Subgroups — #37555769

Order 23571.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14579757	C₃	1	1	yes	maximal, characteristic
2	C₃²⋊(C₉₇⋊C₉)	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊(C₉₇⋊C₉)	C₃	1	1	yes	maximal, characteristic
4	C₃²⋊C₉×C₉₇	C₃	1	1	yes	maximal, characteristic
5	#14579757	—	3	3	no	—
6	C₃×C₉₇⋊C₉	—	3	1	no	—
7	C₃×C₉₇⋊C₉	—	3	1	no	—
8	C₃×C₉₇⋊C₉	—	3	1	no	—
9	C₃³×C₉₇	C₃²	1	1	yes	characteristic
10	C₃×C₉×C₉₇	—	3	1	no	—
11	C₃²×C₉₇	—	3	1	no	—
12	C₃²×C₉₇	—	3	1	no	—
13	C₃²×C₉₇	—	3	1	no	—
14	C₃²×C₉₇	—	3	1	no	—
15	C₃²×C₉₇	H(3,𝔽₃)	1	1	yes	characteristic
16	C₉×C₉₇	—	9	1	no	—
17	C₉₇⋊C₉	—	3	3	no	—
18	C₉₇⋊C₉	—	9	1	no	—
19	C₉₇⋊C₉	—	9	1	no	—
20	C₃×C₉₇	—	9	1	no	—
21	C₃×C₉₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
22	C₃×C₉₇	#731	1	1	yes	characteristic
23	C₃×C₉₇	#731	1	1	yes	characteristic
24	C₃×C₉₇	#733	1	1	yes	characteristic
25	#6858	—	97	1	no	maximal, Sylow
26	C₉₇	#6858	1	1	yes	characteristic, complemented, Sylow
27	C₃×3₋¹⁺²	—	97	1	no	—
28	C₃²⋊C₉	—	97	1	no	—
29	C₃²⋊C₉	—	97	1	no	—
30	C₃²⋊C₉	C₉₇⋊C₃	1	1	yes	characteristic
31	C₃³	C₃×C₉₇⋊C₃	1	1	yes	characteristic
32	3₋¹⁺²	—	291	3	no	—
33	C₃×C₉	—	3	1	no	—
34	C₃×C₉	—	291	1	no	—
35	C₃×C₉	—	291	1	no	—
36	C₃×C₉	—	291	1	no	—
37	C₃²	—	3	1	no	—
38	C₃²	—	3	1	no	—
39	C₃²	—	3	1	no	—
40	C₃²	—	3	1	no	—
41	C₃²	C₃²⋊(C₉₇⋊C₃)	1	1	yes	characteristic
42	C₉	—	9	1	no	—
43	C₉	—	291	3	no	—
44	C₉	—	873	1	no	—
45	C₉	—	873	1	no	—
46	C₃	—	9	1	no	—
47	C₃	#21463358	1	1	yes	characteristic
48	C₃	#21463362	1	1	yes	characteristic
49	C₃	#21463363	1	1	yes	characteristic
50	C₃	#21463366	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	C₃²⋊(C₉₇⋊C₃)
commutator, prime radical, cosocle	C₃³×C₉₇	C₃²
Frattini	C₃³	C₃×C₉₇⋊C₃
Fitting	C₃²⋊C₉×C₉₇	C₃
socle	C₃²×C₉₇	H(3,𝔽₃)
hypercenter	C₃²⋊C₉	C₉₇⋊C₃
hypocenter	C₉₇	#6858
LCS	G ▷ C₃³×C₉₇ ▷ C₃²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
26. C₉₇	#6858	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⋊C₉×C₉₇	3
#6858	97