Subgroups — #41×Dic₄₂₁ #41003311

Order 26944.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₄₂₁	C₂	1	2	yes	maximal, complemented
2	D₄×Dic₄₂₁	C₂	1	6	yes	maximal, complemented
3	#27212325	C₂	1	3	yes	maximal, complemented
4	C₂×C₄×Dic₄₂₁	C₂	1	3	yes	maximal, complemented
5	C₂×#41×C₄₂₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₂₁	C₂²	1	6	yes	—
7	C₄⋊Dic₄₂₁	C₂²	1	6	yes	—
8	C₂²⋊Dic₄₂₁	C₂²	1	6	yes	—
9	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
10	C₄×Dic₄₂₁	C₂²	1	3	yes	—
11	C₄×Dic₄₂₁	C₂²	1	6	yes	—
12	#41×C₄₂₁	C₄	1	8	yes	complemented
13	C₂×Q₈×C₄₂₁	C₂²	1	1	yes	characteristic
14	C₂×D₄×C₄₂₁	C₂²	1	3	yes	—
15	C₂²×C₄×C₄₂₁	C₂²	1	3	yes	—
16	C₂×Dic₄₂₁	—	2	12	no	—
17	C₂×Dic₄₂₁	C₂³	1	6	yes	—
18	C₂×Dic₄₂₁	C₂³	1	2	yes	—
19	C₂³×C₄₂₁	C₂³	1	3	yes	—
20	Q₈×C₄₂₁	C₂×C₄	1	4	yes	complemented
21	D₄×C₄₂₁	C₂×C₄	1	12	yes	complemented
22	C₂×C₄×C₄₂₁	C₂×C₄	1	12	yes	complemented
23	C₂×C₄×C₄₂₁	C₂³	1	3	yes	—
24	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄₂₁	—	2	6	no	—
26	C₂²×C₄₂₁	C₂²×C₄	1	6	yes	—
27	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
28	C₄×C₄₂₁	C₂²×C₄	1	2	yes	—
29	C₄×C₄₂₁	C₂²×C₄	1	6	yes	—
30	Dic₄₂₁	—	2	6	no	—
31	Dic₄₂₁	#41	1	4	yes	direct product factor
32	C₂×C₄₂₁	—	2	6	no	—
33	C₂×C₄₂₁	C₂³×C₄	1	1	yes	characteristic
34	C₂×C₄₂₁	C₂×#41	1	2	yes	—
35	C₄₂₁	C₄×#41	1	1	yes	characteristic, complemented, Sylow
36	C₄×#41	—	421	1	no	maximal, Sylow
37	C₂×#41	D₄₂₁	1	1	yes	characteristic
38	C₄×Q₈	—	421	2	no	—
39	C₄×D₄	—	421	6	no	—
40	#117	—	421	3	no	—
41	C₂×C₄²	—	421	3	no	—
42	#41	Dic₄₂₁	1	8	yes	direct product factor
43	C₂×Q₈	C₂×D₄₂₁	1	1	yes	characteristic
44	C₂×D₄	C₂×D₄₂₁	1	3	yes	—
45	C₂²×C₄	—	421	6	no	—
46	C₂²×C₄	C₂×D₄₂₁	1	3	yes	—
47	C₄⋊C₄	—	421	6	no	—
48	C₂²⋊C₄	—	421	6	no	—
49	C₄²	—	421	6	no	—
50	C₄²	—	421	1	no	—
51	C₄²	—	421	3	no	—
52	C₂³	C₂²×D₄₂₁	1	3	yes	—
53	Q₈	C₂×Dic₄₂₁	1	4	yes	complemented
54	D₄	C₂×Dic₄₂₁	1	12	yes	complemented
55	C₂×C₄	—	842	12	no	—
56	C₂×C₄	—	421	6	no	—
57	C₂×C₄	—	421	2	no	—
58	C₂×C₄	C₂×Dic₄₂₁	1	12	yes	complemented
59	C₂×C₄	C₂²×D₄₂₁	1	3	yes	—
60	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
61	C₂²	—	2	6	no	—
62	C₂²	C₂²×Dic₄₂₁	1	6	yes	—
63	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
64	C₄	—	842	6	no	—
65	C₄	—	421	4	no	—
66	C₄	C₂²×Dic₄₂₁	1	2	yes	—
67	C₄	C₂²×Dic₄₂₁	1	6	yes	—
68	C₂	—	2	6	no	—
69	C₂	C₂³×Dic₄₂₁	1	1	yes	characteristic
70	C₂	#41×D₄₂₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄₂₁
commutator	C₂×C₄₂₁	C₂³×C₄
Frattini	C₂²	C₂³×D₄₂₁
Fitting, prime radical	C₂×#41×C₄₂₁	C₂
socle, cosocle	C₂²×C₄₂₁	C₂⁴
hypercenter	C₂×#41	D₄₂₁
hypocenter	C₄₂₁	C₄×#41
LCS	G ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₄₂₁	C₂	semidirect
2. D₄×Dic₄₂₁	C₂	semidirect
3. #27212325	C₂	semidirect
4. C₂×C₄×Dic₄₂₁	C₂	semidirect
12. #41×C₄₂₁	C₄	semidirect
20. Q₈×C₄₂₁	C₂×C₄	semidirect
21. D₄×C₄₂₁	C₂×C₄	semidirect
22. C₂×C₄×C₄₂₁	C₂×C₄	semidirect
31. Dic₄₂₁	#41	direct (N × H)
35. C₄₂₁	C₄×#41	semidirect
42. #41	Dic₄₂₁	direct (N × H)
53. Q₈	C₂×Dic₄₂₁	semidirect
54. D₄	C₂×Dic₄₂₁	semidirect
58. C₂×C₄	C₂×Dic₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₄₂₁	2
D₄×Dic₄₂₁	2
#27212325	2
C₂×C₄×Dic₄₂₁	2
C₂×#41×C₄₂₁	2
C₄×#41	421

Subgroups — #41×Dic421 #41003311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #41×Dic₄₂₁ #41003311