Subgroups — #37555916

Order 23571.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14579755	C₃	1	1	yes	maximal, characteristic
2	#21463352	C₃	1	2	yes	maximal, complemented
3	(C₉₇⋊C₉)⋊C₉	C₃	1	6	yes	maximal, complemented
4	(C₃×C₉₇⋊C₃)⋊C₉	C₃	1	3	yes	maximal
5	C₉²×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃²×C₉₇⋊C₃	C₃²	1	1	yes	characteristic
7	#14579755	—	3	3	no	—
8	C₃×C₉₇⋊C₉	C₃²	1	6	yes	—
9	C₃×C₉₇⋊C₉	C₃²	1	2	yes	—
10	C₃×C₉×C₉₇	C₃²	1	3	yes	—
11	C₃×C₉×C₉₇	C₃²	1	1	yes	characteristic
12	C₃²×C₉₇	C₃³	1	1	yes	characteristic
13	C₃×C₉₇⋊C₃	—	3	1	no	—
14	C₃×C₉₇⋊C₃	—	3	3	no	—
15	C₉×C₉₇	—	3	3	no	—
16	C₉×C₉₇	3₋¹⁺²	1	3	yes	—
17	C₉₇⋊C₉	—	3	2	no	—
18	C₉₇⋊C₉	—	3	6	no	—
19	C₃×C₉₇	C₃×3₋¹⁺²	1	1	yes	characteristic
20	C₃×C₉₇	#737	1	3	yes	—
21	C₉₇⋊C₃	—	9	1	no	—
22	#6899	—	97	1	no	maximal, Sylow
23	C₉₇	#6899	1	1	yes	characteristic, complemented, Sylow
24	C₃×3₋¹⁺²	—	97	1	no	—
25	C₉⋊C₉	—	97	6	no	—
26	C₉⋊C₉	—	97	2	no	—
27	C₃²⋊C₉	—	97	3	no	—
28	C₉²	C₉₇⋊C₃	1	1	yes	characteristic, complemented
29	C₃³	—	97	1	no	—
30	3₋¹⁺²	—	291	3	no	—
31	C₃×C₉	—	97	6	no	—
32	C₃×C₉	—	97	2	no	—
33	C₃×C₉	C₃×C₉₇⋊C₃	1	3	yes	—
34	C₃×C₉	C₃×C₉₇⋊C₃	1	1	yes	characteristic
35	C₃²	—	291	1	no	—
36	C₃²	—	291	3	no	—
37	C₃²	C₃²×C₉₇⋊C₃	1	1	yes	characteristic
38	C₉	—	3	3	no	—
39	C₉	—	291	2	no	—
40	C₉	—	291	6	no	—
41	C₉	#14579755	1	3	yes	—
42	C₃	—	873	1	no	—
43	C₃	C₃×#14579755	1	1	yes	characteristic
44	C₃	#21463377	1	3	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	#6899
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
2. #21463352	C₃	semidirect
3. (C₉₇⋊C₉)⋊C₉	C₃	semidirect
5. C₉²×C₉₇	C₃	semidirect
23. C₉₇	#6899	semidirect
28. C₉²	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×#14579755	3
#21463352	3
(C₉₇⋊C₉)⋊C₉	3
(C₃×C₉₇⋊C₃)⋊C₉	3
C₉²×C₉₇	3
#6899	97