Subgroups — #37555886

Order 23571.

156 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, complemented
2	C₃×C₃²⋊(C₉₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	C₃²×C₉₇⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	(C₃×C₉₇⋊C₃)⋊C₉	C₃	1	9	yes	maximal, complemented
5	C₃×3₋¹⁺²×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃²×C₉₇⋊C₃	C₃²	1	3	yes	—
7	#14579757	—	3	3	no	—
8	C₃²⋊(C₉₇⋊C₃)	—	3	3	no	—
9	C₃×C₉₇⋊C₉	—	3	3	no	—
10	C₃×C₉₇⋊C₉	C₃²	1	3	yes	—
11	C₃×C₉₇⋊C₉	C₃²	1	3	yes	—
12	C₃³×C₉₇	C₃²	1	1	yes	characteristic
13	3₋¹⁺²×C₉₇	—	3	3	no	—
14	C₃×C₉×C₉₇	C₃²	1	3	yes	—
15	C₃²×C₉₇	—	3	3	no	—
16	C₃²×C₉₇	H(3,𝔽₃)	1	3	yes	—
17	C₃²×C₉₇	C₃³	1	1	yes	characteristic
18	C₃×C₉₇⋊C₃	—	3	3	no	—
19	C₃×C₉₇⋊C₃	—	3	9	no	—
20	C₉×C₉₇	—	3	3	no	—
21	C₉₇⋊C₉	—	3	3	no	—
22	C₉₇⋊C₉	—	3	3	no	—
23	C₃×C₉₇	—	3	3	no	—
24	C₃×C₉₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
25	C₃×C₉₇	#737	1	3	yes	—
26	C₉₇⋊C₃	—	9	3	no	—
27	#6893	—	97	1	no	maximal, Sylow
28	C₉₇	#6893	1	1	yes	characteristic, complemented, Sylow
29	C₃×3₋¹⁺²	—	97	1	no	—
30	C₃×3₋¹⁺²	C₉₇⋊C₃	1	1	yes	characteristic, complemented
31	C₃×H(3,𝔽₃)	—	97	1	no	—
32	C₃²×C₉	—	97	1	no	—
33	C₃²⋊C₉	—	97	9	no	—
34	C₃³	—	97	3	no	—
35	C₃³	C₃×C₉₇⋊C₃	1	1	yes	characteristic
36	3₋¹⁺²	—	3	3	no	—
37	3₋¹⁺²	—	291	3	no	—
38	H(3,𝔽₃)	—	291	3	no	—
39	C₃×C₉	—	291	3	no	—
40	C₃×C₉	—	97	3	no	—
41	C₃×C₉	—	97	3	no	—
42	C₃×C₉	C₃×C₉₇⋊C₃	1	3	yes	—
43	C₃²	—	3	3	no	—
44	C₃²	—	291	3	no	—
45	C₃²	—	291	9	no	—
46	C₃²	C₃²⋊(C₉₇⋊C₃)	1	3	yes	—
47	C₃²	C₃²×C₉₇⋊C₃	1	1	yes	characteristic
48	C₉	—	3	3	no	—
49	C₉	—	291	3	no	—
50	C₉	—	291	3	no	—
51	C₃	—	3	3	no	—
52	C₃	—	873	3	no	—
53	C₃	C₃×C₃²⋊(C₉₇⋊C₃)	1	1	yes	characteristic
54	C₃	#21463378	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₉₇⋊C₃
commutator, socle, cosocle	C₃²×C₉₇	C₃³
Fitting	C₃×3₋¹⁺²×C₉₇	C₃
prime radical	C₃×C₃²⋊(C₉₇⋊C₃)	C₃
hypercenter	C₃×3₋¹⁺²	C₉₇⋊C₃
hypocenter	C₉₇	#6893
LCS	G ▷ C₃²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. C₃×#14579757	C₃	semidirect
3. C₃²×C₉₇⋊C₉	C₃	semidirect
4. (C₃×C₉₇⋊C₃)⋊C₉	C₃	semidirect
5. C₃×3₋¹⁺²×C₉₇	C₃	semidirect
28. C₉₇	#6893	semidirect
30. C₃×3₋¹⁺²	C₉₇⋊C₃	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
C₃×C₃²⋊(C₉₇⋊C₃)	3
C₃²×C₉₇⋊C₉	3
(C₃×C₉₇⋊C₃)⋊C₉	3
C₃×3₋¹⁺²×C₉₇	3
#6893	97