Subgroups — #37555884

Order 23571.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14579756	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#14579755	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃²⋊(C₉₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
4	C₃×C₉×C₉₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃²⋊(C₉₇⋊C₉)	C₃	1	2	yes	maximal, complemented
6	(C₃×C₉₇⋊C₃)⋊C₉	C₃	1	2	yes	maximal, complemented
7	(C₃×C₉₇⋊C₃)⋊C₉	C₃	1	2	yes	maximal, complemented
8	(C₃×C₉₇⋊C₃)⋊C₉	C₃	1	2	yes	maximal, complemented
9	C₃²⋊C₉×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃²×C₉₇⋊C₃	C₃²	1	2	yes	—
11	C₃²×C₉₇⋊C₃	C₃²	1	1	yes	characteristic
12	#14579756	—	3	3	no	—
13	#14579755	—	3	3	no	—
14	C₃²⋊(C₉₇⋊C₃)	—	3	3	no	—
15	C₃×C₉₇⋊C₉	C₃²	1	2	yes	—
16	C₃×C₉₇⋊C₉	C₃²	1	2	yes	—
17	C₃×C₉₇⋊C₉	C₃²	1	2	yes	—
18	C₉×C₉₇⋊C₃	—	3	3	no	—
19	C₃³×C₉₇	C₃²	1	1	yes	characteristic
20	C₃×C₉×C₉₇	C₃²	1	1	yes	characteristic
21	C₃×C₉×C₉₇	C₃²	1	1	yes	characteristic
22	C₃×C₉×C₉₇	C₃²	1	1	yes	characteristic
23	C₃²×C₉₇	—	3	1	no	—
24	C₃²×C₉₇	—	3	1	no	—
25	C₃²×C₉₇	—	3	2	no	—
26	C₃²×C₉₇	C₃³	1	1	yes	characteristic
27	C₃×C₉₇⋊C₃	—	3	1	no	—
28	C₃×C₉₇⋊C₃	—	3	2	no	—
29	C₃×C₉₇⋊C₃	—	3	2	no	—
30	C₃×C₉₇⋊C₃	—	3	2	no	—
31	C₃×C₉₇⋊C₃	—	3	4	no	—
32	C₃×C₉₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
33	C₉×C₉₇	—	3	1	no	—
34	C₉×C₉₇	—	3	1	no	—
35	C₉×C₉₇	—	3	1	no	—
36	C₉₇⋊C₉	—	3	2	no	—
37	C₉₇⋊C₉	—	3	2	no	—
38	C₉₇⋊C₉	—	3	2	no	—
39	C₃×C₉₇	—	9	1	no	—
40	C₃×C₉₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
41	C₃×C₉₇	#737	1	1	yes	characteristic
42	C₃×C₉₇	#737	1	2	yes	—
43	C₉₇⋊C₃	—	3	3	no	—
44	C₉₇⋊C₃	—	9	2	no	—
45	#6893	—	97	1	no	maximal, Sylow
46	C₉₇	#6893	1	1	yes	characteristic, complemented, Sylow
47	C₃×3₋¹⁺²	—	97	1	no	—
48	C₃×3₋¹⁺²	—	97	1	no	—
49	C₃×H(3,𝔽₃)	—	97	1	no	—
50	C₃²×C₉	—	97	1	no	—
51	C₃²⋊C₉	—	97	2	no	—
52	C₃²⋊C₉	—	97	2	no	—
53	C₃²⋊C₉	—	97	2	no	—
54	C₃²⋊C₉	—	97	2	no	—
55	C₃²⋊C₉	C₉₇⋊C₃	1	1	yes	characteristic, complemented
56	C₃³	—	97	2	no	—
57	C₃³	—	97	1	no	—
58	C₃³	C₃×C₉₇⋊C₃	1	1	yes	characteristic
59	3₋¹⁺²	—	291	3	no	—
60	3₋¹⁺²	—	291	3	no	—
61	H(3,𝔽₃)	—	291	3	no	—
62	C₃×C₉	—	291	3	no	—
63	C₃×C₉	—	97	2	no	—
64	C₃×C₉	—	97	2	no	—
65	C₃×C₉	—	97	2	no	—
66	C₃×C₉	C₃×C₉₇⋊C₃	1	1	yes	characteristic
67	C₃×C₉	C₃×C₉₇⋊C₃	1	1	yes	characteristic
68	C₃×C₉	C₃×C₉₇⋊C₃	1	1	yes	characteristic
69	C₃²	—	3	1	no	—
70	C₃²	—	3	1	no	—
71	C₃²	—	3	2	no	—
72	C₃²	—	291	1	no	—
73	C₃²	—	291	2	no	—
74	C₃²	—	291	2	no	—
75	C₃²	—	291	2	no	—
76	C₃²	—	291	4	no	—
77	C₃²	—	97	3	no	—
78	C₃²	C₃²×C₉₇⋊C₃	1	1	yes	characteristic
79	C₉	—	3	1	no	—
80	C₉	—	3	1	no	—
81	C₉	—	3	1	no	—
82	C₉	—	291	2	no	—
83	C₉	—	291	2	no	—
84	C₉	—	291	2	no	—
85	C₃	—	9	1	no	—
86	C₃	—	291	3	no	—
87	C₃	—	873	2	no	—
88	C₃	H(3,𝔽₃)×C₉₇⋊C₃	1	1	yes	characteristic
89	C₃	#21463377	1	1	yes	characteristic
90	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	C₃×C₃²⋊(C₉₇⋊C₃)	C₃
hypercenter	C₃²⋊C₉	C₉₇⋊C₃
hypocenter	C₉₇	#6893
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. C₃×#14579756	C₃	semidirect
2. C₃×#14579755	C₃	semidirect
4. C₃×C₉×C₉₇⋊C₃	C₃	semidirect
5. C₃²⋊(C₉₇⋊C₉)	C₃	semidirect
6. (C₃×C₉₇⋊C₃)⋊C₉	C₃	semidirect
7. (C₃×C₉₇⋊C₃)⋊C₉	C₃	semidirect
8. (C₃×C₉₇⋊C₃)⋊C₉	C₃	semidirect
9. C₃²⋊C₉×C₉₇	C₃	semidirect
46. C₉₇	#6893	semidirect
55. C₃²⋊C₉	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×#14579756	3
C₃×#14579755	3
C₃×C₃²⋊(C₉₇⋊C₃)	3
C₃×C₉×C₉₇⋊C₃	3
C₃²⋊(C₉₇⋊C₉)	3
(C₃×C₉₇⋊C₃)⋊C₉	3
C₃²⋊C₉×C₉₇	3
#6893	97