Subgroups — 2₋¹⁺⁴×D₄₂₁ #41003672

Order 26944.

794 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212414	C₂	1	5	yes	maximal, complemented
2	#27212413	C₂	1	10	yes	maximal, complemented
3	#41×D₄₂₁	C₂	1	10	yes	maximal, complemented
4	C₂×Q₈×D₄₂₁	C₂	1	5	yes	maximal, complemented
5	2₋¹⁺⁴×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#20269772	C₂²	1	10	yes	complemented
7	#20269771	C₂²	1	30	yes	complemented
8	#20269770	C₂²	1	30	yes	complemented
9	C₂×Dic₈₄₂	C₂²	1	15	yes	complemented
10	Q₈×D₄₂₁	C₂²	1	10	yes	—
11	Q₈×D₄₂₁	C₂²	1	20	yes	complemented
12	D₄×D₄₂₁	C₂²	1	10	yes	—
13	C₂×C₄×D₄₂₁	C₂²	1	15	yes	—
14	#41×C₄₂₁	C₂²	1	10	yes	complemented
15	C₂×Q₈×C₄₂₁	C₂²	1	5	yes	complemented
16	C₂²×D₄₂₁	C₂³	1	5	yes	—
17	Dic₈₄₂	C₂³	1	30	yes	—
18	D₁₆₈₄	C₂³	1	10	yes	—
19	C₂²⋊D₄₂₁	C₂³	1	20	yes	—
20	C₂×Dic₄₂₁	C₂³	1	15	yes	—
21	C₄×D₄₂₁	C₂³	1	10	yes	—
22	C₄×D₄₂₁	C₂³	1	30	yes	—
23	Q₈×C₄₂₁	C₂³	1	10	yes	—
24	D₄×C₄₂₁	C₂³	1	10	yes	—
25	C₂×C₄×C₄₂₁	C₂³	1	15	yes	—
26	C₂²×C₄₂₁	C₂⁴	1	5	yes	—
27	C₂×D₄₂₁	—	2	10	no	—
28	C₂×D₄₂₁	C₂⁴	1	5	yes	—
29	C₂×D₄₂₁	C₂⁴	1	1	yes	characteristic
30	C₄×C₄₂₁	C₂⁴	1	10	yes	—
31	Dic₄₂₁	C₂⁴	1	10	yes	—
32	C₂×C₄₂₁	—	2	5	no	—
33	C₂×C₄₂₁	C₂⁵	1	1	yes	characteristic
34	D₄₂₁	—	2	5	no	—
35	D₄₂₁	2₋¹⁺⁴	1	2	yes	direct product factor
36	C₄₂₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
37	C₂×2₋¹⁺⁴	—	421	1	no	maximal, Sylow
38	2₋¹⁺⁴	—	421	10	no	—
39	2₋¹⁺⁴	—	421	5	no	—
40	2₋¹⁺⁴	D₄₂₁	1	1	yes	characteristic, direct product factor
41	C₂×#41	—	421	10	no	—
42	C₂²×Q₈	—	421	5	no	—
43	#41	—	421	10	no	—
44	#41	—	421	30	no	—
45	#41	—	421	30	no	—
46	#41	C₂×D₄₂₁	1	10	yes	complemented
47	C₂×Q₈	—	421	10	no	—
48	C₂×Q₈	—	421	20	no	—
49	C₂×Q₈	—	421	15	no	—
50	C₂×Q₈	C₂×D₄₂₁	1	5	yes	complemented
51	C₂×D₄	—	421	10	no	—
52	C₂²×C₄	—	421	15	no	—
53	C₂³	—	421	5	no	—
54	Q₈	—	421	30	no	—
55	Q₈	C₂²×D₄₂₁	1	10	yes	—
56	D₄	—	421	20	no	—
57	D₄	—	421	10	no	—
58	D₄	C₂²×D₄₂₁	1	10	yes	—
59	C₂×C₄	—	421	15	no	—
60	C₂×C₄	—	421	10	no	—
61	C₂×C₄	—	421	30	no	—
62	C₂×C₄	C₂²×D₄₂₁	1	15	yes	—
63	C₂²	—	842	10	no	—
64	C₂²	—	421	5	no	—
65	C₂²	—	421	1	no	—
66	C₂²	C₂³×D₄₂₁	1	5	yes	—
67	C₄	—	421	10	no	—
68	C₄	C₂³×D₄₂₁	1	10	yes	—
69	C₂	—	2	5	no	—
70	C₂	—	842	5	no	—
71	C₂	—	421	2	no	—
72	C₂	C₂⁴×D₄₂₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₄₂₁
commutator, socle, cosocle	C₂×C₄₂₁	C₂⁵
Fitting	2₋¹⁺⁴×C₄₂₁	C₂
hypercenter	2₋¹⁺⁴	D₄₂₁
hypocenter	C₄₂₁	C₂×2₋¹⁺⁴
LCS	G ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂ ◁ 2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. #27212414	C₂	semidirect
2. #27212413	C₂	semidirect
3. #41×D₄₂₁	C₂	semidirect
4. C₂×Q₈×D₄₂₁	C₂	semidirect
5. 2₋¹⁺⁴×C₄₂₁	C₂	semidirect
6. #20269772	C₂²	semidirect
7. #20269771	C₂²	semidirect
8. #20269770	C₂²	semidirect
9. C₂×Dic₈₄₂	C₂²	semidirect
11. Q₈×D₄₂₁	C₂²	semidirect
14. #41×C₄₂₁	C₂²	semidirect
15. C₂×Q₈×C₄₂₁	C₂²	semidirect
35. D₄₂₁	2₋¹⁺⁴	direct (N × H)
36. C₄₂₁	C₂×2₋¹⁺⁴	semidirect
40. 2₋¹⁺⁴	D₄₂₁	direct (N × H)
46. #41	C₂×D₄₂₁	semidirect
50. C₂×Q₈	C₂×D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
#27212414	2
#27212413	2
#41×D₄₂₁	2
C₂×Q₈×D₄₂₁	2
2₋¹⁺⁴×C₄₂₁	2
C₂×2₋¹⁺⁴	421

Subgroups — 2−1+4×D421 #41003672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 2₋¹⁺⁴×D₄₂₁ #41003672