Group #20290891 — subgroups

Subgroups — #20290891

Order 6784.

260 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 74 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	C₄×#11719936	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#6157	C₂	1	1	yes	maximal, characteristic
3	#16811493	C₂	1	4	yes	maximal
4	#16811178	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719936	C₂²	1	2	yes	—
6	C₂×C₄⋊#6157	—	2	1	no	—
7	C₂×C₄×#6157	C₂²	1	2	yes	—
8	#13133272	—	2	1	no	—
9	C₄×#88324	—	2	1	no	—
10	(C₂×#6157)⋊C₄	—	2	1	no	—
11	#13133217	C₄	1	1	yes	characteristic, complemented
12	#13133208	C₄	1	2	yes	complemented
13	#13133197	—	2	1	no	—
14	#13133193	—	2	1	no	—
15	Q₈×Dic₅₃	—	2	1	no	—
16	#13133182	—	2	1	no	—
17	#13133175	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	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
23	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
24	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
25	C₂×Dic₁₀₆	—	2	1	no	—
26	C₂×#88324	—	2	2	no	—
27	C₄⋊#6157	—	2	4	no	—
28	C₄×#6157	—	2	4	no	—
29	#11719936	—	2	2	no	—
30	#11719936	Q₈	1	4	yes	complemented
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
33	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	#11719921	—	2	2	no	—
35	C₄⋊Dic₅₃	—	2	1	no	—
36	C₄⋊Dic₅₃	—	2	2	no	—
37	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
38	#11719919	—	2	2	no	—
39	C₄×Dic₅₃	—	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	—	2	4	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	—	4	1	no	—
58	C₄×D₅₃	C₂×Q₈	1	2	yes	—
59	C₄×D₅₃	#41	1	2	yes	—
60	Dic₁₀₆	—	4	2	no	—
61	#88324	—	2	4	no	—
62	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×D₅₃	—	2	1	no	—
64	C₂×D₅₃	—	2	1	no	—
65	C₂×D₅₃	#124	1	1	yes	characteristic
66	C₂×D₅₃	C₄⋊Q₈	1	1	yes	characteristic
67	#6157	—	4	4	no	—
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	—	4	2	no	—
70	C₄×C₅₃	#104	1	2	yes	—
71	C₄×C₅₃	C₄×Q₈	1	2	yes	—
72	Dic₅₃	—	2	1	no	—
73	Dic₅₃	—	2	1	no	—
74	Dic₅₃	—	4	2	no	—
75	Dic₅₃	C₂²⋊Q₈	1	2	yes	—
76	#1986	—	53	1	no	maximal, Sylow
77	C₂×C₅₃	#391	1	1	yes	characteristic
78	C₂×C₅₃	C₂×#104	1	1	yes	characteristic
79	C₂×C₅₃	#421	1	1	yes	characteristic
80	D₅₃	—	2	1	no	—
81	D₅₃	—	2	1	no	—
82	#533	—	53	1	no	—
83	C₄×M₁₆	—	53	1	no	—
84	C₄×C₄⋊C₄	—	53	1	no	—
85	#339	—	53	4	no	—
86	C₅₃	#1986	1	1	yes	characteristic, complemented, Sylow
87	C₂×M₁₆	—	53	2	no	—
88	C₄⋊Q₈	—	53	1	no	—
89	C₄⋊Q₈	#6157	1	1	yes	characteristic, complemented
90	#125	—	53	2	no	—
91	C₂²⋊Q₈	—	106	1	no	—
92	C₂²⋊Q₈	—	106	1	no	—
93	C₄×Q₈	—	106	1	no	—
94	C₄×Q₈	—	106	1	no	—
95	#117	—	53	2	no	—
96	C₂×C₄⋊C₄	—	106	1	no	—
97	C₂×C₄²	—	53	1	no	—
98	C₂×C₄²	—	53	2	no	—
99	#97	—	106	1	no	—
100	C₄×C₈	—	106	1	no	—
101	#95	—	106	1	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	106	1	no	—
104	C₂²×C₄	—	53	1	no	—
105	C₂²×C₄	—	53	1	no	—
106	C₂²×C₄	—	53	1	no	—
107	C₂²×C₄	—	106	2	no	—
108	M₁₆	—	53	4	no	—
109	M₁₆	—	106	2	no	—
110	C₂×C₈	—	106	2	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	53	2	no	—
113	C₄⋊C₄	—	106	2	no	—
114	C₄⋊C₄	—	106	2	no	—
115	C₄⋊C₄	—	106	4	no	—
116	C₄⋊C₄	—	106	1	no	—
117	C₄⋊C₄	C₂×#6157	1	2	yes	—
118	C₂²⋊C₄	—	106	2	no	—
119	C₄²	—	53	1	no	—
120	C₄²	—	53	1	no	—
121	C₄²	—	53	1	no	—
122	C₄²	—	106	2	no	—
123	C₄²	—	106	4	no	—
124	C₄²	C₂×#6157	1	1	yes	characteristic
125	C₂³	—	53	1	no	—
126	Q₈	—	4	2	no	—
127	Q₈	—	212	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	53	1	no	—
130	C₂×C₄	—	53	2	no	—
131	C₂×C₄	—	53	2	no	—
132	C₂×C₄	—	53	1	no	—
133	C₂×C₄	—	53	1	no	—
134	C₂×C₄	—	106	2	no	—
135	C₂×C₄	—	106	4	no	—
136	C₂×C₄	—	106	2	no	—
137	C₂×C₄	—	212	1	no	—
138	C₂×C₄	—	212	2	no	—
139	C₂×C₄	—	212	2	no	—
140	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
141	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
142	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
143	C₈	—	106	4	no	—
144	C₂²	—	53	1	no	—
145	C₂²	—	53	1	no	—
146	C₂²	—	106	1	no	—
147	C₂²	—	106	1	no	—
148	C₂²	C₂×C₂²⋊#6157	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	4	2	no	—
151	C₄	—	53	2	no	—
152	C₄	—	106	1	no	—
153	C₄	—	106	1	no	—
154	C₄	—	212	4	no	—
155	C₄	—	212	2	no	—
156	C₄	#13133281	1	2	yes	—
157	C₄	Q₈×#6157	1	2	yes	—
158	C₂	—	106	1	no	—
159	C₂	—	106	1	no	—
160	C₂	(C₂×#6157)⋊Q₈	1	1	yes	characteristic
161	C₂	C₂×#13133281	1	1	yes	characteristic
162	C₂	#16811622	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#11719936

C₂

yes

maximal, characteristic

C₄×C₄⋊#6157

C₂

yes

maximal, characteristic

#16811493

C₂

yes

maximal

#16811178

C₂

yes

maximal, characteristic

C₂×#11719936

C₂²

yes

—

C₂×C₄⋊#6157

—

C₂×C₄×#6157

C₂²

yes

—

—

—

—

C₄

yes

characteristic, complemented

#13133208

C₄

yes

complemented

#13133197

—

#13133193

—

Q₈×Dic₅₃

—

#13133182

—

#13133175

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

—

C₂×Q₈×C₅₃

—

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

—

—

—

—

Q₈

yes

complemented

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

C₄×Dic₅₃

—

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

C₂×Q₈

yes

—

C₄×D₅₃

#41

yes

—

Dic₁₀₆

—

#88324

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#124

yes

characteristic

C₂×D₅₃

C₄⋊Q₈

yes

characteristic

#6157

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#104

yes

—

C₄×C₅₃

C₄×Q₈

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂²⋊Q₈

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

D₅₃

—

#533

—

C₄×M₁₆

—

C₄×C₄⋊C₄

—

#339

—

C₅₃

#1986

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

#6157

yes

characteristic, complemented

#125

—

C₂²⋊Q₈

—

106

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

106

—

C₄×Q₈

—

106

—

#117

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

C₂×C₄²

—

#97

—

106

—

100

C₄×C₈

—

106

—

101

#95

—

106

—

102

C₂×Q₈

—

103

C₂×Q₈

—

106

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

106

—

108

M₁₆

—

109

M₁₆

—

106

—

110

C₂×C₈

—

106

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

106

—

114

C₄⋊C₄

—

106

—

115

C₄⋊C₄

—

106

—

116

C₄⋊C₄

—

106

—

117

C₄⋊C₄

C₂×#6157

yes

—

118

C₂²⋊C₄

—

106

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

106

—

123

C₄²

—

106

—

124

C₄²

C₂×#6157

yes

characteristic

125

C₂³

—

126

Q₈

—

127

Q₈

—

212

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

106

—

135

C₂×C₄

—

106

—

136

C₂×C₄

—

106

—

137

C₂×C₄

—

212

—

138

C₂×C₄

—

212

—

139

C₂×C₄

—

212

—

140

C₂×C₄

C₂²⋊#6157

yes

characteristic

141

C₂×C₄

C₂²⋊#6157

yes

characteristic

142

C₂×C₄

C₂²×#6157

yes

characteristic

143

C₈

—

106

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

106

—

147

C₂²

—

106

—

148

C₂²

C₂×C₂²⋊#6157

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

106

—

153

C₄

—

106

—

154

C₄

—

212

—

155

C₄

—

212

—

156

C₄

#13133281

yes

—

157

C₄

Q₈×#6157

yes

—

158

C₂

—

106

—

159

C₂

—

106

—

160

C₂

(C₂×#6157)⋊Q₈

yes

characteristic

161

C₂

C₂×#13133281

yes

characteristic

162

C₂

#16811622

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	#1986
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
11. #13133217	C₄	semidirect
12. #13133208	C₄	semidirect
19. C₄⋊Q₈×C₅₃	C₄	semidirect
30. #11719936	Q₈	semidirect
86. C₅₃	#1986	semidirect
89. C₄⋊Q₈	#6157	semidirect

H ≅ G/N

type

11. #13133217

C₄

semidirect

12. #13133208

C₄

semidirect

19. C₄⋊Q₈×C₅₃

C₄

semidirect

30. #11719936

Q₈

semidirect

86. C₅₃

#1986

semidirect

89. C₄⋊Q₈

#6157

semidirect

Subgroups — #20290891

Conjugacy classes of subgroups

Complements