Subgroups — C₂×#20269974 #27213686

Order 13504.

498 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₈₄₄	C₂	1	6	yes	maximal, complemented
2	#20269974	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄²×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₂₁₁	C₂²	1	4	yes	—
5	C₂×D₈₄₄	—	2	24	no	—
6	C₂×D₈₄₄	C₂²	1	24	yes	complemented
7	C₂²×C₄×C₂₁₁	C₂²	1	3	yes	—
8	C₄²×C₂₁₁	C₂²	1	4	yes	complemented
9	C₂²×D₂₁₁	—	2	24	no	—
10	C₂²×D₂₁₁	C₂³	1	8	yes	—
11	D₈₄₄	—	2	48	no	—
12	C₂³×C₂₁₁	C₂³	1	1	yes	characteristic
13	C₂×C₄×C₂₁₁	D₄	1	12	yes	complemented
14	C₂×C₄×C₂₁₁	C₂³	1	6	yes	—
15	C₂²×C₂₁₁	C₂×D₄	1	6	yes	—
16	C₂²×C₂₁₁	C₂⁴	1	1	yes	characteristic
17	C₂×D₂₁₁	—	2	24	no	—
18	C₂×D₂₁₁	—	4	16	no	—
19	C₄×C₂₁₁	C₂×D₄	1	12	yes	complemented
20	C₂×C₂₁₁	#127	1	4	yes	complemented
21	C₂×C₂₁₁	C₂²×D₄	1	3	yes	—
22	D₂₁₁	—	4	8	no	—
23	C₂₁₁	C₂×#127	1	1	yes	characteristic, complemented, Sylow
24	C₂×#127	—	211	1	no	maximal, Sylow
25	C₂²×D₄	—	211	6	no	—
26	#127	—	211	8	no	—
27	C₂×C₄²	D₂₁₁	1	1	yes	characteristic, complemented
28	C₂⁴	—	211	4	no	—
29	C₂×D₄	—	422	24	no	—
30	C₂×D₄	—	211	24	no	—
31	C₂²×C₄	C₂×D₂₁₁	1	3	yes	—
32	C₄²	C₂×D₂₁₁	1	4	yes	complemented
33	C₂³	—	422	24	no	—
34	C₂³	—	211	8	no	—
35	C₂³	C₂²×D₂₁₁	1	1	yes	characteristic
36	D₄	—	422	48	no	—
37	C₂×C₄	D₈₄₄	1	12	yes	complemented
38	C₂×C₄	C₂²×D₂₁₁	1	6	yes	—
39	C₂²	—	844	16	no	—
40	C₂²	—	422	24	no	—
41	C₂²	C₂×D₈₄₄	1	6	yes	—
42	C₂²	C₂³×D₂₁₁	1	1	yes	characteristic
43	C₄	C₂×D₈₄₄	1	12	yes	complemented
44	C₂	—	844	8	no	—
45	C₂	#20269974	1	4	yes	direct product factor
46	C₂	C₂²×D₈₄₄	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₁₁
commutator, cosocle	C₂²×C₂₁₁	C₂⁴
Frattini	C₂²	C₂³×D₂₁₁
Fitting	C₂×C₄²×C₂₁₁	C₂
socle	C₂³×C₂₁₁	C₂³
hypercenter	C₂×C₄²	D₂₁₁
hypocenter	C₂₁₁	C₂×#127
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂²×D₈₄₄	C₂	semidirect
2. #20269974	C₂	direct (N × H)
3. C₂×C₄²×C₂₁₁	C₂	semidirect
6. C₂×D₈₄₄	C₂²	semidirect
8. C₄²×C₂₁₁	C₂²	semidirect
13. C₂×C₄×C₂₁₁	D₄	semidirect
19. C₄×C₂₁₁	C₂×D₄	semidirect
20. C₂×C₂₁₁	#127	semidirect
23. C₂₁₁	C₂×#127	semidirect
27. C₂×C₄²	D₂₁₁	semidirect
32. C₄²	C₂×D₂₁₁	semidirect
37. C₂×C₄	D₈₄₄	semidirect
43. C₄	C₂×D₈₄₄	semidirect
45. C₂	#20269974	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×D₈₄₄	2
#20269974	2
C₂×C₄²×C₂₁₁	2
C₂×#127	211

Subgroups — C2×#20269974 #27213686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20269974 #27213686