Group #20285097 — subgroups

Subgroups — #20285097

Order 6784.

380 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13133217	C₂	1	1	yes	maximal, characteristic
2	#16810993	C₂	1	4	yes	maximal, complemented
3	#16810975	C₂	1	4	yes	maximal, complemented
4	#16810631	C₂	1	4	yes	maximal
5	C₄×#11719917	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#533	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
8	C₂×#11719917	C₂²	1	2	yes	—
9	#13133217	—	2	2	no	—
10	#13133217	C₂²	1	2	yes	—
11	#13133217	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₅₃	—	2	2	no	—
13	#13133140	C₂²	1	16	yes	—
14	#13133128	C₂²	1	2	yes	—
15	C₄×#88322	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
17	#125×C₅₃	C₂²	1	2	yes	—
18	C₂²⋊Q₈×C₅₃	—	2	2	no	—
19	C₄×Q₈×C₅₃	—	2	2	no	—
20	#117×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×Q₈×C₅₃	—	2	2	no	—
23	C₂²×C₄×C₅₃	D₄	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅₃	—	2	2	no	—
26	C₂×Dic₁₀₆	—	2	8	no	—
27	C₂×Dic₁₀₆	—	2	2	no	—
28	C₂×Dic₁₀₆	C₂³	1	4	yes	—
29	C₄⋊C₄×C₅₃	—	2	4	no	—
30	C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
32	C₂²⋊C₄×C₅₃	—	2	2	no	—
33	C₄²×C₅₃	—	2	2	no	—
34	C₄²×C₅₃	D₄	1	2	yes	—
35	C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₄⋊Dic₅₃	—	2	4	no	—
38	C₄⋊Dic₅₃	—	2	4	no	—
39	#11719917	—	2	4	no	—
40	C₂×#88322	C₂³	1	4	yes	—
41	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₅₃	—	4	4	no	—
43	C₂×C₄×C₅₃	—	2	4	no	—
44	C₂×C₄×C₅₃	—	2	4	no	—
45	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₅₃	#41	1	4	yes	—
49	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₅₃	—	2	4	no	—
51	C₂×Dic₅₃	—	4	4	no	—
52	Dic₁₀₆	—	2	8	no	—
53	Dic₁₀₆	—	4	4	no	—
54	#88322	—	2	4	no	—
55	C₂²×C₅₃	—	2	2	no	—
56	C₂²×C₅₃	#124	1	2	yes	—
57	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₅₃	—	2	2	no	—
59	C₄×C₅₃	—	4	4	no	—
60	C₄×C₅₃	#124	1	2	yes	—
61	C₄×C₅₃	#137	1	4	yes	—
62	C₄×C₅₃	C₂×#41	1	2	yes	—
63	Dic₅₃	—	4	4	no	—
64	#3139	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	—	2	2	no	—
66	C₂×C₅₃	C₂×#124	1	1	yes	characteristic
67	C₂×C₅₃	C₂×#137	1	2	yes	—
68	#533	D₅₃	1	1	yes	characteristic
69	C₂×C₄⋊Q₈	—	53	1	no	—
70	#491	—	53	4	no	—
71	#487	—	53	4	no	—
72	#419	—	53	4	no	—
73	C₄×M₁₆	—	53	1	no	—
74	C₅₃	#3139	1	1	yes	characteristic, complemented, Sylow
75	C₂²×Q₈	—	53	2	no	—
76	C₂×M₁₆	—	53	2	no	—
77	C₄⋊Q₈	—	53	2	no	—
78	C₄⋊Q₈	—	53	2	no	—
79	C₄⋊Q₈	—	106	2	no	—
80	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
81	#125	C₂×D₅₃	1	2	yes	—
82	C₂²⋊Q₈	—	2	2	no	—
83	C₄×Q₈	—	2	2	no	—
84	#117	C₂×D₅₃	1	2	yes	—
85	C₂×C₄⋊C₄	—	106	2	no	—
86	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
87	#103	—	53	16	no	—
88	#97	—	53	2	no	—
89	C₄×C₈	—	53	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	53	4	no	—
92	C₂×Q₈	—	106	2	no	—
93	C₂×Q₈	—	106	8	no	—
94	C₂²×C₄	—	106	2	no	—
95	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
96	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
97	M₁₆	—	106	4	no	—
98	C₂×C₈	—	53	4	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	106	4	no	—
102	C₄⋊C₄	—	106	4	no	—
103	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
104	C₂²⋊C₄	—	2	2	no	—
105	C₄²	—	2	2	no	—
106	C₄²	C₂²⋊D₅₃	1	2	yes	—
107	C₄²	C₂²×D₅₃	1	1	yes	characteristic
108	C₄²	C₂²×D₅₃	1	1	yes	characteristic
109	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	106	8	no	—
112	Q₈	—	212	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	106	4	no	—
116	C₂×C₄	—	212	4	no	—
117	C₂×C₄	#11719949	1	4	yes	—
118	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
121	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
122	C₈	—	106	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#13133206	1	2	yes	—
125	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	4	4	no	—
128	C₄	—	212	4	no	—
129	C₄	#13133206	1	2	yes	—
130	C₄	#13133247	1	4	yes	—
131	C₄	C₂×#11719949	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#13133206	1	1	yes	characteristic
134	C₂	C₂×#13133247	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133217

C₂

yes

maximal, characteristic

#16810993

C₂

yes

maximal, complemented

#16810975

C₂

yes

maximal, complemented

#16810631

C₂

yes

maximal

C₄×#11719917

C₂

yes

maximal, characteristic

C₅₃×#533

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

C₂×#11719917

C₂²

yes

—

#13133217

—

#13133217

C₂²

yes

—

#13133217

C₂²

yes

—

C₂×C₄⋊Dic₅₃

—

#13133140

C₂²

yes

—

#13133128

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

#125×C₅₃

C₂²

yes

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719917

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

#124

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#124

yes

—

C₄×C₅₃

#137

yes

—

C₄×C₅₃

C₂×#41

yes

—

Dic₅₃

—

#3139

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#124

yes

characteristic

C₂×C₅₃

C₂×#137

yes

—

#533

D₅₃

yes

characteristic

C₂×C₄⋊Q₈

—

#491

—

#487

—

#419

—

C₄×M₁₆

—

C₅₃

#3139

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

C₂×D₅₃

yes

—

#125

C₂×D₅₃

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#103

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

106

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

106

—

102

C₄⋊C₄

—

106

—

103

C₄⋊C₄

C₂²×D₅₃

yes

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

C₂²⋊D₅₃

yes

—

107

C₄²

C₂²×D₅₃

yes

characteristic

108

C₄²

C₂²×D₅₃

yes

characteristic

109

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

110

Q₈

—

111

Q₈

—

106

—

112

Q₈

—

212

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

106

—

116

C₂×C₄

—

212

—

117

C₂×C₄

#11719949

yes

—

118

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

119

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

121

C₂×C₄

C₂³×D₅₃

yes

characteristic

122

C₈

—

106

—

123

C₂²

—

124

C₂²

#13133206

yes

—

125

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

212

—

129

C₄

#13133206

yes

—

130

C₄

#13133247

yes

—

131

C₄

C₂×#11719949

yes

—

132

C₂

—

133

C₂

C₂×#13133206

yes

characteristic

134

C₂

C₂×#13133247

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#533	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#533	D₅₃
hypocenter	C₅₃	#3139
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#533

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #16810993	C₂	semidirect
3. #16810975	C₂	semidirect
74. C₅₃	#3139	semidirect

H ≅ G/N

type

2. #16810993

C₂

semidirect

3. #16810975

C₂

semidirect

74. C₅₃

#3139

semidirect

Subgroups — #20285097

Conjugacy classes of subgroups

Complements