Subgroups — #72249124

Order 3402.

920 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133672	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×D₇×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×#4430⋊C₂	C₃	1	12	yes	maximal, complemented
4	#11765711	C₃	1	27	yes	maximal, complemented
5	C₃×C₇×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
6	C₃×#4430	C₂×C₃	1	12	yes	complemented
7	#10588062	C₂×C₃	1	27	yes	complemented
8	C₂×3₋¹⁺⁴	—	7	1	no	maximal, Hall
9	C₃³×D₇	C₃²	1	4	yes	—
10	D₇×H(3,𝔽₃)	C₃²	1	9	yes	—
11	#4430⋊C₂	C₃²	1	9	yes	—
12	#4430⋊C₂	C₃²	1	72	yes	complemented
13	C₃×D₇⋊C₉	C₃²	1	36	yes	complemented
14	3₋¹⁺⁴	—	7	1	no	Sylow
15	C₃³×C₇	C₂×C₃²	1	4	yes	—
16	C₇×H(3,𝔽₃)	C₂×C₃²	1	9	yes	—
17	#4430	C₂×C₃²	1	9	yes	—
18	#4430	C₂×C₃²	1	72	yes	complemented
19	C₃×C₇⋊C₉	C₂×C₃²	1	36	yes	complemented
20	C₂×#737	—	7	27	no	—
21	C₂×C₃×3₋¹⁺²	—	7	12	no	—
22	C₂×C₃×H(3,𝔽₃)	—	21	1	no	—
23	C₃²×D₇	—	9	4	no	—
24	C₃²×D₇	C₃³	1	12	yes	—
25	C₃²×D₇	C₃³	1	1	yes	characteristic
26	D₇⋊C₉	C₃³	1	27	yes	—
27	#737	—	7	27	no	—
28	C₃×3₋¹⁺²	—	7	12	no	—
29	C₃×H(3,𝔽₃)	AGL(1,𝔽₇)	1	1	yes	characteristic
30	C₃²×C₇	—	9	4	no	—
31	C₃²×C₇	C₂×C₃³	1	12	yes	—
32	C₃²×C₇	C₂×C₃³	1	1	yes	characteristic
33	C₇⋊C₉	C₂×C₃³	1	27	yes	—
34	C₂×C₃³	—	21	4	no	—
35	C₂×3₋¹⁺²	—	7	9	no	—
36	C₂×3₋¹⁺²	—	7	72	no	—
37	C₂×H(3,𝔽₃)	—	21	9	no	—
38	C₂×C₃×C₉	—	7	36	no	—
39	C₃×D₇	—	3	1	no	—
40	C₃×D₇	—	3	12	no	—
41	C₃×D₇	C₃⁴	1	1	yes	characteristic
42	C₃³	C₃×AGL(1,𝔽₇)	1	4	yes	—
43	3₋¹⁺²	—	7	9	no	—
44	3₋¹⁺²	—	7	72	no	—
45	H(3,𝔽₃)	C₃×AGL(1,𝔽₇)	1	9	yes	—
46	C₃×C₉	—	7	36	no	—
47	C₃×C₇	—	3	1	no	—
48	C₃×C₇	—	3	12	no	—
49	C₃×C₇	C₂×C₃⁴	1	1	yes	characteristic
50	C₂×C₃²	—	21	1	no	—
51	C₂×C₃²	—	21	12	no	—
52	C₂×C₃²	—	63	4	no	—
53	C₂×C₉	—	7	27	no	—
54	D₇	3₋¹⁺⁴	1	1	yes	characteristic, complemented, Hall
55	C₃²	—	9	4	no	—
56	C₃²	C₃²×AGL(1,𝔽₇)	1	12	yes	—
57	C₃²	C₃²×AGL(1,𝔽₇)	1	1	yes	characteristic
58	C₉	—	7	27	no	—
59	C₇	C₂×3₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₃	—	21	1	no	—
61	C₂×C₃	—	21	1	no	—
62	C₂×C₃	—	63	12	no	—
63	C₃	—	3	1	no	—
64	C₃	—	3	12	no	—
65	C₃	C₃³×AGL(1,𝔽₇)	1	1	yes	characteristic
66	C₂	—	21	1	no	Sylow

	subgroup	G/H
center, Frattini	C₃	C₃³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₃×C₇	C₂×C₃⁴
Fitting	C₃×C₇×H(3,𝔽₃)	C₂×C₃
prime radical	C₃×D₇×H(3,𝔽₃)	C₃
hypercenter	C₃×H(3,𝔽₃)	AGL(1,𝔽₇)
hypocenter	C₇	C₂×3₋¹⁺⁴
LCS	G ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. #13133672	C₂	semidirect
3. C₃×#4430⋊C₂	C₃	semidirect
4. #11765711	C₃	semidirect
6. C₃×#4430	C₂×C₃	semidirect
7. #10588062	C₂×C₃	semidirect
12. #4430⋊C₂	C₃²	semidirect
13. C₃×D₇⋊C₉	C₃²	semidirect
18. #4430	C₂×C₃²	semidirect
19. C₃×C₇⋊C₉	C₂×C₃²	semidirect
54. D₇	3₋¹⁺⁴	semidirect
59. C₇	C₂×3₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
#13133672	2
C₃×D₇×H(3,𝔽₃)	3
C₃×#4430⋊C₂	3
#11765711	3
C₂×3₋¹⁺⁴	7