Group #20287576 — subgroups

Subgroups — Q₈×#11719913 #20287576

Order 6784.

370 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 149 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	Q₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#16811021	C₂	1	3	yes	maximal, complemented
4	#16810914	C₂	1	3	yes	maximal
5	#16810909	C₂	1	3	yes	maximal, complemented
6	C₄×#11719913	C₂	1	3	yes	maximal
7	C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₅₃	—	2	1	no	—
9	C₂×#11719913	C₂²	1	1	yes	characteristic
10	#13133217	C₂²	1	3	yes	—
11	#13133217	C₂²	1	3	yes	—
12	#13133216	—	2	3	no	—
13	#13133197	—	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	#13133149	C₂²	1	6	yes	—
19	#13133142	C₂²	1	3	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	Q₈×D₅₃	—	4	4	no	—
26	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
27	C₂×C₄×D₅₃	—	2	3	no	—
28	C₂×Dic₁₀₆	—	2	3	no	—
29	C₂×Dic₁₀₆	—	2	3	no	—
30	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₂³	1	3	yes	—
32	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₅₃	D₄	1	3	yes	—
34	C₄²×C₅₃	C₂³	1	3	yes	—
35	#11719921	—	2	3	no	—
36	C₄⋊Dic₅₃	—	2	3	no	—
37	C₄⋊Dic₅₃	—	2	3	no	—
38	C₄⋊Dic₅₃	C₂³	1	3	yes	—
39	C₄⋊Dic₅₃	C₂³	1	3	yes	—
40	#11719919	—	2	3	no	—
41	C₄×Dic₅₃	—	2	3	no	—
42	#11719913	Q₈	1	4	yes	direct product factor
43	C₂²×D₅₃	—	2	1	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	3	no	—
51	D₂₁₂	—	2	1	no	—
52	D₂₁₂	C₂×Q₈	1	2	yes	—
53	C₄×D₅₃	—	4	6	no	—
54	Dic₁₀₆	—	2	1	no	—
55	Dic₁₀₆	—	4	6	no	—
56	Dic₁₀₆	—	8	3	no	—
57	Dic₁₀₆	C₂×Q₈	1	2	yes	complemented
58	C₈×C₅₃	—	2	3	no	—
59	C₈×C₅₃	C₂×Q₈	1	2	yes	complemented
60	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
61	C₂×D₅₃	—	2	2	no	—
62	C₂×D₅₃	—	4	1	no	—
63	C₂×D₅₃	—	4	1	no	—
64	C₄×C₅₃	—	2	3	no	—
65	C₄×C₅₃	C₂×SD₁₆	1	6	yes	—
66	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
67	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
68	Dic₅₃	—	2	2	no	—
69	Dic₅₃	—	4	3	no	—
70	Dic₅₃	—	4	3	no	—
71	Q₈×SD₁₆	—	53	1	no	maximal, Sylow
72	C₂×C₅₃	D₄×Q₈	1	1	yes	characteristic
73	C₂×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
74	C₂×C₅₃	#577	1	1	yes	characteristic
75	D₅₃	—	4	2	no	—
76	Q₈²	—	53	1	no	—
77	D₄×Q₈	—	53	1	no	—
78	#498	—	53	3	no	—
79	#476	—	53	3	no	—
80	#475	—	53	3	no	—
81	C₈×Q₈	D₅₃	1	1	yes	characteristic, complemented
82	C₄×SD₁₆	—	53	3	no	—
83	C₅₃	Q₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
84	C₂²×Q₈	—	106	1	no	—
85	C₂×SD₁₆	—	53	1	no	—
86	C₄⋊Q₈	—	53	3	no	—
87	C₄⋊Q₈	—	53	3	no	—
88	C₄⋊Q₈	—	106	3	no	—
89	C₂²⋊Q₈	—	106	3	no	—
90	C₄×Q₈	—	53	3	no	—
91	C₄×Q₈	—	106	1	no	—
92	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
93	C₄×D₄	—	53	3	no	—
94	#106	—	53	3	no	—
95	#106	—	53	6	no	—
96	C₄⋊C₈	C₂×D₅₃	1	3	yes	—
97	#103	—	53	3	no	—
98	#102	—	53	3	no	—
99	C₄×C₈	C₂×D₅₃	1	3	yes	—
100	C₂×Q₈	—	53	1	no	—
101	C₂×Q₈	—	106	3	no	—
102	C₂×Q₈	—	106	3	no	—
103	C₂×Q₈	—	212	4	no	—
104	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
105	C₂×D₄	—	53	1	no	—
106	C₂²×C₄	—	106	3	no	—
107	SD₁₆	—	53	4	no	—
108	C₂×C₈	C₂²×D₅₃	1	3	yes	—
109	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
110	C₄⋊C₄	—	53	3	no	—
111	C₄⋊C₄	—	53	3	no	—
112	C₄⋊C₄	—	106	3	no	—
113	C₄⋊C₄	—	106	3	no	—
114	C₄⋊C₄	—	106	3	no	—
115	C₄⋊C₄	D₂₁₂	1	3	yes	—
116	C₂²⋊C₄	—	106	3	no	—
117	C₄²	—	106	3	no	—
118	C₄²	C₂²×D₅₃	1	3	yes	—
119	C₂³	—	106	1	no	—
120	Q₈	—	53	2	no	—
121	Q₈	—	106	1	no	—
122	Q₈	—	424	3	no	—
123	Q₈	—	212	6	no	—
124	Q₈	#11719913	1	4	yes	direct product factor
125	D₄	—	53	2	no	—
126	D₄	—	106	1	no	—
127	C₂×C₄	—	106	1	no	—
128	C₂×C₄	—	106	3	no	—
129	C₂×C₄	—	106	3	no	—
130	C₂×C₄	—	212	6	no	—
131	C₂×C₄	C₂×D₂₁₂	1	3	yes	—
132	C₂×C₄	C₂×D₂₁₂	1	3	yes	—
133	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
134	C₈	—	2	3	no	—
135	C₈	Q₈×D₅₃	1	2	yes	complemented
136	C₂²	—	106	2	no	—
137	C₂²	—	212	1	no	—
138	C₂²	—	212	1	no	—
139	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
140	C₄	—	2	3	no	—
141	C₄	—	106	2	no	—
142	C₄	—	212	3	no	—
143	C₄	—	212	3	no	—
144	C₄	C₂×#11719913	1	6	yes	—
145	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
146	C₄	#13133264	1	1	yes	characteristic
147	C₂	—	212	2	no	—
148	C₂	Q₈×D₂₁₂	1	1	yes	characteristic
149	C₂	C₂²×#11719913	1	1	yes	characteristic
150	C₂	#16811435	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₀₆

C₂

yes

maximal, characteristic, complemented

Q₈×D₂₁₂

C₂

yes

maximal, characteristic

#16811021

C₂

yes

maximal, complemented

#16810914

C₂

yes

maximal

#16810909

C₂

yes

maximal, complemented

C₄×#11719913

C₂

yes

maximal

C₈×Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₅₃

—

C₂×#11719913

C₂²

yes

characteristic

#13133217

C₂²

yes

—

#13133217

C₂²

yes

—

#13133216

—

#13133197

—

Q₈×Dic₅₃

—

C₄×Dic₁₀₆

C₂²

yes

—

C₄×D₂₁₂

C₂²

yes

—

#13133149

C₂²

yes

—

#13133149

C₂²

yes

—

#13133142

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

Q₈×D₅₃

—

C₂×D₂₁₂

C₂³

yes

characteristic

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

#11719913

Q₈

yes

direct product factor

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

C₂×Q₈

yes

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

C₂×Q₈

yes

complemented

C₈×C₅₃

—

C₈×C₅₃

C₂×Q₈

yes

complemented

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

—

C₄×C₅₃

C₂²×Q₈

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Q₈×SD₁₆

—

maximal, Sylow

C₂×C₅₃

D₄×Q₈

yes

characteristic

C₂×C₅₃

C₂²×SD₁₆

yes

characteristic

C₂×C₅₃

#577

yes

characteristic

D₅₃

—

Q₈²

—

D₄×Q₈

—

#498

—

#476

—

#475

—

C₈×Q₈

D₅₃

yes

characteristic, complemented

C₄×SD₁₆

—

C₅₃

Q₈×SD₁₆

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

C₄×Q₈

—

106

—

C₄×Q₈

C₂×D₅₃

yes

characteristic

C₄×D₄

—

#106

—

#106

—

C₄⋊C₈

C₂×D₅₃

yes

—

#103

—

#102

—

C₄×C₈

C₂×D₅₃

yes

—

100

C₂×Q₈

—

101

C₂×Q₈

—

106

—

102

C₂×Q₈

—

106

—

103

C₂×Q₈

—

212

—

104

C₂×Q₈

D₂₁₂

yes

characteristic

105

C₂×D₄

—

106

C₂²×C₄

—

106

—

107

SD₁₆

—

108

C₂×C₈

C₂²×D₅₃

yes

—

109

C₂×C₈

C₂²×D₅₃

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

106

—

113

C₄⋊C₄

—

106

—

114

C₄⋊C₄

—

106

—

115

C₄⋊C₄

D₂₁₂

yes

—

116

C₂²⋊C₄

—

106

—

117

C₄²

—

106

—

118

C₄²

C₂²×D₅₃

yes

—

119

C₂³

—

106

—

120

Q₈

—

121

Q₈

—

106

—

122

Q₈

—

424

—

123

Q₈

—

212

—

124

Q₈

#11719913

yes

direct product factor

125

D₄

—

126

D₄

—

106

—

127

C₂×C₄

—

106

—

128

C₂×C₄

—

106

—

129

C₂×C₄

—

106

—

130

C₂×C₄

—

212

—

131

C₂×C₄

C₂×D₂₁₂

yes

—

132

C₂×C₄

C₂×D₂₁₂

yes

—

133

C₂×C₄

C₂³×D₅₃

yes

characteristic

134

C₈

—

135

C₈

Q₈×D₅₃

yes

complemented

136

C₂²

—

106

—

137

C₂²

—

212

—

138

C₂²

—

212

—

139

C₂²

C₂²×D₂₁₂

yes

characteristic

140

C₄

—

141

C₄

—

106

—

142

C₄

—

212

—

143

C₄

—

212

—

144

C₄

C₂×#11719913

yes

—

145

C₄

C₂×Q₈×D₅₃

yes

characteristic

146

C₄

#13133264

yes

characteristic

147

C₂

—

212

—

148

C₂

Q₈×D₂₁₂

yes

characteristic

149

C₂

C₂²×#11719913

yes

characteristic

150

C₂

#16811435

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	Q₈×SD₁₆
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

Q₈×SD₁₆

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. #16811021	C₂	semidirect
5. #16810909	C₂	semidirect
7. C₈×Q₈×C₅₃	C₂	semidirect
42. #11719913	Q₈	direct (N × H)
44. Q₈×C₅₃	SD₁₆	semidirect
57. Dic₁₀₆	C₂×Q₈	semidirect
59. C₈×C₅₃	C₂×Q₈	semidirect
81. C₈×Q₈	D₅₃	semidirect
83. C₅₃	Q₈×SD₁₆	semidirect
124. Q₈	#11719913	direct (N × H)
135. C₈	Q₈×D₅₃	semidirect

H ≅ G/N

type

1. Q₈×Dic₁₀₆

C₂

semidirect

3. #16811021

C₂

semidirect

5. #16810909

C₂

semidirect

7. C₈×Q₈×C₅₃

C₂

semidirect

42. #11719913

Q₈

direct (N × H)

44. Q₈×C₅₃

SD₁₆

semidirect

57. Dic₁₀₆

C₂×Q₈

semidirect

59. C₈×C₅₃

C₂×Q₈

semidirect

81. C₈×Q₈

D₅₃

semidirect

83. C₅₃

Q₈×SD₁₆

semidirect

124. Q₈

#11719913

direct (N × H)

135. C₈

Q₈×D₅₃

semidirect

Subgroups — Q₈×#11719913 #20287576

Conjugacy classes of subgroups

Complements

Subgroups — Q8×#11719913 #20287576

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×#11719913 #20287576