Subgroups — C₉₇×C₃²⋊C₂₇ #37555705

Order 23571.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₉×C₉₇	C₃	1	1	yes	maximal, characteristic
2	C₃×C₂₇×C₉₇	C₃	1	3	yes	maximal, complemented
3	C₃³×C₉₇	C₉	1	1	yes	characteristic
4	C₃×C₉×C₉₇	—	3	1	no	—
5	C₃×C₉×C₉₇	—	3	2	no	—
6	C₃×C₉×C₉₇	C₉	1	2	yes	—
7	C₃×C₉×C₉₇	C₃²	1	1	yes	characteristic
8	C₂₇×C₉₇	—	3	3	no	—
9	C₃²×C₉₇	—	3	1	no	—
10	C₃²×C₉₇	—	3	2	no	—
11	C₃²×C₉₇	C₂₇	1	3	yes	complemented
12	C₃²×C₉₇	C₃×C₉	1	1	yes	characteristic
13	C₉×C₉₇	—	3	2	no	—
14	C₉×C₉₇	H(3,𝔽₃)	1	1	yes	characteristic
15	C₉×C₉₇	3₋¹⁺²	1	2	yes	—
16	C₃×C₉₇	—	3	3	no	—
17	C₃×C₉₇	C₃²⋊C₉	1	1	yes	characteristic
18	C₃×C₉₇	C₃×C₂₇	1	1	yes	characteristic
19	C₃×C₉₇	C₂₇⋊C₃	1	2	yes	—
20	C₃²⋊C₂₇	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₉₇	C₃²⋊C₂₇	1	1	yes	characteristic, direct product factor, Sylow
22	C₃²×C₉	C₃×C₉₇	1	1	yes	characteristic
23	C₃×C₂₇	C₃×C₉₇	1	3	yes	complemented
24	C₃³	C₉×C₉₇	1	1	yes	characteristic
25	C₃×C₉	—	3	1	no	—
26	C₃×C₉	—	3	2	no	—
27	C₃×C₉	C₉×C₉₇	1	2	yes	—
28	C₃×C₉	C₃²×C₉₇	1	1	yes	characteristic
29	C₂₇	—	3	3	no	—
30	C₃²	—	3	1	no	—
31	C₃²	—	3	2	no	—
32	C₃²	C₂₇×C₉₇	1	3	yes	complemented
33	C₃²	C₃×C₉×C₉₇	1	1	yes	characteristic
34	C₉	—	3	2	no	—
35	C₉	H(3,𝔽₃)×C₉₇	1	1	yes	characteristic
36	C₉	3₋¹⁺²×C₉₇	1	2	yes	—
37	C₃	—	3	3	no	—
38	C₃	C₃²⋊C₉×C₉₇	1	1	yes	characteristic
39	C₃	C₃×C₂₇×C₉₇	1	1	yes	characteristic
40	C₃	C₂₇⋊C₃×C₉₇	1	2	yes	—

	subgroup	G/H
center	C₃×C₉×C₉₇	C₃²
commutator	C₃	C₃×C₂₇×C₉₇
Frattini, cosocle	C₃×C₉	C₃²×C₉₇
socle	C₃²×C₉₇	C₃×C₉
prime radical	C₃³×C₉₇	C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉×C₉₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₂₇×C₉₇	C₃	semidirect
11. C₃²×C₉₇	C₂₇	semidirect
20. C₃²⋊C₂₇	C₉₇	direct (N × H)
21. C₉₇	C₃²⋊C₂₇	direct (N × H)
23. C₃×C₂₇	C₃×C₉₇	semidirect
32. C₃²	C₂₇×C₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₉×C₉₇	3
C₃×C₂₇×C₉₇	3
C₃²⋊C₂₇	97

Subgroups — C97×C32⋊C27 #37555705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×C₃²⋊C₂₇ #37555705