Subgroups — #37555882

Order 23571.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14579756	C₃	1	3	yes	maximal, complemented
2	C₃×C₃²⋊(C₉₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	C₃²×C₉₇⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²⋊(C₉₇⋊C₉)	C₃	1	1	yes	maximal, characteristic, complemented
5	(C₃×C₉₇⋊C₃)⋊C₉	C₃	1	6	yes	maximal, complemented
6	C₃²⋊C₉×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃²×C₉₇⋊C₃	C₃²	1	3	yes	—
8	#14579756	—	3	9	no	—
9	C₃²⋊(C₉₇⋊C₃)	—	3	3	no	—
10	C₃×C₉₇⋊C₉	—	3	3	no	—
11	C₃×C₉₇⋊C₉	C₃²	1	2	yes	—
12	C₃×C₉₇⋊C₉	C₃²	1	3	yes	—
13	C₃×C₉₇⋊C₉	C₃²	1	1	yes	characteristic
14	C₃³×C₉₇	C₃²	1	1	yes	characteristic
15	C₃×C₉×C₉₇	C₃²	1	3	yes	—
16	C₃²×C₉₇	—	3	1	no	—
17	C₃²×C₉₇	—	3	2	no	—
18	C₃²×C₉₇	3₋¹⁺²	1	3	yes	complemented
19	C₃²×C₉₇	C₃³	1	1	yes	characteristic
20	C₃×C₉₇⋊C₃	—	3	3	no	—
21	C₃×C₉₇⋊C₃	—	3	3	no	—
22	C₃×C₉₇⋊C₃	—	3	6	no	—
23	C₉×C₉₇	—	3	3	no	—
24	C₉₇⋊C₉	—	3	2	no	—
25	C₉₇⋊C₉	—	3	3	no	—
26	C₉₇⋊C₉	H(3,𝔽₃)	1	3	yes	complemented
27	C₃×C₉₇	—	3	3	no	—
28	C₃×C₉₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
29	C₃×C₉₇	C₃×3₋¹⁺²	1	1	yes	characteristic
30	C₃×C₉₇	#737	1	2	yes	—
31	C₉₇⋊C₃	—	9	3	no	—
32	C₉⋊H(3,𝔽₃)	—	97	1	no	maximal, Sylow
33	C₉₇	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
34	C₃×3₋¹⁺²	—	97	3	no	—
35	C₃×H(3,𝔽₃)	—	97	1	no	—
36	C₃²×C₉	—	97	1	no	—
37	C₃²⋊C₉	—	97	1	no	—
38	C₃²⋊C₉	—	97	6	no	—
39	C₃²⋊C₉	C₉₇⋊C₃	1	1	yes	characteristic, complemented
40	C₃³	—	97	3	no	—
41	C₃³	C₃×C₉₇⋊C₃	1	1	yes	characteristic
42	3₋¹⁺²	—	291	9	no	—
43	H(3,𝔽₃)	—	291	3	no	—
44	C₃×C₉	—	291	3	no	—
45	C₃×C₉	—	97	2	no	—
46	C₃×C₉	—	97	3	no	—
47	C₃×C₉	—	97	1	no	—
48	C₃×C₉	C₃×C₉₇⋊C₃	1	3	yes	—
49	C₃²	—	3	1	no	—
50	C₃²	—	3	2	no	—
51	C₃²	—	291	3	no	—
52	C₃²	—	291	3	no	—
53	C₃²	—	291	6	no	—
54	C₃²	#14579756	1	3	yes	complemented
55	C₃²	C₃²×C₉₇⋊C₃	1	1	yes	characteristic
56	C₉	—	3	3	no	—
57	C₉	—	291	2	no	—
58	C₉	—	291	3	no	—
59	C₉	—	97	3	no	—
60	C₃	—	3	3	no	—
61	C₃	—	873	3	no	—
62	C₃	H(3,𝔽₃)×C₉₇⋊C₃	1	1	yes	characteristic
63	C₃	C₃×#14579756	1	1	yes	characteristic
64	C₃	#21463378	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₉₇⋊C₃
commutator, socle, cosocle	C₃²×C₉₇	C₃³
Fitting	C₃²⋊C₉×C₉₇	C₃
prime radical	C₃×C₃²⋊(C₉₇⋊C₃)	C₃
hypercenter	C₃²⋊C₉	C₉₇⋊C₃
hypocenter	C₉₇	C₉⋊H(3,𝔽₃)
LCS	G ▷ C₃²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
1. C₃×#14579756	C₃	semidirect
3. C₃²×C₉₇⋊C₉	C₃	semidirect
4. C₃²⋊(C₉₇⋊C₉)	C₃	semidirect
5. (C₃×C₉₇⋊C₃)⋊C₉	C₃	semidirect
6. C₃²⋊C₉×C₉₇	C₃	semidirect
18. C₃²×C₉₇	3₋¹⁺²	semidirect
26. C₉₇⋊C₉	H(3,𝔽₃)	semidirect
33. C₉₇	C₉⋊H(3,𝔽₃)	semidirect
39. C₃²⋊C₉	C₉₇⋊C₃	semidirect
54. C₃²	#14579756	semidirect

Maximal subgroups

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

subgroup	index
C₃×#14579756	3
C₃×C₃²⋊(C₉₇⋊C₃)	3
C₃²×C₉₇⋊C₉	3
C₃²⋊(C₉₇⋊C₉)	3
(C₃×C₉₇⋊C₃)⋊C₉	3
C₃²⋊C₉×C₉₇	3
C₉⋊H(3,𝔽₃)	97