Subgroups — #37555883

Order 23571.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14579757	C₃	1	2	yes	maximal, complemented
2	C₃²×C₉₇⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²⋊(C₉₇⋊C₉)	C₃	1	9	yes	maximal, complemented
4	C₃×H(3,𝔽₃)×C₉₇	C₃	1	1	yes	maximal, characteristic
5	#14579757	—	3	6	no	—
6	C₃×C₉₇⋊C₉	—	3	3	no	—
7	C₃×C₉₇⋊C₉	C₃²	1	3	yes	—
8	C₃×C₉₇⋊C₉	C₃²	1	6	yes	—
9	C₃³×C₉₇	C₃²	1	3	yes	—
10	C₃³×C₉₇	C₃²	1	1	yes	characteristic
11	H(3,𝔽₃)×C₉₇	—	3	3	no	—
12	C₃²×C₉₇	—	3	3	no	—
13	C₃²×C₉₇	—	3	3	no	—
14	C₃²×C₉₇	—	3	9	no	—
15	C₃²×C₉₇	H(3,𝔽₃)	1	3	yes	—
16	C₃²×C₉₇	C₃³	1	1	yes	characteristic
17	C₉₇⋊C₉	—	3	3	no	—
18	C₉₇⋊C₉	—	3	6	no	—
19	C₃×C₉₇	—	3	3	no	—
20	C₃×C₉₇	—	9	3	no	—
21	C₃×C₉₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
22	C₃×C₉₇	#737	1	3	yes	—
23	#6893	—	97	1	no	maximal, Sylow
24	C₉₇	#6893	1	1	yes	characteristic, complemented, Sylow
25	C₃×3₋¹⁺²	—	97	2	no	—
26	C₃×H(3,𝔽₃)	C₉₇⋊C₃	1	1	yes	characteristic
27	C₃²×C₉	—	97	1	no	—
28	C₃²⋊C₉	—	97	9	no	—
29	C₃³	C₃×C₉₇⋊C₃	1	3	yes	—
30	C₃³	C₃×C₉₇⋊C₃	1	1	yes	characteristic
31	3₋¹⁺²	—	291	6	no	—
32	H(3,𝔽₃)	—	3	3	no	—
33	C₃×C₉	—	291	3	no	—
34	C₃×C₉	—	97	3	no	—
35	C₃×C₉	—	97	6	no	—
36	C₃²	—	3	3	no	—
37	C₃²	—	3	3	no	—
38	C₃²	—	3	9	no	—
39	C₃²	C₃²⋊(C₉₇⋊C₃)	1	3	yes	—
40	C₃²	C₃²×C₉₇⋊C₃	1	1	yes	characteristic
41	C₉	—	291	3	no	—
42	C₉	—	291	6	no	—
43	C₃	—	3	3	no	—
44	C₃	—	9	3	no	—
45	C₃	C₃×C₃²⋊(C₉₇⋊C₃)	1	1	yes	characteristic
46	C₃	#21463376	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₉₇⋊C₃
commutator, socle, cosocle	C₃²×C₉₇	C₃³
Fitting, prime radical	C₃×H(3,𝔽₃)×C₉₇	C₃
hypercenter	C₃×H(3,𝔽₃)	C₉₇⋊C₃
hypocenter	C₉₇	#6893
LCS	G ▷ C₃²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₃² ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃×#14579757	C₃	semidirect
2. C₃²×C₉₇⋊C₉	C₃	semidirect
3. C₃²⋊(C₉₇⋊C₉)	C₃	semidirect
24. C₉₇	#6893	semidirect

Maximal subgroups

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

subgroup	index
C₃×#14579757	3
C₃²×C₉₇⋊C₉	3
C₃²⋊(C₉₇⋊C₉)	3
C₃×H(3,𝔽₃)×C₉₇	3
#6893	97