Subgroups — C₂²×#20297744 #43273078

Order 28736.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20297744	C₂	1	12	yes	maximal, direct product factor
2	C₂²×#16813318	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄×D₄₄₉	C₂	1	1	yes	maximal, characteristic
4	#20297744	C₂²	1	16	yes	direct product factor
5	C₂×#16813318	C₂²	1	12	yes	complemented
6	C₂³×D₄₄₉	C₄	1	1	yes	characteristic
7	C₂²×Dic₄₄₉	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄₄₉	C₄	1	6	yes	—
9	C₂×C₄×D₄₄₉	C₂²	1	6	yes	—
10	C₂²×C₄×C₄₄₉	C₄	1	1	yes	characteristic
11	#16813318	C₂³	1	8	yes	complemented
12	C₂²×D₄₄₉	C₈	1	8	yes	complemented
13	C₂²×D₄₄₉	C₂×C₄	1	6	yes	—
14	C₂×Dic₄₄₉	C₂×C₄	1	3	yes	—
15	C₂×Dic₄₄₉	C₂³	1	3	yes	—
16	C₄×D₄₄₉	C₂×C₄	1	12	yes	—
17	C₄×D₄₄₉	C₂³	1	4	yes	—
18	C₂³×C₄₄₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₄₄₉	C₂×C₄	1	6	yes	—
20	C₂²×C₄₄₉	C₂×C₈	1	4	yes	complemented
21	C₂²×C₄₄₉	C₂²×C₄	1	3	yes	—
22	C₂×D₄₄₉	C₂×C₈	1	24	yes	complemented
23	C₂×D₄₄₉	C₂²×C₄	1	4	yes	—
24	C₄×C₄₄₉	C₂²×C₄	1	4	yes	—
25	Dic₄₄₉	C₂²×C₄	1	3	yes	—
26	Dic₄₄₉	C₂⁴	1	1	yes	characteristic
27	C₂×C₄₄₉	C₂²×C₈	1	6	yes	complemented
28	C₂×C₄₄₉	C₂³×C₄	1	1	yes	characteristic
29	D₄₄₉	C₂²×C₈	1	8	yes	complemented
30	C₄₄₉	C₂³×C₈	1	1	yes	characteristic, complemented, Sylow
31	C₂³×C₈	—	449	1	no	maximal, Sylow
32	C₂³×C₄	—	449	1	no	—
33	C₂²×C₈	—	449	2	no	—
34	C₂²×C₈	—	449	12	no	—
35	C₂⁴	—	449	1	no	—
36	C₂²×C₄	—	449	1	no	—
37	C₂²×C₄	—	449	6	no	—
38	C₂²×C₄	—	449	6	no	—
39	C₂²×C₄	#14267798	1	1	yes	characteristic
40	C₂×C₈	—	449	12	no	—
41	C₂×C₈	—	449	16	no	—
42	C₂³	—	449	8	no	—
43	C₂³	—	449	6	no	—
44	C₂³	C₂×#14267798	1	1	yes	characteristic
45	C₂×C₄	—	449	3	no	—
46	C₂×C₄	—	449	3	no	—
47	C₂×C₄	—	449	12	no	—
48	C₂×C₄	—	449	4	no	—
49	C₂×C₄	C₂×#14267798	1	6	yes	—
50	C₈	—	449	8	no	—
51	C₂²	—	449	24	no	—
52	C₂²	—	449	4	no	—
53	C₂²	#20297744	1	4	yes	direct product factor
54	C₂²	C₂²×#14267798	1	3	yes	—
55	C₄	—	449	1	no	—
56	C₄	—	449	3	no	—
57	C₄	C₂²×#14267798	1	4	yes	—
58	C₂	—	449	8	no	—
59	C₂	C₂×#20297744	1	6	yes	direct product factor
60	C₂	C₂³×#14267798	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	#14267798
commutator, hypocenter	C₄₄₉	C₂³×C₈
Frattini	C₂	C₂³×#14267798
Fitting	C₂²×C₄×C₄₄₉	C₄
socle	C₂³×C₄₄₉	C₂×C₄
prime radical	C₂³×D₄₄₉	C₄
cosocle	Dic₄₄₉	C₂⁴
LCS	G ▷ C₄₄₉
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#20297744	C₂	direct (N × H)
2. C₂²×#16813318	C₂	semidirect
4. #20297744	C₂²	direct (N × H)
5. C₂×#16813318	C₂²	semidirect
11. #16813318	C₂³	semidirect
12. C₂²×D₄₄₉	C₈	semidirect
20. C₂²×C₄₄₉	C₂×C₈	semidirect
22. C₂×D₄₄₉	C₂×C₈	semidirect
27. C₂×C₄₄₉	C₂²×C₈	semidirect
29. D₄₄₉	C₂²×C₈	semidirect
30. C₄₄₉	C₂³×C₈	semidirect
53. C₂²	#20297744	direct (N × H)
59. C₂	C₂×#20297744	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#20297744	2
C₂²×#16813318	2
C₂²×C₄×D₄₄₉	2
C₂³×C₈	449

Subgroups — C22×#20297744 #43273078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#20297744 #43273078