Subgroups — C₂²×D₄×C₉⋊C₉ #72268934

Order 2592.

2006 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₉⋊C₉	C₂	1	2	yes	maximal, complemented
2	C₂×D₄×C₉⋊C₉	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄×C₉⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃×D₄×C₉	C₃	1	3	yes	maximal
5	C₂²×C₃×D₄×C₉	C₃	1	1	yes	maximal, characteristic
6	C₂³×C₉⋊C₉	—	2	8	no	—
7	C₂³×C₉⋊C₉	C₂²	1	1	yes	characteristic
8	C₂³×C₉⋊C₉	C₂²	1	12	yes	complemented
9	D₄×C₉⋊C₉	C₂²	1	16	yes	direct product factor
10	C₂×C₄×C₉⋊C₉	C₂²	1	6	yes	complemented
11	C₂⁴×C₃×C₉	C₂×C₃	1	6	yes	—
12	C₂⁴×C₃×C₉	C₂×C₃	1	2	yes	—
13	C₂×C₃×D₄×C₉	C₂×C₃	1	36	yes	—
14	C₂×C₃×D₄×C₉	C₂×C₃	1	12	yes	—
15	C₂²×C₃×C₄×C₉	C₂×C₃	1	3	yes	—
16	C₂²×C₃×C₄×C₉	C₂×C₃	1	1	yes	characteristic
17	C₂²×C₉⋊C₉	—	2	24	no	—
18	C₂²×C₉⋊C₉	D₄	1	4	yes	direct product factor
19	C₂²×C₉⋊C₉	C₂³	1	3	yes	—
20	C₂²×C₉⋊C₉	C₂³	1	8	yes	complemented
21	C₄×C₉⋊C₉	C₂³	1	4	yes	complemented
22	C₂²×C₃²×D₄	C₃²	1	1	yes	characteristic
23	C₂²×D₄×C₉	—	3	3	no	—
24	C₂²×D₄×C₉	C₉	1	3	yes	complemented
25	C₂³×C₃×C₉	—	2	8	no	—
26	C₂³×C₃×C₉	—	2	24	no	—
27	C₂³×C₃×C₉	C₂²×C₃	1	36	yes	—
28	C₂³×C₃×C₉	C₂²×C₃	1	12	yes	—
29	C₂³×C₃×C₉	C₂²×C₃	1	3	yes	—
30	C₂³×C₃×C₉	C₂²×C₃	1	1	yes	characteristic
31	C₃×D₄×C₉	C₂²×C₃	1	48	yes	—
32	C₃×D₄×C₉	C₂²×C₃	1	16	yes	—
33	C₂×C₃×C₄×C₉	C₂²×C₃	1	18	yes	—
34	C₂×C₃×C₄×C₉	C₂²×C₃	1	6	yes	—
35	C₂×C₉⋊C₉	—	2	8	no	—
36	C₂×C₉⋊C₉	C₂×D₄	1	6	yes	direct product factor
37	C₂×C₉⋊C₉	C₂⁴	1	1	yes	characteristic
38	C₂⁴×C₃²	C₂×C₃²	1	2	yes	—
39	C₂×C₃²×D₄	C₂×C₃²	1	12	yes	—
40	C₂²×C₃²×C₄	C₂×C₃²	1	1	yes	characteristic
41	C₂⁴×C₉	—	3	6	no	—
42	C₂⁴×C₉	C₂×C₉	1	6	yes	complemented
43	C₂×D₄×C₉	—	3	36	no	—
44	C₂×D₄×C₉	C₂×C₉	1	36	yes	complemented
45	C₂²×C₄×C₉	—	3	3	no	—
46	C₂²×C₄×C₉	C₂×C₉	1	3	yes	complemented
47	C₂²×C₃×C₉	—	2	24	no	—
48	C₂²×C₃×C₉	—	2	72	no	—
49	C₂²×C₃×C₉	C₃×D₄	1	12	yes	—
50	C₂²×C₃×C₉	C₃×D₄	1	4	yes	—
51	C₂²×C₃×C₉	C₂³×C₃	1	24	yes	—
52	C₂²×C₃×C₉	C₂³×C₃	1	8	yes	—
53	C₂²×C₃×C₉	C₂³×C₃	1	9	yes	—
54	C₂²×C₃×C₉	C₂³×C₃	1	3	yes	—
55	C₃×C₄×C₉	C₂³×C₃	1	12	yes	—
56	C₃×C₄×C₉	C₂³×C₃	1	4	yes	—
57	C₂²×C₃×D₄	C₃×C₉	1	1	yes	characteristic
58	C₂²×C₃×D₄	3₋¹⁺²	1	3	yes	—
59	C₉⋊C₉	C₂²×D₄	1	1	yes	characteristic, direct product factor, Sylow
60	C₂³×C₃²	—	2	8	no	—
61	C₂³×C₃²	C₂²×C₃²	1	12	yes	—
62	C₂³×C₃²	C₂²×C₃²	1	1	yes	characteristic
63	C₃²×D₄	C₂²×C₃²	1	16	yes	—
64	C₂×C₃²×C₄	C₂²×C₃²	1	6	yes	—
65	C₂³×C₉	—	2	24	no	—
66	C₂³×C₉	—	3	3	no	—
67	C₂³×C₉	—	3	36	no	—
68	C₂³×C₉	—	6	24	no	—
69	C₂³×C₉	C₂²×C₉	1	3	yes	—
70	C₂³×C₉	C₂²×C₉	1	36	yes	complemented
71	D₄×C₉	—	3	48	no	—
72	D₄×C₉	C₂²×C₉	1	48	yes	complemented
73	C₂×C₄×C₉	—	3	18	no	—
74	C₂×C₄×C₉	C₂²×C₉	1	18	yes	complemented
75	C₂×C₃×C₉	—	2	8	no	—
76	C₂×C₃×C₉	—	2	24	no	—
77	C₂×C₃×C₉	C₂×C₃×D₄	1	18	yes	—
78	C₂×C₃×C₉	C₂×C₃×D₄	1	6	yes	—
79	C₂×C₃×C₉	C₂⁴×C₃	1	3	yes	—
80	C₂×C₃×C₉	C₂⁴×C₃	1	1	yes	characteristic
81	C₂⁴×C₃	C₂×C₃×C₉	1	2	yes	—
82	C₂⁴×C₃	C₂×3₋¹⁺²	1	6	yes	—
83	C₂×C₃×D₄	C₂×C₃×C₉	1	12	yes	—
84	C₂×C₃×D₄	C₂×3₋¹⁺²	1	36	yes	—
85	C₂²×C₃×C₄	C₂×C₃×C₉	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₂×3₋¹⁺²	1	3	yes	—
87	C₂²×C₃²	—	2	24	no	—
88	C₂²×C₃²	C₃²×D₄	1	4	yes	—
89	C₂²×C₃²	C₂³×C₃²	1	8	yes	—
90	C₂²×C₃²	C₂³×C₃²	1	3	yes	—
91	C₃²×C₄	C₂³×C₃²	1	4	yes	—
92	C₂²×C₉	—	2	72	no	—
93	C₂²×C₉	—	3	12	no	—
94	C₂²×C₉	—	3	9	no	—
95	C₂²×C₉	—	3	24	no	—
96	C₂²×C₉	—	6	72	no	—
97	C₂²×C₉	D₄×C₉	1	12	yes	complemented
98	C₂²×C₉	C₂³×C₉	1	9	yes	—
99	C₂²×C₉	C₂³×C₉	1	24	yes	complemented
100	C₄×C₉	—	3	12	no	—
101	C₄×C₉	C₂³×C₉	1	12	yes	complemented
102	C₂²×D₄	C₉⋊C₉	1	1	yes	characteristic, direct product factor, Sylow
103	C₃×C₉	C₂²×C₃×D₄	1	3	yes	—
104	C₃×C₉	C₂²×C₃×D₄	1	1	yes	characteristic
105	C₂³×C₃	—	2	8	no	—
106	C₂³×C₃	—	2	24	no	—
107	C₂³×C₃	C₂²×C₃×C₉	1	1	yes	characteristic
108	C₂³×C₃	C₂²×C₃×C₉	1	12	yes	—
109	C₂³×C₃	C₂²×3₋¹⁺²	1	3	yes	—
110	C₂³×C₃	C₂²×3₋¹⁺²	1	36	yes	—
111	C₃×D₄	C₂²×C₃×C₉	1	16	yes	—
112	C₃×D₄	C₂²×3₋¹⁺²	1	48	yes	—
113	C₂×C₃×C₄	C₂²×C₃×C₉	1	6	yes	—
114	C₂×C₃×C₄	C₂²×3₋¹⁺²	1	18	yes	—
115	C₂×C₃²	—	2	8	no	—
116	C₂×C₃²	C₂×C₃²×D₄	1	6	yes	—
117	C₂×C₃²	C₂⁴×C₃²	1	1	yes	characteristic
118	C₂×C₉	—	2	24	no	—
119	C₂×C₉	—	3	3	no	—
120	C₂×C₉	—	3	18	no	—
121	C₂×C₉	—	6	24	no	—
122	C₂×C₉	C₂×D₄×C₉	1	18	yes	complemented
123	C₂×C₉	C₂⁴×C₉	1	3	yes	—
124	C₂⁴	C₂×C₉⋊C₉	1	2	yes	complemented
125	C₂×D₄	C₂×C₉⋊C₉	1	12	yes	direct product factor
126	C₂²×C₄	C₂×C₉⋊C₉	1	1	yes	characteristic, complemented
127	C₂²×C₃	—	2	24	no	—
128	C₂²×C₃	—	2	72	no	—
129	C₂²×C₃	C₃×D₄×C₉	1	4	yes	—
130	C₂²×C₃	D₄×3₋¹⁺²	1	12	yes	—
131	C₂²×C₃	C₂³×C₃×C₉	1	3	yes	—
132	C₂²×C₃	C₂³×C₃×C₉	1	8	yes	—
133	C₂²×C₃	C₂³×3₋¹⁺²	1	9	yes	—
134	C₂²×C₃	C₂³×3₋¹⁺²	1	24	yes	—
135	C₃×C₄	C₂³×C₃×C₉	1	4	yes	—
136	C₃×C₄	C₂³×3₋¹⁺²	1	12	yes	—
137	C₃²	C₂²×C₃²×D₄	1	1	yes	characteristic
138	C₉	—	3	3	no	—
139	C₉	C₂²×D₄×C₉	1	3	yes	complemented
140	C₂³	—	2	8	no	—
141	C₂³	C₂²×C₉⋊C₉	1	1	yes	characteristic
142	C₂³	C₂²×C₉⋊C₉	1	12	yes	complemented
143	D₄	C₂²×C₉⋊C₉	1	16	yes	direct product factor
144	C₂×C₄	C₂²×C₉⋊C₉	1	6	yes	complemented
145	C₂×C₃	—	2	8	no	—
146	C₂×C₃	—	2	24	no	—
147	C₂×C₃	C₂×C₃×D₄×C₉	1	6	yes	—
148	C₂×C₃	C₂×D₄×3₋¹⁺²	1	18	yes	—
149	C₂×C₃	C₂⁴×C₃×C₉	1	1	yes	characteristic
150	C₂×C₃	C₂⁴×3₋¹⁺²	1	3	yes	—
151	C₂²	—	2	24	no	—
152	C₂²	D₄×C₉⋊C₉	1	4	yes	direct product factor
153	C₂²	C₂³×C₉⋊C₉	1	3	yes	—
154	C₂²	C₂³×C₉⋊C₉	1	8	yes	complemented
155	C₄	C₂³×C₉⋊C₉	1	4	yes	complemented
156	C₃	C₂²×C₃×D₄×C₉	1	1	yes	characteristic
157	C₃	C₂²×D₄×3₋¹⁺²	1	3	yes	—
158	C₂	—	2	8	no	—
159	C₂	C₂×D₄×C₉⋊C₉	1	6	yes	direct product factor
160	C₂	C₂⁴×C₉⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃²	C₂²×C₃²
commutator	C₂×C₃	C₂⁴×C₃×C₉
Frattini, cosocle	C₂×C₃²	C₂⁴×C₃²
prime radical	C₂²×C₃²×D₄	C₃²
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₉⋊C₉	C₂	semidirect
2. C₂×D₄×C₉⋊C₉	C₂	direct (N × H)
3. C₂²×C₄×C₉⋊C₉	C₂	semidirect
8. C₂³×C₉⋊C₉	C₂²	semidirect
9. D₄×C₉⋊C₉	C₂²	direct (N × H)
10. C₂×C₄×C₉⋊C₉	C₂²	semidirect
18. C₂²×C₉⋊C₉	D₄	direct (N × H)
20. C₂²×C₉⋊C₉	C₂³	semidirect
21. C₄×C₉⋊C₉	C₂³	semidirect
24. C₂²×D₄×C₉	C₉	semidirect
36. C₂×C₉⋊C₉	C₂×D₄	direct (N × H)
42. C₂⁴×C₉	C₂×C₉	semidirect
44. C₂×D₄×C₉	C₂×C₉	semidirect
46. C₂²×C₄×C₉	C₂×C₉	semidirect
59. C₉⋊C₉	C₂²×D₄	direct (N × H)
70. C₂³×C₉	C₂²×C₉	semidirect
72. D₄×C₉	C₂²×C₉	semidirect
74. C₂×C₄×C₉	C₂²×C₉	semidirect
97. C₂²×C₉	D₄×C₉	semidirect
99. C₂²×C₉	C₂³×C₉	semidirect
101. C₄×C₉	C₂³×C₉	semidirect
102. C₂²×D₄	C₉⋊C₉	direct (N × H)
122. C₂×C₉	C₂×D₄×C₉	semidirect
124. C₂⁴	C₂×C₉⋊C₉	semidirect
125. C₂×D₄	C₂×C₉⋊C₉	direct (N × H)
126. C₂²×C₄	C₂×C₉⋊C₉	semidirect
139. C₉	C₂²×D₄×C₉	semidirect
142. C₂³	C₂²×C₉⋊C₉	semidirect
143. D₄	C₂²×C₉⋊C₉	direct (N × H)
144. C₂×C₄	C₂²×C₉⋊C₉	semidirect
152. C₂²	D₄×C₉⋊C₉	direct (N × H)
154. C₂²	C₂³×C₉⋊C₉	semidirect
155. C₄	C₂³×C₉⋊C₉	semidirect
159. C₂	C₂×D₄×C₉⋊C₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₉⋊C₉	2
C₂×D₄×C₉⋊C₉	2
C₂²×C₄×C₉⋊C₉	2
C₂²×C₃×D₄×C₉	3

Subgroups — C22×D4×C9⋊C9 #72268934

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₄×C₉⋊C₉ #72268934