Subgroups — #43243366

Order 28096.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₄₃₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×C₄₃₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₄₃₉	—	2	1	no	—
5	C₂×C₄₃₉⋊C₈	—	2	1	no	—
6	C₂×C₄₃₉⋊C₈	—	2	1	no	—
7	C₂×C₄₃₉⋊C₈	C₄	1	2	yes	complemented
8	C₄⋊Dic₄₃₉	—	2	1	no	—
9	C₄⋊Dic₄₃₉	C₄	1	2	yes	complemented
10	C₂²×C₄×C₄₃₉	—	2	1	no	—
11	C₂²×C₄×C₄₃₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₄₃₉	—	2	1	no	—
13	C₄⋊C₄×C₄₃₉	C₄	1	2	yes	complemented
14	C₂×Dic₄₃₉	—	2	2	no	—
15	C₂×Dic₄₃₉	—	4	1	no	—
16	C₂×Dic₄₃₉	—	4	1	no	—
17	C₄₃₉⋊C₈	—	2	2	no	—
18	C₂³×C₄₃₉	D₄	1	1	yes	characteristic
19	C₂×C₄×C₄₃₉	—	2	2	no	—
20	C₂×C₄×C₄₃₉	—	4	1	no	—
21	C₂×C₄×C₄₃₉	—	4	1	no	—
22	C₂×C₄×C₄₃₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₄₃₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₄₃₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₄₃₉	D₄	1	1	yes	characteristic
26	C₂×C₄×C₄₃₉	D₄	1	1	yes	characteristic
27	C₂×C₄×C₄₃₉	Q₈	1	1	yes	characteristic
28	C₂²×C₄₃₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₄₃₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄₃₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄₃₉	D₈	1	1	yes	characteristic
32	C₂²×C₄₃₉	SD₁₆	1	1	yes	characteristic
33	C₂²×C₄₃₉	SD₁₆	1	1	yes	characteristic
34	C₂²×C₄₃₉	Q₁₆	1	1	yes	characteristic
35	C₄×C₄₃₉	—	4	2	no	—
36	C₄×C₄₃₉	C₄²	1	1	yes	characteristic
37	C₄×C₄₃₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×C₄₃₉	C₄⋊C₄	1	1	yes	characteristic
39	C₄×C₄₃₉	C₄⋊C₄	1	1	yes	characteristic
40	Dic₄₃₉	—	4	2	no	—
41	C₂×C₄₃₉	#95	1	1	yes	characteristic
42	C₂×C₄₃₉	#102	1	1	yes	characteristic
43	C₂×C₄₃₉	#102	1	1	yes	characteristic
44	C₂×C₄₃₉	#103	1	1	yes	characteristic
45	C₂×C₄₃₉	#103	1	1	yes	characteristic
46	C₂×C₄₃₉	#106	1	1	yes	characteristic
47	C₂×C₄₃₉	#107	1	1	yes	characteristic
48	C₄₃₉	#340	1	1	yes	characteristic, complemented, Sylow
49	#340	—	439	1	no	maximal, Sylow
50	C₂²×C₈	—	439	1	no	—
51	C₂×C₄⋊C₄	—	439	1	no	—
52	C₂×C₄⋊C₄	D₄₃₉	1	1	yes	characteristic
53	C₂²×C₄	—	2	1	no	—
54	C₂²×C₄	—	878	1	no	—
55	C₂²×C₄	C₂×D₄₃₉	1	1	yes	characteristic
56	C₂×C₈	—	878	1	no	—
57	C₂×C₈	—	878	1	no	—
58	C₂×C₈	—	439	2	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	878	1	no	—
61	C₄⋊C₄	—	439	2	no	—
62	C₄⋊C₄	Dic₄₃₉	1	2	yes	complemented
63	C₂³	C₂²⋊D₄₃₉	1	1	yes	characteristic
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	4	1	no	—
66	C₂×C₄	—	4	1	no	—
67	C₂×C₄	—	878	2	no	—
68	C₂×C₄	—	1756	1	no	—
69	C₂×C₄	—	1756	1	no	—
70	C₂×C₄	C₄×D₄₃₉	1	1	yes	characteristic
71	C₂×C₄	C₄×D₄₃₉	1	1	yes	characteristic
72	C₂×C₄	C₂×Dic₄₃₉	1	1	yes	characteristic
73	C₂×C₄	C₂²⋊D₄₃₉	1	1	yes	characteristic
74	C₂×C₄	D₁₇₅₆	1	1	yes	characteristic
75	C₂×C₄	Dic₈₇₈	1	1	yes	characteristic
76	C₈	—	878	2	no	—
77	C₂²	(C₂×D₄₃₉)⋊C₄	1	1	yes	characteristic
78	C₂²	C₂²⋊Dic₄₃₉	1	1	yes	characteristic
79	C₂²	Dic₄₃₉⋊C₄	1	1	yes	characteristic
80	C₂²	#20296314	1	1	yes	characteristic
81	C₂²	#20296316	1	1	yes	characteristic
82	C₂²	#20296317	1	1	yes	characteristic
83	C₂²	#20296319	1	1	yes	characteristic
84	C₄	—	4	2	no	—
85	C₄	—	1756	2	no	—
86	C₄	C₄×Dic₄₃₉	1	1	yes	characteristic
87	C₄	(C₂×D₄₃₉)⋊C₄	1	1	yes	characteristic
88	C₄	Dic₄₃₉⋊C₄	1	1	yes	characteristic
89	C₄	C₄⋊Dic₄₃₉	1	1	yes	characteristic
90	C₂	C₄₃₉⋊#95	1	1	yes	characteristic
91	C₂	#28380663	1	1	yes	characteristic
92	C₂	#28380664	1	1	yes	characteristic
93	C₂	#28380666	1	1	yes	characteristic
94	C₂	#28380667	1	1	yes	characteristic
95	C₂	#28380674	1	1	yes	characteristic
96	C₂	#28380676	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₄₃₉
commutator	C₄×C₄₃₉	C₄²
Frattini	C₂²×C₄	C₂×D₄₃₉
Fitting	C₂×C₄⋊C₄×C₄₃₉	C₂
socle, prime radical	C₂³×C₄₃₉	D₄
cosocle	C₂²×C₄×C₄₃₉	C₂²
hypercenter	C₂×C₄⋊C₄	D₄₃₉
hypocenter	C₄₃₉	#340
LCS	G ▷ C₄×C₄₃₉ ▷ C₂×C₄₃₉ ▷ C₄₃₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. C₂×C₄₃₉⋊C₈	C₄	semidirect
9. C₄⋊Dic₄₃₉	C₄	semidirect
13. C₄⋊C₄×C₄₃₉	C₄	semidirect
48. C₄₃₉	#340	semidirect
62. C₄⋊C₄	Dic₄₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄₃₉⋊C₈	2
C₂×C₄⋊Dic₄₃₉	2
C₂×C₄⋊C₄×C₄₃₉	2
#340	439