Subgroups — #41004822

Order 26973.

238 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582550	C₃	1	1	yes	maximal, characteristic, complemented
2	#22582468	C₃	1	2	yes	maximal, complemented
3	C₃₇×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
5	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
6	C₃²⋊#66524	—	3	2	no	—
7	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
8	C₃×H(3,𝔽₃)×C₃₇	—	3	1	no	—
9	C₃×H(3,𝔽₃)×C₃₇	—	3	3	no	—
10	C₃³×C₃₇	—	3	1	no	—
11	C₃³×C₃₇	—	3	1	no	—
12	C₃³×C₃₇	—	3	3	no	—
13	C₃³×C₃₇	—	3	2	no	—
14	C₃³×C₃₇	—	9	3	no	—
15	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
16	C₃²×C₃₇⋊C₃	—	3	3	no	—
17	H(3,𝔽₃)×C₃₇	—	9	9	no	—
18	H(3,𝔽₃)×C₃₇	—	9	1	no	—
19	H(3,𝔽₃)×C₃₇	—	9	2	no	—
20	C₃²⋊(C₃₇⋊C₃)	—	9	9	no	—
21	C₃²⋊(C₃₇⋊C₃)	—	9	1	no	—
22	C₃²⋊(C₃₇⋊C₃)	—	9	2	no	—
23	C₃×#66524	—	9	2	no	—
24	#10624474	—	37	1	no	maximal, Sylow
25	C₃²×C₃₇	—	3	3	no	—
26	C₃²×C₃₇	—	3	3	no	—
27	C₃²×C₃₇	—	9	1	no	—
28	C₃²×C₃₇	—	9	6	no	—
29	C₃²×C₃₇	—	9	2	no	—
30	C₃²×C₃₇	—	9	2	no	—
31	C₃²×C₃₇	—	9	9	no	—
32	C₃²×C₃₇	—	9	1	no	—
33	C₃²×C₃₇	—	9	3	no	—
34	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
35	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
36	C₃²×C₃₇	#732	1	2	yes	—
37	C₃×C₃₇⋊C₃	—	9	9	no	—
38	C₃×C₃₇⋊C₃	—	9	3	no	—
39	#66524	—	27	2	no	—
40	C₃²⋊H(3,𝔽₃)	—	37	1	no	—
41	C₃²⋊H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic, complemented
42	#6866	—	37	2	no	—
43	C₃×C₃₇	—	3	2	no	—
44	C₃×C₃₇	—	3	1	no	—
45	C₃×C₃₇	—	3	1	no	—
46	C₃×C₃₇	—	9	3	no	—
47	C₃×C₃₇	—	27	3	no	—
48	C₃×C₃₇	#6856	1	1	yes	characteristic
49	C₃₇⋊C₃	—	27	3	no	—
50	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
51	C₃×H(3,𝔽₃)	—	3	3	no	—
52	C₃×H(3,𝔽₃)	—	3	1	no	—
53	C₃×H(3,𝔽₃)	—	111	3	no	—
54	C₃×H(3,𝔽₃)	—	111	1	no	—
55	C₃²⋊C₉	—	111	2	no	—
56	C₃₇	#10624474	1	1	yes	characteristic, complemented, Sylow
57	C₃³	—	3	1	no	—
58	C₃³	—	3	1	no	—
59	C₃³	—	3	3	no	—
60	C₃³	—	3	2	no	—
61	C₃³	—	9	3	no	—
62	C₃³	—	111	3	no	—
63	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
64	H(3,𝔽₃)	—	9	9	no	—
65	H(3,𝔽₃)	—	9	1	no	—
66	H(3,𝔽₃)	—	9	2	no	—
67	H(3,𝔽₃)	—	333	9	no	—
68	H(3,𝔽₃)	—	333	1	no	—
69	H(3,𝔽₃)	—	333	2	no	—
70	C₃×C₉	—	333	2	no	—
71	C₃²	—	3	3	no	—
72	C₃²	—	3	3	no	—
73	C₃²	—	9	1	no	—
74	C₃²	—	9	6	no	—
75	C₃²	—	9	2	no	—
76	C₃²	—	9	2	no	—
77	C₃²	—	9	9	no	—
78	C₃²	—	9	1	no	—
79	C₃²	—	9	3	no	—
80	C₃²	—	333	9	no	—
81	C₃²	—	333	3	no	—
82	C₃²	#15688512	1	1	yes	characteristic
83	C₃²	#15688513	1	1	yes	characteristic
84	C₃²	#15688518	1	2	yes	—
85	C₉	—	999	2	no	—
86	C₃	—	3	2	no	—
87	C₃	—	3	1	no	—
88	C₃	—	3	1	no	—
89	C₃	—	9	3	no	—
90	C₃	—	27	3	no	—
91	C₃	—	999	3	no	—
92	C₃	#22582442	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582442
commutator, cosocle	C₃⁴×C₃₇	C₃²
Frattini	C₃⁴	C₃×C₃₇⋊C₃
Fitting	C₃₇×C₃²⋊H(3,𝔽₃)	C₃
socle	C₃×C₃₇	#6856
hypercenter	C₃²⋊H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624474
LCS	G ▷ C₃⁴×C₃₇ ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃⁴ ◁ C₃²⋊H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. #22582550	C₃	semidirect
2. #22582468	C₃	semidirect
3. C₃₇×C₃²⋊H(3,𝔽₃)	C₃	semidirect
41. C₃²⋊H(3,𝔽₃)	C₃₇⋊C₃	semidirect
56. C₃₇	#10624474	semidirect

Maximal subgroups

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

subgroup	index
#22582550	3
#22582468	3
C₃₇×C₃²⋊H(3,𝔽₃)	3
#10624474	37