Subgroups — #41000694

Order 26816.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₄₁₉	C₂	1	1	yes	maximal, characteristic
2	#27211651	C₂	1	2	yes	maximal, complemented
3	#27211642	C₂	1	2	yes	maximal, complemented
4	C₂²⋊(C₄₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₂²⋊Q₈×C₄₁₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₁₉	—	2	1	no	—
7	C₂×C₄₁₉⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₄₁₉	—	2	1	no	—
9	C₄⋊Dic₄₁₉	C₂²	1	2	yes	—
10	C₂×Q₈×C₄₁₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₄₁₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₄₁₉	—	2	1	no	—
13	C₄⋊C₄×C₄₁₉	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₄₁₉	—	2	1	no	—
15	C₂×Dic₄₁₉	—	2	2	no	—
16	C₂×Dic₄₁₉	—	4	2	no	—
17	C₄₁₉⋊C₈	—	2	2	no	—
18	C₂³×C₄₁₉	D₄	1	1	yes	characteristic
19	Q₈×C₄₁₉	—	2	2	no	—
20	C₂×C₄×C₄₁₉	—	2	2	no	—
21	C₂×C₄×C₄₁₉	—	2	1	no	—
22	C₂×C₄×C₄₁₉	—	2	1	no	—
23	C₂×C₄×C₄₁₉	D₄	1	1	yes	characteristic
24	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
25	C₂²×C₄₁₉	—	2	1	no	—
26	C₂²×C₄₁₉	—	2	1	no	—
27	C₂²×C₄₁₉	Q₁₆	1	2	yes	—
28	C₂²×C₄₁₉	C₂×D₄	1	1	yes	characteristic
29	C₄×C₄₁₉	—	2	1	no	—
30	C₄×C₄₁₉	—	4	1	no	—
31	C₄×C₄₁₉	—	4	1	no	—
32	C₄×C₄₁₉	#41	1	2	yes	—
33	Dic₄₁₉	—	4	2	no	—
34	C₂×C₄₁₉	—	2	2	no	—
35	C₂×C₄₁₉	#123	1	1	yes	characteristic
36	C₂×C₄₁₉	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₄₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
38	C₄₁₉	#484	1	1	yes	characteristic, complemented, Sylow
39	#484	—	419	1	no	maximal, Sylow
40	C₂²⋊Q₈	D₄₁₉	1	1	yes	characteristic
41	C₂×C₄⋊C₄	—	419	1	no	—
42	#107	—	419	2	no	—
43	#103	—	419	2	no	—
44	C₂²⋊C₈	—	419	1	no	—
45	C₂×Q₈	C₂×D₄₁₉	1	1	yes	characteristic
46	C₂²×C₄	—	838	1	no	—
47	C₂²×C₄	C₂×D₄₁₉	1	1	yes	characteristic
48	C₂×C₈	—	419	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	—	838	1	no	—
51	C₄⋊C₄	—	419	2	no	—
52	C₄⋊C₄	C₂×D₄₁₉	1	1	yes	characteristic
53	C₂²⋊C₄	—	2	1	no	—
54	C₂³	C₂²⋊D₄₁₉	1	1	yes	characteristic
55	Q₈	—	2	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	838	2	no	—
60	C₂×C₄	—	1676	2	no	—
61	C₂×C₄	C₂²⋊D₄₁₉	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₄₁₉	1	1	yes	characteristic
63	C₈	—	838	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	#20269401	1	2	yes	—
67	C₂²	C₂×C₂²⋊D₄₁₉	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	4	1	no	—
70	C₄	—	4	1	no	—
71	C₄	—	1676	2	no	—
72	C₄	#20269410	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#27211699	1	1	yes	characteristic
75	C₂	C₂×#20269401	1	1	yes	characteristic
76	C₂	#27211743	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₁₉
commutator, cosocle	C₂×C₄×C₄₁₉	C₂³
Frattini	C₂×C₄	C₂²×D₄₁₉
Fitting	C₂²⋊Q₈×C₄₁₉	C₂
socle	C₂²×C₄₁₉	C₂×D₄
prime radical	C₂³×C₄₁₉	D₄
hypercenter	C₂²⋊Q₈	D₄₁₉
hypocenter	C₄₁₉	#484
LCS	G ▷ C₂×C₄×C₄₁₉ ▷ C₂×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #27211651	C₂	semidirect
3. #27211642	C₂	semidirect
38. C₄₁₉	#484	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₄₁₉	2
#27211651	2
#27211642	2
C₂²⋊(C₄₁₉⋊C₈)	2
C₂²⋊Q₈×C₄₁₉	2
#484	419