Subgroups — #41002623

Order 26944.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212372	C₂	1	1	yes	maximal, characteristic
2	#27212306	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#20269771	C₄	1	1	yes	characteristic
5	#20269755	C₄	1	1	yes	characteristic
6	#20269754	—	2	1	no	—
7	C₈×D₄₂₁	—	2	1	no	—
8	#20269746	—	2	1	no	—
9	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
10	C₁₆×C₄₂₁	—	2	1	no	—
11	Dic₈₄₂	C₈	1	1	yes	characteristic
12	D₁₆₈₄	C₈	1	1	yes	characteristic
13	C₂²⋊D₄₂₁	—	2	1	no	—
14	C₄×D₄₂₁	—	2	1	no	—
15	#16809890	—	2	1	no	—
16	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
17	C₈×C₄₂₁	D₄	1	1	yes	characteristic
18	C₈×C₄₂₁	D₄	1	1	yes	characteristic
19	C₂²×C₄₂₁	M₁₆	1	1	yes	characteristic
20	C₂×D₄₂₁	—	2	1	no	—
21	C₄×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₄₂₁	C₂×C₈	1	1	yes	characteristic
23	Dic₄₂₁	—	2	1	no	—
24	C₂×C₄₂₁	—	2	1	no	—
25	C₂×C₄₂₁	C₂²⋊C₈	1	1	yes	characteristic
26	D₄₂₁	—	4	1	no	—
27	C₄₂₁	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	421	1	no	maximal, Sylow
29	#131	—	421	1	no	—
30	M₃₂	—	421	1	no	—
31	C₂×C₁₆	D₄₂₁	1	1	yes	characteristic, complemented
32	#41	—	421	1	no	—
33	M₁₆	—	842	1	no	—
34	M₁₆	—	421	1	no	—
35	C₂×C₈	—	842	1	no	—
36	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	—	842	1	no	—
39	Q₈	—	421	1	no	—
40	D₄	—	842	1	no	—
41	D₄	—	421	1	no	—
42	C₂×C₄	—	842	1	no	—
43	C₂×C₄	C₄×D₄₂₁	1	1	yes	characteristic
44	C₈	—	842	1	no	—
45	C₈	C₂²⋊D₄₂₁	1	1	yes	characteristic
46	C₈	D₁₆₈₄	1	1	yes	characteristic
47	C₂²	—	842	1	no	—
48	C₂²	#20269754	1	1	yes	characteristic
49	C₄	—	842	1	no	—
50	C₄	#20269748	1	1	yes	characteristic
51	C₄	C₈×D₄₂₁	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	1684	1	no	—
54	C₂	#27212278	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₁₆₈₄
commutator	C₄×C₄₂₁	C₂×C₈
Frattini	C₂×C₈	C₂×D₄₂₁
Fitting	C₂×C₁₆×C₄₂₁	C₂
socle	C₂×C₄₂₁	C₂²⋊C₈
prime radical	#20269771	C₄
cosocle	C₂×C₈×C₄₂₁	C₂²
hypercenter	C₂×C₁₆	D₄₂₁
hypocenter	C₄₂₁	#350
LCS	G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. #27212306	C₂	semidirect
3. C₂×C₁₆×C₄₂₁	C₂	semidirect
27. C₄₂₁	#350	semidirect
31. C₂×C₁₆	D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
#27212372	2
#27212306	2
C₂×C₁₆×C₄₂₁	2
#350	421