Subgroups — #37555894

Order 23571.

124 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	C₃²×C₉₇⋊C₉	C₃	1	1	yes	maximal, characteristic
3	#21463352	C₃	1	3	yes	maximal, complemented
4	(C₉₇⋊C₉)⋊C₉	C₃	1	3	yes	maximal, complemented
5	(C₉₇⋊C₉)⋊C₉	C₃	1	3	yes	maximal, complemented
6	C₃²⋊(C₉₇⋊C₉)	C₃	1	1	yes	maximal, characteristic
7	C₃²⋊C₉×C₉₇	C₃	1	1	yes	maximal, characteristic
8	#14579757	—	3	3	no	—
9	C₃×C₉₇⋊C₉	—	3	3	no	—
10	C₃×C₉₇⋊C₉	C₃²	1	3	yes	—
11	C₃×C₉₇⋊C₉	C₃²	1	1	yes	characteristic
12	C₃×C₉₇⋊C₉	C₃²	1	3	yes	—
13	C₃×C₉₇⋊C₉	C₃²	1	1	yes	characteristic
14	C₃×C₉₇⋊C₉	C₃²	1	1	yes	characteristic
15	C₃³×C₉₇	C₃²	1	1	yes	characteristic
16	C₃×C₉×C₉₇	C₃²	1	3	yes	—
17	C₃²×C₉₇	—	3	1	no	—
18	C₃²×C₉₇	—	3	1	no	—
19	C₃²×C₉₇	—	3	1	no	—
20	C₃²×C₉₇	3₋¹⁺²	1	3	yes	—
21	C₃²×C₉₇	C₃³	1	1	yes	characteristic
22	C₉×C₉₇	—	3	3	no	—
23	C₉₇⋊C₉	—	3	1	no	—
24	C₉₇⋊C₉	—	3	1	no	—
25	C₉₇⋊C₉	—	3	3	no	—
26	C₉₇⋊C₉	—	3	3	no	—
27	C₉₇⋊C₉	3₋¹⁺²	1	3	yes	complemented
28	C₃×C₉₇	—	3	3	no	—
29	C₃×C₉₇	C₃×3₋¹⁺²	1	1	yes	characteristic
30	C₃×C₉₇	C₃×3₋¹⁺²	1	1	yes	characteristic
31	C₃×C₉₇	#737	1	1	yes	characteristic
32	C₃×C₉₇	#737	1	1	yes	characteristic
33	#6894	—	97	1	no	maximal, Sylow
34	C₉₇	#6894	1	1	yes	characteristic, complemented, Sylow
35	C₃×3₋¹⁺²	—	97	1	no	—
36	C₃²×C₉	—	97	1	no	—
37	C₉⋊C₉	—	97	3	no	—
38	C₉⋊C₉	—	97	3	no	—
39	C₉⋊C₉	—	97	3	no	—
40	C₃²⋊C₉	—	97	1	no	—
41	C₃²⋊C₉	C₉₇⋊C₃	1	1	yes	characteristic
42	C₃³	C₃×C₉₇⋊C₃	1	1	yes	characteristic
43	3₋¹⁺²	—	291	3	no	—
44	C₃×C₉	—	291	3	no	—
45	C₃×C₉	—	97	3	no	—
46	C₃×C₉	—	97	1	no	—
47	C₃×C₉	—	97	1	no	—
48	C₃×C₉	—	97	3	no	—
49	C₃×C₉	—	97	1	no	—
50	C₃×C₉	C₃×C₉₇⋊C₃	1	3	yes	—
51	C₃²	—	3	1	no	—
52	C₃²	—	3	1	no	—
53	C₃²	—	3	1	no	—
54	C₃²	#14579755	1	3	yes	—
55	C₃²	C₃²×C₉₇⋊C₃	1	1	yes	characteristic
56	C₉	—	3	3	no	—
57	C₉	—	291	1	no	—
58	C₉	—	291	1	no	—
59	C₉	—	291	3	no	—
60	C₉	—	291	3	no	—
61	C₉	—	97	3	no	—
62	C₃	—	3	3	no	—
63	C₃	3₋¹⁺²×C₉₇⋊C₃	1	1	yes	characteristic
64	C₃	C₃×#14579755	1	1	yes	characteristic
65	C₃	#21463376	1	1	yes	characteristic
66	C₃	#21463378	1	1	yes	characteristic

	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	#6894
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
3. #21463352	C₃	semidirect
4. (C₉₇⋊C₉)⋊C₉	C₃	semidirect
5. (C₉₇⋊C₉)⋊C₉	C₃	semidirect
27. C₉₇⋊C₉	3₋¹⁺²	semidirect
34. C₉₇	#6894	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
C₃²×C₉₇⋊C₉	3
#21463352	3
(C₉₇⋊C₉)⋊C₉	3
C₃²⋊(C₉₇⋊C₉)	3
C₃²⋊C₉×C₉₇	3
#6894	97