Group #20274036 — subgroups

Subgroups — #20274036

Order 6784.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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	#16811371	C₂	1	1	yes	maximal, characteristic
2	#16810985	C₂	1	2	yes	maximal, complemented
3	#16810365	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810362	C₂	1	2	yes	maximal, complemented
5	C₅₃×#491	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133205	C₂²	1	2	yes	—
7	#13133202	—	2	2	no	—
8	#13133195	—	2	2	no	—
9	#13133192	—	2	2	no	—
10	#13133140	—	2	2	no	—
11	#13133137	—	2	2	no	—
12	#13133128	C₂²	1	2	yes	—
13	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	#125×C₅₃	C₂²	1	1	yes	characteristic
15	#103×C₅₃	—	2	2	no	—
16	#97×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	2	1	no	—
18	C₂×D₂₁₂	—	2	2	no	—
19	C₂×D₂₁₂	D₄	1	2	yes	—
20	C₂×C₄×D₅₃	—	2	2	no	—
21	C₂×Dic₁₀₆	D₄	1	2	yes	—
22	C₂×C₈×C₅₃	—	2	1	no	—
23	C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
26	C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	#11719921	—	2	2	no	—
28	#11719921	—	2	2	no	—
29	#11719921	—	2	2	no	—
30	#11719919	—	2	2	no	—
31	C₂×#88322	—	2	2	no	—
32	C₂²×D₅₃	—	2	2	no	—
33	Q₈×C₅₃	—	4	2	no	—
34	C₂×C₄×C₅₃	—	2	2	no	—
35	C₂×C₄×C₅₃	—	4	1	no	—
36	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₅₃	—	2	2	no	—
39	D₂₁₂	—	4	2	no	—
40	D₂₁₂	—	8	2	no	—
41	C₄×D₅₃	—	8	2	no	—
42	Dic₁₀₆	—	4	2	no	—
43	C₈×C₅₃	—	4	1	no	—
44	#88322	—	4	2	no	—
45	C₂²×C₅₃	#120	1	1	yes	characteristic
46	C₂×D₅₃	—	4	2	no	—
47	C₂×D₅₃	—	4	2	no	—
48	C₂×D₅₃	—	4	2	no	—
49	C₄×C₅₃	—	2	1	no	—
50	C₄×C₅₃	—	2	2	no	—
51	C₄×C₅₃	—	4	2	no	—
52	C₄×C₅₃	—	8	1	no	—
53	Dic₅₃	—	4	2	no	—
54	#1662	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	#454	1	1	yes	characteristic
56	C₂×C₅₃	#455	1	2	yes	—
57	D₅₃	—	8	2	no	—
58	#562	—	53	1	no	—
59	#491	D₅₃	1	1	yes	characteristic, complemented
60	#489	—	53	2	no	—
61	#330	—	53	1	no	—
62	#329	—	53	2	no	—
63	C₅₃	#1662	1	1	yes	characteristic, complemented, Sylow
64	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
65	#125	C₂×D₅₃	1	1	yes	characteristic
66	#124	—	53	2	no	—
67	#123	—	106	2	no	—
68	C₂²⋊Q₈	—	106	2	no	—
69	#121	—	106	2	no	—
70	#103	—	2	2	no	—
71	#103	—	106	2	no	—
72	#102	—	106	2	no	—
73	#97	—	53	2	no	—
74	#97	C₂×D₅₃	1	1	yes	characteristic
75	C₂×Q₈	—	2	1	no	—
76	C₂×Q₈	—	53	2	no	—
77	C₂×D₄	—	53	2	no	—
78	C₂×D₄	—	106	2	no	—
79	C₂²×C₄	—	106	2	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	106	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄⋊C₄	—	106	2	no	—
85	C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
86	C₂²⋊C₄	—	106	2	no	—
87	C₂²⋊C₄	—	106	2	no	—
88	C₂²⋊C₄	—	106	2	no	—
89	C₄²	C₂²×D₅₃	1	1	yes	characteristic
90	C₂³	—	106	2	no	—
91	Q₈	—	4	2	no	—
92	Q₈	—	212	2	no	—
93	D₄	—	424	2	no	—
94	D₄	—	212	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	—	106	2	no	—
98	C₂×C₄	—	424	2	no	—
99	C₂×C₄	D₄×D₅₃	1	2	yes	—
100	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
101	C₈	—	4	1	no	—
102	C₈	—	212	2	no	—
103	C₂²	—	212	2	no	—
104	C₂²	—	212	2	no	—
105	C₂²	—	212	2	no	—
106	C₂²	#13133185	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	2	no	—
109	C₄	—	4	2	no	—
110	C₄	—	8	1	no	—
111	C₄	—	212	2	no	—
112	C₂	—	424	2	no	—
113	C₂	#16810795	1	1	yes	characteristic
114	C₂	#16810799	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811371

C₂

yes

maximal, characteristic

#16810985

C₂

yes

maximal, complemented

#16810365

C₂

yes

maximal, characteristic, complemented

#16810362

C₂

yes

maximal, complemented

C₅₃×#491

C₂

yes

maximal, characteristic, complemented

#13133205

C₂²

yes

—

—

—

—

—

—

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#125×C₅₃

C₂²

yes

characteristic

#103×C₅₃

—

#97×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

D₄

yes

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

D₄

yes

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

—

—

—

—

C₂²×D₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

#120

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

#562

—

#491

D₅₃

yes

characteristic, complemented

#489

—

#330

—

#329

—

C₅₃

#1662

yes

characteristic, complemented, Sylow

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

#125

C₂×D₅₃

yes

characteristic

#124

—

#123

—

106

—

C₂²⋊Q₈

—

106

—

#121

—

106

—

#103

—

#103

—

106

—

#102

—

106

—

#97

—

#97

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

106

—

C₂²×C₄

—

106

—

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

C₂²⋊D₅₃

yes

—

C₂²⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

106

—

Q₈

—

Q₈

—

212

—

D₄

—

424

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

424

—

C₂×C₄

D₄×D₅₃

yes

—

100

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

101

C₈

—

102

C₈

—

212

—

103

C₂²

—

212

—

104

C₂²

—

212

—

105

C₂²

—

212

—

106

C₂²

#13133185

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

212

—

112

C₂

—

424

—

113

C₂

#16810795

yes

characteristic

114

C₂

#16810799

yes

—

	subgroup	G/H
center	C₂²	#13133185
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#491	C₂
socle	C₂²×C₅₃	#120
hypercenter	#491	D₅₃
hypocenter	C₅₃	#1662
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #491

subgroup

G/H

center

C₂²

#13133185

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#491

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #491

N	H ≅ G/N	type
2. #16810985	C₂	semidirect
3. #16810365	C₂	semidirect
4. #16810362	C₂	semidirect
5. C₅₃×#491	C₂	semidirect
59. #491	D₅₃	semidirect
63. C₅₃	#1662	semidirect

H ≅ G/N

type

2. #16810985

C₂

semidirect

3. #16810365

C₂

semidirect

4. #16810362

C₂

semidirect

5. C₅₃×#491

C₂

semidirect

59. #491

D₅₃

semidirect

63. C₅₃

#1662

semidirect

Subgroups — #20274036

Conjugacy classes of subgroups

Complements