Subgroups — C₈₉×#9353 #36387201

Order 22784.

472 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3039	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2990	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1997	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#1548	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1489	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#1309	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1299	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#137×C₈₉	C₄	1	2	yes	—
9	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
11	#462×C₈₉	—	2	1	no	—
12	#462×C₈₉	—	2	1	no	—
13	#461×C₈₉	—	2	1	no	—
14	#461×C₈₉	—	2	1	no	—
15	#446×C₈₉	—	2	1	no	—
16	#435×C₈₉	—	2	2	no	—
17	C₈×D₄×C₈₉	—	2	1	no	—
18	#432×C₈₉	—	2	1	no	—
19	#431×C₈₉	C₂²	1	1	yes	characteristic
20	#423×C₈₉	C₂²	1	1	yes	characteristic
21	#423×C₈₉	C₂²	1	1	yes	characteristic
22	#423×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
24	#419×C₈₉	C₄	1	2	yes	—
25	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
26	#343×C₈₉	—	2	2	no	—
27	#333×C₈₉	—	2	1	no	—
28	#333×C₈₉	—	2	1	no	—
29	Q₈⋊C₈×C₈₉	—	2	1	no	—
30	D₄⋊C₈×C₈₉	—	2	1	no	—
31	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×Q₈×C₈₉	C₂×C₄	1	2	yes	—
33	#137×C₈₉	—	4	4	no	—
34	C₂×Q₁₆×C₈₉	—	2	2	no	—
35	C₂×SD₁₆×C₈₉	—	2	2	no	—
36	C₂×M₁₆×C₈₉	—	2	1	no	—
37	C₂×M₁₆×C₈₉	—	2	1	no	—
38	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
39	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
40	C₂²×C₈×C₈₉	—	2	2	no	—
41	C₄⋊Q₈×C₈₉	—	2	1	no	—
42	C₄⋊Q₈×C₈₉	—	2	1	no	—
43	C₄⋊Q₈×C₈₉	—	2	1	no	—
44	C₄⋊Q₈×C₈₉	—	2	1	no	—
45	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
46	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
47	C₄×D₄×C₈₉	—	2	1	no	—
48	C₄×D₄×C₈₉	—	2	1	no	—
49	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
50	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
51	#117×C₈₉	—	2	1	no	—
52	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
55	C₂×C₄²×C₈₉	—	2	1	no	—
56	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₈₉	—	2	2	no	—
58	C₄⋊C₈×C₈₉	—	2	1	no	—
59	C₄⋊C₈×C₈₉	—	2	1	no	—
60	C₄⋊C₈×C₈₉	—	2	1	no	—
61	C₄⋊C₈×C₈₉	—	2	1	no	—
62	C₄⋊C₈×C₈₉	—	2	1	no	—
63	C₄⋊C₈×C₈₉	—	2	1	no	—
64	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
65	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
66	#103×C₈₉	—	2	2	no	—
67	#103×C₈₉	—	2	2	no	—
68	C₂²⋊C₈×C₈₉	—	2	2	no	—
69	C₂²⋊C₈×C₈₉	—	2	1	no	—
70	#97×C₈₉	—	2	2	no	—
71	#97×C₈₉	—	2	1	no	—
72	C₄×C₈×C₈₉	—	2	1	no	—
73	C₄×C₈×C₈₉	—	2	1	no	—
74	#41×C₈₉	—	2	2	no	—
75	#41×C₈₉	—	4	1	no	—
76	C₂×Q₈×C₈₉	—	2	2	no	—
77	C₂×Q₈×C₈₉	—	2	2	no	—
78	C₂×Q₈×C₈₉	—	4	2	no	—
79	C₂×Q₈×C₈₉	—	4	2	no	—
80	C₂×Q₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×D₄×C₈₉	—	2	1	no	—
82	C₂×D₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	—	2	1	no	—
84	C₂²×C₄×C₈₉	—	2	1	no	—
85	C₂²×C₄×C₈₉	—	2	1	no	—
86	C₂²×C₄×C₈₉	—	2	2	no	—
87	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
90	Q₁₆×C₈₉	—	4	4	no	—
91	SD₁₆×C₈₉	—	4	4	no	—
92	M₁₆×C₈₉	—	2	2	no	—
93	M₁₆×C₈₉	—	4	1	no	—
94	M₁₆×C₈₉	—	4	2	no	—
95	M₁₆×C₈₉	—	4	2	no	—
96	M₁₆×C₈₉	—	8	1	no	—
97	C₂×C₈×C₈₉	—	2	2	no	—
98	C₂×C₈×C₈₉	—	2	1	no	—
99	C₂×C₈×C₈₉	—	2	1	no	—
100	C₂×C₈×C₈₉	—	2	2	no	—
101	C₂×C₈×C₈₉	—	4	1	no	—
102	C₂×C₈×C₈₉	—	4	1	no	—
103	C₂×C₈×C₈₉	—	8	2	no	—
104	C₄⋊C₄×C₈₉	—	2	1	no	—
105	C₄⋊C₄×C₈₉	—	2	1	no	—
106	C₄⋊C₄×C₈₉	—	4	1	no	—
107	C₄⋊C₄×C₈₉	—	4	1	no	—
108	C₄⋊C₄×C₈₉	—	4	1	no	—
109	C₄⋊C₄×C₈₉	—	4	1	no	—
110	C₄⋊C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
111	C₄⋊C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
112	C₂²⋊C₄×C₈₉	—	2	1	no	—
113	C₂²⋊C₄×C₈₉	—	2	1	no	—
114	C₂²⋊C₄×C₈₉	—	2	1	no	—
115	C₄²×C₈₉	—	2	1	no	—
116	C₄²×C₈₉	—	2	1	no	—
117	C₄²×C₈₉	—	2	1	no	—
118	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₈₉	#41	1	1	yes	characteristic
122	C₂³×C₈₉	—	2	1	no	—
123	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
124	Q₈×C₈₉	—	2	1	no	—
125	Q₈×C₈₉	—	2	1	no	—
126	Q₈×C₈₉	—	4	2	no	—
127	Q₈×C₈₉	—	4	2	no	—
128	Q₈×C₈₉	—	8	2	no	—
129	D₄×C₈₉	—	2	1	no	—
130	D₄×C₈₉	—	4	2	no	—
131	D₄×C₈₉	#100	1	2	yes	—
132	C₂×C₄×C₈₉	—	2	1	no	—
133	C₂×C₄×C₈₉	—	2	1	no	—
134	C₂×C₄×C₈₉	—	2	1	no	—
135	C₂×C₄×C₈₉	—	2	1	no	—
136	C₂×C₄×C₈₉	—	2	1	no	—
137	C₂×C₄×C₈₉	—	2	1	no	—
138	C₂×C₄×C₈₉	—	2	1	no	—
139	C₂×C₄×C₈₉	—	2	1	no	—
140	C₂×C₄×C₈₉	—	2	1	no	—
141	C₂×C₄×C₈₉	—	4	2	no	—
142	C₂×C₄×C₈₉	—	4	1	no	—
143	C₂×C₄×C₈₉	—	4	1	no	—
144	C₂×C₄×C₈₉	—	4	1	no	—
145	C₂×C₄×C₈₉	—	4	1	no	—
146	C₂×C₄×C₈₉	—	4	2	no	—
147	C₂×C₄×C₈₉	—	8	2	no	—
148	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
151	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
152	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
153	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
154	C₈×C₈₉	—	4	2	no	—
155	C₈×C₈₉	—	4	2	no	—
156	C₈×C₈₉	—	4	1	no	—
157	C₈×C₈₉	—	4	2	no	—
158	C₈×C₈₉	—	4	1	no	—
159	C₈×C₈₉	—	8	1	no	—
160	C₂²×C₈₉	—	2	1	no	—
161	C₂²×C₈₉	—	2	1	no	—
162	C₂²×C₈₉	—	2	2	no	—
163	C₂²×C₈₉	—	2	1	no	—
164	C₂²×C₈₉	—	4	1	no	—
165	C₂²×C₈₉	—	4	1	no	—
166	C₂²×C₈₉	#386	1	1	yes	characteristic
167	C₄×C₈₉	—	2	1	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	2	1	no	—
170	C₄×C₈₉	—	4	1	no	—
171	C₄×C₈₉	—	4	1	no	—
172	C₄×C₈₉	—	4	1	no	—
173	C₄×C₈₉	—	8	2	no	—
174	C₄×C₈₉	C₂×#100	1	1	yes	characteristic
175	C₄×C₈₉	#413	1	1	yes	characteristic
176	C₄×C₈₉	#455	1	1	yes	characteristic
177	C₄×C₈₉	#456	1	1	yes	characteristic
178	C₄×C₈₉	#471	1	1	yes	characteristic
179	C₄×C₈₉	#473	1	1	yes	characteristic
180	#9353	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
181	C₂×C₈₉	—	4	1	no	—
182	C₂×C₈₉	—	4	2	no	—
183	C₂×C₈₉	#1882	1	1	yes	characteristic
184	C₂×C₈₉	#1900	1	1	yes	characteristic
185	C₂×C₈₉	#1908	1	1	yes	characteristic
186	#3039	C₂×C₈₉	1	1	yes	characteristic
187	#2990	C₂×C₈₉	1	1	yes	characteristic
188	#1997	C₂×C₈₉	1	1	yes	characteristic, complemented
189	#1548	C₂×C₈₉	1	1	yes	characteristic, complemented
190	#1489	C₂×C₈₉	1	1	yes	characteristic
191	#1309	C₂×C₈₉	1	1	yes	characteristic, complemented
192	#1299	C₂×C₈₉	1	1	yes	characteristic, complemented
193	C₈₉	#9353	1	1	yes	characteristic, direct product factor, Sylow
194	C₂×#137	C₄×C₈₉	1	2	yes	—
195	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
196	C₄×#41	C₂²×C₈₉	1	1	yes	characteristic
197	#462	—	2	1	no	—
198	#462	—	2	1	no	—
199	#461	—	2	1	no	—
200	#461	—	2	1	no	—
201	#446	—	2	1	no	—
202	#435	—	2	2	no	—
203	C₈×D₄	—	2	1	no	—
204	#432	—	2	1	no	—
205	#431	C₂²×C₈₉	1	1	yes	characteristic
206	#423	C₂²×C₈₉	1	1	yes	characteristic
207	#423	C₂²×C₈₉	1	1	yes	characteristic
208	#423	C₂²×C₈₉	1	1	yes	characteristic
209	C₂×C₄⋊C₈	—	2	1	no	—
210	#419	C₄×C₈₉	1	2	yes	—
211	C₄×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
212	#343	—	2	2	no	—
213	#333	—	2	1	no	—
214	#333	—	2	1	no	—
215	Q₈⋊C₈	—	2	1	no	—
216	D₄⋊C₈	—	2	1	no	—
217	C₂×#41	C₂×C₄×C₈₉	1	1	yes	characteristic
218	C₂²×Q₈	C₂×C₄×C₈₉	1	2	yes	—
219	#137	—	4	4	no	—
220	C₂×Q₁₆	—	2	2	no	—
221	C₂×SD₁₆	—	2	2	no	—
222	C₂×M₁₆	—	2	1	no	—
223	C₂×M₁₆	—	2	1	no	—
224	C₂×M₁₆	C₂×C₄×C₈₉	1	2	yes	—
225	C₂×M₁₆	D₄×C₈₉	1	2	yes	—
226	C₂²×C₈	—	2	2	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
232	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
233	C₄×D₄	—	2	1	no	—
234	C₄×D₄	—	2	1	no	—
235	C₄×D₄	D₄×C₈₉	1	1	yes	characteristic
236	C₄×D₄	D₄×C₈₉	1	1	yes	characteristic
237	#117	—	2	1	no	—
238	#117	C₂×C₄×C₈₉	1	1	yes	characteristic
239	C₂×C₄⋊C₄	—	2	1	no	—
240	C₂×C₄⋊C₄	—	2	1	no	—
241	C₂×C₄²	—	2	1	no	—
242	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
243	C₄⋊C₈	—	2	2	no	—
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	—	2	1	no	—
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	D₄×C₈₉	1	1	yes	characteristic
251	C₄⋊C₈	D₄×C₈₉	1	1	yes	characteristic
252	#103	—	2	2	no	—
253	#103	—	2	2	no	—
254	C₂²⋊C₈	—	2	2	no	—
255	C₂²⋊C₈	—	2	1	no	—
256	#97	—	2	2	no	—
257	#97	—	2	1	no	—
258	C₄×C₈	—	2	1	no	—
259	C₄×C₈	—	2	1	no	—
260	#41	—	2	2	no	—
261	#41	—	4	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	4	2	no	—
265	C₂×Q₈	—	4	2	no	—
266	C₂×Q₈	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
275	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
276	Q₁₆	—	4	4	no	—
277	SD₁₆	—	4	4	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	4	1	no	—
280	M₁₆	—	4	2	no	—
281	M₁₆	—	4	2	no	—
282	M₁₆	—	8	1	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	4	1	no	—
289	C₂×C₈	—	8	2	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
297	C₄⋊C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
298	C₂²⋊C₄	—	2	1	no	—
299	C₂²⋊C₄	—	2	1	no	—
300	C₂²⋊C₄	—	2	1	no	—
301	C₄²	—	2	1	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	2	1	no	—
304	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
305	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
306	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
307	C₄²	#41×C₈₉	1	1	yes	characteristic
308	C₂³	—	2	1	no	—
309	C₂³	C₄×D₄×C₈₉	1	1	yes	characteristic
310	Q₈	—	2	1	no	—
311	Q₈	—	2	1	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	8	2	no	—
315	D₄	—	2	1	no	—
316	D₄	—	4	2	no	—
317	D₄	#100×C₈₉	1	2	yes	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	8	2	no	—
334	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
335	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
336	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
337	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
338	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
339	C₂×C₄	#123×C₈₉	1	1	yes	characteristic
340	C₈	—	4	2	no	—
341	C₈	—	4	2	no	—
342	C₈	—	4	1	no	—
343	C₈	—	4	2	no	—
344	C₈	—	4	1	no	—
345	C₈	—	8	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	#386×C₈₉	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	1	no	—
359	C₄	—	8	2	no	—
360	C₄	C₂×#100×C₈₉	1	1	yes	characteristic
361	C₄	#413×C₈₉	1	1	yes	characteristic
362	C₄	#455×C₈₉	1	1	yes	characteristic
363	C₄	#456×C₈₉	1	1	yes	characteristic
364	C₄	#471×C₈₉	1	1	yes	characteristic
365	C₄	#473×C₈₉	1	1	yes	characteristic
366	C₂	—	4	1	no	—
367	C₂	—	4	2	no	—
368	C₂	C₈₉×#1882	1	1	yes	characteristic
369	C₂	C₈₉×#1900	1	1	yes	characteristic
370	C₂	C₈₉×#1908	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	#386
commutator	C₂²×C₄	C₂²×C₄×C₈₉
Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
prime radical	C₂×#41×C₈₉	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₈₉×#1997	C₂	semidirect
4. C₈₉×#1548	C₂	semidirect
6. C₈₉×#1309	C₂	semidirect
7. C₈₉×#1299	C₂	semidirect
180. #9353	C₈₉	direct (N × H)
188. #1997	C₂×C₈₉	semidirect
189. #1548	C₂×C₈₉	semidirect
191. #1309	C₂×C₈₉	semidirect
192. #1299	C₂×C₈₉	semidirect
193. C₈₉	#9353	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈₉×#3039	2
C₈₉×#2990	2
C₈₉×#1997	2
C₈₉×#1548	2
C₈₉×#1489	2
C₈₉×#1309	2
C₈₉×#1299	2
#9353	89

Subgroups — C89×#9353 #36387201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#9353 #36387201