Subgroups — C₉₇×#6856 #37555696

Order 23571.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×H(3,𝔽₃)×C₉₇	C₃	1	2	yes	maximal, complemented
2	C₃²⋊C₉×C₉₇	C₃	1	2	yes	maximal, complemented
3	C₃³×C₉₇	—	3	2	no	—
4	C₃³×C₉₇	C₃²	1	1	yes	characteristic
5	H(3,𝔽₃)×C₉₇	—	3	6	no	—
6	C₃×C₉×C₉₇	—	3	2	no	—
7	C₃²×C₉₇	—	3	2	no	—
8	C₃²×C₉₇	—	3	6	no	—
9	C₃²×C₉₇	—	3	2	no	—
10	C₃²×C₉₇	—	9	2	no	—
11	C₃²×C₉₇	—	9	4	no	—
12	C₃²×C₉₇	H(3,𝔽₃)	1	1	yes	characteristic
13	C₉×C₉₇	—	9	2	no	—
14	C₃×C₉₇	—	9	1	no	—
15	C₃×C₉₇	—	9	6	no	—
16	C₃×C₉₇	AΣL(1,𝔽₂₇)	1	2	yes	—
17	C₃×C₉₇	#732	1	2	yes	—
18	#6856	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₉₇	#6856	1	1	yes	characteristic, direct product factor, Sylow
20	C₃×H(3,𝔽₃)	C₃×C₉₇	1	2	yes	complemented
21	C₃²⋊C₉	C₃×C₉₇	1	2	yes	complemented
22	C₃³	—	3	2	no	—
23	C₃³	C₃²×C₉₇	1	1	yes	characteristic
24	H(3,𝔽₃)	—	3	6	no	—
25	C₃×C₉	—	3	2	no	—
26	C₃²	—	3	2	no	—
27	C₃²	—	3	6	no	—
28	C₃²	—	3	2	no	—
29	C₃²	—	9	2	no	—
30	C₃²	—	9	4	no	—
31	C₃²	H(3,𝔽₃)×C₉₇	1	1	yes	characteristic
32	C₉	—	9	2	no	—
33	C₃	—	9	1	no	—
34	C₃	—	9	6	no	—
35	C₃	AΣL(1,𝔽₂₇)×C₉₇	1	2	yes	—
36	C₃	#732×C₉₇	1	2	yes	—

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