Subgroups — C₉₇×#6911 #37555751

Order 23571.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×3₋¹⁺²×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×H(3,𝔽₃)×C₉₇	C₃	1	3	yes	maximal, complemented
3	#732×C₉₇	C₃	1	9	yes	maximal, complemented
4	C₃³×C₉₇	—	3	3	no	—
5	C₃³×C₉₇	C₃²	1	1	yes	characteristic
6	3₋¹⁺²×C₉₇	—	3	3	no	—
7	H(3,𝔽₃)×C₉₇	—	3	6	no	—
8	H(3,𝔽₃)×C₉₇	C₃²	1	9	yes	complemented
9	C₃×C₉×C₉₇	C₃²	1	3	yes	complemented
10	C₃²×C₉₇	—	3	9	no	—
11	C₃²×C₉₇	—	9	1	no	—
12	C₃²×C₉₇	—	9	9	no	—
13	C₃²×C₉₇	H(3,𝔽₃)	1	3	yes	—
14	C₃²×C₉₇	C₃³	1	1	yes	characteristic
15	C₉×C₉₇	—	3	3	no	—
16	C₃×C₉₇	—	3	3	no	—
17	C₃×C₉₇	—	3	1	no	—
18	C₃×C₉₇	—	9	9	no	—
19	C₃×C₉₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
20	#6911	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₉₇	#6911	1	1	yes	characteristic, direct product factor, Sylow
22	C₃×3₋¹⁺²	C₃×C₉₇	1	1	yes	characteristic, complemented
23	C₃×H(3,𝔽₃)	C₃×C₉₇	1	3	yes	complemented
24	#732	C₃×C₉₇	1	9	yes	complemented
25	C₃³	—	3	3	no	—
26	C₃³	C₃²×C₉₇	1	1	yes	characteristic
27	3₋¹⁺²	—	3	3	no	—
28	H(3,𝔽₃)	—	3	6	no	—
29	H(3,𝔽₃)	C₃²×C₉₇	1	9	yes	complemented
30	C₃×C₉	C₃²×C₉₇	1	3	yes	complemented
31	C₃²	—	3	9	no	—
32	C₃²	—	9	1	no	—
33	C₃²	—	9	9	no	—
34	C₃²	H(3,𝔽₃)×C₉₇	1	3	yes	—
35	C₃²	C₃³×C₉₇	1	1	yes	characteristic
36	C₉	—	3	3	no	—
37	C₃	—	3	3	no	—
38	C₃	—	3	1	no	—
39	C₃	—	9	9	no	—
40	C₃	C₃×H(3,𝔽₃)×C₉₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃×C₉₇	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₃³×C₉₇
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉₇ ◁ C₃³×C₉₇ ◁ G

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

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