Subgroups — C₈₉×#10164 #36388012

Order 22784.

498 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3227	C₂	1	1	yes	maximal, characteristic
2	C₂×#500×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3141	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1997	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1548	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#1492	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	#532×C₈₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
11	#500×C₈₉	—	2	2	no	—
12	#500×C₈₉	—	2	2	no	—
13	#498×C₈₉	—	2	1	no	—
14	#489×C₈₉	—	2	2	no	—
15	#487×C₈₉	—	2	1	no	—
16	#486×C₈₉	—	2	1	no	—
17	#486×C₈₉	—	2	1	no	—
18	#481×C₈₉	—	2	2	no	—
19	#476×C₈₉	—	2	2	no	—
20	#431×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×#107×C₈₉	—	2	2	no	—
22	#423×C₈₉	C₂²	1	1	yes	characteristic
23	#417×C₈₉	—	2	2	no	—
24	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	#343×C₈₉	—	2	2	no	—
27	#333×C₈₉	—	2	2	no	—
28	D₄⋊C₈×C₈₉	—	2	2	no	—
29	C₂×#41×C₈₉	—	2	1	no	—
30	C₂²×Q₈×C₈₉	—	2	1	no	—
31	C₂²×Q₈×C₈₉	—	2	1	no	—
32	C₂×M₁₆×C₈₉	—	2	1	no	—
33	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
34	C₂²×C₈×C₈₉	—	2	1	no	—
35	C₄⋊Q₈×C₈₉	—	2	1	no	—
36	C₄⋊Q₈×C₈₉	—	2	1	no	—
37	C₄⋊Q₈×C₈₉	—	2	1	no	—
38	C₄⋊Q₈×C₈₉	—	2	1	no	—
39	C₄⋊Q₈×C₈₉	—	2	1	no	—
40	C₄⋊Q₈×C₈₉	—	2	1	no	—
41	C₄⋊Q₈×C₈₉	—	2	1	no	—
42	C₄⋊Q₈×C₈₉	D₄	1	1	yes	characteristic
43	C₄⋊Q₈×C₈₉	D₄	1	2	yes	—
44	#127×C₈₉	D₄	1	1	yes	characteristic
45	#124×C₈₉	—	2	1	no	—
46	#121×C₈₉	—	2	1	no	—
47	#121×C₈₉	—	2	1	no	—
48	C₄×D₄×C₈₉	—	2	1	no	—
49	C₄×D₄×C₈₉	—	2	1	no	—
50	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
51	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
52	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
53	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
54	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
55	#107×C₈₉	—	2	8	no	—
56	#106×C₈₉	—	2	4	no	—
57	C₄⋊C₈×C₈₉	—	2	1	no	—
58	C₄⋊C₈×C₈₉	—	2	1	no	—
59	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
60	#103×C₈₉	—	2	4	no	—
61	#102×C₈₉	—	2	4	no	—
62	#102×C₈₉	—	2	4	no	—
63	C₂²⋊C₈×C₈₉	—	2	1	no	—
64	#97×C₈₉	—	2	1	no	—
65	C₄×C₈×C₈₉	—	2	1	no	—
66	C₄×C₈×C₈₉	—	2	1	no	—
67	C₄×C₈×C₈₉	—	2	1	no	—
68	C₄×C₈×C₈₉	D₄	1	2	yes	—
69	#41×C₈₉	—	4	2	no	—
70	#41×C₈₉	—	4	2	no	—
71	C₂×Q₈×C₈₉	—	2	1	no	—
72	C₂×Q₈×C₈₉	—	2	2	no	—
73	C₂×Q₈×C₈₉	—	4	1	no	—
74	C₂×Q₈×C₈₉	—	4	1	no	—
75	C₂×Q₈×C₈₉	—	4	1	no	—
76	C₂×Q₈×C₈₉	—	8	2	no	—
77	C₂×Q₈×C₈₉	—	8	2	no	—
78	C₂×D₄×C₈₉	—	2	1	no	—
79	C₂×D₄×C₈₉	—	2	1	no	—
80	C₂×D₄×C₈₉	—	2	1	no	—
81	C₂×D₄×C₈₉	—	4	1	no	—
82	C₂²×C₄×C₈₉	—	2	2	no	—
83	C₂²×C₄×C₈₉	—	4	1	no	—
84	C₂²×C₄×C₈₉	—	4	1	no	—
85	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
88	M₁₆×C₈₉	—	2	2	no	—
89	M₁₆×C₈₉	—	2	2	no	—
90	M₁₆×C₈₉	—	4	2	no	—
91	C₂×C₈×C₈₉	—	2	4	no	—
92	C₂×C₈×C₈₉	—	2	2	no	—
93	C₂×C₈×C₈₉	—	2	2	no	—
94	C₂×C₈×C₈₉	—	2	2	no	—
95	C₂×C₈×C₈₉	—	4	1	no	—
96	C₄⋊C₄×C₈₉	—	2	2	no	—
97	C₄⋊C₄×C₈₉	—	2	2	no	—
98	C₄⋊C₄×C₈₉	—	2	1	no	—
99	C₄⋊C₄×C₈₉	—	2	2	no	—
100	C₄⋊C₄×C₈₉	—	2	2	no	—
101	C₄⋊C₄×C₈₉	—	2	1	no	—
102	C₄⋊C₄×C₈₉	—	4	1	no	—
103	C₄⋊C₄×C₈₉	—	4	1	no	—
104	C₄⋊C₄×C₈₉	—	4	1	no	—
105	C₄⋊C₄×C₈₉	—	4	1	no	—
106	C₂²⋊C₄×C₈₉	—	4	1	no	—
107	C₂²⋊C₄×C₈₉	—	4	1	no	—
108	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₈₉	—	4	1	no	—
113	C₂³×C₈₉	#121	1	1	yes	characteristic
114	Q₈×C₈₉	—	4	2	no	—
115	Q₈×C₈₉	—	8	2	no	—
116	Q₈×C₈₉	—	8	2	no	—
117	Q₈×C₈₉	—	8	2	no	—
118	Q₈×C₈₉	—	8	2	no	—
119	D₄×C₈₉	—	4	2	no	—
120	D₄×C₈₉	—	4	2	no	—
121	D₄×C₈₉	—	4	2	no	—
122	D₄×C₈₉	—	8	2	no	—
123	C₂×C₄×C₈₉	—	2	2	no	—
124	C₂×C₄×C₈₉	—	2	2	no	—
125	C₂×C₄×C₈₉	—	4	1	no	—
126	C₂×C₄×C₈₉	—	4	2	no	—
127	C₂×C₄×C₈₉	—	4	2	no	—
128	C₂×C₄×C₈₉	—	8	2	no	—
129	C₂×C₄×C₈₉	—	8	1	no	—
130	C₂×C₄×C₈₉	—	8	1	no	—
131	C₂×C₄×C₈₉	—	8	2	no	—
132	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
133	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
134	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
135	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
136	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
137	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
138	C₂×C₄×C₈₉	#135	1	2	yes	—
139	C₂×C₄×C₈₉	#137	1	2	yes	—
140	C₈×C₈₉	—	2	4	no	—
141	C₈×C₈₉	—	4	2	no	—
142	C₈×C₈₉	—	4	2	no	—
143	C₈×C₈₉	—	8	1	no	—
144	C₂²×C₈₉	—	2	1	no	—
145	C₂²×C₈₉	—	2	1	no	—
146	C₂²×C₈₉	—	8	1	no	—
147	C₂²×C₈₉	—	8	1	no	—
148	C₂²×C₈₉	—	8	1	no	—
149	C₂²×C₈₉	#392	1	1	yes	characteristic
150	C₂²×C₈₉	#464	1	2	yes	—
151	C₄×C₈₉	—	2	1	no	—
152	C₄×C₈₉	—	2	1	no	—
153	C₄×C₈₉	—	8	1	no	—
154	C₄×C₈₉	—	8	2	no	—
155	C₄×C₈₉	—	8	2	no	—
156	C₄×C₈₉	#452	1	2	yes	—
157	C₄×C₈₉	#456	1	2	yes	—
158	C₄×C₈₉	#472	1	2	yes	—
159	C₄×C₈₉	#495	1	1	yes	characteristic
160	C₄×C₈₉	#497	1	1	yes	characteristic
161	#10164	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂×C₈₉	—	2	2	no	—
163	C₂×C₈₉	—	16	1	no	—
164	C₂×C₈₉	#2005	1	1	yes	characteristic
165	C₂×C₈₉	#2015	1	1	yes	characteristic
166	C₂×C₈₉	#2018	1	1	yes	characteristic
167	#3227	C₂×C₈₉	1	1	yes	characteristic
168	C₂×#500	C₂×C₈₉	1	1	yes	characteristic, complemented
169	#3141	C₂×C₈₉	1	1	yes	characteristic
170	#1997	C₂×C₈₉	1	1	yes	characteristic, complemented
171	#1548	C₂×C₈₉	1	1	yes	characteristic, complemented
172	#1492	C₂×C₈₉	1	1	yes	characteristic
173	#1278	C₂×C₈₉	1	1	yes	characteristic, complemented
174	C₈₉	#10164	1	1	yes	characteristic, direct product factor, Sylow
175	#532	C₂²×C₈₉	1	1	yes	characteristic
176	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
177	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
178	#500	—	2	2	no	—
179	#500	—	2	2	no	—
180	#498	—	2	1	no	—
181	#489	—	2	2	no	—
182	#487	—	2	1	no	—
183	#486	—	2	1	no	—
184	#486	—	2	1	no	—
185	#481	—	2	2	no	—
186	#476	—	2	2	no	—
187	#431	C₂²×C₈₉	1	1	yes	characteristic
188	C₂×#107	—	2	2	no	—
189	#423	C₂²×C₈₉	1	1	yes	characteristic
190	#417	—	2	2	no	—
191	C₄×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
192	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
193	#343	—	2	2	no	—
194	#333	—	2	2	no	—
195	D₄⋊C₈	—	2	2	no	—
196	C₂×#41	—	2	1	no	—
197	C₂²×Q₈	—	2	1	no	—
198	C₂²×Q₈	—	2	1	no	—
199	C₂×M₁₆	—	2	1	no	—
200	C₂×M₁₆	D₄×C₈₉	1	2	yes	—
201	C₂²×C₈	—	2	1	no	—
202	C₄⋊Q₈	—	2	1	no	—
203	C₄⋊Q₈	—	2	1	no	—
204	C₄⋊Q₈	—	2	1	no	—
205	C₄⋊Q₈	—	2	1	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	D₄×C₈₉	1	2	yes	—
210	C₄⋊Q₈	D₄×C₈₉	1	1	yes	characteristic
211	#127	D₄×C₈₉	1	1	yes	characteristic
212	#124	—	2	1	no	—
213	#121	—	2	1	no	—
214	#121	—	2	1	no	—
215	C₄×D₄	—	2	1	no	—
216	C₄×D₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
221	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
222	#107	—	2	8	no	—
223	#106	—	2	4	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	D₄×C₈₉	1	2	yes	—
227	#103	—	2	4	no	—
228	#102	—	2	4	no	—
229	#102	—	2	4	no	—
230	C₂²⋊C₈	—	2	1	no	—
231	#97	—	2	1	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	D₄×C₈₉	1	2	yes	—
236	#41	—	4	2	no	—
237	#41	—	4	2	no	—
238	C₂×Q₈	—	2	1	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	8	2	no	—
244	C₂×Q₈	—	8	2	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
254	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
255	M₁₆	—	2	2	no	—
256	M₁₆	—	2	2	no	—
257	M₁₆	—	4	2	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	4	1	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₂²⋊C₄	—	4	1	no	—
274	C₂²⋊C₄	—	4	1	no	—
275	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
276	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
277	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
278	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
279	C₂³	—	4	1	no	—
280	C₂³	#121×C₈₉	1	1	yes	characteristic
281	Q₈	—	4	2	no	—
282	Q₈	—	8	2	no	—
283	Q₈	—	8	2	no	—
284	Q₈	—	8	2	no	—
285	Q₈	—	8	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	8	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	8	2	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
300	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
301	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
302	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
303	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
304	C₂×C₄	#127×C₈₉	1	1	yes	characteristic
305	C₂×C₄	#135×C₈₉	1	2	yes	—
306	C₂×C₄	#137×C₈₉	1	2	yes	—
307	C₈	—	2	4	no	—
308	C₈	—	4	2	no	—
309	C₈	—	4	2	no	—
310	C₈	—	8	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	8	1	no	—
314	C₂²	—	8	1	no	—
315	C₂²	—	8	1	no	—
316	C₂²	#392×C₈₉	1	1	yes	characteristic
317	C₂²	#464×C₈₉	1	2	yes	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	8	1	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	#452×C₈₉	1	2	yes	—
324	C₄	#456×C₈₉	1	2	yes	—
325	C₄	#472×C₈₉	1	2	yes	—
326	C₄	#495×C₈₉	1	1	yes	characteristic
327	C₄	#497×C₈₉	1	1	yes	characteristic
328	C₂	—	2	2	no	—
329	C₂	—	16	1	no	—
330	C₂	C₈₉×#2005	1	1	yes	characteristic
331	C₂	C₈₉×#2015	1	1	yes	characteristic
332	C₂	C₈₉×#2018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
prime radical	#532×C₈₉	C₂²
LCS	G ▷ C₂×C₄² ▷ 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
2. C₂×#500×C₈₉	C₂	semidirect
4. C₈₉×#1997	C₂	semidirect
5. C₈₉×#1548	C₂	semidirect
7. C₈₉×#1278	C₂	semidirect
161. #10164	C₈₉	direct (N × H)
168. C₂×#500	C₂×C₈₉	semidirect
170. #1997	C₂×C₈₉	semidirect
171. #1548	C₂×C₈₉	semidirect
173. #1278	C₂×C₈₉	semidirect
174. C₈₉	#10164	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×#3227	2
C₂×#500×C₈₉	2
C₈₉×#3141	2
C₈₉×#1997	2
C₈₉×#1548	2
C₈₉×#1492	2
C₈₉×#1278	2
#10164	89

Subgroups — C89×#10164 #36388012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#10164 #36388012