Group #20290229 — subgroups

Subgroups — #20290229

Order 6784.

240 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 70 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	#16811641	C₂	1	1	yes	maximal, characteristic
2	#16811628	C₂	1	1	yes	maximal, characteristic
3	#16811483	C₂	1	4	yes	maximal, complemented
4	C₄⋊Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719936	—	2	1	no	—
6	#13133284	—	2	1	no	—
7	#13133284	C₂²	1	2	yes	—
8	#13133284	C₂²	1	2	yes	—
9	#13133274	—	2	1	no	—
10	C₄×#88324	—	2	1	no	—
11	C₂×Q₈×D₅₃	C₄	1	2	yes	—
12	#13133217	C₄	1	1	yes	characteristic
13	#13133216	—	2	1	no	—
14	#13133216	—	2	1	no	—
15	#13133215	C₂²	1	2	yes	—
16	C₄⋊C₄×D₅₃	—	2	1	no	—
17	C₄⋊C₄×D₅₃	—	2	1	no	—
18	C₄²×D₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
20	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
21	Q₈×D₅₃	—	4	4	no	—
22	C₂×C₄×D₅₃	—	2	2	no	—
23	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
25	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
26	C₂×Dic₁₀₆	—	2	2	no	—
27	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
28	C₂×#88324	—	2	2	no	—
29	C₂×#88324	—	2	2	no	—
30	#11719936	—	4	2	no	—
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄⋊Dic₅₃	—	2	1	no	—
35	C₄⋊Dic₅₃	—	2	1	no	—
36	#11719919	—	2	2	no	—
37	#11719919	—	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	—	4	4	no	—
54	C₄×D₅₃	SD₁₆	1	2	yes	—
55	C₄×D₅₃	Q₁₆	1	2	yes	—
56	Dic₁₀₆	—	4	4	no	—
57	Dic₁₀₆	—	4	2	no	—
58	#88324	—	4	2	no	—
59	#88324	—	4	2	no	—
60	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×D₅₃	—	2	1	no	—
62	C₂×D₅₃	—	2	1	no	—
63	C₂×D₅₃	#103	1	2	yes	—
64	C₄×C₅₃	—	2	1	no	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	4	2	no	—
67	C₄×C₅₃	#103	1	2	yes	—
68	Dic₅₃	—	4	2	no	—
69	Dic₅₃	#100	1	2	yes	—
70	Dic₅₃	C₂×SD₁₆	1	1	yes	characteristic
71	Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
72	Dic₅₃	#137	1	2	yes	—
73	#1548	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	C₂×#100	1	1	yes	characteristic
75	C₂×C₅₃	C₂×#103	1	1	yes	characteristic
76	C₂×C₅₃	#419	1	1	yes	characteristic
77	D₅₃	—	2	2	no	—
78	C₂×C₄⋊Q₈	—	53	1	no	—
79	#431	—	53	1	no	—
80	#423	—	53	1	no	—
81	#333	—	53	4	no	—
82	C₅₃	#1548	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₈	—	53	2	no	—
84	C₂×M₁₆	—	106	1	no	—
85	C₄⋊Q₈	—	53	2	no	—
86	C₄⋊Q₈	—	53	1	no	—
87	C₄⋊Q₈	—	106	1	no	—
88	C₄⋊Q₈	—	106	1	no	—
89	C₄⋊Q₈	#6157	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	106	1	no	—
91	C₂×C₄⋊C₄	—	106	1	no	—
92	C₂×C₄²	—	53	1	no	—
93	C₄⋊C₈	—	53	2	no	—
94	C₄⋊C₈	—	53	2	no	—
95	C₄⋊C₈	—	106	1	no	—
96	C₂²⋊C₈	—	106	1	no	—
97	C₄×C₈	—	106	1	no	—
98	C₂×Q₈	—	53	2	no	—
99	C₂×Q₈	—	106	2	no	—
100	C₂×Q₈	—	212	4	no	—
101	C₂×Q₈	C₂×#6157	1	2	yes	—
102	C₂²×C₄	—	53	1	no	—
103	C₂²×C₄	—	53	1	no	—
104	C₂²×C₄	—	53	1	no	—
105	C₂²×C₄	—	106	2	no	—
106	M₁₆	—	212	2	no	—
107	C₂×C₈	—	106	2	no	—
108	C₂×C₈	—	106	2	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	106	2	no	—
112	C₄⋊C₄	—	106	2	no	—
113	C₄⋊C₄	—	106	1	no	—
114	C₄⋊C₄	—	106	1	no	—
115	C₄²	—	53	1	no	—
116	C₄²	—	53	1	no	—
117	C₄²	—	53	1	no	—
118	C₄²	C₂×#6157	1	1	yes	characteristic
119	C₂³	—	53	1	no	—
120	Q₈	—	4	2	no	—
121	Q₈	—	212	2	no	—
122	Q₈	—	212	4	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	53	1	no	—
125	C₂×C₄	—	53	1	no	—
126	C₂×C₄	—	53	1	no	—
127	C₂×C₄	—	53	2	no	—
128	C₂×C₄	—	53	2	no	—
129	C₂×C₄	—	106	2	no	—
130	C₂×C₄	—	106	2	no	—
131	C₂×C₄	—	106	2	no	—
132	C₂×C₄	—	212	4	no	—
133	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
134	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
135	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
136	C₈	—	212	2	no	—
137	C₈	—	212	2	no	—
138	C₂²	—	53	2	no	—
139	C₂²	—	106	1	no	—
140	C₂²	—	106	1	no	—
141	C₂²	C₂×C₂²⋊#6157	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	4	2	no	—
145	C₄	—	53	2	no	—
146	C₄	—	53	1	no	—
147	C₄	—	53	1	no	—
148	C₄	—	53	2	no	—
149	C₄	—	212	2	no	—
150	C₄	#13133280	1	2	yes	—
151	C₂	—	106	2	no	—
152	C₂	D₅₃⋊#100	1	1	yes	characteristic
153	C₂	C₂×#13133280	1	1	yes	characteristic
154	C₂	#16811616	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811641

C₂

yes

maximal, characteristic

#16811628

C₂

yes

maximal, characteristic

#16811483

C₂

yes

maximal, complemented

C₄⋊Q₈×D₅₃

C₂

yes

maximal, characteristic

C₂×#11719936

—

#13133284

—

#13133284

C₂²

yes

—

#13133284

C₂²

yes

—

#13133274

—

C₄×#88324

—

C₂×Q₈×D₅₃

C₄

yes

—

#13133217

C₄

yes

—

—

C₂²

yes

—

C₄⋊C₄×D₅₃

—

C₄⋊C₄×D₅₃

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₄

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

—

Q₈×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×#88324

—

C₂×#88324

—

#11719936

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

#11719919

—

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

—

C₄×D₅₃

SD₁₆

yes

—

C₄×D₅₃

Q₁₆

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

#88324

—

#88324

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#103

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#103

yes

—

Dic₅₃

—

Dic₅₃

#100

yes

—

Dic₅₃

C₂×SD₁₆

yes

characteristic

Dic₅₃

C₂×Q₁₆

yes

characteristic

Dic₅₃

#137

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂×C₄⋊Q₈

—

#431

—

#423

—

#333

—

C₅₃

#1548

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

#6157

yes

characteristic

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

106

—

C₂×Q₈

—

C₂×Q₈

—

106

—

100

C₂×Q₈

—

212

—

101

C₂×Q₈

C₂×#6157

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

—

106

M₁₆

—

212

—

107

C₂×C₈

—

106

—

108

C₂×C₈

—

106

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

106

—

112

C₄⋊C₄

—

106

—

113

C₄⋊C₄

—

106

—

114

C₄⋊C₄

—

106

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

C₂×#6157

yes

characteristic

119

C₂³

—

120

Q₈

—

121

Q₈

—

212

—

122

Q₈

—

212

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

106

—

130

C₂×C₄

—

106

—

131

C₂×C₄

—

106

—

132

C₂×C₄

—

212

—

133

C₂×C₄

C₂²⋊#6157

yes

characteristic

134

C₂×C₄

C₂²⋊#6157

yes

characteristic

135

C₂×C₄

C₂²×#6157

yes

characteristic

136

C₈

—

212

—

137

C₈

—

212

—

138

C₂²

—

139

C₂²

—

106

—

140

C₂²

—

106

—

141

C₂²

C₂×C₂²⋊#6157

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

212

—

150

C₄

#13133280

yes

—

151

C₂

—

106

—

152

C₂

D₅₃⋊#100

yes

characteristic

153

C₂

C₂×#13133280

yes

characteristic

154

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	#1548
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×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 ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
3. #16811483	C₂	semidirect
82. C₅₃	#1548	semidirect

H ≅ G/N

type

3. #16811483

C₂

semidirect

82. C₅₃

#1548

semidirect

Subgroups — #20290229

Conjugacy classes of subgroups

Complements