Subgroups — #37555799

Order 23571.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14579757	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₉₇⋊C₉	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊(C₉₇⋊C₉)	C₃	1	1	yes	maximal, characteristic
4	C₃²⋊C₉×C₉₇	C₃	1	1	yes	maximal, characteristic
5	#14579757	—	3	1	no	—
6	#14579757	—	3	1	no	—
7	#14579757	C₉	1	3	yes	complemented
8	C₃×C₉₇⋊C₉	—	3	3	no	—
9	C₃×C₉₇⋊C₉	—	3	1	no	—
10	C₃×C₉₇⋊C₉	—	3	1	no	—
11	C₃×C₉₇⋊C₉	—	3	1	no	—
12	C₃³×C₉₇	C₃²	1	1	yes	characteristic
13	C₃×C₉×C₉₇	—	3	1	no	—
14	C₃²×C₉₇	—	3	1	no	—
15	C₃²×C₉₇	—	3	1	no	—
16	C₃²×C₉₇	—	3	1	no	—
17	C₃²×C₉₇	C₃×C₉	1	1	yes	characteristic
18	C₃²×C₉₇	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃²×C₉₇	3₋¹⁺²	1	1	yes	characteristic
20	C₃²×C₉₇	3₋¹⁺²	1	1	yes	characteristic
21	C₉×C₉₇	—	9	1	no	—
22	C₉₇⋊C₉	—	3	3	no	—
23	C₉₇⋊C₉	—	3	3	no	—
24	C₉₇⋊C₉	—	9	1	no	—
25	C₃×C₉₇	—	3	1	no	—
26	C₃×C₉₇	—	3	1	no	—
27	C₃×C₉₇	—	3	1	no	—
28	C₃×C₉₇	C₃²⋊C₉	1	1	yes	characteristic
29	C₃×C₉₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
30	C₃×C₉₇	#731	1	1	yes	characteristic
31	C₃×C₉₇	#733	1	1	yes	characteristic
32	3₋¹⁺²⋊C₉	—	97	1	no	maximal, Sylow
33	C₉₇	3₋¹⁺²⋊C₉	1	1	yes	characteristic, complemented, Sylow
34	C₃×3₋¹⁺²	—	97	1	no	—
35	C₃²×C₉	—	97	1	no	—
36	C₃²⋊C₉	—	97	1	no	—
37	C₃²⋊C₉	C₉₇⋊C₃	1	1	yes	characteristic
38	C₃³	C₃×C₉₇⋊C₃	1	1	yes	characteristic
39	3₋¹⁺²	—	291	1	no	—
40	3₋¹⁺²	—	291	1	no	—
41	3₋¹⁺²	—	97	3	no	—
42	C₃×C₉	—	3	1	no	—
43	C₃×C₉	—	291	1	no	—
44	C₃×C₉	—	291	1	no	—
45	C₃×C₉	—	291	3	no	—
46	C₃×C₉	—	291	1	no	—
47	C₃²	—	3	1	no	—
48	C₃²	—	3	1	no	—
49	C₃²	—	3	1	no	—
50	C₃²	C₉×C₉₇⋊C₃	1	1	yes	characteristic
51	C₃²	C₃²⋊(C₉₇⋊C₃)	1	1	yes	characteristic
52	C₃²	#14579755	1	1	yes	characteristic
53	C₃²	#14579756	1	1	yes	characteristic
54	C₉	—	9	1	no	—
55	C₉	—	291	3	no	—
56	C₉	—	291	3	no	—
57	C₉	—	873	1	no	—
58	C₃	—	3	1	no	—
59	C₃	—	3	1	no	—
60	C₃	—	3	1	no	—
61	C₃	(C₃×C₉₇⋊C₃)⋊C₉	1	1	yes	characteristic
62	C₃	#21463358	1	1	yes	characteristic
63	C₃	#21463363	1	1	yes	characteristic
64	C₃	#21463366	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	C₃²⋊(C₉₇⋊C₃)
commutator	C₃²×C₉₇	C₃×C₉
Frattini	C₃³	C₃×C₉₇⋊C₃
Fitting	C₃²⋊C₉×C₉₇	C₃
socle	C₃²×C₉₇	H(3,𝔽₃)
prime radical, cosocle	C₃³×C₉₇	C₃²
hypercenter	C₃²⋊C₉	C₉₇⋊C₃
hypocenter	C₉₇	3₋¹⁺²⋊C₉
LCS	G ▷ C₃²×C₉₇ ▷ C₃×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
7. #14579757	C₉	semidirect
33. C₉₇	3₋¹⁺²⋊C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×#14579757	3
C₃²×C₉₇⋊C₉	3
C₃²⋊(C₉₇⋊C₉)	3
C₃²⋊C₉×C₉₇	3
3₋¹⁺²⋊C₉	97