Group #20289936 — subgroups

Subgroups — #20289936

Order 6784.

178 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 78 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	#16811640	C₂	1	1	yes	maximal, characteristic
2	#16811639	C₂	1	1	yes	maximal, characteristic
3	#88324⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#88324⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811464	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#88324	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810673	C₂	1	1	yes	maximal, characteristic
8	#13133284	C₄	1	2	yes	—
9	C₄⋊#88324	C₄	1	2	yes	—
10	C₂²⋊#88324	C₄	1	2	yes	—
11	C₂²⋊#88324	C₄	1	2	yes	—
12	C₄×#88324	C₂²	1	1	yes	characteristic
13	C₄×#88324	C₂²	1	1	yes	characteristic
14	C₄×#88324	C₂²	1	1	yes	characteristic
15	C₄×#88324	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
17	#13133146	C₄	1	1	yes	characteristic
18	#13133146	C₄	1	1	yes	characteristic
19	C₂²⋊#88322	C₄	1	1	yes	characteristic
20	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
21	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
22	C₂²⋊C₈×C₅₃	C₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×#88324	C₂×C₄	1	2	yes	—
27	C₂×#88324	C₂×C₄	1	2	yes	—
28	C₂×#88324	C₂×C₄	1	2	yes	—
29	C₂×#88324	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
36	C₂×#88322	C₂×C₄	1	1	yes	characteristic
37	C₂×#88322	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₅₃	C₄²	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	2	1	no	—
41	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₅₃	—	2	2	no	—
44	C₂×Dic₅₃	—	2	2	no	—
45	C₂×Dic₅₃	C₄²	1	1	yes	characteristic
46	C₂×Dic₅₃	C₄²	1	1	yes	characteristic
47	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
49	#88324	—	2	2	no	—
50	#88324	—	2	2	no	—
51	#88324	—	2	2	no	—
52	#88324	—	2	2	no	—
53	C₈×C₅₃	—	2	1	no	—
54	C₈×C₅₃	—	2	1	no	—
55	#88322	—	2	1	no	—
56	#88322	—	2	1	no	—
57	C₂²×C₅₃	—	2	1	no	—
58	C₂²×C₅₃	—	2	1	no	—
59	C₂²×C₅₃	—	2	1	no	—
60	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
61	C₄×C₅₃	—	2	1	no	—
62	C₄×C₅₃	#131	1	1	yes	characteristic
63	C₄×C₅₃	#131	1	1	yes	characteristic
64	Dic₅₃	—	2	1	no	—
65	Dic₅₃	—	2	1	no	—
66	Dic₅₃	#131	1	2	yes	—
67	Dic₅₃	#131	1	2	yes	—
68	#1454	—	53	1	no	maximal, Sylow
69	C₂×C₅₃	—	4	1	no	—
70	C₂×C₅₃	#405	1	1	yes	characteristic
71	C₂×C₅₃	#405	1	1	yes	characteristic
72	C₂×C₅₃	#405	1	1	yes	characteristic
73	#431	—	53	1	no	—
74	#431	—	53	1	no	—
75	#431	—	53	1	no	—
76	#322	—	53	1	no	—
77	#322	—	53	1	no	—
78	#322	—	53	1	no	—
79	C₈²	—	53	1	no	—
80	C₅₃	#1454	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₄²	—	53	1	no	—
82	C₄⋊C₈	—	53	2	no	—
83	C₄⋊C₈	—	53	2	no	—
84	C₄⋊C₈	—	53	1	no	—
85	C₄⋊C₈	—	53	1	no	—
86	C₂²⋊C₈	—	53	2	no	—
87	C₂²⋊C₈	—	53	2	no	—
88	C₂²⋊C₈	—	53	1	no	—
89	C₂²⋊C₈	#6157	1	1	yes	characteristic
90	C₄×C₈	—	53	1	no	—
91	C₄×C₈	—	53	1	no	—
92	C₄×C₈	—	53	1	no	—
93	C₄×C₈	—	53	1	no	—
94	C₄×C₈	—	53	1	no	—
95	C₄×C₈	—	53	1	no	—
96	C₂²×C₄	—	53	1	no	—
97	C₂²×C₄	—	53	1	no	—
98	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
99	C₂×C₈	—	53	2	no	—
100	C₂×C₈	—	53	2	no	—
101	C₂×C₈	—	53	2	no	—
102	C₂×C₈	—	53	2	no	—
103	C₂×C₈	—	53	1	no	—
104	C₂×C₈	—	53	1	no	—
105	C₂×C₈	C₂×#6157	1	1	yes	characteristic
106	C₂×C₈	C₂×#6157	1	1	yes	characteristic
107	C₄²	—	53	1	no	—
108	C₄²	—	53	1	no	—
109	C₄²	—	53	1	no	—
110	C₄²	—	53	1	no	—
111	C₂³	C₄×#6157	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	53	1	no	—
115	C₂×C₄	—	53	1	no	—
116	C₂×C₄	—	53	1	no	—
117	C₂×C₄	—	53	1	no	—
118	C₂×C₄	—	106	2	no	—
119	C₂×C₄	—	106	2	no	—
120	C₂×C₄	C₄×#6157	1	1	yes	characteristic
121	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
122	C₈	—	2	1	no	—
123	C₈	—	2	1	no	—
124	C₈	—	106	2	no	—
125	C₈	—	106	2	no	—
126	C₈	—	106	2	no	—
127	C₈	—	106	2	no	—
128	C₈	—	106	1	no	—
129	C₈	—	106	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	C₂×C₄×#6157	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	53	2	no	—
136	C₄	—	53	2	no	—
137	C₄	—	106	1	no	—
138	C₄	—	106	1	no	—
139	C₄	#13133304	1	1	yes	characteristic
140	C₄	#13133304	1	1	yes	characteristic
141	C₂	—	4	1	no	—
142	C₂	#16811586	1	1	yes	characteristic
143	C₂	#16811587	1	1	yes	characteristic
144	C₂	#16811589	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811640

C₂

yes

maximal, characteristic

#16811639

C₂

yes

maximal, characteristic

#88324⋊C₈

C₂

yes

maximal, characteristic, complemented

#88324⋊C₈

C₂

yes

maximal, characteristic, complemented

#16811464

C₂

yes

maximal, characteristic, complemented

C₈×#88324

C₂

yes

maximal, characteristic, complemented

#16810673

C₂

yes

maximal, characteristic

#13133284

C₄

yes

—

C₄⋊#88324

C₄

yes

—

C₂²⋊#88324

C₄

yes

—

C₂²⋊#88324

C₄

yes

—

C₄×#88324

C₂²

yes

characteristic

C₄×#88324

C₂²

yes

characteristic

C₄×#88324

C₂²

yes

characteristic

C₄×#88324

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133146

C₄

yes

characteristic

#13133146

C₄

yes

characteristic

C₂²⋊#88322

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

C₂²⋊C₈×C₅₃

C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂×#88324

C₂×C₄

yes

—

C₂×#88324

C₂×C₄

yes

—

C₂×#88324

C₂×C₄

yes

—

C₂×#88324

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂³×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄²

yes

characteristic

C₂×Dic₅₃

C₄²

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

—

—

—

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#131

yes

—

Dic₅₃

#131

yes

—

#1454

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

—

—

—

C₈²

—

C₅₃

#1454

yes

characteristic, complemented, Sylow

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

#6157

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#6157

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

C₂×#6157

yes

characteristic

106

C₂×C₈

C₂×#6157

yes

characteristic

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₂³

C₄×#6157

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

106

—

119

C₂×C₄

—

106

—

120

C₂×C₄

C₄×#6157

yes

characteristic

121

C₂×C₄

C₂²×#6157

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

—

106

—

125

C₈

—

106

—

126

C₈

—

106

—

127

C₈

—

106

—

128

C₈

—

106

—

129

C₈

—

106

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

C₂×C₄×#6157

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

106

—

138

C₄

—

106

—

139

C₄

#13133304

yes

characteristic

140

C₄

#13133304

yes

characteristic

141

C₂

—

142

C₂

#16811586

yes

characteristic

143

C₂

#16811587

yes

characteristic

144

C₂

#16811589

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂×C₄²
Fitting	C₂²⋊C₈×C₅₃	C₄
hypercenter	C₂²⋊C₈	#6157
hypocenter	C₅₃	#1454
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#6157

commutator, socle

C₂²×C₅₃

C₂×C₄²

Fitting

C₂²⋊C₈×C₅₃

C₄

C₂²⋊C₈

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
3. #88324⋊C₈	C₂	semidirect
4. #88324⋊C₈	C₂	semidirect
5. #16811464	C₂	semidirect
6. C₈×#88324	C₂	semidirect
80. C₅₃	#1454	semidirect

H ≅ G/N

type

3. #88324⋊C₈

C₂

semidirect

4. #88324⋊C₈

C₂

semidirect

5. #16811464

C₂

semidirect

6. C₈×#88324

C₂

semidirect

80. C₅₃

#1454

semidirect

Subgroups — #20289936

Conjugacy classes of subgroups

Complements