Subgroups — C₂×C₄×C₉⋊C₁₆ #11795805

Order 1152.

324 conjugacy classes of subgroups.

The subgroup lattice has 126 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₉⋊C₁₆	C₂	1	2	yes	maximal
3	C₄×C₉⋊C₁₆	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₃⋊C₁₆	—	3	1	no	maximal
5	C₂²×C₈×C₉	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
7	C₂×C₄²×C₉	C₄	1	1	yes	characteristic
8	C₄×C₈×C₉	C₄	1	2	yes	—
9	C₄×C₈×C₉	C₂²	1	2	yes	—
10	C₂×C₉⋊C₁₆	C₄	1	8	yes	direct product factor
11	C₂×C₉⋊C₁₆	C₂²	1	4	yes	—
12	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
13	C₂²×C₃⋊C₁₆	—	3	2	no	—
14	C₄×C₃⋊C₁₆	—	3	4	no	—
15	C₂²×C₄×C₉	C₈	1	2	yes	—
16	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₉	C₈	1	4	yes	—
18	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₉	C₂×C₄	1	4	yes	—
20	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
22	C₄²×C₉	C₈	1	2	yes	—
23	C₄²×C₉	C₂×C₄	1	2	yes	—
24	C₉⋊C₁₆	C₂×C₄	1	8	yes	direct product factor
25	C₂×C₄×C₁₆	—	9	1	no	Sylow
26	C₂²×C₃×C₈	Dic₃	1	1	yes	characteristic
27	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
28	C₂×C₃×C₄²	Dic₃	1	1	yes	characteristic
29	C₃×C₄×C₈	Dic₃	1	2	yes	—
30	C₃×C₄×C₈	C₂×D₃	1	2	yes	—
31	C₂×C₃⋊C₁₆	—	3	8	no	—
32	C₂×C₃⋊C₁₆	—	3	4	no	—
33	C₂³×C₉	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₉	C₁₆	1	8	yes	complemented
35	C₂×C₄×C₉	C₄²	1	2	yes	—
36	C₂×C₄×C₉	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×C₉	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₉	C₂×C₈	1	4	yes	—
39	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₉	C₄²	1	2	yes	—
41	C₈×C₉	C₂×C₈	1	4	yes	—
42	C₈×C₉	C₂²×C₄	1	2	yes	—
43	C₂²×C₁₆	—	9	2	no	—
44	C₂×C₄×C₈	D₉	1	1	yes	characteristic
45	C₄×C₁₆	—	9	4	no	—
46	C₂²×C₃×C₄	C₃⋊C₈	1	2	yes	—
47	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
48	C₂×C₃×C₈	C₃⋊C₈	1	4	yes	—
49	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
50	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
51	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
52	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
53	C₃×C₄²	C₃⋊C₈	1	2	yes	—
54	C₃×C₄²	C₂×Dic₃	1	2	yes	—
55	C₃⋊C₁₆	—	3	8	no	—
56	C₂²×C₉	C₄×C₈	1	2	yes	—
57	C₂²×C₉	C₂×C₁₆	1	4	yes	—
58	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
59	C₄×C₉	C₄×C₈	1	2	yes	—
60	C₄×C₉	C₂×C₁₆	1	8	yes	complemented
61	C₄×C₉	C₂×C₄²	1	1	yes	characteristic
62	C₄×C₉	C₂²×C₈	1	1	yes	characteristic
63	C₂²×C₈	Dic₉	1	1	yes	characteristic
64	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
65	C₂×C₄²	Dic₉	1	1	yes	characteristic
66	C₂×C₁₆	—	9	8	no	—
67	C₂×C₁₆	—	9	4	no	—
68	C₄×C₈	Dic₉	1	2	yes	—
69	C₄×C₈	C₂×D₉	1	2	yes	—
70	C₂³×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
71	C₂×C₃×C₄	C₃⋊C₁₆	1	8	yes	—
72	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
73	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
74	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
75	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
76	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
77	C₃×C₈	C₂×C₃⋊C₈	1	4	yes	—
78	C₃×C₈	C₄×Dic₃	1	2	yes	—
79	C₃×C₈	C₂×C₄×D₃	1	2	yes	—
80	C₂×C₉	C₄×C₁₆	1	4	yes	complemented
81	C₂×C₉	C₂×C₄×C₈	1	1	yes	characteristic
82	C₂×C₉	C₂²×C₁₆	1	2	yes	—
83	C₂²×C₄	C₉⋊C₈	1	2	yes	—
84	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
85	C₂×C₈	C₉⋊C₈	1	4	yes	—
86	C₂×C₈	C₄×D₉	1	4	yes	—
87	C₂×C₈	C₂×Dic₉	1	2	yes	—
88	C₂×C₈	C₂×Dic₉	1	1	yes	characteristic
89	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
90	C₄²	C₉⋊C₈	1	2	yes	—
91	C₄²	C₂×Dic₉	1	2	yes	—
92	C₁₆	—	9	8	no	—
93	C₂²×C₃	C₄×C₃⋊C₈	1	2	yes	—
94	C₂²×C₃	C₂×C₃⋊C₁₆	1	4	yes	—
95	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
96	C₃×C₄	C₄×C₃⋊C₈	1	2	yes	—
97	C₃×C₄	C₂×C₃⋊C₁₆	1	8	yes	—
98	C₃×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
99	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
100	C₉	C₂×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
101	C₂³	C₂×C₉⋊C₈	1	1	yes	characteristic
102	C₂×C₄	C₉⋊C₁₆	1	8	yes	direct product factor
103	C₂×C₄	C₂×C₉⋊C₈	1	1	yes	characteristic
104	C₂×C₄	C₂×C₉⋊C₈	1	2	yes	—
105	C₂×C₄	C₂×C₉⋊C₈	1	4	yes	—
106	C₂×C₄	C₄×Dic₉	1	2	yes	—
107	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
108	C₈	C₂×C₉⋊C₈	1	4	yes	—
109	C₈	C₄×Dic₉	1	2	yes	—
110	C₈	C₂×C₄×D₉	1	2	yes	—
111	C₂×C₃	C₄×C₃⋊C₁₆	1	4	yes	—
112	C₂×C₃	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
113	C₂×C₃	C₂²×C₃⋊C₁₆	1	2	yes	—
114	C₂²	C₄×C₉⋊C₈	1	2	yes	—
115	C₂²	C₂×C₉⋊C₁₆	1	4	yes	—
116	C₂²	C₂²×C₉⋊C₈	1	1	yes	characteristic
117	C₄	C₄×C₉⋊C₈	1	2	yes	—
118	C₄	C₂×C₉⋊C₁₆	1	8	yes	direct product factor
119	C₄	C₂²×C₉⋊C₈	1	1	yes	characteristic
120	C₄	C₂×C₄×Dic₉	1	1	yes	characteristic
121	C₃	C₂×C₄×C₃⋊C₁₆	1	1	yes	characteristic
122	C₂	C₄×C₉⋊C₁₆	1	4	yes	direct product factor
123	C₂	C₂×C₄×C₉⋊C₈	1	1	yes	characteristic
124	C₂	C₂²×C₉⋊C₁₆	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄×C₈	D₉
commutator, hypocenter	C₉	C₂×C₄×C₁₆
Frattini	C₂×C₃×C₈	C₂²×D₃
Fitting	C₂×C₄×C₈×C₉	C₂
socle, prime radical	C₂³×C₃	C₂×C₃⋊C₈
cosocle	C₂×C₈×C₉	C₂³
LCS	G ▷ C₉
UCS	C₁ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄×C₉⋊C₁₆	C₂	direct (N × H)
10. C₂×C₉⋊C₁₆	C₄	direct (N × H)
24. C₉⋊C₁₆	C₂×C₄	direct (N × H)
34. C₂×C₄×C₉	C₁₆	semidirect
60. C₄×C₉	C₂×C₁₆	semidirect
80. C₂×C₉	C₄×C₁₆	semidirect
100. C₉	C₂×C₄×C₁₆	semidirect
102. C₂×C₄	C₉⋊C₁₆	direct (N × H)
118. C₄	C₂×C₉⋊C₁₆	direct (N × H)
122. C₂	C₄×C₉⋊C₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×C₉	2
C₂²×C₉⋊C₁₆	2
C₄×C₉⋊C₁₆	2
C₂×C₄×C₃⋊C₁₆	3

Subgroups — C2×C4×C9⋊C16 #11795805

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₉⋊C₁₆ #11795805