Subgroups — #41003754

Order 26944.

56 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	#27212437	C₂	1	1	yes	maximal, characteristic, complemented
3	#27212385	C₂	1	1	yes	maximal, characteristic
4	#20269781	—	2	1	no	—
5	#20269774	—	2	1	no	—
6	#20269770	—	2	1	no	—
7	Dic₁₆₈₄	C₄	1	1	yes	characteristic
8	#20269759	—	2	1	no	—
9	C₈×D₄₂₁	C₂²	1	1	yes	characteristic
10	D₈×C₄₂₁	C₄	1	1	yes	characteristic
11	#16809897	—	4	1	no	—
12	Dic₈₄₂	—	2	1	no	—
13	C₂²⋊D₄₂₁	—	4	1	no	—
14	C₂×Dic₄₂₁	—	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	SD₁₆	1	1	yes	characteristic
21	C₄×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
22	Dic₄₂₁	—	4	1	no	—
23	Dic₄₂₁	D₈	1	1	yes	characteristic
24	C₂×C₄₂₁	—	8	1	no	—
25	C₂×C₄₂₁	#102	1	1	yes	characteristic
26	D₄₂₁	—	2	1	no	—
27	C₄₂₁	#359	1	1	yes	characteristic, complemented, Sylow
28	#359	—	421	1	no	maximal, Sylow
29	#135	—	421	1	no	—
30	C₂×C₁₆	—	421	1	no	—
31	#108	—	421	1	no	—
32	#41	—	842	1	no	—
33	Q₁₆	—	421	1	no	—
34	SD₁₆	—	842	1	no	—
35	D₈	#13132997	1	1	yes	characteristic
36	M₁₆	—	842	1	no	—
37	C₂×C₈	—	421	1	no	—
38	C₁₆	—	842	1	no	—
39	Q₈	—	842	1	no	—
40	D₄	—	2	1	no	—
41	D₄	—	1684	1	no	—
42	C₂×C₄	—	1684	1	no	—
43	C₂×C₄	—	421	1	no	—
44	C₈	—	1684	1	no	—
45	C₈	—	421	1	no	—
46	C₈	C₂×#13132997	1	1	yes	characteristic
47	C₂²	—	4	1	no	—
48	C₂²	—	421	1	no	—
49	C₄	—	1684	1	no	—
50	C₄	—	421	1	no	—
51	C₄	C₂²⋊#13132997	1	1	yes	characteristic
52	C₂	—	8	1	no	—
53	C₂	—	842	1	no	—
54	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	#27212385	C₂
cosocle	C₈×D₄₂₁	C₂²
hypercenter	D₈	#13132997
hypocenter	C₄₂₁	#359
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. #27212437	C₂	semidirect
27. C₄₂₁	#359	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
#27212437	2
#27212385	2
#359	421