Subgroups — C₂²×C₉×C₅₁₂ #72196153

Order 18432.

312 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	C₂²×C₉×C₂₅₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₅₁₂	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×C₅₁₂	C₃	1	1	yes	maximal, characteristic
4	C₂²×C₉×C₁₂₈	C₄	1	1	yes	characteristic
5	C₂×C₉×C₂₅₆	C₄	1	3	yes	—
6	C₂×C₉×C₂₅₆	C₂²	1	3	yes	—
7	C₉×C₅₁₂	C₂²	1	4	yes	direct product factor
8	C₂²×C₃×C₂₅₆	C₂×C₃	1	1	yes	characteristic
9	C₂×C₃×C₅₁₂	C₂×C₃	1	6	yes	—
10	C₂²×C₉×C₆₄	C₈	1	1	yes	characteristic
11	C₂×C₉×C₁₂₈	C₈	1	3	yes	—
12	C₂×C₉×C₁₂₈	C₂×C₄	1	3	yes	—
13	C₉×C₂₅₆	C₂×C₄	1	3	yes	—
14	C₉×C₂₅₆	C₂³	1	1	yes	characteristic
15	C₂²×C₅₁₂	C₉	1	1	yes	characteristic, direct product factor, Sylow
16	C₂²×C₃×C₁₂₈	C₃×C₄	1	1	yes	characteristic
17	C₂×C₃×C₂₅₆	C₃×C₄	1	3	yes	—
18	C₂×C₃×C₂₅₆	C₂²×C₃	1	3	yes	—
19	C₃×C₅₁₂	C₂²×C₃	1	4	yes	—
20	C₂²×C₂₅₆	C₂×C₉	1	1	yes	characteristic
21	C₂×C₅₁₂	C₂×C₉	1	6	yes	direct product factor
22	C₂²×C₉×C₃₂	C₁₆	1	1	yes	characteristic
23	C₂×C₉×C₆₄	C₁₆	1	3	yes	—
24	C₂×C₉×C₆₄	C₂×C₈	1	3	yes	—
25	C₉×C₁₂₈	C₂×C₈	1	3	yes	—
26	C₉×C₁₂₈	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₆₄	C₃×C₈	1	1	yes	characteristic
28	C₂×C₃×C₁₂₈	C₃×C₈	1	3	yes	—
29	C₂×C₃×C₁₂₈	C₂×C₃×C₄	1	3	yes	—
30	C₃×C₂₅₆	C₂×C₃×C₄	1	3	yes	—
31	C₃×C₂₅₆	C₂³×C₃	1	1	yes	characteristic
32	C₂²×C₉×C₁₆	C₃₂	1	1	yes	characteristic
33	C₂×C₉×C₃₂	C₃₂	1	3	yes	—
34	C₂×C₉×C₃₂	C₂×C₁₆	1	3	yes	—
35	C₉×C₆₄	C₂×C₁₆	1	3	yes	—
36	C₉×C₆₄	C₂²×C₈	1	1	yes	characteristic
37	C₂²×C₁₂₈	C₄×C₉	1	1	yes	characteristic
38	C₂×C₂₅₆	C₄×C₉	1	3	yes	—
39	C₂×C₂₅₆	C₂²×C₉	1	3	yes	—
40	C₅₁₂	C₂²×C₉	1	4	yes	direct product factor
41	C₂²×C₃×C₃₂	C₃×C₁₆	1	1	yes	characteristic
42	C₂×C₃×C₆₄	C₃×C₁₆	1	3	yes	—
43	C₂×C₃×C₆₄	C₂×C₃×C₈	1	3	yes	—
44	C₃×C₁₂₈	C₂×C₃×C₈	1	3	yes	—
45	C₃×C₁₂₈	C₂²×C₃×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₉	C₆₄	1	1	yes	characteristic
47	C₂×C₉×C₁₆	C₆₄	1	3	yes	—
48	C₂×C₉×C₁₆	C₂×C₃₂	1	3	yes	—
49	C₉×C₃₂	C₂×C₃₂	1	3	yes	—
50	C₉×C₃₂	C₂²×C₁₆	1	1	yes	characteristic
51	C₂²×C₆₄	C₈×C₉	1	1	yes	characteristic
52	C₂×C₁₂₈	C₈×C₉	1	3	yes	—
53	C₂×C₁₂₈	C₂×C₄×C₉	1	3	yes	—
54	C₂₅₆	C₂×C₄×C₉	1	3	yes	—
55	C₂₅₆	C₂³×C₉	1	1	yes	characteristic
56	C₂²×C₃×C₁₆	C₃×C₃₂	1	1	yes	characteristic
57	C₂×C₃×C₃₂	C₃×C₃₂	1	3	yes	—
58	C₂×C₃×C₃₂	C₂×C₃×C₁₆	1	3	yes	—
59	C₃×C₆₄	C₂×C₃×C₁₆	1	3	yes	—
60	C₃×C₆₄	C₂²×C₃×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₉	C₁₂₈	1	1	yes	characteristic
62	C₂×C₈×C₉	C₁₂₈	1	3	yes	—
63	C₂×C₈×C₉	C₂×C₆₄	1	3	yes	—
64	C₉×C₁₆	C₂×C₆₄	1	3	yes	—
65	C₉×C₁₆	C₂²×C₃₂	1	1	yes	characteristic
66	C₂²×C₃₂	C₉×C₁₆	1	1	yes	characteristic
67	C₂×C₆₄	C₉×C₁₆	1	3	yes	—
68	C₂×C₆₄	C₂×C₈×C₉	1	3	yes	—
69	C₁₂₈	C₂×C₈×C₉	1	3	yes	—
70	C₁₂₈	C₂²×C₄×C₉	1	1	yes	characteristic
71	C₂²×C₃×C₈	C₃×C₆₄	1	1	yes	characteristic
72	C₂×C₃×C₁₆	C₃×C₆₄	1	3	yes	—
73	C₂×C₃×C₁₆	C₂×C₃×C₃₂	1	3	yes	—
74	C₃×C₃₂	C₂×C₃×C₃₂	1	3	yes	—
75	C₃×C₃₂	C₂²×C₃×C₁₆	1	1	yes	characteristic
76	C₂³×C₉	C₂₅₆	1	1	yes	characteristic
77	C₂×C₄×C₉	C₂₅₆	1	3	yes	—
78	C₂×C₄×C₉	C₂×C₁₂₈	1	3	yes	—
79	C₈×C₉	C₂×C₁₂₈	1	3	yes	—
80	C₈×C₉	C₂²×C₆₄	1	1	yes	characteristic
81	C₂²×C₁₆	C₉×C₃₂	1	1	yes	characteristic
82	C₂×C₃₂	C₉×C₃₂	1	3	yes	—
83	C₂×C₃₂	C₂×C₉×C₁₆	1	3	yes	—
84	C₆₄	C₂×C₉×C₁₆	1	3	yes	—
85	C₆₄	C₂²×C₈×C₉	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₃×C₁₂₈	1	1	yes	characteristic
87	C₂×C₃×C₈	C₃×C₁₂₈	1	3	yes	—
88	C₂×C₃×C₈	C₂×C₃×C₆₄	1	3	yes	—
89	C₃×C₁₆	C₂×C₃×C₆₄	1	3	yes	—
90	C₃×C₁₆	C₂²×C₃×C₃₂	1	1	yes	characteristic
91	C₂²×C₉	C₅₁₂	1	4	yes	direct product factor
92	C₂²×C₉	C₂×C₂₅₆	1	3	yes	—
93	C₄×C₉	C₂×C₂₅₆	1	3	yes	—
94	C₄×C₉	C₂²×C₁₂₈	1	1	yes	characteristic
95	C₂²×C₈	C₉×C₆₄	1	1	yes	characteristic
96	C₂×C₁₆	C₉×C₆₄	1	3	yes	—
97	C₂×C₁₆	C₂×C₉×C₃₂	1	3	yes	—
98	C₃₂	C₂×C₉×C₃₂	1	3	yes	—
99	C₃₂	C₂²×C₉×C₁₆	1	1	yes	characteristic
100	C₂³×C₃	C₃×C₂₅₆	1	1	yes	characteristic
101	C₂×C₃×C₄	C₃×C₂₅₆	1	3	yes	—
102	C₂×C₃×C₄	C₂×C₃×C₁₂₈	1	3	yes	—
103	C₃×C₈	C₂×C₃×C₁₂₈	1	3	yes	—
104	C₃×C₈	C₂²×C₃×C₆₄	1	1	yes	characteristic
105	C₂×C₉	C₂×C₅₁₂	1	6	yes	direct product factor
106	C₂×C₉	C₂²×C₂₅₆	1	1	yes	characteristic
107	C₂²×C₄	C₉×C₁₂₈	1	1	yes	characteristic
108	C₂×C₈	C₉×C₁₂₈	1	3	yes	—
109	C₂×C₈	C₂×C₉×C₆₄	1	3	yes	—
110	C₁₆	C₂×C₉×C₆₄	1	3	yes	—
111	C₁₆	C₂²×C₉×C₃₂	1	1	yes	characteristic
112	C₂²×C₃	C₃×C₅₁₂	1	4	yes	—
113	C₂²×C₃	C₂×C₃×C₂₅₆	1	3	yes	—
114	C₃×C₄	C₂×C₃×C₂₅₆	1	3	yes	—
115	C₃×C₄	C₂²×C₃×C₁₂₈	1	1	yes	characteristic
116	C₉	C₂²×C₅₁₂	1	1	yes	characteristic, direct product factor, Sylow
117	C₂³	C₉×C₂₅₆	1	1	yes	characteristic
118	C₂×C₄	C₉×C₂₅₆	1	3	yes	—
119	C₂×C₄	C₂×C₉×C₁₂₈	1	3	yes	—
120	C₈	C₂×C₉×C₁₂₈	1	3	yes	—
121	C₈	C₂²×C₉×C₆₄	1	1	yes	characteristic
122	C₂×C₃	C₂×C₃×C₅₁₂	1	6	yes	—
123	C₂×C₃	C₂²×C₃×C₂₅₆	1	1	yes	characteristic
124	C₂²	C₉×C₅₁₂	1	4	yes	direct product factor
125	C₂²	C₂×C₉×C₂₅₆	1	3	yes	—
126	C₄	C₂×C₉×C₂₅₆	1	3	yes	—
127	C₄	C₂²×C₉×C₁₂₈	1	1	yes	characteristic
128	C₃	C₂²×C₃×C₅₁₂	1	1	yes	characteristic
129	C₂	C₂×C₉×C₅₁₂	1	6	yes	direct product factor
130	C₂	C₂²×C₉×C₂₅₆	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₂₅₆	C₂³×C₃
socle, prime radical	C₂³×C₃	C₃×C₂₅₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×C₅₁₂	C₂	direct (N × H)
7. C₉×C₅₁₂	C₂²	direct (N × H)
15. C₂²×C₅₁₂	C₉	direct (N × H)
21. C₂×C₅₁₂	C₂×C₉	direct (N × H)
40. C₅₁₂	C₂²×C₉	direct (N × H)
91. C₂²×C₉	C₅₁₂	direct (N × H)
105. C₂×C₉	C₂×C₅₁₂	direct (N × H)
116. C₉	C₂²×C₅₁₂	direct (N × H)
124. C₂²	C₉×C₅₁₂	direct (N × H)
129. C₂	C₂×C₉×C₅₁₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₉×C₂₅₆	2
C₂×C₉×C₅₁₂	2
C₂²×C₃×C₅₁₂	3

Subgroups — C22×C9×C512 #72196153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₉×C₅₁₂ #72196153