Group #20274307 — subgroups

Subgroups — #20274307

Order 6784.

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

normal subgroups only

This group has 224 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	#16811028	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811000	C₂	1	1	yes	maximal, characteristic
3	#16810996	C₂	1	2	yes	maximal, complemented
4	#16810970	C₂	1	2	yes	maximal, complemented
5	C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719913	—	2	4	no	—
7	C₂×D₄₂₄	—	2	2	no	—
8	#13133217	C₂²	1	2	yes	—
9	#13133214	C₂²	1	2	yes	—
10	#13133151	—	2	2	no	—
11	#13133139	—	2	4	no	—
12	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×D₂₁₂	—	2	2	no	—
16	C₂×D₂₁₂	—	2	2	no	—
17	C₂×D₂₁₂	—	2	2	no	—
18	C₂×Dic₁₀₆	—	2	2	no	—
19	C₂×C₈×C₅₃	D₄	1	2	yes	—
20	C₂×C₈×C₅₃	D₄	1	2	yes	—
21	C₂×C₈×C₅₃	D₄	1	2	yes	—
22	C₄²×C₅₃	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₅₃	—	2	2	no	—
24	C₄⋊Dic₅₃	—	2	2	no	—
25	D₄₂₄	—	4	4	no	—
26	#11719913	—	4	8	no	—
27	C₂²×D₅₃	—	4	2	no	—
28	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₅₃	—	4	2	no	—
32	D₂₁₂	—	4	4	no	—
33	D₂₁₂	—	4	4	no	—
34	D₂₁₂	—	4	4	no	—
35	Dic₁₀₆	—	4	4	no	—
36	C₈×C₅₃	—	2	2	no	—
37	C₈×C₅₃	D₈	1	4	yes	complemented
38	C₈×C₅₃	SD₁₆	1	4	yes	complemented
39	C₂²×C₅₃	#127	1	1	yes	characteristic
40	C₂×D₅₃	—	8	2	no	—
41	C₂×D₅₃	—	8	2	no	—
42	C₂×D₅₃	—	8	2	no	—
43	C₄×C₅₃	C₂×D₈	1	2	yes	—
44	C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
45	C₄×C₅₃	#135	1	2	yes	—
46	Dic₅₃	—	8	2	no	—
47	#1706	—	53	1	no	maximal, Sylow
48	C₂×C₅₃	#492	1	1	yes	characteristic
49	C₂×C₅₃	#493	1	1	yes	characteristic
50	C₂×C₅₃	#495	1	1	yes	characteristic
51	D₅₃	—	16	2	no	—
52	#500	—	53	1	no	—
53	#493	—	53	1	no	—
54	#492	—	53	2	no	—
55	#486	—	53	2	no	—
56	C₈²	D₅₃	1	1	yes	characteristic, complemented
57	C₅₃	#1706	1	1	yes	characteristic, complemented, Sylow
58	C₂×SD₁₆	—	106	4	no	—
59	C₂×D₈	—	106	2	no	—
60	C₄⋊Q₈	—	53	2	no	—
61	#127	—	53	2	no	—
62	#107	—	106	2	no	—
63	#102	—	106	4	no	—
64	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
65	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
66	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
67	C₂×Q₈	—	106	2	no	—
68	C₂×D₄	—	106	2	no	—
69	C₂×D₄	—	106	2	no	—
70	C₂×D₄	—	106	2	no	—
71	SD₁₆	—	212	8	no	—
72	D₈	—	212	4	no	—
73	C₂×C₈	D₂₁₂	1	2	yes	—
74	C₂×C₈	D₂₁₂	1	2	yes	—
75	C₂×C₈	D₂₁₂	1	2	yes	—
76	C₄⋊C₄	—	106	2	no	—
77	C₄⋊C₄	—	106	2	no	—
78	C₄²	C₂²×D₅₃	1	1	yes	characteristic
79	C₂³	—	212	2	no	—
80	Q₈	—	212	4	no	—
81	D₄	—	212	4	no	—
82	D₄	—	212	4	no	—
83	D₄	—	212	4	no	—
84	C₂×C₄	—	212	2	no	—
85	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
86	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
87	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
88	C₈	—	2	2	no	—
89	C₈	#11719913	1	4	yes	complemented
90	C₈	D₄₂₄	1	4	yes	complemented
91	C₂²	—	424	2	no	—
92	C₂²	—	424	2	no	—
93	C₂²	—	424	2	no	—
94	C₂²	#13133214	1	1	yes	characteristic
95	C₄	—	424	2	no	—
96	C₄	C₂×D₄₂₄	1	2	yes	—
97	C₄	C₂×#11719913	1	2	yes	—
98	C₄	#13133239	1	2	yes	—
99	C₂	—	848	2	no	—
100	C₂	#16810996	1	1	yes	characteristic
101	C₂	#16811000	1	1	yes	characteristic
102	C₂	#16811007	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811028

C₂

yes

maximal, characteristic, complemented

#16811000

C₂

yes

maximal, characteristic

#16810996

C₂

yes

maximal, complemented

#16810970

C₂

yes

maximal, complemented

C₈²×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#11719913

—

C₂×D₄₂₄

—

#13133217

C₂²

yes

—

#13133214

C₂²

yes

—

#13133151

—

#13133139

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

D₄₂₄

—

#11719913

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

D₈

yes

complemented

C₈×C₅₃

SD₁₆

yes

complemented

C₂²×C₅₃

#127

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

C₂×D₈

yes

—

C₄×C₅₃

C₂×SD₁₆

yes

—

C₄×C₅₃

#135

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

—

—

—

—

C₈²

D₅₃

yes

characteristic, complemented

C₅₃

#1706

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

106

—

C₂×D₈

—

106

—

C₄⋊Q₈

—

#127

—

#107

—

106

—

#102

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

SD₁₆

—

212

—

D₈

—

212

—

C₂×C₈

D₂₁₂

yes

—

C₂×C₈

D₂₁₂

yes

—

C₂×C₈

D₂₁₂

yes

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

212

—

Q₈

—

212

—

D₄

—

212

—

D₄

—

212

—

D₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₈

—

C₈

#11719913

yes

complemented

C₈

D₄₂₄

yes

complemented

C₂²

—

424

—

C₂²

—

424

—

C₂²

—

424

—

C₂²

#13133214

yes

characteristic

C₄

—

424

—

C₄

C₂×D₄₂₄

yes

—

C₄

C₂×#11719913

yes

—

C₄

#13133239

yes

—

C₂

—

848

—

100

C₂

#16810996

yes

characteristic

101

C₂

#16811000

yes

characteristic

102

C₂

#16811007

yes

characteristic

	subgroup	G/H
center	C₂²	#13133214
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₈²×C₅₃	C₂
socle	C₂²×C₅₃	#127
hypercenter	C₈²	D₅₃
hypocenter	C₅₃	#1706
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#13133214

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₈²×C₅₃

C₂

socle

C₂²×C₅₃

#127

hypercenter

C₈²

D₅₃

hypocenter

C₅₃

#1706

LCS

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
1. #16811028	C₂	semidirect
3. #16810996	C₂	semidirect
4. #16810970	C₂	semidirect
5. C₈²×C₅₃	C₂	semidirect
37. C₈×C₅₃	D₈	semidirect
38. C₈×C₅₃	SD₁₆	semidirect
56. C₈²	D₅₃	semidirect
57. C₅₃	#1706	semidirect
89. C₈	#11719913	semidirect
90. C₈	D₄₂₄	semidirect

H ≅ G/N

type

1. #16811028

C₂

semidirect

3. #16810996

C₂

semidirect

4. #16810970

C₂

semidirect

5. C₈²×C₅₃

C₂

semidirect

37. C₈×C₅₃

D₈

semidirect

38. C₈×C₅₃

SD₁₆

semidirect

56. C₈²

D₅₃

semidirect

57. C₅₃

#1706

semidirect

89. C₈

#11719913

semidirect

90. C₈

D₄₂₄

semidirect

Subgroups — #20274307

Conjugacy classes of subgroups

Complements