Subgroups — #43243646

Order 28096.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28380702	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₄₃₉	C₂	1	1	yes	maximal, characteristic
3	#28380677	C₂	1	2	yes	maximal, complemented
4	#28380675	C₂	1	2	yes	maximal, complemented
5	C₂×M₁₆×C₄₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₃₉	—	2	1	no	—
7	C₄⋊Dic₄₃₉	—	2	1	no	—
8	C₄⋊Dic₄₃₉	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₄₃₉	C₂²	1	2	yes	—
10	C₄⋊Dic₄₃₉	C₂²	1	1	yes	characteristic
11	C₂²⋊Dic₄₃₉	—	2	1	no	—
12	C₄×Dic₄₃₉	—	2	1	no	—
13	C₂²×C₄×C₄₃₉	C₂²	1	1	yes	characteristic
14	M₁₆×C₄₃₉	C₄	1	4	yes	complemented
15	C₂×C₈×C₄₃₉	C₂²	1	2	yes	—
16	C₂×Dic₄₃₉	—	2	1	no	—
17	C₂×Dic₄₃₉	—	2	1	no	—
18	C₂×Dic₄₃₉	—	2	2	no	—
19	C₂×Dic₄₃₉	—	4	2	no	—
20	C₂³×C₄₃₉	D₄	1	1	yes	characteristic
21	C₂×C₄×C₄₃₉	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₄₃₉	D₄	1	1	yes	characteristic
23	C₂×C₄×C₄₃₉	Q₈	1	2	yes	—
24	C₂×C₄×C₄₃₉	C₂³	1	1	yes	characteristic
25	C₈×C₄₃₉	C₂×C₄	1	4	yes	complemented
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	C₄⋊C₄	1	2	yes	—
31	C₄×C₄₃₉	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₄₃₉	C₂×Q₈	1	1	yes	characteristic
33	Dic₄₃₉	—	4	1	no	—
34	Dic₄₃₉	—	4	2	no	—
35	Dic₄₃₉	—	4	1	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	D₄₃₉	1	1	yes	characteristic
43	#117	—	439	1	no	—
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	Dic₄₃₉	1	4	yes	complemented
50	C₂×C₈	C₂×D₄₃₉	1	2	yes	—
51	C₄⋊C₄	—	878	1	no	—
52	C₄⋊C₄	—	439	1	no	—
53	C₄⋊C₄	—	439	2	no	—
54	C₄⋊C₄	—	439	1	no	—
55	C₂²⋊C₄	—	878	1	no	—
56	C₄²	—	878	1	no	—
57	C₂³	D₁₇₅₆	1	1	yes	characteristic
58	C₂×C₄	—	878	1	no	—
59	C₂×C₄	—	878	1	no	—
60	C₂×C₄	—	878	2	no	—
61	C₂×C₄	—	1756	2	no	—
62	C₂×C₄	C₂×Dic₄₃₉	1	2	yes	—
63	C₂×C₄	D₁₇₅₆	1	1	yes	characteristic
64	C₂×C₄	Dic₈₇₈	1	2	yes	—
65	C₂×C₄	C₂²×D₄₃₉	1	1	yes	characteristic
66	C₈	C₂×Dic₄₃₉	1	4	yes	complemented
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	C₄⋊Dic₄₃₉	1	2	yes	—
70	C₂²	C₂×D₁₇₅₆	1	1	yes	characteristic
71	C₄	—	1756	2	no	—
72	C₄	—	1756	1	no	—
73	C₄	—	1756	1	no	—
74	C₄	C₄⋊Dic₄₃₉	1	2	yes	—
75	C₄	C₂²×Dic₄₃₉	1	1	yes	characteristic
76	C₄	C₂×Dic₈₇₈	1	1	yes	characteristic
77	C₂	—	2	2	no	—
78	C₂	C₂×C₄⋊Dic₄₃₉	1	1	yes	characteristic
79	C₂	#28380771	1	1	yes	characteristic
80	C₂	#28380776	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₇₅₆
commutator	C₄×C₄₃₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₃₉
Fitting	C₂×M₁₆×C₄₃₉	C₂
socle	C₂²×C₄₃₉	C₂×D₄
prime radical	C₂³×C₄₃₉	D₄
cosocle	C₂×C₄×C₄₃₉	C₂³
hypercenter	C₂×M₁₆	D₄₃₉
hypocenter	C₄₃₉	#428
LCS	G ▷ C₄×C₄₃₉ ▷ C₂×C₄₃₉ ▷ C₄₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #28380677	C₂	semidirect
4. #28380675	C₂	semidirect
14. M₁₆×C₄₃₉	C₄	semidirect
25. C₈×C₄₃₉	C₂×C₄	semidirect
40. C₄₃₉	#428	semidirect
49. M₁₆	Dic₄₃₉	semidirect
66. C₈	C₂×Dic₄₃₉	semidirect

Maximal subgroups

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

subgroup	index
#28380702	2
C₂×C₄⋊Dic₄₃₉	2
#28380677	2
#28380675	2
C₂×M₁₆×C₄₃₉	2
#428	439