Group #20273440 — subgroups

Subgroups — #20273440

Order 6784.

228 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 77 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	#16810680	C₂	1	1	yes	maximal, characteristic
3	#16810371	C₂	1	4	yes	maximal, complemented
4	C₅₃×#432	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₀₆	C₄	1	2	yes	—
6	#13133217	—	2	1	no	—
7	#13133217	—	2	1	no	—
8	#13133217	C₄	1	2	yes	—
9	#13133217	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₅₃	—	2	1	no	—
11	C₂×C₄⋊Dic₅₃	—	2	1	no	—
12	C₄⋊#88322	C₂²	1	2	yes	—
13	C₂²⋊#88322	—	2	1	no	—
14	#13133128	—	2	1	no	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
17	C₂²⋊C₈×C₅₃	—	2	1	no	—
18	#97×C₅₃	—	2	1	no	—
19	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×Dic₅₃	—	2	2	no	—
23	C₂×Dic₁₀₆	—	2	2	no	—
24	C₂×Dic₁₀₆	—	4	4	no	—
25	C₂×Dic₁₀₆	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₅₃	—	2	2	no	—
27	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₅₃	D₄	1	1	yes	characteristic
29	C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₅₃	—	2	2	no	—
32	C₄⋊Dic₅₃	—	2	2	no	—
33	C₄⋊Dic₅₃	—	2	2	no	—
34	C₄⋊Dic₅₃	—	2	2	no	—
35	C₂×#88322	—	2	2	no	—
36	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₅₃	—	2	4	no	—
47	C₂×Dic₅₃	—	4	4	no	—
48	Dic₁₀₆	—	4	4	no	—
49	Dic₁₀₆	—	4	4	no	—
50	C₈×C₅₃	—	4	2	no	—
51	#88322	—	4	2	no	—
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	—	2	1	no	—
54	C₂²×C₅₃	#100	1	2	yes	—
55	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₄×C₅₃	—	2	1	no	—
57	C₄×C₅₃	—	2	1	no	—
58	C₄×C₅₃	—	2	1	no	—
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	#137	1	2	yes	—
61	C₄×C₅₃	#137	1	2	yes	—
62	Dic₅₃	—	4	4	no	—
63	#1558	—	53	1	no	maximal, Sylow
64	C₂×C₅₃	—	2	2	no	—
65	C₂×C₅₃	C₂×#100	1	1	yes	characteristic
66	C₂×C₅₃	#419	1	1	yes	characteristic
67	C₂×C₅₃	#419	1	1	yes	characteristic
68	C₂×C₄⋊Q₈	—	53	1	no	—
69	#432	—	53	1	no	—
70	#432	D₅₃	1	1	yes	characteristic
71	#333	—	53	4	no	—
72	C₅₃	#1558	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	53	2	no	—
74	C₄⋊Q₈	—	53	2	no	—
75	C₄⋊Q₈	—	53	2	no	—
76	C₄⋊Q₈	—	106	1	no	—
77	C₄⋊Q₈	—	106	1	no	—
78	C₂×C₄⋊C₄	—	106	1	no	—
79	C₂×C₄⋊C₄	—	106	1	no	—
80	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
81	C₄⋊C₈	—	53	2	no	—
82	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
83	C₂²⋊C₈	—	2	1	no	—
84	C₂²⋊C₈	—	106	1	no	—
85	#97	—	2	1	no	—
86	#97	—	106	1	no	—
87	C₂×Q₈	—	53	4	no	—
88	C₂×Q₈	—	106	2	no	—
89	C₂×Q₈	—	212	4	no	—
90	C₂²×C₄	—	106	2	no	—
91	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
92	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
93	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	106	2	no	—
96	C₄⋊C₄	—	106	2	no	—
97	C₄⋊C₄	—	106	2	no	—
98	C₄⋊C₄	—	106	2	no	—
99	C₄⋊C₄	—	106	2	no	—
100	C₄²	C₄×D₅₃	1	1	yes	characteristic
101	C₄²	D₂₁₂	1	1	yes	characteristic
102	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
103	C₄²	C₂²×D₅₃	1	1	yes	characteristic
104	C₂³	#11719921	1	1	yes	characteristic
105	Q₈	—	212	4	no	—
106	Q₈	—	212	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	106	4	no	—
111	C₂×C₄	—	212	4	no	—
112	C₂×C₄	#11719921	1	1	yes	characteristic
113	C₂×C₄	#11719921	1	1	yes	characteristic
114	C₂×C₄	#11719921	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
116	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
118	C₈	—	4	2	no	—
119	C₈	—	212	2	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	#13133133	1	2	yes	—
123	C₂²	C₂×#11719921	1	1	yes	characteristic
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	212	4	no	—
129	C₄	#13133247	1	2	yes	—
130	C₄	#13133250	1	2	yes	—
131	C₂	—	2	2	no	—
132	C₂	C₂×#13133133	1	1	yes	characteristic
133	C₂	#16810631	1	1	yes	characteristic
134	C₂	#16810633	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133217

C₂

yes

maximal, characteristic

#16810680

C₂

yes

maximal, characteristic

#16810371

C₂

yes

maximal, complemented

C₅₃×#432

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₄

yes

—

#13133217

—

#13133217

—

#13133217

C₄

yes

—

#13133217

C₂²

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₄⋊#88322

C₂²

yes

—

C₂²⋊#88322

—

#13133128

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#100

yes

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#137

yes

—

C₄×C₅₃

#137

yes

—

Dic₅₃

—

#1558

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊Q₈

—

#432

—

#432

D₅₃

yes

characteristic

#333

—

C₅₃

#1558

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

106

—

#97

—

#97

—

106

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

212

—

C₂²×C₄

—

106

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

100

C₄²

C₄×D₅₃

yes

characteristic

101

C₄²

D₂₁₂

yes

characteristic

102

C₄²

C₂²⋊D₅₃

yes

characteristic

103

C₄²

C₂²×D₅₃

yes

characteristic

104

C₂³

#11719921

yes

characteristic

105

Q₈

—

212

—

106

Q₈

—

212

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

106

—

111

C₂×C₄

—

212

—

112

C₂×C₄

#11719921

yes

characteristic

113

C₂×C₄

#11719921

yes

characteristic

114

C₂×C₄

#11719921

yes

characteristic

115

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

116

C₂×C₄

C₂×D₂₁₂

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

118

C₈

—

119

C₈

—

212

—

120

C₂²

—

121

C₂²

—

122

C₂²

#13133133

yes

—

123

C₂²

C₂×#11719921

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

212

—

129

C₄

#13133247

yes

—

130

C₄

#13133250

yes

—

131

C₂

—

132

C₂

C₂×#13133133

yes

characteristic

133

C₂

#16810631

yes

characteristic

134

C₂

#16810633

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11719921
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#432	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#432	D₅₃
hypocenter	C₅₃	#1558
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₂×#11719921

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#432

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
3. #16810371	C₂	semidirect
72. C₅₃	#1558	semidirect

H ≅ G/N

type

3. #16810371

C₂

semidirect

72. C₅₃

#1558

semidirect

Subgroups — #20273440

Conjugacy classes of subgroups

Complements