Group #20274763 — subgroups

Subgroups — #20274763

Order 6784.

252 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₂×#13133152	C₂	1	2	yes	maximal, complemented
3	#16810607	C₂	1	1	yes	maximal, characteristic
4	#16810381	C₂	1	2	yes	maximal, complemented
5	C₅₃×#413	C₂	1	1	yes	maximal, characteristic
6	#13133224	—	2	4	no	—
7	C₂×#11719917	C₄	1	2	yes	—
8	C₂×#11719917	C₂²	1	2	yes	—
9	C₂×#11719917	C₂²	1	1	yes	characteristic
10	C₂²×#88322	C₄	1	2	yes	—
11	C₂²×#88322	C₂²	1	1	yes	characteristic
12	#13133152	—	2	4	no	—
13	#13133136	—	2	1	no	—
14	#13133134	—	2	1	no	—
15	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
17	#101×C₅₃	—	2	1	no	—
18	#100×C₅₃	—	2	1	no	—
19	#41×C₅₃	—	2	2	no	—
20	#41×C₅₃	—	2	2	no	—
21	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂×D₄×C₅₃	Q₈	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	M₁₆×C₅₃	—	2	2	no	—
27	M₁₆×C₅₃	—	2	2	no	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	#11719917	—	2	2	no	—
30	#11719917	—	2	4	no	—
31	#11719917	—	2	2	no	—
32	#11719917	—	2	2	no	—
33	C₂×#88322	—	2	4	no	—
34	C₂×#88322	—	2	2	no	—
35	C₂×#88322	C₂×C₄	1	2	yes	—
36	C₂×#88322	C₂×C₄	1	1	yes	characteristic
37	C₂×#88322	C₂×C₄	1	1	yes	characteristic
38	C₂×#88322	D₄	1	4	yes	—
39	C₂³×C₅₃	C₄⋊C₄	1	2	yes	—
40	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	Q₈×C₅₃	—	2	2	no	—
42	D₄×C₅₃	—	2	4	no	—
43	D₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#41	1	2	yes	—
51	C₈×C₅₃	—	4	2	no	—
52	#88322	—	2	2	no	—
53	#88322	—	4	2	no	—
54	#88322	C₂²⋊C₄	1	2	yes	—
55	#88322	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₅₃	—	2	2	no	—
57	C₂²×C₅₃	—	4	1	no	—
58	C₂²×C₅₃	—	4	2	no	—
59	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₅₃	—	2	2	no	—
62	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₅₃	#117	1	1	yes	characteristic
64	C₄×C₅₃	#121	1	2	yes	—
65	#1813	—	53	1	no	maximal, Sylow
66	C₂×C₅₃	—	2	1	no	—
67	C₂×C₅₃	—	2	2	no	—
68	C₂×C₅₃	—	4	2	no	—
69	C₂×C₅₃	#380	1	1	yes	characteristic
70	C₂×#131	—	53	1	no	—
71	C₂×#108	—	53	2	no	—
72	#413	—	53	1	no	—
73	#413	D₅₃	1	1	yes	characteristic
74	#338	—	53	2	no	—
75	C₅₃	#1813	1	1	yes	characteristic, complemented, Sylow
76	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
77	#131	—	106	4	no	—
78	C₂×M₁₆	—	53	2	no	—
79	C₂×M₁₆	—	53	1	no	—
80	C₂×M₁₆	—	53	2	no	—
81	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
82	C₂²×C₈	—	53	1	no	—
83	C₂²×C₈	—	53	2	no	—
84	#108	—	106	4	no	—
85	#101	—	2	1	no	—
86	#101	—	106	1	no	—
87	#100	—	2	1	no	—
88	#100	—	106	1	no	—
89	#41	—	2	2	no	—
90	#41	—	2	2	no	—
91	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
92	C₂×D₄	Dic₁₀₆	1	2	yes	—
93	C₂×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
94	C₂²×C₄	C₄×D₅₃	1	2	yes	—
95	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
96	M₁₆	—	2	2	no	—
97	M₁₆	—	2	2	no	—
98	M₁₆	—	106	4	no	—
99	M₁₆	—	106	2	no	—
100	M₁₆	—	106	2	no	—
101	M₁₆	—	106	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	53	1	no	—
104	C₂×C₈	—	53	1	no	—
105	C₂×C₈	—	53	4	no	—
106	C₂×C₈	—	53	2	no	—
107	C₂×C₈	—	106	4	no	—
108	C₂×C₈	—	106	2	no	—
109	C₂³	#11719919	1	2	yes	—
110	C₂³	C₂×Dic₁₀₆	1	1	yes	characteristic
111	Q₈	—	2	2	no	—
112	D₄	—	2	4	no	—
113	D₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	#11719919	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
118	C₂×C₄	D₄×D₅₃	1	2	yes	—
119	C₂×C₄	#11719947	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
121	C₈	—	4	2	no	—
122	C₈	—	53	2	no	—
123	C₈	—	53	2	no	—
124	C₈	—	106	2	no	—
125	C₈	—	212	2	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	4	1	no	—
128	C₂²	—	4	2	no	—
129	C₂²	C₂×#11719919	1	1	yes	characteristic
130	C₂²	#13133194	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
133	C₄	#13133173	1	1	yes	characteristic
134	C₄	#13133189	1	2	yes	—
135	C₂	—	2	1	no	—
136	C₂	—	2	2	no	—
137	C₂	—	4	2	no	—
138	C₂	#16810481	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133224

C₂

yes

maximal, characteristic

C₂×#13133152

C₂

yes

maximal, complemented

#16810607

C₂

yes

maximal, characteristic

#16810381

C₂

yes

maximal, complemented

C₅₃×#413

C₂

yes

maximal, characteristic

#13133224

—

C₂×#11719917

C₄

yes

—

C₂×#11719917

C₂²

yes

—

C₂×#11719917

C₂²

yes

characteristic

C₂²×#88322

C₄

yes

—

C₂²×#88322

C₂²

yes

—

—

—

C₂×#41×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

—

#101×C₅₃

—

#100×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

D₄

yes

—

C₂³×C₅₃

C₄⋊C₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₈×C₅₃

—

#88322

—

#88322

—

#88322

C₂²⋊C₄

yes

—

#88322

C₂²⋊C₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#1813

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

—

—

—

D₅₃

yes

characteristic

#338

—

C₅₃

#1813

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

characteristic

#131

—

106

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

#108

—

106

—

#101

—

#101

—

106

—

#100

—

#100

—

106

—

#41

—

#41

—

C₂×Q₈

C₂²⋊D₅₃

yes

characteristic

C₂×D₄

Dic₁₀₆

yes

—

C₂×D₄

C₂²⋊D₅₃

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

100

M₁₆

—

106

—

101

M₁₆

—

106

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

106

—

108

C₂×C₈

—

106

—

109

C₂³

#11719919

yes

—

110

C₂³

C₂×Dic₁₀₆

yes

characteristic

111

Q₈

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

#11719919

yes

—

117

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

118

C₂×C₄

D₄×D₅₃

yes

—

119

C₂×C₄

#11719947

yes

—

120

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

106

—

125

C₈

—

212

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×#11719919

yes

characteristic

130

C₂²

#13133194

yes

—

131

C₄

—

132

C₄

C₂²⋊C₄×D₅₃

yes

characteristic

133

C₄

#13133173

yes

characteristic

134

C₄

#13133189

yes

—

135

C₂

—

136

C₂

—

137

C₂

—

138

C₂

#16810481

yes

characteristic

	subgroup	G/H
center	C₄	#13133173
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#413	C₂
socle	C₂×C₅₃	#380
hypercenter	#413	D₅₃
hypocenter	C₅₃	#1813
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

#13133173

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#413

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
2. C₂×#13133152	C₂	semidirect
4. #16810381	C₂	semidirect
75. C₅₃	#1813	semidirect

H ≅ G/N

type

2. C₂×#13133152

C₂

semidirect

4. #16810381

C₂

semidirect

75. C₅₃

#1813

semidirect

Subgroups — #20274763

Conjugacy classes of subgroups

Complements