Subgroups — #37555911

Order 23571.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21463352	C₃	1	1	yes	maximal, characteristic, complemented
2	#21463351	C₃	1	1	yes	maximal, characteristic, complemented
3	(C₉₇⋊C₉)⋊C₉	C₃	1	2	yes	maximal, complemented
4	(C₉₇⋊C₉)⋊C₉	C₃	1	2	yes	maximal, complemented
5	(C₉₇⋊C₉)⋊C₉	C₃	1	2	yes	maximal, complemented
6	C₃²⋊(C₉₇⋊C₉)	C₃	1	2	yes	maximal
7	C₃²⋊(C₉₇⋊C₉)	C₃	1	1	yes	maximal, characteristic
8	C₉×C₉₇⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃²⋊C₉×C₉₇	C₃	1	1	yes	maximal, characteristic
10	C₃×C₉₇⋊C₉	C₃²	1	2	yes	—
11	C₃×C₉₇⋊C₉	C₃²	1	1	yes	characteristic
12	C₃×C₉₇⋊C₉	C₃²	1	2	yes	—
13	C₃×C₉₇⋊C₉	C₃²	1	2	yes	—
14	C₃×C₉₇⋊C₉	C₃²	1	2	yes	—
15	C₃³×C₉₇	C₃²	1	1	yes	characteristic
16	C₃×C₉×C₉₇	C₃²	1	1	yes	characteristic
17	C₃×C₉×C₉₇	C₃²	1	1	yes	characteristic
18	C₃×C₉×C₉₇	C₃²	1	1	yes	characteristic
19	C₃²×C₉₇	—	3	1	no	—
20	C₃²×C₉₇	—	3	1	no	—
21	C₃²×C₉₇	—	3	2	no	—
22	C₃²×C₉₇	C₃³	1	1	yes	characteristic
23	C₉×C₉₇	—	3	1	no	—
24	C₉×C₉₇	—	3	1	no	—
25	C₉×C₉₇	—	3	1	no	—
26	C₉₇⋊C₉	—	3	1	no	—
27	C₉₇⋊C₉	—	3	2	no	—
28	C₉₇⋊C₉	—	3	2	no	—
29	C₉₇⋊C₉	—	3	2	no	—
30	C₉₇⋊C₉	—	3	2	no	—
31	C₃×C₉₇	—	9	1	no	—
32	C₃×C₉₇	#737	1	1	yes	characteristic
33	C₃×C₉₇	#737	1	1	yes	characteristic
34	C₃×C₉₇	#737	1	2	yes	—
35	#6898	—	97	1	no	maximal, Sylow
36	C₉₇	#6898	1	1	yes	characteristic, complemented, Sylow
37	C₉⋊C₉	—	97	2	no	—
38	C₉⋊C₉	—	97	2	no	—
39	C₉⋊C₉	—	97	2	no	—
40	C₉⋊C₉	—	97	1	no	—
41	C₉⋊C₉	—	97	1	no	—
42	C₃²⋊C₉	—	97	1	no	—
43	C₃²⋊C₉	—	97	2	no	—
44	C₃²⋊C₉	C₉₇⋊C₃	1	1	yes	characteristic
45	C₉²	—	97	1	no	—
46	C₃³	C₃×C₉₇⋊C₃	1	1	yes	characteristic
47	C₃×C₉	—	97	1	no	—
48	C₃×C₉	—	97	2	no	—
49	C₃×C₉	—	97	2	no	—
50	C₃×C₉	—	97	2	no	—
51	C₃×C₉	—	97	2	no	—
52	C₃×C₉	C₃×C₉₇⋊C₃	1	1	yes	characteristic
53	C₃×C₉	C₃×C₉₇⋊C₃	1	1	yes	characteristic
54	C₃×C₉	C₃×C₉₇⋊C₃	1	1	yes	characteristic
55	C₃²	—	3	1	no	—
56	C₃²	—	3	1	no	—
57	C₃²	—	3	2	no	—
58	C₃²	C₃²×C₉₇⋊C₃	1	1	yes	characteristic
59	C₉	—	3	1	no	—
60	C₉	—	3	1	no	—
61	C₉	—	3	1	no	—
62	C₉	—	291	1	no	—
63	C₉	—	291	2	no	—
64	C₉	—	291	2	no	—
65	C₉	—	291	2	no	—
66	C₉	—	291	2	no	—
67	C₃	—	9	1	no	—
68	C₃	#21463376	1	1	yes	characteristic
69	C₃	#21463377	1	1	yes	characteristic
70	C₃	#21463378	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₉₇⋊C₃
commutator, socle, cosocle	C₃²×C₉₇	C₃³
Fitting	C₃²⋊C₉×C₉₇	C₃
prime radical	C₃³×C₉₇	C₃²
hypercenter	C₃²⋊C₉	C₉₇⋊C₃
hypocenter	C₉₇	#6898
LCS	G ▷ C₃²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
1. #21463352	C₃	semidirect
2. #21463351	C₃	semidirect
3. (C₉₇⋊C₉)⋊C₉	C₃	semidirect
4. (C₉₇⋊C₉)⋊C₉	C₃	semidirect
5. (C₉₇⋊C₉)⋊C₉	C₃	semidirect
8. C₉×C₉₇⋊C₉	C₃	semidirect
36. C₉₇	#6898	semidirect

Maximal subgroups

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

subgroup	index
#21463352	3
#21463351	3
(C₉₇⋊C₉)⋊C₉	3
C₃²⋊(C₉₇⋊C₉)	3
C₉×C₉₇⋊C₉	3
C₃²⋊C₉×C₉₇	3
#6898	97