Subgroups — #41002963

Order 26944.

104 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	#27212372	C₂	1	2	yes	maximal, complemented
3	#27212302	C₂	1	1	yes	maximal, characteristic, complemented
4	#27212294	C₂	1	2	yes	maximal, complemented
5	#97×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#20269771	C₂²	1	2	yes	—
7	Dic₁₆₈₄	C₄	1	1	yes	characteristic, complemented
8	#20269760	C₄	1	2	yes	complemented
9	D₃₃₆₈	C₄	1	1	yes	characteristic, complemented
10	#20269755	C₂²	1	2	yes	—
11	#20269754	—	2	2	no	—
12	C₈×D₄₂₁	—	2	2	no	—
13	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
15	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
16	Dic₈₄₂	C₂×C₄	1	2	yes	—
17	D₁₆₈₄	C₂×C₄	1	2	yes	—
18	C₂²⋊D₄₂₁	—	2	2	no	—
19	C₄×D₄₂₁	—	2	2	no	—
20	#16809890	—	2	2	no	—
21	C₂×C₄×C₄₂₁	—	2	1	no	—
22	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
23	C₈×C₄₂₁	C₂×C₄	1	1	yes	characteristic
24	C₈×C₄₂₁	C₂×C₄	1	1	yes	characteristic
25	C₈×C₄₂₁	D₄	1	2	yes	—
26	C₂²×C₄₂₁	#41	1	1	yes	characteristic
27	C₂×D₄₂₁	—	2	2	no	—
28	C₄×C₄₂₁	—	4	1	no	—
29	C₄×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
31	Dic₄₂₁	—	2	2	no	—
32	C₂×C₄₂₁	—	2	1	no	—
33	C₂×C₄₂₁	C₄×D₄	1	1	yes	characteristic
34	D₄₂₁	—	4	2	no	—
35	C₄₂₁	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	421	1	no	maximal, Sylow
37	#135	—	421	1	no	—
38	#131	—	421	2	no	—
39	#108	—	421	1	no	—
40	#104	—	421	2	no	—
41	#97	D₄₂₁	1	1	yes	characteristic, complemented
42	#41	—	421	2	no	—
43	Q₁₆	—	421	1	no	—
44	SD₁₆	—	421	2	no	—
45	D₈	—	421	1	no	—
46	M₁₆	—	842	2	no	—
47	M₁₆	—	421	2	no	—
48	C₂×C₈	—	842	2	no	—
49	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
50	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
51	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
52	Q₈	—	421	2	no	—
53	D₄	—	842	2	no	—
54	D₄	—	421	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	842	2	no	—
57	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
58	C₈	—	842	2	no	—
59	C₈	C₄×D₄₂₁	1	1	yes	characteristic
60	C₈	C₄×D₄₂₁	1	1	yes	characteristic
61	C₈	D₁₆₈₄	1	2	yes	—
62	C₂²	—	842	2	no	—
63	C₂²	#20269771	1	1	yes	characteristic
64	C₄	—	4	1	no	—
65	C₄	—	842	2	no	—
66	C₄	C₂×C₄×D₄₂₁	1	1	yes	characteristic
67	C₄	C₂×D₁₆₈₄	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	1684	2	no	—
70	C₂	C₄×D₁₆₈₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₁₆₈₄
commutator	C₄×C₄₂₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#97×C₄₂₁	C₂
socle	C₂×C₄₂₁	C₄×D₄
prime radical	#27212388	C₂
cosocle	C₂×C₄×C₄₂₁	C₂³
hypercenter	#97	D₄₂₁
hypocenter	C₄₂₁	#444
LCS	G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #97

Complements

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

N	H ≅ G/N	type
2. #27212372	C₂	semidirect
3. #27212302	C₂	semidirect
4. #27212294	C₂	semidirect
5. #97×C₄₂₁	C₂	semidirect
7. Dic₁₆₈₄	C₄	semidirect
8. #20269760	C₄	semidirect
9. D₃₃₆₈	C₄	semidirect
35. C₄₂₁	#444	semidirect
41. #97	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
#27212372	2
#27212302	2
#27212294	2
#97×C₄₂₁	2
#444	421