Group #20274518 — subgroups

Subgroups — #20274518

Order 6784.

284 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 113 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×#13133224	C₂	1	1	yes	maximal, characteristic
2	#16810597	C₂	1	1	yes	maximal, characteristic
3	#16810581	C₂	1	2	yes	maximal, complemented
4	#16810395	C₂	1	2	yes	maximal, complemented
5	C₅₃×#413	C₂	1	1	yes	maximal, characteristic
6	#13133224	—	2	2	no	—
7	#13133224	—	2	2	no	—
8	C₂×#11719917	C₄	1	2	yes	complemented
9	C₂×#11719917	C₂²	1	1	yes	characteristic
10	C₂²×#88322	C₄	1	2	yes	complemented
11	C₂²×#88322	C₂²	1	1	yes	characteristic
12	#13133176	C₂²	1	2	yes	—
13	#13133132	C₄	1	4	yes	—
14	#13133127	—	2	2	no	—
15	C₈×Dic₅₃	—	2	2	no	—
16	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
18	#101×C₅₃	C₄	1	2	yes	complemented
19	#100×C₅₃	C₄	1	2	yes	complemented
20	#41×C₅₃	—	2	2	no	—
21	#41×C₅₃	—	2	2	no	—
22	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	M₁₆×C₅₃	—	2	2	no	—
28	M₁₆×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₈×C₅₃	—	2	2	no	—
30	C₂²⋊Dic₅₃	C₂×C₄	1	4	yes	—
31	C₄⋊Dic₅₃	—	2	2	no	—
32	C₄×Dic₅₃	—	2	2	no	—
33	#11719917	—	2	2	no	—
34	#11719917	—	2	4	no	—
35	C₂×#88322	—	2	2	no	—
36	C₂×#88322	—	2	2	no	—
37	C₂×#88322	C₂×C₄	1	2	yes	—
38	C₂×#88322	C₂×C₄	1	1	yes	characteristic
39	C₂×#88322	C₂×C₄	1	1	yes	characteristic
40	C₂×#88322	D₄	1	2	yes	—
41	C₂×#88322	D₄	1	2	yes	—
42	C₂³×C₅₃	C₄²	1	2	yes	—
43	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	Q₈×C₅₃	—	2	2	no	—
45	D₄×C₅₃	—	2	4	no	—
46	D₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	C₄²	1	2	yes	—
50	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₅₃	#41	1	2	yes	—
54	C₂×Dic₅₃	—	2	4	no	—
55	C₈×C₅₃	—	2	4	no	—
56	#88322	—	2	2	no	—
57	#88322	C₂²⋊C₄	1	1	yes	characteristic
58	#88322	C₂²⋊C₄	1	1	yes	characteristic
59	#88322	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₅₃	—	2	2	no	—
61	C₂²×C₅₃	—	4	1	no	—
62	C₂²×C₅₃	—	4	2	no	—
63	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
64	C₂²×C₅₃	C₄×D₄	1	2	yes	—
65	C₄×C₅₃	—	2	2	no	—
66	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₅₃	#117	1	1	yes	characteristic
68	C₄×C₅₃	C₄×D₄	1	2	yes	—
69	Dic₅₃	—	4	4	no	—
70	#1757	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	—	2	2	no	—
72	C₂×C₅₃	—	2	1	no	—
73	C₂×C₅₃	—	4	2	no	—
74	C₂×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×#131	—	53	1	no	—
76	#413	D₅₃	1	1	yes	characteristic
77	#410	—	53	1	no	—
78	#405	—	53	2	no	—
79	#344	—	53	2	no	—
80	C₅₃	#1757	1	1	yes	characteristic, complemented, Sylow
81	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
82	#131	—	106	2	no	—
83	#131	—	106	2	no	—
84	C₂×M₁₆	—	53	2	no	—
85	C₂×M₁₆	—	53	1	no	—
86	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
87	C₂²×C₈	—	53	1	no	—
88	C₂²×C₈	—	53	2	no	—
89	#117	—	53	2	no	—
90	#101	Dic₅₃	1	2	yes	complemented
91	#100	Dic₅₃	1	2	yes	complemented
92	#99	—	53	4	no	—
93	#97	—	106	2	no	—
94	C₄×C₈	—	106	2	no	—
95	#41	—	2	2	no	—
96	#41	—	2	2	no	—
97	C₂×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
98	C₂×D₄	C₄×D₅₃	1	2	yes	—
99	C₂×D₄	C₂×Dic₅₃	1	1	yes	characteristic
100	C₂²×C₄	C₄×D₅₃	1	2	yes	—
101	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
102	M₁₆	—	2	2	no	—
103	M₁₆	—	106	4	no	—
104	M₁₆	—	106	2	no	—
105	M₁₆	C₂×Dic₅₃	1	4	yes	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	53	1	no	—
108	C₂×C₈	—	53	1	no	—
109	C₂×C₈	—	53	2	no	—
110	C₂×C₈	—	53	2	no	—
111	C₂×C₈	—	53	2	no	—
112	C₂×C₈	—	106	2	no	—
113	C₂×C₈	—	106	2	no	—
114	C₄⋊C₄	—	106	2	no	—
115	C₂²⋊C₄	—	53	4	no	—
116	C₄²	—	106	2	no	—
117	C₂³	C₄×Dic₅₃	1	2	yes	—
118	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
119	Q₈	—	2	2	no	—
120	D₄	—	2	4	no	—
121	D₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	106	4	no	—
125	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
126	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
127	C₂×C₄	D₄×D₅₃	1	2	yes	—
128	C₂×C₄	#11719947	1	2	yes	—
129	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
130	C₈	—	2	4	no	—
131	C₈	—	53	1	no	—
132	C₈	—	53	1	no	—
133	C₈	—	53	2	no	—
134	C₈	—	106	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	4	1	no	—
137	C₂²	—	4	2	no	—
138	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
139	C₂²	#13133177	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	212	4	no	—
142	C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
143	C₄	#13133173	1	1	yes	characteristic
144	C₄	D₄×Dic₅₃	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	C₂²⋊C₄×Dic₅₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133224

C₂

yes

maximal, characteristic

#16810597

C₂

yes

maximal, characteristic

#16810581

C₂

yes

maximal, complemented

#16810395

C₂

yes

maximal, complemented

C₅₃×#413

C₂

yes

maximal, characteristic

#13133224

—

#13133224

—

C₂×#11719917

C₄

yes

complemented

C₂×#11719917

C₂²

yes

characteristic

C₂²×#88322

C₄

yes

complemented

C₂²×#88322

C₂²

yes

characteristic

#13133176

C₂²

yes

—

#13133132

C₄

yes

—

#13133127

—

C₈×Dic₅₃

—

C₂×#41×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

—

#101×C₅₃

C₄

yes

complemented

#100×C₅₃

C₄

yes

complemented

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂²⋊Dic₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

D₄

yes

—

C₂×#88322

D₄

yes

—

C₂³×C₅₃

C₄²

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

#88322

C₂²⋊C₄

yes

characteristic

#88322

C₂²⋊C₄

yes

characteristic

#88322

C₂²⋊C₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

—

Dic₅₃

—

#1757

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₄×C₂²⋊C₄

yes

characteristic

C₂×#131

—

#413

D₅₃

yes

—

—

—

C₅₃

#1757

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

characteristic

#131

—

106

—

#131

—

106

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

#101

Dic₅₃

yes

complemented

#100

Dic₅₃

yes

complemented

#99

—

#97

—

106

—

C₄×C₈

—

106

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₅₃

yes

characteristic

C₂×D₄

C₄×D₅₃

yes

—

C₂×D₄

C₂×Dic₅₃

yes

characteristic

100

C₂²×C₄

C₄×D₅₃

yes

—

101

C₂²×C₄

C₂²×D₅₃

yes

characteristic

102

M₁₆

—

103

M₁₆

—

106

—

104

M₁₆

—

106

—

105

M₁₆

C₂×Dic₅₃

yes

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

106

—

113

C₂×C₈

—

106

—

114

C₄⋊C₄

—

106

—

115

C₂²⋊C₄

—

116

C₄²

—

106

—

117

C₂³

C₄×Dic₅₃

yes

—

118

C₂³

C₂×C₄×D₅₃

yes

characteristic

119

Q₈

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

106

—

125

C₂×C₄

C₄×Dic₅₃

yes

—

126

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

127

C₂×C₄

D₄×D₅₃

yes

—

128

C₂×C₄

#11719947

yes

—

129

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

106

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂×C₄×Dic₅₃

yes

characteristic

139

C₂²

#13133177

yes

—

140

C₄

—

141

C₄

—

212

—

142

C₄

C₂²⋊C₄×D₅₃

yes

characteristic

143

C₄

#13133173

yes

characteristic

144

C₄

D₄×Dic₅₃

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

C₂²⋊C₄×Dic₅₃

yes

characteristic

	subgroup	G/H
center	C₄	C₂²⋊C₄×D₅₃
commutator	C₂²×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#413	C₂
socle	C₂×C₅₃	C₄×C₂²⋊C₄
hypercenter	#413	D₅₃
hypocenter	C₅₃	#1757
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

C₂²⋊C₄×D₅₃

commutator

C₂²×C₅₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#413

C₂

socle

C₂×C₅₃

C₄×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
3. #16810581	C₂	semidirect
4. #16810395	C₂	semidirect
8. C₂×#11719917	C₄	semidirect
10. C₂²×#88322	C₄	semidirect
18. #101×C₅₃	C₄	semidirect
19. #100×C₅₃	C₄	semidirect
80. C₅₃	#1757	semidirect
90. #101	Dic₅₃	semidirect
91. #100	Dic₅₃	semidirect

H ≅ G/N

type

3. #16810581

C₂

semidirect

4. #16810395

C₂

semidirect

8. C₂×#11719917

C₄

semidirect

10. C₂²×#88322

C₄

semidirect

18. #101×C₅₃

C₄

semidirect

19. #100×C₅₃

C₄

semidirect

80. C₅₃

#1757

semidirect

90. #101

Dic₅₃

semidirect

91. #100

Dic₅₃

semidirect

Subgroups — #20274518

Conjugacy classes of subgroups

Complements