Subgroups — #14520253

Order 1968.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#11758315	C₂	1	2	yes	maximal, complemented
3	C₂×Dic₁₂₃	C₂	1	2	yes	maximal, complemented
4	C₄×D₁₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₂₄₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#10620308	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₄₁	C₂²	1	2	yes	—
8	C₂×D₁₂₃	C₂²	1	1	yes	characteristic
9	C₃×C₄×C₄₁	C₂²	1	1	yes	characteristic
10	Dic₁₂₃	C₂²	1	2	yes	—
11	Dic₁₂₃	C₂²	1	1	yes	characteristic
12	D₄×C₄₁	D₃	1	1	yes	characteristic, complemented
13	C₂²⋊D₄₁	—	3	2	no	—
14	C₂×Dic₄₁	—	3	2	no	—
15	C₄×D₄₁	—	3	1	no	—
16	Dic₈₂	—	3	1	no	—
17	C₂×C₃×C₄₁	—	2	2	no	—
18	C₂×C₃×C₄₁	C₂³	1	1	yes	characteristic
19	D₁₂₃	—	2	1	no	—
20	C₂²×C₄₁	C₂×D₃	1	2	yes	—
21	C₂×D₄₁	—	3	1	no	—
22	C₄×C₄₁	C₂×D₃	1	1	yes	characteristic
23	Dic₄₁	—	3	2	no	—
24	Dic₄₁	—	3	1	no	—
25	C₃×C₄₁	#41	1	1	yes	characteristic, complemented, Hall
26	C₂×C₄₁	—	2	2	no	—
27	C₂×C₄₁	C₂²×D₃	1	1	yes	characteristic
28	D₄₁	—	6	1	no	—
29	#237	—	41	1	no	maximal, Hall
30	C₄₁	#237	1	1	yes	characteristic, complemented, Sylow
31	C₃×D₄	D₄₁	1	1	yes	characteristic, complemented
32	#67	—	41	2	no	—
33	C₂×Dic₃	—	41	2	no	—
34	C₄×D₃	—	41	1	no	—
35	Dic₆	—	41	1	no	—
36	#41	—	123	1	no	Sylow
37	C₂²×C₃	C₂×D₄₁	1	2	yes	—
38	C₂×D₃	—	41	1	no	—
39	C₃×C₄	C₂×D₄₁	1	1	yes	characteristic
40	Dic₃	—	41	2	no	—
41	Dic₃	—	41	1	no	—
42	Q₈	—	123	1	no	—
43	D₄	—	123	2	no	—
44	D₄	D₁₂₃	1	1	yes	characteristic, complemented
45	C₂×C₄	—	123	2	no	—
46	C₂×C₄	—	123	1	no	—
47	C₂×C₃	—	2	2	no	—
48	C₂×C₃	C₂²×D₄₁	1	1	yes	characteristic
49	D₃	—	82	1	no	—
50	C₂²	—	123	1	no	—
51	C₂²	C₂×D₁₂₃	1	2	yes	—
52	C₄	—	123	1	no	—
53	C₄	—	123	2	no	—
54	C₄	C₂×D₁₂₃	1	1	yes	characteristic
55	C₃	#10620308	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	2	2	no	—
57	C₂	—	246	1	no	—
58	C₂	C₂²×D₁₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×D₁₂₃
commutator, socle, cosocle	C₂×C₃×C₄₁	C₂³
Fitting	C₃×D₄×C₄₁	C₂
hypercenter	D₄	D₁₂₃
hypocenter	C₃×C₄₁	#41
LCS	G ▷ C₂×C₃×C₄₁ ▷ C₃×C₄₁
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. C₃×D₄×C₄₁	C₂	semidirect
2. #11758315	C₂	semidirect
3. C₂×Dic₁₂₃	C₂	semidirect
4. C₄×D₁₂₃	C₂	semidirect
5. Dic₂₄₆	C₂	semidirect
12. D₄×C₄₁	D₃	semidirect
25. C₃×C₄₁	#41	semidirect
30. C₄₁	#237	semidirect
31. C₃×D₄	D₄₁	semidirect
44. D₄	D₁₂₃	semidirect
55. C₃	#10620308	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₄×C₄₁	2
#11758315	2
C₂×Dic₁₂₃	2
C₄×D₁₂₃	2
Dic₂₄₆	2
#10620308	3
#237	41