Subgroups — #31774009

Order 17344.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×Dic₂₇₁	C₂	1	2	yes	maximal, complemented
2	#22576092	C₂	1	1	yes	maximal, characteristic, complemented
3	#22576081	C₂	1	2	yes	maximal, complemented
4	#22576070	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈×C₂₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₇₁	—	2	2	no	—
7	C₂×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₇₁	C₂²	1	2	yes	—
9	C₂²⋊Dic₂₇₁	—	2	2	no	—
10	C₄×Dic₂₇₁	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₂₇₁	C₂²	1	2	yes	—
12	D₈×C₂₇₁	C₄	1	4	yes	complemented
13	C₂×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₇₁	—	2	2	no	—
15	C₂×Dic₂₇₁	—	4	4	no	—
16	C₂×Dic₂₇₁	D₄	1	2	yes	—
17	C₂₇₁⋊C₈	—	2	1	no	—
18	C₂³×C₂₇₁	—	2	2	no	—
19	D₄×C₂₇₁	—	2	2	no	—
20	D₄×C₂₇₁	C₂×C₄	1	4	yes	complemented
21	C₂×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
22	C₈×C₂₇₁	C₂×C₄	1	2	yes	complemented
23	C₂²×C₂₇₁	—	2	4	no	—
24	C₂²×C₂₇₁	—	4	2	no	—
25	C₂²×C₂₇₁	—	4	2	no	—
26	C₂²×C₂₇₁	C₂×D₄	1	1	yes	characteristic
27	C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₂₇₁	#41	1	1	yes	characteristic
29	Dic₂₇₁	—	2	2	no	—
30	Dic₂₇₁	—	4	2	no	—
31	C₂×C₂₇₁	—	4	4	no	—
32	C₂×C₂₇₁	C₄×D₄	1	1	yes	characteristic
33	C₂×C₂₇₁	AGL(1,ℤ/8)	1	2	yes	—
34	C₂₇₁	#442	1	1	yes	characteristic, complemented, Sylow
35	#442	—	271	1	no	maximal, Sylow
36	C₂×D₈	D₂₇₁	1	1	yes	characteristic
37	C₄×D₄	—	271	2	no	—
38	#106	—	271	1	no	—
39	#102	—	271	2	no	—
40	#97	—	271	1	no	—
41	C₂×D₄	C₂×D₂₇₁	1	2	yes	—
42	C₂²×C₄	—	542	2	no	—
43	D₈	Dic₂₇₁	1	4	yes	complemented
44	C₂×C₈	—	271	1	no	—
45	C₂×C₈	C₂×D₂₇₁	1	1	yes	characteristic
46	C₄⋊C₄	—	271	2	no	—
47	C₂²⋊C₄	—	542	2	no	—
48	C₄²	—	271	1	no	—
49	C₂³	—	2	2	no	—
50	D₄	—	2	2	no	—
51	D₄	C₂×Dic₂₇₁	1	4	yes	complemented
52	C₂×C₄	—	271	2	no	—
53	C₂×C₄	—	542	2	no	—
54	C₂×C₄	—	1084	4	no	—
55	C₂×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
56	C₈	—	542	1	no	—
57	C₈	C₂×Dic₂₇₁	1	2	yes	complemented
58	C₂²	—	2	4	no	—
59	C₂²	—	4	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	D₄×D₂₇₁	1	1	yes	characteristic
62	C₄	—	542	2	no	—
63	C₄	—	1084	2	no	—
64	C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
65	C₄	#16864653	1	1	yes	characteristic
66	C₂	—	4	4	no	—
67	C₂	D₄×Dic₂₇₁	1	1	yes	characteristic
68	C₂	#22576184	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₇₁
commutator	C₄×C₂₇₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₇₁
Fitting, prime radical	C₂×D₈×C₂₇₁	C₂
socle	C₂²×C₂₇₁	C₂×D₄
cosocle	C₂×C₄×C₂₇₁	C₂³
hypercenter	C₂×D₈	D₂₇₁
hypocenter	C₂₇₁	#442
LCS	G ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. D₄×Dic₂₇₁	C₂	semidirect
2. #22576092	C₂	semidirect
3. #22576081	C₂	semidirect
4. #22576070	C₂	semidirect
12. D₈×C₂₇₁	C₄	semidirect
20. D₄×C₂₇₁	C₂×C₄	semidirect
22. C₈×C₂₇₁	C₂×C₄	semidirect
34. C₂₇₁	#442	semidirect
43. D₈	Dic₂₇₁	semidirect
51. D₄	C₂×Dic₂₇₁	semidirect
57. C₈	C₂×Dic₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
D₄×Dic₂₇₁	2
#22576092	2
#22576081	2
#22576070	2
C₂×D₈×C₂₇₁	2
#442	271