Subgroups — #46838639

Order 34688.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31774302	C₂	1	2	yes	maximal
2	#31773723	C₂	1	4	yes	maximal
3	C₄×C₁₆×C₂₇₁	C₂	1	1	yes	maximal, characteristic
4	#22576160	—	2	2	no	—
5	#22576094	—	2	2	no	—
6	#22576094	C₂²	1	4	yes	—
7	C₂×C₁₆×C₂₇₁	C₂²	1	2	yes	—
8	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
9	C₂×Dic₅₄₂	—	4	2	no	—
10	C₄⋊Dic₂₇₁	—	2	4	no	—
11	C₄⋊Dic₂₇₁	—	4	2	no	—
12	C₂×C₈×C₂₇₁	D₄	1	1	yes	characteristic
13	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
14	C₄²×C₂₇₁	D₄	1	1	yes	characteristic
15	C₁₆×C₂₇₁	Q₈	1	4	yes	complemented
16	Dic₅₄₂	—	8	4	no	—
17	C₂×Dic₂₇₁	—	4	4	no	—
18	C₂×C₄×C₂₇₁	D₈	1	1	yes	characteristic
19	C₂×C₄×C₂₇₁	D₈	1	1	yes	characteristic
20	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
21	C₈×C₂₇₁	Q₁₆	1	2	yes	—
22	C₈×C₂₇₁	C₂×Q₈	1	2	yes	—
23	C₂²×C₂₇₁	C₂×D₈	1	1	yes	characteristic
24	C₄×C₂₇₁	D₁₆	1	2	yes	—
25	C₄×C₂₇₁	Q₃₂	1	2	yes	—
26	C₄×C₂₇₁	C₄⋊Q₈	1	1	yes	characteristic
27	C₄×C₂₇₁	C₂×Q₁₆	1	1	yes	characteristic
28	Dic₂₇₁	—	8	4	no	—
29	C₂×C₂₇₁	#500	1	1	yes	characteristic
30	C₂×C₂₇₁	C₂×D₁₆	1	1	yes	characteristic
31	C₂×C₂₇₁	C₂×Q₃₂	1	1	yes	characteristic
32	C₂₇₁	#2252	1	1	yes	characteristic, complemented, Sylow
33	#2252	—	271	1	no	maximal, Sylow
34	#500	—	271	2	no	—
35	#366	—	271	4	no	—
36	C₄×C₁₆	D₂₇₁	1	1	yes	characteristic
37	C₄⋊Q₈	—	542	2	no	—
38	C₂×C₁₆	C₂×D₂₇₁	1	2	yes	—
39	#107	—	271	4	no	—
40	#107	—	542	2	no	—
41	C₄×C₈	C₂×D₂₇₁	1	1	yes	characteristic
42	C₂×Q₈	—	1084	2	no	—
43	C₂×C₈	D₁₀₈₄	1	1	yes	characteristic
44	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
45	C₄⋊C₄	—	542	4	no	—
46	C₄⋊C₄	—	1084	2	no	—
47	C₄²	D₁₀₈₄	1	1	yes	characteristic
48	C₁₆	Dic₅₄₂	1	4	yes	complemented
49	Q₈	—	2168	4	no	—
50	C₂×C₄	—	1084	4	no	—
51	C₂×C₄	D₂₁₆₈	1	1	yes	characteristic
52	C₂×C₄	D₂₁₆₈	1	1	yes	characteristic
53	C₂×C₄	C₂×D₁₀₈₄	1	1	yes	characteristic
54	C₈	Dic₁₀₈₄	1	2	yes	—
55	C₈	C₂×Dic₅₄₂	1	2	yes	—
56	C₂²	C₂×D₂₁₆₈	1	1	yes	characteristic
57	C₄	—	2168	4	no	—
58	C₄	D₄₃₃₆	1	2	yes	—
59	C₄	Dic₂₁₆₈	1	2	yes	—
60	C₄	#22576160	1	1	yes	characteristic
61	C₄	C₂×Dic₁₀₈₄	1	1	yes	characteristic
62	C₂	#31774302	1	1	yes	characteristic
63	C₂	C₂×D₄₃₃₆	1	1	yes	characteristic
64	C₂	C₂×Dic₂₁₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₁₆₈
commutator, cosocle	C₂×C₈×C₂₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₂₇₁
Fitting	C₄×C₁₆×C₂₇₁	C₂
socle, prime radical	C₂²×C₂₇₁	C₂×D₈
hypercenter	C₄×C₁₆	D₂₇₁
hypocenter	C₂₇₁	#2252
LCS	G ▷ C₂×C₈×C₂₇₁ ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
15. C₁₆×C₂₇₁	Q₈	semidirect
32. C₂₇₁	#2252	semidirect
48. C₁₆	Dic₅₄₂	semidirect

Maximal subgroups

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

subgroup	index
#31774302	2
#31773723	2
C₄×C₁₆×C₂₇₁	2
#2252	271