Subgroups — #41001577

Order 26848.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27211955	C₂	1	1	yes	maximal, characteristic, complemented
2	#27211945	C₂	1	1	yes	maximal, characteristic, complemented
3	#27211943	C₂	1	1	yes	maximal, characteristic, complemented
4	#27211942	C₂	1	1	yes	maximal, characteristic, complemented
5	#27211940	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#41×C₈₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₁₆₇₈	C₂²	1	1	yes	characteristic
9	D₃₃₅₆	C₂²	1	1	yes	characteristic
10	C₂²⋊D₈₃₉	—	2	1	no	—
11	C₄×D₈₃₉	—	2	1	no	—
12	C₈₃₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₈₃₉⋊C₈	C₂²	1	1	yes	characteristic
14	Q₈×C₈₃₉	C₂²	1	1	yes	characteristic
15	D₄×C₈₃₉	—	2	1	no	—
16	D₄×C₈₃₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈₃₉	—	2	1	no	—
18	C₂×C₄×C₈₃₉	C₂²	1	1	yes	characteristic
19	C₂²×C₈₃₉	—	2	1	no	—
20	C₂²×C₈₃₉	D₄	1	1	yes	characteristic
21	C₂×D₈₃₉	—	2	1	no	—
22	C₄×C₈₃₉	—	2	1	no	—
23	C₄×C₈₃₉	D₄	1	1	yes	characteristic
24	C₄×C₈₃₉	C₂³	1	1	yes	characteristic
25	Dic₈₃₉	—	2	1	no	—
26	C₂×C₈₃₉	—	2	1	no	—
27	C₂×C₈₃₉	—	4	1	no	—
28	C₂×C₈₃₉	C₂×D₄	1	1	yes	characteristic
29	D₈₃₉	—	4	1	no	—
30	C₈₃₉	#135	1	1	yes	characteristic, complemented, Sylow
31	#135	—	839	1	no	maximal, Sylow
32	#41	—	839	1	no	—
33	#41	D₈₃₉	1	1	yes	characteristic, complemented
34	Q₁₆	—	839	1	no	—
35	SD₁₆	—	839	1	no	—
36	SD₁₆	—	839	1	no	—
37	D₈	—	839	1	no	—
38	C₂×C₈	—	839	1	no	—
39	Q₈	—	839	1	no	—
40	Q₈	C₂×D₈₃₉	1	1	yes	characteristic
41	D₄	—	2	1	no	—
42	D₄	—	839	1	no	—
43	D₄	—	1678	1	no	—
44	D₄	C₂×D₈₃₉	1	1	yes	characteristic
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	1678	1	no	—
47	C₂×C₄	C₂×D₈₃₉	1	1	yes	characteristic
48	C₈	—	839	1	no	—
49	C₈	—	839	1	no	—
50	C₂²	—	2	1	no	—
51	C₂²	—	1678	1	no	—
52	C₂²	C₂²⋊D₈₃₉	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	1678	1	no	—
55	C₄	C₂²⋊D₈₃₉	1	1	yes	characteristic
56	C₄	C₂²×D₈₃₉	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	—	4	1	no	—
59	C₂	—	3356	1	no	—
60	C₂	C₂×C₂²⋊D₈₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊D₈₃₉
commutator, cosocle	C₄×C₈₃₉	C₂³
Frattini	C₄	C₂²×D₈₃₉
Fitting	#41×C₈₃₉	C₂
socle	C₂×C₈₃₉	C₂×D₄
hypercenter	#41	D₈₃₉
hypocenter	C₈₃₉	#135
LCS	G ▷ C₄×C₈₃₉ ▷ C₂×C₈₃₉ ▷ C₈₃₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #27211955	C₂	semidirect
2. #27211945	C₂	semidirect
3. #27211943	C₂	semidirect
4. #27211942	C₂	semidirect
5. #27211940	C₂	semidirect
6. C₂×C₈₃₉⋊C₈	C₂	semidirect
7. #41×C₈₃₉	C₂	semidirect
30. C₈₃₉	#135	semidirect
33. #41	D₈₃₉	semidirect

Maximal subgroups

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

subgroup	index
#27211955	2
#27211945	2
#27211943	2
#27211942	2
#27211940	2
C₂×C₈₃₉⋊C₈	2
#41×C₈₃₉	2
#135	839