Subgroups — #41003281

Order 26944.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212388	C₂	1	1	yes	maximal, characteristic
2	#27212312	C₂	1	2	yes	maximal, complemented
3	#27212309	C₂	1	2	yes	maximal, complemented
4	#27212306	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₁₆×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#20269771	—	2	1	no	—
7	Dic₁₆₈₄	C₂²	1	1	yes	characteristic
8	#20269760	—	2	1	no	—
9	D₃₃₆₈	C₂²	1	1	yes	characteristic
10	#20269746	C₂²	1	2	yes	—
11	C₂×Q₈×C₄₂₁	—	2	1	no	—
12	Q₁₆×C₄₂₁	—	2	1	no	—
13	Q₁₆×C₄₂₁	C₂²	1	2	yes	—
14	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
15	Dic₈₄₂	—	2	1	no	—
16	D₁₆₈₄	—	2	1	no	—
17	C₂²⋊D₄₂₁	—	4	1	no	—
18	C₄×D₄₂₁	—	4	1	no	—
19	Q₈×C₄₂₁	—	2	2	no	—
20	Q₈×C₄₂₁	—	4	1	no	—
21	C₂×C₄×C₄₂₁	—	4	1	no	—
22	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
23	C₈×C₄₂₁	D₄	1	1	yes	characteristic
24	C₈×C₄₂₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄₂₁	D₈	1	1	yes	characteristic
26	C₂×D₄₂₁	—	4	1	no	—
27	C₄×C₄₂₁	—	4	2	no	—
28	C₄×C₄₂₁	D₈	1	1	yes	characteristic
29	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
30	Dic₄₂₁	—	4	1	no	—
31	C₂×C₄₂₁	—	2	1	no	—
32	C₂×C₄₂₁	C₂×D₈	1	1	yes	characteristic
33	D₄₂₁	—	8	1	no	—
34	C₄₂₁	#510	1	1	yes	characteristic, complemented, Sylow
35	#510	—	421	1	no	maximal, Sylow
36	#135	—	421	1	no	—
37	C₂×Q₁₆	D₄₂₁	1	1	yes	characteristic, complemented
38	Q₃₂	—	421	2	no	—
39	SD₃₂	—	421	2	no	—
40	M₃₂	—	421	1	no	—
41	#41	—	842	1	no	—
42	C₂×Q₈	—	2	1	no	—
43	Q₁₆	—	2	1	no	—
44	Q₁₆	—	421	1	no	—
45	Q₁₆	C₂×D₄₂₁	1	2	yes	—
46	SD₁₆	—	842	1	no	—
47	D₈	—	421	1	no	—
48	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
49	C₁₆	—	421	2	no	—
50	Q₈	—	2	2	no	—
51	Q₈	—	4	1	no	—
52	Q₈	—	842	1	no	—
53	D₄	—	842	1	no	—
54	D₄	—	1684	1	no	—
55	C₂×C₄	—	4	1	no	—
56	C₂×C₄	—	1684	1	no	—
57	C₂×C₄	C₂²⋊D₄₂₁	1	1	yes	characteristic
58	C₈	C₂²⋊D₄₂₁	1	1	yes	characteristic
59	C₈	C₂²×D₄₂₁	1	1	yes	characteristic
60	C₂²	—	1684	1	no	—
61	C₂²	#20269756	1	1	yes	characteristic
62	C₄	—	4	2	no	—
63	C₄	—	1684	1	no	—
64	C₄	#20269756	1	1	yes	characteristic
65	C₄	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
66	C₂	—	2	1	no	—
67	C₂	—	3368	1	no	—
68	C₂	C₂×#20269756	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#20269756
commutator, cosocle	C₈×C₄₂₁	C₂³
Frattini	C₈	C₂²×D₄₂₁
Fitting	C₂×Q₁₆×C₄₂₁	C₂
socle	C₂×C₄₂₁	C₂×D₈
prime radical	#27212388	C₂
hypercenter	C₂×Q₁₆	D₄₂₁
hypocenter	C₄₂₁	#510
LCS	G ▷ C₈×C₄₂₁ ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #27212312	C₂	semidirect
3. #27212309	C₂	semidirect
4. #27212306	C₂	semidirect
5. C₂×Q₁₆×C₄₂₁	C₂	semidirect
34. C₄₂₁	#510	semidirect
37. C₂×Q₁₆	D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
#27212388	2
#27212312	2
#27212309	2
#27212306	2
C₂×Q₁₆×C₄₂₁	2
#510	421