Subgroups — C₃×#731×C₉₇ #37555745

Order 23571.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×3₋¹⁺²×C₉₇	C₃	1	2	yes	maximal, complemented
2	C₃×H(3,𝔽₃)×C₉₇	C₃	1	1	yes	maximal, characteristic
3	C₃²×C₉×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented
4	#731×C₉₇	C₃	1	9	yes	maximal, direct product factor
5	C₃³×C₉₇	—	3	1	no	—
6	C₃³×C₉₇	C₃²	1	1	yes	characteristic
7	3₋¹⁺²×C₉₇	—	3	4	no	—
8	3₋¹⁺²×C₉₇	C₃²	1	6	yes	complemented
9	H(3,𝔽₃)×C₉₇	—	3	2	no	—
10	H(3,𝔽₃)×C₉₇	C₃²	1	3	yes	—
11	C₃×C₉×C₉₇	—	3	2	no	—
12	C₃×C₉×C₉₇	—	3	1	no	—
13	C₃×C₉×C₉₇	—	3	2	no	—
14	C₃×C₉×C₉₇	C₃²	1	3	yes	complemented
15	C₃²×C₉₇	—	3	3	no	—
16	C₃²×C₉₇	—	3	3	no	—
17	C₃²×C₉₇	—	9	3	no	—
18	C₃²×C₉₇	H(3,𝔽₃)	1	2	yes	—
19	C₃²×C₉₇	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃²×C₉₇	C₃³	1	1	yes	characteristic
21	C₉×C₉₇	—	3	3	no	—
22	C₉×C₉₇	—	3	6	no	—
23	C₃×C₉₇	—	3	2	no	—
24	C₃×C₉₇	—	3	1	no	—
25	C₃×C₉₇	—	9	3	no	—
26	C₃×C₉₇	#731	1	3	yes	direct product factor
27	C₃×C₉₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
28	C₃×#731	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₉₇	C₃×#731	1	1	yes	characteristic, direct product factor, Sylow
30	C₃×3₋¹⁺²	C₃×C₉₇	1	2	yes	complemented
31	C₃×H(3,𝔽₃)	C₃×C₉₇	1	1	yes	characteristic
32	C₃²×C₉	C₃×C₉₇	1	1	yes	characteristic, complemented
33	#731	C₃×C₉₇	1	9	yes	direct product factor
34	C₃³	—	3	1	no	—
35	C₃³	C₃²×C₉₇	1	1	yes	characteristic
36	3₋¹⁺²	—	3	4	no	—
37	3₋¹⁺²	C₃²×C₉₇	1	6	yes	complemented
38	H(3,𝔽₃)	—	3	2	no	—
39	H(3,𝔽₃)	C₃²×C₉₇	1	3	yes	—
40	C₃×C₉	—	3	2	no	—
41	C₃×C₉	—	3	1	no	—
42	C₃×C₉	—	3	2	no	—
43	C₃×C₉	C₃²×C₉₇	1	3	yes	complemented
44	C₃²	—	3	3	no	—
45	C₃²	—	3	3	no	—
46	C₃²	—	9	3	no	—
47	C₃²	H(3,𝔽₃)×C₉₇	1	2	yes	—
48	C₃²	H(3,𝔽₃)×C₉₇	1	1	yes	characteristic
49	C₃²	C₃³×C₉₇	1	1	yes	characteristic
50	C₉	—	3	3	no	—
51	C₉	—	3	6	no	—
52	C₃	—	3	2	no	—
53	C₃	—	3	1	no	—
54	C₃	—	9	3	no	—
55	C₃	#731×C₉₇	1	3	yes	direct product factor
56	C₃	C₃×H(3,𝔽₃)×C₉₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²×C₉₇	H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₃³×C₉₇
prime radical	C₃×H(3,𝔽₃)×C₉₇	C₃
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₉₇ ◁ C₃³×C₉₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×3₋¹⁺²×C₉₇	C₃	semidirect
3. C₃²×C₉×C₉₇	C₃	semidirect
4. #731×C₉₇	C₃	direct (N × H)
8. 3₋¹⁺²×C₉₇	C₃²	semidirect
14. C₃×C₉×C₉₇	C₃²	semidirect
26. C₃×C₉₇	#731	direct (N × H)
28. C₃×#731	C₉₇	direct (N × H)
29. C₉₇	C₃×#731	direct (N × H)
30. C₃×3₋¹⁺²	C₃×C₉₇	semidirect
32. C₃²×C₉	C₃×C₉₇	semidirect
33. #731	C₃×C₉₇	direct (N × H)
37. 3₋¹⁺²	C₃²×C₉₇	semidirect
43. C₃×C₉	C₃²×C₉₇	semidirect
55. C₃	#731×C₉₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×3₋¹⁺²×C₉₇	3
C₃×H(3,𝔽₃)×C₉₇	3
C₃²×C₉×C₉₇	3
#731×C₉₇	3
C₃×#731	97

Subgroups — C3×#731×C97 #37555745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#731×C₉₇ #37555745