Subgroups — #14520200

Order 1968.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	Dic₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₁₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#11758297	C₂	1	2	yes	maximal, complemented
5	Dic₃×D₄₁	C₂	1	2	yes	maximal, complemented
6	#10620310	—	3	1	no	maximal, Hall
7	C₂×D₁₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₃×D₄₁	C₂²	1	2	yes	—
9	C₃×C₄×C₄₁	C₂²	1	1	yes	characteristic
10	Dic₁₂₃	C₂²	1	1	yes	characteristic
11	Dic₃×C₄₁	C₂²	1	2	yes	—
12	Q₈×C₄₁	—	3	1	no	—
13	D₁₆₄	—	3	2	no	—
14	D₁₆₄	D₃	1	1	yes	characteristic, complemented
15	C₄×D₄₁	—	3	2	no	—
16	C₄×D₄₁	—	3	1	no	—
17	C₂×C₃×C₄₁	C₂³	1	1	yes	characteristic
18	D₁₂₃	—	2	1	no	—
19	C₃×D₄₁	—	2	2	no	—
20	C₂×D₄₁	—	3	1	no	—
21	C₂×D₄₁	C₂×D₃	1	2	yes	—
22	C₄×C₄₁	—	3	2	no	—
23	C₄×C₄₁	C₂×D₃	1	1	yes	characteristic
24	Dic₄₁	—	3	1	no	—
25	C₃×C₄₁	#41	1	1	yes	characteristic, complemented, Hall
26	C₂×C₄₁	C₂²×D₃	1	1	yes	characteristic
27	D₄₁	—	2	2	no	—
28	D₄₁	—	6	1	no	—
29	#237	—	41	1	no	maximal, Hall
30	C₄₁	#237	1	1	yes	characteristic, complemented, Sylow
31	C₃×D₄	—	41	1	no	—
32	#67	—	41	2	no	—
33	C₂×Dic₃	—	41	2	no	—
34	C₄×D₃	—	41	1	no	—
35	Dic₆	D₄₁	1	1	yes	characteristic, complemented
36	#41	—	123	1	no	Sylow
37	C₂²×C₃	—	41	2	no	—
38	C₂×D₃	—	41	1	no	—
39	C₃×C₄	C₂×D₄₁	1	1	yes	characteristic
40	Dic₃	—	41	1	no	—
41	Dic₃	C₂×D₄₁	1	2	yes	—
42	Q₈	—	3	1	no	—
43	D₄	—	41	1	no	—
44	D₄	—	123	2	no	—
45	C₂×C₄	—	123	2	no	—
46	C₂×C₄	—	123	1	no	—
47	C₂×C₃	—	82	2	no	—
48	C₂×C₃	C₂²×D₄₁	1	1	yes	characteristic
49	D₃	—	82	1	no	—
50	C₂²	—	41	2	no	—
51	C₂²	—	123	1	no	—
52	C₄	—	3	2	no	—
53	C₄	—	123	1	no	—
54	C₄	D₃×D₄₁	1	1	yes	characteristic
55	C₃	#10620310	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	82	2	no	—
57	C₂	—	246	1	no	—
58	C₂	C₂×D₃×D₄₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×D₄₁
commutator, socle, cosocle	C₂×C₃×C₄₁	C₂³
Fitting	C₃×C₄×C₄₁	C₂²
hypercenter	C₄	D₃×D₄₁
hypocenter	C₃×C₄₁	#41
LCS	G ▷ C₂×C₃×C₄₁ ▷ C₃×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₄×D₁₂₃	C₂	semidirect
2. Dic₆×C₄₁	C₂	semidirect
3. C₃×D₁₆₄	C₂	semidirect
4. #11758297	C₂	semidirect
5. Dic₃×D₄₁	C₂	semidirect
14. D₁₆₄	D₃	semidirect
25. C₃×C₄₁	#41	semidirect
30. C₄₁	#237	semidirect
35. Dic₆	D₄₁	semidirect
55. C₃	#10620310	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₁₂₃	2
Dic₆×C₄₁	2
C₃×D₁₆₄	2
#11758297	2
Dic₃×D₄₁	2
#10620310	3
#237	41