Subgroups — #14517731

Order 1944.

432 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×C₂₇	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₃⋊Dic₂₇)	C₂	1	2	yes	maximal
3	#10619985	—	3	1	no	maximal
4	#10619657	—	3	12	no	maximal
5	C₂×C₃²×C₂₇	C₂²	1	1	yes	characteristic
6	C₃²×C₄×C₉	D₃	1	1	yes	characteristic
7	C₃⋊(C₃⋊Dic₉)	—	3	2	no	—
8	C₃×C₄×C₂₇	D₃	1	12	yes	—
9	C₃⋊Dic₂₇	—	3	24	no	—
10	C₃²×C₂₇	Q₈	1	1	yes	characteristic, complemented, Sylow
11	#6308	—	9	1	no	—
12	#6226	—	9	8	no	—
13	#6226	—	9	4	no	—
14	Dic₅₄	—	9	9	no	—
15	C₂×C₃²×C₉	C₂×D₃	1	1	yes	characteristic
16	C₂×C₃×C₂₇	C₂×D₃	1	12	yes	—
17	C₃³×C₄	D₉	1	1	yes	characteristic
18	C₃⋊(C₃⋊Dic₃)	—	9	2	no	—
19	C₃×C₄×C₉	D₉	1	8	yes	—
20	C₃×C₄×C₉	C₃⋊D₃	1	4	yes	—
21	C₃⋊Dic₉	—	9	16	no	—
22	C₃⋊Dic₉	—	9	8	no	—
23	C₄×C₂₇	C₃⋊D₃	1	9	yes	—
24	Dic₂₇	—	9	18	no	—
25	C₃²×C₉	Dic₆	1	1	yes	characteristic
26	C₃×C₂₇	Dic₆	1	12	yes	complemented
27	#634	—	27	9	no	—
28	#634	—	27	4	no	—
29	Dic₁₈	—	27	8	no	—
30	Dic₁₈	—	27	1	no	—
31	C₂×C₃³	C₂×D₉	1	1	yes	characteristic
32	C₂×C₃×C₉	C₂×D₉	1	8	yes	—
33	C₂×C₃×C₉	C₂×C₃⋊D₃	1	4	yes	—
34	C₂×C₂₇	C₂×C₃⋊D₃	1	9	yes	—
35	C₃²×C₄	D₂₇	1	9	yes	—
36	C₃²×C₄	C₃⋊D₉	1	4	yes	—
37	C₃⋊Dic₃	—	27	18	no	—
38	C₃⋊Dic₃	—	27	8	no	—
39	C₄×C₉	C₃⋊D₉	1	8	yes	—
40	C₄×C₉	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
41	Dic₉	—	27	16	no	—
42	Dic₉	—	27	2	no	—
43	C₃³	Dic₁₈	1	1	yes	characteristic
44	C₃×C₉	Dic₁₈	1	8	yes	—
45	C₃×C₉	#634	1	4	yes	—
46	C₂₇	#634	1	9	yes	complemented
47	Dic₆	—	81	12	no	—
48	Dic₆	—	81	1	no	—
49	C₂×C₃²	C₂×D₂₇	1	9	yes	—
50	C₂×C₃²	C₂×C₃⋊D₉	1	4	yes	—
51	C₂×C₉	C₂×C₃⋊D₉	1	8	yes	—
52	C₂×C₉	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
53	C₃×C₄	C₃⋊D₂₇	1	12	yes	—
54	C₃×C₄	C₃⋊(C₃⋊D₉)	1	1	yes	characteristic
55	Dic₃	—	81	24	no	—
56	Dic₃	—	81	2	no	—
57	C₃²	Dic₅₄	1	9	yes	complemented
58	C₃²	#6226	1	4	yes	—
59	C₉	#6226	1	8	yes	—
60	C₉	#6308	1	1	yes	characteristic
61	Q₈	—	243	1	no	Sylow
62	C₂×C₃	C₂×C₃⋊D₂₇	1	12	yes	—
63	C₂×C₃	C₂×C₃⋊(C₃⋊D₉)	1	1	yes	characteristic
64	C₄	—	243	2	no	—
65	C₄	C₃⋊(C₃⋊D₂₇)	1	1	yes	characteristic
66	C₃	#10619657	1	12	yes	complemented
67	C₃	#10619985	1	1	yes	characteristic
68	C₂	C₂×C₃⋊(C₃⋊D₂₇)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₃⋊(C₃⋊D₂₇)
commutator, cosocle	C₂×C₃²×C₂₇	C₂²
Frattini	C₂×C₉	C₂×C₃⋊(C₃⋊D₃)
Fitting	C₃²×C₄×C₂₇	C₂
socle, prime radical	C₂×C₃³	C₂×D₉
hypercenter	C₄	C₃⋊(C₃⋊D₂₇)
hypocenter	C₃²×C₂₇	Q₈
LCS	G ▷ C₂×C₃²×C₂₇ ▷ C₃²×C₂₇
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
10. C₃²×C₂₇	Q₈	semidirect
26. C₃×C₂₇	Dic₆	semidirect
46. C₂₇	#634	semidirect
57. C₃²	Dic₅₄	semidirect
66. C₃	#10619657	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×C₂₇	2
C₃⋊(C₃⋊Dic₂₇)	2
#10619985	3
#10619657	3