Subgroups — C₄²×D₃×C₇×C₉ #72219181

Order 6048.

708 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄²×D₃×C₇	C₃	1	1	yes	maximal, characteristic
2	C₄×C₇×C₉×Dic₃	C₂	1	3	yes	maximal, complemented
3	C₂×C₄×D₃×C₇×C₉	C₂	1	3	yes	maximal
4	C₃×C₄²×C₇×C₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄²×C₇×C₉	—	3	1	no	maximal
6	C₂²×D₃×C₇×C₉	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄×C₇×C₉	C₂²	1	3	yes	—
8	C₂×C₇×C₉×Dic₃	C₂²	1	3	yes	—
9	C₄×D₃×C₇×C₉	C₄	1	12	yes	direct product factor
10	C₂×C₃×C₄×D₃×C₇	C₂×C₃	1	3	yes	—
11	C₃²×C₄²×C₇	C₂×C₃	1	1	yes	characteristic
12	C₃×C₄×C₇×Dic₃	C₂×C₃	1	3	yes	—
13	C₂²×C₄×C₇×C₉	—	3	3	no	—
14	C₄²×C₇×C₉	—	2	1	no	—
15	C₄²×C₇×C₉	—	3	3	no	—
16	C₄²×C₇×C₉	D₃	1	1	yes	characteristic, direct product factor
17	C₄²×D₃×C₉	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂²×C₃×C₇×C₉	C₂³	1	1	yes	characteristic
19	C₂×D₃×C₇×C₉	C₂×C₄	1	6	yes	—
20	C₃×C₄×C₇×C₉	C₂×C₄	1	6	yes	complemented
21	C₇×C₉×Dic₃	C₂×C₄	1	6	yes	complemented
22	C₂×C₃×C₄²×C₇	—	3	1	no	—
23	C₄²×D₃×C₇	C₉	1	1	yes	characteristic, direct product factor
24	C₂²×C₃×D₃×C₇	C₂²×C₃	1	1	yes	characteristic
25	C₂×C₃²×C₄×C₇	C₂²×C₃	1	3	yes	—
26	C₂×C₃×C₇×Dic₃	C₂²×C₃	1	3	yes	—
27	C₃×C₄×D₃×C₇	C₃×C₄	1	12	yes	—
28	C₂³×C₇×C₉	—	3	1	no	—
29	C₂×C₄×C₇×C₉	—	2	3	no	—
30	C₂×C₄×C₇×C₉	—	3	3	no	—
31	C₂×C₄×C₇×C₉	—	3	12	no	—
32	C₂×C₄×C₇×C₉	C₂×D₃	1	3	yes	—
33	C₂×C₄×D₃×C₉	C₂×C₇	1	3	yes	—
34	C₃×C₄²×C₉	C₂×C₇	1	1	yes	characteristic, complemented
35	C₄×C₉×Dic₃	C₂×C₇	1	3	yes	complemented
36	C₂×C₃×C₇×C₉	C₂²×C₄	1	3	yes	—
37	D₃×C₇×C₉	C₄²	1	4	yes	direct product factor
38	C₂²×C₃×C₄×C₇	—	3	3	no	—
39	C₂×C₄×D₃×C₇	C₂×C₉	1	3	yes	—
40	C₃×C₄²×C₇	—	2	1	no	—
41	C₃×C₄²×C₇	—	3	3	no	—
42	C₃×C₄²×C₇	C₂×C₉	1	1	yes	characteristic, complemented
43	C₃×C₄²×C₇	C₃×D₃	1	1	yes	characteristic
44	C₄×C₇×Dic₃	C₂×C₉	1	3	yes	complemented
45	C₃×C₄²×D₃	C₃×C₇	1	1	yes	characteristic
46	C₂×C₄²×C₉	—	3	1	no	—
47	C₂²×C₃²×C₇	C₂³×C₃	1	1	yes	characteristic
48	C₂×C₃×D₃×C₇	C₂×C₃×C₄	1	6	yes	—
49	C₃²×C₄×C₇	C₂×C₃×C₄	1	6	yes	—
50	C₃×C₇×Dic₃	C₂×C₃×C₄	1	6	yes	—
51	C₂²×C₇×C₉	—	2	1	no	—
52	C₂²×C₇×C₉	—	3	6	no	—
53	C₂²×C₇×C₉	C₂²×D₃	1	1	yes	characteristic
54	C₄×C₇×C₉	—	2	6	no	—
55	C₄×C₇×C₉	—	3	6	no	—
56	C₄×C₇×C₉	C₄×D₃	1	6	yes	direct product factor
57	C₂×C₄²×C₇	—	3	1	no	Hall
58	C₂²×D₃×C₉	C₂²×C₇	1	1	yes	characteristic
59	C₂×C₃×C₄×C₉	C₂²×C₇	1	3	yes	—
60	C₂×C₉×Dic₃	C₂²×C₇	1	3	yes	—
61	C₄×D₃×C₉	C₄×C₇	1	12	yes	direct product factor
62	C₃×C₇×C₉	C₂×C₄²	1	1	yes	characteristic, complemented, Hall
63	C₂³×C₃×C₇	—	3	1	no	—
64	C₂²×D₃×C₇	C₂²×C₉	1	1	yes	characteristic
65	C₂×C₃×C₄×C₇	—	2	3	no	—
66	C₂×C₃×C₄×C₇	—	3	3	no	—
67	C₂×C₃×C₄×C₇	—	3	12	no	—
68	C₂×C₃×C₄×C₇	C₂²×C₉	1	3	yes	—
69	C₂×C₃×C₄×C₇	C₂×C₃×D₃	1	3	yes	—
70	C₂×C₇×Dic₃	C₂²×C₉	1	3	yes	—
71	C₄×D₃×C₇	C₄×C₉	1	12	yes	direct product factor
72	C₂×C₃×C₄×D₃	C₂×C₃×C₇	1	3	yes	—
73	C₃²×C₄²	C₂×C₃×C₇	1	1	yes	characteristic
74	C₃×C₄×Dic₃	C₂×C₃×C₇	1	3	yes	—
75	C₂²×C₄×C₉	—	3	3	no	—
76	C₄²×C₉	—	2	1	no	—
77	C₄²×C₉	—	3	3	no	—
78	C₄²×C₉	D₃×C₇	1	1	yes	characteristic, direct product factor
79	C₂×C₃²×C₇	C₂²×C₃×C₄	1	3	yes	—
80	C₃×D₃×C₇	C₃×C₄²	1	4	yes	—
81	C₂×C₇×C₉	—	2	3	no	—
82	C₂×C₇×C₉	—	3	4	no	—
83	C₂×C₇×C₉	C₂×C₄×D₃	1	3	yes	—
84	C₂²×C₄×C₇	—	3	3	no	—
85	C₄²×C₇	—	3	3	no	—
86	C₄²×C₇	D₃×C₉	1	1	yes	characteristic, direct product factor
87	C₂²×C₃×C₉	C₂³×C₇	1	1	yes	characteristic
88	C₂×D₃×C₉	C₂×C₄×C₇	1	6	yes	—
89	C₃×C₄×C₉	C₂×C₄×C₇	1	6	yes	complemented
90	C₉×Dic₃	C₂×C₄×C₇	1	6	yes	complemented
91	C₂×C₃×C₄²	—	3	1	no	—
92	C₄²×D₃	C₇×C₉	1	1	yes	characteristic, direct product factor
93	C₂²×C₃×C₇	—	2	1	no	—
94	C₂²×C₃×C₇	—	3	6	no	—
95	C₂²×C₃×C₇	C₂³×C₉	1	1	yes	characteristic
96	C₂²×C₃×C₇	C₂²×C₃×D₃	1	1	yes	characteristic
97	C₂×D₃×C₇	C₂×C₄×C₉	1	6	yes	—
98	C₃×C₄×C₇	—	2	6	no	—
99	C₃×C₄×C₇	—	3	6	no	—
100	C₃×C₄×C₇	C₂×C₄×C₉	1	6	yes	complemented
101	C₃×C₄×C₇	C₃×C₄×D₃	1	6	yes	—
102	C₇×Dic₃	C₂×C₄×C₉	1	6	yes	complemented
103	C₂²×C₃×D₃	C₂²×C₃×C₇	1	1	yes	characteristic
104	C₂×C₃²×C₄	C₂²×C₃×C₇	1	3	yes	—
105	C₂×C₃×Dic₃	C₂²×C₃×C₇	1	3	yes	—
106	C₃×C₄×D₃	C₃×C₄×C₇	1	12	yes	—
107	C₂³×C₉	—	3	1	no	—
108	C₂×C₄×C₉	—	2	3	no	—
109	C₂×C₄×C₉	—	3	3	no	—
110	C₂×C₄×C₉	—	3	12	no	—
111	C₂×C₄×C₉	C₂×D₃×C₇	1	3	yes	—
112	C₃²×C₇	C₂×C₃×C₄²	1	1	yes	characteristic
113	C₇×C₉	—	2	1	no	—
114	C₇×C₉	C₄²×D₃	1	1	yes	characteristic, direct product factor
115	C₂³×C₇	—	3	1	no	—
116	C₂×C₄×C₇	—	3	3	no	—
117	C₂×C₄×C₇	—	3	12	no	—
118	C₂×C₄×C₇	C₂×D₃×C₉	1	3	yes	—
119	C₂×C₃×C₉	C₂²×C₄×C₇	1	3	yes	—
120	D₃×C₉	C₄²×C₇	1	4	yes	direct product factor
121	C₂²×C₃×C₄	—	3	3	no	—
122	C₂×C₄×D₃	C₂×C₇×C₉	1	3	yes	—
123	C₃×C₄²	—	2	1	no	—
124	C₃×C₄²	—	3	3	no	—
125	C₃×C₄²	C₂×C₇×C₉	1	1	yes	characteristic, complemented
126	C₃×C₄²	C₃×D₃×C₇	1	1	yes	characteristic
127	C₄×Dic₃	C₂×C₇×C₉	1	3	yes	complemented
128	C₂×C₃×C₇	—	2	3	no	—
129	C₂×C₃×C₇	—	3	4	no	—
130	C₂×C₃×C₇	C₂²×C₄×C₉	1	3	yes	—
131	C₂×C₃×C₇	C₂×C₃×C₄×D₃	1	3	yes	—
132	D₃×C₇	C₄²×C₉	1	4	yes	direct product factor
133	C₂²×C₃²	C₂³×C₃×C₇	1	1	yes	characteristic
134	C₂×C₃×D₃	C₂×C₃×C₄×C₇	1	6	yes	—
135	C₃²×C₄	C₂×C₃×C₄×C₇	1	6	yes	—
136	C₃×Dic₃	C₂×C₃×C₄×C₇	1	6	yes	—
137	C₂²×C₉	—	2	1	no	—
138	C₂²×C₉	—	3	6	no	—
139	C₂²×C₉	C₂²×D₃×C₇	1	1	yes	characteristic
140	C₄×C₉	—	2	6	no	—
141	C₄×C₉	—	3	6	no	—
142	C₄×C₉	C₄×D₃×C₇	1	6	yes	direct product factor
143	C₂×C₄²	—	3	1	no	Sylow
144	C₂²×C₇	—	3	6	no	—
145	C₂²×C₇	C₂²×D₃×C₉	1	1	yes	characteristic
146	C₄×C₇	—	3	6	no	—
147	C₄×C₇	C₄×D₃×C₉	1	6	yes	direct product factor
148	C₃×C₉	C₂×C₄²×C₇	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₃	—	3	1	no	—
150	C₂²×D₃	C₂²×C₇×C₉	1	1	yes	characteristic
151	C₂×C₃×C₄	—	2	3	no	—
152	C₂×C₃×C₄	—	3	3	no	—
153	C₂×C₃×C₄	—	3	12	no	—
154	C₂×C₃×C₄	C₂²×C₇×C₉	1	3	yes	—
155	C₂×C₃×C₄	C₂×C₃×D₃×C₇	1	3	yes	—
156	C₂×Dic₃	C₂²×C₇×C₉	1	3	yes	—
157	C₄×D₃	C₄×C₇×C₉	1	12	yes	direct product factor
158	C₃×C₇	—	2	1	no	—
159	C₃×C₇	C₂×C₄²×C₉	1	1	yes	characteristic, complemented
160	C₃×C₇	C₃×C₄²×D₃	1	1	yes	characteristic
161	C₂×C₃²	C₂²×C₃×C₄×C₇	1	3	yes	—
162	C₃×D₃	C₃×C₄²×C₇	1	4	yes	—
163	C₂×C₉	—	2	3	no	—
164	C₂×C₉	—	3	4	no	—
165	C₂×C₉	C₂×C₄×D₃×C₇	1	3	yes	—
166	C₂²×C₄	—	3	3	no	—
167	C₄²	—	3	3	no	—
168	C₄²	D₃×C₇×C₉	1	1	yes	characteristic, direct product factor
169	C₂×C₇	—	3	4	no	—
170	C₂×C₇	C₂×C₄×D₃×C₉	1	3	yes	—
171	C₂²×C₃	—	2	1	no	—
172	C₂²×C₃	—	3	6	no	—
173	C₂²×C₃	C₂³×C₇×C₉	1	1	yes	characteristic
174	C₂²×C₃	C₂²×C₃×D₃×C₇	1	1	yes	characteristic
175	C₂×D₃	C₂×C₄×C₇×C₉	1	6	yes	—
176	C₃×C₄	—	2	6	no	—
177	C₃×C₄	—	3	6	no	—
178	C₃×C₄	C₂×C₄×C₇×C₉	1	6	yes	complemented
179	C₃×C₄	C₃×C₄×D₃×C₇	1	6	yes	—
180	Dic₃	C₂×C₄×C₇×C₉	1	6	yes	complemented
181	C₃²	C₂×C₃×C₄²×C₇	1	1	yes	characteristic
182	C₉	—	2	1	no	—
183	C₉	C₄²×D₃×C₇	1	1	yes	characteristic, direct product factor
184	C₂³	—	3	1	no	—
185	C₂×C₄	—	3	3	no	—
186	C₂×C₄	—	3	12	no	—
187	C₂×C₄	C₂×D₃×C₇×C₉	1	3	yes	—
188	C₇	C₄²×D₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
189	C₂×C₃	—	2	3	no	—
190	C₂×C₃	—	3	4	no	—
191	C₂×C₃	C₂²×C₄×C₇×C₉	1	3	yes	—
192	C₂×C₃	C₂×C₃×C₄×D₃×C₇	1	3	yes	—
193	D₃	C₄²×C₇×C₉	1	4	yes	direct product factor
194	C₂²	—	3	6	no	—
195	C₂²	C₂²×D₃×C₇×C₉	1	1	yes	characteristic
196	C₄	—	3	6	no	—
197	C₄	C₄×D₃×C₇×C₉	1	6	yes	direct product factor
198	C₃	—	2	1	no	—
199	C₃	C₂×C₄²×C₇×C₉	1	1	yes	characteristic, complemented
200	C₃	C₃×C₄²×D₃×C₇	1	1	yes	characteristic
201	C₂	—	3	4	no	—
202	C₂	C₂×C₄×D₃×C₇×C₉	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₄²×C₇×C₉	D₃
commutator, hypocenter	C₃	C₂×C₄²×C₇×C₉
Frattini	C₂²×C₃	C₂²×C₃×D₃×C₇
Fitting	C₃×C₄²×C₇×C₉	C₂
socle	C₂²×C₃²×C₇	C₂³×C₃
prime radical	C₂²×C₃×D₃×C₇	C₂²×C₃
cosocle	C₂²×C₃²	C₂³×C₃×C₇
LCS	G ▷ C₃
UCS	C₁ ◁ C₄²×C₇×C₉

Complements

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

N	H ≅ G/N	type
2. C₄×C₇×C₉×Dic₃	C₂	semidirect
4. C₃×C₄²×C₇×C₉	C₂	semidirect
9. C₄×D₃×C₇×C₉	C₄	direct (N × H)
16. C₄²×C₇×C₉	D₃	direct (N × H)
17. C₄²×D₃×C₉	C₇	direct (N × H)
20. C₃×C₄×C₇×C₉	C₂×C₄	semidirect
21. C₇×C₉×Dic₃	C₂×C₄	semidirect
23. C₄²×D₃×C₇	C₉	direct (N × H)
34. C₃×C₄²×C₉	C₂×C₇	semidirect
35. C₄×C₉×Dic₃	C₂×C₇	semidirect
37. D₃×C₇×C₉	C₄²	direct (N × H)
42. C₃×C₄²×C₇	C₂×C₉	semidirect
44. C₄×C₇×Dic₃	C₂×C₉	semidirect
56. C₄×C₇×C₉	C₄×D₃	direct (N × H)
61. C₄×D₃×C₉	C₄×C₇	direct (N × H)
62. C₃×C₇×C₉	C₂×C₄²	semidirect
71. C₄×D₃×C₇	C₄×C₉	direct (N × H)
78. C₄²×C₉	D₃×C₇	direct (N × H)
86. C₄²×C₇	D₃×C₉	direct (N × H)
89. C₃×C₄×C₉	C₂×C₄×C₇	semidirect
90. C₉×Dic₃	C₂×C₄×C₇	semidirect
92. C₄²×D₃	C₇×C₉	direct (N × H)
100. C₃×C₄×C₇	C₂×C₄×C₉	semidirect
102. C₇×Dic₃	C₂×C₄×C₉	semidirect
114. C₇×C₉	C₄²×D₃	direct (N × H)
120. D₃×C₉	C₄²×C₇	direct (N × H)
125. C₃×C₄²	C₂×C₇×C₉	semidirect
127. C₄×Dic₃	C₂×C₇×C₉	semidirect
132. D₃×C₇	C₄²×C₉	direct (N × H)
142. C₄×C₉	C₄×D₃×C₇	direct (N × H)
147. C₄×C₇	C₄×D₃×C₉	direct (N × H)
148. C₃×C₉	C₂×C₄²×C₇	semidirect
157. C₄×D₃	C₄×C₇×C₉	direct (N × H)
159. C₃×C₇	C₂×C₄²×C₉	semidirect
168. C₄²	D₃×C₇×C₉	direct (N × H)
178. C₃×C₄	C₂×C₄×C₇×C₉	semidirect
180. Dic₃	C₂×C₄×C₇×C₉	semidirect
183. C₉	C₄²×D₃×C₇	direct (N × H)
188. C₇	C₄²×D₃×C₉	direct (N × H)
193. D₃	C₄²×C₇×C₉	direct (N × H)
197. C₄	C₄×D₃×C₇×C₉	direct (N × H)
199. C₃	C₂×C₄²×C₇×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄²×D₃×C₇	3
C₄×C₇×C₉×Dic₃	2
C₂×C₄×D₃×C₇×C₉	2
C₃×C₄²×C₇×C₉	2
C₂×C₄²×C₇×C₉	3
C₄²×D₃×C₉	7

Subgroups — C42×D3×C7×C9 #72219181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×D₃×C₇×C₉ #72219181