Subgroups — C₂×#27212363 #41003380

Order 26944.

498 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₆₈₄	C₂	1	6	yes	maximal, complemented
2	#27212363	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄²×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₄₂₁	C₂²	1	4	yes	—
5	C₂×D₁₆₈₄	—	2	24	no	—
6	C₂×D₁₆₈₄	C₂²	1	24	yes	complemented
7	C₂²×C₄×C₄₂₁	C₂²	1	3	yes	—
8	C₄²×C₄₂₁	C₂²	1	4	yes	complemented
9	C₂²×D₄₂₁	—	2	24	no	—
10	C₂²×D₄₂₁	C₂³	1	8	yes	—
11	D₁₆₈₄	—	2	48	no	—
12	C₂³×C₄₂₁	C₂³	1	1	yes	characteristic
13	C₂×C₄×C₄₂₁	D₄	1	12	yes	complemented
14	C₂×C₄×C₄₂₁	C₂³	1	6	yes	—
15	C₂²×C₄₂₁	C₂×D₄	1	6	yes	—
16	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
17	C₂×D₄₂₁	—	2	24	no	—
18	C₂×D₄₂₁	—	4	16	no	—
19	C₄×C₄₂₁	C₂×D₄	1	12	yes	complemented
20	C₂×C₄₂₁	#127	1	4	yes	complemented
21	C₂×C₄₂₁	C₂²×D₄	1	3	yes	—
22	D₄₂₁	—	4	8	no	—
23	C₄₂₁	C₂×#127	1	1	yes	characteristic, complemented, Sylow
24	C₂×#127	—	421	1	no	maximal, Sylow
25	C₂²×D₄	—	421	6	no	—
26	#127	—	421	8	no	—
27	C₂×C₄²	D₄₂₁	1	1	yes	characteristic, complemented
28	C₂⁴	—	421	4	no	—
29	C₂×D₄	—	842	24	no	—
30	C₂×D₄	—	421	24	no	—
31	C₂²×C₄	C₂×D₄₂₁	1	3	yes	—
32	C₄²	C₂×D₄₂₁	1	4	yes	complemented
33	C₂³	—	842	24	no	—
34	C₂³	—	421	8	no	—
35	C₂³	C₂²×D₄₂₁	1	1	yes	characteristic
36	D₄	—	842	48	no	—
37	C₂×C₄	D₁₆₈₄	1	12	yes	complemented
38	C₂×C₄	C₂²×D₄₂₁	1	6	yes	—
39	C₂²	—	842	24	no	—
40	C₂²	—	1684	16	no	—
41	C₂²	C₂×D₁₆₈₄	1	6	yes	—
42	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
43	C₄	C₂×D₁₆₈₄	1	12	yes	complemented
44	C₂	—	1684	8	no	—
45	C₂	#27212363	1	4	yes	direct product factor
46	C₂	C₂²×D₁₆₈₄	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₄₂₁
commutator, cosocle	C₂²×C₄₂₁	C₂⁴
Frattini	C₂²	C₂³×D₄₂₁
Fitting	C₂×C₄²×C₄₂₁	C₂
socle	C₂³×C₄₂₁	C₂³
hypercenter	C₂×C₄²	D₄₂₁
hypocenter	C₄₂₁	C₂×#127
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₆₈₄	C₂	semidirect
2. #27212363	C₂	direct (N × H)
3. C₂×C₄²×C₄₂₁	C₂	semidirect
6. C₂×D₁₆₈₄	C₂²	semidirect
8. C₄²×C₄₂₁	C₂²	semidirect
13. C₂×C₄×C₄₂₁	D₄	semidirect
19. C₄×C₄₂₁	C₂×D₄	semidirect
20. C₂×C₄₂₁	#127	semidirect
23. C₄₂₁	C₂×#127	semidirect
27. C₂×C₄²	D₄₂₁	semidirect
32. C₄²	C₂×D₄₂₁	semidirect
37. C₂×C₄	D₁₆₈₄	semidirect
43. C₄	C₂×D₁₆₈₄	semidirect
45. C₂	#27212363	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₁₆₈₄	2
#27212363	2
C₂×C₄²×C₄₂₁	2
C₂×#127	421

Subgroups — C2×#27212363 #41003380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27212363 #41003380