Subgroups — #41000711

Order 26816.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27211718	C₂	1	1	yes	maximal, characteristic
2	#27211642	C₂	1	4	yes	maximal
3	C₄×C₄₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₄₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₈₃₈	—	2	1	no	—
6	C₂×C₄₁₉⋊C₈	C₂²	1	2	yes	—
7	C₄⋊Dic₄₁₉	—	2	1	no	—
8	C₄⋊Dic₄₁₉	C₂²	1	2	yes	—
9	C₂×Q₈×C₄₁₉	C₂²	1	2	yes	—
10	C₄⋊C₄×C₄₁₉	—	2	1	no	—
11	C₄⋊C₄×C₄₁₉	—	2	1	no	—
12	C₄²×C₄₁₉	C₂²	1	1	yes	characteristic
13	Dic₈₃₈	—	4	2	no	—
14	C₂×Dic₄₁₉	—	2	2	no	—
15	C₄₁₉⋊C₈	—	2	2	no	—
16	Q₈×C₄₁₉	—	2	4	no	—
17	C₂×C₄×C₄₁₉	—	2	2	no	—
18	C₂×C₄×C₄₁₉	D₄	1	1	yes	characteristic
19	C₂×C₄×C₄₁₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
21	C₂²×C₄₁₉	C₂×D₄	1	1	yes	characteristic
22	C₄×C₄₁₉	—	4	2	no	—
23	C₄×C₄₁₉	SD₁₆	1	2	yes	—
24	C₄×C₄₁₉	Q₁₆	1	2	yes	—
25	C₄×C₄₁₉	#41	1	2	yes	—
26	Dic₄₁₉	—	4	2	no	—
27	C₂×C₄₁₉	#124	1	1	yes	characteristic
28	C₂×C₄₁₉	C₂×SD₁₆	1	1	yes	characteristic
29	C₂×C₄₁₉	C₂×Q₁₆	1	1	yes	characteristic
30	C₄₁₉	#487	1	1	yes	characteristic, complemented, Sylow
31	#487	—	419	1	no	maximal, Sylow
32	C₄⋊Q₈	—	419	1	no	—
33	C₄⋊Q₈	D₄₁₉	1	1	yes	characteristic
34	#103	—	419	4	no	—
35	C₄×C₈	—	419	1	no	—
36	C₂×Q₈	—	838	1	no	—
37	C₂×Q₈	C₂×D₄₁₉	1	2	yes	—
38	C₂×C₈	—	419	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	2	1	no	—
41	C₄⋊C₄	—	838	1	no	—
42	C₄⋊C₄	—	419	2	no	—
43	C₄²	C₂×D₄₁₉	1	1	yes	characteristic
44	Q₈	—	2	4	no	—
45	Q₈	—	1676	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	838	2	no	—
48	C₂×C₄	C₂²⋊D₄₁₉	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₄₁₉	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₄₁₉	1	1	yes	characteristic
51	C₈	—	838	2	no	—
52	C₂²	C₂×C₂²⋊D₄₁₉	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	1676	2	no	—
55	C₄	#20269398	1	2	yes	—
56	C₄	#20269401	1	2	yes	—
57	C₄	#20269410	1	2	yes	—
58	C₂	#27211705	1	1	yes	characteristic
59	C₂	C₂×#20269398	1	1	yes	characteristic
60	C₂	C₂×#20269401	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₁₉
commutator, cosocle	C₂×C₄×C₄₁₉	C₂³
Frattini	C₂×C₄	C₂²×D₄₁₉
Fitting	C₄⋊Q₈×C₄₁₉	C₂
socle, prime radical	C₂²×C₄₁₉	C₂×D₄
hypercenter	C₄⋊Q₈	D₄₁₉
hypocenter	C₄₁₉	#487
LCS	G ▷ C₂×C₄×C₄₁₉ ▷ C₂×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
30. C₄₁₉	#487	semidirect

Maximal subgroups

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

subgroup	index
#27211718	2
#27211642	2
C₄×C₄₁₉⋊C₈	2
C₄⋊Q₈×C₄₁₉	2
#487	419