Group #20273266 — subgroups

Subgroups — #20273266

Order 6784.

208 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 77 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	#16810680	C₂	1	1	yes	maximal, characteristic
2	#16810491	C₂	1	1	yes	maximal, characteristic
3	#16810360	C₂	1	4	yes	maximal
4	C₅₃×#533	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#88322	—	2	1	no	—
6	C₂²⋊#88322	C₂²	1	2	yes	—
7	#13133128	—	2	1	no	—
8	#13133124	—	2	1	no	—
9	#13133124	—	2	1	no	—
10	#13133124	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
12	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
13	#125×C₅₃	—	2	1	no	—
14	C₂²⋊Q₈×C₅₃	—	2	1	no	—
15	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
16	C₄×Q₈×C₅₃	—	2	1	no	—
17	C₄×Q₈×C₅₃	C₄	1	2	yes	complemented
18	#117×C₅₃	—	2	1	no	—
19	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×Q₈×C₅₃	—	2	1	no	—
21	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅₃	—	2	2	no	—
26	C₄⋊C₄×C₅₃	—	2	1	no	—
27	C₄⋊C₄×C₅₃	—	2	1	no	—
28	C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²⋊C₄×C₅₃	—	2	2	no	—
33	C₄²×C₅₃	—	2	2	no	—
34	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₅₃	D₄	1	1	yes	characteristic
37	C₄²×C₅₃	D₄	1	1	yes	characteristic
38	C₂×#88322	—	2	2	no	—
39	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₅₃	—	4	2	no	—
41	Q₈×C₅₃	—	4	2	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	4	1	no	—
46	C₂×C₄×C₅₃	—	4	1	no	—
47	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₅₃	—	4	2	no	—
54	C₂×Dic₅₃	—	4	2	no	—
55	C₂×Dic₅₃	—	4	2	no	—
56	#88322	—	4	2	no	—
57	C₂²×C₅₃	—	2	1	no	—
58	C₂²×C₅₃	—	2	1	no	—
59	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₅₃	#137	1	2	yes	—
61	C₄×C₅₃	—	2	1	no	—
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	—	2	1	no	—
64	C₄×C₅₃	—	2	1	no	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	2	1	no	—
67	C₄×C₅₃	—	4	2	no	—
68	C₄×C₅₃	—	4	2	no	—
69	Dic₅₃	—	8	2	no	—
70	#1515	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	—	2	2	no	—
72	C₂×C₅₃	#410	1	1	yes	characteristic
73	C₂×C₅₃	#419	1	1	yes	characteristic
74	C₂×C₅₃	#421	1	1	yes	characteristic
75	#533	D₅₃	1	1	yes	characteristic
76	#432	—	53	1	no	—
77	#383	—	53	1	no	—
78	#328	—	53	4	no	—
79	C₅₃	#1515	1	1	yes	characteristic, complemented, Sylow
80	C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic, complemented
81	C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic, complemented
82	#125	—	2	1	no	—
83	C₂²⋊Q₈	—	2	1	no	—
84	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	—
85	C₄×Q₈	—	2	1	no	—
86	C₄×Q₈	Dic₅₃	1	2	yes	complemented
87	#117	—	2	1	no	—
88	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
89	C₄⋊C₈	—	106	1	no	—
90	C₂²⋊C₈	—	53	2	no	—
91	#97	—	106	1	no	—
92	#95	—	53	2	no	—
93	#95	—	106	1	no	—
94	#95	—	106	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
97	C₂²×C₄	—	106	2	no	—
98	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
101	C₂×C₈	—	106	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
111	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
112	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
113	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
114	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
115	Q₈	—	4	2	no	—
116	Q₈	—	4	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	212	2	no	—
123	C₂×C₄	—	212	2	no	—
124	C₂×C₄	—	212	2	no	—
125	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
126	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
127	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
128	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
129	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
131	C₈	—	212	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
135	C₂²	#13133248	1	2	yes	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	4	2	no	—
143	C₄	—	4	2	no	—
144	C₄	—	424	2	no	—
145	C₂	—	2	2	no	—
146	C₂	#16810597	1	1	yes	characteristic
147	C₂	#16810632	1	1	yes	characteristic
148	C₂	#16810640	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810680

C₂

yes

maximal, characteristic

#16810491

C₂

yes

maximal, characteristic

#16810360

C₂

yes

maximal

C₅₃×#533

C₂

yes

maximal, characteristic

C₄⋊#88322

—

C₂²⋊#88322

C₂²

yes

—

—

—

—

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₄

yes

characteristic, complemented

C₄⋊Q₈×C₅₃

C₄

yes

characteristic, complemented

#125×C₅₃

—

C₂²⋊Q₈×C₅₃

—

C₂²⋊Q₈×C₅₃

C₂²

yes

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

C₄

yes

complemented

#117×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₂×#88322

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×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×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#137

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

#1515

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

—

—

—

C₅₃

#1515

yes

characteristic, complemented, Sylow

C₄⋊Q₈

Dic₅₃

yes

characteristic, complemented

C₄⋊Q₈

Dic₅₃

yes

characteristic, complemented

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂×D₅₃

yes

—

C₄×Q₈

—

C₄×Q₈

Dic₅₃

yes

complemented

#117

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

#97

—

106

—

#95

—

#95

—

106

—

#95

—

106

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

100

C₂²×C₄

C₂²×D₅₃

yes

characteristic

101

C₂×C₈

—

106

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂×Dic₅₃

yes

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₄²

C₂×Dic₅₃

yes

characteristic

111

C₄²

C₂×Dic₅₃

yes

characteristic

112

C₄²

C₂²⋊D₅₃

yes

characteristic

113

C₄²

C₂²⋊D₅₃

yes

characteristic

114

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

115

Q₈

—

116

Q₈

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

212

—

123

C₂×C₄

—

212

—

124

C₂×C₄

—

212

—

125

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

126

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

127

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

128

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

129

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

131

C₈

—

212

—

132

C₂²

—

133

C₂²

—

134

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

135

C₂²

#13133248

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

424

—

145

C₂

—

146

C₂

#16810597

yes

characteristic

147

C₂

#16810632

yes

characteristic

148

C₂

#16810640

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#533	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#533	D₅₃
hypocenter	C₅₃	#1515
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#533

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
11. C₄⋊Q₈×C₅₃	C₄	semidirect
12. C₄⋊Q₈×C₅₃	C₄	semidirect
17. C₄×Q₈×C₅₃	C₄	semidirect
79. C₅₃	#1515	semidirect
80. C₄⋊Q₈	Dic₅₃	semidirect
81. C₄⋊Q₈	Dic₅₃	semidirect
86. C₄×Q₈	Dic₅₃	semidirect

H ≅ G/N

type

11. C₄⋊Q₈×C₅₃

C₄

semidirect

12. C₄⋊Q₈×C₅₃

C₄

semidirect

17. C₄×Q₈×C₅₃

C₄

semidirect

79. C₅₃

#1515

semidirect

80. C₄⋊Q₈

Dic₅₃

semidirect

81. C₄⋊Q₈

Dic₅₃

semidirect

86. C₄×Q₈

Dic₅₃

semidirect

Subgroups — #20273266

Conjugacy classes of subgroups

Complements