Group #20290829 — subgroups

Subgroups — #20290829

Order 6784.

200 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 74 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	2	yes	maximal
2	#16811627	C₂	1	1	yes	maximal, characteristic
3	C₄×#11719936	C₂	1	1	yes	maximal, characteristic
4	#16811499	C₂	1	2	yes	maximal
5	C₄⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719936	C₂²	1	2	yes	—
7	C₂×#11719936	C₂²	1	1	yes	characteristic
8	C₂×#11719936	C₂²	1	1	yes	characteristic
9	#13133288	—	2	4	no	—
10	C₄⋊#88324	—	2	1	no	—
11	C₄⋊#88324	—	2	1	no	—
12	#13133272	—	2	1	no	—
13	C₄×#88324	—	2	1	no	—
14	C₂×C₈×D₅₃	C₂²	1	2	yes	—
15	C₄²×D₅₃	C₂²	1	1	yes	characteristic
16	C₄⋊#88322	C₄	1	1	yes	characteristic
17	#13133146	C₄	1	2	yes	—
18	C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
19	C₂×C₄×D₅₃	Q₈	1	1	yes	characteristic
20	C₂×C₄×D₅₃	Q₈	1	1	yes	characteristic
21	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
22	C₂×#88324	—	2	1	no	—
23	C₂×#88324	—	2	1	no	—
24	C₂×#88324	—	2	2	no	—
25	#11719936	—	2	1	no	—
26	#11719936	—	2	1	no	—
27	#11719936	—	2	2	no	—
28	#11719936	—	2	2	no	—
29	#11719936	D₄	1	2	yes	—
30	#11719936	Q₈	1	2	yes	—
31	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₅₃	D₄	1	1	yes	characteristic
35	C₄×Dic₅₃	D₄	1	1	yes	characteristic
36	C₂×#88322	C₂×C₄	1	2	yes	—
37	C₈×D₅₃	—	2	2	no	—
38	C₈×D₅₃	—	2	2	no	—
39	C₂²×D₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
46	C₄×D₅₃	—	2	2	no	—
47	C₄×D₅₃	—	4	1	no	—
48	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
49	C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₄×D₅₃	#41	1	1	yes	characteristic
51	C₄×D₅₃	#41	1	1	yes	characteristic
52	#88324	—	2	2	no	—
53	#88324	—	2	2	no	—
54	#88324	—	4	2	no	—
55	C₈×C₅₃	—	2	2	no	—
56	#88322	—	2	2	no	—
57	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×D₅₃	—	2	1	no	—
59	C₂×D₅₃	—	2	1	no	—
60	C₂×D₅₃	#125	1	1	yes	characteristic
61	C₂×D₅₃	C₄⋊Q₈	1	1	yes	characteristic
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	#108	1	2	yes	complemented
64	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
65	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
66	Dic₅₃	—	2	1	no	—
67	Dic₅₃	—	2	1	no	—
68	Dic₅₃	#121	1	1	yes	characteristic
69	Dic₅₃	C₂²⋊Q₈	1	1	yes	characteristic
70	#1950	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	#389	1	1	yes	characteristic
72	C₂×C₅₃	C₂×#108	1	1	yes	characteristic
73	C₂×C₅₃	#430	1	1	yes	characteristic
74	D₅₃	—	2	1	no	—
75	D₅₃	—	2	1	no	—
76	C₂×#108	—	53	2	no	—
77	#423	—	53	1	no	—
78	C₂×C₄⋊C₈	—	53	1	no	—
79	C₄×M₁₆	—	53	1	no	—
80	#341	—	53	2	no	—
81	C₅₃	#1950	1	1	yes	characteristic, complemented, Sylow
82	C₂×M₁₆	—	53	1	no	—
83	C₂×M₁₆	—	53	1	no	—
84	C₂×M₁₆	—	53	2	no	—
85	C₂²×C₈	—	53	2	no	—
86	C₂×C₄²	—	53	1	no	—
87	#108	—	106	4	no	—
88	C₄⋊C₈	—	53	2	no	—
89	C₄⋊C₈	—	53	1	no	—
90	C₄⋊C₈	—	106	1	no	—
91	C₄⋊C₈	—	106	1	no	—
92	C₄⋊C₈	#6157	1	1	yes	characteristic
93	#97	—	106	1	no	—
94	C₄×C₈	—	106	1	no	—
95	C₂²×C₄	—	53	1	no	—
96	C₂²×C₄	—	53	1	no	—
97	C₂²×C₄	—	53	1	no	—
98	M₁₆	—	53	2	no	—
99	M₁₆	—	53	2	no	—
100	M₁₆	—	106	1	no	—
101	M₁₆	—	106	1	no	—
102	M₁₆	—	106	2	no	—
103	M₁₆	—	106	2	no	—
104	C₂×C₈	—	53	2	no	—
105	C₂×C₈	—	106	2	no	—
106	C₂×C₈	—	106	2	no	—
107	C₂×C₈	—	106	1	no	—
108	C₂×C₈	—	106	1	no	—
109	C₂×C₈	—	106	2	no	—
110	C₂×C₈	C₂×#6157	1	2	yes	—
111	C₄²	—	53	1	no	—
112	C₄²	—	53	1	no	—
113	C₄²	—	53	1	no	—
114	C₄²	C₂×#6157	1	1	yes	characteristic
115	C₂³	—	53	1	no	—
116	C₂×C₄	—	53	1	no	—
117	C₂×C₄	—	53	1	no	—
118	C₂×C₄	—	53	1	no	—
119	C₂×C₄	—	53	1	no	—
120	C₂×C₄	—	53	1	no	—
121	C₂×C₄	—	53	1	no	—
122	C₂×C₄	—	53	1	no	—
123	C₂×C₄	—	106	2	no	—
124	C₂×C₄	—	212	1	no	—
125	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
126	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
127	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	106	2	no	—
130	C₈	—	106	2	no	—
131	C₈	—	106	2	no	—
132	C₈	—	212	2	no	—
133	C₂²	—	53	1	no	—
134	C₂²	—	53	1	no	—
135	C₂²	—	106	1	no	—
136	C₂²	—	106	1	no	—
137	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	53	1	no	—
140	C₄	—	53	1	no	—
141	C₄	—	106	1	no	—
142	C₄	—	106	1	no	—
143	C₄	#13133288	1	2	yes	complemented
144	C₄	D₄×#6157	1	1	yes	characteristic
145	C₄	Q₈×#6157	1	1	yes	characteristic
146	C₂	—	106	1	no	—
147	C₂	—	106	1	no	—
148	C₂	#16811574	1	1	yes	characteristic
149	C₂	C₂×#13133288	1	1	yes	characteristic
150	C₂	#16811638	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133288

C₂

yes

maximal

#16811627

C₂

yes

maximal, characteristic

C₄×#11719936

C₂

yes

maximal, characteristic

#16811499

C₂

yes

maximal

C₄⋊C₈×D₅₃

C₂

yes

maximal, characteristic

C₂×#11719936

C₂²

yes

—

C₂×#11719936

C₂²

yes

characteristic

C₂×#11719936

C₂²

yes

—

—

—

—

—

C₂×C₈×D₅₃

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊#88322

C₄

yes

characteristic

#13133146

C₄

yes

—

C₄⋊C₈×C₅₃

C₄

yes

characteristic

C₂×C₄×D₅₃

Q₈

yes

characteristic

C₂×C₄×D₅₃

Q₈

yes

characteristic

C₂×C₄×D₅₃

C₂³

yes

—

—

—

—

—

—

—

D₄

yes

—

#11719936

Q₈

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

—

C₈×D₅₃

—

C₈×D₅₃

—

C₂²×D₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

C₂×Q₈

yes

C₄×D₅₃

yes

C₄×D₅₃

yes

—

—

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#125

yes

characteristic

C₂×D₅₃

C₄⋊Q₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#108

yes

complemented

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#121

yes

characteristic

Dic₅₃

C₂²⋊Q₈

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

D₅₃

—

C₂×#108

—

#423

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

#341

—

C₅₃

#1950

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

#108

—

106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₄⋊C₈

#6157

yes

characteristic

#97

—

106

—

C₄×C₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

100

M₁₆

—

106

—

101

M₁₆

—

106

—

102

M₁₆

—

106

—

103

M₁₆

—

106

—

104

C₂×C₈

—

105

C₂×C₈

—

106

—

106

C₂×C₈

—

106

—

107

C₂×C₈

—

106

—

108

C₂×C₈

—

106

—

109

C₂×C₈

—

106

—

110

C₂×C₈

C₂×#6157

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂×#6157

yes

characteristic

115

C₂³

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

106

—

124

C₂×C₄

—

212

—

125

C₂×C₄

C₄⋊#6157

yes

characteristic

126

C₂×C₄

C₄⋊#6157

yes

characteristic

127

C₂×C₄

C₂²×#6157

yes

characteristic

128

C₈

—

129

C₈

—

106

—

130

C₈

—

106

—

131

C₈

—

106

—

132

C₈

—

212

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

106

—

136

C₂²

—

106

—

137

C₂²

C₂×C₄⋊#6157

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

106

—

142

C₄

—

106

—

143

C₄

#13133288

yes

complemented

144

C₄

D₄×#6157

yes

characteristic

145

C₄

Q₈×#6157

yes

characteristic

146

C₂

—

106

—

147

C₂

—

106

—

148

C₂

#16811574

yes

characteristic

149

C₂

C₂×#13133288

yes

characteristic

150

C₂

#16811638

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄⋊C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₄⋊C₈	#6157
hypocenter	C₅₃	#1950
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₄⋊C₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₄⋊C₄

C₄⋊C₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
63. C₄×C₅₃	#108	semidirect
81. C₅₃	#1950	semidirect
143. C₄	#13133288	semidirect

H ≅ G/N

type

63. C₄×C₅₃

#108

semidirect

81. C₅₃

#1950

semidirect

143. C₄

#13133288

semidirect

Subgroups — #20290829

Conjugacy classes of subgroups

Complements