Subgroups — C₃²⋊C₉×C₉₇⋊C₃ #37555849

Order 23571.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₉₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃×C₉×C₉₇⋊C₃	C₃	1	3	yes	maximal, complemented
3	C₃²⋊(C₉₇⋊C₉)	C₃	1	2	yes	maximal, complemented
4	(C₃×C₉₇⋊C₃)⋊C₉	C₃	1	6	yes	maximal, complemented
5	C₃²⋊C₉×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃²×C₉₇⋊C₃	—	3	3	no	—
7	C₃²×C₉₇⋊C₃	—	3	6	no	—
8	C₃²×C₉₇⋊C₃	C₉	1	9	yes	complemented
9	C₃²×C₉₇⋊C₃	C₃²	1	2	yes	—
10	C₃²×C₉₇⋊C₃	C₃²	1	1	yes	characteristic
11	C₃×C₉₇⋊C₉	C₃²	1	6	yes	complemented
12	C₉×C₉₇⋊C₃	—	3	9	no	—
13	C₃³×C₉₇	C₃²	1	1	yes	characteristic
14	C₃×C₉×C₉₇	C₃²	1	3	yes	complemented
15	C₃²×C₉₇	—	3	1	no	—
16	C₃²×C₉₇	—	3	2	no	—
17	C₃²×C₉₇	C₃×C₉	1	3	yes	complemented
18	C₃²×C₉₇	C₃³	1	1	yes	characteristic
19	C₃×C₉₇⋊C₃	—	3	2	no	—
20	C₃×C₉₇⋊C₃	—	3	4	no	—
21	C₃×C₉₇⋊C₃	—	3	27	no	—
22	C₃×C₉₇⋊C₃	C₃×C₉	1	3	yes	—
23	C₃×C₉₇⋊C₃	C₃×C₉	1	6	yes	complemented
24	C₃×C₉₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
25	C₃×C₉₇⋊C₃	3₋¹⁺²	1	6	yes	—
26	C₉×C₉₇	—	3	3	no	—
27	C₉₇⋊C₉	—	3	6	no	—
28	C₃×C₉₇	—	3	3	no	—
29	C₃×C₉₇	C₃²×C₉	1	1	yes	characteristic
30	C₃×C₉₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
31	C₃×C₉₇	C₃×3₋¹⁺²	1	2	yes	—
32	C₉₇⋊C₃	—	3	6	no	—
33	C₉₇⋊C₃	C₃²⋊C₉	1	9	yes	direct product factor
34	C₃×C₃²⋊C₉	—	97	1	no	maximal, Sylow
35	C₉₇	C₃×C₃²⋊C₉	1	1	yes	characteristic, complemented, Sylow
36	C₃⁴	—	97	1	no	—
37	C₃²×C₉	—	97	3	no	—
38	C₃²⋊C₉	—	97	2	no	—
39	C₃²⋊C₉	—	97	6	no	—
40	C₃²⋊C₉	C₉₇⋊C₃	1	1	yes	characteristic, direct product factor
41	C₃³	—	291	3	no	—
42	C₃³	—	291	6	no	—
43	C₃³	—	97	9	no	—
44	C₃³	—	97	2	no	—
45	C₃³	—	97	1	no	—
46	C₃³	C₃×C₉₇⋊C₃	1	1	yes	characteristic
47	C₃×C₉	—	291	9	no	—
48	C₃×C₉	—	97	6	no	—
49	C₃×C₉	C₃×C₉₇⋊C₃	1	3	yes	complemented
50	C₃²	—	3	1	no	—
51	C₃²	—	3	2	no	—
52	C₃²	—	291	27	no	—
53	C₃²	—	291	2	no	—
54	C₃²	—	291	4	no	—
55	C₃²	—	97	6	no	—
56	C₃²	—	97	3	no	—
57	C₃²	—	97	3	no	—
58	C₃²	—	97	6	no	—
59	C₃²	C₉×C₉₇⋊C₃	1	3	yes	complemented
60	C₃²	C₃²×C₉₇⋊C₃	1	1	yes	characteristic
61	C₉	—	3	3	no	—
62	C₉	—	291	6	no	—
63	C₃	—	3	3	no	—
64	C₃	—	291	6	no	—
65	C₃	—	97	9	no	—
66	C₃	C₃×C₉×C₉₇⋊C₃	1	1	yes	characteristic
67	C₃	H(3,𝔽₃)×C₉₇⋊C₃	1	1	yes	characteristic
68	C₃	3₋¹⁺²×C₉₇⋊C₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₉₇⋊C₃
commutator	C₃×C₉₇	C₃²×C₉
Fitting	C₃²⋊C₉×C₉₇	C₃
socle, cosocle	C₃²×C₉₇	C₃³
prime radical	C₃³×C₉₇⋊C₃	C₃
hypercenter	C₃²⋊C₉	C₉₇⋊C₃
hypocenter	C₉₇	C₃×C₃²⋊C₉
LCS	G ▷ C₃×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
2. C₃×C₉×C₉₇⋊C₃	C₃	semidirect
3. C₃²⋊(C₉₇⋊C₉)	C₃	semidirect
4. (C₃×C₉₇⋊C₃)⋊C₉	C₃	semidirect
5. C₃²⋊C₉×C₉₇	C₃	semidirect
8. C₃²×C₉₇⋊C₃	C₉	semidirect
11. C₃×C₉₇⋊C₉	C₃²	semidirect
14. C₃×C₉×C₉₇	C₃²	semidirect
17. C₃²×C₉₇	C₃×C₉	semidirect
23. C₃×C₉₇⋊C₃	C₃×C₉	semidirect
33. C₉₇⋊C₃	C₃²⋊C₉	direct (N × H)
35. C₉₇	C₃×C₃²⋊C₉	semidirect
40. C₃²⋊C₉	C₉₇⋊C₃	direct (N × H)
49. C₃×C₉	C₃×C₉₇⋊C₃	semidirect
59. C₃²	C₉×C₉₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₉₇⋊C₃	3
C₃×C₉×C₉₇⋊C₃	3
C₃²⋊(C₉₇⋊C₉)	3
(C₃×C₉₇⋊C₃)⋊C₉	3
C₃²⋊C₉×C₉₇	3
C₃×C₃²⋊C₉	97

Subgroups — C32⋊C9×C97⋊C3 #37555849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²⋊C₉×C₉₇⋊C₃ #37555849