Subgroups — #14520105

Order 1968.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₃×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×Dic₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#10620281	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₄₁	C₂²	1	1	yes	characteristic
6	C₂×D₁₂₃	—	2	1	no	—
7	C₂×D₁₂₃	—	2	1	no	—
8	C₂×D₁₂₃	C₄	1	2	yes	complemented
9	C₃×Dic₄₁	—	2	1	no	—
10	Dic₃×C₄₁	—	2	1	no	—
11	C₂²×D₄₁	—	3	1	no	—
12	C₂×C₄×C₄₁	—	3	1	no	—
13	C₂×Dic₄₁	D₃	1	1	yes	characteristic, complemented
14	C₂×C₃×C₄₁	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃×C₄₁	D₄	1	1	yes	characteristic
16	C₂×C₃×C₄₁	D₄	1	1	yes	characteristic
17	D₁₂₃	—	2	2	no	—
18	C₂²×C₄₁	C₂×D₃	1	1	yes	characteristic
19	C₂×D₄₁	—	3	2	no	—
20	C₂×D₄₁	—	6	1	no	—
21	C₂×D₄₁	—	6	1	no	—
22	C₄×C₄₁	—	6	1	no	—
23	Dic₄₁	—	2	1	no	—
24	C₃×C₄₁	C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
25	C₂×C₄₁	C₄×D₃	1	1	yes	characteristic
26	C₂×C₄₁	D₁₂	1	1	yes	characteristic
27	C₂×C₄₁	#67	1	1	yes	characteristic
28	D₄₁	—	6	2	no	—
29	(C₂×D₃)⋊C₄	—	41	1	no	maximal, Hall
30	C₄₁	(C₂×D₃)⋊C₄	1	1	yes	characteristic, complemented, Sylow
31	C₂²×D₃	—	41	1	no	—
32	C₂×C₃×C₄	—	41	1	no	—
33	C₂×Dic₃	D₄₁	1	1	yes	characteristic, complemented
34	C₂²⋊C₄	—	123	1	no	Sylow
35	C₂²×C₃	C₂×D₄₁	1	1	yes	characteristic
36	C₂×D₃	—	41	2	no	—
37	C₂×D₃	—	82	1	no	—
38	C₂×D₃	—	82	1	no	—
39	C₃×C₄	—	82	1	no	—
40	Dic₃	—	2	1	no	—
41	C₂³	—	123	1	no	—
42	C₂×C₄	—	3	1	no	—
43	C₂×C₄	—	41	1	no	—
44	C₂×C₃	C₄×D₄₁	1	1	yes	characteristic
45	C₂×C₃	D₁₆₄	1	1	yes	characteristic
46	C₂×C₃	C₂²⋊D₄₁	1	1	yes	characteristic
47	D₃	—	82	2	no	—
48	C₂²	—	123	2	no	—
49	C₂²	—	246	1	no	—
50	C₂²	—	246	1	no	—
51	C₂²	D₃×D₄₁	1	1	yes	characteristic
52	C₄	—	6	1	no	—
53	C₄	—	82	1	no	—
54	C₃	#10620281	1	1	yes	characteristic, complemented, Sylow
55	C₂	—	246	2	no	—
56	C₂	#11758295	1	1	yes	characteristic
57	C₂	#11758297	1	1	yes	characteristic
58	C₂	#11758298	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₃×D₄₁
commutator	C₂×C₃×C₄₁	C₂×C₄
Fitting, socle, cosocle	C₂²×C₃×C₄₁	C₂²
prime radical	C₂²×D₁₂₃	C₂
hypocenter	C₃×C₄₁	C₂²⋊C₄
LCS	G ▷ C₂×C₃×C₄₁ ▷ C₃×C₄₁
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₃×C₄₁	C₂	semidirect
3. C₂×C₃×Dic₄₁	C₂	semidirect
8. C₂×D₁₂₃	C₄	semidirect
13. C₂×Dic₄₁	D₃	semidirect
24. C₃×C₄₁	C₂²⋊C₄	semidirect
30. C₄₁	(C₂×D₃)⋊C₄	semidirect
33. C₂×Dic₃	D₄₁	semidirect
54. C₃	#10620281	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₂₃	2
C₂×Dic₃×C₄₁	2
C₂×C₃×Dic₄₁	2
#10620281	3
(C₂×D₃)⋊C₄	41