Group #20290823 — subgroups

Subgroups — #20290823

Order 6784.

208 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 70 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×#13133288	C₂	1	1	yes	maximal, characteristic
2	#16811630	C₂	1	2	yes	maximal
3	C₂×C₈×#6157	C₂	1	1	yes	maximal, characteristic
4	#16811493	C₂	1	2	yes	maximal
5	#16810660	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719936	C₂²	1	2	yes	—
7	C₂×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×#6157	C₂²	1	1	yes	characteristic
9	#13133288	—	2	2	no	—
10	#13133284	—	2	2	no	—
11	C₄⋊#88324	—	2	2	no	—
12	C₈×#6157	—	2	2	no	—
13	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
14	#13133175	C₂²	1	2	yes	—
15	#13133151	C₄	1	1	yes	characteristic, complemented
16	#13133148	C₄	1	2	yes	complemented
17	#107×C₅₃	C₄	1	1	yes	characteristic, complemented
18	C₂²×#6157	—	2	1	no	—
19	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
20	C₂×#88324	—	2	2	no	—
21	C₂×#88324	D₄	1	1	yes	characteristic
22	C₂×#88324	D₄	1	1	yes	characteristic
23	C₄×#6157	—	2	1	no	—
24	C₄×#6157	Q₈	1	2	yes	—
25	#11719936	—	2	2	no	—
26	#11719936	—	2	2	no	—
27	D₅₃⋊C₈	—	2	1	no	—
28	D₅₃⋊C₈	—	2	1	no	—
29	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
31	#11719921	—	2	2	no	—
32	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
33	C₄×Dic₅₃	—	2	2	no	—
34	C₂×#88322	C₂×C₄	1	1	yes	characteristic
35	C₈×D₅₃	D₄	1	2	yes	—
36	C₈×D₅₃	Q₈	1	2	yes	—
37	C₂²×D₅₃	#41	1	1	yes	characteristic
38	C₂×#6157	—	2	2	no	—
39	C₂×#6157	—	2	2	no	—
40	C₂×#6157	—	2	2	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₅₃	—	2	2	no	—
44	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
45	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
46	C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₄×D₅₃	#41	1	1	yes	characteristic
49	#88324	—	2	1	no	—
50	#88324	—	2	1	no	—
51	#88324	—	4	2	no	—
52	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
53	#88322	C₄⋊C₄	1	2	yes	—
54	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
55	C₂×D₅₃	—	2	1	no	—
56	C₂×D₅₃	—	2	1	no	—
57	C₂×D₅₃	C₂²⋊Q₈	1	1	yes	characteristic
58	C₂×D₅₃	#125	1	1	yes	characteristic
59	#6157	—	2	4	no	—
60	C₄×C₅₃	—	4	2	no	—
61	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
63	Dic₅₃	—	4	2	no	—
64	Dic₅₃	#121	1	1	yes	characteristic
65	Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
66	#1945	—	53	1	no	maximal, Sylow
67	C₂×C₅₃	#389	1	1	yes	characteristic
68	C₂×C₅₃	#443	1	1	yes	characteristic
69	C₂×C₅₃	#443	1	1	yes	characteristic
70	D₅₃	—	2	1	no	—
71	D₅₃	—	2	1	no	—
72	C₂×#108	—	53	1	no	—
73	#427	—	53	1	no	—
74	#424	—	53	2	no	—
75	C₂×C₄×C₈	—	53	1	no	—
76	#339	—	53	2	no	—
77	C₅₃	#1945	1	1	yes	characteristic, complemented, Sylow
78	C₂×M₁₆	—	53	2	no	—
79	C₂²×C₈	—	53	1	no	—
80	C₂²×C₈	—	53	1	no	—
81	#117	—	53	2	no	—
82	C₂×C₄²	—	53	1	no	—
83	#108	—	106	2	no	—
84	#107	—	53	1	no	—
85	#107	#6157	1	1	yes	characteristic, complemented
86	#106	—	53	2	no	—
87	C₄⋊C₈	—	106	2	no	—
88	C₄⋊C₈	—	106	2	no	—
89	C₄×C₈	—	106	2	no	—
90	C₂²×C₄	—	53	1	no	—
91	C₂²×C₄	—	106	1	no	—
92	M₁₆	—	106	2	no	—
93	M₁₆	—	106	2	no	—
94	C₂×C₈	—	53	1	no	—
95	C₂×C₈	—	53	2	no	—
96	C₂×C₈	—	53	2	no	—
97	C₂×C₈	—	53	1	no	—
98	C₂×C₈	—	53	1	no	—
99	C₂×C₈	—	106	1	no	—
100	C₂×C₈	—	106	1	no	—
101	C₂×C₈	—	106	2	no	—
102	C₂×C₈	C₂×#6157	1	1	yes	characteristic
103	C₄⋊C₄	—	53	2	no	—
104	C₄⋊C₄	C₂×#6157	1	2	yes	—
105	C₂²⋊C₄	—	106	2	no	—
106	C₄²	—	53	2	no	—
107	C₄²	—	106	2	no	—
108	C₄²	—	106	1	no	—
109	C₂³	—	53	1	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	53	1	no	—
112	C₂×C₄	—	53	1	no	—
113	C₂×C₄	—	53	1	no	—
114	C₂×C₄	—	53	1	no	—
115	C₂×C₄	—	53	1	no	—
116	C₂×C₄	—	106	2	no	—
117	C₂×C₄	—	106	2	no	—
118	C₂×C₄	—	106	2	no	—
119	C₂×C₄	—	106	2	no	—
120	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
121	C₈	—	53	2	no	—
122	C₈	—	106	1	no	—
123	C₈	—	106	1	no	—
124	C₈	—	212	2	no	—
125	C₈	C₄⋊#6157	1	2	yes	—
126	C₂²	—	53	1	no	—
127	C₂²	—	53	1	no	—
128	C₂²	—	106	1	no	—
129	C₂²	—	106	1	no	—
130	C₂²	D₄×#6157	1	1	yes	characteristic
131	C₄	—	4	2	no	—
132	C₄	—	53	1	no	—
133	C₄	—	53	1	no	—
134	C₄	—	106	4	no	—
135	C₄	—	212	2	no	—
136	C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
137	C₄	Q₈×#6157	1	1	yes	characteristic
138	C₂	—	106	1	no	—
139	C₂	—	106	1	no	—
140	C₂	#16811574	1	1	yes	characteristic
141	C₂	#16811665	1	1	yes	characteristic
142	C₂	#16811667	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133288

C₂

yes

maximal, characteristic

#16811630

C₂

yes

maximal

C₂×C₈×#6157

C₂

yes

maximal, characteristic

#16811493

C₂

yes

maximal

#16810660

C₂

yes

maximal, characteristic

C₂×#11719936

C₂²

yes

—

C₂×D₅₃⋊C₈

C₂²

yes

characteristic

C₂×C₄×#6157

C₂²

yes

—

—

—

—

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133175

C₂²

yes

—

#13133151

C₄

yes

characteristic, complemented

#13133148

C₄

yes

complemented

#107×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×#88324

—

C₂×#88324

D₄

yes

characteristic

C₂×#88324

D₄

yes

characteristic

C₄×#6157

—

C₄×#6157

Q₈

yes

—

#11719936

—

#11719936

—

D₅₃⋊C₈

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

#11719921

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

—

C₂×#88322

C₂×C₄

yes

characteristic

C₈×D₅₃

D₄

yes

—

C₈×D₅₃

Q₈

yes

—

C₂²×D₅₃

yes

—

—

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

C₂×Q₈

yes

characteristic

C₄×D₅₃

C₂×Q₈

yes

C₄×D₅₃

yes

—

—

—

C₈×C₅₃

C₄⋊C₄

yes

—

#88322

C₄⋊C₄

yes

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂²⋊Q₈

yes

C₂×D₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

#121

yes

characteristic

Dic₅₃

C₄⋊Q₈

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

D₅₃

—

C₂×#108

—

#427

—

#424

—

C₂×C₄×C₈

—

#339

—

C₅₃

#1945

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

C₂×C₄²

—

#108

—

106

—

#107

—

#107

#6157

yes

characteristic, complemented

#106

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₄×C₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

100

C₂×C₈

—

106

—

101

C₂×C₈

—

106

—

102

C₂×C₈

C₂×#6157

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

C₂×#6157

yes

—

105

C₂²⋊C₄

—

106

—

106

C₄²

—

107

C₄²

—

106

—

108

C₄²

—

106

—

109

C₂³

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

106

—

117

C₂×C₄

—

106

—

118

C₂×C₄

—

106

—

119

C₂×C₄

—

106

—

120

C₂×C₄

C₂²×#6157

yes

characteristic

121

C₈

—

122

C₈

—

106

—

123

C₈

—

106

—

124

C₈

—

212

—

125

C₈

C₄⋊#6157

yes

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

106

—

129

C₂²

—

106

—

130

C₂²

D₄×#6157

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

106

—

135

C₄

—

212

—

136

C₄

C₂×C₄⋊#6157

yes

characteristic

137

C₄

Q₈×#6157

yes

characteristic

138

C₂

—

106

—

139

C₂

—

106

—

140

C₂

#16811574

yes

characteristic

141

C₂

#16811665

yes

characteristic

142

C₂

#16811667

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	#107×C₅₃	C₄
socle	C₂²×C₅₃	C₄×D₄
hypercenter	#107	#6157
hypocenter	C₅₃	#1945
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

#107×C₅₃

C₄

socle

C₂²×C₅₃

C₄×D₄

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
15. #13133151	C₄	semidirect
16. #13133148	C₄	semidirect
17. #107×C₅₃	C₄	semidirect
77. C₅₃	#1945	semidirect
85. #107	#6157	semidirect

H ≅ G/N

type

15. #13133151

C₄

semidirect

16. #13133148

C₄

semidirect

17. #107×C₅₃

C₄

semidirect

77. C₅₃

#1945

semidirect

85. #107

#6157

semidirect

Subgroups — #20290823

Conjugacy classes of subgroups

Complements