Subgroups — #41007853

Order 26973.

730 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688530	C₃	1	1	yes	maximal, characteristic
2	#22582575	C₃	1	8	yes	maximal, complemented
3	#22582573	C₃	1	16	yes	maximal, complemented
4	#22582572	C₃	1	8	yes	maximal, complemented
5	H(3,𝔽₃)×#66524	C₃	1	2	yes	maximal
6	C₃×#15688506	C₃	1	4	yes	maximal, complemented
7	C₉×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
8	#15688530	—	3	9	no	—
9	C₃×#11758608	C₃²	1	8	yes	—
10	C₃²×#66524	C₃²	1	4	yes	—
11	C₃²×#66524	C₃²	1	8	yes	—
12	#15688506	C₃²	1	1	yes	characteristic
13	#15688506	C₃²	1	8	yes	—
14	#15688506	C₃²	1	16	yes	—
15	#15688506	C₃²	1	8	yes	—
16	#15688504	C₃²	1	48	yes	—
17	C₃²⋊#66524	C₃²	1	16	yes	—
18	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
19	C₃²×C₉×C₃₇	C₃²	1	4	yes	—
20	C₃₇×C₃²⋊C₉	C₃²	1	8	yes	—
21	C₃³×C₃₇	C₃³	1	4	yes	—
22	H(3,𝔽₃)×C₃₇	3₋¹⁺²	1	9	yes	—
23	C₃×C₉×C₃₇	—	3	12	no	—
24	C₃×C₉×C₃₇	C₃³	1	8	yes	—
25	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
26	#11758608	—	3	24	no	—
27	C₃×#66524	—	3	12	no	—
28	C₃×#66524	—	9	8	no	—
29	C₃×#66524	C₃³	1	16	yes	—
30	C₃×#66524	C₃³	1	8	yes	—
31	C₃×#66524	C₃³	1	1	yes	characteristic
32	C₃×#66524	C₃³	1	2	yes	—
33	#10624900	—	37	1	no	maximal, Sylow
34	C₃²×C₃₇	—	3	4	no	—
35	C₃²×C₃₇	—	3	8	no	—
36	C₃²×C₃₇	C₃×3₋¹⁺²	1	12	yes	—
37	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
38	C₉×C₃₇	—	3	8	no	—
39	C₉×C₃₇	#737	1	3	yes	complemented
40	#66524	—	3	1	no	—
41	#66524	—	3	8	no	—
42	#66524	—	3	2	no	—
43	#66524	—	3	16	no	—
44	C₃×#737	—	37	1	no	—
45	#6895	—	37	8	no	—
46	#6894	—	37	8	no	—
47	#6894	—	37	16	no	—
48	C₉×H(3,𝔽₃)	—	37	2	no	—
49	C₉×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
50	C₃×C₉⋊C₉	—	37	4	no	—
51	C₃×C₃₇	—	3	12	no	—
52	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
53	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
54	C₃×C₃₇	3₋¹⁺⁴	1	2	yes	—
55	#737	—	111	9	no	—
56	C₃×3₋¹⁺²	—	37	8	no	—
57	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
58	C₃²×C₉	—	37	8	no	—
59	C₃²×C₉	—	37	4	no	—
60	C₃²×C₉	C₃×C₃₇⋊C₃	1	4	yes	—
61	C₉⋊C₉	—	37	48	no	—
62	C₉⋊C₉	—	37	1	no	—
63	C₉⋊C₉	—	37	8	no	—
64	C₉⋊C₉	—	37	16	no	—
65	C₉⋊C₉	—	37	8	no	—
66	C₃²⋊C₉	—	37	16	no	—
67	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	8	yes	—
68	C₃₇	#10624900	1	1	yes	characteristic, complemented, Sylow
69	C₃³	C₃²×C₃₇⋊C₃	1	4	yes	—
70	3₋¹⁺²	—	111	24	no	—
71	H(3,𝔽₃)	#11758606	1	9	yes	—
72	C₃×C₉	—	3	12	no	—
73	C₃×C₉	—	37	16	no	—
74	C₃×C₉	—	37	1	no	—
75	C₃×C₉	—	37	2	no	—
76	C₃×C₉	—	37	8	no	—
77	C₃×C₉	—	333	8	no	—
78	C₃×C₉	—	111	12	no	—
79	C₃×C₉	C₃²×C₃₇⋊C₃	1	8	yes	—
80	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
81	C₃²	—	3	4	no	—
82	C₃²	—	3	8	no	—
83	C₃²	C₃×#11758606	1	12	yes	—
84	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
85	C₉	—	3	8	no	—
86	C₉	—	111	1	no	—
87	C₉	—	111	8	no	—
88	C₉	—	111	2	no	—
89	C₉	—	111	16	no	—
90	C₉	#15688530	1	3	yes	complemented
91	C₃	—	3	12	no	—
92	C₃	C₃²×#11758606	1	1	yes	characteristic
93	C₃	C₃×#15688530	1	1	yes	characteristic
94	C₃	#22582677	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₉×H(3,𝔽₃)×C₃₇	C₃
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₃²
hypercenter	C₉×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624900
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₉×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
2. #22582575	C₃	semidirect
3. #22582573	C₃	semidirect
4. #22582572	C₃	semidirect
6. C₃×#15688506	C₃	semidirect
39. C₉×C₃₇	#737	semidirect
68. C₃₇	#10624900	semidirect
90. C₉	#15688530	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688530	3
#22582575	3
#22582573	3
#22582572	3
H(3,𝔽₃)×#66524	3
C₃×#15688506	3
C₉×H(3,𝔽₃)×C₃₇	3
#10624900	37