Group #20275109 — subgroups

Subgroups — #20275109

Order 6784.

324 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16811101	C₂	1	1	yes	maximal, characteristic
2	#16810632	C₂	1	2	yes	maximal
3	C₂×#13133136	C₂	1	1	yes	maximal, characteristic
4	#16810383	C₂	1	2	yes	maximal
5	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	Q₈×Dic₅₃	—	2	4	no	—
8	#13133176	—	2	2	no	—
9	#13133176	C₂²	1	2	yes	—
10	#13133141	—	2	4	no	—
11	#13133136	—	2	2	no	—
12	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₅₃	—	2	2	no	—
15	C₄⋊Q₈×C₅₃	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
17	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×Q₈×C₅₃	—	2	2	no	—
19	C₂×Q₈×C₅₃	—	2	1	no	—
20	C₂×Q₈×C₅₃	—	2	2	no	—
21	C₂×Q₈×C₅₃	—	2	2	no	—
22	C₂×Q₈×C₅₃	—	2	2	no	—
23	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×Q₈×C₅₃	D₄	1	2	yes	—
26	C₂²×C₄×C₅₃	—	2	1	no	—
27	C₂²×C₄×C₅₃	—	2	1	no	—
28	C₂²×C₄×C₅₃	D₄	1	2	yes	—
29	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₅₃	—	2	8	no	—
31	C₄²×C₅₃	—	2	1	no	—
32	C₄²×C₅₃	C₂×C₄	1	2	yes	—
33	C₂²⋊Dic₅₃	—	2	2	no	—
34	C₄⋊Dic₅₃	—	2	4	no	—
35	C₄⋊Dic₅₃	D₄	1	4	yes	—
36	C₄×Dic₅₃	—	2	4	no	—
37	C₄×Dic₅₃	—	2	2	no	—
38	#11719917	—	2	2	no	—
39	#11719917	—	2	2	no	—
40	C₂×#88322	—	2	2	no	—
41	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₅₃	—	2	4	no	—
43	Q₈×C₅₃	—	2	4	no	—
44	Q₈×C₅₃	—	2	4	no	—
45	Q₈×C₅₃	—	4	2	no	—
46	C₂×C₄×C₅₃	—	2	4	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	4	1	no	—
52	C₂×C₄×C₅₃	—	4	1	no	—
53	C₂×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
54	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#41	1	2	yes	—
58	C₂×Dic₅₃	—	2	4	no	—
59	#88322	—	4	2	no	—
60	C₂²×C₅₃	—	2	2	no	—
61	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₅₃	#120	1	1	yes	characteristic
63	C₂²×C₅₃	#123	1	1	yes	characteristic
64	C₄×C₅₃	—	2	4	no	—
65	C₄×C₅₃	—	2	4	no	—
66	C₄×C₅₃	—	4	2	no	—
67	C₄×C₅₃	C₄×D₄	1	2	yes	—
68	C₄×C₅₃	#121	1	2	yes	—
69	Dic₅₃	—	4	4	no	—
70	#1886	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	—	2	1	no	—
72	C₂×C₅₃	—	2	1	no	—
73	C₂×C₅₃	#386	1	1	yes	characteristic
74	C₂×C₅₃	#456	1	2	yes	—
75	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
76	#519	—	53	1	no	—
77	#419	—	53	2	no	—
78	C₂×#101	—	53	1	no	—
79	#339	—	53	2	no	—
80	C₅₃	#1886	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
82	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
83	C₂×M₁₆	—	53	2	no	—
84	C₄⋊Q₈	—	2	2	no	—
85	C₄⋊Q₈	Dic₅₃	1	4	yes	complemented
86	C₄×Q₈	—	106	4	no	—
87	#117	—	53	2	no	—
88	#117	—	106	2	no	—
89	C₂×C₄⋊C₄	—	2	2	no	—
90	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
91	#103	—	106	4	no	—
92	#101	—	106	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
99	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
100	C₂×Q₈	C₂²⋊D₅₃	1	2	yes	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
104	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
105	M₁₆	—	106	2	no	—
106	M₁₆	—	106	2	no	—
107	C₂×C₈	—	106	2	no	—
108	C₄⋊C₄	—	2	8	no	—
109	C₄⋊C₄	—	53	4	no	—
110	C₄⋊C₄	—	106	4	no	—
111	C₂²⋊C₄	—	106	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	106	4	no	—
114	C₄²	—	106	2	no	—
115	C₄²	C₂×Dic₅₃	1	2	yes	—
116	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
117	Q₈	—	2	4	no	—
118	Q₈	—	2	4	no	—
119	Q₈	—	2	4	no	—
120	Q₈	—	4	2	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	106	4	no	—
129	C₂×C₄	C₂²⋊Dic₅₃	1	4	yes	—
130	C₂×C₄	D₄×D₅₃	1	2	yes	—
131	C₂×C₄	#11719947	1	2	yes	—
132	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
134	C₈	—	212	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
137	C₂²	#13133185	1	1	yes	characteristic
138	C₂²	#13133198	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	2	4	no	—
141	C₄	—	4	2	no	—
142	C₄	—	212	4	no	—
143	C₄	D₄×Dic₅₃	1	2	yes	—
144	C₄	#13133189	1	2	yes	—
145	C₂	—	2	1	no	—
146	C₂	—	2	1	no	—
147	C₂	#16810503	1	1	yes	characteristic
148	C₂	#16810805	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811101

C₂

yes

maximal, characteristic

#16810632

C₂

yes

maximal

C₂×#13133136

C₂

yes

maximal, characteristic

#16810383

C₂

yes

maximal

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

Q₈×Dic₅₃

—

#13133176

—

#13133176

C₂²

yes

—

#13133141

—

#13133136

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

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

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

#121

yes

—

Dic₅₃

—

#1886

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#386

yes

characteristic

C₂×C₅₃

#456

yes

—

C₂×C₄⋊Q₈

D₅₃

yes

—

—

—

—

C₅₃

#1886

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₅₃

yes

complemented

C₄×Q₈

—

106

—

#117

—

#117

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#103

—

106

—

#101

—

106

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂×Q₈

C₂×Dic₅₃

yes

—

100

C₂×Q₈

C₂²⋊D₅₃

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²⋊D₅₃

yes

—

104

C₂²×C₄

C₂²×D₅₃

yes

characteristic

105

M₁₆

—

106

—

106

M₁₆

—

106

—

107

C₂×C₈

—

106

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

106

—

111

C₂²⋊C₄

—

106

—

112

C₄²

—

113

C₄²

—

106

—

114

C₄²

—

106

—

115

C₄²

C₂×Dic₅₃

yes

—

116

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

117

Q₈

—

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

106

—

129

C₂×C₄

C₂²⋊Dic₅₃

yes

—

130

C₂×C₄

D₄×D₅₃

yes

—

131

C₂×C₄

#11719947

yes

—

132

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

134

C₈

—

212

—

135

C₂²

—

136

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

137

C₂²

#13133185

yes

characteristic

138

C₂²

#13133198

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

212

—

143

C₄

D₄×Dic₅₃

yes

—

144

C₄

#13133189

yes

—

145

C₂

—

146

C₂

—

147

C₂

#16810503

yes

characteristic

148

C₂

#16810805

yes

—

	subgroup	G/H
center	C₂²	#13133185
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
socle	C₂²×C₅₃	#120
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#1886
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

#13133185

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

socle

C₂²×C₅₃

#120

hypercenter

C₂×C₄⋊Q₈

D₅₃

hypocenter

C₅₃

#1886

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
15. C₄⋊Q₈×C₅₃	C₄	semidirect
80. C₅₃	#1886	semidirect
85. C₄⋊Q₈	Dic₅₃	semidirect

H ≅ G/N

type

15. C₄⋊Q₈×C₅₃

C₄

semidirect

80. C₅₃

#1886

semidirect

85. C₄⋊Q₈

Dic₅₃

semidirect

Subgroups — #20275109

Conjugacy classes of subgroups

Complements