Group #20277551 — subgroups

Subgroups — Q₈×C₂²⋊Dic₅₃ #20277551

Order 6784.

820 conjugacy classes of subgroups; only the 333 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 333 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	C₂×Q₈×Dic₅₃	C₂	1	2	yes	maximal, complemented
2	#16810530	C₂	1	6	yes	maximal, complemented
3	#16810483	C₂	1	3	yes	maximal
4	C₄×C₂²⋊Dic₅₃	C₂	1	3	yes	maximal
5	C₂³×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₅₃	—	2	8	no	—
7	C₂×C₄⋊Dic₅₃	C₂²	1	6	yes	—
8	C₂×C₂²⋊Dic₅₃	C₂²	1	3	yes	—
9	C₂×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₅₃	C₂²	1	6	yes	—
11	#13133124	C₂²	1	6	yes	—
12	#13133124	C₂²	1	6	yes	—
13	C₂²×Q₈×C₅₃	—	2	8	no	—
14	C₂²×Q₈×C₅₃	C₄	1	8	yes	complemented
15	C₂²×Q₈×C₅₃	C₂²	1	3	yes	—
16	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₅₃	C₂²	1	3	yes	—
18	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
19	C₂×Q₈×C₅₃	—	2	32	no	—
20	C₂×Q₈×C₅₃	—	2	12	no	—
21	C₂×Q₈×C₅₃	C₂×C₄	1	12	yes	—
22	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	—
23	C₂×Q₈×C₅₃	D₄	1	8	yes	—
24	C₂²×C₄×C₅₃	—	2	12	no	—
25	C₂²×C₄×C₅₃	C₂×C₄	1	12	yes	—
26	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
27	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
28	C₂²×Dic₅₃	C₂³	1	6	yes	—
29	C₂²×Dic₅₃	C₂³	1	2	yes	—
30	C₂²⋊Dic₅₃	—	2	6	no	—
31	C₂²⋊Dic₅₃	Q₈	1	4	yes	direct product factor
32	C₄⋊Dic₅₃	—	2	12	no	—
33	C₄×Dic₅₃	—	2	12	no	—
34	C₂³×C₅₃	—	2	2	no	—
35	C₂³×C₅₃	—	2	2	no	—
36	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
37	C₂³×C₅₃	C₂×Q₈	1	2	yes	—
38	C₂³×C₅₃	#41	1	2	yes	—
39	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
40	Q₈×C₅₃	—	2	24	no	—
41	Q₈×C₅₃	C₂²⋊C₄	1	16	yes	complemented
42	C₂×C₄×C₅₃	—	2	6	no	—
43	C₂×C₄×C₅₃	—	2	24	no	—
44	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
45	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
46	C₂×C₄×C₅₃	C₂×D₄	1	12	yes	—
47	C₂×Dic₅₃	—	2	2	no	—
48	C₂×Dic₅₃	—	2	2	no	—
49	C₂×Dic₅₃	—	2	6	no	—
50	C₂×Dic₅₃	—	2	6	no	—
51	C₂×Dic₅₃	—	2	6	no	—
52	C₂×Dic₅₃	C₂×Q₈	1	4	yes	complemented
53	C₂²×C₅₃	—	2	2	no	—
54	C₂²×C₅₃	—	2	2	no	—
55	C₂²×C₅₃	—	2	4	no	—
56	C₂²×C₅₃	—	2	4	no	—
57	C₂²×C₅₃	C₄×Q₈	1	4	yes	complemented
58	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
59	C₂²×C₅₃	C₂²×D₄	1	2	yes	—
60	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
61	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
62	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
63	C₄×C₅₃	—	2	6	no	—
64	C₄×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
65	Dic₅₃	—	2	4	no	—
66	Dic₅₃	—	4	6	no	—
67	Q₈×C₂²⋊C₄	—	53	1	no	maximal, Sylow
68	C₂×C₅₃	—	2	4	no	—
69	C₂×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
71	C₂×C₅₃	#519	1	1	yes	characteristic
72	C₂×C₅₃	#548	1	2	yes	—
73	C₂×C₅₃	D₄×Q₈	1	2	yes	—
74	C₂³×Q₈	D₅₃	1	1	yes	characteristic
75	C₂×C₄×Q₈	—	53	2	no	—
76	#391	—	53	6	no	—
77	#380	—	53	3	no	—
78	C₄×C₂²⋊C₄	—	53	3	no	—
79	C₅₃	Q₈×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
80	C₂²×Q₈	—	2	8	no	—
81	C₂²×Q₈	Dic₅₃	1	8	yes	complemented
82	C₂²×Q₈	C₂×D₅₃	1	3	yes	—
83	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
84	C₂³×C₄	C₂×D₅₃	1	3	yes	—
85	C₄×Q₈	—	106	8	no	—
86	C₂×C₄⋊C₄	—	53	6	no	—
87	C₂×C₂²⋊C₄	—	53	1	no	—
88	C₂×C₂²⋊C₄	—	53	3	no	—
89	C₂×C₄²	—	53	6	no	—
90	#95	—	53	6	no	—
91	#95	—	53	6	no	—
92	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
93	C₂×Q₈	—	2	32	no	—
94	C₂×Q₈	—	2	12	no	—
95	C₂×Q₈	C₂×Dic₅₃	1	12	yes	—
96	C₂×Q₈	C₂×Dic₅₃	1	4	yes	—
97	C₂×Q₈	C₂²⋊D₅₃	1	8	yes	—
98	C₂²×C₄	—	2	12	no	—
99	C₂²×C₄	—	53	2	no	—
100	C₂²×C₄	—	53	6	no	—
101	C₂²×C₄	C₂×Dic₅₃	1	12	yes	—
102	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
103	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
104	C₄⋊C₄	—	106	12	no	—
105	C₂²⋊C₄	—	53	4	no	—
106	C₂²⋊C₄	—	106	6	no	—
107	C₄²	—	106	12	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	Q₈×D₅₃	1	2	yes	—
111	C₂³	#11719949	1	2	yes	—
112	C₂³	C₂²×Dic₅₃	1	2	yes	—
113	C₂³	C₂³×D₅₃	1	1	yes	characteristic
114	Q₈	—	2	24	no	—
115	Q₈	C₂²⋊Dic₅₃	1	16	yes	direct product factor
116	C₂×C₄	—	2	6	no	—
117	C₂×C₄	—	2	24	no	—
118	C₂×C₄	—	53	4	no	—
119	C₂×C₄	—	106	2	no	—
120	C₂×C₄	—	106	2	no	—
121	C₂×C₄	—	106	6	no	—
122	C₂×C₄	—	106	6	no	—
123	C₂×C₄	—	106	6	no	—
124	C₂×C₄	C₂²×Dic₅₃	1	6	yes	—
125	C₂×C₄	C₂²×Dic₅₃	1	6	yes	—
126	C₂×C₄	C₂×C₂²⋊D₅₃	1	12	yes	—
127	C₂²	—	2	4	no	—
128	C₂²	—	2	4	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	2	no	—
131	C₂²	Q₈×Dic₅₃	1	4	yes	complemented
132	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
133	C₂²	C₂²×C₂²⋊D₅₃	1	2	yes	—
134	C₂²	C₂×Q₈×D₅₃	1	1	yes	characteristic
135	C₂²	C₂×#11719949	1	1	yes	characteristic
136	C₂²	#13133265	1	2	yes	—
137	C₄	—	2	6	no	—
138	C₄	—	106	4	no	—
139	C₄	—	212	6	no	—
140	C₄	C₂×C₂²⋊Dic₅₃	1	12	yes	—
141	C₂	—	2	4	no	—
142	C₂	C₂²×C₂²⋊Dic₅₃	1	1	yes	characteristic
143	C₂	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
144	C₂	#16811101	1	1	yes	characteristic
145	C₂	#16811284	1	2	yes	—
146	C₂	Q₈×C₂²⋊D₅₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₅₃

C₂

yes

maximal, complemented

#16810530

C₂

yes

maximal, complemented

#16810483

C₂

yes

maximal

C₄×C₂²⋊Dic₅₃

C₂

yes

maximal

C₂³×Q₈×C₅₃

C₂

yes

maximal, characteristic

Q₈×Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

C₄

yes

complemented

C₂²×Q₈×C₅₃

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₂³×C₄×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

Q₈

yes

direct product factor

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²×C₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

—

C₂³×C₅₃

#41

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

C₂²⋊C₄

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×Q₈

yes

complemented

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×Q₈

yes

complemented

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂²×C₅₃

C₂²×D₄

yes

—

C₂²×C₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

characteristic

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

—

Dic₅₃

—

Dic₅₃

—

Q₈×C₂²⋊C₄

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₅₃

C₂×C₄×Q₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

D₄×Q₈

yes

—

C₂³×Q₈

D₅₃

yes

characteristic

C₂×C₄×Q₈

—

#391

—

#380

—

C₄×C₂²⋊C₄

—

C₅₃

Q₈×C₂²⋊C₄

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

Dic₅₃

yes

complemented

C₂²×Q₈

C₂×D₅₃

yes

—

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₂³×C₄

C₂×D₅₃

yes

—

C₄×Q₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

#95

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂×Q₈

C₂²⋊D₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂×Dic₅₃

yes

—

102

C₂²×C₄

C₂²×D₅₃

yes

—

103

C₂²×C₄

C₂²×D₅₃

yes

—

104

C₄⋊C₄

—

106

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

—

106

—

107

C₄²

—

106

—

108

C₂³

—

109

C₂³

—

110

C₂³

Q₈×D₅₃

yes

—

111

C₂³

#11719949

yes

—

112

C₂³

C₂²×Dic₅₃

yes

—

113

C₂³

C₂³×D₅₃

yes

characteristic

114

Q₈

—

115

Q₈

C₂²⋊Dic₅₃

yes

direct product factor

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

106

—

120

C₂×C₄

—

106

—

121

C₂×C₄

—

106

—

122

C₂×C₄

—

106

—

123

C₂×C₄

—

106

—

124

C₂×C₄

C₂²×Dic₅₃

yes

—

125

C₂×C₄

C₂²×Dic₅₃

yes

—

126

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

Q₈×Dic₅₃

yes

complemented

132

C₂²

C₂³×Dic₅₃

yes

characteristic

133

C₂²

C₂²×C₂²⋊D₅₃

yes

—

134

C₂²

C₂×Q₈×D₅₃

yes

characteristic

135

C₂²

C₂×#11719949

yes

characteristic

136

C₂²

#13133265

yes

—

137

C₄

—

138

C₄

—

106

—

139

C₄

—

212

—

140

C₄

C₂×C₂²⋊Dic₅₃

yes

—

141

C₂

—

142

C₂

C₂²×C₂²⋊Dic₅₃

yes

characteristic

143

C₂

C₂×Q₈×Dic₅₃

yes

characteristic

144

C₂

#16811101

yes

characteristic

145

C₂

#16811284

yes

—

146

C₂

Q₈×C₂²⋊D₅₃

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₄
Fitting	C₂³×Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂³×Q₈	D₅₃
hypocenter	C₅₃	Q₈×C₂²⋊C₄
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator

C₂²×C₅₃

C₂³×C₄

Fitting

C₂³×Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂⁴

hypercenter

C₂³×Q₈

D₅₃

hypocenter

C₅₃

Q₈×C₂²⋊C₄

LCS

G ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂³×Q₈

N	H ≅ G/N	type
1. C₂×Q₈×Dic₅₃	C₂	semidirect
2. #16810530	C₂	semidirect
14. C₂²×Q₈×C₅₃	C₄	semidirect
31. C₂²⋊Dic₅₃	Q₈	direct (N × H)
41. Q₈×C₅₃	C₂²⋊C₄	semidirect
52. C₂×Dic₅₃	C₂×Q₈	semidirect
57. C₂²×C₅₃	C₄×Q₈	semidirect
79. C₅₃	Q₈×C₂²⋊C₄	semidirect
81. C₂²×Q₈	Dic₅₃	semidirect
115. Q₈	C₂²⋊Dic₅₃	direct (N × H)
131. C₂²	Q₈×Dic₅₃	semidirect

H ≅ G/N

type

1. C₂×Q₈×Dic₅₃

C₂

semidirect

2. #16810530

C₂

semidirect

14. C₂²×Q₈×C₅₃

C₄

semidirect

31. C₂²⋊Dic₅₃

Q₈

direct (N × H)

41. Q₈×C₅₃

C₂²⋊C₄

semidirect

52. C₂×Dic₅₃

C₂×Q₈

semidirect

57. C₂²×C₅₃

C₄×Q₈

semidirect

79. C₅₃

Q₈×C₂²⋊C₄

semidirect

81. C₂²×Q₈

Dic₅₃

semidirect

115. Q₈

C₂²⋊Dic₅₃

direct (N × H)

131. C₂²

Q₈×Dic₅₃

semidirect

Subgroups — Q₈×C₂²⋊Dic₅₃ #20277551

Conjugacy classes of subgroups

Complements

Subgroups — Q8×C22⋊Dic53 #20277551

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×C₂²⋊Dic₅₃ #20277551