Subgroups — #37555902

Order 23571.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14579756	C₃	1	1	yes	maximal, characteristic
2	C₃×#14579755	C₃	1	1	yes	maximal, characteristic
3	#21463352	C₃	1	2	yes	maximal, complemented
4	#21463351	C₃	1	2	yes	maximal, complemented
5	(C₉₇⋊C₉)⋊C₉	C₃	1	4	yes	maximal, complemented
6	(C₃×C₉₇⋊C₃)⋊C₉	C₃	1	2	yes	maximal
7	C₉²×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₉₇⋊C₃	C₃²	1	1	yes	characteristic
9	#14579756	—	3	3	no	—
10	#14579755	—	3	3	no	—
11	C₃×C₉₇⋊C₉	C₃²	1	4	yes	—
12	C₃×C₉₇⋊C₉	C₃²	1	2	yes	—
13	C₃×C₉₇⋊C₉	C₃²	1	2	yes	—
14	C₃×C₉×C₉₇	C₃²	1	2	yes	—
15	C₃×C₉×C₉₇	C₃²	1	1	yes	characteristic
16	C₃×C₉×C₉₇	C₃²	1	1	yes	characteristic
17	C₃²×C₉₇	C₃³	1	1	yes	characteristic
18	C₃×C₉₇⋊C₃	—	3	1	no	—
19	C₃×C₉₇⋊C₃	—	3	1	no	—
20	C₃×C₉₇⋊C₃	—	3	2	no	—
21	C₉×C₉₇	—	3	2	no	—
22	C₉×C₉₇	3₋¹⁺²	1	3	yes	complemented
23	C₉×C₉₇	3₋¹⁺²	1	3	yes	complemented
24	C₉₇⋊C₉	—	3	2	no	—
25	C₉₇⋊C₉	—	3	2	no	—
26	C₉₇⋊C₉	—	3	4	no	—
27	C₃×C₉₇	C₃×3₋¹⁺²	1	1	yes	characteristic
28	C₃×C₉₇	C₃×3₋¹⁺²	1	1	yes	characteristic
29	C₃×C₉₇	#737	1	2	yes	—
30	C₉₇⋊C₃	—	9	1	no	—
31	#6896	—	97	1	no	maximal, Sylow
32	C₉₇	#6896	1	1	yes	characteristic, complemented, Sylow
33	C₃×3₋¹⁺²	—	97	1	no	—
34	C₃×3₋¹⁺²	—	97	1	no	—
35	C₉⋊C₉	—	97	4	no	—
36	C₉⋊C₉	—	97	2	no	—
37	C₉⋊C₉	—	97	2	no	—
38	C₃²⋊C₉	—	97	2	no	—
39	C₉²	C₉₇⋊C₃	1	1	yes	characteristic, complemented
40	C₃³	—	97	1	no	—
41	3₋¹⁺²	—	291	3	no	—
42	3₋¹⁺²	—	291	3	no	—
43	C₃×C₉	—	97	4	no	—
44	C₃×C₉	—	97	2	no	—
45	C₃×C₉	—	97	2	no	—
46	C₃×C₉	C₃×C₉₇⋊C₃	1	2	yes	—
47	C₃×C₉	C₃×C₉₇⋊C₃	1	1	yes	characteristic
48	C₃×C₉	C₃×C₉₇⋊C₃	1	1	yes	characteristic
49	C₃²	—	291	1	no	—
50	C₃²	—	291	1	no	—
51	C₃²	—	291	2	no	—
52	C₃²	C₃²×C₉₇⋊C₃	1	1	yes	characteristic
53	C₉	—	3	2	no	—
54	C₉	—	291	2	no	—
55	C₉	—	291	2	no	—
56	C₉	—	291	4	no	—
57	C₉	#14579755	1	3	yes	complemented
58	C₉	#14579756	1	3	yes	complemented
59	C₃	—	873	1	no	—
60	C₃	C₃×#14579755	1	1	yes	characteristic
61	C₃	C₃×#14579756	1	1	yes	characteristic
62	C₃	#21463377	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₉₇⋊C₃
commutator, socle, cosocle	C₃²×C₉₇	C₃³
Fitting	C₉²×C₉₇	C₃
prime radical	C₃²×C₉₇⋊C₃	C₃²
hypercenter	C₉²	C₉₇⋊C₃
hypocenter	C₉₇	#6896
LCS	G ▷ C₃²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₃² ◁ C₉²

Complements

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

N	H ≅ G/N	type
3. #21463352	C₃	semidirect
4. #21463351	C₃	semidirect
5. (C₉₇⋊C₉)⋊C₉	C₃	semidirect
7. C₉²×C₉₇	C₃	semidirect
22. C₉×C₉₇	3₋¹⁺²	semidirect
23. C₉×C₉₇	3₋¹⁺²	semidirect
32. C₉₇	#6896	semidirect
39. C₉²	C₉₇⋊C₃	semidirect
57. C₉	#14579755	semidirect
58. C₉	#14579756	semidirect

Maximal subgroups

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

subgroup	index
C₃×#14579756	3
C₃×#14579755	3
#21463352	3
#21463351	3
(C₉₇⋊C₉)⋊C₉	3
(C₃×C₉₇⋊C₃)⋊C₉	3
C₉²×C₉₇	3
#6896	97