Subgroups — #43243655

Order 28096.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₄₃₉	C₂	1	1	yes	maximal, characteristic
2	Dic₄₃₉⋊C₈	C₂	1	2	yes	maximal, complemented
3	(C₂×D₄₃₉)⋊C₈	C₂	1	2	yes	maximal
4	C₄×C₄₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₈×C₄₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄₃₉	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄₃₉	C₄	1	1	yes	characteristic
8	C₂×C₄×D₄₃₉	C₂²	1	1	yes	characteristic
9	C₂×C₄₃₉⋊C₈	C₂²	1	2	yes	—
10	C₄×Dic₄₃₉	C₄	1	1	yes	characteristic
11	C₄×Dic₄₃₉	C₄	1	1	yes	characteristic
12	C₄×Dic₄₃₉	C₂²	1	1	yes	characteristic
13	C₂×C₈×C₄₃₉	C₂²	1	2	yes	—
14	C₄²×C₄₃₉	C₂²	1	1	yes	characteristic
15	C₂²×D₄₃₉	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₄₃₉	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₄₃₉	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₄₃₉	C₂×C₄	1	1	yes	characteristic
19	C₄×D₄₃₉	—	2	2	no	—
20	C₄×D₄₃₉	—	2	2	no	—
21	C₄×D₄₃₉	C₈	1	4	yes	complemented
22	C₄₃₉⋊C₈	—	2	2	no	—
23	C₂×C₄×C₄₃₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₄₃₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₄₃₉	C₂³	1	1	yes	characteristic
26	C₈×C₄₃₉	—	2	2	no	—
27	C₂²×C₄₃₉	C₂²×C₄	1	1	yes	characteristic
28	C₂×D₄₃₉	—	2	1	no	—
29	C₂×D₄₃₉	—	2	1	no	—
30	C₂×D₄₃₉	C₂×C₈	1	2	yes	—
31	C₄×C₄₃₉	C₂×C₈	1	2	yes	—
32	C₄×C₄₃₉	M₁₆	1	2	yes	—
33	C₄×C₄₃₉	#41	1	2	yes	—
34	Dic₄₃₉	—	2	1	no	—
35	Dic₄₃₉	—	2	1	no	—
36	Dic₄₃₉	C₂×C₈	1	2	yes	—
37	C₂×C₄₃₉	#117	1	1	yes	characteristic
38	C₂×C₄₃₉	C₂²×C₈	1	1	yes	characteristic
39	C₂×C₄₃₉	C₂×M₁₆	1	1	yes	characteristic
40	D₄₃₉	—	2	2	no	—
41	C₄₃₉	#431	1	1	yes	characteristic, complemented, Sylow
42	#431	—	439	1	no	maximal, Sylow
43	C₂×C₄²	—	439	1	no	—
44	C₄⋊C₈	—	439	2	no	—
45	C₂²⋊C₈	—	439	2	no	—
46	C₄×C₈	—	439	1	no	—
47	C₄×C₈	D₄₃₉	1	1	yes	characteristic, complemented
48	C₂²×C₄	—	439	1	no	—
49	C₂²×C₄	—	439	1	no	—
50	C₂²×C₄	—	439	1	no	—
51	C₂×C₈	—	439	2	no	—
52	C₂×C₈	C₂×D₄₃₉	1	2	yes	—
53	C₄²	—	439	1	no	—
54	C₄²	—	439	1	no	—
55	C₄²	—	439	1	no	—
56	C₄²	C₂×D₄₃₉	1	1	yes	characteristic
57	C₂³	—	439	1	no	—
58	C₂×C₄	—	878	2	no	—
59	C₂×C₄	—	878	2	no	—
60	C₂×C₄	—	439	1	no	—
61	C₂×C₄	—	439	1	no	—
62	C₂×C₄	—	439	1	no	—
63	C₂×C₄	—	439	4	no	—
64	C₂×C₄	C₄×D₄₃₉	1	1	yes	characteristic
65	C₂×C₄	C₄×D₄₃₉	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₄₃₉	1	1	yes	characteristic
67	C₈	—	2	2	no	—
68	C₈	—	878	2	no	—
69	C₂²	—	878	1	no	—
70	C₂²	—	878	1	no	—
71	C₂²	—	439	2	no	—
72	C₂²	C₂×C₄×D₄₃₉	1	1	yes	characteristic
73	C₄	—	878	1	no	—
74	C₄	—	878	1	no	—
75	C₄	—	439	2	no	—
76	C₄	C₈×D₄₃₉	1	2	yes	—
77	C₄	#20296312	1	2	yes	—
78	C₄	#20296329	1	2	yes	—
79	C₂	—	878	2	no	—
80	C₂	#28380700	1	1	yes	characteristic
81	C₂	C₂×C₈×D₄₃₉	1	1	yes	characteristic
82	C₂	C₂×#20296312	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₄₃₉
commutator	C₂×C₄₃₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×D₄₃₉
Fitting	C₄×C₈×C₄₃₉	C₂
socle	C₂²×C₄₃₉	C₂²×C₄
prime radical	C₂²×D₄₃₉	C₂×C₄
cosocle	C₂×C₄×C₄₃₉	C₂³
hypercenter	C₄×C₈	D₄₃₉
hypocenter	C₄₃₉	#431
LCS	G ▷ C₂×C₄₃₉ ▷ C₄₃₉
UCS	C₁ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. Dic₄₃₉⋊C₈	C₂	semidirect
4. C₄×C₄₃₉⋊C₈	C₂	semidirect
5. C₄×C₈×C₄₃₉	C₂	semidirect
21. C₄×D₄₃₉	C₈	semidirect
41. C₄₃₉	#431	semidirect
47. C₄×C₈	D₄₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₄²×D₄₃₉	2
Dic₄₃₉⋊C₈	2
(C₂×D₄₃₉)⋊C₈	2
C₄×C₄₃₉⋊C₈	2
C₄×C₈×C₄₃₉	2
#431	439