Subgroups — C₉×#1853 #11766574

Order 1152.

540 conjugacy classes of subgroups.

The subgroup lattice has 291 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#106	C₂	1	2	yes	maximal, complemented
3	C₂×C₉×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₉×#380	C₂	1	1	yes	maximal, characteristic
5	C₉×#340	C₂	1	2	yes	maximal, complemented
6	C₃×#1853	C₃	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₉	—	2	1	no	—
8	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₉	C₂²	1	2	yes	—
10	C₂×C₉×C₄⋊C₄	—	2	1	no	—
11	C₂×C₉×C₄⋊C₄	—	2	4	no	—
12	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
16	C₉×#106	—	2	4	no	—
17	C₉×C₂²⋊C₈	C₄	1	4	yes	complemented
18	C₉×#95	—	2	1	no	—
19	C₂²×C₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×#106	C₂×C₃	1	2	yes	—
21	C₂×C₃×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
22	C₃×#380	C₂×C₃	1	1	yes	characteristic
23	C₃×#340	C₂×C₃	1	2	yes	—
24	C₂⁴×C₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₉	—	2	2	no	—
26	C₂²×C₄×C₉	—	2	2	no	—
27	C₂²×C₄×C₉	—	2	4	no	—
28	C₂²×C₄×C₉	—	2	2	no	—
29	C₂²×C₄×C₉	—	2	1	no	—
30	C₂²×C₄×C₉	—	2	1	no	—
31	C₂²×C₄×C₉	—	4	2	no	—
32	C₂²×C₄×C₉	—	4	2	no	—
33	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	Q₈	1	2	yes	—
36	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₉	—	2	2	no	—
38	C₂×C₈×C₉	—	2	2	no	—
39	C₂×C₈×C₉	C₂×C₄	1	4	yes	complemented
40	C₉×C₄⋊C₄	—	2	4	no	—
41	C₉×C₄⋊C₄	—	2	2	no	—
42	C₉×C₄⋊C₄	—	2	2	no	—
43	C₉×C₄⋊C₄	—	2	2	no	—
44	C₉×C₄⋊C₄	D₄	1	4	yes	—
45	C₉×C₂²⋊C₄	—	4	2	no	—
46	#1853	C₉	1	1	yes	characteristic, direct product factor, Sylow
47	C₂³×C₃×C₄	—	2	1	no	—
48	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
49	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
50	C₂×C₃×C₄⋊C₄	—	2	1	no	—
51	C₂×C₃×C₄⋊C₄	—	2	4	no	—
52	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
55	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
56	C₃×#106	—	2	4	no	—
57	C₃×C₂²⋊C₈	C₃×C₄	1	4	yes	—
58	C₃×#95	—	2	1	no	—
59	C₂³×C₉	—	2	1	no	—
60	C₂³×C₉	—	2	1	no	—
61	C₂³×C₉	—	2	1	no	—
62	C₂³×C₉	—	2	1	no	—
63	C₂³×C₉	C₄⋊C₄	1	2	yes	—
64	C₂³×C₉	SD₁₆	1	2	yes	—
65	C₂³×C₉	SD₁₆	1	2	yes	—
66	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₉	—	2	1	no	—
68	C₂×C₄×C₉	—	2	1	no	—
69	C₂×C₄×C₉	—	2	4	no	—
70	C₂×C₄×C₉	—	2	4	no	—
71	C₂×C₄×C₉	—	2	4	no	—
72	C₂×C₄×C₉	—	4	8	no	—
73	C₂×C₄×C₉	—	4	2	no	—
74	C₂×C₄×C₉	—	4	2	no	—
75	C₂×C₄×C₉	—	4	1	no	—
76	C₂×C₄×C₉	—	4	1	no	—
77	C₂×C₄×C₉	—	4	1	no	—
78	C₂×C₄×C₉	—	4	1	no	—
79	C₂×C₄×C₉	—	4	1	no	—
80	C₂×C₄×C₉	—	4	1	no	—
81	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₉	#41	1	2	yes	—
86	C₈×C₉	—	2	4	no	—
87	C₂²×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
88	C₂×#106	C₂×C₉	1	2	yes	complemented
89	C₂×C₂²⋊C₈	C₂×C₉	1	1	yes	characteristic
90	#380	C₂×C₉	1	1	yes	characteristic
91	#340	C₂×C₉	1	2	yes	complemented
92	C₂⁴×C₃	C₃×D₄	1	1	yes	characteristic
93	C₂²×C₃×C₄	—	2	2	no	—
94	C₂²×C₃×C₄	—	2	2	no	—
95	C₂²×C₃×C₄	—	2	4	no	—
96	C₂²×C₃×C₄	—	2	2	no	—
97	C₂²×C₃×C₄	—	2	1	no	—
98	C₂²×C₃×C₄	—	2	1	no	—
99	C₂²×C₃×C₄	—	4	2	no	—
100	C₂²×C₃×C₄	—	4	2	no	—
101	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
102	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
103	C₂²×C₃×C₄	C₃×Q₈	1	2	yes	—
104	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
105	C₂×C₃×C₈	—	2	2	no	—
106	C₂×C₃×C₈	—	2	2	no	—
107	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
108	C₃×C₄⋊C₄	—	2	4	no	—
109	C₃×C₄⋊C₄	—	2	2	no	—
110	C₃×C₄⋊C₄	—	2	2	no	—
111	C₃×C₄⋊C₄	—	2	2	no	—
112	C₃×C₄⋊C₄	C₃×D₄	1	4	yes	—
113	C₃×C₂²⋊C₄	—	4	2	no	—
114	C₂²×C₉	—	2	2	no	—
115	C₂²×C₉	—	2	2	no	—
116	C₂²×C₉	—	2	2	no	—
117	C₂²×C₉	—	2	2	no	—
118	C₂²×C₉	—	2	2	no	—
119	C₂²×C₉	—	2	2	no	—
120	C₂²×C₉	#106	1	4	yes	complemented
121	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₉	#120	1	1	yes	characteristic
123	C₂²×C₉	#123	1	1	yes	characteristic
124	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
125	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
126	C₂²×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂²×C₉	#137	1	1	yes	characteristic
128	C₄×C₉	—	2	2	no	—
129	C₄×C₉	—	4	4	no	—
130	C₄×C₉	—	8	1	no	—
131	C₄×C₉	—	8	1	no	—
132	C₄×C₉	C₄×D₄	1	2	yes	—
133	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
134	C₂³×C₄	—	2	1	no	—
135	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
136	C₂²×C₈	C₂²×C₉	1	2	yes	—
137	C₂×C₄⋊C₄	—	2	1	no	—
138	C₂×C₄⋊C₄	—	2	4	no	—
139	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
142	C₂×C₂²⋊C₄	—	2	1	no	—
143	#106	—	2	4	no	—
144	C₂²⋊C₈	C₄×C₉	1	4	yes	complemented
145	#95	—	2	1	no	—
146	C₂³×C₃	—	2	1	no	—
147	C₂³×C₃	—	2	1	no	—
148	C₂³×C₃	—	2	1	no	—
149	C₂³×C₃	—	2	1	no	—
150	C₂³×C₃	C₃×C₄⋊C₄	1	2	yes	—
151	C₂³×C₃	C₃×SD₁₆	1	2	yes	—
152	C₂³×C₃	C₃×SD₁₆	1	2	yes	—
153	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	2	4	no	—
157	C₂×C₃×C₄	—	2	4	no	—
158	C₂×C₃×C₄	—	2	4	no	—
159	C₂×C₃×C₄	—	4	8	no	—
160	C₂×C₃×C₄	—	4	2	no	—
161	C₂×C₃×C₄	—	4	2	no	—
162	C₂×C₃×C₄	—	4	1	no	—
163	C₂×C₃×C₄	—	4	1	no	—
164	C₂×C₃×C₄	—	4	1	no	—
165	C₂×C₃×C₄	—	4	1	no	—
166	C₂×C₃×C₄	—	4	1	no	—
167	C₂×C₃×C₄	—	4	1	no	—
168	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
169	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
171	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
172	C₂×C₃×C₄	C₃×#41	1	2	yes	—
173	C₃×C₈	—	2	4	no	—
174	C₂×C₉	—	2	4	no	—
175	C₂×C₉	#385	1	1	yes	characteristic
176	C₂×C₉	C₂×#106	1	1	yes	characteristic
177	C₂×C₉	#428	1	1	yes	characteristic
178	C₂×C₉	#448	1	1	yes	characteristic
179	C₂×C₉	#450	1	1	yes	characteristic
180	C₂×C₉	#481	1	1	yes	characteristic
181	C₂×C₉	#483	1	1	yes	characteristic
182	C₂⁴	D₄×C₉	1	1	yes	characteristic
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
192	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
193	C₂²×C₄	Q₈×C₉	1	2	yes	—
194	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	C₂×C₄×C₉	1	4	yes	complemented
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	D₄×C₉	1	4	yes	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	—	2	2	no	—
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	C₃×#106	1	4	yes	—
211	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
212	C₂²×C₃	C₃×#120	1	1	yes	characteristic
213	C₂²×C₃	C₃×#123	1	1	yes	characteristic
214	C₂²×C₃	C₂×C₃×SD₁₆	1	1	yes	characteristic
215	C₂²×C₃	C₂×C₃×SD₁₆	1	1	yes	characteristic
216	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₂²×C₃	C₃×#137	1	1	yes	characteristic
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	4	4	no	—
220	C₃×C₄	—	8	1	no	—
221	C₃×C₄	—	8	1	no	—
222	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
223	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
224	C₉	#1853	1	1	yes	characteristic, direct product factor, Sylow
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	C₉×C₄⋊C₄	1	2	yes	—
230	C₂³	C₉×SD₁₆	1	2	yes	—
231	C₂³	C₉×SD₁₆	1	2	yes	—
232	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	C₉×C₄⋊C₄	1	2	yes	—
248	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
249	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
250	C₂×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
251	C₂×C₄	C₉×#41	1	2	yes	—
252	C₈	—	2	4	no	—
253	C₂×C₃	—	2	4	no	—
254	C₂×C₃	C₃×#385	1	1	yes	characteristic
255	C₂×C₃	C₂×C₃×#106	1	1	yes	characteristic
256	C₂×C₃	C₃×#428	1	1	yes	characteristic
257	C₂×C₃	C₃×#448	1	1	yes	characteristic
258	C₂×C₃	C₃×#450	1	1	yes	characteristic
259	C₂×C₃	C₃×#481	1	1	yes	characteristic
260	C₂×C₃	C₃×#483	1	1	yes	characteristic
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	C₉×#106	1	4	yes	complemented
268	C₂²	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
269	C₂²	C₉×#120	1	1	yes	characteristic
270	C₂²	C₉×#123	1	1	yes	characteristic
271	C₂²	C₂×C₉×SD₁₆	1	1	yes	characteristic
272	C₂²	C₂×C₉×SD₁₆	1	1	yes	characteristic
273	C₂²	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
274	C₂²	C₉×#137	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	—	8	1	no	—
278	C₄	—	8	1	no	—
279	C₄	C₄×D₄×C₉	1	2	yes	—
280	C₄	C₉×C₂²⋊Q₈	1	2	yes	—
281	C₃	C₃×#1853	1	1	yes	characteristic
282	C₂	—	2	4	no	—
283	C₂	C₉×#385	1	1	yes	characteristic
284	C₂	C₂×C₉×#106	1	1	yes	characteristic
285	C₂	C₉×#428	1	1	yes	characteristic
286	C₂	C₉×#448	1	1	yes	characteristic
287	C₂	C₉×#450	1	1	yes	characteristic
288	C₂	C₉×#481	1	1	yes	characteristic
289	C₂	C₉×#483	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂³×C₃	C₂×C₃×D₄
prime radical	C₂⁴×C₃	C₃×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂³×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×#106	C₂	semidirect
5. C₉×#340	C₂	semidirect
17. C₉×C₂²⋊C₈	C₄	semidirect
39. C₂×C₈×C₉	C₂×C₄	semidirect
46. #1853	C₉	direct (N × H)
88. C₂×#106	C₂×C₉	semidirect
91. #340	C₂×C₉	semidirect
120. C₂²×C₉	#106	semidirect
144. C₂²⋊C₈	C₄×C₉	semidirect
197. C₂×C₈	C₂×C₄×C₉	semidirect
224. C₉	#1853	direct (N × H)
267. C₂²	C₉×#106	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×C₄⋊C₄	2
C₂×C₉×#106	2
C₂×C₉×C₂²⋊C₈	2
C₉×#380	2
C₉×#340	2
C₃×#1853	3

Subgroups — C9×#1853 #11766574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1853 #11766574