Subgroups — #46834437

Order 34688.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#31773650	C₂	1	1	yes	maximal, characteristic
3	#335×C₂₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₇₁⋊C₁₆	C₄	1	1	yes	characteristic
5	C₂×C₂₇₁⋊C₁₆	C₄	1	1	yes	characteristic
6	C₄⋊(C₂₇₁⋊C₈)	—	2	1	no	—
7	C₄⋊C₈×C₂₇₁	—	2	1	no	—
8	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
9	C₂×C₂₇₁⋊C₈	—	4	1	no	—
10	C₂₇₁⋊C₁₆	—	2	1	no	—
11	C₂₇₁⋊C₁₆	C₈	1	2	yes	complemented
12	C₂×C₈×C₂₇₁	—	4	1	no	—
13	C₂×C₈×C₂₇₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₂₇₁	Q₈	1	1	yes	characteristic
15	C₄²×C₂₇₁	D₄	1	1	yes	characteristic
16	C₂₇₁⋊C₈	—	8	1	no	—
17	C₂×C₄×C₂₇₁	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₇₁	D₈	1	1	yes	characteristic
19	C₂×C₄×C₂₇₁	Q₁₆	1	1	yes	characteristic
20	C₈×C₂₇₁	—	2	1	no	—
21	C₈×C₂₇₁	—	8	1	no	—
22	C₈×C₂₇₁	C₂×C₈	1	1	yes	characteristic
23	C₈×C₂₇₁	M₁₆	1	1	yes	characteristic
24	C₂²×C₂₇₁	#107	1	1	yes	characteristic
25	C₄×C₂₇₁	—	2	1	no	—
26	C₄×C₂₇₁	C₄⋊C₈	1	1	yes	characteristic
27	C₄×C₂₇₁	#108	1	1	yes	characteristic
28	C₄×C₂₇₁	SD₃₂	1	1	yes	characteristic
29	C₄×C₂₇₁	SD₃₂	1	1	yes	characteristic
30	C₂×C₂₇₁	#335	1	1	yes	characteristic
31	C₂×C₂₇₁	#367	1	1	yes	characteristic
32	C₂×C₂₇₁	#368	1	1	yes	characteristic
33	C₂₇₁	#1372	1	1	yes	characteristic, complemented, Sylow
34	#1372	—	271	1	no	maximal, Sylow
35	C₄×C₁₆	—	271	1	no	—
36	#335	—	271	1	no	—
37	#335	D₂₇₁	1	1	yes	characteristic
38	C₂×C₁₆	—	271	1	no	—
39	C₂×C₁₆	—	271	1	no	—
40	C₄⋊C₈	—	2	1	no	—
41	C₄⋊C₈	—	542	1	no	—
42	C₄×C₈	C₂×D₂₇₁	1	1	yes	characteristic
43	C₂×C₈	—	4	1	no	—
44	C₂×C₈	—	1084	1	no	—
45	C₂×C₈	C₄×D₂₇₁	1	1	yes	characteristic
46	C₂×C₈	Dic₅₄₂	1	1	yes	characteristic
47	C₄²	C₂²⋊D₂₇₁	1	1	yes	characteristic
48	C₁₆	—	271	2	no	—
49	C₁₆	—	542	1	no	—
50	C₂×C₄	Dic₂₇₁⋊C₄	1	1	yes	characteristic
51	C₂×C₄	#16864639	1	1	yes	characteristic
52	C₂×C₄	#16864644	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	—	8	1	no	—
55	C₈	—	2168	1	no	—
56	C₈	C₈×D₂₇₁	1	1	yes	characteristic
57	C₈	#16864637	1	1	yes	characteristic
58	C₂²	#22576093	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	Dic₂₇₁⋊C₈	1	1	yes	characteristic
61	C₄	#22576095	1	1	yes	characteristic
62	C₄	#22576103	1	1	yes	characteristic
63	C₄	#22576104	1	1	yes	characteristic
64	C₂	#31773649	1	1	yes	characteristic
65	C₂	#31773724	1	1	yes	characteristic
66	C₂	#31773726	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#16864639
commutator	C₈×C₂₇₁	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₇₁
Fitting	#335×C₂₇₁	C₂
socle, prime radical	C₂²×C₂₇₁	#107
cosocle	C₄×C₈×C₂₇₁	C₂²
hypercenter	#335	D₂₇₁
hypocenter	C₂₇₁	#1372
LCS	G ▷ C₈×C₂₇₁ ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
11. C₂₇₁⋊C₁₆	C₈	semidirect
33. C₂₇₁	#1372	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂₇₁⋊C₁₆	2
#31773650	2
#335×C₂₇₁	2
#1372	271