Subgroups — C₉×#1850 #11766571

Order 1152.

522 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₉×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄×C₉×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₉×#336	C₂	1	2	yes	maximal, complemented
6	C₃×#1850	C₃	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₉	—	2	2	no	—
9	C₂²×C₈×C₉	—	2	4	no	—
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₉	C₂²	1	2	yes	—
13	C₂×C₉×C₂²⋊C₄	C₄	1	1	yes	characteristic
14	C₂×C₉×C₂²⋊C₄	C₄	1	1	yes	characteristic
15	C₂×C₄²×C₉	C₂²	1	2	yes	—
16	C₉×C₄⋊C₈	—	2	4	no	—
17	C₉×C₂²⋊C₈	—	2	2	no	—
18	C₉×#95	C₄	1	2	yes	—
19	C₂³×C₃×C₈	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×C₄⋊C₈	C₂×C₃	1	2	yes	—
21	C₂×C₃×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
22	C₃×C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
23	C₃×#336	C₂×C₃	1	2	yes	—
24	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₉	—	2	2	no	—
26	C₂²×C₄×C₉	—	2	2	no	—
27	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₉	D₄	1	2	yes	—
31	C₂²×C₄×C₉	Q₈	1	2	yes	—
32	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₉	—	2	8	no	—
34	C₂×C₈×C₉	—	2	2	no	—
35	C₂×C₈×C₉	—	2	2	no	—
36	C₂×C₈×C₉	—	2	1	no	—
37	C₂×C₈×C₉	—	2	1	no	—
38	C₂×C₈×C₉	—	2	2	no	—
39	C₂×C₈×C₉	—	2	2	no	—
40	C₂×C₈×C₉	D₄	1	4	yes	—
41	C₉×C₂²⋊C₄	—	2	2	no	—
42	C₉×C₂²⋊C₄	C₈	1	4	yes	complemented
43	C₄²×C₉	—	2	2	no	—
44	C₄²×C₉	—	2	2	no	—
45	#1850	C₉	1	1	yes	characteristic, direct product factor, Sylow
46	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
47	C₂²×C₃×C₈	—	2	2	no	—
48	C₂²×C₃×C₈	—	2	4	no	—
49	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
50	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
51	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
52	C₂×C₃×C₂²⋊C₄	C₃×C₄	1	1	yes	characteristic
53	C₂×C₃×C₂²⋊C₄	C₃×C₄	1	1	yes	characteristic
54	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
55	C₃×C₄⋊C₈	—	2	4	no	—
56	C₃×C₂²⋊C₈	—	2	2	no	—
57	C₃×#95	C₃×C₄	1	2	yes	—
58	C₂³×C₉	—	2	2	no	—
59	C₂³×C₉	—	2	2	no	—
60	C₂³×C₉	C₄⋊C₄	1	2	yes	—
61	C₂³×C₉	C₂×C₈	1	2	yes	—
62	C₂³×C₉	M₁₆	1	2	yes	—
63	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₉	—	2	2	no	—
65	C₂×C₄×C₉	—	2	2	no	—
66	C₂×C₄×C₉	—	2	2	no	—
67	C₂×C₄×C₉	—	2	2	no	—
68	C₂×C₄×C₉	—	2	4	no	—
69	C₂×C₄×C₉	—	2	2	no	—
70	C₂×C₄×C₉	—	2	2	no	—
71	C₂×C₄×C₉	—	2	2	no	—
72	C₂×C₄×C₉	—	2	2	no	—
73	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₉	C₂×C₈	1	4	yes	complemented
75	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₉	#41	1	2	yes	—
79	C₈×C₉	—	2	4	no	—
80	C₈×C₉	—	4	1	no	—
81	C₈×C₉	—	4	1	no	—
82	C₂³×C₈	C₂×C₉	1	1	yes	characteristic
83	C₂×C₄⋊C₈	C₂×C₉	1	2	yes	complemented
84	C₂×C₂²⋊C₈	C₂×C₉	1	1	yes	characteristic
85	C₄×C₂²⋊C₄	C₂×C₉	1	1	yes	characteristic
86	#336	C₂×C₉	1	2	yes	complemented
87	C₂⁴×C₃	C₂×C₃×C₄	1	1	yes	characteristic
88	C₂²×C₃×C₄	—	2	2	no	—
89	C₂²×C₃×C₄	—	2	2	no	—
90	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
91	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
93	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
94	C₂²×C₃×C₄	C₃×Q₈	1	2	yes	—
95	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
96	C₂×C₃×C₈	—	2	8	no	—
97	C₂×C₃×C₈	—	2	2	no	—
98	C₂×C₃×C₈	—	2	2	no	—
99	C₂×C₃×C₈	—	2	1	no	—
100	C₂×C₃×C₈	—	2	1	no	—
101	C₂×C₃×C₈	—	2	2	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
104	C₃×C₂²⋊C₄	—	2	2	no	—
105	C₃×C₂²⋊C₄	C₃×C₈	1	4	yes	—
106	C₃×C₄²	—	2	2	no	—
107	C₃×C₄²	—	2	2	no	—
108	C₂²×C₉	—	2	2	no	—
109	C₂²×C₉	—	2	2	no	—
110	C₂²×C₉	—	2	4	no	—
111	C₂²×C₉	—	2	4	no	—
112	C₂²×C₉	C₄⋊C₈	1	4	yes	complemented
113	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₉	C₄×D₄	1	2	yes	—
115	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
116	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
117	C₂²×C₉	#131	1	2	yes	—
118	C₄×C₉	—	2	2	no	—
119	C₄×C₉	—	2	4	no	—
120	C₄×C₉	—	4	2	no	—
121	C₄×C₉	#120	1	1	yes	characteristic
122	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
123	C₄×C₉	#123	1	1	yes	characteristic
124	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
128	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
129	C₂²×C₈	C₂²×C₉	1	2	yes	—
130	C₂×C₂²⋊C₄	C₄×C₉	1	1	yes	characteristic
131	C₂×C₂²⋊C₄	C₄×C₉	1	1	yes	characteristic
132	C₂×C₄²	C₂²×C₉	1	2	yes	—
133	C₄⋊C₈	—	2	4	no	—
134	C₂²⋊C₈	—	2	2	no	—
135	#95	C₄×C₉	1	2	yes	—
136	C₂³×C₃	—	2	2	no	—
137	C₂³×C₃	—	2	2	no	—
138	C₂³×C₃	C₃×C₄⋊C₄	1	2	yes	—
139	C₂³×C₃	C₂×C₃×C₈	1	2	yes	—
140	C₂³×C₃	C₃×M₁₆	1	2	yes	—
141	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	4	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
152	C₂×C₃×C₄	C₂×C₃×C₈	1	4	yes	—
153	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
154	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
156	C₂×C₃×C₄	C₃×#41	1	2	yes	—
157	C₃×C₈	—	2	4	no	—
158	C₃×C₈	—	4	1	no	—
159	C₃×C₈	—	4	1	no	—
160	C₂×C₉	—	2	4	no	—
161	C₂×C₉	#385	1	1	yes	characteristic
162	C₂×C₉	C₂×C₄⋊C₈	1	1	yes	characteristic
163	C₂×C₉	#424	1	1	yes	characteristic
164	C₂×C₉	C₈×D₄	1	2	yes	—
165	C₂×C₉	#435	1	2	yes	—
166	C₂⁴	C₂×C₄×C₉	1	1	yes	characteristic
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
170	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
172	C₂²×C₄	D₄×C₉	1	2	yes	—
173	C₂²×C₄	Q₈×C₉	1	2	yes	—
174	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
175	C₂×C₈	—	2	8	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	D₄×C₉	1	4	yes	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	C₈×C₉	1	4	yes	complemented
185	C₄²	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	2	4	no	—
190	C₂²×C₃	—	2	4	no	—
191	C₂²×C₃	C₃×C₄⋊C₈	1	4	yes	—
192	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
193	C₂²×C₃	C₃×C₄×D₄	1	2	yes	—
194	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
195	C₂²×C₃	C₂×C₃×M₁₆	1	1	yes	characteristic
196	C₂²×C₃	C₃×#131	1	2	yes	—
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	2	4	no	—
199	C₃×C₄	—	4	2	no	—
200	C₃×C₄	C₃×#120	1	1	yes	characteristic
201	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
202	C₃×C₄	C₃×#123	1	1	yes	characteristic
203	C₉	#1850	1	1	yes	characteristic, direct product factor, Sylow
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	C₉×C₄⋊C₄	1	2	yes	—
207	C₂³	C₂×C₈×C₉	1	2	yes	—
208	C₂³	C₉×M₁₆	1	2	yes	—
209	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	C₉×C₄⋊C₄	1	2	yes	—
220	C₂×C₄	C₂×C₈×C₉	1	4	yes	complemented
221	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
222	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
223	C₂×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
224	C₂×C₄	C₉×#41	1	2	yes	—
225	C₈	—	2	4	no	—
226	C₈	—	4	1	no	—
227	C₈	—	4	1	no	—
228	C₂×C₃	—	2	4	no	—
229	C₂×C₃	C₃×#385	1	1	yes	characteristic
230	C₂×C₃	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
231	C₂×C₃	C₃×#424	1	1	yes	characteristic
232	C₂×C₃	C₃×C₈×D₄	1	2	yes	—
233	C₂×C₃	C₃×#435	1	2	yes	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	4	no	—
238	C₂²	C₉×C₄⋊C₈	1	4	yes	complemented
239	C₂²	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
240	C₂²	C₄×D₄×C₉	1	2	yes	—
241	C₂²	C₂²×C₈×C₉	1	1	yes	characteristic
242	C₂²	C₂×C₉×M₁₆	1	1	yes	characteristic
243	C₂²	C₉×#131	1	2	yes	—
244	C₄	—	2	2	no	—
245	C₄	—	2	4	no	—
246	C₄	—	4	2	no	—
247	C₄	C₉×#120	1	1	yes	characteristic
248	C₄	C₉×C₂²⋊Q₈	1	2	yes	—
249	C₄	C₉×#123	1	1	yes	characteristic
250	C₃	C₃×#1850	1	1	yes	characteristic
251	C₂	—	2	4	no	—
252	C₂	C₉×#385	1	1	yes	characteristic
253	C₂	C₂×C₉×C₄⋊C₈	1	1	yes	characteristic
254	C₂	C₉×#424	1	1	yes	characteristic
255	C₂	C₈×D₄×C₉	1	2	yes	—
256	C₂	C₉×#435	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₉	C₂³
commutator	C₂²	C₂²×C₈×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂³×C₃	C₂²×C₃×C₄
prime radical	C₂⁴×C₃	C₂×C₃×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×C₄⋊C₈	C₂	semidirect
5. C₉×#336	C₂	semidirect
42. C₉×C₂²⋊C₄	C₈	semidirect
45. #1850	C₉	direct (N × H)
74. C₂×C₄×C₉	C₂×C₈	semidirect
83. C₂×C₄⋊C₈	C₂×C₉	semidirect
86. #336	C₂×C₉	semidirect
112. C₂²×C₉	C₄⋊C₈	semidirect
184. C₂²⋊C₄	C₈×C₉	semidirect
203. C₉	#1850	direct (N × H)
220. C₂×C₄	C₂×C₈×C₉	semidirect
238. C₂²	C₉×C₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₈×C₉	2
C₂×C₉×C₄⋊C₈	2
C₂×C₉×C₂²⋊C₈	2
C₄×C₉×C₂²⋊C₄	2
C₉×#336	2
C₃×#1850	3

Subgroups — C9×#1850 #11766571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1850 #11766571