Subgroups — #41003245

Order 26944.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212366	C₂	1	2	yes	maximal
2	#27212300	C₂	1	4	yes	maximal
3	C₄×C₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₈₄₂	—	2	2	no	—
5	C₄⋊Dic₄₂₁	—	2	2	no	—
6	C₄⋊Dic₄₂₁	C₂²	1	4	yes	—
7	C₂×C₈×C₄₂₁	C₂²	1	2	yes	—
8	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
9	Dic₈₄₂	—	4	4	no	—
10	C₂×Dic₄₂₁	—	2	4	no	—
11	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
12	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
13	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
14	C₈×C₄₂₁	Q₈	1	4	yes	complemented
15	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
16	C₄×C₄₂₁	D₈	1	2	yes	—
17	C₄×C₄₂₁	Q₁₆	1	2	yes	—
18	C₄×C₄₂₁	C₂×Q₈	1	2	yes	—
19	Dic₄₂₁	—	4	4	no	—
20	C₂×C₄₂₁	C₄⋊Q₈	1	1	yes	characteristic
21	C₂×C₄₂₁	C₂×D₈	1	1	yes	characteristic
22	C₂×C₄₂₁	C₂×Q₁₆	1	1	yes	characteristic
23	C₄₂₁	#500	1	1	yes	characteristic, complemented, Sylow
24	#500	—	421	1	no	maximal, Sylow
25	C₄⋊Q₈	—	421	2	no	—
26	#107	—	421	4	no	—
27	C₄×C₈	D₄₂₁	1	1	yes	characteristic
28	C₂×Q₈	—	842	2	no	—
29	C₂×C₈	C₂×D₄₂₁	1	2	yes	—
30	C₄⋊C₄	—	842	2	no	—
31	C₄⋊C₄	—	421	4	no	—
32	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
33	Q₈	—	1684	4	no	—
34	C₂×C₄	—	842	4	no	—
35	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
36	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
37	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
38	C₈	Dic₈₄₂	1	4	yes	complemented
39	C₂²	C₂×D₁₆₈₄	1	1	yes	characteristic
40	C₄	—	1684	4	no	—
41	C₄	D₃₃₆₈	1	2	yes	—
42	C₄	Dic₁₆₈₄	1	2	yes	—
43	C₄	C₂×Dic₈₄₂	1	2	yes	—
44	C₂	#27212366	1	1	yes	characteristic
45	C₂	C₂×D₃₃₆₈	1	1	yes	characteristic
46	C₂	C₂×Dic₁₆₈₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₆₈₄
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	C₄×C₈×C₄₂₁	C₂
socle, prime radical	C₂²×C₄₂₁	C₂×D₄
hypercenter	C₄×C₈	D₄₂₁
hypocenter	C₄₂₁	#500
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
14. C₈×C₄₂₁	Q₈	semidirect
23. C₄₂₁	#500	semidirect
38. C₈	Dic₈₄₂	semidirect

Maximal subgroups

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

subgroup	index
#27212366	2
#27212300	2
C₄×C₈×C₄₂₁	2
#500	421