Group #20274477 — subgroups

Subgroups — #20274477

Order 6784.

412 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 165 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×C₂²⋊#88322	C₂	1	2	yes	maximal
2	C₂×#13133128	C₂	1	2	yes	maximal, complemented
3	#16810378	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	C₂²	1	4	yes	—
6	C₂²⋊#88322	C₄	1	8	yes	complemented
7	#13133128	—	2	4	no	—
8	C₂³×C₄×C₅₃	C₄	1	2	yes	—
9	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₅₃	—	2	4	no	—
11	C₂×C₄²×C₅₃	C₄	1	2	yes	—
12	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	—	2	4	no	—
16	C₂²×C₄×C₅₃	—	2	4	no	—
17	C₂²×C₄×C₅₃	—	2	4	no	—
18	C₂²×C₄×C₅₃	—	2	2	no	—
19	C₂²×C₄×C₅₃	—	2	2	no	—
20	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₄²×C₅₃	—	2	4	no	—
26	C₄²×C₅₃	—	2	2	no	—
27	C₄²×C₅₃	D₄	1	4	yes	—
28	C₂×#88322	—	2	4	no	—
29	C₂×#88322	—	2	4	no	—
30	C₂×#88322	C₂×C₄	1	8	yes	complemented
31	C₂³×C₅₃	—	2	1	no	—
32	C₂³×C₅₃	—	2	1	no	—
33	C₂³×C₅₃	—	2	2	no	—
34	C₂³×C₅₃	C₄²	1	2	yes	—
35	C₂³×C₅₃	M₁₆	1	4	yes	—
36	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	—	2	16	no	—
38	C₂×C₄×C₅₃	—	2	1	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	2	1	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	8	no	—
45	C₂×C₄×C₅₃	C₄²	1	2	yes	—
46	C₂×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
47	C₂×C₄×C₅₃	M₁₆	1	4	yes	—
48	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₅₃	#41	1	2	yes	—
51	#88322	—	2	8	no	—
52	C₂²×C₅₃	—	2	2	no	—
53	C₂²×C₅₃	—	2	2	no	—
54	C₂²×C₅₃	—	2	4	no	—
55	C₂²×C₅₃	—	2	4	no	—
56	C₂²×C₅₃	#97	1	4	yes	complemented
57	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
58	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₅₃	#117	1	1	yes	characteristic
60	C₂²×C₅₃	C₂×M₁₆	1	2	yes	—
61	C₂²×C₅₃	C₂×M₁₆	1	2	yes	—
62	C₄×C₅₃	—	2	2	no	—
63	C₄×C₅₃	—	2	4	no	—
64	C₄×C₅₃	—	2	4	no	—
65	C₄×C₅₃	C₄×D₄	1	4	yes	—
66	#1749	—	53	1	no	maximal, Sylow
67	C₂×C₅₃	—	2	4	no	—
68	C₂×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₅₃	C₂×#97	1	1	yes	characteristic
70	C₂×C₅₃	C₄×M₁₆	1	1	yes	characteristic
71	C₂×C₅₃	#407	1	2	yes	—
72	C₂×C₅₃	#432	1	2	yes	—
73	C₂²×C₄²	D₅₃	1	1	yes	characteristic
74	C₂×C₂²⋊C₈	—	53	2	no	—
75	C₂×#97	—	53	2	no	—
76	#336	—	53	2	no	—
77	C₅₃	#1749	1	1	yes	characteristic, complemented, Sylow
78	C₂³×C₄	Dic₅₃	1	2	yes	—
79	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
80	C₂²×C₈	—	53	4	no	—
81	C₂×C₄²	—	2	4	no	—
82	C₂×C₄²	Dic₅₃	1	2	yes	—
83	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
84	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
85	C₂²⋊C₈	—	53	8	no	—
86	#97	—	106	4	no	—
87	C₂⁴	C₂×Dic₅₃	1	1	yes	characteristic
88	C₂²×C₄	—	2	4	no	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	2	4	no	—
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	C₄×D₅₃	1	4	yes	—
94	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
95	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
96	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
97	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
98	C₂×C₈	—	53	8	no	—
99	C₂×C₈	—	106	4	no	—
100	C₂×C₈	—	106	4	no	—
101	C₄²	—	2	2	no	—
102	C₄²	—	2	4	no	—
103	C₄²	C₂²⋊D₅₃	1	4	yes	—
104	C₂³	—	2	1	no	—
105	C₂³	—	2	1	no	—
106	C₂³	—	2	2	no	—
107	C₂³	#11719917	1	4	yes	—
108	C₂³	C₄×Dic₅₃	1	2	yes	—
109	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
110	C₂×C₄	—	2	16	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	8	no	—
118	C₂×C₄	#11719917	1	4	yes	—
119	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
120	C₂×C₄	C₂²⋊Dic₅₃	1	4	yes	—
121	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
122	C₂×C₄	#11719945	1	2	yes	—
123	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
124	C₈	—	106	8	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	4	no	—
127	C₂²	—	2	4	no	—
128	C₂²	—	2	2	no	—
129	C₂²	#13133128	1	4	yes	complemented
130	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
131	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
132	C₂²	#13133176	1	1	yes	characteristic
133	C₂²	C₂×#11719917	1	2	yes	—
134	C₂²	C₂×#11719917	1	2	yes	—
135	C₄	—	2	4	no	—
136	C₄	—	2	4	no	—
137	C₄	—	2	2	no	—
138	C₄	C₄×C₂²⋊D₅₃	1	4	yes	—
139	C₂	—	2	4	no	—
140	C₂	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
141	C₂	C₂×#13133128	1	1	yes	characteristic
142	C₂	C₄×#11719917	1	1	yes	characteristic
143	C₂	#16810587	1	2	yes	—
144	C₂	#16810680	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊#88322

C₂

yes

maximal

C₂×#13133128

C₂

yes

maximal, complemented

#16810378

C₂

yes

maximal, complemented

C₂²×C₄²×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂²⋊#88322

C₄

yes

complemented

#13133128

—

C₂³×C₄×C₅₃

C₄

yes

—

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

complemented

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₄²

yes

—

C₂³×C₅₃

M₁₆

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#97

yes

complemented

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#117

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

—

C₂²×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

#1749

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₄×C₂²⋊C₄

yes

characteristic

C₂×C₅₃

C₂×#97

yes

characteristic

C₂×C₅₃

C₄×M₁₆

yes

characteristic

C₂×C₅₃

#407

yes

—

C₂×C₅₃

#432

yes

—

C₂²×C₄²

D₅₃

yes

characteristic

C₂×C₂²⋊C₈

—

C₂×#97

—

#336

—

C₅₃

#1749

yes

characteristic, complemented, Sylow

C₂³×C₄

Dic₅₃

yes

—

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂²⋊C₈

—

#97

—

106

—

C₂⁴

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

100

C₂×C₈

—

106

—

101

C₄²

—

102

C₄²

—

103

C₄²

C₂²⋊D₅₃

yes

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

#11719917

yes

—

108

C₂³

C₄×Dic₅₃

yes

—

109

C₂³

C₂²×Dic₅₃

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

#11719917

yes

—

119

C₂×C₄

C₄×Dic₅₃

yes

—

120

C₂×C₄

C₂²⋊Dic₅₃

yes

—

121

C₂×C₄

C₂×C₄×D₅₃

yes

—

122

C₂×C₄

#11719945

yes

—

123

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

124

C₈

—

106

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#13133128

yes

complemented

130

C₂²

C₂×C₄×Dic₅₃

yes

characteristic

131

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

132

C₂²

#13133176

yes

characteristic

133

C₂²

C₂×#11719917

yes

—

134

C₂²

C₂×#11719917

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

C₄×C₂²⋊D₅₃

yes

—

139

C₂

—

140

C₂

C₄×C₂²⋊Dic₅₃

yes

characteristic

141

C₂

C₂×#13133128

yes

characteristic

142

C₂

C₄×#11719917

yes

characteristic

143

C₂

#16810587

yes

—

144

C₂

#16810680

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄²
Fitting	C₂²×C₄²×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂²×C₄²	D₅₃
hypocenter	C₅₃	#1749
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂×C₄²

Fitting

C₂²×C₄²×C₅₃

C₂

socle

C₂³×C₅₃

C₂²×C₄

hypercenter

C₂²×C₄²

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

N	H ≅ G/N	type
2. C₂×#13133128	C₂	semidirect
3. #16810378	C₂	semidirect
6. C₂²⋊#88322	C₄	semidirect
30. C₂×#88322	C₂×C₄	semidirect
56. C₂²×C₅₃	#97	semidirect
77. C₅₃	#1749	semidirect
129. C₂²	#13133128	semidirect

H ≅ G/N

type

2. C₂×#13133128

C₂

semidirect

3. #16810378

C₂

semidirect

6. C₂²⋊#88322

C₄

semidirect

30. C₂×#88322

C₂×C₄

semidirect

56. C₂²×C₅₃

#97

semidirect

77. C₅₃

#1749

semidirect

129. C₂²

#13133128

semidirect

Subgroups — #20274477

Conjugacy classes of subgroups

Complements