Subgroups — C₉×#5070 #13142969

Order 1728.

462 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#127	C₂	1	1	yes	maximal, characteristic
2	C₉×#869	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#832	C₂	1	4	yes	maximal, complemented
4	C₄×C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#5070	C₃	1	1	yes	maximal, characteristic
6	C₉×#486	—	3	1	no	maximal
7	C₂×C₃×D₄×C₉	—	2	1	no	—
8	C₂×C₃×D₄×C₉	—	2	1	no	—
9	C₂×C₃×D₄×C₉	C₂²	1	2	yes	—
10	C₂×C₉×Dic₆	—	2	1	no	—
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₉×C₄⋊Dic₃	—	2	1	no	—
13	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
15	C₃²×#127	C₂×C₃	1	1	yes	characteristic
16	C₃×#869	C₂×C₃	1	1	yes	characteristic
17	C₃×#832	C₂×C₃	1	4	yes	—
18	C₃×C₄×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
19	C₉×C₄⋊Q₈	—	3	1	no	—
20	C₉×#127	—	2	1	no	—
21	C₉×#127	D₃	1	1	yes	characteristic
22	C₉×#102	—	3	4	no	—
23	C₄×C₈×C₉	—	3	1	no	—
24	C₂³×C₃×C₉	—	2	2	no	—
25	C₃×D₄×C₉	—	2	4	no	—
26	C₃×D₄×C₉	—	4	2	no	—
27	C₃×D₄×C₉	—	4	2	no	—
28	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂×C₉×Dic₃	—	2	2	no	—
32	C₉×Dic₆	—	4	2	no	—
33	C₉×C₃⋊C₈	—	2	2	no	—
34	C₃×#486	—	3	1	no	—
35	#5070	C₉	1	1	yes	characteristic, direct product factor
36	C₂×C₃²×D₄	—	2	1	no	—
37	C₂×C₃²×D₄	—	2	1	no	—
38	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	—
39	C₂×C₃×Dic₆	—	2	1	no	—
40	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
41	C₃×C₄⋊Dic₃	—	2	1	no	—
42	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
43	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
44	C₂×Q₈×C₉	—	6	1	no	—
45	C₂×D₄×C₉	—	2	2	no	—
46	C₂×D₄×C₉	—	2	2	no	—
47	C₂×D₄×C₉	—	2	2	no	—
48	C₂×D₄×C₉	—	2	1	no	—
49	C₂×D₄×C₉	—	2	1	no	—
50	C₂×D₄×C₉	C₂×D₃	1	2	yes	—
51	C₂×C₈×C₉	—	3	2	no	—
52	C₉×C₄⋊C₄	—	3	2	no	—
53	C₉×C₄⋊C₄	—	6	1	no	—
54	C₄²×C₉	—	2	1	no	—
55	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
56	C₂²×C₃×C₉	—	4	2	no	—
57	C₂²×C₃×C₉	—	4	2	no	—
58	C₂²×C₃×C₉	—	4	2	no	—
59	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
60	C₃×C₄×C₉	D₈	1	2	yes	—
61	C₃×C₄×C₉	SD₁₆	1	2	yes	—
62	C₃×C₄×C₉	#41	1	2	yes	—
63	C₉×Dic₃	—	4	2	no	—
64	C₃×C₄⋊Q₈	—	3	1	no	—
65	C₃×#127	—	2	1	no	—
66	C₃×#127	C₂×C₉	1	1	yes	characteristic
67	C₃×#127	C₃×D₃	1	1	yes	characteristic
68	#869	C₂×C₉	1	1	yes	characteristic, complemented
69	C₃×#102	—	3	4	no	—
70	C₃×C₄×C₈	—	3	1	no	—
71	#832	C₂×C₉	1	4	yes	complemented
72	C₄×C₃⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
73	C₂³×C₃²	—	2	2	no	—
74	C₃²×D₄	—	2	4	no	—
75	C₃²×D₄	—	4	2	no	—
76	C₃²×D₄	—	4	2	no	—
77	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
80	C₂×C₃×Dic₃	—	2	2	no	—
81	C₃×Dic₆	—	4	2	no	—
82	C₂³×C₉	—	2	4	no	—
83	C₂³×C₉	—	2	2	no	—
84	C₃×C₃⋊C₈	—	2	2	no	—
85	Q₈×C₉	—	12	2	no	—
86	D₄×C₉	—	2	4	no	—
87	D₄×C₉	—	4	4	no	—
88	D₄×C₉	—	4	4	no	—
89	D₄×C₉	—	4	4	no	—
90	D₄×C₉	—	4	2	no	—
91	D₄×C₉	—	4	2	no	—
92	C₂×C₄×C₉	—	2	1	no	—
93	C₂×C₄×C₉	—	2	1	no	—
94	C₂×C₄×C₉	—	2	1	no	—
95	C₂×C₄×C₉	—	6	2	no	—
96	C₂×C₄×C₉	#67	1	1	yes	characteristic
97	C₂×C₄×C₉	#67	1	1	yes	characteristic
98	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
99	C₈×C₉	—	6	2	no	—
100	#486	—	3	1	no	Sylow
101	C₂×C₃×C₉	—	8	2	no	—
102	C₂×C₃×C₉	#124	1	1	yes	characteristic
103	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
104	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₃×Q₈	—	6	1	no	—
106	C₂×C₃×D₄	—	2	2	no	—
107	C₂×C₃×D₄	—	2	2	no	—
108	C₂×C₃×D₄	—	2	1	no	—
109	C₂×C₃×D₄	—	2	1	no	—
110	C₂×C₃×D₄	—	2	1	no	—
111	C₂×C₃×D₄	—	2	1	no	—
112	C₂×C₃×D₄	—	2	2	no	—
113	C₂×C₃×D₄	C₂²×C₉	1	2	yes	—
114	C₂×C₃×D₄	C₂×C₃×D₃	1	2	yes	—
115	C₂×Dic₆	—	2	1	no	—
116	C₂×C₃×C₈	—	3	2	no	—
117	C₃×C₄⋊C₄	—	3	2	no	—
118	C₃×C₄⋊C₄	—	6	1	no	—
119	C₃×C₄²	—	2	1	no	—
120	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
121	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
122	C₄⋊Dic₃	—	2	1	no	—
123	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
124	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
125	C₂²×C₃²	—	4	2	no	—
126	C₂²×C₃²	—	4	2	no	—
127	C₂²×C₃²	—	4	2	no	—
128	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
129	C₃²×C₄	C₃×D₈	1	2	yes	—
130	C₃²×C₄	C₃×SD₁₆	1	2	yes	—
131	C₃²×C₄	C₃×#41	1	2	yes	—
132	C₃×Dic₃	—	4	2	no	—
133	C₂²×C₉	—	2	1	no	—
134	C₂²×C₉	—	4	4	no	—
135	C₂²×C₉	—	4	4	no	—
136	C₂²×C₉	—	4	4	no	—
137	C₂²×C₉	—	4	2	no	—
138	C₂²×C₉	—	4	2	no	—
139	C₂²×C₉	—	4	2	no	—
140	C₂²×C₉	C₂×#67	1	1	yes	characteristic
141	C₄×C₉	—	2	2	no	—
142	C₄×C₉	—	2	2	no	—
143	C₄×C₉	—	2	2	no	—
144	C₄×C₉	—	12	2	no	—
145	C₄×C₉	#213	1	2	yes	—
146	C₄×C₉	#214	1	2	yes	—
147	C₄×C₉	#237	1	2	yes	—
148	C₄⋊Q₈	—	3	1	no	—
149	#127	D₃×C₉	1	1	yes	characteristic
150	#102	—	3	4	no	—
151	C₄×C₈	—	3	1	no	—
152	C₃×C₉	#486	1	1	yes	characteristic, complemented, Sylow
153	C₂³×C₃	—	2	4	no	—
154	C₂³×C₃	—	2	2	no	—
155	C₂³×C₃	—	2	2	no	—
156	C₃×Q₈	—	12	2	no	—
157	C₃×D₄	—	2	4	no	—
158	C₃×D₄	—	2	4	no	—
159	C₃×D₄	—	4	4	no	—
160	C₃×D₄	—	4	4	no	—
161	C₃×D₄	—	4	4	no	—
162	C₃×D₄	—	4	2	no	—
163	C₃×D₄	—	4	2	no	—
164	C₃×D₄	—	4	2	no	—
165	C₃×D₄	—	4	2	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	6	2	no	—
170	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
171	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
173	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
174	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
176	C₂×Dic₃	—	2	2	no	—
177	Dic₆	—	4	2	no	—
178	C₃×C₈	—	6	2	no	—
179	C₃⋊C₈	—	2	2	no	—
180	C₂×C₃²	—	8	2	no	—
181	C₂×C₃²	C₃×#124	1	1	yes	characteristic
182	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
183	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
184	C₂×C₉	—	2	1	no	—
185	C₂×C₉	—	2	1	no	—
186	C₂×C₉	—	2	1	no	—
187	C₂×C₉	—	8	4	no	—
188	C₂×C₉	—	8	2	no	—
189	C₂×C₉	C₂×#213	1	1	yes	characteristic
190	C₂×C₉	C₂×#214	1	1	yes	characteristic
191	C₂×C₉	#936	1	1	yes	characteristic
192	C₂×Q₈	—	6	1	no	—
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	—	2	1	no	—
195	C₂×D₄	C₂×D₃×C₉	1	2	yes	—
196	C₂×C₈	—	3	2	no	—
197	C₄⋊C₄	—	3	2	no	—
198	C₄⋊C₄	—	6	1	no	—
199	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	—	4	4	no	—
202	C₂²×C₃	—	4	4	no	—
203	C₂²×C₃	—	4	4	no	—
204	C₂²×C₃	—	4	2	no	—
205	C₂²×C₃	—	4	2	no	—
206	C₂²×C₃	—	4	2	no	—
207	C₂²×C₃	—	4	2	no	—
208	C₂²×C₃	—	4	2	no	—
209	C₂²×C₃	—	4	2	no	—
210	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
211	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	2	2	no	—
215	C₃×C₄	—	12	2	no	—
216	C₃×C₄	C₉×D₈	1	2	yes	—
217	C₃×C₄	C₉×SD₁₆	1	2	yes	—
218	C₃×C₄	C₉×#41	1	2	yes	—
219	C₃×C₄	C₃×#213	1	2	yes	—
220	C₃×C₄	C₃×#214	1	2	yes	—
221	C₃×C₄	C₃×#237	1	2	yes	—
222	Dic₃	—	4	2	no	—
223	C₃²	C₃×#486	1	1	yes	characteristic
224	C₉	—	2	1	no	—
225	C₉	#5070	1	1	yes	characteristic, direct product factor
226	C₂³	—	2	2	no	—
227	Q₈	—	12	2	no	—
228	D₄	—	2	4	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	2	no	—
231	C₂×C₄	—	6	2	no	—
232	C₂×C₄	C₉×#67	1	1	yes	characteristic
233	C₂×C₄	C₉×#67	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
235	C₈	—	6	2	no	—
236	C₂×C₃	—	2	1	no	—
237	C₂×C₃	—	2	1	no	—
238	C₂×C₃	—	2	1	no	—
239	C₂×C₃	—	8	4	no	—
240	C₂×C₃	—	8	2	no	—
241	C₂×C₃	—	8	2	no	—
242	C₂×C₃	C₉×#124	1	1	yes	characteristic
243	C₂×C₃	C₂×C₉×D₈	1	1	yes	characteristic
244	C₂×C₃	C₂×C₉×SD₁₆	1	1	yes	characteristic
245	C₂×C₃	C₂×C₃×#213	1	1	yes	characteristic
246	C₂×C₃	C₂×C₃×#214	1	1	yes	characteristic
247	C₂×C₃	C₃×#936	1	1	yes	characteristic
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	C₂×C₉×#67	1	1	yes	characteristic
252	C₄	—	12	2	no	—
253	C₄	C₉×#213	1	2	yes	—
254	C₄	C₉×#214	1	2	yes	—
255	C₄	C₉×#237	1	2	yes	—
256	C₃	—	2	1	no	—
257	C₃	C₉×#486	1	1	yes	characteristic, complemented
258	C₃	C₃×#5070	1	1	yes	characteristic
259	C₂	—	8	2	no	—
260	C₂	C₂×C₉×#213	1	1	yes	characteristic
261	C₂	C₂×C₉×#214	1	1	yes	characteristic
262	C₂	C₉×#936	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×#67
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#127	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₃²×#127	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#127	D₃
hypocenter	C₃	C₉×#486
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₉×#127

Complements

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

N	H ≅ G/N	type
2. C₉×#869	C₂	semidirect
3. C₉×#832	C₂	semidirect
4. C₄×C₉×C₃⋊C₈	C₂	semidirect
35. #5070	C₉	direct (N × H)
68. #869	C₂×C₉	semidirect
71. #832	C₂×C₉	semidirect
72. C₄×C₃⋊C₈	C₂×C₉	semidirect
152. C₃×C₉	#486	semidirect
225. C₉	#5070	direct (N × H)
257. C₃	C₉×#486	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#127	2
C₉×#869	2
C₉×#832	2
C₄×C₉×C₃⋊C₈	2
C₃×#5070	3
C₉×#486	3

Subgroups — C9×#5070 #13142969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5070 #13142969