Subgroups — C₄×#20297720 #43271960

Order 28736.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20297720	C₂	1	2	yes	maximal
2	#28385251	C₂	1	2	yes	maximal, complemented
3	C₄×#16813311	C₂	1	2	yes	maximal, complemented
4	C₂×C₄²×C₄₄₉	C₂	1	1	yes	maximal, characteristic
5	#20297720	C₄	1	8	yes	direct product factor
6	C₂×#16813311	C₂²	1	4	yes	—
7	C₂²×C₄×C₄₄₉	C₄	1	2	yes	—
8	C₂²×C₄×C₄₄₉	C₂²	1	1	yes	characteristic
9	C₄²×C₄₄₉	C₄	1	2	yes	—
10	C₄²×C₄₄₉	C₂²	1	1	yes	characteristic
11	C₄²×C₄₄₉	C₂²	1	1	yes	characteristic
12	#16813311	C₂×C₄	1	8	yes	complemented
13	C₂³×C₄₄₉	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₄₄₉	—	2	4	no	—
15	C₂×C₄×C₄₄₉	C₂×C₄	1	4	yes	—
16	C₂×C₄×C₄₄₉	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₄₄₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₄₄₉	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₄₄₉	C₂³	1	1	yes	characteristic
20	C₂²×C₄₄₉	—	2	2	no	—
21	C₂²×C₄₄₉	C₄²	1	2	yes	—
22	C₂²×C₄₄₉	C₂²×C₄	1	1	yes	characteristic
23	C₄×C₄₄₉	—	2	2	no	—
24	C₄×C₄₄₉	C₄²	1	2	yes	—
25	C₄×C₄₄₉	M₁₆	1	4	yes	complemented
26	C₄×C₄₄₉	C₂²×C₄	1	2	yes	—
27	C₂×C₄₄₉	—	2	2	no	—
28	C₂×C₄₄₉	C₂×C₄²	1	1	yes	characteristic
29	C₂×C₄₄₉	C₂×M₁₆	1	2	yes	—
30	C₄₄₉	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
31	C₄×M₁₆	—	449	1	no	maximal, Sylow
32	C₂×M₁₆	—	449	2	no	—
33	C₂×C₄²	D₄₄₉	1	1	yes	characteristic
34	#97	—	449	2	no	—
35	C₄×C₈	—	449	2	no	—
36	C₂²×C₄	Dic₄₄₉	1	2	yes	—
37	C₂²×C₄	C₂×D₄₄₉	1	1	yes	characteristic
38	M₁₆	—	449	8	no	—
39	C₂×C₈	—	449	4	no	—
40	C₄²	Dic₄₄₉	1	2	yes	—
41	C₄²	C₂×D₄₄₉	1	1	yes	characteristic
42	C₄²	C₂×D₄₄₉	1	1	yes	characteristic
43	C₂³	C₂×Dic₄₄₉	1	1	yes	characteristic
44	C₂×C₄	—	2	4	no	—
45	C₂×C₄	C₄×D₄₄₉	1	4	yes	—
46	C₂×C₄	C₂×Dic₄₄₉	1	2	yes	—
47	C₂×C₄	C₂×Dic₄₄₉	1	1	yes	characteristic
48	C₂×C₄	C₂×Dic₄₄₉	1	2	yes	—
49	C₂×C₄	C₂²×D₄₄₉	1	1	yes	characteristic
50	C₈	—	449	8	no	—
51	C₂²	—	2	2	no	—
52	C₂²	C₄×Dic₄₄₉	1	2	yes	—
53	C₂²	C₂²×Dic₄₄₉	1	1	yes	characteristic
54	C₄	—	2	2	no	—
55	C₄	C₄×Dic₄₄₉	1	2	yes	—
56	C₄	#20297720	1	4	yes	direct product factor
57	C₄	C₂×C₄×D₄₄₉	1	2	yes	—
58	C₂	—	2	2	no	—
59	C₂	C₂×C₄×Dic₄₄₉	1	1	yes	characteristic
60	C₂	C₂×#20297720	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₄₄₉
commutator	C₂×C₄₄₉	C₂×C₄²
Frattini	C₂×C₄	C₂²×D₄₄₉
Fitting	C₂×C₄²×C₄₄₉	C₂
socle	C₂²×C₄₄₉	C₂²×C₄
prime radical	C₂³×C₄₄₉	C₂×C₄
cosocle	C₂×C₄×C₄₄₉	C₂³
hypercenter	C₂×C₄²	D₄₄₉
hypocenter	C₄₄₉	C₄×M₁₆
LCS	G ▷ C₂×C₄₄₉ ▷ C₄₄₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. #28385251	C₂	semidirect
3. C₄×#16813311	C₂	semidirect
5. #20297720	C₄	direct (N × H)
12. #16813311	C₂×C₄	semidirect
25. C₄×C₄₄₉	M₁₆	semidirect
30. C₄₄₉	C₄×M₁₆	semidirect
56. C₄	#20297720	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×#20297720	2
#28385251	2
C₄×#16813311	2
C₂×C₄²×C₄₄₉	2
C₄×M₁₆	449

Subgroups — C4×#20297720 #43271960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#20297720 #43271960