₅₃	—	2	1	no	—
19	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	#103×C₅₃	—	2	1	no	—
23	C₂×Q₈×C₅₃	—	2	1	no	—
24	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂×D₂₁₂	—	2	1	no	—
26	C₂×D₂₁₂	—	2	1	no	—
27	C₂×D₂₁₂	—	2	1	no	—
28	Q₁₆×C₅₃	—	4	2	no	—
29	C₂×C₈×C₅₃	—	2	1	no	—
30	C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
34	C₄²×C₅₃	—	2	1	no	—
35	C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	#11719924	—	4	2	no	—
37	C₂×#88322	—	2	1	no	—
38	C₂×#88322	—	2	1	no	—
39	C₂×#88322	D₄	1	1	yes	characteristic
40	C₂×#88322	D₄	1	1	yes	characteristic
41	C₂²×D₅₃	—	4	1	no	—
42	Q₈×C₅₃	—	2	1	no	—
43	Q₈×C₅₃	—	4	1	no	—
44	Q₈×C₅₃	—	4	1	no	—
45	Q₈×C₅₃	D₈	1	2	yes	—
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	—	2	1	no	—
48	C₂×C₄×C₅₃	—	4	1	no	—
49	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
52	D₂₁₂	—	4	1	no	—
53	D₂₁₂	—	4	1	no	—
54	D₂₁₂	—	4	1	no	—
55	D₂₁₂	—	4	1	no	—
56	D₂₁₂	—	8	1	no	—
57	C₈×C₅₃	—	4	1	no	—
58	#88322	—	2	1	no	—
59	#88322	—	2	1	no	—
60	#88322	—	4	1	no	—
61	#88322	—	4	1	no	—
62	C₂²×C₅₃	#121	1	1	yes	characteristic
63	C₂×D₅₃	—	8	1	no	—
64	C₂×D₅₃	—	8	1	no	—
65	C₂×D₅₃	—	8	1	no	—
66	C₄×C₅₃	—	2	2	no	—
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	—	4	1	no	—
69	C₄×C₅₃	—	8	1	no	—
70	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
71	C₄×C₅₃	#135	1	1	yes	characteristic
72	C₄×C₅₃	#135	1	1	yes	characteristic
73	C₄×C₅₃	#137	1	1	yes	characteristic
74	#1682	—	53	1	no	maximal, Sylow
75	C₂×C₅₃	#464	1	1	yes	characteristic
76	C₂×C₅₃	#466	1	1	yes	characteristic
77	C₂×C₅₃	#471	1	1	yes	characteristic
78	D₅₃	—	16	1	no	—
79	#486	—	53	1	no	—
80	#462	D₅₃	1	1	yes	characteristic, complemented
81	#460	—	53	1	no	—
82	C₈×Q₈	—	53	1	no	—
83	#335	—	53	1	no	—
84	#332	—	53	1	no	—
85	#331	—	53	1	no	—
86	C₅₃	#1682	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₁₆	—	2	1	no	—
88	C₂×SD₁₆	—	106	1	no	—
89	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
90	#127	—	53	1	no	—
91	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
92	C₄⋊C₈	—	53	1	no	—
93	C₄⋊C₈	—	53	1	no	—
94	C₄⋊C₈	—	106	1	no	—
95	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
96	#103	—	2	1	no	—
97	#102	—	106	1	no	—
98	#102	—	106	1	no	—
99	#102	—	106	1	no	—
100	C₄×C₈	—	53	1	no	—
101	C₄×C₈	—	106	1	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
104	C₂×D₄	—	106	1	no	—
105	C₂×D₄	—	106	1	no	—
106	C₂×D₄	—	106	1	no	—
107	Q₁₆	—	4	2	no	—
108	SD₁₆	—	212	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	53	1	no	—
111	C₂×C₈	—	53	1	no	—
112	C₂×C₈	—	106	1	no	—
113	C₂×C₈	—	106	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
118	C₄²	—	2	1	no	—
119	C₄²	C₂²×D₅₃	1	1	yes	characteristic
120	C₂³	—	212	1	no	—
121	Q₈	—	2	1	no	—
122	Q₈	—	4	1	no	—
123	Q₈	—	4	1	no	—
124	Q₈	#11719922	1	2	yes	—
125	D₄	—	424	1	no	—
126	D₄	—	212	1	no	—
127	D₄	—	212	1	no	—
128	D₄	—	212	1	no	—
129	D₄	—	212	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
134	C₂×C₄	#11719947	1	1	yes	characteristic
135	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
136	C₈	—	4	1	no	—
137	C₈	—	106	1	no	—
138	C₈	—	106	1	no	—
139	C₈	—	212	1	no	—
140	C₈	—	212	1	no	—
141	C₂²	—	424	1	no	—
142	C₂²	—	424	1	no	—
143	C₂²	—	424	1	no	—
144	C₂²	#13133189	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	1	no	—
147	C₄	—	4	1	no	—
148	C₄	—	8	1	no	—
149	C₄	C₂×#11719922	1	1	yes	characteristic
150	C₄	#13133237	1	1	yes	characteristic
151	C₄	#13133240	1	1	yes	characteristic
152	C₄	#13133249	1	1	yes	characteristic
153	C₂	—	848	1	no	—
154	C₂	#16810847	1	1	yes	characteristic
155	C₂	#16810857	1	1	yes	characteristic
156	C₂	#16810888	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810971

C₂

yes

maximal, characteristic

#16810823

C₂

yes

maximal, characteristic

Q₈×#88322

C₂

yes

maximal, characteristic, complemented

#16810375

C₂

yes

maximal, characteristic, complemented

#16810369

C₂

yes

maximal, characteristic, complemented

#16810366

C₂

yes

maximal, characteristic

C₅₃×#462

C₂

yes

maximal, characteristic, complemented

C₂×#11719924

—

#13133214

C₂²

yes

characteristic

C₄⋊#88322

—

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×Q₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

#103×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

—

—

—

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

—

—

—

—

C₂²×C₅₃

#121

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×D₈

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#486

—

#462

D₅₃

yes

characteristic, complemented

#460

—

C₈×Q₈

—

#335

—

#332

—

#331

—

C₅₃

#1682

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×SD₁₆

—

106

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

#127

—

C₄×Q₈

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

C₂×D₅₃

yes

characteristic

#103

—

#102

—

106

—

#102

—

106

—

#102

—

106

—

100

C₄×C₈

—

101

C₄×C₈

—

106

—

102

C₂×Q₈

—

103

C₂×Q₈

C₂²⋊D₅₃

yes

characteristic

104

C₂×D₄

—

106

—

105

C₂×D₄

—

106

—

106

C₂×D₄

—

106

—

107

Q₁₆

—

108

SD₁₆

—

212

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

106

—

113

C₂×C₈

—

106

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

118

C₄²

—

119

C₄²

C₂²×D₅₃

yes

characteristic

120

C₂³

—

212

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

#11719922

yes

—

125

D₄

—

424

—

126

D₄

—

212

—

127

D₄

—

212

—

128

D₄

—

212

—

129

D₄

—

212

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

D₄×D₅₃

yes

characteristic

134

C₂×C₄

#11719947

yes

characteristic

135

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

136

C₈

—

137

C₈

—

106

—

138

C₈

—

106

—

139

C₈

—

212

—

140

C₈

—

212

—

141

C₂²

—

424

—

142

C₂²

—

424

—

143

C₂²

—

424

—

144

C₂²

#13133189

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

C₂×#11719922

yes

characteristic

150

C₄

#13133237

yes

characteristic

151

C₄

#13133240

yes

characteristic

152

C₄

#13133249

yes

characteristic

153

C₂

—

848

—

154

C₂

#16810847

yes

characteristic

155

C₂

#16810857

yes

characteristic

156

C₂

#16810888

yes

characteristic

	subgroup	G/H
center	C₂²	#13133189
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#462	C₂
socle	C₂²×C₅₃	#121
hypercenter	#462	D₅₃
hypocenter	C₅₃	#1682
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

subgroup

G/H

center

C₂²

#13133189

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#462

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #462

N	H ≅ G/N	type
3. Q₈×#88322	C₂	semidirect
4. #16810375	C₂	semidirect
5. #16810369	C₂	semidirect
7. C₅₃×#462	C₂	semidirect
80. #462	D₅₃	semidirect
86. C₅₃	#1682	semidirect

H ≅ G/N

type

3. Q₈×#88322

C₂

semidirect

4. #16810375

C₂

semidirect

5. #16810369

C₂

semidirect

7. C₅₃×#462

C₂

semidirect

80. #462

D₅₃

semidirect

86. C₅₃

#1682

semidirect

Subgroups — #20274161

Conjugacy classes of subgroups

Complements