Subgroups — #43243410

Order 28096.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28380765	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₃₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#108×C₄₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#20296329	—	2	1	no	—
5	Dic₁₇₅₆	C₄	1	1	yes	characteristic
6	#20296318	—	2	1	no	—
7	D₃₅₁₂	C₄	1	1	yes	characteristic
8	C₄₃₉⋊C₁₆	—	2	1	no	—
9	M₁₆×C₄₃₉	—	2	1	no	—
10	C₂×C₈×C₄₃₉	C₂²	1	1	yes	characteristic
11	Dic₈₇₈	—	2	1	no	—
12	D₁₇₅₆	—	2	1	no	—
13	C₂²⋊D₄₃₉	—	4	1	no	—
14	C₄×D₄₃₉	—	4	1	no	—
15	C₂×C₄×C₄₃₉	D₄	1	1	yes	characteristic
16	C₈×C₄₃₉	—	4	1	no	—
17	C₈×C₄₃₉	C₂×C₄	1	1	yes	characteristic
18	C₈×C₄₃₉	D₄	1	1	yes	characteristic
19	C₂²×C₄₃₉	SD₁₆	1	1	yes	characteristic
20	C₂×D₄₃₉	—	4	1	no	—
21	C₄×C₄₃₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₄₃₉	D₈	1	1	yes	characteristic
23	Dic₄₃₉	—	4	1	no	—
24	C₂×C₄₃₉	—	2	1	no	—
25	C₂×C₄₃₉	#102	1	1	yes	characteristic
26	D₄₃₉	—	8	1	no	—
27	C₄₃₉	#359	1	1	yes	characteristic, complemented, Sylow
28	#359	—	439	1	no	maximal, Sylow
29	#135	—	439	1	no	—
30	C₂×C₁₆	—	439	1	no	—
31	#108	D₄₃₉	1	1	yes	characteristic, complemented
32	#41	—	878	1	no	—
33	Q₁₆	—	439	1	no	—
34	SD₁₆	—	878	1	no	—
35	D₈	—	439	1	no	—
36	M₁₆	—	2	1	no	—
37	C₂×C₈	C₂×D₄₃₉	1	1	yes	characteristic
38	C₁₆	—	878	1	no	—
39	Q₈	—	878	1	no	—
40	D₄	—	878	1	no	—
41	D₄	—	1756	1	no	—
42	C₂×C₄	—	1756	1	no	—
43	C₂×C₄	C₂²⋊D₄₃₉	1	1	yes	characteristic
44	C₈	—	4	1	no	—
45	C₈	C₄×D₄₃₉	1	1	yes	characteristic
46	C₈	D₁₇₅₆	1	1	yes	characteristic
47	C₂²	—	1756	1	no	—
48	C₂²	#20296316	1	1	yes	characteristic
49	C₄	—	1756	1	no	—
50	C₄	(C₂×D₄₃₉)⋊C₄	1	1	yes	characteristic
51	C₄	#20296314	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	3512	1	no	—
54	C₂	#28380663	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#20296314
commutator	C₈×C₄₃₉	C₂×C₄
Frattini	C₂×C₈	C₂×D₄₃₉
Fitting	#108×C₄₃₉	C₂
socle	C₂×C₄₃₉	#102
prime radical	#28380765	C₂
cosocle	C₂×C₈×C₄₃₉	C₂²
hypercenter	#108	D₄₃₉
hypocenter	C₄₃₉	#359
LCS	G ▷ C₈×C₄₃₉ ▷ C₄×C₄₃₉ ▷ C₂×C₄₃₉ ▷ C₄₃₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₃₉⋊C₁₆	C₂	semidirect
3. #108×C₄₃₉	C₂	semidirect
27. C₄₃₉	#359	semidirect
31. #108	D₄₃₉	semidirect

Maximal subgroups

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

subgroup	index
#28380765	2
C₂×C₄₃₉⋊C₁₆	2
#108×C₄₃₉	2
#359	439