Subgroups — C₉×#2317 #11767038

Order 1152.

900 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#117	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₉×#389	C₂	1	4	yes	maximal, complemented
5	C₉×#385	C₂	1	4	yes	maximal
6	C₉×#384	C₂	1	2	yes	maximal, complemented
7	C₉×#382	C₂	1	2	yes	maximal, complemented
8	C₃×#2317	C₃	1	1	yes	maximal, characteristic
9	C₂³×C₄×C₉	C₂²	1	2	yes	—
10	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₉×#117	—	2	4	no	—
12	C₉×#117	C₄	1	8	yes	complemented
13	C₂×C₉×C₄⋊C₄	—	2	4	no	—
14	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₉×C₄⋊C₄	C₂²	1	8	yes	—
19	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₄²×C₉	C₂²	1	2	yes	—
23	C₂×C₄²×C₉	C₂²	1	2	yes	—
24	C₉×#95	C₂²	1	8	yes	—
25	C₂×C₃×#117	C₂×C₃	1	1	yes	characteristic
26	C₂×C₃×#117	C₂×C₃	1	1	yes	characteristic
27	C₂²×C₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
28	C₃×#389	C₂×C₃	1	4	yes	—
29	C₃×#385	C₂×C₃	1	4	yes	—
30	C₃×#384	C₂×C₃	1	2	yes	—
31	C₃×#382	C₂×C₃	1	2	yes	—
32	C₂⁴×C₉	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₉	—	2	4	no	—
34	C₂²×C₄×C₉	—	4	4	no	—
35	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
36	C₂²×C₄×C₉	D₄	1	4	yes	—
37	C₂²×C₄×C₉	Q₈	1	4	yes	—
38	C₂²×C₄×C₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₉	C₂³	1	4	yes	—
41	C₂²×C₄×C₉	C₂³	1	4	yes	—
42	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
44	C₉×C₄⋊C₄	—	2	4	no	—
45	C₉×C₄⋊C₄	—	2	4	no	—
46	C₉×C₄⋊C₄	—	2	2	no	—
47	C₉×C₄⋊C₄	—	2	2	no	—
48	C₉×C₄⋊C₄	—	4	8	no	—
49	C₉×C₄⋊C₄	C₂×C₄	1	8	yes	complemented
50	C₉×C₂²⋊C₄	—	2	4	no	—
51	C₉×C₂²⋊C₄	C₂×C₄	1	8	yes	—
52	C₄²×C₉	—	2	4	no	—
53	C₄²×C₉	C₂×C₄	1	8	yes	complemented
54	#2317	C₉	1	1	yes	characteristic, direct product factor, Sylow
55	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
56	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
57	C₃×#117	—	2	4	no	—
58	C₃×#117	C₃×C₄	1	8	yes	—
59	C₂×C₃×C₄⋊C₄	—	2	4	no	—
60	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
62	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
64	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	—
65	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
66	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
67	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
68	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
69	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
70	C₃×#95	C₂²×C₃	1	8	yes	—
71	C₂³×C₉	—	2	2	no	—
72	C₂³×C₉	—	2	2	no	—
73	C₂³×C₉	C₂²×C₄	1	2	yes	—
74	C₂³×C₉	C₂×D₄	1	2	yes	—
75	C₂³×C₉	C₂×Q₈	1	2	yes	—
76	C₂³×C₉	C₂⁴	1	1	yes	characteristic
77	C₂×C₄×C₉	—	2	4	no	—
78	C₂×C₄×C₉	—	2	4	no	—
79	C₂×C₄×C₉	—	2	2	no	—
80	C₂×C₄×C₉	—	2	2	no	—
81	C₂×C₄×C₉	—	2	4	no	—
82	C₂×C₄×C₉	—	2	4	no	—
83	C₂×C₄×C₉	—	2	8	no	—
84	C₂×C₄×C₉	—	2	4	no	—
85	C₂×C₄×C₉	—	4	8	no	—
86	C₂×C₄×C₉	C₄⋊C₄	1	16	yes	complemented
87	C₂×C₄×C₉	C₂²×C₄	1	8	yes	—
88	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
93	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
94	C₂×#117	C₂×C₉	1	1	yes	characteristic
95	C₂×#117	C₂×C₉	1	1	yes	characteristic
96	C₂²×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
97	#389	C₂×C₉	1	4	yes	complemented
98	#385	C₂×C₉	1	4	yes	—
99	#384	C₂×C₉	1	2	yes	complemented
100	#382	C₂×C₉	1	2	yes	complemented
101	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
102	C₂²×C₃×C₄	—	2	4	no	—
103	C₂²×C₃×C₄	—	4	4	no	—
104	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
105	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
106	C₂²×C₃×C₄	C₃×Q₈	1	4	yes	—
107	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
108	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
109	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
110	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
111	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
113	C₃×C₄⋊C₄	—	2	4	no	—
114	C₃×C₄⋊C₄	—	2	4	no	—
115	C₃×C₄⋊C₄	—	2	2	no	—
116	C₃×C₄⋊C₄	—	2	2	no	—
117	C₃×C₄⋊C₄	—	4	8	no	—
118	C₃×C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
119	C₃×C₂²⋊C₄	—	2	4	no	—
120	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	—
121	C₃×C₄²	—	2	4	no	—
122	C₃×C₄²	C₂×C₃×C₄	1	8	yes	—
123	C₂²×C₉	—	2	4	no	—
124	C₂²×C₉	—	2	2	no	—
125	C₂²×C₉	—	2	2	no	—
126	C₂²×C₉	—	2	4	no	—
127	C₂²×C₉	C₂×C₄⋊C₄	1	4	yes	—
128	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
129	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
130	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
131	C₂²×C₉	H(5,𝔽₂)	1	2	yes	—
132	C₂²×C₉	2₋¹⁺⁴	1	2	yes	—
133	C₄×C₉	—	2	8	no	—
134	C₄×C₉	—	4	2	no	—
135	C₄×C₉	—	4	4	no	—
136	C₄×C₉	—	4	2	no	—
137	C₄×C₉	C₂×C₄⋊C₄	1	4	yes	—
138	C₄×C₉	C₂×C₄⋊C₄	1	4	yes	—
139	C₂³×C₄	C₂²×C₉	1	2	yes	—
140	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
141	#117	—	2	4	no	—
142	#117	C₄×C₉	1	8	yes	complemented
143	C₂×C₄⋊C₄	—	2	4	no	—
144	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
145	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
146	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂²×C₉	1	8	yes	—
149	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
150	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
151	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
152	C₂×C₄²	C₂²×C₉	1	2	yes	—
153	C₂×C₄²	C₂²×C₉	1	2	yes	—
154	#95	C₂²×C₉	1	8	yes	—
155	C₂³×C₃	—	2	2	no	—
156	C₂³×C₃	—	2	2	no	—
157	C₂³×C₃	C₂²×C₃×C₄	1	2	yes	—
158	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
159	C₂³×C₃	C₂×C₃×Q₈	1	2	yes	—
160	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
161	C₂×C₃×C₄	—	2	4	no	—
162	C₂×C₃×C₄	—	2	4	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	2	4	no	—
167	C₂×C₃×C₄	—	2	8	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	4	8	no	—
170	C₂×C₃×C₄	C₃×C₄⋊C₄	1	16	yes	—
171	C₂×C₃×C₄	C₂²×C₃×C₄	1	8	yes	—
172	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
173	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
174	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
175	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
176	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
177	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
178	C₂×C₉	—	2	4	no	—
179	C₂×C₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₉	#520	1	2	yes	—
181	C₂×C₉	#535	1	1	yes	characteristic
182	C₂×C₉	#536	1	1	yes	characteristic
183	C₂×C₉	#544	1	2	yes	—
184	C₂⁴	C₂³×C₉	1	1	yes	characteristic
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	C₂×C₄×C₉	1	4	yes	—
188	C₂²×C₄	D₄×C₉	1	4	yes	—
189	C₂²×C₄	Q₈×C₉	1	4	yes	—
190	C₂²×C₄	C₂³×C₉	1	2	yes	—
191	C₂²×C₄	C₂³×C₉	1	4	yes	—
192	C₂²×C₄	C₂³×C₉	1	2	yes	—
193	C₂²×C₄	C₂³×C₉	1	4	yes	—
194	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	4	8	no	—
201	C₄⋊C₄	C₂×C₄×C₉	1	8	yes	complemented
202	C₂²⋊C₄	—	2	4	no	—
203	C₂²⋊C₄	C₂×C₄×C₉	1	8	yes	—
204	C₄²	—	2	4	no	—
205	C₄²	C₂×C₄×C₉	1	8	yes	complemented
206	C₂²×C₃	—	2	4	no	—
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	4	no	—
210	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	4	yes	—
211	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
212	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
213	C₂²×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
214	C₂²×C₃	C₃×H(5,𝔽₂)	1	2	yes	—
215	C₂²×C₃	C₃×2₋¹⁺⁴	1	2	yes	—
216	C₃×C₄	—	2	8	no	—
217	C₃×C₄	—	4	4	no	—
218	C₃×C₄	—	4	2	no	—
219	C₃×C₄	—	4	2	no	—
220	C₃×C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
221	C₃×C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
222	C₉	#2317	1	1	yes	characteristic, direct product factor, Sylow
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	C₂²×C₄×C₉	1	2	yes	—
226	C₂³	C₂×D₄×C₉	1	2	yes	—
227	C₂³	C₂×Q₈×C₉	1	2	yes	—
228	C₂³	C₂⁴×C₉	1	1	yes	characteristic
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	C₉×C₄⋊C₄	1	16	yes	complemented
239	C₂×C₄	C₂²×C₄×C₉	1	8	yes	—
240	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
241	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
242	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
243	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
244	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
245	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
246	C₂×C₃	—	2	4	no	—
247	C₂×C₃	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
248	C₂×C₃	C₃×#520	1	2	yes	—
249	C₂×C₃	C₃×#535	1	1	yes	characteristic
250	C₂×C₃	C₃×#536	1	1	yes	characteristic
251	C₂×C₃	C₃×#544	1	2	yes	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	4	no	—
256	C₂²	C₂×C₉×C₄⋊C₄	1	4	yes	—
257	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
258	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
259	C₂²	C₂²×Q₈×C₉	1	1	yes	characteristic
260	C₂²	C₉×H(5,𝔽₂)	1	2	yes	—
261	C₂²	C₉×2₋¹⁺⁴	1	2	yes	—
262	C₄	—	2	8	no	—
263	C₄	—	4	4	no	—
264	C₄	—	4	2	no	—
265	C₄	—	4	2	no	—
266	C₄	C₂×C₉×C₄⋊C₄	1	4	yes	—
267	C₄	C₂×C₉×C₄⋊C₄	1	4	yes	—
268	C₃	C₃×#2317	1	1	yes	characteristic
269	C₂	—	2	4	no	—
270	C₂	C₂²×C₉×C₄⋊C₄	1	1	yes	characteristic
271	C₂	C₉×#520	1	2	yes	—
272	C₂	C₉×#535	1	1	yes	characteristic
273	C₂	C₉×#536	1	1	yes	characteristic
274	C₂	C₉×#544	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₉
Frattini, socle, cosocle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂³×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₉×#389	C₂	semidirect
6. C₉×#384	C₂	semidirect
7. C₉×#382	C₂	semidirect
12. C₉×#117	C₄	semidirect
49. C₉×C₄⋊C₄	C₂×C₄	semidirect
53. C₄²×C₉	C₂×C₄	semidirect
54. #2317	C₉	direct (N × H)
86. C₂×C₄×C₉	C₄⋊C₄	semidirect
97. #389	C₂×C₉	semidirect
99. #384	C₂×C₉	semidirect
100. #382	C₂×C₉	semidirect
142. #117	C₄×C₉	semidirect
201. C₄⋊C₄	C₂×C₄×C₉	semidirect
205. C₄²	C₂×C₄×C₉	semidirect
222. C₉	#2317	direct (N × H)
238. C₂×C₄	C₉×C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#117	2
C₂²×C₉×C₄⋊C₄	2
C₉×#389	2
C₉×#385	2
C₉×#384	2
C₉×#382	2
C₃×#2317	3

Subgroups — C9×#2317 #11767038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2317 #11767038