Subgroups — #37555857

Order 23571.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×C₉₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	(C₉₇⋊C₉)⋊C₉	C₃	1	6	yes	maximal, complemented
3	(C₃×C₉₇⋊C₃)⋊C₉	C₃	1	3	yes	maximal
4	C₉×C₉₇⋊C₉	C₃	1	2	yes	maximal, complemented
5	C₉²×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃²×C₉₇⋊C₃	C₃²	1	1	yes	characteristic
7	C₃×C₉₇⋊C₉	C₃²	1	6	yes	—
8	C₃×C₉₇⋊C₉	C₃²	1	2	yes	—
9	C₉×C₉₇⋊C₃	C₉	1	9	yes	complemented
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	3	no	—
14	C₃×C₉₇⋊C₃	C₃×C₉	1	3	yes	—
15	C₉×C₉₇	—	3	3	no	—
16	C₉×C₉₇	C₃×C₉	1	3	yes	—
17	C₉₇⋊C₉	—	3	6	no	—
18	C₉₇⋊C₉	C₃×C₉	1	6	yes	—
19	C₃×C₉₇	C₃²×C₉	1	1	yes	characteristic
20	C₃×C₉₇	#737	1	3	yes	—
21	C₉₇⋊C₃	—	3	3	no	—
22	#6887	—	97	1	no	maximal, Sylow
23	C₉₇	#6887	1	1	yes	characteristic, complemented, Sylow
24	C₃²×C₉	—	97	1	no	—
25	C₉⋊C₉	—	97	6	no	—
26	C₃²⋊C₉	—	97	3	no	—
27	C₉²	—	97	2	no	—
28	C₉²	C₉₇⋊C₃	1	1	yes	characteristic, complemented
29	C₃³	—	97	1	no	—
30	C₃×C₉	—	97	6	no	—
31	C₃×C₉	—	97	2	no	—
32	C₃×C₉	—	97	9	no	—
33	C₃×C₉	C₃×C₉₇⋊C₃	1	3	yes	—
34	C₃×C₉	C₃×C₉₇⋊C₃	1	1	yes	characteristic
35	C₃²	—	291	3	no	—
36	C₃²	—	97	3	no	—
37	C₃²	C₃²×C₉₇⋊C₃	1	1	yes	characteristic
38	C₉	—	3	3	no	—
39	C₉	—	291	6	no	—
40	C₉	—	97	6	no	—
41	C₉	C₉×C₉₇⋊C₃	1	3	yes	—
42	C₃	—	291	3	no	—
43	C₃	C₃×C₉×C₉₇⋊C₃	1	1	yes	characteristic
44	C₃	#21463377	1	3	yes	—

	subgroup	G/H
center	C₃×C₉	C₃×C₉₇⋊C₃
commutator	C₃×C₉₇	C₃²×C₉
Frattini	C₃²	C₃²×C₉₇⋊C₃
Fitting	C₉²×C₉₇	C₃
socle, cosocle	C₃²×C₉₇	C₃³
prime radical	C₃²×C₉₇⋊C₃	C₃²
hypercenter	C₉²	C₉₇⋊C₃
hypocenter	C₉₇	#6887
LCS	G ▷ C₃×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₃×C₉ ◁ C₉²

Complements

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

N	H ≅ G/N	type
2. (C₉₇⋊C₉)⋊C₉	C₃	semidirect
4. C₉×C₉₇⋊C₉	C₃	semidirect
5. C₉²×C₉₇	C₃	semidirect
9. C₉×C₉₇⋊C₃	C₉	semidirect
23. C₉₇	#6887	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×C₉×C₉₇⋊C₃	3
(C₉₇⋊C₉)⋊C₉	3
(C₃×C₉₇⋊C₃)⋊C₉	3
C₉×C₉₇⋊C₉	3
C₉²×C₉₇	3
#6887	97