Subgroups — #43243647

Order 28096.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20296313	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₄₃₉	C₂	1	1	yes	maximal, characteristic
3	#28380676	C₂	1	2	yes	maximal, complemented
4	#28380674	C₂	1	2	yes	maximal, complemented
5	#117×C₄₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₃₉	—	2	1	no	—
7	#20296313	C₄	1	4	yes	complemented
8	C₂×C₄₃₉⋊C₈	C₂²	1	2	yes	—
9	C₄⋊Dic₄₃₉	—	2	1	no	—
10	C₄⋊Dic₄₃₉	C₂²	1	2	yes	—
11	C₂²×C₄×C₄₃₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₄₃₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₄₃₉	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₄₃₉	—	2	1	no	—
15	C₄²×C₄₃₉	—	2	1	no	—
16	C₂×Dic₄₃₉	—	2	2	no	—
17	C₂×Dic₄₃₉	—	4	2	no	—
18	C₄₃₉⋊C₈	C₂×C₄	1	4	yes	complemented
19	C₂³×C₄₃₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₄₃₉	—	2	1	no	—
21	C₂×C₄×C₄₃₉	—	2	1	no	—
22	C₂×C₄×C₄₃₉	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₄₃₉	D₄	1	1	yes	characteristic
24	C₂×C₄×C₄₃₉	Q₈	1	2	yes	—
25	C₂×C₄×C₄₃₉	C₂³	1	1	yes	characteristic
26	C₂²×C₄₃₉	—	2	1	no	—
27	C₂²×C₄₃₉	—	2	1	no	—
28	C₂²×C₄₃₉	C₄⋊C₄	1	2	yes	—
29	C₂²×C₄₃₉	C₂×D₄	1	1	yes	characteristic
30	C₄×C₄₃₉	—	4	1	no	—
31	C₄×C₄₃₉	—	4	1	no	—
32	C₄×C₄₃₉	C₄⋊C₄	1	2	yes	—
33	C₄×C₄₃₉	C₂²×C₄	1	1	yes	characteristic
34	C₄×C₄₃₉	C₂×Q₈	1	1	yes	characteristic
35	Dic₄₃₉	—	4	2	no	—
36	C₂×C₄₃₉	—	2	2	no	—
37	C₂×C₄₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
39	C₂×C₄₃₉	#137	1	1	yes	characteristic
40	C₄₃₉	#428	1	1	yes	characteristic, complemented, Sylow
41	#428	—	439	1	no	maximal, Sylow
42	C₂×M₁₆	—	439	1	no	—
43	#117	D₄₃₉	1	1	yes	characteristic
44	C₂×C₄⋊C₄	—	439	1	no	—
45	#107	—	439	2	no	—
46	#106	—	439	2	no	—
47	C₂²×C₄	—	878	1	no	—
48	C₂²×C₄	C₂×D₄₃₉	1	1	yes	characteristic
49	M₁₆	—	439	4	no	—
50	C₂×C₈	—	439	2	no	—
51	C₄⋊C₄	—	878	1	no	—
52	C₄⋊C₄	—	439	2	no	—
53	C₄⋊C₄	C₂×D₄₃₉	1	1	yes	characteristic
54	C₄⋊C₄	C₂×D₄₃₉	1	1	yes	characteristic
55	C₂²⋊C₄	—	2	1	no	—
56	C₄²	—	2	1	no	—
57	C₂³	C₂²⋊D₄₃₉	1	1	yes	characteristic
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	878	2	no	—
61	C₂×C₄	—	1756	2	no	—
62	C₂×C₄	C₄×D₄₃₉	1	2	yes	—
63	C₂×C₄	C₂²⋊D₄₃₉	1	1	yes	characteristic
64	C₂×C₄	Dic₈₇₈	1	2	yes	—
65	C₂×C₄	C₂²×D₄₃₉	1	1	yes	characteristic
66	C₈	—	439	4	no	—
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	Dic₄₃₉⋊C₄	1	2	yes	—
70	C₂²	C₂×C₂²⋊D₄₃₉	1	1	yes	characteristic
71	C₄	—	4	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	1756	2	no	—
74	C₄	Dic₄₃₉⋊C₄	1	2	yes	—
75	C₄	C₂×C₄×D₄₃₉	1	1	yes	characteristic
76	C₄	C₂×Dic₈₇₈	1	1	yes	characteristic
77	C₂	—	2	2	no	—
78	C₂	C₂×Dic₄₃₉⋊C₄	1	1	yes	characteristic
79	C₂	#28380768	1	1	yes	characteristic
80	C₂	#28380773	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₃₉
commutator	C₄×C₄₃₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₃₉
Fitting	#117×C₄₃₉	C₂
socle	C₂²×C₄₃₉	C₂×D₄
prime radical	C₂³×C₄₃₉	D₄
cosocle	C₂×C₄×C₄₃₉	C₂³
hypercenter	#117	D₄₃₉
hypocenter	C₄₃₉	#428
LCS	G ▷ C₄×C₄₃₉ ▷ C₂×C₄₃₉ ▷ C₄₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. #28380676	C₂	semidirect
4. #28380674	C₂	semidirect
7. #20296313	C₄	semidirect
18. C₄₃₉⋊C₈	C₂×C₄	semidirect
40. C₄₃₉	#428	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20296313	2
C₂×C₄⋊Dic₄₃₉	2
#28380676	2
#28380674	2
#117×C₄₃₉	2
#428	439