Subgroups — #41003752

Order 26944.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄₂₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#27212435	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×D₄₂₁	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#13132997	—	2	1	no	—
5	#20269774	—	2	1	no	—
6	D₄×D₄₂₁	—	2	1	no	—
7	D₃₃₆₈	C₄	1	1	yes	characteristic, complemented
8	#20269756	—	2	1	no	—
9	C₈×D₄₂₁	C₂²	1	1	yes	characteristic
10	D₈×C₄₂₁	C₄	1	1	yes	characteristic, complemented
11	C₂×#13132997	—	4	1	no	—
12	C₂²×D₄₂₁	—	4	1	no	—
13	D₁₆₈₄	—	2	1	no	—
14	C₂²⋊D₄₂₁	—	4	1	no	—
15	C₄×D₄₂₁	D₄	1	1	yes	characteristic
16	#16809890	D₄	1	1	yes	characteristic
17	D₄×C₄₂₁	—	2	1	no	—
18	C₈×C₄₂₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄₂₁	—	4	1	no	—
20	C₂×D₄₂₁	—	4	1	no	—
21	C₂×D₄₂₁	—	8	1	no	—
22	C₂×D₄₂₁	—	8	1	no	—
23	C₂×D₄₂₁	D₈	1	1	yes	characteristic
24	#13132997	—	8	1	no	—
25	C₄×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
26	Dic₄₂₁	SD₁₆	1	1	yes	characteristic
27	C₂×C₄₂₁	—	8	1	no	—
28	C₂×C₄₂₁	#102	1	1	yes	characteristic
29	D₄₂₁	—	8	1	no	—
30	D₄₂₁	D₁₆	1	1	yes	characteristic
31	D₄₂₁	SD₃₂	1	1	yes	characteristic
32	C₄₂₁	#357	1	1	yes	characteristic, complemented, Sylow
33	#357	—	421	1	no	maximal, Sylow
34	C₂×D₈	—	421	1	no	—
35	C₂×C₁₆	—	421	1	no	—
36	#107	—	421	1	no	—
37	C₂×D₄	—	842	1	no	—
38	D₈	—	842	1	no	—
39	D₈	—	421	1	no	—
40	D₈	#13132997	1	1	yes	characteristic, complemented
41	C₂×C₈	—	421	1	no	—
42	C₄⋊C₄	—	842	1	no	—
43	C₁₆	—	842	1	no	—
44	C₂³	—	1684	1	no	—
45	D₄	—	2	1	no	—
46	D₄	—	842	1	no	—
47	D₄	—	1684	1	no	—
48	C₂×C₄	—	1684	1	no	—
49	C₂×C₄	—	421	1	no	—
50	C₈	—	421	1	no	—
51	C₈	C₂×#13132997	1	1	yes	characteristic
52	C₂²	—	4	1	no	—
53	C₂²	—	3368	1	no	—
54	C₂²	—	3368	1	no	—
55	C₂²	—	1684	1	no	—
56	C₂²	—	421	1	no	—
57	C₄	—	3368	1	no	—
58	C₄	—	421	1	no	—
59	C₄	C₂²⋊#13132997	1	1	yes	characteristic
60	C₂	—	8	1	no	—
61	C₂	—	3368	1	no	—
62	C₂	—	421	1	no	—
63	C₂	—	421	1	no	—
64	C₂	#27212428	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27212428
commutator	C₈×C₄₂₁	C₂×C₄
Frattini	C₈	C₂×#13132997
Fitting	D₈×C₄₂₁	C₄
socle	C₂×C₄₂₁	#102
prime radical	D₈×D₄₂₁	C₂
cosocle	C₈×D₄₂₁	C₂²
hypercenter	D₈	#13132997
hypocenter	C₄₂₁	#357
LCS	G ▷ C₈×C₄₂₁ ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. D₄₂₁⋊C₁₆	C₂	semidirect
2. #27212435	C₂	semidirect
7. D₃₃₆₈	C₄	semidirect
10. D₈×C₄₂₁	C₄	semidirect
32. C₄₂₁	#357	semidirect
40. D₈	#13132997	semidirect

Maximal subgroups

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

subgroup	index
D₄₂₁⋊C₁₆	2
#27212435	2
D₈×D₄₂₁	2
#357	421