Subgroups — #41005216

Order 26973.

332 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃²⋊#66524	C₃	1	1	yes	maximal, characteristic
2	C₃×C₃²⋊#66524	C₃	1	2	yes	maximal
3	#22582441	C₃	1	9	yes	maximal, complemented
4	C₃×C₃₇×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
5	C₃²×#66524	C₃²	1	3	yes	—
6	C₃²×#66524	C₃²	1	6	yes	—
7	C₃²⋊#66524	—	3	3	no	—
8	C₃²⋊#66524	—	3	6	no	—
9	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
10	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
11	C₃₇×C₃²⋊C₉	—	3	3	no	—
12	C₃³×C₃₇	—	3	3	no	—
13	C₃³×C₃₇	—	3	3	no	—
14	C₃³×C₃₇	—	3	6	no	—
15	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
16	C₃³×C₃₇	C₃³	1	1	yes	characteristic
17	C₃×C₉×C₃₇	—	3	3	no	—
18	C₃×C₉×C₃₇	—	3	9	no	—
19	C₃×#66524	—	3	3	no	—
20	C₃×#66524	—	3	6	no	—
21	C₃×#66524	—	3	9	no	—
22	C₃×#66524	—	3	18	no	—
23	#10624568	—	37	1	no	maximal, Sylow
24	C₃²×C₃₇	—	3	3	no	—
25	C₃²×C₃₇	—	3	3	no	—
26	C₃²×C₃₇	—	3	6	no	—
27	C₃²×C₃₇	—	9	9	no	—
28	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
29	C₃²×C₃₇	#737	1	3	yes	—
30	C₃²×C₃₇	#737	1	3	yes	—
31	C₃²×C₃₇	#737	1	6	yes	—
32	C₉×C₃₇	—	9	3	no	—
33	#66524	—	9	3	no	—
34	#66524	—	9	6	no	—
35	C₃×C₃²⋊C₉	—	37	1	no	—
36	C₃×C₃²⋊C₉	—	37	2	no	—
37	C₃×C₃²⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic
38	#6855	—	37	9	no	—
39	C₃×C₃₇	—	9	3	no	—
40	C₃×C₃₇	#6893	1	1	yes	characteristic
41	C₃×C₃₇	#6893	1	1	yes	characteristic
42	C₃×C₃₇	#6893	1	2	yes	—
43	C₃×C₃₇	#6897	1	9	yes	—
44	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
45	C₃²×C₉	—	37	3	no	—
46	C₃²×C₉	—	37	6	no	—
47	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
48	C₃²⋊C₉	—	3	3	no	—
49	C₃²⋊C₉	—	111	3	no	—
50	C₃²⋊C₉	—	111	6	no	—
51	C₃₇	#10624568	1	1	yes	characteristic, complemented, Sylow
52	C₃³	—	3	3	no	—
53	C₃³	—	3	3	no	—
54	C₃³	—	3	6	no	—
55	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
56	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
57	C₃×C₉	—	3	3	no	—
58	C₃×C₉	—	3	9	no	—
59	C₃×C₉	—	111	3	no	—
60	C₃×C₉	—	111	6	no	—
61	C₃×C₉	—	111	9	no	—
62	C₃×C₉	—	111	18	no	—
63	C₃²	—	3	3	no	—
64	C₃²	—	3	3	no	—
65	C₃²	—	3	6	no	—
66	C₃²	—	9	9	no	—
67	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
68	C₃²	#15688530	1	3	yes	—
69	C₃²	#15688531	1	3	yes	—
70	C₃²	#15688532	1	6	yes	—
71	C₉	—	9	3	no	—
72	C₉	—	333	3	no	—
73	C₉	—	333	6	no	—
74	C₃	—	9	3	no	—
75	C₃	#22582562	1	1	yes	characteristic
76	C₃	#22582564	1	1	yes	characteristic
77	C₃	#22582565	1	2	yes	—
78	C₃	#22582585	1	9	yes	—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator, socle, cosocle	C₃³×C₃₇	C₃³
Fitting	C₃×C₃₇×C₃²⋊C₉	C₃
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	C₃×C₃²⋊C₉	C₃₇⋊C₃
hypocenter	C₃₇	#10624568
LCS	G ▷ C₃³×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃×C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
3. #22582441	C₃	semidirect
51. C₃₇	#10624568	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₃²⋊#66524	3
#22582441	3
C₃×C₃₇×C₃²⋊C₉	3
#10624568	37