Subgroups — #14554136

Order 2187.

1622 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#737	C₃	1	1	yes	maximal, characteristic
2	C₃×#6903	C₃	1	3	yes	maximal, complemented
3	#10624684	C₃	1	24	yes	maximal, complemented
4	#10624683	C₃	1	8	yes	maximal, complemented
5	C₃×C₃²⋊C₂₇	C₃	1	4	yes	maximal, complemented
6	C₃×#737	—	3	9	no	—
7	C₃×#737	—	3	18	no	—
8	C₃×#737	C₉	1	16	yes	—
9	C₃×#737	C₉	1	2	yes	—
10	C₃×#737	C₃²	1	8	yes	—
11	C₃×#737	C₃²	1	1	yes	characteristic
12	C₃²×3₋¹⁺²	C₉	1	8	yes	—
13	C₃²×H(3,𝔽₃)	C₉	1	1	yes	characteristic
14	C₃³×C₉	C₃²	1	4	yes	—
15	#6903	—	3	27	no	—
16	C₃×C₂₇⋊C₃	C₃²	1	24	yes	complemented
17	C₃²×C₂₇	C₃²	1	12	yes	complemented
18	C₃²⋊C₂₇	C₃²	1	1	yes	characteristic
19	C₃²⋊C₂₇	C₃²	1	8	yes	—
20	C₃²⋊C₂₇	C₃²	1	24	yes	complemented
21	C₃²⋊C₂₇	C₃²	1	48	yes	complemented
22	C₃⁴	C₃×C₉	1	4	yes	—
23	#737	—	3	144	no	—
24	#737	—	3	18	no	—
25	#737	—	3	24	no	—
26	#737	—	3	48	no	—
27	#737	H(3,𝔽₃)	1	9	yes	—
28	#737	3₋¹⁺²	1	18	yes	—
29	C₃×3₋¹⁺²	—	3	72	no	—
30	C₃×3₋¹⁺²	C₃×C₉	1	16	yes	—
31	C₃×3₋¹⁺²	C₃×C₉	1	48	yes	—
32	C₃×3₋¹⁺²	C₃×C₉	1	8	yes	—
33	C₃×H(3,𝔽₃)	—	3	9	no	—
34	C₃×H(3,𝔽₃)	C₃×C₉	1	8	yes	—
35	C₃×H(3,𝔽₃)	C₃×C₉	1	1	yes	characteristic
36	C₃²×C₉	—	3	12	no	—
37	C₃²×C₉	—	3	24	no	—
38	C₃²×C₉	C₃×C₉	1	8	yes	—
39	C₃²×C₉	C₃×C₉	1	8	yes	—
40	C₃²×C₉	C₃×C₉	1	16	yes	—
41	C₃²×C₉	C₃³	1	1	yes	characteristic
42	C₃²×C₉	C₃³	1	8	yes	—
43	C₃²×C₉	C₃³	1	4	yes	—
44	C₂₇⋊C₃	—	3	72	no	—
45	C₃×C₂₇	—	3	36	no	—
46	C₃×C₂₇	C₃³	1	3	yes	—
47	C₃×C₂₇	C₃³	1	24	yes	—
48	C₃³	—	3	12	no	—
49	C₃³	—	3	12	no	—
50	C₃³	—	9	8	no	—
51	C₃³	C₃²×C₉	1	8	yes	—
52	C₃³	C₃²×C₉	1	4	yes	—
53	C₃³	C₃²×C₉	1	1	yes	characteristic
54	3₋¹⁺²	—	3	48	no	—
55	3₋¹⁺²	—	3	144	no	—
56	3₋¹⁺²	C₃²⋊C₉	1	72	yes	—
57	H(3,𝔽₃)	—	3	24	no	—
58	H(3,𝔽₃)	C₃²⋊C₉	1	9	yes	—
59	C₃×C₉	—	3	24	no	—
60	C₃×C₉	—	3	1	no	—
61	C₃×C₉	—	3	8	no	—
62	C₃×C₉	—	3	2	no	—
63	C₃×C₉	—	3	16	no	—
64	C₃×C₉	—	3	24	no	—
65	C₃×C₉	—	3	48	no	—
66	C₃×C₉	C₃²×C₉	1	2	yes	—
67	C₃×C₉	C₃²×C₉	1	16	yes	—
68	C₃×C₉	C₃²×C₉	1	8	yes	—
69	C₃×C₉	C₃×H(3,𝔽₃)	1	12	yes	—
70	C₃×C₉	C₃×3₋¹⁺²	1	24	yes	—
71	C₃×C₉	C₃⁴	1	1	yes	characteristic
72	C₂₇	—	3	3	no	—
73	C₂₇	—	3	24	no	—
74	C₃²	—	3	1	no	—
75	C₃²	—	3	4	no	—
76	C₃²	—	3	2	no	—
77	C₃²	—	3	8	no	—
78	C₃²	—	3	8	no	—
79	C₃²	—	3	8	no	—
80	C₃²	—	3	16	no	—
81	C₃²	—	9	36	no	—
82	C₃²	C₃×C₃²⋊C₉	1	12	yes	—
83	C₃²	#6903	1	3	yes	complemented
84	C₃²	C₃³×C₉	1	1	yes	characteristic
85	C₉	—	3	2	no	—
86	C₉	—	3	16	no	—
87	C₉	C₃×C₃²⋊C₉	1	24	yes	—
88	C₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
89	C₉	C₃²×3₋¹⁺²	1	2	yes	—
90	C₃	—	3	3	no	—
91	C₃	—	3	12	no	—
92	C₃	—	9	8	no	—
93	C₃	C₃²×C₃²⋊C₉	1	1	yes	characteristic
94	C₃	C₃×#6903	1	1	yes	characteristic
95	C₃	#10624918	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₃×C₉	C₃⁴
commutator, socle	C₃²	C₃³×C₉
prime radical	C₃²×H(3,𝔽₃)	C₉
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃×C₉ ◁ G
LECS	G ▷ C₃×C₉ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₃×#6903	C₃	semidirect
3. #10624684	C₃	semidirect
4. #10624683	C₃	semidirect
5. C₃×C₃²⋊C₂₇	C₃	semidirect
16. C₃×C₂₇⋊C₃	C₃²	semidirect
17. C₃²×C₂₇	C₃²	semidirect
20. C₃²⋊C₂₇	C₃²	semidirect
21. C₃²⋊C₂₇	C₃²	semidirect
83. C₃²	#6903	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#737	3
C₃×#6903	3
#10624684	3
#10624683	3
C₃×C₃²⋊C₂₇	3