Subgroups — C₇×C₃⋊(C₇⋊#4223) #37563142

Order 24108.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇²×D₄₁	C₂	1	1	yes	maximal, characteristic
2	C₇×C₇⋊#4223	—	3	1	no	maximal, Hall
3	C₃×C₇²×C₄₁	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇²×D₄₁	D₃	1	1	yes	characteristic
5	C₃⋊(C₇⋊#4223)	C₇	1	1	yes	maximal, characteristic, direct product factor
6	C₇×C₃⋊#4223	—	7	1	no	maximal
7	C₇²×C₄₁	Dic₃	1	1	yes	characteristic, complemented, Hall
8	C₃×C₇×D₄₁	—	2	3	no	—
9	C₃×C₇×D₄₁	D₇	1	1	yes	characteristic
10	C₃×C₇×D₄₁	C₂×C₇	1	1	yes	characteristic
11	C₇⋊#4223	—	3	1	no	—
12	C₇×#4223	—	21	1	no	—
13	C₃×C₇×C₄₁	—	2	3	no	—
14	C₃×C₇×C₄₁	Dic₇	1	1	yes	characteristic, complemented
15	C₃×C₇×C₄₁	C₄×C₇	1	1	yes	characteristic, complemented
16	C₇×Dic₂₁	—	41	1	no	maximal, Hall
17	C₇×D₄₁	—	2	3	no	—
18	C₇×D₄₁	D₃×C₇	1	1	yes	characteristic
19	C₇×D₄₁	D₂₁	1	1	yes	characteristic
20	C₃⋊#4223	—	7	1	no	Hall
21	C₂×C₃×C₇²	—	41	1	no	—
22	C₇×C₄₁	—	2	3	no	—
23	C₇×C₄₁	C₇×Dic₃	1	1	yes	characteristic, complemented
24	C₇×C₄₁	Dic₂₁	1	1	yes	characteristic, complemented
25	C₃×D₄₁	C₇×D₇	1	1	yes	characteristic
26	C₇×Dic₇	—	123	1	no	Hall
27	#4223	—	21	1	no	Hall
28	C₃×C₇²	#4223	1	1	yes	characteristic, complemented, Hall
29	C₃×C₄₁	C₇×Dic₇	1	1	yes	characteristic, complemented, Hall
30	C₂×C₇²	—	41	1	no	—
31	Dic₂₁	—	41	1	no	—
32	C₇×Dic₃	—	287	1	no	—
33	D₄₁	C₇×D₂₁	1	1	yes	characteristic
34	C₇²	C₃⋊#4223	1	1	yes	characteristic, complemented, Sylow
35	C₂×C₃×C₇	—	41	1	no	—
36	C₂×C₃×C₇	—	41	1	no	—
37	C₂×C₃×C₇	—	82	3	no	—
38	C₄₁	C₇×Dic₂₁	1	1	yes	characteristic, complemented, Sylow
39	C₄×C₇	—	861	1	no	—
40	Dic₇	—	123	1	no	—
41	C₃×C₇	—	2	3	no	—
42	C₃×C₇	C₇×#4223	1	1	yes	characteristic, complemented
43	C₃×C₇	C₇⋊#4223	1	1	yes	characteristic, complemented
44	C₂×C₇	—	41	1	no	—
45	C₂×C₇	—	41	1	no	—
46	C₂×C₇	—	82	3	no	—
47	Dic₃	—	287	1	no	Hall
48	C₇	—	2	3	no	—
49	C₇	C₇×C₃⋊#4223	1	1	yes	characteristic, complemented
50	C₇	C₃⋊(C₇⋊#4223)	1	1	yes	characteristic, direct product factor
51	C₂×C₃	—	41	1	no	—
52	C₄	—	861	1	no	Sylow
53	C₃	C₇×C₇⋊#4223	1	1	yes	characteristic, complemented, Sylow
54	C₂	—	41	1	no	—

	subgroup	G/H
center, hypercenter	C₇	C₃⋊(C₇⋊#4223)
commutator, hypocenter	C₃×C₇×C₄₁	C₄×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₄₁	C₄
prime radical	C₃×C₇²×D₄₁	C₂
cosocle	C₃×C₇×D₄₁	C₂×C₇
LCS	G ▷ C₃×C₇×C₄₁
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
3. C₃×C₇²×C₄₁	C₄	semidirect
5. C₃⋊(C₇⋊#4223)	C₇	direct (N × H)
7. C₇²×C₄₁	Dic₃	semidirect
14. C₃×C₇×C₄₁	Dic₇	semidirect
15. C₃×C₇×C₄₁	C₄×C₇	semidirect
23. C₇×C₄₁	C₇×Dic₃	semidirect
24. C₇×C₄₁	Dic₂₁	semidirect
28. C₃×C₇²	#4223	semidirect
29. C₃×C₄₁	C₇×Dic₇	semidirect
34. C₇²	C₃⋊#4223	semidirect
38. C₄₁	C₇×Dic₂₁	semidirect
42. C₃×C₇	C₇×#4223	semidirect
43. C₃×C₇	C₇⋊#4223	semidirect
49. C₇	C₇×C₃⋊#4223	semidirect
50. C₇	C₃⋊(C₇⋊#4223)	direct (N × H)
53. C₃	C₇×C₇⋊#4223	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇²×D₄₁	2
C₇×C₇⋊#4223	3
C₃⋊(C₇⋊#4223)	7
C₇×C₃⋊#4223	7
C₇×Dic₂₁	41

Subgroups — C7×C3⋊(C7⋊#4223) #37563142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₃⋊(C₇⋊#4223) #37563142