Subgroups — C₉₇×#6908 #37555748

Order 23571.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#737×C₉₇	C₃	1	3	yes	maximal, complemented
2	C₃²×C₉×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented
3	#733×C₉₇	C₃	1	2	yes	maximal, complemented
4	#732×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented
5	#731×C₉₇	C₃	1	3	yes	maximal, complemented
6	AΣL(1,𝔽₂₇)×C₉₇	C₃	1	3	yes	maximal, complemented
7	C₃³×C₉₇	C₃²	1	1	yes	characteristic
8	3₋¹⁺²×C₉₇	—	3	6	no	—
9	3₋¹⁺²×C₉₇	C₃²	1	6	yes	—
10	H(3,𝔽₃)×C₉₇	C₃²	1	3	yes	—
11	C₃×C₉×C₉₇	—	3	2	no	—
12	C₃×C₉×C₉₇	—	3	1	no	—
13	C₃×C₉×C₉₇	—	3	3	no	—
14	C₃×C₉×C₉₇	C₃²	1	1	yes	characteristic
15	C₃×C₉×C₉₇	C₃²	1	1	yes	characteristic
16	C₃×C₉×C₉₇	C₃²	1	1	yes	characteristic
17	C₃²×C₉₇	—	3	3	no	—
18	C₃²×C₉₇	—	3	1	no	—
19	C₃²×C₉₇	—	3	3	no	—
20	C₃²×C₉₇	C₃³	1	1	yes	characteristic
21	C₉×C₉₇	—	3	1	no	—
22	C₉×C₉₇	—	3	1	no	—
23	C₉×C₉₇	—	3	6	no	—
24	C₉×C₉₇	H(3,𝔽₃)	1	2	yes	—
25	C₉×C₉₇	H(3,𝔽₃)	1	1	yes	characteristic
26	C₃×C₉₇	—	3	3	no	—
27	C₃×C₉₇	—	3	1	no	—
28	C₃×C₉₇	—	9	3	no	—
29	C₃×C₉₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
30	#6908	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₉₇	#6908	1	1	yes	characteristic, direct product factor, Sylow
32	#737	C₃×C₉₇	1	3	yes	complemented
33	C₃²×C₉	C₃×C₉₇	1	1	yes	characteristic, complemented
34	#733	C₃×C₉₇	1	2	yes	complemented
35	#732	C₃×C₉₇	1	1	yes	characteristic, complemented
36	#731	C₃×C₉₇	1	3	yes	complemented
37	AΣL(1,𝔽₂₇)	C₃×C₉₇	1	3	yes	complemented
38	C₃³	C₃²×C₉₇	1	1	yes	characteristic
39	3₋¹⁺²	—	3	6	no	—
40	3₋¹⁺²	C₃²×C₉₇	1	6	yes	—
41	H(3,𝔽₃)	C₃²×C₉₇	1	3	yes	—
42	C₃×C₉	—	3	1	no	—
43	C₃×C₉	—	3	2	no	—
44	C₃×C₉	—	3	3	no	—
45	C₃×C₉	C₃²×C₉₇	1	1	yes	characteristic
46	C₃×C₉	C₃²×C₉₇	1	1	yes	characteristic
47	C₃×C₉	C₃²×C₉₇	1	1	yes	characteristic
48	C₃²	—	3	3	no	—
49	C₃²	—	3	1	no	—
50	C₃²	—	3	3	no	—
51	C₃²	C₃³×C₉₇	1	1	yes	characteristic
52	C₉	—	3	1	no	—
53	C₉	—	3	1	no	—
54	C₉	—	3	6	no	—
55	C₉	H(3,𝔽₃)×C₉₇	1	1	yes	characteristic
56	C₉	H(3,𝔽₃)×C₉₇	1	2	yes	—
57	C₃	—	3	1	no	—
58	C₃	—	3	3	no	—
59	C₃	—	9	3	no	—
60	C₃	C₃×H(3,𝔽₃)×C₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₉×C₉₇	H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₃³×C₉₇
socle	C₃×C₉₇	C₃×H(3,𝔽₃)
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉×C₉₇ ◁ C₃×C₉×C₉₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #737×C₉₇	C₃	semidirect
2. C₃²×C₉×C₉₇	C₃	semidirect
3. #733×C₉₇	C₃	semidirect
4. #732×C₉₇	C₃	semidirect
5. #731×C₉₇	C₃	semidirect
6. AΣL(1,𝔽₂₇)×C₉₇	C₃	semidirect
30. #6908	C₉₇	direct (N × H)
31. C₉₇	#6908	direct (N × H)
32. #737	C₃×C₉₇	semidirect
33. C₃²×C₉	C₃×C₉₇	semidirect
34. #733	C₃×C₉₇	semidirect
35. #732	C₃×C₉₇	semidirect
36. #731	C₃×C₉₇	semidirect
37. AΣL(1,𝔽₂₇)	C₃×C₉₇	semidirect

Maximal subgroups

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

subgroup	index
#737×C₉₇	3
C₃²×C₉×C₉₇	3
#733×C₉₇	3
#732×C₉₇	3
#731×C₉₇	3
AΣL(1,𝔽₂₇)×C₉₇	3
#6908	97

Subgroups — C97×#6908 #37555748

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#6908 #37555748