Subgroups — #41000272

Order 26816.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×(C₂×D₄₁₉)⋊C₄	C₂	1	3	yes	maximal
2	C₄₁₉⋊#95	C₂	1	3	yes	maximal, complemented
3	C₂²×Q₈×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₄₁₉	C₂²	1	1	yes	characteristic
5	C₂²×Dic₄₁₉	C₂²	1	3	yes	—
6	(C₂×D₄₁₉)⋊C₄	—	2	6	no	—
7	C₂×Q₈×C₄₁₉	—	2	6	no	—
8	C₂²×C₄×C₄₁₉	C₂²	1	3	yes	—
9	C₂²×D₄₁₉	—	2	3	no	—
10	C₂²×D₄₁₉	—	2	3	no	—
11	C₂²×D₄₁₉	Q₈	1	2	yes	complemented
12	C₂×Dic₄₁₉	—	2	3	no	—
13	C₂×Dic₄₁₉	—	2	6	no	—
14	C₂³×C₄₁₉	C₂³	1	1	yes	characteristic
15	Q₈×C₄₁₉	—	2	8	no	—
16	C₂×C₄×C₄₁₉	—	2	6	no	—
17	C₂×C₄×C₄₁₉	D₄	1	6	yes	—
18	C₂²×C₄₁₉	C₂×D₄	1	3	yes	—
19	C₂²×C₄₁₉	C₂×Q₈	1	1	yes	characteristic
20	C₂²×C₄₁₉	#41	1	3	yes	—
21	C₂×D₄₁₉	—	2	6	no	—
22	C₂×D₄₁₉	—	4	1	no	—
23	C₂×D₄₁₉	—	4	3	no	—
24	C₄×C₄₁₉	—	2	6	no	—
25	Dic₄₁₉	—	4	3	no	—
26	C₂×C₄₁₉	#120	1	1	yes	characteristic
27	C₂×C₄₁₉	C₂²⋊Q₈	1	3	yes	—
28	C₂×C₄₁₉	#124	1	3	yes	—
29	D₄₁₉	—	4	2	no	—
30	C₄₁₉	#393	1	1	yes	characteristic, complemented, Sylow
31	#393	—	419	1	no	maximal, Sylow
32	C₂²×Q₈	D₄₁₉	1	1	yes	characteristic, complemented
33	C₂×C₂²⋊C₄	—	419	3	no	—
34	#95	—	419	3	no	—
35	C₂⁴	—	419	1	no	—
36	C₂×Q₈	—	2	6	no	—
37	C₂²×C₄	—	419	3	no	—
38	C₂²×C₄	C₂×D₄₁₉	1	3	yes	—
39	C₂²⋊C₄	—	838	6	no	—
40	C₂³	—	838	3	no	—
41	C₂³	—	838	3	no	—
42	C₂³	—	419	2	no	—
43	C₂³	C₂²×D₄₁₉	1	1	yes	characteristic
44	Q₈	—	2	8	no	—
45	C₂×C₄	—	2	6	no	—
46	C₂×C₄	—	838	3	no	—
47	C₂×C₄	—	838	6	no	—
48	C₂×C₄	C₂²⋊D₄₁₉	1	6	yes	—
49	C₂²	—	838	6	no	—
50	C₂²	—	1676	1	no	—
51	C₂²	—	1676	3	no	—
52	C₂²	C₂×C₂²⋊D₄₁₉	1	3	yes	—
53	C₂²	Q₈×D₄₁₉	1	1	yes	characteristic
54	C₂²	#20269412	1	3	yes	—
55	C₄	—	2	6	no	—
56	C₄	—	1676	3	no	—
57	C₂	—	1676	2	no	—
58	C₂	#27211687	1	1	yes	characteristic
59	C₂	#27211694	1	3	yes	—
60	C₂	#27211707	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₄₁₉
commutator, socle, cosocle	C₂³×C₄₁₉	C₂³
Fitting	C₂²×Q₈×C₄₁₉	C₂
prime radical	C₂³×D₄₁₉	C₂²
hypercenter	C₂²×Q₈	D₄₁₉
hypocenter	C₄₁₉	#393
LCS	G ▷ C₂³×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
2. C₄₁₉⋊#95	C₂	semidirect
3. C₂²×Q₈×C₄₁₉	C₂	semidirect
11. C₂²×D₄₁₉	Q₈	semidirect
30. C₄₁₉	#393	semidirect
32. C₂²×Q₈	D₄₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×(C₂×D₄₁₉)⋊C₄	2
C₄₁₉⋊#95	2
C₂²×Q₈×C₄₁₉	2
#393	419