Group #20285225 — subgroups

Subgroups — #20285225

Order 6784.

356 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 157 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	#16811180	C₂	1	1	yes	maximal, characteristic
2	#16811026	C₂	1	2	yes	maximal, complemented
3	#16811022	C₂	1	4	yes	maximal, complemented
4	#16811019	C₂	1	2	yes	maximal, complemented
5	#16810659	C₂	1	4	yes	maximal
6	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#533	C₂	1	1	yes	maximal, characteristic
8	C₂²×#88322	C₂²	1	2	yes	—
9	#13133217	C₂²	1	2	yes	—
10	#13133209	C₂²	1	2	yes	—
11	#13133196	—	2	2	no	—
12	C₄×Dic₁₀₆	—	2	2	no	—
13	#13133176	C₂²	1	2	yes	—
14	#13133150	C₂²	1	8	yes	—
15	#13133148	C₂²	1	8	yes	—
16	C₄×#88322	C₂²	1	1	yes	characteristic
17	C₄×#88322	C₂²	1	2	yes	—
18	C₄×#88322	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
20	#125×C₅₃	C₂²	1	2	yes	—
21	C₂²⋊Q₈×C₅₃	—	2	2	no	—
22	C₄×Q₈×C₅₃	—	2	2	no	—
23	#117×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₅₃	—	2	2	no	—
26	C₂²×C₄×C₅₃	D₄	1	2	yes	—
27	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×Dic₁₀₆	—	2	2	no	—
29	C₄⋊C₄×C₅₃	—	2	4	no	—
30	C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
32	C₂²⋊C₄×C₅₃	—	2	2	no	—
33	C₄²×C₅₃	—	2	2	no	—
34	C₄²×C₅₃	D₄	1	2	yes	—
35	C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₂²⋊Dic₅₃	—	2	2	no	—
38	C₄⋊Dic₅₃	—	2	2	no	—
39	C₄⋊Dic₅₃	C₂³	1	4	yes	—
40	#11719919	—	2	4	no	—
41	C₄×Dic₅₃	—	2	2	no	—
42	C₂×#88322	Q₈	1	8	yes	complemented
43	C₂×#88322	C₂³	1	2	yes	—
44	C₂×#88322	C₂³	1	2	yes	—
45	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₅₃	—	4	4	no	—
47	C₂×C₄×C₅₃	—	2	4	no	—
48	C₂×C₄×C₅₃	—	2	4	no	—
49	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₅₃	—	2	4	no	—
55	Dic₁₀₆	—	4	4	no	—
56	#88322	C₂×Q₈	1	4	yes	—
57	#88322	C₂×Q₈	1	4	yes	—
58	C₂²×C₅₃	—	2	2	no	—
59	C₂²×C₅₃	C₄⋊Q₈	1	2	yes	—
60	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
61	C₄×C₅₃	—	2	2	no	—
62	C₄×C₅₃	—	4	4	no	—
63	C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
64	C₄×C₅₃	#135	1	4	yes	—
65	C₄×C₅₃	C₂²×Q₈	1	2	yes	—
66	Dic₅₃	—	4	4	no	—
67	#3160	—	53	1	no	maximal, Sylow
68	C₂×C₅₃	—	2	2	no	—
69	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₅₃	C₂×#135	1	2	yes	—
71	#533	—	53	1	no	—
72	#533	D₅₃	1	1	yes	characteristic
73	#500	—	53	2	no	—
74	#499	—	53	4	no	—
75	#498	—	53	2	no	—
76	#427	—	53	4	no	—
77	C₂×C₄×C₈	—	53	1	no	—
78	C₅₃	#3160	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₈	—	53	2	no	—
80	C₄⋊Q₈	—	53	2	no	—
81	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
82	#125	—	53	2	no	—
83	#125	C₂×D₅₃	1	2	yes	—
84	C₂²⋊Q₈	—	2	2	no	—
85	C₂²⋊Q₈	—	106	2	no	—
86	C₄×Q₈	—	2	2	no	—
87	C₄×Q₈	—	106	2	no	—
88	#117	—	53	2	no	—
89	#117	C₂×D₅₃	1	2	yes	—
90	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
91	#107	—	53	8	no	—
92	#106	—	53	8	no	—
93	C₄×C₈	—	53	1	no	—
94	C₄×C₈	—	53	2	no	—
95	C₄×C₈	—	53	1	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	106	2	no	—
98	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
99	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
100	C₂×C₈	—	53	8	no	—
101	C₂×C₈	—	53	2	no	—
102	C₂×C₈	—	53	2	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	53	4	no	—
106	C₄⋊C₄	—	106	4	no	—
107	C₄⋊C₄	—	106	2	no	—
108	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
109	C₂²⋊C₄	—	2	2	no	—
110	C₂²⋊C₄	—	106	2	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	106	2	no	—
113	C₄²	C₂²⋊D₅₃	1	2	yes	—
114	C₄²	C₂²×D₅₃	1	1	yes	characteristic
115	C₄²	C₂²×D₅₃	1	1	yes	characteristic
116	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
117	Q₈	—	4	4	no	—
118	Q₈	—	212	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	106	4	no	—
122	C₂×C₄	Q₈×D₅₃	1	4	yes	—
123	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
124	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
125	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
126	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
127	C₈	—	53	4	no	—
128	C₈	—	53	4	no	—
129	C₂²	—	2	2	no	—
130	C₂²	#13133215	1	2	yes	—
131	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	4	4	no	—
134	C₄	—	212	4	no	—
135	C₄	#13133215	1	2	yes	—
136	C₄	#13133237	1	4	yes	—
137	C₄	C₂×Q₈×D₅₃	1	2	yes	—
138	C₂	—	2	2	no	—
139	C₂	C₂×#13133215	1	1	yes	characteristic
140	C₂	C₂×#13133237	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal, characteristic

#16811026

C₂

yes

maximal, complemented

#16811022

C₂

yes

maximal, complemented

#16811019

C₂

yes

maximal, complemented

#16810659

C₂

yes

maximal

C₂×C₄×#88322

C₂

yes

maximal, characteristic

C₅₃×#533

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

#13133217

C₂²

yes

—

#13133209

C₂²

yes

—

#13133196

—

C₄×Dic₁₀₆

—

#13133176

C₂²

yes

—

#13133150

C₂²

yes

—

#13133148

C₂²

yes

—

C₄×#88322

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

—

C₄×#88322

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

—

#125×C₅₃

C₂²

yes

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

Q₈

yes

complemented

C₂×#88322

C₂³

yes

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

#88322

C₂×Q₈

yes

—

#88322

C₂×Q₈

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄⋊Q₈

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄⋊Q₈

yes

—

C₄×C₅₃

#135

yes

—

C₄×C₅₃

C₂²×Q₈

yes

—

Dic₅₃

—

#3160

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₅₃

C₂×#135

yes

—

#533

—

#533

D₅₃

yes

—

—

—

—

C₂×C₄×C₈

—

C₅₃

#3160

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

—

#125

—

#125

C₂×D₅₃

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

C₄×Q₈

—

106

—

#117

—

#117

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#107

—

#106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

106

—

107

C₄⋊C₄

—

106

—

108

C₄⋊C₄

C₂²×D₅₃

yes

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

106

—

111

C₄²

—

112

C₄²

—

106

—

113

C₄²

C₂²⋊D₅₃

yes

—

114

C₄²

C₂²×D₅₃

yes

characteristic

115

C₄²

C₂²×D₅₃

yes

characteristic

116

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

117

Q₈

—

118

Q₈

—

212

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

106

—

122

C₂×C₄

Q₈×D₅₃

yes

—

123

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

124

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

125

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

126

C₂×C₄

C₂³×D₅₃

yes

characteristic

127

C₈

—

128

C₈

—

129

C₂²

—

130

C₂²

#13133215

yes

—

131

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

212

—

135

C₄

#13133215

yes

—

136

C₄

#13133237

yes

—

137

C₄

C₂×Q₈×D₅₃

yes

—

138

C₂

—

139

C₂

C₂×#13133215

yes

characteristic

140

C₂

C₂×#13133237

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#533	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#533	D₅₃
hypocenter	C₅₃	#3160
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#533

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #16811026	C₂	semidirect
3. #16811022	C₂	semidirect
4. #16811019	C₂	semidirect
42. C₂×#88322	Q₈	semidirect
78. C₅₃	#3160	semidirect

H ≅ G/N

type

2. #16811026

C₂

semidirect

3. #16811022

C₂

semidirect

4. #16811019

C₂

semidirect

42. C₂×#88322

Q₈

semidirect

78. C₅₃

#3160

semidirect

Subgroups — #20285225

Conjugacy classes of subgroups

Complements