Subgroups — #37555889

Order 23571.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₉₇⋊C₉	C₃	1	1	yes	maximal, characteristic
2	(C₉₇⋊C₉)⋊C₉	C₃	1	6	yes	maximal, complemented
3	(C₉₇⋊C₉)⋊C₉	C₃	1	3	yes	maximal, complemented
4	C₃²⋊(C₉₇⋊C₉)	C₃	1	2	yes	maximal
5	C₃×3₋¹⁺²×C₉₇	C₃	1	1	yes	maximal, characteristic
6	C₃×C₉₇⋊C₉	—	3	3	no	—
7	C₃×C₉₇⋊C₉	C₃²	1	6	yes	—
8	C₃×C₉₇⋊C₉	C₃²	1	2	yes	—
9	C₃×C₉₇⋊C₉	C₃²	1	1	yes	characteristic
10	C₃³×C₉₇	C₃²	1	1	yes	characteristic
11	3₋¹⁺²×C₉₇	—	3	3	no	—
12	C₃×C₉×C₉₇	C₃²	1	3	yes	—
13	C₃²×C₉₇	—	3	1	no	—
14	C₃²×C₉₇	—	3	2	no	—
15	C₃²×C₉₇	3₋¹⁺²	1	3	yes	—
16	C₃²×C₉₇	C₃³	1	1	yes	characteristic
17	C₉×C₉₇	—	3	3	no	—
18	C₉₇⋊C₉	—	3	2	no	—
19	C₉₇⋊C₉	—	3	6	no	—
20	C₉₇⋊C₉	3₋¹⁺²	1	3	yes	complemented
21	C₃×C₉₇	—	3	3	no	—
22	C₃×C₉₇	C₃×3₋¹⁺²	1	1	yes	characteristic
23	C₃×C₉₇	C₃×3₋¹⁺²	1	1	yes	characteristic
24	C₃×C₉₇	#737	1	2	yes	—
25	#6894	—	97	1	no	maximal, Sylow
26	C₉₇	#6894	1	1	yes	characteristic, complemented, Sylow
27	C₃×3₋¹⁺²	C₉₇⋊C₃	1	1	yes	characteristic
28	C₃²×C₉	—	97	1	no	—
29	C₉⋊C₉	—	97	6	no	—
30	C₉⋊C₉	—	97	3	no	—
31	C₃²⋊C₉	—	97	2	no	—
32	C₃³	C₃×C₉₇⋊C₃	1	1	yes	characteristic
33	3₋¹⁺²	—	3	3	no	—
34	C₃×C₉	—	291	3	no	—
35	C₃×C₉	—	97	6	no	—
36	C₃×C₉	—	97	2	no	—
37	C₃×C₉	—	97	1	no	—
38	C₃×C₉	C₃×C₉₇⋊C₃	1	3	yes	—
39	C₃²	—	3	1	no	—
40	C₃²	—	3	2	no	—
41	C₃²	#14579757	1	3	yes	—
42	C₃²	C₃²×C₉₇⋊C₃	1	1	yes	characteristic
43	C₉	—	3	3	no	—
44	C₉	—	291	2	no	—
45	C₉	—	291	6	no	—
46	C₉	—	97	3	no	—
47	C₃	—	3	3	no	—
48	C₃	3₋¹⁺²×C₉₇⋊C₃	1	1	yes	characteristic
49	C₃	C₃×#14579757	1	1	yes	characteristic
50	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	#6894
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. (C₉₇⋊C₉)⋊C₉	C₃	semidirect
3. (C₉₇⋊C₉)⋊C₉	C₃	semidirect
20. C₉₇⋊C₉	3₋¹⁺²	semidirect
26. C₉₇	#6894	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₉₇⋊C₉	3
(C₉₇⋊C₉)⋊C₉	3
C₃²⋊(C₉₇⋊C₉)	3
C₃×3₋¹⁺²×C₉₇	3
#6894	97