Subgroups — C₉₇×#6857 #37555697

Order 23571.

92 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, complemented
2	C₃×H(3,𝔽₃)×C₉₇	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊C₉×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₉₇	—	3	1	no	—
5	C₃³×C₉₇	C₃²	1	1	yes	characteristic
6	3₋¹⁺²×C₉₇	—	3	6	no	—
7	H(3,𝔽₃)×C₉₇	—	3	3	no	—
8	C₃×C₉×C₉₇	—	3	2	no	—
9	C₃×C₉×C₉₇	—	3	1	no	—
10	C₃²×C₉₇	—	3	1	no	—
11	C₃²×C₉₇	—	3	3	no	—
12	C₃²×C₉₇	—	3	2	no	—
13	C₃²×C₉₇	—	3	1	no	—
14	C₃²×C₉₇	—	9	2	no	—
15	C₃²×C₉₇	—	9	1	no	—
16	C₃²×C₉₇	H(3,𝔽₃)	1	1	yes	characteristic
17	C₉×C₉₇	—	3	6	no	—
18	C₉×C₉₇	—	9	1	no	—
19	C₃×C₉₇	—	9	1	no	—
20	C₃×C₉₇	—	9	3	no	—
21	C₃×C₉₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
22	C₃×C₉₇	AΣL(1,𝔽₂₇)	1	2	yes	—
23	C₃×C₉₇	#731	1	1	yes	characteristic
24	#6857	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₉₇	#6857	1	1	yes	characteristic, direct product factor, Sylow
26	C₃×3₋¹⁺²	C₃×C₉₇	1	2	yes	complemented
27	C₃×H(3,𝔽₃)	C₃×C₉₇	1	1	yes	characteristic
28	C₃²⋊C₉	C₃×C₉₇	1	1	yes	characteristic, complemented
29	C₃³	—	3	1	no	—
30	C₃³	C₃²×C₉₇	1	1	yes	characteristic
31	3₋¹⁺²	—	3	6	no	—
32	H(3,𝔽₃)	—	3	3	no	—
33	C₃×C₉	—	3	2	no	—
34	C₃×C₉	—	3	1	no	—
35	C₃²	—	3	1	no	—
36	C₃²	—	3	3	no	—
37	C₃²	—	3	2	no	—
38	C₃²	—	3	1	no	—
39	C₃²	—	9	2	no	—
40	C₃²	—	9	1	no	—
41	C₃²	H(3,𝔽₃)×C₉₇	1	1	yes	characteristic
42	C₉	—	3	6	no	—
43	C₉	—	9	1	no	—
44	C₃	—	9	1	no	—
45	C₃	—	9	3	no	—
46	C₃	AΣL(1,𝔽₂₇)×C₉₇	1	2	yes	—
47	C₃	AΣL(1,𝔽₂₇)×C₉₇	1	1	yes	characteristic
48	C₃	#731×C₉₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃×3₋¹⁺²×C₉₇	C₃	semidirect
3. C₃²⋊C₉×C₉₇	C₃	semidirect
24. #6857	C₉₇	direct (N × H)
25. C₉₇	#6857	direct (N × H)
26. C₃×3₋¹⁺²	C₃×C₉₇	semidirect
28. C₃²⋊C₉	C₃×C₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×3₋¹⁺²×C₉₇	3
C₃×H(3,𝔽₃)×C₉₇	3
C₃²⋊C₉×C₉₇	3
#6857	97

Subgroups — C97×#6857 #37555697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#6857 #37555697