Subgroups — #41000768

Order 26816.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27211717	C₂	1	1	yes	maximal, characteristic
2	#27211709	C₂	1	1	yes	maximal, characteristic
3	#27211652	C₂	1	1	yes	maximal, characteristic
4	#27211651	C₂	1	1	yes	maximal, characteristic
5	#27211649	C₂	1	1	yes	maximal, characteristic
6	#27211628	C₂	1	1	yes	maximal, characteristic
7	#106×C₄₁₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₈₃₈	—	2	1	no	—
9	C₂×C₄₁₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₄₁₉	—	2	1	no	—
11	C₄⋊Dic₄₁₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₄₁₉	C₂²	1	1	yes	characteristic
13	Dic₄₁₉⋊C₄	—	2	1	no	—
14	Dic₄₁₉⋊C₄	—	2	1	no	—
15	C₄×Dic₄₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₄₁₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₄₁₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₄₁₉	C₂²	1	1	yes	characteristic
19	Dic₈₃₈	—	4	2	no	—
20	C₂×Dic₄₁₉	—	2	1	no	—
21	C₂×Dic₄₁₉	—	2	1	no	—
22	C₂×Dic₄₁₉	D₄	1	1	yes	characteristic
23	C₂×Dic₄₁₉	D₄	1	1	yes	characteristic
24	C₄₁₉⋊C₈	Q₈	1	2	yes	complemented
25	C₂×C₄×C₄₁₉	—	2	1	no	—
26	C₂×C₄×C₄₁₉	—	2	1	no	—
27	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
28	C₈×C₄₁₉	Q₈	1	2	yes	complemented
29	C₂²×C₄₁₉	C₂×D₄	1	1	yes	characteristic
30	C₄×C₄₁₉	—	4	1	no	—
31	C₄×C₄₁₉	—	4	1	no	—
32	C₄×C₄₁₉	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₄₁₉	C₂×Q₈	1	1	yes	characteristic
34	Dic₄₁₉	—	2	1	no	—
35	Dic₄₁₉	—	2	1	no	—
36	Dic₄₁₉	—	4	1	no	—
37	Dic₄₁₉	—	4	1	no	—
38	C₂×C₄₁₉	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₄₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
40	C₂×C₄₁₉	#137	1	1	yes	characteristic
41	C₄₁₉	#501	1	1	yes	characteristic, complemented, Sylow
42	#501	—	419	1	no	maximal, Sylow
43	C₄⋊Q₈	—	419	1	no	—
44	#125	—	419	1	no	—
45	#107	—	419	1	no	—
46	#107	—	419	1	no	—
47	#106	—	419	1	no	—
48	#106	D₄₁₉	1	1	yes	characteristic
49	#97	—	419	1	no	—
50	C₂×Q₈	—	838	1	no	—
51	C₂×C₈	—	419	1	no	—
52	C₂×C₈	C₂×D₄₁₉	1	1	yes	characteristic
53	C₄⋊C₄	—	838	1	no	—
54	C₄⋊C₄	—	838	1	no	—
55	C₄⋊C₄	—	838	1	no	—
56	C₄⋊C₄	—	419	1	no	—
57	C₄⋊C₄	—	419	1	no	—
58	C₄⋊C₄	C₂×D₄₁₉	1	1	yes	characteristic
59	C₄⋊C₄	C₂×D₄₁₉	1	1	yes	characteristic
60	C₄²	—	419	1	no	—
61	Q₈	—	1676	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	838	1	no	—
65	C₂×C₄	—	838	1	no	—
66	C₂×C₄	—	419	1	no	—
67	C₂×C₄	—	419	1	no	—
68	C₂×C₄	C₂²×D₄₁₉	1	1	yes	characteristic
69	C₈	—	419	2	no	—
70	C₈	Dic₈₃₈	1	2	yes	complemented
71	C₂²	D₄×D₄₁₉	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	838	1	no	—
75	C₄	—	838	1	no	—
76	C₄	—	1676	1	no	—
77	C₄	—	1676	1	no	—
78	C₄	Q₈×D₄₁₉	1	1	yes	characteristic
79	C₄	C₂×Dic₈₃₈	1	1	yes	characteristic
80	C₂	#27211717	1	1	yes	characteristic
81	C₂	#27211745	1	1	yes	characteristic
82	C₂	#27211747	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₁₉
commutator, cosocle	C₂×C₄×C₄₁₉	C₂³
Frattini	C₂×C₄	C₂²×D₄₁₉
Fitting	#106×C₄₁₉	C₂
socle, prime radical	C₂²×C₄₁₉	C₂×D₄
hypercenter	#106	D₄₁₉
hypocenter	C₄₁₉	#501
LCS	G ▷ C₂×C₄×C₄₁₉ ▷ C₂×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
24. C₄₁₉⋊C₈	Q₈	semidirect
28. C₈×C₄₁₉	Q₈	semidirect
41. C₄₁₉	#501	semidirect
70. C₈	Dic₈₃₈	semidirect

Maximal subgroups

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

subgroup	index
#27211717	2
#27211709	2
#27211652	2
#27211651	2
#27211649	2
#27211628	2
#106×C₄₁₉	2
#501	419