Subgroups — #31772675

Order 17312.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575836	C₂	1	3	yes	maximal, complemented
2	#22575835	C₂	1	6	yes	maximal, complemented
3	C₂×Dic₁₀₈₂	C₂	1	3	yes	maximal, complemented
4	Q₈×D₅₄₁	C₂	1	2	yes	maximal, complemented
5	#41×C₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₀₈₂	C₂²	1	6	yes	—
7	Dic₁₀₈₂	C₂²	1	3	yes	—
8	D₂₁₆₄	C₂²	1	1	yes	characteristic
9	C₂²⋊D₅₄₁	C₂²	1	6	yes	—
10	C₂×Dic₅₄₁	C₂²	1	6	yes	—
11	C₄×D₅₄₁	C₂²	1	6	yes	—
12	Q₈×C₅₄₁	C₂²	1	1	yes	characteristic
13	D₄×C₅₄₁	C₂²	1	3	yes	—
14	C₂×C₄×C₅₄₁	C₂²	1	3	yes	—
15	C₂²×C₅₄₁	C₂³	1	3	yes	—
16	C₂×D₅₄₁	C₂³	1	2	yes	—
17	C₄×C₅₄₁	C₂³	1	3	yes	—
18	C₄×C₅₄₁	C₂³	1	1	yes	characteristic
19	Dic₅₄₁	C₂³	1	6	yes	—
20	C₂×C₅₄₁	—	2	3	no	—
21	C₂×C₅₄₁	C₂⁴	1	1	yes	characteristic
22	D₅₄₁	—	2	2	no	—
23	C₅₄₁	2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
24	2₋¹⁺⁴	—	541	1	no	maximal, Sylow
25	#41	—	541	6	no	—
26	#41	—	541	3	no	—
27	#41	D₅₄₁	1	1	yes	characteristic, complemented
28	C₂×Q₈	—	541	2	no	—
29	C₂×Q₈	—	541	3	no	—
30	Q₈	—	541	6	no	—
31	Q₈	—	541	3	no	—
32	Q₈	C₂×D₅₄₁	1	1	yes	characteristic
33	D₄	—	541	6	no	—
34	D₄	—	541	1	no	—
35	D₄	C₂×D₅₄₁	1	3	yes	—
36	C₂×C₄	—	541	6	no	—
37	C₂×C₄	—	541	6	no	—
38	C₂×C₄	C₂×D₅₄₁	1	3	yes	—
39	C₂²	—	541	2	no	—
40	C₂²	C₂²×D₅₄₁	1	3	yes	—
41	C₄	—	541	6	no	—
42	C₄	C₂²×D₅₄₁	1	3	yes	—
43	C₄	C₂²×D₅₄₁	1	1	yes	characteristic
44	C₂	—	2	3	no	—
45	C₂	—	1082	2	no	—
46	C₂	C₂³×D₅₄₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₅₄₁
commutator, socle, cosocle	C₂×C₅₄₁	C₂⁴
Fitting	#41×C₅₄₁	C₂
hypercenter	#41	D₅₄₁
hypocenter	C₅₄₁	2₋¹⁺⁴
LCS	G ▷ C₂×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #22575836	C₂	semidirect
2. #22575835	C₂	semidirect
3. C₂×Dic₁₀₈₂	C₂	semidirect
4. Q₈×D₅₄₁	C₂	semidirect
5. #41×C₅₄₁	C₂	semidirect
23. C₅₄₁	2₋¹⁺⁴	semidirect
27. #41	D₅₄₁	semidirect

Maximal subgroups

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

subgroup	index
#22575836	2
#22575835	2
C₂×Dic₁₀₈₂	2
Q₈×D₅₄₁	2
#41×C₅₄₁	2
2₋¹⁺⁴	541