Group #20274236 — subgroups

Subgroups — #20274236

Order 6784.

154 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16811026	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810913	C₂	1	1	yes	maximal, characteristic
3	#16810701	C₂	1	1	yes	maximal, characteristic
4	#16810373	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810372	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810371	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#476	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719917	—	2	1	no	—
9	#13133217	C₂²	1	1	yes	characteristic
10	#13133150	—	2	1	no	—
11	#13133150	—	2	1	no	—
12	#13133148	—	2	1	no	—
13	C₄⋊#88322	C₂²	1	1	yes	characteristic
14	C₄⋊#88322	C₂²	1	1	yes	characteristic
15	#13133138	—	2	1	no	—
16	C₂²⋊#88322	—	2	1	no	—
17	C₄×#88322	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
20	#106×C₅₃	—	2	1	no	—
21	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	#102×C₅₃	—	2	1	no	—
23	C₂×Q₈×C₅₃	—	2	1	no	—
24	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	—	2	1	no	—
26	C₂×Dic₁₀₆	—	2	1	no	—
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
31	C₂²⋊C₄×C₅₃	—	2	1	no	—
32	C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₅₃	—	2	1	no	—
34	C₄⋊Dic₅₃	—	2	1	no	—
35	#11719917	—	4	2	no	—
36	C₂×#88322	—	2	1	no	—
37	C₂×#88322	—	2	1	no	—
38	C₂×#88322	D₄	1	1	yes	characteristic
39	C₂×#88322	D₄	1	1	yes	characteristic
40	C₂³×C₅₃	—	2	1	no	—
41	Q₈×C₅₃	—	8	1	no	—
42	D₄×C₅₃	—	2	1	no	—
43	D₄×C₅₃	—	2	1	no	—
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	—	4	1	no	—
46	C₂×C₄×C₅₃	—	4	1	no	—
47	C₂×C₄×C₅₃	—	4	1	no	—
48	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
51	C₂×Dic₅₃	—	4	1	no	—
52	Dic₁₀₆	—	8	1	no	—
53	C₈×C₅₃	—	4	1	no	—
54	#88322	—	2	1	no	—
55	#88322	—	4	1	no	—
56	#88322	—	4	1	no	—
57	#88322	SD₁₆	1	2	yes	—
58	C₂²×C₅₃	—	4	1	no	—
59	C₂²×C₅₃	—	4	1	no	—
60	C₂²×C₅₃	—	4	1	no	—
61	C₂²×C₅₃	#121	1	1	yes	characteristic
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	—	4	1	no	—
64	C₄×C₅₃	—	8	1	no	—
65	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₄×C₅₃	#137	1	1	yes	characteristic
67	C₄×C₅₃	#137	1	1	yes	characteristic
68	C₄×C₅₃	#137	1	1	yes	characteristic
69	Dic₅₃	—	8	1	no	—
70	#1695	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	—	8	1	no	—
72	C₂×C₅₃	#461	1	1	yes	characteristic
73	C₂×C₅₃	#470	1	1	yes	characteristic
74	C₂×C₅₃	#472	1	1	yes	characteristic
75	#500	—	53	1	no	—
76	#476	—	53	1	no	—
77	#476	D₅₃	1	1	yes	characteristic
78	#436	—	53	1	no	—
79	#334	—	53	1	no	—
80	#333	—	53	1	no	—
81	#333	—	53	1	no	—
82	C₅₃	#1695	1	1	yes	characteristic, complemented, Sylow
83	C₂×M₁₆	—	106	1	no	—
84	C₄⋊Q₈	—	53	1	no	—
85	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
86	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
87	#107	—	106	1	no	—
88	#107	—	106	1	no	—
89	#106	—	2	1	no	—
90	#106	—	106	1	no	—
91	C₄⋊C₈	—	53	1	no	—
92	C₄⋊C₈	—	53	1	no	—
93	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
94	#102	—	2	1	no	—
95	#102	—	106	1	no	—
96	C₂²⋊C₈	—	106	1	no	—
97	C₄×C₈	—	53	1	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	—	106	1	no	—
100	C₂×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
101	C₂²×C₄	—	2	1	no	—
102	M₁₆	—	212	2	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	53	1	no	—
105	C₂×C₈	—	53	1	no	—
106	C₂×C₈	—	106	1	no	—
107	C₂×C₈	—	106	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	106	1	no	—
111	C₄⋊C₄	—	106	1	no	—
112	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
113	C₂²⋊C₄	—	2	1	no	—
114	C₄²	C₂²×D₅₃	1	1	yes	characteristic
115	C₂³	—	2	1	no	—
116	Q₈	—	8	1	no	—
117	Q₈	—	424	1	no	—
118	D₄	—	2	1	no	—
119	D₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	212	1	no	—
125	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
126	C₂×C₄	#11719947	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
128	C₈	—	4	1	no	—
129	C₈	—	53	2	no	—
130	C₈	—	106	1	no	—
131	C₈	—	212	1	no	—
132	C₈	—	212	1	no	—
133	C₂²	—	4	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	4	1	no	—
136	C₂²	#13133189	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	4	1	no	—
139	C₄	—	8	1	no	—
140	C₄	—	424	1	no	—
141	C₄	SD₁₆×D₅₃	1	1	yes	characteristic
142	C₄	#13133246	1	1	yes	characteristic
143	C₄	#13133247	1	1	yes	characteristic
144	C₄	#13133248	1	1	yes	characteristic
145	C₂	—	8	1	no	—
146	C₂	#16810826	1	1	yes	characteristic
147	C₂	#16810879	1	1	yes	characteristic
148	C₂	#16810891	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811026

C₂

yes

maximal, characteristic, complemented

#16810913

C₂

yes

maximal, characteristic

#16810701

C₂

yes

maximal, characteristic

#16810373

C₂

yes

maximal, characteristic, complemented

#16810372

C₂

yes

maximal, characteristic, complemented

#16810371

C₂

yes

maximal, characteristic, complemented

C₅₃×#476

C₂

yes

maximal, characteristic

C₂×#11719917

—

#13133217

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

C₂²

yes

characteristic

#106×C₅₃

—

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

#102×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

—

—

—

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

C₂³×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

—

—

—

SD₁₆

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1695

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

—

—

—

—

C₅₃

#1695

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

C₄×D₄

C₂×D₅₃

yes

characteristic

#107

—

106

—

#107

—

106

—

#106

—

#106

—

106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

characteristic

#102

—

#102

—

106

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

100

C₂×D₄

C₂²⋊D₅₃

yes

characteristic

101

C₂²×C₄

—

102

M₁₆

—

212

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

106

—

107

C₂×C₈

—

106

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

106

—

111

C₄⋊C₄

—

106

—

112

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

113

C₂²⋊C₄

—

114

C₄²

C₂²×D₅₃

yes

characteristic

115

C₂³

—

116

Q₈

—

117

Q₈

—

424

—

118

D₄

—

119

D₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

212

—

125

C₂×C₄

D₄×D₅₃

yes

characteristic

126

C₂×C₄

#11719947

yes

characteristic

127

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

106

—

131

C₈

—

212

—

132

C₈

—

212

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

#13133189

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

424

—

141

C₄

SD₁₆×D₅₃

yes

characteristic

142

C₄

#13133246

yes

characteristic

143

C₄

#13133247

yes

characteristic

144

C₄

#13133248

yes

characteristic

145

C₂

—

146

C₂

#16810826

yes

characteristic

147

C₂

#16810879

yes

characteristic

148

C₂

#16810891

yes

characteristic

	subgroup	G/H
center	C₂²	#13133189
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#476	C₂
socle	C₂²×C₅₃	#121
hypercenter	#476	D₅₃
hypocenter	C₅₃	#1695
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

subgroup

G/H

center

C₂²

#13133189

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#476

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #476

N	H ≅ G/N	type
1. #16811026	C₂	semidirect
4. #16810373	C₂	semidirect
5. #16810372	C₂	semidirect
6. #16810371	C₂	semidirect
82. C₅₃	#1695	semidirect

H ≅ G/N

type

1. #16811026

C₂

semidirect

4. #16810373

C₂

semidirect

5. #16810372

C₂

semidirect

6. #16810371

C₂

semidirect

82. C₅₃

#1695

semidirect

Subgroups — #20274236

Conjugacy classes of subgroups

Complements