Subgroups — #37555922

Order 23571.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14579757	C₃	1	3	yes	maximal
2	#21463352	C₃	1	1	yes	maximal, characteristic, complemented
3	#21463351	C₃	1	1	yes	maximal, characteristic, complemented
4	(C₉₇⋊C₉)⋊C₉	C₃	1	6	yes	maximal, complemented
5	C₉×C₉₇⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×3₋¹⁺²×C₉₇	C₃	1	1	yes	maximal, characteristic
7	#14579757	—	3	9	no	—
8	C₃×C₉₇⋊C₉	C₃²	1	3	yes	—
9	C₃×C₉₇⋊C₉	C₃²	1	3	yes	—
10	C₃×C₉₇⋊C₉	C₃²	1	3	yes	—
11	C₃³×C₉₇	C₃²	1	1	yes	characteristic
12	3₋¹⁺²×C₉₇	—	3	3	no	—
13	C₃×C₉×C₉₇	C₃²	1	2	yes	—
14	C₃×C₉×C₉₇	C₃²	1	1	yes	characteristic
15	C₃²×C₉₇	—	3	1	no	—
16	C₃²×C₉₇	—	3	3	no	—
17	C₃²×C₉₇	C₃³	1	1	yes	characteristic
18	C₉×C₉₇	—	3	2	no	—
19	C₉×C₉₇	3₋¹⁺²	1	3	yes	complemented
20	C₉₇⋊C₉	—	3	3	no	—
21	C₉₇⋊C₉	—	3	3	no	—
22	C₉₇⋊C₉	3₋¹⁺²	1	9	yes	complemented
23	C₃×C₉₇	—	9	1	no	—
24	C₃×C₉₇	C₃×3₋¹⁺²	1	3	yes	—
25	C₃×C₉₇	C₃×3₋¹⁺²	1	1	yes	characteristic
26	#6900	—	97	1	no	maximal, Sylow
27	C₉₇	#6900	1	1	yes	characteristic, complemented, Sylow
28	C₃×3₋¹⁺²	—	97	3	no	—
29	C₃×3₋¹⁺²	C₉₇⋊C₃	1	1	yes	characteristic
30	C₉⋊C₉	—	97	6	no	—
31	C₉⋊C₉	—	97	1	no	—
32	C₉⋊C₉	—	97	1	no	—
33	C₉²	—	97	1	no	—
34	C₃³	C₃×C₉₇⋊C₃	1	1	yes	characteristic
35	3₋¹⁺²	—	3	3	no	—
36	3₋¹⁺²	—	291	9	no	—
37	C₃×C₉	—	97	3	no	—
38	C₃×C₉	—	97	3	no	—
39	C₃×C₉	—	97	3	no	—
40	C₃×C₉	C₃×C₉₇⋊C₃	1	2	yes	—
41	C₃×C₉	C₃×C₉₇⋊C₃	1	1	yes	characteristic
42	C₃²	—	3	3	no	—
43	C₃²	—	3	1	no	—
44	C₃²	C₃²×C₉₇⋊C₃	1	1	yes	characteristic
45	C₉	—	3	2	no	—
46	C₉	—	291	3	no	—
47	C₉	—	291	3	no	—
48	C₉	—	97	9	no	—
49	C₉	#14579757	1	3	yes	complemented
50	C₃	—	9	1	no	—
51	C₃	3₋¹⁺²×C₉₇⋊C₃	1	3	yes	—
52	C₃	C₃×#14579757	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₉₇⋊C₃
commutator, socle, cosocle	C₃²×C₉₇	C₃³
Fitting	C₃×3₋¹⁺²×C₉₇	C₃
prime radical	C₃³×C₉₇	C₃²
hypercenter	C₃×3₋¹⁺²	C₉₇⋊C₃
hypocenter	C₉₇	#6900
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
2. #21463352	C₃	semidirect
3. #21463351	C₃	semidirect
4. (C₉₇⋊C₉)⋊C₉	C₃	semidirect
5. C₉×C₉₇⋊C₉	C₃	semidirect
19. C₉×C₉₇	3₋¹⁺²	semidirect
22. C₉₇⋊C₉	3₋¹⁺²	semidirect
27. C₉₇	#6900	semidirect
49. C₉	#14579757	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
#21463352	3
#21463351	3
(C₉₇⋊C₉)⋊C₉	3
C₉×C₉₇⋊C₉	3
C₃×3₋¹⁺²×C₉₇	3
#6900	97