Subgroups — C₉×#4999 #13142674

Order 1728.

418 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#208	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₉×C₄⋊(C₃⋊C₈)	C₂	1	4	yes	maximal, complemented
5	C₃×#4999	C₃	1	1	yes	maximal, characteristic
6	C₉×#424	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₃×C₉×C₄⋊C₄	C₄	1	2	yes	—
9	C₃×C₉×C₂²⋊C₄	C₄	1	2	yes	—
10	C₃×C₄²×C₉	C₂²	1	2	yes	—
11	C₉×#208	—	2	2	no	—
12	C₂×C₉×C₃⋊C₈	—	2	2	no	—
13	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
14	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
15	C₃²×#117	C₂×C₃	1	1	yes	characteristic
16	C₂×C₃×#208	C₂×C₃	1	1	yes	characteristic
17	C₂²×C₃×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
18	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	4	yes	—
19	C₂×C₉×M₁₆	—	3	1	no	—
20	C₂²×C₈×C₉	—	3	1	no	—
21	C₉×#117	—	2	1	no	—
22	C₉×#117	D₃	1	1	yes	characteristic
23	C₉×C₄⋊C₈	—	3	4	no	—
24	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
26	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
28	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
29	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₉×C₃⋊C₈	—	2	2	no	—
31	C₉×C₃⋊C₈	—	2	2	no	—
32	C₃×#424	—	3	1	no	—
33	#4999	C₉	1	1	yes	characteristic, direct product factor
34	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
35	C₃²×C₄⋊C₄	C₃×C₄	1	2	yes	—
36	C₃²×C₂²⋊C₄	C₃×C₄	1	2	yes	—
37	C₃²×C₄²	C₂²×C₃	1	2	yes	—
38	C₃×#208	—	2	2	no	—
39	C₂×C₃×C₃⋊C₈	—	2	2	no	—
40	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
41	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
42	C₂²×C₄×C₉	—	2	1	no	—
43	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
44	C₉×M₁₆	—	6	2	no	—
45	C₂×C₈×C₉	—	3	2	no	—
46	C₂×C₈×C₉	—	3	2	no	—
47	C₂×C₈×C₉	—	6	2	no	—
48	C₉×C₄⋊C₄	—	2	2	no	—
49	C₉×C₄⋊C₄	Dic₃	1	2	yes	—
50	C₉×C₂²⋊C₄	—	2	2	no	—
51	C₉×C₂²⋊C₄	Dic₃	1	2	yes	—
52	C₄²×C₉	—	2	2	no	—
53	C₄²×C₉	C₂×D₃	1	2	yes	—
54	C₂²×C₃×C₉	—	2	2	no	—
55	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
56	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
57	C₃×C₄×C₉	—	2	4	no	—
58	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
59	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
60	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₃×M₁₆	—	3	1	no	—
62	C₂²×C₃×C₈	—	3	1	no	—
63	C₃×#117	—	2	1	no	—
64	C₃×#117	C₂×C₉	1	1	yes	characteristic
65	C₃×#117	C₃×D₃	1	1	yes	characteristic
66	C₂×#208	C₂×C₉	1	1	yes	characteristic
67	C₂²×C₃⋊C₈	C₂×C₉	1	1	yes	characteristic
68	C₃×C₄⋊C₈	—	3	4	no	—
69	C₄⋊(C₃⋊C₈)	C₂×C₉	1	4	yes	complemented
70	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
72	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
74	C₂×C₃²×C₄	C₃×Q₈	1	2	yes	—
75	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
76	C₂³×C₉	—	2	1	no	—
77	C₂³×C₉	C₂×Dic₃	1	1	yes	characteristic
78	C₃×C₃⋊C₈	—	2	2	no	—
79	C₃×C₃⋊C₈	—	2	2	no	—
80	C₂×C₄×C₉	—	2	2	no	—
81	C₂×C₄×C₉	—	2	2	no	—
82	C₂×C₄×C₉	—	2	1	no	—
83	C₂×C₄×C₉	—	2	4	no	—
84	C₂×C₄×C₉	—	2	1	no	—
85	C₂×C₄×C₉	Dic₆	1	2	yes	—
86	C₂×C₄×C₉	D₁₂	1	2	yes	—
87	C₂×C₄×C₉	C₂×Dic₃	1	4	yes	—
88	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
89	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
90	C₈×C₉	—	6	2	no	—
91	C₈×C₉	—	6	2	no	—
92	#424	—	3	1	no	Sylow
93	C₂×C₃×C₉	—	2	2	no	—
94	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₃×C₉	#131	1	2	yes	—
96	C₂²×C₃×C₄	—	2	1	no	—
97	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
99	C₃×M₁₆	—	6	2	no	—
100	C₂×C₃×C₈	—	3	2	no	—
101	C₂×C₃×C₈	—	3	2	no	—
102	C₂×C₃×C₈	—	6	2	no	—
103	C₃×C₄⋊C₄	—	2	2	no	—
104	C₃×C₄⋊C₄	C₄×C₉	1	2	yes	—
105	C₃×C₄⋊C₄	C₃×Dic₃	1	2	yes	—
106	C₃×C₂²⋊C₄	—	2	2	no	—
107	C₃×C₂²⋊C₄	C₄×C₉	1	2	yes	—
108	C₃×C₂²⋊C₄	C₃×Dic₃	1	2	yes	—
109	C₃×C₄²	—	2	2	no	—
110	C₃×C₄²	C₂²×C₉	1	2	yes	—
111	C₃×C₄²	C₂×C₃×D₃	1	2	yes	—
112	#208	—	2	2	no	—
113	C₂×C₃⋊C₈	—	2	2	no	—
114	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
115	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
116	C₂²×C₃²	—	2	2	no	—
117	C₂²×C₃²	C₃×C₄⋊C₄	1	2	yes	—
118	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
119	C₃²×C₄	—	2	4	no	—
120	C₃²×C₄	C₃×C₄⋊C₄	1	2	yes	—
121	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
122	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
123	C₂²×C₉	—	2	2	no	—
124	C₂²×C₉	—	2	2	no	—
125	C₂²×C₉	—	2	1	no	—
126	C₂²×C₉	—	4	2	no	—
127	C₂²×C₉	C₄⋊Dic₃	1	2	yes	—
128	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
129	C₄×C₉	—	2	4	no	—
130	C₄×C₉	—	2	2	no	—
131	C₄×C₉	—	2	1	no	—
132	C₄×C₉	—	2	1	no	—
133	C₄×C₉	—	4	4	no	—
134	C₄×C₉	C₄⋊Dic₃	1	2	yes	—
135	C₄×C₉	C₂×Dic₆	1	1	yes	characteristic
136	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
137	C₂×M₁₆	—	3	1	no	—
138	C₂²×C₈	—	3	1	no	—
139	#117	D₃×C₉	1	1	yes	characteristic
140	C₄⋊C₈	—	3	4	no	—
141	C₃×C₉	#424	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₃	—	2	1	no	—
143	C₂³×C₃	C₂×C₄×C₉	1	1	yes	characteristic
144	C₂³×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	C₂×C₄×C₉	1	4	yes	—
151	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
152	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
153	C₂×C₃×C₄	Q₈×C₉	1	2	yes	—
154	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
155	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
156	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
157	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
158	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
160	C₃×C₈	—	6	2	no	—
161	C₃×C₈	—	6	2	no	—
162	C₃⋊C₈	—	2	2	no	—
163	C₃⋊C₈	—	2	2	no	—
164	C₂×C₃²	—	2	2	no	—
165	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₃²	C₃×#131	1	2	yes	—
167	C₂×C₉	—	2	2	no	—
168	C₂×C₉	—	2	1	no	—
169	C₂×C₉	—	2	2	no	—
170	C₂×C₉	—	4	2	no	—
171	C₂×C₉	C₂×C₄⋊Dic₃	1	1	yes	characteristic
172	C₂×C₉	#948	1	2	yes	—
173	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
174	M₁₆	—	6	2	no	—
175	C₂×C₈	—	3	2	no	—
176	C₂×C₈	—	3	2	no	—
177	C₂×C₈	—	6	2	no	—
178	C₄⋊C₄	C₉×Dic₃	1	2	yes	—
179	C₂²⋊C₄	C₉×Dic₃	1	2	yes	—
180	C₄²	C₂×D₃×C₉	1	2	yes	—
181	C₂²×C₃	—	2	2	no	—
182	C₂²×C₃	—	2	2	no	—
183	C₂²×C₃	—	2	1	no	—
184	C₂²×C₃	—	2	2	no	—
185	C₂²×C₃	—	4	2	no	—
186	C₂²×C₃	C₉×C₄⋊C₄	1	2	yes	—
187	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
188	C₂²×C₃	C₃×C₄⋊Dic₃	1	2	yes	—
189	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
190	C₃×C₄	—	2	4	no	—
191	C₃×C₄	—	2	4	no	—
192	C₃×C₄	—	2	1	no	—
193	C₃×C₄	—	2	1	no	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	—	4	4	no	—
196	C₃×C₄	C₉×C₄⋊C₄	1	2	yes	—
197	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
198	C₃×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
199	C₃×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
200	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
201	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
202	C₃²	C₃×#424	1	1	yes	characteristic
203	C₉	—	2	1	no	—
204	C₉	#4999	1	1	yes	characteristic, direct product factor
205	C₂³	C₂×C₉×Dic₃	1	1	yes	characteristic
206	C₂×C₄	C₉×Dic₆	1	2	yes	—
207	C₂×C₄	C₉×D₁₂	1	2	yes	—
208	C₂×C₄	C₂×C₉×Dic₃	1	4	yes	—
209	C₂×C₄	C₂×C₉×Dic₃	1	1	yes	characteristic
210	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
211	C₈	—	6	2	no	—
212	C₈	—	6	2	no	—
213	C₂×C₃	—	2	2	no	—
214	C₂×C₃	—	2	2	no	—
215	C₂×C₃	—	2	1	no	—
216	C₂×C₃	—	2	2	no	—
217	C₂×C₃	—	4	2	no	—
218	C₂×C₃	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
219	C₂×C₃	C₉×#131	1	2	yes	—
220	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
221	C₂×C₃	C₃×#948	1	2	yes	—
222	C₂²	—	2	2	no	—
223	C₂²	C₉×C₄⋊Dic₃	1	2	yes	—
224	C₂²	C₂²×C₉×Dic₃	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	C₉×C₄⋊Dic₃	1	2	yes	—
227	C₄	C₂×C₉×Dic₆	1	1	yes	characteristic
228	C₄	C₂×C₉×D₁₂	1	1	yes	characteristic
229	C₃	—	2	1	no	—
230	C₃	C₉×#424	1	1	yes	characteristic, complemented
231	C₃	C₃×#4999	1	1	yes	characteristic
232	C₂	—	2	2	no	—
233	C₂	C₂×C₉×C₄⋊Dic₃	1	1	yes	characteristic
234	C₂	C₉×#948	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₉	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#117	C₂
socle	C₂²×C₃²	C₂²×C₃×C₄
prime radical	C₂³×C₃²	C₂×C₃×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#117	D₃
hypocenter	C₃	C₉×#424
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₉×#117

Complements

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

N	H ≅ G/N	type
4. C₉×C₄⋊(C₃⋊C₈)	C₂	semidirect
33. #4999	C₉	direct (N × H)
69. C₄⋊(C₃⋊C₈)	C₂×C₉	semidirect
141. C₃×C₉	#424	semidirect
204. C₉	#4999	direct (N × H)
230. C₃	C₉×#424	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#117	2
C₂×C₉×#208	2
C₂²×C₉×C₃⋊C₈	2
C₉×C₄⋊(C₃⋊C₈)	2
C₃×#4999	3
C₉×#424	3

Subgroups — C9×#4999 #13142674

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4999 #13142674