Subgroups — #13135831

Order 1728.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#11720535	C₂	1	2	yes	maximal
3	#10589372	—	3	1	no	maximal
4	#10588183	—	3	3	no	maximal
5	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
6	#88532	—	2	2	no	—
7	C₃×C₉×C₁₆	C₄	1	2	yes	—
8	C₂×C₃²×C₁₆	D₃	1	1	yes	characteristic
9	#63646	—	3	2	no	—
10	C₂×C₉×C₁₆	D₃	1	3	yes	—
11	#63372	—	3	6	no	—
12	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
13	C₃×C₈×C₉	C₂×C₄	1	1	yes	characteristic
14	C₃×C₈×C₉	Q₈	1	1	yes	characteristic
15	C₃⋊(C₉⋊C₈)	—	4	2	no	—
16	#4506	—	9	3	no	—
17	#4506	—	9	1	no	—
18	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
19	#3748	—	6	2	no	—
20	C₃²×C₁₆	Dic₃	1	2	yes	—
21	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
22	#3667	—	6	6	no	—
23	C₉×C₁₆	Dic₃	1	6	yes	—
24	C₂²×C₃×C₉	Q₁₆	1	1	yes	characteristic
25	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
26	C₃×C₄×C₉	D₈	1	1	yes	characteristic
27	C₂×C₃×C₁₆	D₉	1	3	yes	—
28	C₂×C₃×C₁₆	C₃⋊D₃	1	1	yes	characteristic
29	#819	—	9	6	no	—
30	#819	—	9	2	no	—
31	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
32	C₃²×C₈	Dic₆	1	1	yes	characteristic
33	C₃²×C₈	C₂×Dic₃	1	1	yes	characteristic
34	C₃⋊(C₃⋊C₈)	—	12	2	no	—
35	C₂×C₄×C₉	D₁₂	1	3	yes	—
36	C₈×C₉	Dic₆	1	3	yes	—
37	C₈×C₉	C₂×Dic₃	1	3	yes	—
38	C₉⋊C₈	—	12	6	no	—
39	#368	—	27	1	no	Sylow
40	C₂×C₃×C₉	—	2	1	no	—
41	C₂×C₃×C₉	#107	1	1	yes	characteristic
42	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
43	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
44	#208	—	18	6	no	—
45	#208	—	18	2	no	—
46	C₃×C₁₆	Dic₉	1	6	yes	—
47	C₃×C₁₆	C₃⋊Dic₃	1	2	yes	—
48	C₂²×C₃²	Dic₁₂	1	1	yes	characteristic
49	C₃²×C₄	D₂₄	1	1	yes	characteristic
50	C₃²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
51	C₂²×C₉	Dic₁₂	1	3	yes	—
52	C₄×C₉	D₂₄	1	3	yes	—
53	C₄×C₉	C₄⋊Dic₃	1	3	yes	—
54	C₂×C₁₆	C₃⋊D₉	1	1	yes	characteristic
55	#108	—	27	2	no	—
56	C₃×C₉	#368	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₃×C₄	D₃₆	1	3	yes	—
58	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
59	C₃×C₈	Dic₁₈	1	3	yes	—
60	C₃×C₈	C₂×Dic₉	1	3	yes	—
61	C₃×C₈	#634	1	1	yes	characteristic
62	C₃×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
63	C₃⋊C₈	—	36	6	no	—
64	C₃⋊C₈	—	36	2	no	—
65	C₂×C₃²	—	2	1	no	—
66	C₂×C₃²	#818	1	1	yes	characteristic
67	C₂×C₉	—	2	3	no	—
68	C₂×C₉	#818	1	3	yes	—
69	M₁₆	—	54	2	no	—
70	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
71	C₁₆	C₃⋊Dic₉	1	2	yes	—
72	C₂²×C₃	Dic₃₆	1	3	yes	—
73	C₂²×C₃	#3746	1	1	yes	characteristic
74	C₃×C₄	D₇₂	1	3	yes	—
75	C₃×C₄	C₄⋊Dic₉	1	3	yes	—
76	C₃×C₄	#3745	1	1	yes	characteristic
77	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
78	C₃²	#4506	1	1	yes	characteristic
79	C₉	#4506	1	3	yes	complemented
80	C₂×C₄	C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
81	C₈	—	108	2	no	—
82	C₈	#6226	1	1	yes	characteristic
83	C₈	C₂×C₃⋊Dic₉	1	1	yes	characteristic
84	C₂×C₃	—	2	3	no	—
85	C₂×C₃	—	2	1	no	—
86	C₂×C₃	#63378	1	3	yes	—
87	C₂×C₃	#63645	1	1	yes	characteristic
88	C₂²	#88521	1	1	yes	characteristic
89	C₄	#88525	1	1	yes	characteristic
90	C₄	C₄⋊(C₃⋊Dic₉)	1	1	yes	characteristic
91	C₃	#10588183	1	3	yes	complemented
92	C₃	#10589372	1	1	yes	characteristic
93	C₂	—	2	1	no	—
94	C₂	#11720541	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#88525
commutator	C₃×C₈×C₉	C₂×C₄
Frattini	C₂×C₃×C₈	C₂×C₃⋊D₃
Fitting	C₂×C₃×C₉×C₁₆	C₂
socle	C₂×C₃²	#818
prime radical	C₂²×C₃²	Dic₁₂
cosocle	C₂×C₃×C₈×C₉	C₂²
hypercenter	C₂×C₁₆	C₃⋊D₉
hypocenter	C₃×C₉	#368
LCS	G ▷ C₃×C₈×C₉ ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
56. C₃×C₉	#368	semidirect
79. C₉	#4506	semidirect
91. C₃	#10588183	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×C₁₆	2
#11720535	2
#10589372	3
#10588183	3