Group #20290474 — subgroups

Subgroups — #20290474

Order 6784.

284 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 86 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×#13133305	C₂	1	1	yes	maximal, characteristic
2	#16811604	C₂	1	1	yes	maximal, characteristic
3	#16811589	C₂	1	2	yes	maximal, complemented
4	#16811506	C₂	1	2	yes	maximal, complemented
5	#16810606	C₂	1	1	yes	maximal, characteristic
6	#13133305	—	2	2	no	—
7	#13133305	—	2	2	no	—
8	C₂×#11719936	C₄	1	2	yes	complemented
9	C₂×#11719936	C₂²	1	1	yes	characteristic
10	C₂×D₅₃⋊C₈	C₄	1	2	yes	complemented
11	C₂×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
12	#13133296	C₂²	1	2	yes	—
13	#13133276	C₄	1	4	yes	—
14	#13133271	—	2	2	no	—
15	C₈×#6157	—	2	2	no	—
16	C₂×#11719949	C₂²	1	1	yes	characteristic
17	M₁₆×D₅₃	C₂²	1	2	yes	—
18	#13133136	C₄	1	1	yes	characteristic, complemented
19	#13133133	C₄	1	2	yes	complemented
20	#101×C₅₃	C₄	1	1	yes	characteristic, complemented
21	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	#11719949	—	2	2	no	—
23	#11719949	—	2	2	no	—
24	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
25	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
28	C₂²⋊#6157	C₂×C₄	1	4	yes	—
29	C₂×#88324	C₂×C₄	1	2	yes	—
30	C₂×#88324	C₂×C₄	1	1	yes	characteristic
31	C₂×#88324	C₂×C₄	1	1	yes	characteristic
32	C₄⋊#6157	—	2	2	no	—
33	C₄×#6157	—	2	2	no	—
34	#11719936	—	2	2	no	—
35	#11719936	—	2	4	no	—
36	D₅₃⋊C₈	—	2	2	no	—
37	D₅₃⋊C₈	—	2	2	no	—
38	D₅₃⋊C₈	D₄	1	2	yes	—
39	D₅₃⋊C₈	D₄	1	2	yes	—
40	M₁₆×C₅₃	C₂×C₄	1	2	yes	—
41	#11719917	C₂×C₄	1	2	yes	—
42	#11719912	—	2	2	no	—
43	C₈×D₅₃	—	2	2	no	—
44	C₂²×D₅₃	C₄²	1	2	yes	—
45	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×#6157	—	2	4	no	—
47	Q₈×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	C₄²	1	2	yes	—
49	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
51	D₂₁₂	—	2	4	no	—
52	D₂₁₂	—	2	2	no	—
53	C₄×D₅₃	—	2	2	no	—
54	C₄×D₅₃	—	2	2	no	—
55	C₄×D₅₃	C₂×D₄	1	2	yes	—
56	C₄×D₅₃	#41	1	2	yes	—
57	#88324	—	2	2	no	—
58	#88324	C₂²⋊C₄	1	2	yes	—
59	#88324	C₂²⋊C₄	1	1	yes	characteristic
60	#88324	C₂²⋊C₄	1	1	yes	characteristic
61	C₈×C₅₃	—	2	2	no	—
62	#88322	—	2	2	no	—
63	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
64	C₂×D₅₃	—	2	2	no	—
65	C₂×D₅₃	—	4	1	no	—
66	C₂×D₅₃	—	4	2	no	—
67	C₂×D₅₃	C₄×D₄	1	2	yes	—
68	#6157	—	4	4	no	—
69	C₄×C₅₃	—	2	2	no	—
70	C₄×C₅₃	C₄×D₄	1	2	yes	—
71	Dic₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	Dic₅₃	#117	1	1	yes	characteristic
73	#1757	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	—	2	1	no	—
75	C₂×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
76	D₅₃	—	2	2	no	—
77	D₅₃	—	4	2	no	—
78	C₂×#131	—	53	1	no	—
79	#413	—	53	1	no	—
80	#410	—	53	1	no	—
81	#405	—	53	2	no	—
82	#344	—	53	2	no	—
83	C₅₃	#1757	1	1	yes	characteristic, complemented, Sylow
84	C₂×#41	—	53	1	no	—
85	#131	—	106	2	no	—
86	#131	—	106	2	no	—
87	C₂×M₁₆	—	53	1	no	—
88	C₂×M₁₆	—	53	2	no	—
89	C₂×M₁₆	—	53	2	no	—
90	C₂²×C₈	—	53	1	no	—
91	C₂²×C₈	—	53	2	no	—
92	#117	—	53	2	no	—
93	#101	—	53	1	no	—
94	#101	#6157	1	1	yes	characteristic, complemented
95	#100	—	53	2	no	—
96	#99	—	53	4	no	—
97	#97	—	106	2	no	—
98	C₄×C₈	—	106	2	no	—
99	#41	—	106	2	no	—
100	#41	—	106	2	no	—
101	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
102	C₂×D₄	—	53	2	no	—
103	C₂×D₄	—	53	1	no	—
104	C₂²×C₄	—	53	1	no	—
105	C₂²×C₄	—	53	2	no	—
106	M₁₆	—	53	2	no	—
107	M₁₆	—	106	2	no	—
108	M₁₆	—	106	4	no	—
109	M₁₆	—	106	2	no	—
110	M₁₆	C₂×#6157	1	2	yes	—
111	C₂×C₈	—	53	1	no	—
112	C₂×C₈	—	53	1	no	—
113	C₂×C₈	—	53	2	no	—
114	C₂×C₈	—	53	2	no	—
115	C₂×C₈	—	53	2	no	—
116	C₂×C₈	—	106	2	no	—
117	C₂×C₈	—	106	2	no	—
118	C₂×C₈	—	106	2	no	—
119	C₄⋊C₄	—	106	2	no	—
120	C₂²⋊C₄	—	53	4	no	—
121	C₄²	—	106	2	no	—
122	C₂³	—	53	1	no	—
123	C₂³	—	53	2	no	—
124	Q₈	—	2	2	no	—
125	D₄	—	106	4	no	—
126	D₄	—	106	2	no	—
127	C₂×C₄	—	53	1	no	—
128	C₂×C₄	—	53	2	no	—
129	C₂×C₄	—	53	2	no	—
130	C₂×C₄	—	106	2	no	—
131	C₂×C₄	—	106	2	no	—
132	C₂×C₄	—	106	4	no	—
133	C₂×C₄	C₄×#6157	1	2	yes	—
134	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	53	1	no	—
137	C₈	—	53	1	no	—
138	C₈	—	53	2	no	—
139	C₈	—	106	2	no	—
140	C₈	—	106	2	no	—
141	C₂²	—	53	2	no	—
142	C₂²	—	106	2	no	—
143	C₂²	—	212	1	no	—
144	C₂²	—	212	2	no	—
145	C₂²	C₂×C₄×#6157	1	1	yes	characteristic
146	C₄	—	2	2	no	—
147	C₄	—	53	1	no	—
148	C₄	—	53	1	no	—
149	C₄	—	212	4	no	—
150	C₄	D₄×#6157	1	2	yes	—
151	C₂	—	2	1	no	—
152	C₂	—	106	2	no	—
153	C₂	—	212	2	no	—
154	C₂	C₂²⋊C₄×#6157	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133305

C₂

yes

maximal, characteristic

#16811604

C₂

yes

maximal, characteristic

#16811589

C₂

yes

maximal, complemented

#16811506

C₂

yes

maximal, complemented

#16810606

C₂

yes

maximal, characteristic

#13133305

—

#13133305

—

C₂×#11719936

C₄

yes

complemented

C₂×#11719936

C₂²

yes

characteristic

C₂×D₅₃⋊C₈

C₄

yes

complemented

C₂×D₅₃⋊C₈

C₂²

yes

characteristic

#13133296

C₂²

yes

—

#13133276

C₄

yes

—

#13133271

—

C₈×#6157

—

C₂×#11719949

C₂²

yes

characteristic

M₁₆×D₅₃

C₂²

yes

—

#13133136

C₄

yes

characteristic, complemented

#13133133

C₄

yes

complemented

#101×C₅₃

C₄

yes

characteristic, complemented

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

#11719949

—

#11719949

—

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×D₂₁₂

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂²⋊#6157

C₂×C₄

yes

—

C₂×#88324

C₂×C₄

yes

—

C₂×#88324

C₂×C₄

yes

characteristic

C₂×#88324

C₂×C₄

yes

—

—

—

—

D₅₃⋊C₈

—

D₅₃⋊C₈

—

D₅₃⋊C₈

D₄

yes

—

D₅₃⋊C₈

D₄

yes

—

M₁₆×C₅₃

C₂×C₄

yes

—

#11719917

C₂×C₄

yes

—

#11719912

—

C₈×D₅₃

—

C₂²×D₅₃

C₄²

yes

—

C₂²×D₅₃

C₂²×C₄

yes

characteristic

C₂×#6157

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

characteristic

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×D₄

yes

—

C₄×D₅₃

#41

yes

—

#88324

—

#88324

C₂²⋊C₄

yes

—

#88324

C₂²⋊C₄

yes

characteristic

#88324

C₂²⋊C₄

yes

characteristic

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₄×D₄

yes

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

Dic₅₃

C₂×C₂²⋊C₄

yes

Dic₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

C₄×C₂²⋊C₄

yes

characteristic

D₅₃

—

D₅₃

—

—

—

—

—

—

C₅₃

#1757

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

106

—

#131

—

106

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

—

—

yes

characteristic, complemented

#100

—

#99

—

#97

—

106

—

C₄×C₈

—

106

—

#41

—

106

—

100

#41

—

106

—

101

C₂×Q₈

C₂×#6157

yes

characteristic

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

M₁₆

—

107

M₁₆

—

106

—

108

M₁₆

—

106

—

109

M₁₆

—

106

—

110

M₁₆

C₂×#6157

yes

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

106

—

117

C₂×C₈

—

106

—

118

C₂×C₈

—

106

—

119

C₄⋊C₄

—

106

—

120

C₂²⋊C₄

—

121

C₄²

—

106

—

122

C₂³

—

123

C₂³

—

124

Q₈

—

125

D₄

—

106

—

126

D₄

—

106

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

106

—

131

C₂×C₄

—

106

—

132

C₂×C₄

—

106

—

133

C₂×C₄

C₄×#6157

yes

—

134

C₂×C₄

C₂²×#6157

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

106

—

140

C₈

—

106

—

141

C₂²

—

142

C₂²

—

106

—

143

C₂²

—

212

—

144

C₂²

—

212

—

145

C₂²

C₂×C₄×#6157

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

212

—

150

C₄

D₄×#6157

yes

—

151

C₂

—

152

C₂

—

106

—

153

C₂

—

212

—

154

C₂

C₂²⋊C₄×#6157

yes

characteristic

	subgroup	G/H
center	C₂	C₂²⋊C₄×#6157
commutator	C₂²×C₅₃	C₂×C₄²
Frattini	C₂×C₄	C₂²×#6157
Fitting	#101×C₅₃	C₄
socle	C₂×C₅₃	C₄×C₂²⋊C₄
hypercenter	#101	#6157
hypocenter	C₅₃	#1757
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

subgroup

G/H

center

C₂

C₂²⋊C₄×#6157

commutator

C₂²×C₅₃

C₂×C₄²

Frattini

C₂×C₄

C₂²×#6157

Fitting

#101×C₅₃

C₄

socle

C₂×C₅₃

C₄×C₂²⋊C₄

C₅₃

G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

N	H ≅ G/N	type
3. #16811589	C₂	semidirect
4. #16811506	C₂	semidirect
8. C₂×#11719936	C₄	semidirect
10. C₂×D₅₃⋊C₈	C₄	semidirect
18. #13133136	C₄	semidirect
19. #13133133	C₄	semidirect
20. #101×C₅₃	C₄	semidirect
83. C₅₃	#1757	semidirect
94. #101	#6157	semidirect

H ≅ G/N

type

3. #16811589

C₂

semidirect

4. #16811506

C₂

semidirect

8. C₂×#11719936

C₄

semidirect

10. C₂×D₅₃⋊C₈

C₄

semidirect

18. #13133136

C₄

semidirect

19. #13133133

C₄

semidirect

20. #101×C₅₃

C₄

semidirect

83. C₅₃

#1757

semidirect

94. #101

#6157

semidirect

Subgroups — #20290474

Conjugacy classes of subgroups

Complements