Subgroups — C₄⋊(C₃⋊(C₉⋊C₁₆)) #13135741

Order 1728.

250 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊(C₉⋊C₁₆)	C₂	1	2	yes	maximal
3	C₄⋊(C₃⋊(C₃⋊C₁₆))	—	3	1	no	maximal
4	C₄⋊(C₉⋊C₁₆)	—	3	3	no	maximal
5	C₂×C₃×C₈×C₉	C₄	1	1	yes	characteristic
6	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
7	C₃×C₄²×C₉	C₄	1	1	yes	characteristic
8	C₃⋊(C₉⋊C₁₆)	—	2	2	no	—
9	C₃²×C₄×C₈	D₃	1	1	yes	characteristic
10	C₂×C₃⋊(C₃⋊C₁₆)	—	3	2	no	—
11	C₄×C₈×C₉	D₃	1	3	yes	—
12	C₂×C₉⋊C₁₆	—	3	6	no	—
13	C₂×C₃×C₄×C₉	C₈	1	1	yes	characteristic
14	C₂×C₃×C₄×C₉	C₈	1	1	yes	characteristic
15	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
16	C₃×C₈×C₉	—	2	1	no	—
17	C₃×C₈×C₉	D₄	1	1	yes	characteristic
18	C₃×C₈×C₉	Q₈	1	1	yes	characteristic
19	C₄⋊(C₃⋊C₁₆)	—	9	3	no	—
20	C₄⋊(C₃⋊C₁₆)	—	9	1	no	—
21	C₂×C₃²×C₈	Dic₃	1	1	yes	characteristic
22	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
23	C₃²×C₄²	Dic₃	1	1	yes	characteristic
24	C₃⋊(C₃⋊C₁₆)	—	6	2	no	—
25	C₂×C₈×C₉	Dic₃	1	3	yes	—
26	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
27	C₄²×C₉	Dic₃	1	3	yes	—
28	C₉⋊C₁₆	—	6	6	no	—
29	C₂²×C₃×C₉	C₂×C₈	1	1	yes	characteristic
30	C₃×C₄×C₉	—	2	1	no	—
31	C₃×C₄×C₉	C₁₆	1	2	yes	complemented
32	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
33	C₃×C₄×C₉	M₁₆	1	1	yes	characteristic
34	C₃×C₄×C₈	D₉	1	3	yes	—
35	C₃×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
36	C₂×C₃⋊C₁₆	—	9	6	no	—
37	C₂×C₃⋊C₁₆	—	9	2	no	—
38	C₂×C₃²×C₄	C₃⋊C₈	1	1	yes	characteristic
39	C₂×C₃²×C₄	C₃⋊C₈	1	1	yes	characteristic
40	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
41	C₃²×C₈	—	2	1	no	—
42	C₃²×C₈	Dic₆	1	1	yes	characteristic
43	C₃²×C₈	D₁₂	1	1	yes	characteristic
44	C₂×C₄×C₉	C₃⋊C₈	1	3	yes	—
45	C₂×C₄×C₉	C₃⋊C₈	1	3	yes	—
46	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
47	C₈×C₉	—	2	3	no	—
48	C₈×C₉	Dic₆	1	3	yes	—
49	C₈×C₉	D₁₂	1	3	yes	—
50	C₄⋊C₁₆	—	27	1	no	Sylow
51	C₂×C₃×C₉	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₃×C₉	C₂×C₁₆	1	1	yes	characteristic
53	C₂×C₃×C₉	M₃₂	1	1	yes	characteristic
54	C₂×C₃×C₈	Dic₉	1	3	yes	—
55	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
56	C₂×C₃×C₈	C₃⋊Dic₃	1	1	yes	characteristic
57	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
58	C₃×C₄²	Dic₉	1	3	yes	—
59	C₃×C₄²	C₃⋊Dic₃	1	1	yes	characteristic
60	C₃⋊C₁₆	—	18	6	no	—
61	C₃⋊C₁₆	—	18	2	no	—
62	C₂²×C₃²	C₂×C₃⋊C₈	1	1	yes	characteristic
63	C₃²×C₄	—	2	1	no	—
64	C₃²×C₄	C₃⋊C₁₆	1	2	yes	—
65	C₃²×C₄	#208	1	1	yes	characteristic
66	C₃²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
67	C₂²×C₉	C₂×C₃⋊C₈	1	3	yes	—
68	C₄×C₉	—	2	3	no	—
69	C₄×C₉	C₃⋊C₁₆	1	6	yes	complemented
70	C₄×C₉	#208	1	3	yes	—
71	C₄×C₉	C₄⋊Dic₃	1	3	yes	—
72	C₂×C₁₆	—	27	2	no	—
73	C₄×C₈	C₃⋊D₉	1	1	yes	characteristic
74	C₃×C₉	C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₃×C₄	C₉⋊C₈	1	3	yes	—
76	C₂×C₃×C₄	C₉⋊C₈	1	3	yes	—
77	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
78	C₂×C₃×C₄	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
79	C₂×C₃×C₄	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
80	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
81	C₃×C₈	—	2	3	no	—
82	C₃×C₈	—	2	1	no	—
83	C₃×C₈	Dic₁₈	1	3	yes	—
84	C₃×C₈	D₃₆	1	3	yes	—
85	C₃×C₈	#634	1	1	yes	characteristic
86	C₃×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
87	C₂×C₃²	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
88	C₂×C₃²	C₂×C₃⋊C₁₆	1	1	yes	characteristic
89	C₂×C₃²	#812	1	1	yes	characteristic
90	C₂×C₉	C₄⋊(C₃⋊C₈)	1	3	yes	—
91	C₂×C₉	C₂×C₃⋊C₁₆	1	3	yes	—
92	C₂×C₉	#812	1	3	yes	—
93	C₂×C₈	C₃⋊Dic₉	1	1	yes	characteristic
94	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
95	C₄²	C₃⋊Dic₉	1	1	yes	characteristic
96	C₁₆	—	54	2	no	—
97	C₂²×C₃	C₂×C₉⋊C₈	1	3	yes	—
98	C₂²×C₃	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
99	C₃×C₄	—	2	3	no	—
100	C₃×C₄	—	2	1	no	—
101	C₃×C₄	C₉⋊C₁₆	1	6	yes	complemented
102	C₃×C₄	#3667	1	3	yes	—
103	C₃×C₄	C₄⋊Dic₉	1	3	yes	—
104	C₃×C₄	C₃⋊(C₃⋊C₁₆)	1	2	yes	—
105	C₃×C₄	#3748	1	1	yes	characteristic
106	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
107	C₃²	C₄⋊(C₃⋊C₁₆)	1	1	yes	characteristic
108	C₉	C₄⋊(C₃⋊C₁₆)	1	3	yes	complemented
109	C₂×C₄	C₃⋊(C₉⋊C₈)	1	1	yes	characteristic
110	C₂×C₄	C₃⋊(C₉⋊C₈)	1	1	yes	characteristic
111	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
112	C₈	—	2	1	no	—
113	C₈	#6226	1	1	yes	characteristic
114	C₈	C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
115	C₂×C₃	C₄⋊(C₉⋊C₈)	1	3	yes	—
116	C₂×C₃	C₂×C₉⋊C₁₆	1	3	yes	—
117	C₂×C₃	#63371	1	3	yes	—
118	C₂×C₃	C₄⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic
119	C₂×C₃	C₂×C₃⋊(C₃⋊C₁₆)	1	1	yes	characteristic
120	C₂×C₃	#63639	1	1	yes	characteristic
121	C₂²	C₂×C₃⋊(C₉⋊C₈)	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	C₃⋊(C₉⋊C₁₆)	1	2	yes	complemented
124	C₄	#88532	1	1	yes	characteristic
125	C₄	C₄⋊(C₃⋊Dic₉)	1	1	yes	characteristic
126	C₃	C₄⋊(C₉⋊C₁₆)	1	3	yes	complemented
127	C₃	C₄⋊(C₃⋊(C₃⋊C₁₆))	1	1	yes	characteristic
128	C₂	C₄⋊(C₃⋊(C₉⋊C₈))	1	1	yes	characteristic
129	C₂	C₂×C₃⋊(C₉⋊C₁₆)	1	1	yes	characteristic
130	C₂	#11720534	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
31. C₃×C₄×C₉	C₁₆	semidirect
69. C₄×C₉	C₃⋊C₁₆	semidirect
74. C₃×C₉	C₄⋊C₁₆	semidirect
101. C₃×C₄	C₉⋊C₁₆	semidirect
108. C₉	C₄⋊(C₃⋊C₁₆)	semidirect
123. C₄	C₃⋊(C₉⋊C₁₆)	semidirect
126. C₃	C₄⋊(C₉⋊C₁₆)	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₈×C₉	2
C₂×C₃⋊(C₉⋊C₁₆)	2
C₄⋊(C₃⋊(C₃⋊C₁₆))	3
C₄⋊(C₉⋊C₁₆)	3

Subgroups — C4⋊(C3⋊(C9⋊C16)) #13135741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊(C₃⋊(C₉⋊C₁₆)) #13135741