Subgroups — C₂²×C₄×#11757210 #29508777

Order 15424.

498 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11757210	C₂	1	2	yes	maximal
2	C₂×C₄×#11757210	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄×D₂₄₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11757210	C₄	1	16	yes	direct product factor
5	C₂²×#11757210	C₂²	1	12	yes	—
6	C₄×#11757210	C₂²	1	16	yes	direct product factor
7	C₂³×D₂₄₁	C₂²	1	1	yes	characteristic
8	C₂²×Dic₂₄₁	C₄	1	1	yes	characteristic, complemented
9	C₂×C₄×D₂₄₁	C₄	1	6	yes	complemented
10	C₂×C₄×D₂₄₁	C₂²	1	6	yes	—
11	C₂²×C₄×C₂₄₁	C₄	1	1	yes	characteristic, complemented
12	C₂×#11757210	C₂×C₄	1	48	yes	direct product factor
13	C₂×#11757210	C₂³	1	8	yes	—
14	C₂²×D₂₄₁	C₂×C₄	1	3	yes	—
15	C₂²×D₂₄₁	C₂×C₄	1	8	yes	—
16	C₂²×D₂₄₁	C₂³	1	3	yes	—
17	C₂×Dic₂₄₁	C₂×C₄	1	6	yes	complemented
18	C₄×D₂₄₁	C₂×C₄	1	12	yes	complemented
19	C₄×D₂₄₁	C₂³	1	4	yes	—
20	C₂³×C₂₄₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₄₁	C₂×C₄	1	6	yes	complemented
22	C₂²×C₂₄₁	C₄²	1	4	yes	complemented
23	C₂²×C₂₄₁	C₂²×C₄	1	3	yes	—
24	C₂×D₂₄₁	C₄²	1	12	yes	complemented
25	C₂×D₂₄₁	C₂²×C₄	1	3	yes	—
26	C₂×D₂₄₁	C₂²×C₄	1	12	yes	—
27	C₂×D₂₄₁	C₂⁴	1	1	yes	characteristic
28	#11757210	C₂²×C₄	1	16	yes	direct product factor
29	C₄×C₂₄₁	C₂²×C₄	1	4	yes	complemented
30	Dic₂₄₁	C₂²×C₄	1	4	yes	complemented
31	C₂×C₂₄₁	C₂×C₄²	1	6	yes	complemented
32	C₂×C₂₄₁	C₂³×C₄	1	1	yes	characteristic
33	D₂₄₁	C₂×C₄²	1	6	yes	complemented
34	D₂₄₁	C₂³×C₄	1	2	yes	—
35	C₂₄₁	C₂²×C₄²	1	1	yes	characteristic, complemented, Sylow
36	C₂²×C₄²	—	241	1	no	maximal, Sylow
37	C₂³×C₄	—	241	2	no	—
38	C₂³×C₄	—	241	1	no	—
39	C₂×C₄²	—	241	12	no	—
40	C₂⁴	—	241	1	no	—
41	C₂²×C₄	—	241	12	no	—
42	C₂²×C₄	—	241	16	no	—
43	C₂²×C₄	—	241	1	no	—
44	C₂²×C₄	—	241	6	no	—
45	C₂²×C₄	—	241	6	no	—
46	C₂²×C₄	#11757210	1	1	yes	characteristic, direct product factor
47	C₄²	—	241	16	no	—
48	C₂³	—	241	8	no	—
49	C₂³	—	241	3	no	—
50	C₂³	—	241	3	no	—
51	C₂³	C₂×#11757210	1	1	yes	characteristic
52	C₂×C₄	—	241	8	no	—
53	C₂×C₄	—	241	48	no	—
54	C₂×C₄	—	241	6	no	—
55	C₂×C₄	—	241	4	no	—
56	C₂×C₄	—	241	12	no	—
57	C₂×C₄	C₂×#11757210	1	6	yes	direct product factor
58	C₂²	—	241	12	no	—
59	C₂²	—	241	12	no	—
60	C₂²	—	241	1	no	—
61	C₂²	—	241	3	no	—
62	C₂²	C₄×#11757210	1	4	yes	direct product factor
63	C₂²	C₂²×#11757210	1	3	yes	—
64	C₄	—	241	16	no	—
65	C₄	—	241	4	no	—
66	C₄	C₂²×#11757210	1	4	yes	direct product factor
67	C₂	—	241	2	no	—
68	C₂	—	241	6	no	—
69	C₂	C₂×C₄×#11757210	1	6	yes	direct product factor
70	C₂	C₂³×#11757210	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	#11757210
commutator, hypocenter	C₂₄₁	C₂²×C₄²
Frattini	C₂	C₂³×#11757210
Fitting	C₂²×C₄×C₂₄₁	C₄
socle	C₂³×C₂₄₁	C₂×C₄
prime radical	C₂³×D₂₄₁	C₂²
cosocle	C₂×D₂₄₁	C₂⁴
LCS	G ▷ C₂₄₁
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#11757210	C₂	direct (N × H)
4. C₂²×#11757210	C₄	direct (N × H)
6. C₄×#11757210	C₂²	direct (N × H)
8. C₂²×Dic₂₄₁	C₄	semidirect
9. C₂×C₄×D₂₄₁	C₄	semidirect
11. C₂²×C₄×C₂₄₁	C₄	semidirect
12. C₂×#11757210	C₂×C₄	direct (N × H)
17. C₂×Dic₂₄₁	C₂×C₄	semidirect
18. C₄×D₂₄₁	C₂×C₄	semidirect
21. C₂×C₄×C₂₄₁	C₂×C₄	semidirect
22. C₂²×C₂₄₁	C₄²	semidirect
24. C₂×D₂₄₁	C₄²	semidirect
28. #11757210	C₂²×C₄	direct (N × H)
29. C₄×C₂₄₁	C₂²×C₄	semidirect
30. Dic₂₄₁	C₂²×C₄	semidirect
31. C₂×C₂₄₁	C₂×C₄²	semidirect
33. D₂₄₁	C₂×C₄²	semidirect
35. C₂₄₁	C₂²×C₄²	semidirect
46. C₂²×C₄	#11757210	direct (N × H)
57. C₂×C₄	C₂×#11757210	direct (N × H)
62. C₂²	C₄×#11757210	direct (N × H)
66. C₄	C₂²×#11757210	direct (N × H)
69. C₂	C₂×C₄×#11757210	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11757210	2
C₂×C₄×#11757210	2
C₂²×C₄×D₂₄₁	2
C₂²×C₄²	241

Subgroups — C22×C4×#11757210 #29508777

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#11757210 #29508777