Subgroups — C₉²×ΓL(1,𝔽₈) #13133518

Order 1701.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×ΓL(1,𝔽₈)	C₃	1	4	yes	maximal
2	C₇×C₉²	C₃	1	1	yes	maximal, characteristic, complemented
3	C₉×C₇⋊C₉	C₃	1	8	yes	maximal, complemented
4	C₃×C₉²	—	7	1	no	maximal, Sylow
5	C₃²×ΓL(1,𝔽₈)	C₃²	1	1	yes	characteristic
6	C₃×C₇×C₉	C₃²	1	4	yes	—
7	C₃×C₇⋊C₉	C₃²	1	8	yes	—
8	C₉×ΓL(1,𝔽₈)	C₉	1	36	yes	direct product factor
9	C₃²×C₉	—	7	4	no	—
10	C₉²	—	7	8	no	—
11	C₉²	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
12	C₃²×C₇	C₃³	1	1	yes	characteristic
13	C₃×ΓL(1,𝔽₈)	C₃×C₉	1	12	yes	—
14	C₇×C₉	C₃×C₉	1	12	yes	complemented
15	C₇⋊C₉	C₃×C₉	1	24	yes	complemented
16	C₃³	—	7	1	no	—
17	C₃×C₉	—	7	8	no	—
18	C₃×C₉	—	7	36	no	—
19	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	4	yes	—
20	C₃×C₇	C₃²×C₉	1	4	yes	—
21	ΓL(1,𝔽₈)	C₉²	1	9	yes	direct product factor
22	C₃²	—	7	12	no	—
23	C₃²	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
24	C₉	—	7	24	no	—
25	C₉	C₉×ΓL(1,𝔽₈)	1	12	yes	direct product factor
26	C₇	C₃×C₉²	1	1	yes	characteristic, complemented, Sylow
27	C₃	—	7	9	no	—
28	C₃	C₃×C₉×ΓL(1,𝔽₈)	1	4	yes	—

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.