Subgroups — #43243897

Order 28096.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28380742	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₄₃₉	C₂	1	1	yes	maximal, characteristic
3	#28380677	C₂	1	1	yes	maximal, characteristic
4	#28380676	C₂	1	1	yes	maximal, characteristic
5	Dic₄₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#28380667	C₂	1	1	yes	maximal, characteristic
7	#103×C₄₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₈₇₈	—	2	1	no	—
9	C₂×C₄₃₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₄₃₉	—	2	1	no	—
11	C₄⋊Dic₄₃₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₄₃₉	C₂²	1	1	yes	characteristic
13	Dic₄₃₉⋊C₄	—	2	1	no	—
14	C₄×Dic₄₃₉	—	2	1	no	—
15	C₄×Dic₄₃₉	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₄₃₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₄₃₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₄₃₉	C₂²	1	1	yes	characteristic
19	Dic₈₇₈	—	4	2	no	—
20	C₂×Dic₄₃₉	—	2	1	no	—
21	C₂×Dic₄₃₉	—	2	1	no	—
22	C₂×Dic₄₃₉	D₄	1	1	yes	characteristic
23	C₂×Dic₄₃₉	D₄	1	1	yes	characteristic
24	C₄₃₉⋊C₈	—	2	1	no	—
25	Q₈×C₄₃₉	—	2	1	no	—
26	Q₈×C₄₃₉	Q₈	1	2	yes	complemented
27	C₂×C₄×C₄₃₉	—	2	1	no	—
28	C₂×C₄×C₄₃₉	—	2	1	no	—
29	C₂×C₄×C₄₃₉	C₂³	1	1	yes	characteristic
30	C₈×C₄₃₉	—	2	1	no	—
31	C₂²×C₄₃₉	C₂×D₄	1	1	yes	characteristic
32	C₄×C₄₃₉	—	2	2	no	—
33	C₄×C₄₃₉	—	4	1	no	—
34	C₄×C₄₃₉	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₄₃₉	#41	1	1	yes	characteristic
36	Dic₄₃₉	—	2	1	no	—
37	Dic₄₃₉	—	4	1	no	—
38	Dic₄₃₉	—	4	1	no	—
39	Dic₄₃₉	Q₁₆	1	2	yes	—
40	C₂×C₄₃₉	C₂²⋊Q₈	1	1	yes	characteristic
41	C₂×C₄₃₉	C₂×Q₁₆	1	1	yes	characteristic
42	C₂×C₄₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
43	C₄₃₉	#477	1	1	yes	characteristic, complemented, Sylow
44	#477	—	439	1	no	maximal, Sylow
45	C₄⋊Q₈	—	439	1	no	—
46	C₄×Q₈	—	439	1	no	—
47	#107	—	439	1	no	—
48	#107	—	439	1	no	—
49	C₄⋊C₈	—	439	1	no	—
50	#103	—	439	1	no	—
51	#103	D₄₃₉	1	1	yes	characteristic
52	C₂×Q₈	—	878	1	no	—
53	C₂×Q₈	C₂×D₄₃₉	1	1	yes	characteristic
54	C₂×C₈	—	439	1	no	—
55	C₂×C₈	C₂×D₄₃₉	1	1	yes	characteristic
56	C₄⋊C₄	—	878	1	no	—
57	C₄⋊C₄	—	878	1	no	—
58	C₄⋊C₄	—	439	1	no	—
59	C₄⋊C₄	—	439	1	no	—
60	C₄⋊C₄	C₂×D₄₃₉	1	1	yes	characteristic
61	C₄²	—	878	1	no	—
62	C₄²	—	439	1	no	—
63	Q₈	—	2	1	no	—
64	Q₈	—	1756	2	no	—
65	Q₈	Dic₈₇₈	1	2	yes	complemented
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	878	1	no	—
69	C₂×C₄	—	878	1	no	—
70	C₂×C₄	—	439	1	no	—
71	C₂×C₄	—	439	1	no	—
72	C₂×C₄	C₂²×D₄₃₉	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	878	1	no	—
75	C₂²	D₄×D₄₃₉	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	878	1	no	—
79	C₄	—	439	2	no	—
80	C₄	—	1756	1	no	—
81	C₄	—	1756	1	no	—
82	C₄	C₂×Dic₈₇₈	1	1	yes	characteristic
83	C₄	#20296328	1	1	yes	characteristic
84	C₂	#28380720	1	1	yes	characteristic
85	C₂	Q₁₆×D₄₃₉	1	1	yes	characteristic
86	C₂	#28380770	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₃₉
commutator, cosocle	C₂×C₄×C₄₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₄₃₉
Fitting	#103×C₄₃₉	C₂
socle, prime radical	C₂²×C₄₃₉	C₂×D₄
hypercenter	#103	D₄₃₉
hypocenter	C₄₃₉	#477
LCS	G ▷ C₂×C₄×C₄₃₉ ▷ C₂×C₄₃₉ ▷ C₄₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
26. Q₈×C₄₃₉	Q₈	semidirect
43. C₄₃₉	#477	semidirect
65. Q₈	Dic₈₇₈	semidirect

Maximal subgroups

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

subgroup	index
#28380742	2
Q₈×Dic₄₃₉	2
#28380677	2
#28380676	2
Dic₄₃₉⋊C₈	2
#28380667	2
#103×C₄₃₉	2
#477	439