Subgroups — #41000484

Order 26816.

104 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	#27211724	C₂	1	2	yes	maximal, complemented
3	#27211654	C₂	1	1	yes	maximal, characteristic, complemented
4	#27211646	C₂	1	2	yes	maximal, complemented
5	#97×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	#20269395	C₂²	1	2	yes	—
11	#20269394	—	2	2	no	—
12	C₈×D₄₁₉	—	2	2	no	—
13	C₂×C₈×C₄₁₉	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₄₁₉	C₂²	1	1	yes	characteristic
15	C₄²×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	C₄×D₄₁₉	—	2	2	no	—
20	C₄₁₉⋊C₈	—	2	2	no	—
21	C₂×C₄×C₄₁₉	—	2	1	no	—
22	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
23	C₈×C₄₁₉	C₂×C₄	1	1	yes	characteristic
24	C₈×C₄₁₉	C₂×C₄	1	1	yes	characteristic
25	C₈×C₄₁₉	D₄	1	2	yes	—
26	C₂²×C₄₁₉	#41	1	1	yes	characteristic
27	C₂×D₄₁₉	—	2	2	no	—
28	C₄×C₄₁₉	—	4	1	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	C₂×C₄₁₉	—	2	1	no	—
33	C₂×C₄₁₉	C₄×D₄	1	1	yes	characteristic
34	D₄₁₉	—	4	2	no	—
35	C₄₁₉	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	419	1	no	maximal, Sylow
37	#135	—	419	1	no	—
38	#131	—	419	2	no	—
39	#108	—	419	1	no	—
40	#104	—	419	2	no	—
41	#97	D₄₁₉	1	1	yes	characteristic, complemented
42	#41	—	419	2	no	—
43	Q₁₆	—	419	1	no	—
44	SD₁₆	—	419	2	no	—
45	D₈	—	419	1	no	—
46	M₁₆	—	838	2	no	—
47	M₁₆	—	419	2	no	—
48	C₂×C₈	—	838	2	no	—
49	C₂×C₈	C₂×D₄₁₉	1	1	yes	characteristic
50	C₂×C₈	C₂×D₄₁₉	1	1	yes	characteristic
51	C₄²	C₂×D₄₁₉	1	1	yes	characteristic
52	Q₈	—	419	2	no	—
53	D₄	—	838	2	no	—
54	D₄	—	419	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	838	2	no	—
57	C₂×C₄	C₂²×D₄₁₉	1	1	yes	characteristic
58	C₈	—	838	2	no	—
59	C₈	C₄×D₄₁₉	1	1	yes	characteristic
60	C₈	C₄×D₄₁₉	1	1	yes	characteristic
61	C₈	D₁₆₇₆	1	2	yes	—
62	C₂²	—	838	2	no	—
63	C₂²	#20269411	1	1	yes	characteristic
64	C₄	—	4	1	no	—
65	C₄	—	838	2	no	—
66	C₄	C₂×C₄×D₄₁₉	1	1	yes	characteristic
67	C₄	C₂×D₁₆₇₆	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	1676	2	no	—
70	C₂	C₄×D₁₆₇₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₁₆₇₆
commutator	C₄×C₄₁₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₁₉
Fitting	#97×C₄₁₉	C₂
socle	C₂×C₄₁₉	C₄×D₄
prime radical	#27211740	C₂
cosocle	C₂×C₄×C₄₁₉	C₂³
hypercenter	#97	D₄₁₉
hypocenter	C₄₁₉	#444
LCS	G ▷ C₄×C₄₁₉ ▷ C₂×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #97

Complements

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

N	H ≅ G/N	type
2. #27211724	C₂	semidirect
3. #27211654	C₂	semidirect
4. #27211646	C₂	semidirect
5. #97×C₄₁₉	C₂	semidirect
7. Dic₁₆₇₆	C₄	semidirect
8. #20269400	C₄	semidirect
9. D₃₃₅₂	C₄	semidirect
35. C₄₁₉	#444	semidirect
41. #97	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
#27211724	2
#27211654	2
#27211646	2
#97×C₄₁₉	2
#444	419