Subgroups — #46834570

Order 34688.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31773727	C₂	1	1	yes	maximal, characteristic
2	#31773725	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₂₇₁	C₂	1	1	yes	maximal, characteristic
4	#22576096	—	2	1	no	—
5	#22576094	—	2	1	no	—
6	C₂×C₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
7	C₃₂×C₂₇₁	C₄	1	2	yes	complemented
8	#16864638	—	4	1	no	—
9	C₄⋊Dic₂₇₁	—	4	1	no	—
10	C₂×C₈×C₂₇₁	D₄	1	1	yes	characteristic
11	C₁₆×C₂₇₁	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₂₇₁	Q₈	1	1	yes	characteristic
13	C₂×Dic₂₇₁	—	8	1	no	—
14	C₂₇₁⋊C₈	—	8	1	no	—
15	C₂×C₄×C₂₇₁	D₈	1	1	yes	characteristic
16	C₈×C₂₇₁	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₂₇₁	Q₁₆	1	1	yes	characteristic
18	C₂²×C₂₇₁	SD₃₂	1	1	yes	characteristic
19	C₄×C₂₇₁	#107	1	1	yes	characteristic
20	C₄×C₂₇₁	SD₃₂	1	1	yes	characteristic
21	Dic₂₇₁	—	16	1	no	—
22	C₂×C₂₇₁	—	2	1	no	—
23	C₂×C₂₇₁	#367	1	1	yes	characteristic
24	C₂₇₁	#1426	1	1	yes	characteristic, complemented, Sylow
25	#1426	—	271	1	no	maximal, Sylow
26	M₆₄	D₂₇₁	1	1	yes	characteristic
27	#368	—	271	1	no	—
28	#367	—	271	1	no	—
29	C₂×C₁₆	C₂×D₂₇₁	1	1	yes	characteristic
30	#108	—	542	1	no	—
31	#107	—	542	1	no	—
32	C₃₂	Dic₂₇₁	1	2	yes	complemented
33	M₁₆	—	1084	1	no	—
34	C₂×C₈	D₁₀₈₄	1	1	yes	characteristic
35	C₄⋊C₄	—	1084	1	no	—
36	C₁₆	C₂×Dic₂₇₁	1	1	yes	characteristic
37	C₁₆	Dic₅₄₂	1	1	yes	characteristic
38	C₂×C₄	—	2168	1	no	—
39	C₂×C₄	D₂₁₆₈	1	1	yes	characteristic
40	C₈	—	2168	1	no	—
41	C₈	C₄⋊Dic₂₇₁	1	1	yes	characteristic
42	C₈	Dic₁₀₈₄	1	1	yes	characteristic
43	C₂²	#22576105	1	1	yes	characteristic
44	C₄	—	4336	1	no	—
45	C₄	#22576094	1	1	yes	characteristic
46	C₄	#22576105	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#31773725	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#31773725
commutator	C₁₆×C₂₇₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₇₁
Fitting	M₆₄×C₂₇₁	C₂
socle	C₂×C₂₇₁	#367
prime radical	C₂²×C₂₇₁	SD₃₂
cosocle	C₂×C₁₆×C₂₇₁	C₂²
hypercenter	M₆₄	D₂₇₁
hypocenter	C₂₇₁	#1426
LCS	G ▷ C₁₆×C₂₇₁ ▷ C₈×C₂₇₁ ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
7. C₃₂×C₂₇₁	C₄	semidirect
24. C₂₇₁	#1426	semidirect
32. C₃₂	Dic₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
#31773727	2
#31773725	2
M₆₄×C₂₇₁	2
#1426	271