Subgroups — #37555901

Order 23571.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14579757	C₃	1	1	yes	maximal, characteristic
2	#21463352	C₃	1	1	yes	maximal, characteristic, complemented
3	#21463351	C₃	1	1	yes	maximal, characteristic, complemented
4	(C₉₇⋊C₉)⋊C₉	C₃	1	2	yes	maximal, complemented
5	(C₉₇⋊C₉)⋊C₉	C₃	1	2	yes	maximal, complemented
6	(C₉₇⋊C₉)⋊C₉	C₃	1	2	yes	maximal, complemented
7	C₃²⋊(C₉₇⋊C₉)	C₃	1	2	yes	maximal
8	C₉×C₉₇⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃×3₋¹⁺²×C₉₇	C₃	1	1	yes	maximal, characteristic
10	#14579757	—	3	3	no	—
11	C₃×C₉₇⋊C₉	C₃²	1	2	yes	—
12	C₃×C₉₇⋊C₉	C₃²	1	2	yes	—
13	C₃×C₉₇⋊C₉	C₃²	1	2	yes	—
14	C₃×C₉₇⋊C₉	C₃²	1	1	yes	characteristic
15	C₃×C₉₇⋊C₉	C₃²	1	1	yes	characteristic
16	C₃×C₉₇⋊C₉	C₃²	1	1	yes	characteristic
17	C₃³×C₉₇	C₃²	1	1	yes	characteristic
18	3₋¹⁺²×C₉₇	—	3	3	no	—
19	C₃×C₉×C₉₇	C₃²	1	2	yes	—
20	C₃×C₉×C₉₇	C₃²	1	1	yes	characteristic
21	C₃²×C₉₇	—	3	1	no	—
22	C₃²×C₉₇	—	3	1	no	—
23	C₃²×C₉₇	—	3	2	no	—
24	C₃²×C₉₇	C₃³	1	1	yes	characteristic
25	C₉×C₉₇	—	3	2	no	—
26	C₉×C₉₇	3₋¹⁺²	1	3	yes	complemented
27	C₉₇⋊C₉	—	3	1	no	—
28	C₉₇⋊C₉	—	3	1	no	—
29	C₉₇⋊C₉	—	3	2	no	—
30	C₉₇⋊C₉	—	3	2	no	—
31	C₉₇⋊C₉	—	3	2	no	—
32	C₉₇⋊C₉	3₋¹⁺²	1	3	yes	complemented
33	C₃×C₉₇	—	9	1	no	—
34	C₃×C₉₇	C₃×3₋¹⁺²	1	1	yes	characteristic
35	C₃×C₉₇	C₃×3₋¹⁺²	1	1	yes	characteristic
36	C₃×C₉₇	#737	1	2	yes	—
37	#6896	—	97	1	no	maximal, Sylow
38	C₉₇	#6896	1	1	yes	characteristic, complemented, Sylow
39	C₃×3₋¹⁺²	—	97	1	no	—
40	C₃×3₋¹⁺²	C₉₇⋊C₃	1	1	yes	characteristic
41	C₉⋊C₉	—	97	2	no	—
42	C₉⋊C₉	—	97	2	no	—
43	C₉⋊C₉	—	97	2	no	—
44	C₉⋊C₉	—	97	1	no	—
45	C₉⋊C₉	—	97	1	no	—
46	C₃²⋊C₉	—	97	2	no	—
47	C₉²	—	97	1	no	—
48	C₃³	C₃×C₉₇⋊C₃	1	1	yes	characteristic
49	3₋¹⁺²	—	3	3	no	—
50	3₋¹⁺²	—	291	3	no	—
51	C₃×C₉	—	97	2	no	—
52	C₃×C₉	—	97	2	no	—
53	C₃×C₉	—	97	2	no	—
54	C₃×C₉	—	97	1	no	—
55	C₃×C₉	—	97	1	no	—
56	C₃×C₉	—	97	1	no	—
57	C₃×C₉	C₃×C₉₇⋊C₃	1	2	yes	—
58	C₃×C₉	C₃×C₉₇⋊C₃	1	1	yes	characteristic
59	C₃²	—	3	1	no	—
60	C₃²	—	3	1	no	—
61	C₃²	—	3	2	no	—
62	C₃²	C₃²×C₉₇⋊C₃	1	1	yes	characteristic
63	C₉	—	3	2	no	—
64	C₉	—	291	1	no	—
65	C₉	—	291	1	no	—
66	C₉	—	291	2	no	—
67	C₉	—	291	2	no	—
68	C₉	—	291	2	no	—
69	C₉	—	97	3	no	—
70	C₉	#14579757	1	3	yes	complemented
71	C₃	—	9	1	no	—
72	C₃	3₋¹⁺²×C₉₇⋊C₃	1	1	yes	characteristic
73	C₃	C₃×#14579757	1	1	yes	characteristic
74	C₃	#21463378	1	2	yes	—

	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	#6896
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. (C₉₇⋊C₉)⋊C₉	C₃	semidirect
6. (C₉₇⋊C₉)⋊C₉	C₃	semidirect
8. C₉×C₉₇⋊C₉	C₃	semidirect
26. C₉×C₉₇	3₋¹⁺²	semidirect
32. C₉₇⋊C₉	3₋¹⁺²	semidirect
38. C₉₇	#6896	semidirect
70. 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
C₃²⋊(C₉₇⋊C₉)	3
C₉×C₉₇⋊C₉	3
C₃×3₋¹⁺²×C₉₇	3
#6896	97