Subgroups — #41002877

Order 26944.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄₂₁	C₂	1	1	yes	maximal, characteristic
2	#27212324	C₂	1	2	yes	maximal
3	#27212300	C₂	1	1	yes	maximal, characteristic, complemented
4	#27212297	C₂	1	2	yes	maximal, complemented
5	#107×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄₂₁	C₂²	1	1	yes	characteristic
7	C₂×#16809890	C₂²	1	1	yes	characteristic
8	C₈×D₄₂₁	C₄	1	4	yes	complemented
9	C₄⋊Dic₄₂₁	C₂²	1	2	yes	—
10	#20269748	—	2	2	no	—
11	C₄×Dic₄₂₁	—	2	2	no	—
12	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₄₂₁	C₂²	1	2	yes	—
14	C₂²×D₄₂₁	D₄	1	1	yes	characteristic
15	C₂×Dic₄₂₁	—	2	2	no	—
16	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
17	C₄×D₄₂₁	C₂×C₄	1	2	yes	—
18	C₄×D₄₂₁	Q₈	1	2	yes	—
19	#16809890	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₄₂₁	—	2	2	no	—
21	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
22	C₈×C₄₂₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
24	C₂×D₄₂₁	—	2	1	no	—
25	C₂×D₄₂₁	—	2	1	no	—
26	C₂×D₄₂₁	C₄⋊C₄	1	2	yes	—
27	C₄×C₄₂₁	—	4	2	no	—
28	C₄×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₄₂₁	C₂×Q₈	1	1	yes	characteristic
30	Dic₄₂₁	—	4	2	no	—
31	Dic₄₂₁	C₄⋊C₄	1	2	yes	—
32	C₂×C₄₂₁	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄₂₁	#135	1	1	yes	characteristic
34	C₂×C₄₂₁	#135	1	1	yes	characteristic
35	D₄₂₁	—	2	2	no	—
36	C₄₂₁	#427	1	1	yes	characteristic, complemented, Sylow
37	#427	—	421	1	no	maximal, Sylow
38	C₂²×C₈	—	421	1	no	—
39	#117	—	421	2	no	—
40	#107	—	421	1	no	—
41	#107	D₄₂₁	1	1	yes	characteristic, complemented
42	#106	—	421	2	no	—
43	C₂²×C₄	—	421	1	no	—
44	C₂×C₈	—	421	1	no	—
45	C₂×C₈	—	421	4	no	—
46	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
47	C₄⋊C₄	—	421	2	no	—
48	C₄⋊C₄	C₂×D₄₂₁	1	2	yes	—
49	C₂²⋊C₄	—	842	2	no	—
50	C₄²	—	842	2	no	—
51	C₂³	—	421	1	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	842	2	no	—
54	C₂×C₄	—	421	1	no	—
55	C₂×C₄	—	421	2	no	—
56	C₂×C₄	—	421	2	no	—
57	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
58	C₈	—	421	2	no	—
59	C₈	C₄×D₄₂₁	1	2	yes	—
60	C₂²	—	842	1	no	—
61	C₂²	—	842	1	no	—
62	C₂²	—	421	2	no	—
63	C₂²	D₄×D₄₂₁	1	1	yes	characteristic
64	C₄	—	4	2	no	—
65	C₄	—	1684	2	no	—
66	C₄	—	421	2	no	—
67	C₄	C₂×C₄×D₄₂₁	1	1	yes	characteristic
68	C₄	Q₈×D₄₂₁	1	1	yes	characteristic
69	C₂	—	842	2	no	—
70	C₂	C₄⋊C₄×D₄₂₁	1	1	yes	characteristic
71	C₂	#27212385	1	1	yes	characteristic
72	C₂	#27212389	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₂₁
commutator	C₄×C₄₂₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#107×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
prime radical	C₂²×D₄₂₁	D₄
cosocle	C₂×C₄×C₄₂₁	C₂³
hypercenter	#107	D₄₂₁
hypocenter	C₄₂₁	#427
LCS	G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
3. #27212300	C₂	semidirect
4. #27212297	C₂	semidirect
5. #107×C₄₂₁	C₂	semidirect
8. C₈×D₄₂₁	C₄	semidirect
36. C₄₂₁	#427	semidirect
41. #107	D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄₂₁	2
#27212324	2
#27212300	2
#27212297	2
#107×C₄₂₁	2
#427	421