Group #20283782 — subgroups

Subgroups — #20283782

Order 6784.

410 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 205 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	#16810701	C₂	1	4	yes	maximal, complemented
2	D₄×#88322	C₂	1	4	yes	maximal, complemented
3	#16810676	C₂	1	2	yes	maximal, complemented
4	#16810590	C₂	1	4	yes	maximal, complemented
5	C₅₃×#532	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	4	yes	—
7	C₂²×#88322	C₂²	1	4	yes	—
8	C₄⋊#88322	C₂²	1	2	yes	—
9	C₄⋊#88322	C₂²	1	2	yes	—
10	C₂²⋊#88322	C₂²	1	4	yes	—
11	C₂²⋊#88322	C₂²	1	8	yes	—
12	C₄×#88322	C₂²	1	4	yes	—
13	C₂×#41×C₅₃	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
15	#127×C₅₃	C₄	1	1	yes	characteristic
16	#124×C₅₃	C₄	1	2	yes	—
17	#121×C₅₃	C₄	1	4	yes	—
18	C₄×D₄×C₅₃	C₂²	1	4	yes	—
19	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	#41×C₅₃	—	4	2	no	—
21	#41×C₅₃	—	4	2	no	—
22	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×D₄×C₅₃	C₂×C₄	1	4	yes	—
24	C₂×D₄×C₅₃	C₂×C₄	1	4	yes	—
25	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
29	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
30	C₂²⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
31	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₅₃	C₂³	1	2	yes	—
34	#11719917	—	4	4	no	—
35	C₂×#88322	—	4	4	no	—
36	C₂×#88322	C₂³	1	4	yes	—
37	C₂×#88322	C₂³	1	4	yes	—
38	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂³×C₅₃	C₂²×C₄	1	4	yes	—
40	Q₈×C₅₃	—	4	2	no	—
41	D₄×C₅₃	—	2	8	no	—
42	D₄×C₅₃	—	4	2	no	—
43	D₄×C₅₃	—	4	4	no	—
44	C₂×C₄×C₅₃	—	2	4	no	—
45	C₂×C₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	—	2	4	no	—
48	C₂×C₄×C₅₃	—	2	4	no	—
49	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
52	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
54	#88322	—	2	4	no	—
55	#88322	—	2	4	no	—
56	C₂²×C₅₃	—	2	1	no	—
57	C₂²×C₅₃	—	2	4	no	—
58	C₂²×C₅₃	—	2	2	no	—
59	C₂²×C₅₃	—	2	8	no	—
60	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₅₃	—	2	1	no	—
62	C₄×C₅₃	—	2	4	no	—
63	C₄×C₅₃	—	2	2	no	—
64	C₄×C₅₃	#131	1	4	yes	—
65	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
66	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
67	#2979	—	53	1	no	maximal, Sylow
68	C₂×C₅₃	—	4	1	no	—
69	C₂×C₅₃	—	4	4	no	—
70	C₂×C₅₃	#518	1	1	yes	characteristic
71	C₂×C₅₃	C₂×#131	1	2	yes	—
72	#532	D₅₃	1	1	yes	characteristic
73	#436	—	53	4	no	—
74	C₈×D₄	—	53	4	no	—
75	#431	—	53	2	no	—
76	#408	—	53	4	no	—
77	C₅₃	#2979	1	1	yes	characteristic, complemented, Sylow
78	C₂×#41	C₂×D₅₃	1	2	yes	—
79	C₂×M₁₆	—	53	4	no	—
80	C₂²×C₈	—	53	4	no	—
81	C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
82	#127	Dic₅₃	1	1	yes	characteristic
83	#124	Dic₅₃	1	2	yes	—
84	#121	Dic₅₃	1	4	yes	—
85	C₄×D₄	C₂×D₅₃	1	4	yes	—
86	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
87	C₄⋊C₈	—	53	2	no	—
88	C₄⋊C₈	—	53	2	no	—
89	C₂²⋊C₈	—	53	4	no	—
90	C₂²⋊C₈	—	53	8	no	—
91	C₄×C₈	—	53	4	no	—
92	#41	—	4	2	no	—
93	#41	—	4	2	no	—
94	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
95	C₂×D₄	C₂×Dic₅₃	1	4	yes	—
96	C₂×D₄	C₂×Dic₅₃	1	4	yes	—
97	C₂×D₄	C₂×Dic₅₃	1	2	yes	—
98	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
99	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
101	M₁₆	—	212	4	no	—
102	C₂×C₈	—	53	4	no	—
103	C₂×C₈	—	53	4	no	—
104	C₂×C₈	—	212	4	no	—
105	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
106	C₂²⋊C₄	C₂×Dic₅₃	1	8	yes	—
107	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
108	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
109	C₄²	C₂²×D₅₃	1	2	yes	—
110	C₂³	C₂²×Dic₅₃	1	4	yes	—
111	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
112	Q₈	—	4	2	no	—
113	D₄	—	2	8	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
122	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
123	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
124	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
125	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
126	C₈	—	106	4	no	—
127	C₈	—	106	4	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	8	no	—
132	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	4	no	—
135	C₄	—	2	2	no	—
136	C₄	#13133224	1	4	yes	—
137	C₄	#13133262	1	1	yes	characteristic
138	C₄	#13133262	1	1	yes	characteristic
139	C₂	—	4	1	no	—
140	C₂	—	4	4	no	—
141	C₂	#16811098	1	1	yes	characteristic
142	C₂	C₂×#13133224	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810701

C₂

yes

maximal, complemented

D₄×#88322

C₂

yes

maximal, complemented

#16810676

C₂

yes

maximal, complemented

#16810590

C₂

yes

maximal, complemented

C₅₃×#532

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂²×#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

C₂²⋊#88322

C₂²

yes

—

C₂²⋊#88322

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₂×#41×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₄

yes

characteristic

#127×C₅₃

C₄

yes

characteristic

#124×C₅₃

C₄

yes

—

#121×C₅₃

C₄

yes

—

C₄×D₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

#11719917

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂³×C₅₃

C₂²×C₄

yes

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#131

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#518

yes

characteristic

C₂×C₅₃

C₂×#131

yes

—

#532

D₅₃

yes

characteristic

#436

—

C₈×D₄

—

#431

—

#408

—

C₅₃

#2979

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

Dic₅₃

yes

characteristic

#127

Dic₅₃

yes

characteristic

#124

Dic₅₃

yes

—

#121

Dic₅₃

yes

—

C₄×D₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂×D₄

C₂×Dic₅₃

yes

—

C₂×D₄

C₂×Dic₅₃

yes

—

C₂×D₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

100

C₂²×C₄

C₂²×D₅₃

yes

—

101

M₁₆

—

212

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

212

—

105

C₄⋊C₄

C₂×Dic₅₃

yes

—

106

C₂²⋊C₄

C₂×Dic₅₃

yes

—

107

C₄²

C₂×Dic₅₃

yes

characteristic

108

C₄²

C₂×Dic₅₃

yes

characteristic

109

C₄²

C₂²×D₅₃

yes

—

110

C₂³

C₂²×Dic₅₃

yes

—

111

C₂³

C₂²×Dic₅₃

yes

characteristic

112

Q₈

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

122

C₂×C₄

C₂²×Dic₅₃

yes

—

123

C₂×C₄

C₂²×Dic₅₃

yes

—

124

C₂×C₄

C₂²×Dic₅₃

yes

—

125

C₂×C₄

C₂³×D₅₃

yes

characteristic

126

C₈

—

106

—

127

C₈

—

106

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂³×Dic₅₃

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#13133224

yes

—

137

C₄

#13133262

yes

characteristic

138

C₄

#13133262

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

#16811098

yes

characteristic

142

C₂

C₂×#13133224

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#532	C₂
hypercenter	#532	D₅₃
hypocenter	C₅₃	#2979
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₃

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

C₅₃×#532

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
1. #16810701	C₂	semidirect
2. D₄×#88322	C₂	semidirect
3. #16810676	C₂	semidirect
4. #16810590	C₂	semidirect
77. C₅₃	#2979	semidirect

H ≅ G/N

type

1. #16810701

C₂

semidirect

2. D₄×#88322

C₂

semidirect

3. #16810676

C₂

semidirect

4. #16810590

C₂

semidirect

77. C₅₃

#2979

semidirect

Subgroups — #20283782

Conjugacy classes of subgroups

Complements