Subgroups — #46834499

Order 34688.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31773731	C₂	1	1	yes	maximal, characteristic
2	#31773672	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₂₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₇₁⋊C₁₆	C₄	1	1	yes	characteristic
5	C₈×Dic₂₇₁	C₄	1	1	yes	characteristic
6	C₂₇₁⋊C₃₂	C₄	1	2	yes	complemented
7	C₂×C₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
8	C₃₂×C₂₇₁	C₄	1	2	yes	complemented
9	C₂×C₂₇₁⋊C₈	C₈	1	1	yes	characteristic
10	C₄×Dic₂₇₁	C₈	1	1	yes	characteristic
11	C₂₇₁⋊C₁₆	C₈	1	2	yes	—
12	C₂×C₈×C₂₇₁	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₂₇₁	C₂×C₄	1	1	yes	characteristic
14	C₁₆×C₂₇₁	C₂×C₄	1	1	yes	characteristic
15	C₂×Dic₂₇₁	—	2	1	no	—
16	C₂₇₁⋊C₈	—	2	1	no	—
17	C₂×C₄×C₂₇₁	C₂×C₈	1	1	yes	characteristic
18	C₈×C₂₇₁	C₄²	1	1	yes	characteristic
19	C₈×C₂₇₁	C₂×C₈	1	1	yes	characteristic
20	C₂²×C₂₇₁	M₃₂	1	1	yes	characteristic
21	C₄×C₂₇₁	C₄×C₈	1	1	yes	characteristic
22	C₄×C₂₇₁	M₃₂	1	1	yes	characteristic
23	Dic₂₇₁	—	4	1	no	—
24	C₂×C₂₇₁	—	2	1	no	—
25	C₂×C₂₇₁	#346	1	1	yes	characteristic
26	C₂₇₁	#1398	1	1	yes	characteristic, complemented, Sylow
27	#1398	—	271	1	no	maximal, Sylow
28	M₆₄	—	271	1	no	—
29	M₆₄	D₂₇₁	1	1	yes	characteristic
30	#346	—	271	1	no	—
31	C₂×C₁₆	—	271	1	no	—
32	C₂×C₁₆	C₂×D₂₇₁	1	1	yes	characteristic
33	C₄×C₈	—	271	1	no	—
34	C₃₂	—	271	2	no	—
35	C₃₂	Dic₂₇₁	1	2	yes	complemented
36	C₂×C₈	—	271	1	no	—
37	C₂×C₈	C₄×D₂₇₁	1	1	yes	characteristic
38	C₄²	—	271	1	no	—
39	C₁₆	—	271	2	no	—
40	C₁₆	C₄×D₂₇₁	1	1	yes	characteristic
41	C₁₆	C₂×Dic₂₇₁	1	1	yes	characteristic
42	C₂×C₄	—	542	1	no	—
43	C₂×C₄	C₈×D₂₇₁	1	1	yes	characteristic
44	C₈	—	542	1	no	—
45	C₈	C₄×Dic₂₇₁	1	1	yes	characteristic
46	C₈	C₈×D₂₇₁	1	1	yes	characteristic
47	C₂²	#22576099	1	1	yes	characteristic
48	C₄	—	1084	1	no	—
49	C₄	C₈×Dic₂₇₁	1	1	yes	characteristic
50	C₄	#22576099	1	1	yes	characteristic
51	C₂	—	2	1	no	—
52	C₂	#31773672	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄×Dic₂₇₁
commutator	C₄×C₂₇₁	C₄×C₈
Frattini	C₂×C₁₆	C₂×D₂₇₁
Fitting	M₆₄×C₂₇₁	C₂
socle	C₂×C₂₇₁	#346
prime radical	C₂²×C₂₇₁	M₃₂
cosocle	C₂×C₁₆×C₂₇₁	C₂²
hypercenter	M₆₄	D₂₇₁
hypocenter	C₂₇₁	#1398
LCS	G ▷ C₄×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₈ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
6. C₂₇₁⋊C₃₂	C₄	semidirect
8. C₃₂×C₂₇₁	C₄	semidirect
26. C₂₇₁	#1398	semidirect
35. C₃₂	Dic₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
#31773731	2
#31773672	2
M₆₄×C₂₇₁	2
#1398	271