Group #20290899 — subgroups

Subgroups — #20290899

Order 6784.

276 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 78 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	#16811582	C₂	1	1	yes	maximal, characteristic
2	#16811567	C₂	1	1	yes	maximal, characteristic
3	#16811496	C₂	1	4	yes	maximal
4	C₄⋊Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×D₅₃⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊#6157	C₂²	1	2	yes	—
7	#13133272	—	2	1	no	—
8	#13133272	—	2	1	no	—
9	(C₂×#6157)⋊C₄	—	2	1	no	—
10	(C₂×#6157)⋊C₄	—	2	1	no	—
11	C₂×Q₈×D₅₃	—	2	1	no	—
12	#13133217	C₄	1	1	yes	characteristic, complemented
13	#13133216	—	2	1	no	—
14	#13133216	C₄	1	2	yes	complemented
15	#13133215	—	2	1	no	—
16	C₄⋊C₄×D₅₃	—	2	1	no	—
17	C₄⋊C₄×D₅₃	C₂²	1	2	yes	—
18	C₄²×D₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
20	C₂²×#6157	—	2	2	no	—
21	C₂×Q₈×C₅₃	—	2	1	no	—
22	Q₈×D₅₃	—	4	4	no	—
23	C₂×C₄×D₅₃	—	2	2	no	—
24	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
25	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
26	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₀₆	—	2	1	no	—
28	C₂×Dic₁₀₆	—	2	2	no	—
29	C₂×#88324	—	2	2	no	—
30	C₄⋊#6157	—	2	4	no	—
31	D₅₃⋊C₈	—	2	2	no	—
32	D₅₃⋊C₈	Q₈	1	4	yes	complemented
33	C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
35	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄⋊Dic₅₃	—	2	1	no	—
37	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
38	#11719919	—	2	2	no	—
39	#11719919	—	2	2	no	—
40	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄×Dic₅₃	D₄	1	1	yes	characteristic
42	C₄×Dic₅₃	D₄	1	1	yes	characteristic
43	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×#6157	—	4	2	no	—
45	C₂×#6157	—	4	2	no	—
46	C₂×#6157	—	4	2	no	—
47	Q₈×C₅₃	—	4	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×Dic₅₃	—	2	2	no	—
53	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
56	C₄×D₅₃	—	2	2	no	—
57	C₄×D₅₃	—	2	2	no	—
58	C₄×D₅₃	—	4	4	no	—
59	C₄×D₅₃	C₂×Q₈	1	2	yes	—
60	C₄×D₅₃	#41	1	2	yes	—
61	Dic₁₀₆	—	4	4	no	—
62	Dic₁₀₆	—	4	2	no	—
63	#88324	—	2	4	no	—
64	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×D₅₃	#124	1	1	yes	characteristic
66	C₂×D₅₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×D₅₃	AGL(1,ℤ/8)	1	2	yes	—
68	C₂×D₅₃	#137	1	2	yes	—
69	#6157	—	8	2	no	—
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	—	2	1	no	—
72	C₄×C₅₃	—	4	2	no	—
73	C₄×C₅₃	C₄×Q₈	1	2	yes	—
74	Dic₅₃	—	2	1	no	—
75	Dic₅₃	—	2	1	no	—
76	Dic₅₃	—	4	2	no	—
77	Dic₅₃	C₂²⋊Q₈	1	2	yes	—
78	#1993	—	53	1	no	maximal, Sylow
79	C₂×C₅₃	#391	1	1	yes	characteristic
80	C₂×C₅₃	#418	1	1	yes	characteristic
81	C₂×C₅₃	#419	1	1	yes	characteristic
82	D₅₃	#489	1	2	yes	—
83	D₅₃	#501	1	2	yes	—
84	C₂×C₄⋊Q₈	—	53	1	no	—
85	C₂×#97	—	53	1	no	—
86	#382	—	53	1	no	—
87	#340	—	53	4	no	—
88	C₅₃	#1993	1	1	yes	characteristic, complemented, Sylow
89	C₂²×Q₈	—	106	1	no	—
90	C₂²×C₈	—	53	2	no	—
91	C₄⋊Q₈	—	53	2	no	—
92	C₄⋊Q₈	—	53	1	no	—
93	C₄⋊Q₈	—	106	1	no	—
94	C₄⋊Q₈	—	106	1	no	—
95	C₄⋊Q₈	#6157	1	1	yes	characteristic, complemented
96	C₂×C₄⋊C₄	—	53	2	no	—
97	C₂×C₄⋊C₄	—	53	2	no	—
98	C₂×C₄⋊C₄	—	106	1	no	—
99	C₂×C₄²	—	53	1	no	—
100	#97	—	106	1	no	—
101	#97	—	106	1	no	—
102	#95	—	106	1	no	—
103	#95	—	106	1	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	106	2	no	—
106	C₂×Q₈	—	106	1	no	—
107	C₂×Q₈	—	212	4	no	—
108	C₂²×C₄	—	53	1	no	—
109	C₂²×C₄	—	53	1	no	—
110	C₂²×C₄	—	53	1	no	—
111	C₂²×C₄	—	106	2	no	—
112	C₂²×C₄	—	106	2	no	—
113	C₂×C₈	—	53	4	no	—
114	C₂×C₈	—	106	2	no	—
115	C₂×C₈	—	106	2	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	53	2	no	—
118	C₄⋊C₄	—	106	2	no	—
119	C₄⋊C₄	—	106	2	no	—
120	C₄⋊C₄	—	106	4	no	—
121	C₄⋊C₄	—	106	1	no	—
122	C₄⋊C₄	C₂×#6157	1	2	yes	—
123	C₄²	—	53	1	no	—
124	C₄²	—	53	1	no	—
125	C₄²	—	53	1	no	—
126	C₄²	C₂×#6157	1	1	yes	characteristic
127	C₂³	—	53	1	no	—
128	Q₈	—	4	2	no	—
129	Q₈	—	212	2	no	—
130	Q₈	—	212	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	53	1	no	—
133	C₂×C₄	—	53	2	no	—
134	C₂×C₄	—	53	2	no	—
135	C₂×C₄	—	53	1	no	—
136	C₂×C₄	—	53	1	no	—
137	C₂×C₄	—	106	2	no	—
138	C₂×C₄	—	106	2	no	—
139	C₂×C₄	—	106	2	no	—
140	C₂×C₄	—	212	4	no	—
141	C₂×C₄	—	212	2	no	—
142	C₂×C₄	—	212	2	no	—
143	C₂×C₄	—	212	2	no	—
144	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
145	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
146	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
147	C₈	—	106	4	no	—
148	C₂²	—	53	1	no	—
149	C₂²	—	53	1	no	—
150	C₂²	—	53	2	no	—
151	C₂²	—	53	2	no	—
152	C₂²	C₂×C₂²⋊#6157	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	2	no	—
156	C₄	—	53	2	no	—
157	C₄	—	106	1	no	—
158	C₄	—	106	1	no	—
159	C₄	—	424	2	no	—
160	C₄	—	212	2	no	—
161	C₄	Q₈×#6157	1	2	yes	—
162	C₂	—	53	2	no	—
163	C₂	—	53	2	no	—
164	C₂	(C₂×#6157)⋊Q₈	1	1	yes	characteristic
165	C₂	#16811615	1	1	yes	characteristic
166	C₂	#16811616	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811582

C₂

yes

maximal, characteristic

#16811567

C₂

yes

maximal, characteristic

#16811496

C₂

yes

maximal

C₄⋊Q₈×D₅₃

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₈

C₂²

yes

—

C₂×C₄⋊#6157

C₂²

yes

—

—

—

—

—

C₂×Q₈×D₅₃

—

#13133217

C₄

yes

characteristic, complemented

#13133216

—

#13133216

C₄

yes

complemented

#13133215

—

C₄⋊C₄×D₅₃

—

C₄⋊C₄×D₅₃

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

—

C₂×Q₈×C₅₃

—

Q₈×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×#88324

—

C₄⋊#6157

—

D₅₃⋊C₈

—

D₅₃⋊C₈

Q₈

yes

complemented

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₂²×D₅₃

C₂×D₄

yes

—

—

—

Q₈×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×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×Q₈

yes

—

C₄×D₅₃

#41

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

#88324

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

#124

yes

characteristic

C₂×D₅₃

C₄⋊Q₈

yes

characteristic

C₂×D₅₃

AGL(1,ℤ/8)

yes

—

C₂×D₅₃

#137

yes

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×Q₈

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂²⋊Q₈

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

—

D₅₃

#501

yes

—

C₂×C₄⋊Q₈

—

C₂×#97

—

#382

—

#340

—

C₅₃

#1993

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

#6157

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

100

#97

—

106

—

101

#97

—

106

—

102

#95

—

106

—

103

#95

—

106

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

—

106

C₂×Q₈

—

106

—

107

C₂×Q₈

—

212

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

106

—

112

C₂²×C₄

—

106

—

113

C₂×C₈

—

114

C₂×C₈

—

106

—

115

C₂×C₈

—

106

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

106

—

119

C₄⋊C₄

—

106

—

120

C₄⋊C₄

—

106

—

121

C₄⋊C₄

—

106

—

122

C₄⋊C₄

C₂×#6157

yes

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

C₂×#6157

yes

characteristic

127

C₂³

—

128

Q₈

—

129

Q₈

—

212

—

130

Q₈

—

212

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

106

—

138

C₂×C₄

—

106

—

139

C₂×C₄

—

106

—

140

C₂×C₄

—

212

—

141

C₂×C₄

—

212

—

142

C₂×C₄

—

212

—

143

C₂×C₄

—

212

—

144

C₂×C₄

C₂²⋊#6157

yes

characteristic

145

C₂×C₄

C₂²⋊#6157

yes

characteristic

146

C₂×C₄

C₂²×#6157

yes

characteristic

147

C₈

—

106

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂×C₂²⋊#6157

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

106

—

158

C₄

—

106

—

159

C₄

—

424

—

160

C₄

—

212

—

161

C₄

Q₈×#6157

yes

—

162

C₂

—

163

C₂

—

164

C₂

(C₂×#6157)⋊Q₈

yes

characteristic

165

C₂

#16811615

yes

characteristic

166

C₂

#16811616

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄⋊Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₄⋊Q₈	#6157
hypocenter	C₅₃	#1993
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₄⋊Q₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

C₄⋊Q₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
12. #13133217	C₄	semidirect
14. #13133216	C₄	semidirect
19. C₄⋊Q₈×C₅₃	C₄	semidirect
32. D₅₃⋊C₈	Q₈	semidirect
88. C₅₃	#1993	semidirect
95. C₄⋊Q₈	#6157	semidirect

H ≅ G/N

type

12. #13133217

C₄

semidirect

14. #13133216

C₄

semidirect

19. C₄⋊Q₈×C₅₃

C₄

semidirect

32. D₅₃⋊C₈

Q₈

semidirect

88. C₅₃

#1993

semidirect

95. C₄⋊Q₈

#6157

semidirect

Subgroups — #20290899

Conjugacy classes of subgroups

Complements