Subgroups — C₈₉×#32664 #36410512

Order 22784.

650 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2581	C₂	1	2	yes	maximal
2	C₈₉×#2097	C₂	1	2	yes	maximal
3	C₈₉×#2085	C₂	1	2	yes	maximal
4	C₈₉×#2068	C₂	1	2	yes	maximal
5	C₈₉×#1985	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#1971	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1944	C₂	1	1	yes	maximal, characteristic
8	C₈₉×#1878	C₂	1	2	yes	maximal
9	C₈₉×#1863	C₂	1	2	yes	maximal
10	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×#107×C₈₉	C₂²	1	2	yes	—
14	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×#103×C₈₉	C₂²	1	2	yes	—
19	C₂×#103×C₈₉	C₂²	1	2	yes	—
20	C₂×#103×C₈₉	C₂²	1	2	yes	—
21	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
22	#400×C₈₉	—	2	2	no	—
23	#395×C₈₉	—	2	2	no	—
24	#391×C₈₉	C₂²	1	2	yes	—
25	#391×C₈₉	C₂²	1	2	yes	—
26	#389×C₈₉	C₂²	1	2	yes	—
27	#389×C₈₉	C₂²	1	2	yes	—
28	#387×C₈₉	—	2	2	no	—
29	#387×C₈₉	—	2	2	no	—
30	#382×C₈₉	C₂²	1	2	yes	—
31	#340×C₈₉	C₂²	1	2	yes	—
32	#340×C₈₉	C₂²	1	2	yes	—
33	#336×C₈₉	C₂²	1	2	yes	—
34	#336×C₈₉	C₂²	1	2	yes	—
35	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
37	C₂×Q₁₆×C₈₉	—	4	1	no	—
38	C₂×Q₁₆×C₈₉	—	4	1	no	—
39	C₂×Q₁₆×C₈₉	Q₈	1	4	yes	complemented
40	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
41	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₈₉	—	4	1	no	—
44	C₄×Q₈×C₈₉	—	4	1	no	—
45	C₄×Q₈×C₈₉	—	4	2	no	—
46	C₄×Q₈×C₈₉	—	4	2	no	—
47	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
48	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
49	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
50	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
55	C₂×C₄²×C₈₉	—	2	2	no	—
56	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
57	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
58	#107×C₈₉	—	4	2	no	—
59	#107×C₈₉	—	4	1	no	—
60	#106×C₈₉	—	2	2	no	—
61	#106×C₈₉	—	4	1	no	—
62	C₄⋊C₈×C₈₉	—	4	1	no	—
63	#103×C₈₉	—	2	4	no	—
64	#103×C₈₉	—	4	2	no	—
65	#103×C₈₉	—	4	2	no	—
66	#97×C₈₉	—	4	1	no	—
67	#95×C₈₉	—	2	2	no	—
68	#95×C₈₉	—	2	2	no	—
69	#95×C₈₉	—	2	2	no	—
70	#95×C₈₉	—	2	2	no	—
71	#95×C₈₉	—	2	2	no	—
72	#95×C₈₉	—	2	2	no	—
73	#95×C₈₉	—	2	2	no	—
74	#95×C₈₉	—	2	2	no	—
75	C₂×Q₈×C₈₉	—	2	2	no	—
76	C₂×Q₈×C₈₉	—	2	2	no	—
77	C₂×Q₈×C₈₉	—	2	2	no	—
78	C₂×Q₈×C₈₉	—	2	2	no	—
79	C₂×Q₈×C₈₉	—	2	2	no	—
80	C₂×Q₈×C₈₉	—	4	1	no	—
81	C₂×Q₈×C₈₉	—	4	2	no	—
82	C₂×Q₈×C₈₉	C₂×Q₈	1	4	yes	—
83	C₂²×C₄×C₈₉	—	2	2	no	—
84	C₂²×C₄×C₈₉	—	2	2	no	—
85	C₂²×C₄×C₈₉	—	2	2	no	—
86	C₂²×C₄×C₈₉	—	2	2	no	—
87	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
92	Q₁₆×C₈₉	—	4	4	no	—
93	C₂×C₈×C₈₉	—	2	1	no	—
94	C₂×C₈×C₈₉	—	2	1	no	—
95	C₂×C₈×C₈₉	—	2	1	no	—
96	C₂×C₈×C₈₉	—	2	1	no	—
97	C₂×C₈×C₈₉	—	2	2	no	—
98	C₂×C₈×C₈₉	—	2	2	no	—
99	C₂×C₈×C₈₉	—	2	2	no	—
100	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
101	C₂×C₈×C₈₉	#41	1	2	yes	—
102	C₄⋊C₄×C₈₉	—	2	2	no	—
103	C₄⋊C₄×C₈₉	—	2	2	no	—
104	C₄⋊C₄×C₈₉	—	2	2	no	—
105	C₄⋊C₄×C₈₉	—	2	2	no	—
106	C₄⋊C₄×C₈₉	—	2	2	no	—
107	C₄⋊C₄×C₈₉	—	2	2	no	—
108	C₄⋊C₄×C₈₉	—	4	1	no	—
109	C₄⋊C₄×C₈₉	—	4	2	no	—
110	C₄⋊C₄×C₈₉	—	4	4	no	—
111	C₄⋊C₄×C₈₉	—	4	4	no	—
112	C₄⋊C₄×C₈₉	—	4	4	no	—
113	C₄²×C₈₉	—	2	1	no	—
114	C₄²×C₈₉	—	2	1	no	—
115	C₄²×C₈₉	—	2	2	no	—
116	C₄²×C₈₉	—	2	2	no	—
117	C₄²×C₈₉	—	4	4	no	—
118	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
119	Q₈×C₈₉	—	2	4	no	—
120	Q₈×C₈₉	—	2	4	no	—
121	Q₈×C₈₉	—	4	4	no	—
122	Q₈×C₈₉	—	4	2	no	—
123	Q₈×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	2	2	no	—
125	C₂×C₄×C₈₉	—	2	2	no	—
126	C₂×C₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	—	2	1	no	—
128	C₂×C₄×C₈₉	—	2	4	no	—
129	C₂×C₄×C₈₉	—	2	2	no	—
130	C₂×C₄×C₈₉	—	2	2	no	—
131	C₂×C₄×C₈₉	—	2	2	no	—
132	C₂×C₄×C₈₉	—	2	2	no	—
133	C₂×C₄×C₈₉	—	2	4	no	—
134	C₂×C₄×C₈₉	—	4	2	no	—
135	C₂×C₄×C₈₉	—	4	2	no	—
136	C₂×C₄×C₈₉	—	4	2	no	—
137	C₂×C₄×C₈₉	—	4	2	no	—
138	C₂×C₄×C₈₉	—	4	2	no	—
139	C₂×C₄×C₈₉	—	4	2	no	—
140	C₂×C₄×C₈₉	—	4	2	no	—
141	C₂×C₄×C₈₉	—	4	2	no	—
142	C₂×C₄×C₈₉	—	4	2	no	—
143	C₂×C₄×C₈₉	—	4	2	no	—
144	C₂×C₄×C₈₉	—	4	2	no	—
145	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
151	C₈×C₈₉	—	2	2	no	—
152	C₈×C₈₉	—	2	2	no	—
153	C₈×C₈₉	—	4	2	no	—
154	C₂²×C₈₉	#534	1	1	yes	characteristic
155	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
156	C₂²×C₈₉	#550	1	1	yes	characteristic
157	C₂²×C₈₉	#575	1	2	yes	—
158	C₂²×C₈₉	#578	1	2	yes	—
159	C₄×C₈₉	—	4	2	no	—
160	C₄×C₈₉	—	4	1	no	—
161	C₄×C₈₉	—	4	2	no	—
162	C₄×C₈₉	—	4	1	no	—
163	C₄×C₈₉	—	4	4	no	—
164	C₄×C₈₉	—	8	2	no	—
165	C₄×C₈₉	—	8	2	no	—
166	C₄×C₈₉	—	8	2	no	—
167	C₄×C₈₉	#539	1	1	yes	characteristic
168	C₄×C₈₉	#543	1	1	yes	characteristic
169	C₄×C₈₉	#554	1	1	yes	characteristic
170	C₄×C₈₉	#562	1	1	yes	characteristic
171	#32664	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂×C₈₉	#2732	1	1	yes	characteristic
173	C₂×C₈₉	#3197	1	1	yes	characteristic
174	C₂×C₈₉	#3202	1	1	yes	characteristic
175	C₂×C₈₉	#3390	1	2	yes	—
176	C₂×C₈₉	#3402	1	2	yes	—
177	#2581	C₂×C₈₉	1	2	yes	—
178	#2097	C₂×C₈₉	1	2	yes	—
179	#2085	C₂×C₈₉	1	2	yes	—
180	#2068	C₂×C₈₉	1	2	yes	—
181	#1985	C₂×C₈₉	1	1	yes	characteristic
182	#1971	C₂×C₈₉	1	1	yes	characteristic
183	#1944	C₂×C₈₉	1	1	yes	characteristic
184	#1878	C₂×C₈₉	1	2	yes	—
185	#1863	C₂×C₈₉	1	2	yes	—
186	C₈₉	#32664	1	1	yes	characteristic, direct product factor, Sylow
187	C₂²×Q₁₆	C₂²×C₈₉	1	1	yes	characteristic
188	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
189	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
190	C₂×#107	C₂²×C₈₉	1	2	yes	—
191	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
192	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
193	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
194	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
195	C₂×#103	C₂²×C₈₉	1	2	yes	—
196	C₂×#103	C₂²×C₈₉	1	2	yes	—
197	C₂×#103	C₂²×C₈₉	1	2	yes	—
198	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
199	#400	—	2	2	no	—
200	#395	—	2	2	no	—
201	#391	C₂²×C₈₉	1	2	yes	—
202	#391	C₂²×C₈₉	1	2	yes	—
203	#389	C₂²×C₈₉	1	2	yes	—
204	#389	C₂²×C₈₉	1	2	yes	—
205	#387	—	2	2	no	—
206	#387	—	2	2	no	—
207	#382	C₂²×C₈₉	1	2	yes	—
208	#340	C₂²×C₈₉	1	2	yes	—
209	#340	C₂²×C₈₉	1	2	yes	—
210	#336	C₂²×C₈₉	1	2	yes	—
211	#336	C₂²×C₈₉	1	2	yes	—
212	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
213	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
214	C₂×Q₁₆	—	4	1	no	—
215	C₂×Q₁₆	—	4	1	no	—
216	C₂×Q₁₆	Q₈×C₈₉	1	4	yes	complemented
217	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
218	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
219	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
220	C₄×Q₈	—	4	1	no	—
221	C₄×Q₈	—	4	1	no	—
222	C₄×Q₈	—	4	2	no	—
223	C₄×Q₈	—	4	2	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
228	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
229	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
230	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
231	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
232	C₂×C₄²	—	2	2	no	—
233	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
234	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
235	#107	—	4	2	no	—
236	#107	—	4	1	no	—
237	#106	—	2	2	no	—
238	#106	—	4	1	no	—
239	C₄⋊C₈	—	4	1	no	—
240	#103	—	2	4	no	—
241	#103	—	4	2	no	—
242	#103	—	4	2	no	—
243	#97	—	4	1	no	—
244	#95	—	2	2	no	—
245	#95	—	2	2	no	—
246	#95	—	2	2	no	—
247	#95	—	2	2	no	—
248	#95	—	2	2	no	—
249	#95	—	2	2	no	—
250	#95	—	2	2	no	—
251	#95	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	4	2	no	—
259	C₂×Q₈	C₂×Q₈×C₈₉	1	4	yes	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
266	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
267	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
268	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
269	Q₁₆	—	4	4	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	2	1	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	C₂×Q₈×C₈₉	1	2	yes	—
278	C₂×C₈	#41×C₈₉	1	2	yes	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	4	4	no	—
288	C₄⋊C₄	—	4	4	no	—
289	C₄⋊C₄	—	4	4	no	—
290	C₄²	—	2	1	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	4	4	no	—
295	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
296	Q₈	—	2	4	no	—
297	Q₈	—	2	4	no	—
298	Q₈	—	4	4	no	—
299	Q₈	—	4	2	no	—
300	Q₈	—	4	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
323	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
324	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
325	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
326	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
327	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
328	C₈	—	2	2	no	—
329	C₈	—	2	2	no	—
330	C₈	—	4	2	no	—
331	C₂²	#534×C₈₉	1	1	yes	characteristic
332	C₂²	D₄×Q₈×C₈₉	1	1	yes	characteristic
333	C₂²	#550×C₈₉	1	1	yes	characteristic
334	C₂²	#575×C₈₉	1	2	yes	—
335	C₂²	#578×C₈₉	1	2	yes	—
336	C₄	—	4	2	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	2	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	4	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	—	8	2	no	—
344	C₄	#539×C₈₉	1	1	yes	characteristic
345	C₄	#543×C₈₉	1	1	yes	characteristic
346	C₄	#554×C₈₉	1	1	yes	characteristic
347	C₄	#562×C₈₉	1	1	yes	characteristic
348	C₂	C₈₉×#2732	1	1	yes	characteristic
349	C₂	C₈₉×#3197	1	1	yes	characteristic
350	C₂	C₈₉×#3202	1	1	yes	characteristic
351	C₂	C₈₉×#3390	1	2	yes	—
352	C₂	C₈₉×#3402	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂×C₄×Q₈×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
39. C₂×Q₁₆×C₈₉	Q₈	semidirect
171. #32664	C₈₉	direct (N × H)
186. C₈₉	#32664	direct (N × H)
216. C₂×Q₁₆	Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#2581	2
C₈₉×#2097	2
C₈₉×#2085	2
C₈₉×#2068	2
C₈₉×#1985	2
C₈₉×#1971	2
C₈₉×#1944	2
C₈₉×#1878	2
C₈₉×#1863	2
#32664	89

Subgroups — C89×#32664 #36410512

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#32664 #36410512