Subgroups — #41000477

Order 26816.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27211740	C₂	1	1	yes	maximal, characteristic
2	#27211726	C₂	1	2	yes	maximal, complemented
3	#27211644	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#108×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#20269411	C₂²	1	2	yes	—
7	Dic₁₆₇₆	C₄	1	1	yes	characteristic, complemented
8	#20269400	C₄	1	2	yes	complemented
9	D₃₃₅₂	C₄	1	1	yes	characteristic, complemented
10	#20269394	—	2	2	no	—
11	C₂×C₄₁₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₈×D₄₁₉	—	2	2	no	—
13	C₄×Dic₄₁₉	C₂²	1	1	yes	characteristic
14	M₁₆×C₄₁₉	C₂²	1	2	yes	—
15	C₂×C₈×C₄₁₉	C₂²	1	1	yes	characteristic
16	Dic₈₃₈	C₂×C₄	1	2	yes	—
17	D₁₆₇₆	C₂×C₄	1	2	yes	—
18	C₂²⋊D₄₁₉	—	2	2	no	—
19	C₂×Dic₄₁₉	—	2	1	no	—
20	C₄×D₄₁₉	—	2	2	no	—
21	C₄₁₉⋊C₈	D₄	1	1	yes	characteristic
22	C₄₁₉⋊C₈	D₄	1	1	yes	characteristic
23	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
24	C₈×C₄₁₉	—	2	2	no	—
25	C₈×C₄₁₉	C₂×C₄	1	1	yes	characteristic
26	C₈×C₄₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄₁₉	#41	1	1	yes	characteristic
28	C₂×D₄₁₉	—	2	2	no	—
29	C₄×C₄₁₉	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₄₁₉	C₂×D₄	1	1	yes	characteristic
31	Dic₄₁₉	—	2	2	no	—
32	Dic₄₁₉	—	2	2	no	—
33	C₂×C₄₁₉	—	2	1	no	—
34	C₂×C₄₁₉	C₄×D₄	1	1	yes	characteristic
35	D₄₁₉	—	4	2	no	—
36	C₄₁₉	#443	1	1	yes	characteristic, complemented, Sylow
37	#443	—	419	1	no	maximal, Sylow
38	#135	—	419	1	no	—
39	#131	—	419	2	no	—
40	#108	D₄₁₉	1	1	yes	characteristic, complemented
41	#104	—	419	2	no	—
42	C₄×C₈	—	419	1	no	—
43	#41	—	419	2	no	—
44	Q₁₆	—	419	1	no	—
45	SD₁₆	—	419	2	no	—
46	D₈	—	419	1	no	—
47	M₁₆	—	838	2	no	—
48	M₁₆	C₂×D₄₁₉	1	2	yes	—
49	C₂×C₈	—	838	2	no	—
50	C₂×C₈	—	419	1	no	—
51	C₂×C₈	C₂×D₄₁₉	1	1	yes	characteristic
52	C₄²	—	419	1	no	—
53	Q₈	—	419	2	no	—
54	D₄	—	838	2	no	—
55	D₄	—	419	2	no	—
56	C₂×C₄	—	838	1	no	—
57	C₂×C₄	—	838	2	no	—
58	C₂×C₄	C₂²×D₄₁₉	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	—	419	1	no	—
61	C₈	—	419	1	no	—
62	C₈	C₄×D₄₁₉	1	1	yes	characteristic
63	C₈	C₄×D₄₁₉	1	1	yes	characteristic
64	C₂²	—	838	2	no	—
65	C₂²	#20269412	1	1	yes	characteristic
66	C₄	—	838	2	no	—
67	C₄	—	838	2	no	—
68	C₄	C₂×C₄×D₄₁₉	1	1	yes	characteristic
69	C₄	D₄×D₄₁₉	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	1676	2	no	—
72	C₂	#27211680	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₄₁₉
commutator	C₄×C₄₁₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₁₉
Fitting	#108×C₄₁₉	C₂
socle	C₂×C₄₁₉	C₄×D₄
prime radical	#27211740	C₂
cosocle	C₂×C₄×C₄₁₉	C₂³
hypercenter	#108	D₄₁₉
hypocenter	C₄₁₉	#443
LCS	G ▷ C₄×C₄₁₉ ▷ C₂×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
2. #27211726	C₂	semidirect
3. #27211644	C₂	semidirect
4. C₈×Dic₄₁₉	C₂	semidirect
5. #108×C₄₁₉	C₂	semidirect
7. Dic₁₆₇₆	C₄	semidirect
8. #20269400	C₄	semidirect
9. D₃₃₅₂	C₄	semidirect
36. C₄₁₉	#443	semidirect
40. #108	D₄₁₉	semidirect

Maximal subgroups

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

subgroup	index
#27211740	2
#27211726	2
#27211644	2
C₈×Dic₄₁₉	2
#108×C₄₁₉	2
#443	419