Group #20290065 — subgroups

Subgroups — #20290065

Order 6784.

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

normal subgroups only

This group has 232 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	#16811627	C₂	1	1	yes	maximal, characteristic
2	C₄×D₅₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#16811473	C₂	1	4	yes	maximal, complemented
4	#16811178	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719936	—	2	1	no	—
6	C₂×D₅₃⋊C₈	—	2	1	no	—
7	C₄⋊#88324	—	2	1	no	—
8	C₄⋊#88324	C₂²	1	2	yes	—
9	C₄×#88324	—	2	1	no	—
10	C₄×#88324	C₂²	1	2	yes	—
11	#13133217	C₄	1	1	yes	characteristic
12	#13133208	—	2	1	no	—
13	#13133197	C₄	1	2	yes	—
14	#13133193	—	2	1	no	—
15	Q₈×Dic₅₃	—	2	1	no	—
16	#13133182	C₂²	1	2	yes	—
17	#13133175	—	2	1	no	—
18	C₄²×D₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
20	C₂×Q₈×C₅₃	—	2	1	no	—
21	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
23	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
24	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
25	C₂×#88324	—	2	2	no	—
26	C₂×#88324	—	2	2	no	—
27	#11719936	—	4	2	no	—
28	D₅₃⋊C₈	—	2	4	no	—
29	C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
31	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	#11719921	—	2	2	no	—
33	C₄⋊Dic₅₃	—	2	1	no	—
34	C₄⋊Dic₅₃	—	2	1	no	—
35	C₄⋊Dic₅₃	—	2	2	no	—
36	#11719919	—	2	2	no	—
37	C₄×Dic₅₃	—	2	2	no	—
38	C₄×Dic₅₃	D₄	1	1	yes	characteristic
39	C₄×Dic₅₃	D₄	1	1	yes	characteristic
40	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
41	C₂²×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	Q₈×C₅₃	—	4	2	no	—
43	C₂×C₄×C₅₃	—	2	2	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×Dic₅₃	—	2	2	no	—
48	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
51	C₄×D₅₃	—	2	2	no	—
52	C₄×D₅₃	—	2	2	no	—
53	C₄×D₅₃	SD₁₆	1	2	yes	—
54	C₄×D₅₃	Q₁₆	1	2	yes	—
55	Dic₁₀₆	—	2	4	no	—
56	#88324	—	2	4	no	—
57	#88324	—	4	2	no	—
58	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×D₅₃	—	2	1	no	—
60	C₂×D₅₃	—	2	1	no	—
61	C₂×D₅₃	#103	1	2	yes	—
62	C₄×C₅₃	—	4	2	no	—
63	C₄×C₅₃	#103	1	2	yes	—
64	C₄×C₅₃	#104	1	2	yes	—
65	C₄×C₅₃	#131	1	2	yes	—
66	Dic₅₃	—	2	1	no	—
67	Dic₅₃	—	2	1	no	—
68	Dic₅₃	—	4	2	no	—
69	Dic₅₃	C₂×SD₁₆	1	1	yes	characteristic
70	Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
71	#1493	—	53	1	no	maximal, Sylow
72	C₂×C₅₃	#408	1	1	yes	characteristic
73	C₂×C₅₃	C₂×#103	1	1	yes	characteristic
74	C₂×C₅₃	C₂×#104	1	1	yes	characteristic
75	D₅₃	—	2	2	no	—
76	#533	—	53	1	no	—
77	#423	—	53	1	no	—
78	C₂×C₄×C₈	—	53	1	no	—
79	Q₈⋊C₈	—	53	4	no	—
80	C₅₃	#1493	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₁₆	—	106	1	no	—
82	C₂²×C₈	—	106	1	no	—
83	C₄⋊Q₈	—	53	1	no	—
84	C₄⋊Q₈	#6157	1	1	yes	characteristic
85	#125	—	106	1	no	—
86	C₂²⋊Q₈	—	53	2	no	—
87	C₂²⋊Q₈	—	106	1	no	—
88	C₄×Q₈	—	53	2	no	—
89	C₄×Q₈	—	106	1	no	—
90	#117	—	106	1	no	—
91	C₂×C₄²	—	53	1	no	—
92	C₄⋊C₈	—	53	2	no	—
93	C₄⋊C₈	—	106	1	no	—
94	C₄×C₈	—	53	2	no	—
95	C₄×C₈	—	106	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	53	2	no	—
98	C₂²×C₄	—	53	1	no	—
99	C₂²×C₄	—	53	1	no	—
100	C₂²×C₄	—	53	1	no	—
101	M₁₆	—	212	2	no	—
102	C₂×C₈	—	106	2	no	—
103	C₂×C₈	—	106	4	no	—
104	C₂×C₈	—	106	2	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	106	2	no	—
107	C₄⋊C₄	—	106	2	no	—
108	C₄⋊C₄	—	106	1	no	—
109	C₄⋊C₄	—	106	1	no	—
110	C₄⋊C₄	C₂×#6157	1	2	yes	—
111	C₂²⋊C₄	—	106	2	no	—
112	C₄²	—	53	1	no	—
113	C₄²	—	53	1	no	—
114	C₄²	—	53	1	no	—
115	C₄²	—	106	2	no	—
116	C₄²	C₂×#6157	1	1	yes	characteristic
117	C₂³	—	53	1	no	—
118	Q₈	—	4	2	no	—
119	Q₈	—	106	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	53	1	no	—
122	C₂×C₄	—	53	2	no	—
123	C₂×C₄	—	53	2	no	—
124	C₂×C₄	—	53	1	no	—
125	C₂×C₄	—	53	1	no	—
126	C₂×C₄	—	106	2	no	—
127	C₂×C₄	—	106	2	no	—
128	C₂×C₄	—	106	2	no	—
129	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
130	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
131	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
132	C₈	—	106	4	no	—
133	C₈	—	212	2	no	—
134	C₂²	—	53	2	no	—
135	C₂²	—	106	1	no	—
136	C₂²	—	106	1	no	—
137	C₂²	C₂×C₂²⋊#6157	1	1	yes	characteristic
138	C₄	—	4	2	no	—
139	C₄	—	53	1	no	—
140	C₄	—	53	1	no	—
141	C₄	—	106	1	no	—
142	C₄	—	106	1	no	—
143	C₄	—	212	2	no	—
144	C₄	#13133280	1	2	yes	—
145	C₄	#13133281	1	2	yes	—
146	C₄	#13133305	1	2	yes	—
147	C₂	—	106	2	no	—
148	C₂	#16811598	1	1	yes	characteristic
149	C₂	C₂×#13133280	1	1	yes	characteristic
150	C₂	C₂×#13133281	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811627

C₂

yes

maximal, characteristic

C₄×D₅₃⋊C₈

C₂

yes

maximal, characteristic

#16811473

C₂

yes

maximal, complemented

#16811178

C₂

yes

maximal, characteristic

C₂×#11719936

—

C₂×D₅₃⋊C₈

—

C₄⋊#88324

—

C₄⋊#88324

C₂²

yes

—

C₄×#88324

—

C₄×#88324

C₂²

yes

—

#13133217

C₄

yes

characteristic

#13133208

—

#13133197

C₄

yes

—

#13133193

—

Q₈×Dic₅₃

—

#13133182

C₂²

yes

—

#13133175

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₄

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×#88324

—

C₂×#88324

—

#11719936

—

D₅₃⋊C₈

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

C₂²×D₅₃

C₂²⋊C₄

yes

characteristic

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

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

SD₁₆

yes

—

C₄×D₅₃

Q₁₆

yes

—

Dic₁₀₆

—

#88324

—

#88324

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#103

yes

—

C₄×C₅₃

—

C₄×C₅₃

#103

yes

—

C₄×C₅₃

#104

yes

—

C₄×C₅₃

#131

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×SD₁₆

yes

characteristic

Dic₅₃

C₂×Q₁₆

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#533

—

#423

—

C₂×C₄×C₈

—

Q₈⋊C₈

—

C₅₃

#1493

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

#6157

yes

characteristic

#125

—

106

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

C₄×Q₈

—

106

—

#117

—

106

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄×C₈

—

C₄×C₈

—

106

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

M₁₆

—

212

—

102

C₂×C₈

—

106

—

103

C₂×C₈

—

106

—

104

C₂×C₈

—

106

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

106

—

107

C₄⋊C₄

—

106

—

108

C₄⋊C₄

—

106

—

109

C₄⋊C₄

—

106

—

110

C₄⋊C₄

C₂×#6157

yes

—

111

C₂²⋊C₄

—

106

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

106

—

116

C₄²

C₂×#6157

yes

characteristic

117

C₂³

—

118

Q₈

—

119

Q₈

—

106

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

106

—

127

C₂×C₄

—

106

—

128

C₂×C₄

—

106

—

129

C₂×C₄

C₂²⋊#6157

yes

characteristic

130

C₂×C₄

C₂²⋊#6157

yes

characteristic

131

C₂×C₄

C₂²×#6157

yes

characteristic

132

C₈

—

106

—

133

C₈

—

212

—

134

C₂²

—

135

C₂²

—

106

—

136

C₂²

—

106

—

137

C₂²

C₂×C₂²⋊#6157

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

106

—

142

C₄

—

106

—

143

C₄

—

212

—

144

C₄

#13133280

yes

—

145

C₄

#13133281

yes

—

146

C₄

#13133305

yes

—

147

C₂

—

106

—

148

C₂

#16811598

yes

characteristic

149

C₂

C₂×#13133280

yes

characteristic

150

C₂

C₂×#13133281

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	#1493
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
3. #16811473	C₂	semidirect
80. C₅₃	#1493	semidirect

H ≅ G/N

type

3. #16811473

C₂

semidirect

80. C₅₃

#1493

semidirect

Subgroups — #20290065

Conjugacy classes of subgroups

Complements