Subgroups — #127×D₄₂₁ #41003378

Order 26944.

498 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×D₄₂₁	C₂	1	6	yes	maximal, complemented
2	#27212363	C₂	1	1	yes	maximal, characteristic, complemented
3	#27212362	C₂	1	6	yes	maximal, complemented
4	C₄²×D₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#127×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄₂₁	C₂²	1	4	yes	—
7	C₂×D₁₆₈₄	C₂²	1	6	yes	complemented
8	C₂×C₂²⋊D₄₂₁	C₂²	1	12	yes	complemented
9	D₄×D₄₂₁	—	2	24	no	—
10	C₂×C₄×D₄₂₁	C₂²	1	3	yes	—
11	C₄×Dic₄₂₁	C₂²	1	3	yes	complemented
12	C₂×D₄×C₄₂₁	C₂²	1	6	yes	complemented
13	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic, complemented
14	C₂²×D₄₂₁	—	2	24	no	—
15	C₂²×D₄₂₁	C₂³	1	4	yes	—
16	C₂²×D₄₂₁	C₂³	1	1	yes	characteristic
17	D₁₆₈₄	—	2	12	no	—
18	C₂²⋊D₄₂₁	—	2	24	no	—
19	C₂×Dic₄₂₁	C₂³	1	3	yes	—
20	C₄×D₄₂₁	D₄	1	12	yes	complemented
21	C₂³×C₄₂₁	C₂³	1	4	yes	—
22	D₄×C₄₂₁	—	2	12	no	—
23	C₂×C₄×C₄₂₁	C₂³	1	3	yes	—
24	C₂²×C₄₂₁	—	2	12	no	—
25	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
26	C₂×D₄₂₁	—	2	12	no	—
27	C₂×D₄₂₁	—	4	16	no	—
28	C₂×D₄₂₁	C₂×D₄	1	6	yes	—
29	C₄×C₄₂₁	C₂×D₄	1	6	yes	complemented
30	Dic₄₂₁	C₂×D₄	1	6	yes	complemented
31	C₂×C₄₂₁	—	4	4	no	—
32	C₂×C₄₂₁	C₂²×D₄	1	3	yes	—
33	D₄₂₁	—	4	4	no	—
34	D₄₂₁	#127	1	4	yes	direct product factor
35	C₄₂₁	C₂×#127	1	1	yes	characteristic, complemented, Sylow
36	C₂×#127	—	421	1	no	maximal, Sylow
37	C₂²×D₄	—	421	6	no	—
38	#127	—	421	6	no	—
39	#127	—	421	1	no	—
40	#127	D₄₂₁	1	1	yes	characteristic, direct product factor
41	C₂×C₄²	—	421	1	no	—
42	C₂⁴	—	421	4	no	—
43	C₂×D₄	—	842	24	no	—
44	C₂×D₄	—	421	12	no	—
45	C₂×D₄	—	421	6	no	—
46	C₂×D₄	C₂×D₄₂₁	1	6	yes	complemented
47	C₂²×C₄	—	421	3	no	—
48	C₄²	—	421	3	no	—
49	C₄²	C₂×D₄₂₁	1	1	yes	characteristic, complemented
50	C₂³	—	842	24	no	—
51	C₂³	—	421	4	no	—
52	C₂³	—	421	1	no	—
53	C₂³	C₂²×D₄₂₁	1	4	yes	—
54	D₄	—	2	12	no	—
55	D₄	—	842	24	no	—
56	D₄	—	842	12	no	—
57	C₂×C₄	—	421	3	no	—
58	C₂×C₄	—	421	12	no	—
59	C₂×C₄	C₂²×D₄₂₁	1	3	yes	—
60	C₂²	—	2	12	no	—
61	C₂²	—	842	12	no	—
62	C₂²	—	1684	16	no	—
63	C₂²	—	421	6	no	—
64	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
65	C₄	—	421	6	no	—
66	C₄	D₄×D₄₂₁	1	6	yes	complemented
67	C₂	—	4	4	no	—
68	C₂	—	1684	4	no	—
69	C₂	—	421	4	no	—
70	C₂	C₂×D₄×D₄₂₁	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₂₁
commutator, socle, cosocle	C₂²×C₄₂₁	C₂⁴
Fitting	#127×C₄₂₁	C₂
hypercenter	#127	D₄₂₁
hypocenter	C₄₂₁	C₂×#127
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×D₄₂₁	C₂	semidirect
2. #27212363	C₂	semidirect
3. #27212362	C₂	semidirect
4. C₄²×D₄₂₁	C₂	semidirect
5. #127×C₄₂₁	C₂	semidirect
7. C₂×D₁₆₈₄	C₂²	semidirect
8. C₂×C₂²⋊D₄₂₁	C₂²	semidirect
11. C₄×Dic₄₂₁	C₂²	semidirect
12. C₂×D₄×C₄₂₁	C₂²	semidirect
13. C₄²×C₄₂₁	C₂²	semidirect
20. C₄×D₄₂₁	D₄	semidirect
29. C₄×C₄₂₁	C₂×D₄	semidirect
30. Dic₄₂₁	C₂×D₄	semidirect
34. D₄₂₁	#127	direct (N × H)
35. C₄₂₁	C₂×#127	semidirect
40. #127	D₄₂₁	direct (N × H)
46. C₂×D₄	C₂×D₄₂₁	semidirect
49. C₄²	C₂×D₄₂₁	semidirect
66. C₄	D₄×D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×D₄₂₁	2
#27212363	2
#27212362	2
C₄²×D₄₂₁	2
#127×C₄₂₁	2
C₂×#127	421

Subgroups — #127×D421 #41003378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #127×D₄₂₁ #41003378