Subgroups — #41003217

Order 26944.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₃₆₈	C₂	1	4	yes	maximal, complemented
2	#27212363	C₂	1	2	yes	maximal, complemented
3	C₄×C₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₆₈₄	—	2	4	no	—
5	C₂×D₁₆₈₄	C₂²	1	4	yes	—
6	D₃₃₆₈	—	2	8	no	—
7	C₂×C₈×C₄₂₁	C₂²	1	2	yes	—
8	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
9	C₂²×D₄₂₁	—	2	4	no	—
10	D₁₆₈₄	—	2	8	no	—
11	D₁₆₈₄	—	4	8	no	—
12	C₂×C₄×C₄₂₁	D₄	1	2	yes	—
13	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
14	C₈×C₄₂₁	D₄	1	4	yes	complemented
15	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
16	C₂×D₄₂₁	—	4	4	no	—
17	C₂×D₄₂₁	—	4	8	no	—
18	C₄×C₄₂₁	D₈	1	4	yes	complemented
19	C₄×C₄₂₁	C₂×D₄	1	2	yes	—
20	C₂×C₄₂₁	#127	1	1	yes	characteristic
21	C₂×C₄₂₁	C₂×D₈	1	2	yes	—
22	D₄₂₁	—	8	4	no	—
23	C₄₂₁	#493	1	1	yes	characteristic, complemented, Sylow
24	#493	—	421	1	no	maximal, Sylow
25	C₂×D₈	—	421	4	no	—
26	#127	—	421	2	no	—
27	C₄×C₈	D₄₂₁	1	1	yes	characteristic, complemented
28	C₂×D₄	—	842	4	no	—
29	C₂×D₄	—	421	4	no	—
30	D₈	—	842	8	no	—
31	C₂×C₈	C₂×D₄₂₁	1	2	yes	—
32	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
33	C₂³	—	842	4	no	—
34	D₄	—	842	8	no	—
35	D₄	—	1684	8	no	—
36	C₂×C₄	D₁₆₈₄	1	2	yes	—
37	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
38	C₈	D₁₆₈₄	1	4	yes	complemented
39	C₂²	—	1684	4	no	—
40	C₂²	—	1684	8	no	—
41	C₂²	C₂×D₁₆₈₄	1	1	yes	characteristic
42	C₄	D₃₃₆₈	1	4	yes	complemented
43	C₄	C₂×D₁₆₈₄	1	2	yes	—
44	C₂	—	3368	4	no	—
45	C₂	#27212363	1	1	yes	characteristic
46	C₂	C₂×D₃₃₆₈	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₆₈₄
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	C₄×C₈×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
hypercenter	C₄×C₈	D₄₂₁
hypocenter	C₄₂₁	#493
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
1. C₂×D₃₃₆₈	C₂	semidirect
2. #27212363	C₂	semidirect
3. C₄×C₈×C₄₂₁	C₂	semidirect
14. C₈×C₄₂₁	D₄	semidirect
18. C₄×C₄₂₁	D₈	semidirect
23. C₄₂₁	#493	semidirect
27. C₄×C₈	D₄₂₁	semidirect
38. C₈	D₁₆₈₄	semidirect
42. C₄	D₃₃₆₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₃₆₈	2
#27212363	2
C₄×C₈×C₄₂₁	2
#493	421