Subgroups — #31774296

Order 17344.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22576152	C₂	1	1	yes	maximal, characteristic
2	#22576093	C₂	1	2	yes	maximal
3	#22576091	C₂	1	2	yes	maximal
4	C₄×C₂₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#125×C₂₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₇₁	C₂²	1	2	yes	—
9	Dic₂₇₁⋊C₄	—	2	2	no	—
10	C₄⋊C₄×C₂₇₁	—	2	2	no	—
11	C₄⋊C₄×C₂₇₁	C₂²	1	2	yes	—
12	C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₇₁	—	2	2	no	—
14	C₂₇₁⋊C₈	Q₈	1	2	yes	—
15	C₂₇₁⋊C₈	Q₈	1	2	yes	—
16	C₂×C₄×C₂₇₁	—	2	2	no	—
17	C₂×C₄×C₂₇₁	D₄	1	2	yes	—
18	C₂×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
19	C₂²×C₂₇₁	C₂×D₄	1	1	yes	characteristic
20	C₄×C₂₇₁	—	2	2	no	—
21	C₄×C₂₇₁	—	4	2	no	—
22	C₄×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
24	Dic₂₇₁	—	4	2	no	—
25	C₂×C₂₇₁	C₄⋊Q₈	1	1	yes	characteristic
26	C₂×C₂₇₁	#135	1	2	yes	—
27	C₂₇₁	#499	1	1	yes	characteristic, complemented, Sylow
28	#499	—	271	1	no	maximal, Sylow
29	#125	—	271	1	no	—
30	#125	D₂₇₁	1	1	yes	characteristic
31	#107	—	271	2	no	—
32	#106	—	271	2	no	—
33	C₄×C₈	—	271	1	no	—
34	C₂×C₈	—	271	1	no	—
35	C₂×C₈	—	271	1	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	271	2	no	—
38	C₄⋊C₄	—	542	2	no	—
39	C₄⋊C₄	C₂×D₂₇₁	1	2	yes	—
40	C₄²	C₂×D₂₇₁	1	1	yes	characteristic
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	542	2	no	—
43	C₂×C₄	C₂²⋊D₂₇₁	1	2	yes	—
44	C₂×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
45	C₈	—	271	2	no	—
46	C₈	—	271	2	no	—
47	C₂²	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
48	C₄	—	2	2	no	—
49	C₄	—	4	2	no	—
50	C₄	—	1084	2	no	—
51	C₄	Q₈×D₂₇₁	1	1	yes	characteristic
52	C₄	Q₈×D₂₇₁	1	1	yes	characteristic
53	C₂	#22576158	1	1	yes	characteristic
54	C₂	#22576180	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₇₁
commutator, cosocle	C₂×C₄×C₂₇₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₇₁
Fitting	#125×C₂₇₁	C₂
socle, prime radical	C₂²×C₂₇₁	C₂×D₄
hypercenter	#125	D₂₇₁
hypocenter	C₂₇₁	#499
LCS	G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
27. C₂₇₁	#499	semidirect

Maximal subgroups

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

subgroup	index
#22576152	2
#22576093	2
#22576091	2
C₄×C₂₇₁⋊C₈	2
#125×C₂₇₁	2
#499	271