Group #27298810 — subgroups

Subgroups — #27298810

Order 13568.

354 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 115 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	#20276971	C₂	1	4	yes	maximal, complemented
2	#20274333	C₂	1	1	yes	maximal, characteristic, complemented
3	#20274330	C₂	1	1	yes	maximal, characteristic
4	C₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13133162	—	2	8	no	—
6	#16811028	—	2	2	no	—
7	#16811003	C₂²	1	2	yes	—
8	#16811000	—	2	2	no	—
9	#16811000	C₂²	1	2	yes	—
10	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
11	C₈²×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₁₂	—	2	4	no	—
13	C₂×D₄₂₄	—	2	4	no	—
14	C₂×D₄₂₄	—	4	4	no	—
15	#13133217	—	2	2	no	—
16	#13133214	—	2	2	no	—
17	#13133162	—	4	16	no	—
18	#13133151	—	4	4	no	—
19	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
20	C₄×C₈×C₅₃	D₄	1	2	yes	—
21	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
22	C₂×D₂₁₂	—	4	2	no	—
23	C₂×D₂₁₂	—	4	4	no	—
24	C₂×Dic₁₀₆	—	4	2	no	—
25	C₂×C₈×C₅₃	D₈	1	4	yes	—
26	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
27	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
28	C₄⋊Dic₅₃	—	4	4	no	—
29	Dic₂₁₂	—	4	8	no	—
30	D₄₂₄	—	4	8	no	—
31	D₄₂₄	—	8	8	no	—
32	C₁₆×C₅₃	D₈	1	8	yes	complemented
33	C₂²×D₅₃	—	8	2	no	—
34	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
36	C₂×Dic₅₃	—	8	2	no	—
37	D₂₁₂	—	8	4	no	—
38	D₂₁₂	—	8	8	no	—
39	Dic₁₀₆	—	8	4	no	—
40	C₈×C₅₃	SD₃₂	1	8	yes	complemented
41	C₈×C₅₃	C₂×D₈	1	4	yes	—
42	#12517	—	53	1	no	maximal, Sylow
43	C₂²×C₅₃	#493	1	1	yes	characteristic
44	C₂×D₅₃	—	16	2	no	—
45	C₂×D₅₃	—	16	4	no	—
46	C₄×C₅₃	#493	1	2	yes	—
47	C₄×C₅₃	C₂×SD₃₂	1	4	yes	—
48	Dic₅₃	—	16	2	no	—
49	#2248	—	53	4	no	—
50	#1713	—	53	1	no	—
51	#1712	—	53	1	no	—
52	C₈×C₁₆	D₅₃	1	1	yes	characteristic, complemented
53	C₂×C₅₃	#1712	1	1	yes	characteristic
54	C₂×C₅₃	#2248	1	2	yes	—
55	D₅₃	—	32	2	no	—
56	C₂×SD₃₂	—	106	8	no	—
57	#500	—	106	2	no	—
58	#494	—	53	2	no	—
59	#493	—	53	2	no	—
60	#493	—	106	2	no	—
61	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
62	C₈²	C₂×D₅₃	1	1	yes	characteristic
63	C₅₃	#12517	1	1	yes	characteristic, complemented, Sylow
64	C₂×Q₁₆	—	106	4	no	—
65	C₂×D₈	—	106	4	no	—
66	C₂×D₈	—	212	4	no	—
67	C₄⋊Q₈	—	106	2	no	—
68	#127	—	106	2	no	—
69	SD₃₂	—	212	16	no	—
70	C₂×C₁₆	D₂₁₂	1	4	yes	—
71	#107	—	212	4	no	—
72	C₄×C₈	D₂₁₂	1	2	yes	—
73	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
74	C₂×Q₈	—	212	2	no	—
75	C₂×D₄	—	212	2	no	—
76	C₂×D₄	—	212	4	no	—
77	Q₁₆	—	212	8	no	—
78	D₈	—	424	8	no	—
79	D₈	—	212	8	no	—
80	C₂×C₈	D₄₂₄	1	4	yes	—
81	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
82	C₄⋊C₄	—	212	4	no	—
83	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
84	C₁₆	D₄₂₄	1	8	yes	complemented
85	C₂³	—	424	2	no	—
86	Q₈	—	424	4	no	—
87	D₄	—	424	8	no	—
88	D₄	—	424	4	no	—
89	C₂×C₄	—	424	2	no	—
90	C₂×C₄	#13133214	1	1	yes	characteristic
91	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
92	C₈	#13133162	1	8	yes	complemented
93	C₈	C₂×D₄₂₄	1	4	yes	—
94	C₂²	—	848	2	no	—
95	C₂²	—	848	4	no	—
96	C₂²	#16811000	1	1	yes	characteristic
97	C₄	—	848	2	no	—
98	C₄	#16811000	1	2	yes	—
99	C₄	C₂×#13133162	1	4	yes	—
100	C₂	—	1696	2	no	—
101	C₂	#20274330	1	1	yes	characteristic
102	C₂	#20276971	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276971

C₂

yes

maximal, complemented

#20274333

C₂

yes

maximal, characteristic, complemented

#20274330

C₂

yes

maximal, characteristic

C₈×C₁₆×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#13133162

—

#16811028

—

#16811003

C₂²

yes

—

#16811000

—

#16811000

C₂²

yes

—

C₄×C₁₆×C₅₃

C₂²

yes

—

C₈²×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

—

—

—

—

C₂×C₁₆×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

Dic₂₁₂

—

D₄₂₄

—

D₄₂₄

—

C₁₆×C₅₃

D₈

yes

complemented

C₂²×D₅₃

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

Dic₁₀₆

—

C₈×C₅₃

SD₃₂

yes

complemented

C₈×C₅₃

C₂×D₈

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

#493

yes

—

C₄×C₅₃

C₂×SD₃₂

yes

—

Dic₅₃

—

#2248

—

#1713

—

#1712

—

C₈×C₁₆

D₅₃

yes

characteristic, complemented

C₂×C₅₃

#1712

yes

characteristic

C₂×C₅₃

#2248

yes

—

D₅₃

—

C₂×SD₃₂

—

106

—

#500

—

106

—

#494

—

#493

—

#493

—

106

—

C₄×C₁₆

C₂×D₅₃

yes

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#12517

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₂×D₈

—

106

—

C₂×D₈

—

212

—

C₄⋊Q₈

—

106

—

#127

—

106

—

SD₃₂

—

212

—

C₂×C₁₆

D₂₁₂

yes

—

#107

—

212

—

C₄×C₈

D₂₁₂

yes

—

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

212

—

C₂×D₄

—

212

—

C₂×D₄

—

212

—

Q₁₆

—

212

—

D₈

—

424

—

D₈

—

212

—

C₂×C₈

D₄₂₄

yes

—

C₂×C₈

C₂×D₂₁₂

yes

—

C₄⋊C₄

—

212

—

C₄²

C₂×D₂₁₂

yes

characteristic

C₁₆

D₄₂₄

yes

complemented

C₂³

—

424

—

Q₈

—

424

—

D₄

—

424

—

D₄

—

424

—

C₂×C₄

—

424

—

C₂×C₄

#13133214

yes

characteristic

C₂×C₄

C₂×D₄₂₄

yes

—

C₈

#13133162

yes

complemented

C₈

C₂×D₄₂₄

yes

—

C₂²

—

848

—

C₂²

—

848

—

C₂²

#16811000

yes

characteristic

C₄

—

848

—

C₄

#16811000

yes

—

C₄

C₂×#13133162

yes

—

100

C₂

—

1696

—

101

C₂

#20274330

yes

characteristic

102

C₂

#20276971

yes

—

	subgroup	G/H
center	C₂²	#16811000
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₈×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	#493
prime radical	#20274330	C₂
hypercenter	C₈×C₁₆	D₅₃
hypocenter	C₅₃	#12517
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#16811000

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₈×C₁₆×C₅₃

C₂

C₂²×C₅₃

C₂

C₈×C₁₆

D₅₃

hypocenter

C₅₃

#12517

LCS

G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #20276971	C₂	semidirect
2. #20274333	C₂	semidirect
4. C₈×C₁₆×C₅₃	C₂	semidirect
32. C₁₆×C₅₃	D₈	semidirect
40. C₈×C₅₃	SD₃₂	semidirect
52. C₈×C₁₆	D₅₃	semidirect
63. C₅₃	#12517	semidirect
84. C₁₆	D₄₂₄	semidirect
92. C₈	#13133162	semidirect

H ≅ G/N

type

1. #20276971

C₂

semidirect

2. #20274333

C₂

semidirect

4. C₈×C₁₆×C₅₃

C₂

semidirect

32. C₁₆×C₅₃

D₈

semidirect

40. C₈×C₅₃

SD₃₂

semidirect

52. C₈×C₁₆

D₅₃

semidirect

63. C₅₃

#12517

semidirect

84. C₁₆

D₄₂₄

semidirect

92. C₈

#13133162

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20276971	2
#20274333	2
#20274330	2
C₈×C₁₆×C₅₃	2
#12517	53

subgroup

index

C₈×C₁₆×C₅₃

Subgroups — #27298810

Conjugacy classes of subgroups

Complements

Maximal subgroups