Subgroups — #43247502

Order 28352.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28381892	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₄₄₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#20296787	C₄	1	1	yes	characteristic
5	#20296771	C₄	1	1	yes	characteristic
6	#20296770	—	2	1	no	—
7	C₈×D₄₄₃	—	2	1	no	—
8	C₄₄₃⋊C₁₆	—	2	1	no	—
9	C₂×C₈×C₄₄₃	C₂²	1	1	yes	characteristic
10	C₁₆×C₄₄₃	—	2	1	no	—
11	Dic₈₈₆	C₈	1	1	yes	characteristic
12	D₁₇₇₂	C₈	1	1	yes	characteristic
13	C₂²⋊D₄₄₃	—	2	1	no	—
14	C₄×D₄₄₃	—	2	1	no	—
15	C₄₄₃⋊C₈	—	2	1	no	—
16	C₂×C₄×C₄₄₃	C₂×C₄	1	1	yes	characteristic
17	C₈×C₄₄₃	D₄	1	1	yes	characteristic
18	C₈×C₄₄₃	D₄	1	1	yes	characteristic
19	C₂²×C₄₄₃	M₁₆	1	1	yes	characteristic
20	C₂×D₄₄₃	—	2	1	no	—
21	C₄×C₄₄₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₄₄₃	C₂×C₈	1	1	yes	characteristic
23	Dic₄₄₃	—	2	1	no	—
24	C₂×C₄₄₃	—	2	1	no	—
25	C₂×C₄₄₃	C₂²⋊C₈	1	1	yes	characteristic
26	D₄₄₃	—	4	1	no	—
27	C₄₄₃	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	443	1	no	maximal, Sylow
29	#131	—	443	1	no	—
30	M₃₂	D₄₄₃	1	1	yes	characteristic, complemented
31	C₂×C₁₆	—	443	1	no	—
32	#41	—	443	1	no	—
33	M₁₆	—	886	1	no	—
34	M₁₆	—	443	1	no	—
35	C₂×C₈	—	886	1	no	—
36	C₂×C₈	C₂×D₄₄₃	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	—	886	1	no	—
39	Q₈	—	443	1	no	—
40	D₄	—	886	1	no	—
41	D₄	—	443	1	no	—
42	C₂×C₄	—	886	1	no	—
43	C₂×C₄	C₄×D₄₄₃	1	1	yes	characteristic
44	C₈	—	886	1	no	—
45	C₈	C₂²⋊D₄₄₃	1	1	yes	characteristic
46	C₈	D₁₇₇₂	1	1	yes	characteristic
47	C₂²	—	886	1	no	—
48	C₂²	#20296770	1	1	yes	characteristic
49	C₄	—	886	1	no	—
50	C₄	(C₂×D₄₄₃)⋊C₄	1	1	yes	characteristic
51	C₄	C₈×D₄₄₃	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	1772	1	no	—
54	C₂	(C₂×D₄₄₃)⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊D₄₄₃
commutator	C₄×C₄₄₃	C₂×C₈
Frattini	C₂×C₈	C₂×D₄₄₃
Fitting	M₃₂×C₄₄₃	C₂
socle	C₂×C₄₄₃	C₂²⋊C₈
prime radical	#20296787	C₄
cosocle	C₂×C₈×C₄₄₃	C₂²
hypercenter	M₃₂	D₄₄₃
hypocenter	C₄₄₃	#350
LCS	G ▷ C₄×C₄₄₃ ▷ C₂×C₄₄₃ ▷ C₄₄₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₄₃⋊C₁₆	C₂	semidirect
3. M₃₂×C₄₄₃	C₂	semidirect
27. C₄₄₃	#350	semidirect
30. M₃₂	D₄₄₃	semidirect

Maximal subgroups

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

subgroup	index
#28381892	2
C₂×C₄₄₃⋊C₁₆	2
M₃₂×C₄₄₃	2
#350	443