Subgroups — #31823749

Order 17984.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582879	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₅₆₂	C₂	1	1	yes	maximal, characteristic
3	#22582811	C₂	1	2	yes	maximal
4	#22582804	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₂₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₆₂	—	2	1	no	—
7	C₂×Dic₅₆₂	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₈₁	—	2	1	no	—
9	C₄⋊Dic₂₈₁	C₂²	1	2	yes	—
10	C₄⋊Dic₂₈₁	C₂²	1	1	yes	characteristic
11	#17985117	—	2	1	no	—
12	C₄×Dic₂₈₁	—	2	1	no	—
13	C₂×C₈×C₂₈₁	C₂²	1	2	yes	—
14	C₄²×C₂₈₁	C₂²	1	1	yes	characteristic
15	Dic₅₆₂	—	2	1	no	—
16	Dic₅₆₂	—	4	2	no	—
17	Dic₅₆₂	Q₈	1	2	yes	complemented
18	C₂×Dic₂₈₁	—	2	1	no	—
19	C₂×Dic₂₈₁	—	2	1	no	—
20	C₂×Dic₂₈₁	—	2	2	no	—
21	C₂×C₄×C₂₈₁	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₈₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
24	C₈×C₂₈₁	—	2	2	no	—
25	C₂²×C₂₈₁	C₂×D₄	1	1	yes	characteristic
26	C₄×C₂₈₁	—	2	1	no	—
27	C₄×C₂₈₁	SD₁₆	1	2	yes	—
28	C₄×C₂₈₁	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₂₈₁	#41	1	1	yes	characteristic
30	Dic₂₈₁	—	2	2	no	—
31	Dic₂₈₁	—	4	2	no	—
32	Dic₂₈₁	—	4	1	no	—
33	C₂×C₂₈₁	C₂²⋊Q₈	1	1	yes	characteristic
34	C₂×C₂₈₁	C₂×SD₁₆	1	1	yes	characteristic
35	C₂×C₂₈₁	#137	1	1	yes	characteristic
36	C₂₈₁	#475	1	1	yes	characteristic, complemented, Sylow
37	#475	—	281	1	no	maximal, Sylow
38	C₄⋊Q₈	—	281	1	no	—
39	C₄×Q₈	—	281	1	no	—
40	#106	—	281	2	no	—
41	C₄⋊C₈	D₂₈₁	1	1	yes	characteristic
42	#103	—	281	2	no	—
43	C₂×Q₈	—	281	1	no	—
44	C₂×Q₈	—	562	1	no	—
45	C₂×C₈	C₂×D₂₈₁	1	2	yes	—
46	C₄⋊C₄	—	281	2	no	—
47	C₄⋊C₄	—	281	1	no	—
48	C₄⋊C₄	—	562	1	no	—
49	C₄⋊C₄	—	562	1	no	—
50	C₄²	—	562	1	no	—
51	C₄²	C₂×D₂₈₁	1	1	yes	characteristic
52	Q₈	—	281	2	no	—
53	Q₈	—	562	1	no	—
54	Q₈	—	1124	2	no	—
55	C₂×C₄	—	562	1	no	—
56	C₂×C₄	—	562	1	no	—
57	C₂×C₄	—	562	2	no	—
58	C₂×C₄	D₁₁₂₄	1	1	yes	characteristic
59	C₂×C₄	D₁₁₂₄	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
61	C₈	—	2	2	no	—
62	C₂²	C₂×D₁₁₂₄	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	562	2	no	—
65	C₄	—	1124	2	no	—
66	C₄	—	1124	1	no	—
67	C₄	#17985127	1	2	yes	—
68	C₄	Q₈×D₂₈₁	1	1	yes	characteristic
69	C₄	#17985137	1	1	yes	characteristic
70	C₂	#22582859	1	1	yes	characteristic
71	C₂	C₂×#17985127	1	1	yes	characteristic
72	C₂	#22582912	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₁₂₄
commutator, cosocle	C₂×C₄×C₂₈₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₈₁
Fitting	C₄⋊C₈×C₂₈₁	C₂
socle, prime radical	C₂²×C₂₈₁	C₂×D₄
hypercenter	C₄⋊C₈	D₂₈₁
hypocenter	C₂₈₁	#475
LCS	G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
17. Dic₅₆₂	Q₈	semidirect
36. C₂₈₁	#475	semidirect

Maximal subgroups

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

subgroup	index
#22582879	2
C₄×Dic₅₆₂	2
#22582811	2
#22582804	2
C₄⋊C₈×C₂₈₁	2
#475	281