Group #20291287 — subgroups

Subgroups — #20291287

Order 6784.

556 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 282 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×#13133305	C₂	1	2	yes	maximal
2	#16811588	C₂	1	6	yes	maximal, complemented
3	C₄×#11719936	C₂	1	3	yes	maximal, complemented
4	#16811582	C₂	1	3	yes	maximal, complemented
5	C₄×#11719949	C₂	1	1	yes	maximal, characteristic
6	#13133305	C₄	1	16	yes	complemented
7	C₂×#11719936	C₂²	1	6	yes	—
8	C₂×D₅₃⋊C₈	C₂²	1	6	yes	—
9	#13133272	C₂²	1	1	yes	characteristic
10	#13133272	C₂²	1	3	yes	—
11	#13133272	C₂²	1	6	yes	—
12	C₄×#88324	C₂²	1	6	yes	—
13	C₂×#11719949	C₂²	1	1	yes	characteristic
14	Q₈×Dic₅₃	C₄	1	1	yes	characteristic
15	C₄×D₂₁₂	C₄	1	3	yes	—
16	#13133179	C₄	1	3	yes	—
17	#13133175	C₂²	1	3	yes	—
18	C₄²×D₅₃	C₂²	1	3	yes	—
19	C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
20	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	#11719949	C₂×C₄	1	8	yes	—
22	C₂×D₂₁₂	C₂×C₄	1	3	yes	—
23	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	—
24	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	—
25	C₂×C₄×D₅₃	C₂³	1	3	yes	—
26	C₂×#88324	C₂³	1	6	yes	—
27	C₂×#88324	C₂³	1	2	yes	—
28	#11719936	C₂×C₄	1	24	yes	complemented
29	D₅₃⋊C₈	C₂×C₄	1	24	yes	complemented
30	C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
31	C₄²×C₅₃	C₂×C₄	1	3	yes	—
32	#11719921	C₂×C₄	1	6	yes	—
33	C₄⋊Dic₅₃	C₂×C₄	1	3	yes	—
34	C₄×Dic₅₃	C₂×C₄	1	3	yes	—
35	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
36	C₄×Dic₅₃	C₂³	1	3	yes	—
37	C₂²×D₅₃	C₂²×C₄	1	3	yes	—
38	Q₈×C₅₃	C₄²	1	4	yes	—
39	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
41	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
42	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₅₃	C₂²×C₄	1	3	yes	—
44	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
45	D₂₁₂	C₄²	1	12	yes	—
46	C₄×D₅₃	—	4	3	no	—
47	C₄×D₅₃	—	4	3	no	—
48	C₄×D₅₃	C₂²×C₄	1	12	yes	—
49	#88324	C₂²×C₄	1	12	yes	—
50	#88324	C₂²×C₄	1	4	yes	—
51	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
52	C₂×D₅₃	—	2	3	no	—
53	C₂×D₅₃	—	2	3	no	—
54	C₂×D₅₃	C₂×C₄²	1	6	yes	—
55	C₄×C₅₃	—	2	1	no	—
56	C₄×C₅₃	—	2	3	no	—
57	C₄×C₅₃	C₂×C₄²	1	6	yes	—
58	Dic₅₃	—	2	1	no	—
59	Dic₅₃	—	2	3	no	—
60	Dic₅₃	C₂³×C₄	1	2	yes	—
61	#2875	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
63	C₂×C₅₃	#568	1	1	yes	characteristic
64	C₂×C₅₃	#568	1	1	yes	characteristic
65	D₅₃	—	2	6	no	—
66	C₂×#131	—	53	2	no	—
67	C₄×#41	—	53	1	no	—
68	#405	—	53	6	no	—
69	C₄×M₁₆	—	53	3	no	—
70	C₂×#97	—	53	3	no	—
71	C₅₃	#2875	1	1	yes	characteristic, complemented, Sylow
72	C₂×#41	—	53	1	no	—
73	#131	—	53	16	no	—
74	C₂×M₁₆	—	53	6	no	—
75	C₂²×C₈	—	53	6	no	—
76	C₄×Q₈	—	53	1	no	—
77	C₄×Q₈	#6157	1	1	yes	characteristic
78	C₄×D₄	—	53	3	no	—
79	C₄×D₄	—	53	3	no	—
80	#117	—	53	3	no	—
81	C₂×C₄²	—	53	3	no	—
82	#97	—	53	1	no	—
83	#97	—	53	3	no	—
84	#97	—	53	6	no	—
85	C₄×C₈	—	53	6	no	—
86	#41	—	53	8	no	—
87	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
88	C₂×D₄	—	53	3	no	—
89	C₂²×C₄	—	53	3	no	—
90	C₂²×C₄	—	53	3	no	—
91	C₂²×C₄	—	53	3	no	—
92	M₁₆	—	53	24	no	—
93	C₂×C₈	—	53	6	no	—
94	C₂×C₈	—	53	2	no	—
95	C₂×C₈	—	53	24	no	—
96	C₄⋊C₄	—	53	3	no	—
97	C₄⋊C₄	C₂×#6157	1	3	yes	—
98	C₂²⋊C₄	—	53	6	no	—
99	C₄²	—	53	3	no	—
100	C₄²	—	53	1	no	—
101	C₄²	—	53	3	no	—
102	C₄²	C₂×#6157	1	3	yes	—
103	C₂³	—	53	3	no	—
104	Q₈	C₄×#6157	1	4	yes	—
105	D₄	—	53	12	no	—
106	C₂×C₄	—	53	1	no	—
107	C₂×C₄	—	53	12	no	—
108	C₂×C₄	—	53	1	no	—
109	C₂×C₄	—	53	3	no	—
110	C₂×C₄	—	212	3	no	—
111	C₂×C₄	—	212	3	no	—
112	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
113	C₂×C₄	C₂²×#6157	1	3	yes	—
114	C₂×C₄	C₂²×#6157	1	3	yes	—
115	C₈	—	53	4	no	—
116	C₈	—	53	12	no	—
117	C₂²	—	53	6	no	—
118	C₂²	—	106	3	no	—
119	C₂²	—	106	3	no	—
120	C₂²	C₂³×#6157	1	1	yes	characteristic
121	C₄	—	2	3	no	—
122	C₄	—	2	1	no	—
123	C₄	—	53	2	no	—
124	C₄	—	106	1	no	—
125	C₄	—	106	3	no	—
126	C₄	C₂×C₄×#6157	1	6	yes	—
127	C₂	—	106	6	no	—
128	C₂	C₂²×C₄×#6157	1	1	yes	characteristic
129	C₂	#16811702	1	1	yes	characteristic
130	C₂	#16811703	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133305

C₂

yes

maximal

#16811588

C₂

yes

maximal, complemented

C₄×#11719936

C₂

yes

maximal, complemented

#16811582

C₂

yes

maximal, complemented

C₄×#11719949

C₂

yes

maximal, characteristic

#13133305

C₄

yes

complemented

C₂×#11719936

C₂²

yes

—

C₂×D₅₃⋊C₈

C₂²

yes

—

#13133272

C₂²

yes

characteristic

#13133272

C₂²

yes

—

#13133272

C₂²

yes

—

C₄×#88324

C₂²

yes

—

C₂×#11719949

C₂²

yes

characteristic

Q₈×Dic₅₃

C₄

yes

characteristic

C₄×D₂₁₂

C₄

yes

—

#13133179

C₄

yes

—

#13133175

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

—

C₄×Q₈×C₅₃

C₄

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

#11719949

C₂×C₄

yes

—

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

C₂×#88324

C₂³

yes

—

C₂×#88324

C₂³

yes

—

#11719936

C₂×C₄

yes

complemented

D₅₃⋊C₈

C₂×C₄

yes

complemented

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

#11719921

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂³

yes

characteristic

C₄×Dic₅₃

C₂³

yes

—

C₂²×D₅₃

C₂²×C₄

yes

—

Q₈×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂⁴

yes

characteristic

D₂₁₂

C₄²

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂²×C₄

yes

—

#88324

C₂²×C₄

yes

—

#88324

C₂²×C₄

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂×C₄²

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂³×C₄

yes

—

#2875

—

maximal, Sylow

C₂×C₅₃

C₂²×C₄²

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂×#131

—

C₄×#41

—

#405

—

C₄×M₁₆

—

C₂×#97

—

C₅₃

#2875

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

C₂×M₁₆

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

#6157

yes

characteristic

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

#97

—

#97

—

#97

—

C₄×C₈

—

#41

—

C₂×Q₈

C₂×#6157

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#6157

yes

—

C₂²⋊C₄

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂×#6157

yes

—

103

C₂³

—

104

Q₈

C₄×#6157

yes

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

212

—

111

C₂×C₄

—

212

—

112

C₂×C₄

C₂²×#6157

yes

characteristic

113

C₂×C₄

C₂²×#6157

yes

—

114

C₂×C₄

C₂²×#6157

yes

—

115

C₈

—

116

C₈

—

117

C₂²

—

118

C₂²

—

106

—

119

C₂²

—

106

—

120

C₂²

C₂³×#6157

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

106

—

125

C₄

—

106

—

126

C₄

C₂×C₄×#6157

yes

—

127

C₂

—

106

—

128

C₂

C₂²×C₄×#6157

yes

characteristic

129

C₂

#16811702

yes

characteristic

130

C₂

#16811703

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6157
commutator	C₂×C₅₃	C₂²×C₄²
Fitting	C₄×Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂³×C₄
hypercenter	C₄×Q₈	#6157
hypocenter	C₅₃	#2875
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#6157

commutator

C₂×C₅₃

C₂²×C₄²

Fitting

C₄×Q₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂³×C₄

C₄×Q₈

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
2. #16811588	C₂	semidirect
3. C₄×#11719936	C₂	semidirect
4. #16811582	C₂	semidirect
6. #13133305	C₄	semidirect
28. #11719936	C₂×C₄	semidirect
29. D₅₃⋊C₈	C₂×C₄	semidirect
71. C₅₃	#2875	semidirect

H ≅ G/N

type

2. #16811588

C₂

semidirect

3. C₄×#11719936

C₂

semidirect

4. #16811582

C₂

semidirect

6. #13133305

C₄

semidirect

28. #11719936

C₂×C₄

semidirect

29. D₅₃⋊C₈

C₂×C₄

semidirect

71. C₅₃

#2875

semidirect

Subgroups — #20291287

Conjugacy classes of subgroups

Complements