Subgroups — #41002953

Order 26944.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₃₆₈	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₆₈₄	C₂	1	2	yes	maximal, complemented
3	#27212298	C₂	1	1	yes	maximal, characteristic, complemented
4	#27212288	C₂	1	2	yes	maximal, complemented
5	#97×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆₈₄	C₂²	1	2	yes	—
7	C₂×C₄×D₄₂₁	—	2	2	no	—
8	D₃₃₆₈	C₄	1	4	yes	complemented
9	C₄⋊Dic₄₂₁	C₂²	1	2	yes	—
10	#20269748	—	2	2	no	—
11	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
13	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
14	C₂²×D₄₂₁	—	2	2	no	—
15	D₁₆₈₄	—	2	2	no	—
16	D₁₆₈₄	C₂×C₄	1	4	yes	complemented
17	C₂×Dic₄₂₁	—	2	2	no	—
18	C₄×D₄₂₁	—	4	4	no	—
19	C₂×C₄×C₄₂₁	D₄	1	2	yes	—
20	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
21	C₈×C₄₂₁	—	2	1	no	—
22	C₈×C₄₂₁	C₂×C₄	1	2	yes	complemented
23	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
24	C₂×D₄₂₁	—	2	4	no	—
25	C₂×D₄₂₁	—	4	2	no	—
26	C₂×D₄₂₁	—	4	2	no	—
27	C₄×C₄₂₁	—	2	2	no	—
28	C₄×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₄₂₁	#41	1	1	yes	characteristic
30	Dic₄₂₁	—	4	2	no	—
31	C₂×C₄₂₁	C₄×D₄	1	1	yes	characteristic
32	C₂×C₄₂₁	AGL(1,ℤ/8)	1	2	yes	—
33	D₄₂₁	—	4	4	no	—
34	C₄₂₁	#442	1	1	yes	characteristic, complemented, Sylow
35	#442	—	421	1	no	maximal, Sylow
36	C₂×D₈	—	421	1	no	—
37	C₄×D₄	—	421	2	no	—
38	#106	—	421	1	no	—
39	#102	—	421	2	no	—
40	#97	D₄₂₁	1	1	yes	characteristic, complemented
41	C₂×D₄	—	421	2	no	—
42	C₂²×C₄	—	842	2	no	—
43	D₈	—	421	4	no	—
44	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
45	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
46	C₄⋊C₄	—	421	2	no	—
47	C₂²⋊C₄	—	842	2	no	—
48	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
49	C₂³	—	842	2	no	—
50	D₄	—	842	2	no	—
51	D₄	—	421	4	no	—
52	C₂×C₄	—	842	2	no	—
53	C₂×C₄	—	1684	4	no	—
54	C₂×C₄	D₁₆₈₄	1	2	yes	—
55	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	C₄×D₄₂₁	1	2	yes	complemented
58	C₂²	—	842	4	no	—
59	C₂²	—	1684	2	no	—
60	C₂²	—	1684	2	no	—
61	C₂²	C₂×D₁₆₈₄	1	1	yes	characteristic
62	C₄	—	2	2	no	—
63	C₄	—	1684	2	no	—
64	C₄	C₂×C₄×D₄₂₁	1	1	yes	characteristic
65	C₄	#20269771	1	1	yes	characteristic
66	C₂	—	1684	4	no	—
67	C₂	C₄×D₁₆₈₄	1	1	yes	characteristic
68	C₂	#27212394	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₆₈₄
commutator	C₄×C₄₂₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#97×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
prime radical	C₂×D₃₃₆₈	C₂
cosocle	C₂×C₄×C₄₂₁	C₂³
hypercenter	#97	D₄₂₁
hypocenter	C₄₂₁	#442
LCS	G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
2. C₄×D₁₆₈₄	C₂	semidirect
3. #27212298	C₂	semidirect
4. #27212288	C₂	semidirect
5. #97×C₄₂₁	C₂	semidirect
8. D₃₃₆₈	C₄	semidirect
16. D₁₆₈₄	C₂×C₄	semidirect
22. C₈×C₄₂₁	C₂×C₄	semidirect
34. C₄₂₁	#442	semidirect
40. #97	D₄₂₁	semidirect
57. C₈	C₄×D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₃₆₈	2
C₄×D₁₆₈₄	2
#27212298	2
#27212288	2
#97×C₄₂₁	2
#442	421