Group #20287387 — subgroups

Subgroups — #20287387

Order 6784.

392 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811295	C₂	1	1	yes	maximal, characteristic
3	#16810970	C₂	1	3	yes	maximal, complemented
4	#16810914	C₂	1	3	yes	maximal, complemented
5	#16810824	C₂	1	3	yes	maximal, complemented
6	C₄×#11719913	C₂	1	3	yes	maximal, complemented
7	C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719949	—	2	1	no	—
9	C₂×#11719913	C₂²	1	3	yes	—
10	#13133217	C₂²	1	3	yes	—
11	#13133216	—	2	3	no	—
12	#13133214	C₂²	1	3	yes	—
13	#13133192	—	2	3	no	—
14	Q₈×Dic₅₃	—	2	1	no	—
15	C₄×Dic₁₀₆	C₂²	1	3	yes	—
16	C₄×D₂₁₂	C₂²	1	3	yes	—
17	#13133149	C₂²	1	3	yes	—
18	#13133142	C₂²	1	1	yes	characteristic
19	#13133139	C₂²	1	6	yes	—
20	#13133139	C₂²	1	3	yes	—
21	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₃	C₂²	1	3	yes	—
23	C₄×C₈×C₅₃	C₂²	1	3	yes	—
24	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
25	#11719949	—	4	4	no	—
26	C₂×D₂₁₂	—	2	3	no	—
27	C₂×D₂₁₂	—	2	3	no	—
28	C₂×D₂₁₂	C₂³	1	3	yes	—
29	C₂×C₄×D₅₃	—	2	3	no	—
30	C₂×Dic₁₀₆	—	2	3	no	—
31	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂³	1	3	yes	—
34	C₄⋊C₄×C₅₃	D₄	1	3	yes	—
35	C₄²×C₅₃	C₂³	1	3	yes	—
36	#11719921	—	2	3	no	—
37	C₄⋊Dic₅₃	—	2	3	no	—
38	C₄⋊Dic₅₃	C₂³	1	3	yes	—
39	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
40	#11719919	—	2	3	no	—
41	C₄×Dic₅₃	—	2	3	no	—
42	#11719913	—	2	6	no	—
43	C₂²×D₅₃	—	2	3	no	—
44	Q₈×C₅₃	SD₁₆	1	4	yes	complemented
45	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
46	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
47	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₅₃	—	2	1	no	—
49	C₂×Dic₅₃	—	2	3	no	—
50	C₂×Dic₅₃	—	2	1	no	—
51	D₂₁₂	—	2	3	no	—
52	D₂₁₂	—	2	3	no	—
53	D₂₁₂	—	4	6	no	—
54	D₂₁₂	—	8	3	no	—
55	C₄×D₅₃	—	4	6	no	—
56	Dic₁₀₆	—	2	1	no	—
57	Dic₁₀₆	—	8	3	no	—
58	Dic₁₀₆	#41	1	2	yes	complemented
59	C₈×C₅₃	—	2	3	no	—
60	C₈×C₅₃	—	2	1	no	—
61	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
62	C₂×D₅₃	—	4	3	no	—
63	C₂×D₅₃	—	4	3	no	—
64	C₂×D₅₃	—	4	3	no	—
65	C₄×C₅₃	—	2	3	no	—
66	C₄×C₅₃	C₂×SD₁₆	1	6	yes	—
67	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
68	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
69	Dic₅₃	—	2	2	no	—
70	Dic₅₃	—	4	1	no	—
71	Dic₅₃	—	4	3	no	—
72	#3362	—	53	1	no	maximal, Sylow
73	C₂×C₅₃	#548	1	1	yes	characteristic
74	C₂×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
75	C₂×C₅₃	#576	1	1	yes	characteristic
76	D₅₃	—	8	3	no	—
77	Q₈²	—	53	1	no	—
78	#550	—	53	1	no	—
79	#486	—	53	3	no	—
80	#476	—	53	3	no	—
81	#460	—	53	3	no	—
82	C₈×Q₈	D₅₃	1	1	yes	characteristic, complemented
83	C₄×SD₁₆	—	53	3	no	—
84	C₅₃	#3362	1	1	yes	characteristic, complemented, Sylow
85	C₂×#41	—	106	1	no	—
86	C₂×SD₁₆	—	53	3	no	—
87	C₄⋊Q₈	—	53	3	no	—
88	C₄⋊Q₈	—	106	3	no	—
89	#127	—	53	3	no	—
90	#121	—	106	3	no	—
91	C₄×Q₈	—	53	3	no	—
92	C₄×Q₈	—	106	1	no	—
93	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
94	C₄×D₄	—	53	3	no	—
95	#106	—	53	3	no	—
96	C₄⋊C₈	C₂×D₅₃	1	3	yes	—
97	#103	—	53	1	no	—
98	#102	—	53	3	no	—
99	#102	—	53	6	no	—
100	C₄×C₈	C₂×D₅₃	1	3	yes	—
101	#41	—	212	4	no	—
102	C₂×Q₈	—	53	1	no	—
103	C₂×Q₈	—	106	3	no	—
104	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
105	C₂×D₄	—	53	3	no	—
106	C₂×D₄	—	106	3	no	—
107	C₂×D₄	—	106	3	no	—
108	C₂²×C₄	—	106	3	no	—
109	SD₁₆	—	106	6	no	—
110	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
111	C₂×C₈	C₂²×D₅₃	1	3	yes	—
112	C₄⋊C₄	—	53	3	no	—
113	C₄⋊C₄	—	53	1	no	—
114	C₄⋊C₄	—	106	3	no	—
115	C₄⋊C₄	—	106	3	no	—
116	C₄⋊C₄	D₂₁₂	1	3	yes	—
117	C₂²⋊C₄	—	106	3	no	—
118	C₄²	—	106	3	no	—
119	C₄²	C₂²×D₅₃	1	3	yes	—
120	C₂³	—	106	3	no	—
121	Q₈	—	53	2	no	—
122	Q₈	—	106	1	no	—
123	Q₈	—	424	3	no	—
124	Q₈	#11719913	1	4	yes	complemented
125	D₄	—	106	3	no	—
126	D₄	—	106	3	no	—
127	D₄	—	424	3	no	—
128	D₄	—	212	6	no	—
129	C₂×C₄	—	106	1	no	—
130	C₂×C₄	—	106	3	no	—
131	C₂×C₄	—	106	1	no	—
132	C₂×C₄	—	212	6	no	—
133	C₂×C₄	C₂×D₂₁₂	1	3	yes	—
134	C₂×C₄	C₂×D₂₁₂	1	3	yes	—
135	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
136	C₈	—	2	3	no	—
137	C₈	—	2	1	no	—
138	C₂²	—	212	3	no	—
139	C₂²	—	212	3	no	—
140	C₂²	—	212	3	no	—
141	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
142	C₄	—	2	3	no	—
143	C₄	—	106	2	no	—
144	C₄	—	212	1	no	—
145	C₄	—	212	3	no	—
146	C₄	C₂×#11719913	1	6	yes	—
147	C₄	#41×D₅₃	1	1	yes	characteristic
148	C₄	#13133265	1	1	yes	characteristic
149	C₂	—	424	3	no	—
150	C₂	#16811285	1	1	yes	characteristic
151	C₂	C₂²×#11719913	1	1	yes	characteristic
152	C₂	#16811428	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₀₆

C₂

yes

maximal, characteristic, complemented

#16811295

C₂

yes

maximal, characteristic

#16810970

C₂

yes

maximal, complemented

#16810914

C₂

yes

maximal, complemented

#16810824

C₂

yes

maximal, complemented

C₄×#11719913

C₂

yes

maximal, complemented

C₈×Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#11719949

—

C₂×#11719913

C₂²

yes

—

#13133217

C₂²

yes

—

#13133216

—

#13133214

C₂²

yes

—

#13133192

—

Q₈×Dic₅₃

—

C₄×Dic₁₀₆

C₂²

yes

—

C₄×D₂₁₂

C₂²

yes

—

#13133149

C₂²

yes

—

#13133142

C₂²

yes

characteristic

#13133139

C₂²

yes

—

#13133139

C₂²

yes

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

#11719949

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂³

yes

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

characteristic

#11719919

—

C₄×Dic₅₃

—

#11719913

—

C₂²×D₅₃

—

Q₈×C₅₃

SD₁₆

yes

complemented

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

#41

yes

complemented

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

—

C₄×C₅₃

C₂×#41

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂²×SD₁₆

yes

characteristic

C₂×C₅₃

#576

yes

characteristic

D₅₃

—

Q₈²

—

—

—

—

—

C₈×Q₈

D₅₃

yes

characteristic, complemented

C₄×SD₁₆

—

C₅₃

#3362

yes

characteristic, complemented, Sylow

C₂×#41

—

106

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

#127

—

#121

—

106

—

C₄×Q₈

—

C₄×Q₈

—

106

—

C₄×Q₈

C₂×D₅₃

yes

characteristic

C₄×D₄

—

#106

—

C₄⋊C₈

C₂×D₅₃

yes

—

#103

—

#102

—

#102

—

100

C₄×C₈

C₂×D₅₃

yes

—

101

#41

—

212

—

102

C₂×Q₈

—

103

C₂×Q₈

—

106

—

104

C₂×Q₈

D₂₁₂

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

106

—

107

C₂×D₄

—

106

—

108

C₂²×C₄

—

106

—

109

SD₁₆

—

106

—

110

C₂×C₈

C₂²×D₅₃

yes

characteristic

111

C₂×C₈

C₂²×D₅₃

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

106

—

115

C₄⋊C₄

—

106

—

116

C₄⋊C₄

D₂₁₂

yes

—

117

C₂²⋊C₄

—

106

—

118

C₄²

—

106

—

119

C₄²

C₂²×D₅₃

yes

—

120

C₂³

—

106

—

121

Q₈

—

122

Q₈

—

106

—

123

Q₈

—

424

—

124

Q₈

#11719913

yes

complemented

125

D₄

—

106

—

126

D₄

—

106

—

127

D₄

—

424

—

128

D₄

—

212

—

129

C₂×C₄

—

106

—

130

C₂×C₄

—

106

—

131

C₂×C₄

—

106

—

132

C₂×C₄

—

212

—

133

C₂×C₄

C₂×D₂₁₂

yes

—

134

C₂×C₄

C₂×D₂₁₂

yes

—

135

C₂×C₄

C₂³×D₅₃

yes

characteristic

136

C₈

—

137

C₈

—

138

C₂²

—

212

—

139

C₂²

—

212

—

140

C₂²

—

212

—

141

C₂²

C₂²×D₂₁₂

yes

characteristic

142

C₄

—

143

C₄

—

106

—

144

C₄

—

212

—

145

C₄

—

212

—

146

C₄

C₂×#11719913

yes

—

147

C₄

#41×D₅₃

yes

characteristic

148

C₄

#13133265

yes

characteristic

149

C₂

—

424

—

150

C₂

#16811285

yes

characteristic

151

C₂

C₂²×#11719913

yes

characteristic

152

C₂

#16811428

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₈×Q₈	D₅₃
hypocenter	C₅₃	#3362
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₈×Q₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂²×D₄

hypercenter

C₈×Q₈

D₅₃

hypocenter

C₅₃

#3362

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. Q₈×Dic₁₀₆	C₂	semidirect
3. #16810970	C₂	semidirect
4. #16810914	C₂	semidirect
5. #16810824	C₂	semidirect
6. C₄×#11719913	C₂	semidirect
7. C₈×Q₈×C₅₃	C₂	semidirect
44. Q₈×C₅₃	SD₁₆	semidirect
58. Dic₁₀₆	#41	semidirect
82. C₈×Q₈	D₅₃	semidirect
84. C₅₃	#3362	semidirect
124. Q₈	#11719913	semidirect

H ≅ G/N

type

1. Q₈×Dic₁₀₆

C₂

semidirect

3. #16810970

C₂

semidirect

4. #16810914

C₂

semidirect

5. #16810824

C₂

semidirect

6. C₄×#11719913

C₂

semidirect

7. C₈×Q₈×C₅₃

C₂

semidirect

44. Q₈×C₅₃

SD₁₆

semidirect

58. Dic₁₀₆

#41

semidirect

82. C₈×Q₈

D₅₃

semidirect

84. C₅₃

#3362

semidirect

124. Q₈

#11719913

semidirect

Subgroups — #20287387

Conjugacy classes of subgroups

Complements