Subgroups — #41006153

Order 26973.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582553	C₃	1	2	yes	maximal
2	#22582552	C₃	1	2	yes	maximal
3	C₃₇⋊#6862	C₃	1	4	yes	maximal, complemented
4	C₃₇⋊#6862	C₃	1	4	yes	maximal, complemented
5	C₃₇×#6862	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
7	C₃×#11758605	—	3	2	no	—
8	C₃×#11758606	—	3	2	no	—
9	C₃²⋊#66524	C₃²	1	4	yes	—
10	C₃²⋊#66524	C₃²	1	4	yes	—
11	#15688502	—	3	4	no	—
12	#15688502	—	3	4	no	—
13	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
14	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
15	C₃³×C₃₇	C₃³	1	1	yes	characteristic
16	C₃²×C₃₇⋊C₃	—	9	2	no	—
17	C₃²×C₃₇⋊C₃	—	9	2	no	—
18	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	2	yes	—
19	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃×C₉×C₃₇	—	3	2	no	—
21	C₃×C₉×C₃₇	—	3	2	no	—
22	C₃×#66524	—	3	4	no	—
23	C₃×#66524	—	3	4	no	—
24	#11758606	—	9	6	no	—
25	#11758605	—	9	6	no	—
26	#10624675	—	37	1	no	maximal, Sylow
27	C₃²×C₃₇	—	3	2	no	—
28	C₃²×C₃₇	—	3	2	no	—
29	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
30	C₃×C₃₇⋊C₃	—	3	2	no	—
31	C₃×C₃₇⋊C₃	—	3	2	no	—
32	C₃×C₃₇⋊C₃	—	3	4	no	—
33	C₃×C₃₇⋊C₃	—	3	4	no	—
34	C₃×C₃₇⋊C₃	—	9	9	no	—
35	C₉×C₃₇	—	9	2	no	—
36	C₉×C₃₇	—	9	2	no	—
37	#66524	—	9	4	no	—
38	#66524	—	9	4	no	—
39	#6891	—	37	2	no	—
40	#6891	—	37	2	no	—
41	#6862	—	37	4	no	—
42	#6862	—	37	4	no	—
43	#6862	C₃₇⋊C₃	1	1	yes	characteristic, complemented
44	C₃×C₃₇	—	9	1	no	—
45	C₃×C₃₇	#6910	1	2	yes	—
46	C₃×C₃₇	#6910	1	2	yes	—
47	C₃₇⋊C₃	—	9	1	no	—
48	C₃₇⋊C₃	—	9	2	no	—
49	C₃⁴	—	37	1	no	—
50	C₃×3₋¹⁺²	—	111	2	no	—
51	C₃×3₋¹⁺²	—	111	2	no	—
52	C₃²⋊C₉	—	37	4	no	—
53	C₃²⋊C₉	—	37	4	no	—
54	C₃²⋊C₉	—	111	4	no	—
55	C₃²⋊C₉	—	111	4	no	—
56	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
57	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
58	C₃₇	#10624675	1	1	yes	characteristic, complemented, Sylow
59	C₃³	—	37	2	no	—
60	C₃³	—	37	1	no	—
61	C₃³	—	333	2	no	—
62	C₃³	—	333	2	no	—
63	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
64	3₋¹⁺²	—	333	6	no	—
65	3₋¹⁺²	—	333	6	no	—
66	C₃×C₉	—	3	2	no	—
67	C₃×C₉	—	3	2	no	—
68	C₃×C₉	—	111	4	no	—
69	C₃×C₉	—	111	4	no	—
70	C₃²	—	3	2	no	—
71	C₃²	—	3	2	no	—
72	C₃²	—	333	9	no	—
73	C₃²	—	111	2	no	—
74	C₃²	—	111	4	no	—
75	C₃²	—	111	2	no	—
76	C₃²	—	111	4	no	—
77	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
78	C₉	—	9	2	no	—
79	C₉	—	9	2	no	—
80	C₉	—	333	4	no	—
81	C₉	—	333	4	no	—
82	C₃	—	9	1	no	—
83	C₃	—	333	1	no	—
84	C₃	—	333	2	no	—
85	C₃	#22582640	1	2	yes	—
86	C₃	#22582641	1	2	yes	—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6862	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6862	C₃₇⋊C₃
hypocenter	C₃₇	#10624675
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6862

Complements

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

N	H ≅ G/N	type
3. C₃₇⋊#6862	C₃	semidirect
4. C₃₇⋊#6862	C₃	semidirect
5. C₃₇×#6862	C₃	semidirect
43. #6862	C₃₇⋊C₃	semidirect
58. C₃₇	#10624675	semidirect

Maximal subgroups

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

subgroup	index
#22582553	3
#22582552	3
C₃₇⋊#6862	3
C₃₇×#6862	3
#10624675	37