Subgroups — #41003136

Order 26944.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212366	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₈₄₂	C₂	1	1	yes	maximal, characteristic
3	#27212300	C₂	1	2	yes	maximal
4	#27212291	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₈₄₂	—	2	1	no	—
7	C₂×Dic₈₄₂	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₄₂₁	—	2	1	no	—
9	C₄⋊Dic₄₂₁	C₂²	1	2	yes	—
10	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
11	#20269750	—	2	1	no	—
12	C₄×Dic₄₂₁	—	2	1	no	—
13	C₂×C₈×C₄₂₁	C₂²	1	2	yes	—
14	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
15	Dic₈₄₂	—	2	1	no	—
16	Dic₈₄₂	—	4	2	no	—
17	Dic₈₄₂	Q₈	1	2	yes	complemented
18	C₂×Dic₄₂₁	—	2	1	no	—
19	C₂×Dic₄₂₁	—	2	1	no	—
20	C₂×Dic₄₂₁	—	2	2	no	—
21	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
22	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
24	C₈×C₄₂₁	—	2	2	no	—
25	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
26	C₄×C₄₂₁	—	2	1	no	—
27	C₄×C₄₂₁	Q₁₆	1	2	yes	—
28	C₄×C₄₂₁	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₄₂₁	#41	1	1	yes	characteristic
30	Dic₄₂₁	—	2	2	no	—
31	Dic₄₂₁	—	4	1	no	—
32	Dic₄₂₁	—	4	2	no	—
33	C₂×C₄₂₁	C₂²⋊Q₈	1	1	yes	characteristic
34	C₂×C₄₂₁	C₂×Q₁₆	1	1	yes	characteristic
35	C₂×C₄₂₁	AGL(1,ℤ/8)	1	1	yes	characteristic
36	C₄₂₁	#477	1	1	yes	characteristic, complemented, Sylow
37	#477	—	421	1	no	maximal, Sylow
38	C₄⋊Q₈	—	421	1	no	—
39	C₄×Q₈	—	421	1	no	—
40	#107	—	421	2	no	—
41	C₄⋊C₈	D₄₂₁	1	1	yes	characteristic
42	#103	—	421	2	no	—
43	C₂×Q₈	—	842	1	no	—
44	C₂×Q₈	—	421	1	no	—
45	C₂×C₈	C₂×D₄₂₁	1	2	yes	—
46	C₄⋊C₄	—	842	1	no	—
47	C₄⋊C₄	—	842	1	no	—
48	C₄⋊C₄	—	421	2	no	—
49	C₄⋊C₄	—	421	1	no	—
50	C₄²	—	842	1	no	—
51	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
52	Q₈	—	842	1	no	—
53	Q₈	—	1684	2	no	—
54	Q₈	—	421	2	no	—
55	C₂×C₄	—	842	1	no	—
56	C₂×C₄	—	842	1	no	—
57	C₂×C₄	—	842	2	no	—
58	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
59	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
61	C₈	—	2	2	no	—
62	C₂²	C₂×D₁₆₈₄	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	842	2	no	—
65	C₄	—	1684	2	no	—
66	C₄	—	1684	1	no	—
67	C₄	Dic₁₆₈₄	1	2	yes	—
68	C₄	Q₈×D₄₂₁	1	1	yes	characteristic
69	C₄	#20269770	1	1	yes	characteristic
70	C₂	#27212346	1	1	yes	characteristic
71	C₂	C₂×Dic₁₆₈₄	1	1	yes	characteristic
72	C₂	#27212394	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₆₈₄
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	C₄⋊C₈×C₄₂₁	C₂
socle, prime radical	C₂²×C₄₂₁	C₂×D₄
hypercenter	C₄⋊C₈	D₄₂₁
hypocenter	C₄₂₁	#477
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
17. Dic₈₄₂	Q₈	semidirect
36. C₄₂₁	#477	semidirect

Maximal subgroups

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

subgroup	index
#27212366	2
C₄×Dic₈₄₂	2
#27212300	2
#27212291	2
C₄⋊C₈×C₄₂₁	2
#477	421