Subgroups — #41003908

Order 26944.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#13132997	C₂	1	8	yes	maximal, complemented
2	#27212445	C₂	1	6	yes	maximal
3	C₂×Q₈×D₄₂₁	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#13132997	C₂²	1	12	yes	—
5	C₂²⋊#13132997	C₂²	1	4	yes	—
6	C₄×#13132997	C₂²	1	12	yes	—
7	C₂×Dic₈₄₂	C₄	1	3	yes	complemented
8	Q₈×D₄₂₁	C₄	1	4	yes	complemented
9	Q₈×D₄₂₁	C₂²	1	4	yes	—
10	C₂×C₄×D₄₂₁	C₂²	1	3	yes	—
11	C₂×Q₈×C₄₂₁	C₄	1	1	yes	characteristic, complemented
12	C₂×#13132997	C₂³	1	8	yes	—
13	C₂²×D₄₂₁	C₂³	1	1	yes	characteristic
14	Dic₈₄₂	C₂×C₄	1	12	yes	—
15	C₂×Dic₄₂₁	C₂×C₄	1	3	yes	—
16	C₄×D₄₂₁	C₂×C₄	1	6	yes	—
17	C₄×D₄₂₁	C₂³	1	6	yes	—
18	Q₈×C₄₂₁	C₂×C₄	1	4	yes	—
19	C₂×C₄×C₄₂₁	C₂×C₄	1	3	yes	—
20	C₂²×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
21	C₂×D₄₂₁	—	2	2	no	—
22	C₂×D₄₂₁	C₂²×C₄	1	1	yes	characteristic
23	C₂×D₄₂₁	C₂⁴	1	1	yes	characteristic
24	#13132997	—	2	8	no	—
25	C₄×C₄₂₁	C₂²×C₄	1	6	yes	—
26	Dic₄₂₁	C₂²×C₄	1	6	yes	—
27	C₂×C₄₂₁	—	2	1	no	—
28	C₂×C₄₂₁	C₂³×C₄	1	1	yes	characteristic
29	D₄₂₁	—	2	1	no	—
30	D₄₂₁	2₋¹⁺⁴	1	2	yes	—
31	C₄₂₁	#519	1	1	yes	characteristic, complemented, Sylow
32	#519	—	421	1	no	maximal, Sylow
33	C₂²×Q₈	—	421	1	no	—
34	C₄×Q₈	—	421	8	no	—
35	#117	—	421	6	no	—
36	C₂×Q₈	—	421	4	no	—
37	C₂×Q₈	—	421	4	no	—
38	C₂×Q₈	—	421	3	no	—
39	C₂×Q₈	#13132997	1	1	yes	characteristic, complemented
40	C₂²×C₄	—	421	3	no	—
41	C₄⋊C₄	—	421	12	no	—
42	C₂²⋊C₄	—	421	4	no	—
43	C₄²	—	421	12	no	—
44	C₂³	—	421	1	no	—
45	Q₈	—	421	12	no	—
46	Q₈	C₂×#13132997	1	4	yes	—
47	C₂×C₄	—	421	8	no	—
48	C₂×C₄	—	421	3	no	—
49	C₂×C₄	—	421	6	no	—
50	C₂×C₄	—	421	6	no	—
51	C₂×C₄	C₂×#13132997	1	3	yes	—
52	C₂²	—	842	2	no	—
53	C₂²	—	421	1	no	—
54	C₂²	—	421	1	no	—
55	C₂²	C₂²×#13132997	1	1	yes	characteristic
56	C₄	—	842	8	no	—
57	C₄	—	421	6	no	—
58	C₄	C₂²×#13132997	1	6	yes	—
59	C₂	—	2	1	no	—
60	C₂	—	842	1	no	—
61	C₂	—	421	2	no	—
62	C₂	C₂³×#13132997	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#13132997
commutator, socle	C₂×C₄₂₁	C₂³×C₄
Fitting	C₂×Q₈×C₄₂₁	C₄
prime radical	C₂²×D₄₂₁	C₂³
cosocle	C₂×D₄₂₁	C₂⁴
hypercenter	C₂×Q₈	#13132997
hypocenter	C₄₂₁	#519
LCS	G ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#13132997	C₂	semidirect
7. C₂×Dic₈₄₂	C₄	semidirect
8. Q₈×D₄₂₁	C₄	semidirect
11. C₂×Q₈×C₄₂₁	C₄	semidirect
31. C₄₂₁	#519	semidirect
39. C₂×Q₈	#13132997	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#13132997	2
#27212445	2
C₂×Q₈×D₄₂₁	2
#519	421