Subgroups — #41002649

Order 26944.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212388	C₂	1	1	yes	maximal, characteristic
2	#27212306	C₂	1	1	yes	maximal, characteristic, complemented
3	#106×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#20269771	—	2	1	no	—
5	Dic₁₆₈₄	C₄	1	1	yes	characteristic, complemented
6	#20269760	—	2	1	no	—
7	D₃₃₆₈	C₄	1	1	yes	characteristic, complemented
8	#20269746	—	2	1	no	—
9	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₄₂₁	—	2	1	no	—
11	Dic₈₄₂	—	2	1	no	—
12	D₁₆₈₄	—	2	1	no	—
13	C₂²⋊D₄₂₁	—	4	1	no	—
14	C₄×D₄₂₁	—	4	1	no	—
15	C₂×C₄×C₄₂₁	—	4	1	no	—
16	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
17	C₈×C₄₂₁	C₂×C₄	1	1	yes	characteristic
18	C₈×C₄₂₁	D₄	1	1	yes	characteristic
19	C₂²×C₄₂₁	D₈	1	1	yes	characteristic
20	C₂×D₄₂₁	—	4	1	no	—
21	C₄×C₄₂₁	—	8	1	no	—
22	C₄×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₄₂₁	SD₁₆	1	1	yes	characteristic
24	Dic₄₂₁	—	4	1	no	—
25	C₂×C₄₂₁	—	2	1	no	—
26	C₂×C₄₂₁	#102	1	1	yes	characteristic
27	D₄₂₁	—	8	1	no	—
28	C₄₂₁	#360	1	1	yes	characteristic, complemented, Sylow
29	#360	—	421	1	no	maximal, Sylow
30	#135	—	421	1	no	—
31	M₃₂	—	421	1	no	—
32	#106	D₄₂₁	1	1	yes	characteristic, complemented
33	#41	—	842	1	no	—
34	Q₁₆	—	421	1	no	—
35	SD₁₆	—	842	1	no	—
36	D₈	—	421	1	no	—
37	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
38	C₄⋊C₄	—	2	1	no	—
39	C₁₆	—	842	1	no	—
40	Q₈	—	842	1	no	—
41	D₄	—	842	1	no	—
42	D₄	—	1684	1	no	—
43	C₂×C₄	—	4	1	no	—
44	C₂×C₄	—	1684	1	no	—
45	C₂×C₄	C₂²⋊D₄₂₁	1	1	yes	characteristic
46	C₈	C₄×D₄₂₁	1	1	yes	characteristic
47	C₈	D₁₆₈₄	1	1	yes	characteristic
48	C₂²	—	1684	1	no	—
49	C₂²	#20269756	1	1	yes	characteristic
50	C₄	—	8	1	no	—
51	C₄	—	1684	1	no	—
52	C₄	#20269748	1	1	yes	characteristic
53	C₄	#20269758	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	3368	1	no	—
56	C₂	#27212286	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27212286
commutator	C₈×C₄₂₁	C₂×C₄
Frattini	C₂×C₈	C₂×D₄₂₁
Fitting	#106×C₄₂₁	C₂
socle	C₂×C₄₂₁	#102
prime radical	#27212388	C₂
cosocle	C₂×C₈×C₄₂₁	C₂²
hypercenter	#106	D₄₂₁
hypocenter	C₄₂₁	#360
LCS	G ▷ C₈×C₄₂₁ ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #106

Complements

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

N	H ≅ G/N	type
2. #27212306	C₂	semidirect
3. #106×C₄₂₁	C₂	semidirect
5. Dic₁₆₈₄	C₄	semidirect
7. D₃₃₆₈	C₄	semidirect
28. C₄₂₁	#360	semidirect
32. #106	D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
#27212388	2
#27212306	2
#106×C₄₂₁	2
#360	421