Subgroups — C₃²×#4174 #13091573

Order 1458.

1388 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×AΣL(1,𝔽₂₇)	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃⁴×D₃	—	3	1	no	maximal
3	C₃²×#272	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#4174	C₃	1	12	yes	maximal, direct product factor
5	C₃⁵	D₃	1	1	yes	characteristic, complemented
6	C₃²×3₋¹⁺²	—	2	1	no	—
7	C₃²×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃×AΣL(1,𝔽₂₇)	—	2	12	no	—
9	C₃×AΣL(1,𝔽₂₇)	C₂×C₃	1	12	yes	complemented
10	C₂×C₃⁴	—	9	1	no	—
11	C₃³×D₃	—	3	27	no	—
12	C₃³×D₃	—	3	1	no	—
13	C₃³×D₃	—	3	12	no	—
14	C₃³×D₃	—	9	1	no	—
15	C₃×#272	C₃²	1	4	yes	complemented
16	#4174	C₃²	1	9	yes	direct product factor
17	C₃⁴	—	3	1	no	—
18	C₃⁴	—	3	27	no	—
19	C₃⁴	—	3	1	no	—
20	C₃⁴	—	6	4	no	—
21	C₃⁴	C₃×D₃	1	1	yes	characteristic
22	C₃⁴	C₃×D₃	1	12	yes	complemented
23	C₃×3₋¹⁺²	—	2	12	no	—
24	C₃×3₋¹⁺²	—	6	8	no	—
25	C₃×H(3,𝔽₃)	—	2	4	no	—
26	C₃×H(3,𝔽₃)	—	6	4	no	—
27	C₃×H(3,𝔽₃)	C₂×C₃²	1	4	yes	complemented
28	C₃²×C₉	—	6	1	no	—
29	AΣL(1,𝔽₂₇)	—	2	36	no	—
30	AΣL(1,𝔽₂₇)	C₂×C₃²	1	9	yes	complemented
31	C₂×C₃³	—	9	27	no	—
32	C₂×C₃³	—	9	12	no	—
33	C₂×C₃³	—	9	1	no	—
34	C₃²×D₃	—	3	108	no	—
35	C₃²×D₃	—	3	9	no	—
36	C₃²×D₃	—	3	4	no	—
37	C₃²×D₃	—	3	9	no	—
38	C₃²×D₃	—	9	4	no	—
39	C₃²×D₃	—	9	9	no	—
40	#272	C₃³	1	1	yes	characteristic, complemented
41	C₃³	—	2	4	no	—
42	C₃³	—	3	27	no	—
43	C₃³	—	3	12	no	—
44	C₃³	—	3	108	no	—
45	C₃³	—	3	9	no	—
46	C₃³	—	3	4	no	—
47	C₃³	—	6	12	no	—
48	C₃³	—	6	108	no	—
49	C₃³	—	6	4	no	—
50	C₃³	—	9	9	no	—
51	C₃³	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
52	C₃³	C₃²×D₃	1	4	yes	—
53	C₃³	C₃²×D₃	1	9	yes	complemented
54	3₋¹⁺²	—	2	9	no	—
55	3₋¹⁺²	—	6	24	no	—
56	H(3,𝔽₃)	—	2	4	no	—
57	H(3,𝔽₃)	—	6	4	no	—
58	H(3,𝔽₃)	—	6	8	no	—
59	H(3,𝔽₃)	C₂×C₃³	1	1	yes	characteristic, complemented
60	C₃×C₉	—	6	12	no	—
61	C₂×C₃²	—	9	108	no	—
62	C₂×C₃²	—	9	9	no	—
63	C₂×C₃²	—	9	9	no	—
64	C₂×C₃²	—	9	4	no	—
65	C₃×D₃	—	3	27	no	—
66	C₃×D₃	—	3	12	no	—
67	C₃×D₃	—	3	1	no	—
68	C₃×D₃	—	9	1	no	—
69	C₃×D₃	—	9	12	no	—
70	C₃²	—	2	4	no	—
71	C₃²	—	3	27	no	—
72	C₃²	—	3	108	no	—
73	C₃²	—	3	9	no	—
74	C₃²	—	3	12	no	—
75	C₃²	—	3	1	no	—
76	C₃²	—	6	108	no	—
77	C₃²	—	6	12	no	—
78	C₃²	—	6	4	no	—
79	C₃²	—	9	12	no	—
80	C₃²	—	18	12	no	—
81	C₃²	#4174	1	9	yes	direct product factor
82	C₃²	C₃×(C₃⋊D₃)⋊C₃	1	4	yes	—
83	C₃²	C₃³×D₃	1	1	yes	characteristic
84	C₉	—	6	9	no	—
85	C₂×C₃	—	9	27	no	—
86	C₂×C₃	—	9	12	no	—
87	C₂×C₃	—	9	1	no	—
88	D₃	—	3	1	no	—
89	D₃	—	9	1	no	—
90	C₃	—	3	1	no	—
91	C₃	—	3	27	no	—
92	C₃	—	6	4	no	—
93	C₃	—	9	1	no	—
94	C₃	—	18	4	no	—
95	C₃	C₃×#4174	1	12	yes	direct product factor
96	C₃	C₃²×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
97	C₂	—	9	1	no	Sylow

	subgroup	G/H
center, socle, hypercenter	C₃³	(C₃⋊D₃)⋊C₃
commutator, cosocle, hypocenter	H(3,𝔽₃)	C₂×C₃³
Frattini	C₃²	C₃³×D₃
Fitting	C₃²×AΣL(1,𝔽₂₇)	C₂
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃³

Complements

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

N	H ≅ G/N	type
1. C₃²×AΣL(1,𝔽₂₇)	C₂	semidirect
3. C₃²×#272	C₃	semidirect
4. C₃×#4174	C₃	direct (N × H)
5. C₃⁵	D₃	semidirect
7. C₃²×H(3,𝔽₃)	C₂×C₃	semidirect
9. C₃×AΣL(1,𝔽₂₇)	C₂×C₃	semidirect
15. C₃×#272	C₃²	semidirect
16. #4174	C₃²	direct (N × H)
22. C₃⁴	C₃×D₃	semidirect
27. C₃×H(3,𝔽₃)	C₂×C₃²	semidirect
30. AΣL(1,𝔽₂₇)	C₂×C₃²	semidirect
40. #272	C₃³	semidirect
53. C₃³	C₃²×D₃	semidirect
59. H(3,𝔽₃)	C₂×C₃³	semidirect
81. C₃²	#4174	direct (N × H)
95. C₃	C₃×#4174	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×AΣL(1,𝔽₂₇)	2
C₃⁴×D₃	3
C₃²×#272	3
C₃×#4174	3

Subgroups — C32×#4174 #13091573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4174 #13091573