Subgroups — #41002964

Order 26944.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₈₄₂	C₂	1	1	yes	maximal, characteristic
2	#27212295	C₂	1	2	yes	maximal
3	C₄×#16809890	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₄₂₁	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₈₄₂	C₄	1	1	yes	characteristic
7	C₂×#16809890	C₂²	1	1	yes	characteristic
8	C₂×#16809890	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₄₂₁	C₄	1	1	yes	characteristic
10	#20269750	C₄	1	2	yes	—
11	C₄×Dic₄₂₁	C₂²	1	2	yes	—
12	C₂×C₈×C₄₂₁	C₂²	1	2	yes	—
13	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
14	Dic₈₄₂	C₈	1	4	yes	complemented
15	C₂×Dic₄₂₁	C₂×C₄	1	2	yes	—
16	C₂×Dic₄₂₁	C₂×C₄	1	2	yes	—
17	#16809890	—	2	1	no	—
18	#16809890	Q₈	1	2	yes	complemented
19	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
22	C₈×C₄₂₁	—	2	2	no	—
23	C₂²×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₄₂₁	—	2	1	no	—
25	C₄×C₄₂₁	C₂×C₈	1	2	yes	—
26	C₄×C₄₂₁	C₂×Q₈	1	1	yes	characteristic
27	C₄×C₄₂₁	#41	1	1	yes	characteristic
28	Dic₄₂₁	—	2	2	no	—
29	Dic₄₂₁	C₂×C₈	1	4	yes	—
30	C₂×C₄₂₁	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₄₂₁	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₄₂₁	#131	1	1	yes	characteristic
33	C₄₂₁	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
34	C₈×Q₈	—	421	1	no	maximal, Sylow
35	C₄×Q₈	—	421	1	no	—
36	C₄⋊C₈	—	421	2	no	—
37	C₄⋊C₈	D₄₂₁	1	1	yes	characteristic
38	C₄×C₈	—	421	2	no	—
39	C₄×C₈	—	421	1	no	—
40	C₂×Q₈	—	421	1	no	—
41	C₂×C₈	—	421	1	no	—
42	C₂×C₈	—	421	1	no	—
43	C₂×C₈	C₂×D₄₂₁	1	2	yes	—
44	C₄⋊C₄	—	421	2	no	—
45	C₄⋊C₄	—	421	1	no	—
46	C₄²	—	421	2	no	—
47	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
48	Q₈	—	421	4	no	—
49	C₂×C₄	—	421	2	no	—
50	C₂×C₄	—	421	2	no	—
51	C₂×C₄	C₄×D₄₂₁	1	1	yes	characteristic
52	C₂×C₄	C₄×D₄₂₁	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
54	C₈	—	2	2	no	—
55	C₈	—	842	1	no	—
56	C₈	—	421	2	no	—
57	C₂²	C₂×C₄×D₄₂₁	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	842	2	no	—
60	C₄	—	421	4	no	—
61	C₄	C₈×D₄₂₁	1	2	yes	—
62	C₄	Q₈×D₄₂₁	1	1	yes	characteristic
63	C₄	#20269770	1	1	yes	characteristic
64	C₂	#27212331	1	1	yes	characteristic
65	C₂	C₂×C₈×D₄₂₁	1	1	yes	characteristic
66	C₂	#27212374	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₄₂₁
commutator	C₂×C₄₂₁	C₂²×C₈
Fitting	C₄⋊C₈×C₄₂₁	C₂
socle, prime radical	C₂²×C₄₂₁	C₂²×C₄
cosocle	C₂×C₄×C₄₂₁	C₂³
hypercenter	C₄⋊C₈	D₄₂₁
hypocenter	C₄₂₁	C₈×Q₈
LCS	G ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
14. Dic₈₄₂	C₈	semidirect
18. #16809890	Q₈	semidirect
33. C₄₂₁	C₈×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₈₄₂	2
#27212295	2
C₄×#16809890	2
C₈×Dic₄₂₁	2
C₄⋊C₈×C₄₂₁	2
C₈×Q₈	421