Subgroups — #41002645

Order 26944.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₆₈₄	C₂	1	1	yes	maximal, characteristic
2	#27212300	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×C₄₂₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₈₄₂	—	2	1	no	—
5	Dic₁₆₈₄	—	2	1	no	—
6	Dic₁₆₈₄	C₄	1	2	yes	complemented
7	C₄⋊Dic₄₂₁	—	2	1	no	—
8	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
9	C₁₆×C₄₂₁	—	2	1	no	—
10	Dic₈₄₂	—	2	2	no	—
11	Dic₈₄₂	—	4	1	no	—
12	C₂×Dic₄₂₁	—	4	1	no	—
13	C₂×Dic₄₂₁	—	4	1	no	—
14	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
15	C₈×C₄₂₁	C₂×C₄	1	1	yes	characteristic
16	C₈×C₄₂₁	D₄	1	1	yes	characteristic
17	C₂²×C₄₂₁	D₈	1	1	yes	characteristic
18	C₄×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₄₂₁	SD₁₆	1	1	yes	characteristic
20	Dic₄₂₁	—	4	2	no	—
21	Dic₄₂₁	—	8	1	no	—
22	C₂×C₄₂₁	#102	1	1	yes	characteristic
23	C₂×C₄₂₁	SD₃₂	1	1	yes	characteristic
24	C₂×C₄₂₁	Q₃₂	1	1	yes	characteristic
25	C₄₂₁	#358	1	1	yes	characteristic, complemented, Sylow
26	#358	—	421	1	no	maximal, Sylow
27	C₂×Q₁₆	—	421	1	no	—
28	C₂×C₁₆	D₄₂₁	1	1	yes	characteristic
29	#107	—	421	1	no	—
30	C₂×Q₈	—	842	1	no	—
31	Q₁₆	—	842	1	no	—
32	Q₁₆	—	421	2	no	—
33	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
34	C₄⋊C₄	—	842	1	no	—
35	C₁₆	—	2	1	no	—
36	Q₈	—	842	2	no	—
37	Q₈	—	1684	1	no	—
38	C₂×C₄	—	1684	1	no	—
39	C₂×C₄	—	1684	1	no	—
40	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
41	C₈	C₄×D₄₂₁	1	1	yes	characteristic
42	C₈	C₂²⋊D₄₂₁	1	1	yes	characteristic
43	C₂²	D₃₃₆₈	1	1	yes	characteristic
44	C₄	—	3368	1	no	—
45	C₄	—	1684	2	no	—
46	C₄	#20269748	1	1	yes	characteristic
47	C₄	#20269760	1	1	yes	characteristic
48	C₂	#27212288	1	1	yes	characteristic
49	C₂	#27212311	1	1	yes	characteristic
50	C₂	Dic₃₃₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃₃₆₈
commutator	C₈×C₄₂₁	C₂×C₄
Frattini	C₂×C₈	C₂×D₄₂₁
Fitting	C₂×C₁₆×C₄₂₁	C₂
socle, prime radical	C₂²×C₄₂₁	D₈
cosocle	C₂×C₈×C₄₂₁	C₂²
hypercenter	C₂×C₁₆	D₄₂₁
hypocenter	C₄₂₁	#358
LCS	G ▷ C₈×C₄₂₁ ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
6. Dic₁₆₈₄	C₄	semidirect
25. C₄₂₁	#358	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₆₈₄	2
#27212300	2
C₂×C₁₆×C₄₂₁	2
#358	421