Subgroups — C₉₇×#6860 #37555700

Order 23571.

70 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
2	C₃²⋊C₉×C₉₇	C₃	1	2	yes	maximal
3	C₃³×C₉₇	C₃²	1	1	yes	characteristic
4	3₋¹⁺²×C₉₇	—	3	6	no	—
5	C₃×C₉×C₉₇	—	3	2	no	—
6	C₃×C₉×C₉₇	—	3	2	no	—
7	C₃²×C₉₇	—	3	2	no	—
8	C₃²×C₉₇	—	3	2	no	—
9	C₃²×C₉₇	H(3,𝔽₃)	1	1	yes	characteristic
10	C₉×C₉₇	—	3	6	no	—
11	C₉×C₉₇	—	9	2	no	—
12	C₃×C₉₇	—	9	1	no	—
13	C₃×C₉₇	AΣL(1,𝔽₂₇)	1	2	yes	—
14	C₃×C₉₇	#733	1	2	yes	—
15	#6860	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₉₇	#6860	1	1	yes	characteristic, direct product factor, Sylow
17	C₃×3₋¹⁺²	C₃×C₉₇	1	2	yes	—
18	C₃²⋊C₉	C₃×C₉₇	1	2	yes	—
19	C₃³	C₃²×C₉₇	1	1	yes	characteristic
20	3₋¹⁺²	—	3	6	no	—
21	C₃×C₉	—	3	2	no	—
22	C₃×C₉	—	3	2	no	—
23	C₃²	—	3	2	no	—
24	C₃²	—	3	2	no	—
25	C₃²	H(3,𝔽₃)×C₉₇	1	1	yes	characteristic
26	C₉	—	3	6	no	—
27	C₉	—	9	2	no	—
28	C₃	—	9	1	no	—
29	C₃	AΣL(1,𝔽₂₇)×C₉₇	1	2	yes	—
30	C₃	#733×C₉₇	1	2	yes	—

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

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

N	H ≅ G/N	type
15. #6860	C₉₇	direct (N × H)
16. C₉₇	#6860	direct (N × H)

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