Subgroups — C₈₉×#19165 #36397013

Order 22784.

674 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3165	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈₉×#3101	C₂	1	1	yes	maximal, characteristic
3	C₂×#476×C₈₉	C₂	1	1	yes	maximal, characteristic
4	D₄×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1695	C₂	1	2	yes	maximal, complemented
6	C₈₉×#1691	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1675	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1671	C₂	1	2	yes	maximal, complemented
9	C₈₉×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1550	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1548	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₈₉	—	2	1	no	—
13	#533×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
16	#501×C₈₉	—	2	2	no	—
17	#500×C₈₉	C₂²	1	2	yes	complemented
18	#498×C₈₉	C₂²	1	2	yes	complemented
19	#483×C₈₉	—	2	1	no	—
20	#476×C₈₉	—	2	1	no	—
21	#476×C₈₉	—	2	1	no	—
22	#476×C₈₉	C₂²	1	2	yes	—
23	#476×C₈₉	C₂²	1	2	yes	—
24	#476×C₈₉	C₂²	1	1	yes	characteristic
25	#476×C₈₉	C₂²	1	1	yes	characteristic
26	#461×C₈₉	C₂²	1	1	yes	characteristic
27	#461×C₈₉	C₂²	1	1	yes	characteristic
28	#450×C₈₉	—	2	1	no	—
29	C₄×SD₁₆×C₈₉	—	2	1	no	—
30	C₄×SD₁₆×C₈₉	—	2	1	no	—
31	#436×C₈₉	C₂²	1	2	yes	—
32	#435×C₈₉	—	2	2	no	—
33	C₈×D₄×C₈₉	C₂²	1	2	yes	—
34	#431×C₈₉	C₂²	1	1	yes	characteristic
35	#428×C₈₉	—	2	1	no	—
36	#428×C₈₉	—	2	1	no	—
37	C₂×#106×C₈₉	—	2	1	no	—
38	#423×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
40	C₂×#102×C₈₉	—	2	1	no	—
41	#407×C₈₉	—	2	1	no	—
42	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
43	#335×C₈₉	C₂²	1	2	yes	complemented
44	#334×C₈₉	C₂²	1	2	yes	complemented
45	#333×C₈₉	C₂²	1	2	yes	complemented
46	#333×C₈₉	C₂²	1	2	yes	complemented
47	#333×C₈₉	C₂²	1	2	yes	complemented
48	#332×C₈₉	C₂²	1	2	yes	complemented
49	C₂²×Q₈×C₈₉	—	2	1	no	—
50	C₂²×D₄×C₈₉	D₄	1	1	yes	characteristic
51	C₂³×C₄×C₈₉	—	2	1	no	—
52	C₂×SD₁₆×C₈₉	—	2	1	no	—
53	C₂×SD₁₆×C₈₉	—	2	1	no	—
54	C₂×M₁₆×C₈₉	—	2	1	no	—
55	C₂×M₁₆×C₈₉	—	2	2	no	—
56	C₂×M₁₆×C₈₉	—	4	4	no	—
57	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
58	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₈₉	—	2	1	no	—
60	C₂²×C₈×C₈₉	—	2	2	no	—
61	C₄⋊Q₈×C₈₉	—	2	1	no	—
62	C₄⋊Q₈×C₈₉	—	2	1	no	—
63	C₄⋊Q₈×C₈₉	—	2	1	no	—
64	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
66	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
67	#125×C₈₉	—	2	1	no	—
68	C₂²⋊Q₈×C₈₉	—	2	1	no	—
69	C₂²⋊Q₈×C₈₉	—	2	1	no	—
70	C₄×Q₈×C₈₉	—	2	1	no	—
71	C₄×Q₈×C₈₉	—	2	1	no	—
72	C₄×D₄×C₈₉	—	2	1	no	—
73	C₄×D₄×C₈₉	—	2	1	no	—
74	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
75	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
76	C₄×D₄×C₈₉	C₂³	1	1	yes	characteristic
77	C₄×D₄×C₈₉	C₂³	1	1	yes	characteristic
78	#117×C₈₉	—	2	1	no	—
79	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
80	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
81	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
82	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
83	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
84	#107×C₈₉	—	2	2	no	—
85	#107×C₈₉	—	2	2	no	—
86	#106×C₈₉	—	2	2	no	—
87	#106×C₈₉	—	2	2	no	—
88	#106×C₈₉	—	2	2	no	—
89	#106×C₈₉	—	2	2	no	—
90	#106×C₈₉	—	2	1	no	—
91	#106×C₈₉	—	2	1	no	—
92	C₄⋊C₈×C₈₉	—	2	1	no	—
93	C₄⋊C₈×C₈₉	—	2	1	no	—
94	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
95	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
96	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
97	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
98	#103×C₈₉	—	2	1	no	—
99	#103×C₈₉	—	2	1	no	—
100	#102×C₈₉	—	2	2	no	—
101	#102×C₈₉	—	2	2	no	—
102	#102×C₈₉	—	2	1	no	—
103	#102×C₈₉	—	2	1	no	—
104	C₂²⋊C₈×C₈₉	—	2	1	no	—
105	C₂²⋊C₈×C₈₉	—	2	2	no	—
106	C₂²⋊C₈×C₈₉	—	2	2	no	—
107	#97×C₈₉	D₄	1	2	yes	—
108	C₄×C₈×C₈₉	—	2	1	no	—
109	C₄×C₈×C₈₉	C₂³	1	2	yes	—
110	C₂⁴×C₈₉	—	2	1	no	—
111	C₂×Q₈×C₈₉	—	2	1	no	—
112	C₂×Q₈×C₈₉	—	2	1	no	—
113	C₂×Q₈×C₈₉	—	2	2	no	—
114	C₂×Q₈×C₈₉	—	4	1	no	—
115	C₂×Q₈×C₈₉	—	8	2	no	—
116	C₂×D₄×C₈₉	—	2	2	no	—
117	C₂×D₄×C₈₉	—	2	1	no	—
118	C₂×D₄×C₈₉	—	4	1	no	—
119	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₈₉	—	2	1	no	—
122	C₂²×C₄×C₈₉	—	2	1	no	—
123	C₂²×C₄×C₈₉	—	2	1	no	—
124	C₂²×C₄×C₈₉	—	2	1	no	—
125	C₂²×C₄×C₈₉	—	2	1	no	—
126	C₂²×C₄×C₈₉	—	4	2	no	—
127	C₂²×C₄×C₈₉	—	4	2	no	—
128	C₂²×C₄×C₈₉	—	4	1	no	—
129	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
132	SD₁₆×C₈₉	—	4	2	no	—
133	SD₁₆×C₈₉	—	4	2	no	—
134	M₁₆×C₈₉	—	2	2	no	—
135	M₁₆×C₈₉	—	4	2	no	—
136	M₁₆×C₈₉	—	4	4	no	—
137	M₁₆×C₈₉	SD₁₆	1	4	yes	complemented
138	C₂×C₈×C₈₉	—	2	1	no	—
139	C₂×C₈×C₈₉	—	2	1	no	—
140	C₂×C₈×C₈₉	—	2	2	no	—
141	C₂×C₈×C₈₉	—	2	2	no	—
142	C₂×C₈×C₈₉	—	4	2	no	—
143	C₂×C₈×C₈₉	—	4	4	no	—
144	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
145	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
146	C₄⋊C₄×C₈₉	—	2	1	no	—
147	C₄⋊C₄×C₈₉	—	2	2	no	—
148	C₄⋊C₄×C₈₉	—	2	1	no	—
149	C₄⋊C₄×C₈₉	—	2	1	no	—
150	C₄⋊C₄×C₈₉	—	2	2	no	—
151	C₄⋊C₄×C₈₉	—	2	1	no	—
152	C₄⋊C₄×C₈₉	—	2	1	no	—
153	C₄⋊C₄×C₈₉	—	4	1	no	—
154	C₄⋊C₄×C₈₉	—	4	1	no	—
155	C₄⋊C₄×C₈₉	—	4	1	no	—
156	C₄⋊C₄×C₈₉	—	4	1	no	—
157	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
158	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
159	C₂²⋊C₄×C₈₉	—	2	1	no	—
160	C₂²⋊C₄×C₈₉	—	2	1	no	—
161	C₂²⋊C₄×C₈₉	—	2	1	no	—
162	C₂²⋊C₄×C₈₉	—	2	1	no	—
163	C₂²⋊C₄×C₈₉	—	4	1	no	—
164	C₄²×C₈₉	—	4	1	no	—
165	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
166	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
167	C₄²×C₈₉	#41	1	1	yes	characteristic
168	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
169	C₂³×C₈₉	—	2	1	no	—
170	C₂³×C₈₉	—	2	1	no	—
171	C₂³×C₈₉	—	4	2	no	—
172	C₂³×C₈₉	—	4	2	no	—
173	C₂³×C₈₉	—	8	1	no	—
174	C₂³×C₈₉	#121	1	1	yes	characteristic
175	Q₈×C₈₉	—	4	1	no	—
176	Q₈×C₈₉	—	4	1	no	—
177	Q₈×C₈₉	—	8	1	no	—
178	Q₈×C₈₉	—	8	2	no	—
179	Q₈×C₈₉	—	8	2	no	—
180	D₄×C₈₉	—	2	1	no	—
181	D₄×C₈₉	—	2	1	no	—
182	D₄×C₈₉	—	2	1	no	—
183	D₄×C₈₉	—	4	2	no	—
184	D₄×C₈₉	#137	1	2	yes	—
185	C₂×C₄×C₈₉	—	2	1	no	—
186	C₂×C₄×C₈₉	—	2	1	no	—
187	C₂×C₄×C₈₉	—	2	1	no	—
188	C₂×C₄×C₈₉	—	2	1	no	—
189	C₂×C₄×C₈₉	—	4	1	no	—
190	C₂×C₄×C₈₉	—	4	2	no	—
191	C₂×C₄×C₈₉	—	4	1	no	—
192	C₂×C₄×C₈₉	—	4	1	no	—
193	C₂×C₄×C₈₉	—	4	1	no	—
194	C₂×C₄×C₈₉	—	4	1	no	—
195	C₂×C₄×C₈₉	—	4	2	no	—
196	C₂×C₄×C₈₉	—	4	2	no	—
197	C₂×C₄×C₈₉	—	4	1	no	—
198	C₂×C₄×C₈₉	—	4	1	no	—
199	C₂×C₄×C₈₉	—	4	2	no	—
200	C₂×C₄×C₈₉	—	8	2	no	—
201	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
202	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
203	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
204	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
205	C₂×C₄×C₈₉	#137	1	2	yes	—
206	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
207	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
209	C₈×C₈₉	—	2	2	no	—
210	C₈×C₈₉	—	4	2	no	—
211	C₈×C₈₉	—	4	1	no	—
212	C₈×C₈₉	—	4	2	no	—
213	C₈×C₈₉	—	4	1	no	—
214	C₈×C₈₉	C₂×SD₁₆	1	4	yes	—
215	C₂²×C₈₉	—	2	1	no	—
216	C₂²×C₈₉	—	2	1	no	—
217	C₂²×C₈₉	—	2	2	no	—
218	C₂²×C₈₉	—	4	1	no	—
219	C₂²×C₈₉	—	4	1	no	—
220	C₂²×C₈₉	—	4	1	no	—
221	C₂²×C₈₉	—	4	1	no	—
222	C₂²×C₈₉	—	4	1	no	—
223	C₂²×C₈₉	—	8	4	no	—
224	C₂²×C₈₉	#461	1	2	yes	—
225	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
226	C₄×C₈₉	—	2	1	no	—
227	C₄×C₈₉	—	2	1	no	—
228	C₄×C₈₉	—	2	1	no	—
229	C₄×C₈₉	—	4	1	no	—
230	C₄×C₈₉	—	4	1	no	—
231	C₄×C₈₉	—	8	1	no	—
232	C₄×C₈₉	—	8	2	no	—
233	C₄×C₈₉	—	8	1	no	—
234	C₄×C₈₉	#461	1	2	yes	—
235	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
236	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
237	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
238	C₄×C₈₉	#575	1	1	yes	characteristic
239	#19165	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
240	C₂×C₈₉	—	2	2	no	—
241	C₂×C₈₉	—	4	2	no	—
242	C₂×C₈₉	—	8	1	no	—
243	C₂×C₈₉	C₂×#461	1	1	yes	characteristic
244	C₂×C₈₉	#3056	1	1	yes	characteristic
245	C₂×C₈₉	#3068	1	1	yes	characteristic
246	#3165	C₂×C₈₉	1	1	yes	characteristic, complemented
247	#3101	C₂×C₈₉	1	1	yes	characteristic
248	C₂×#476	C₂×C₈₉	1	1	yes	characteristic
249	D₄×M₁₆	C₂×C₈₉	1	1	yes	characteristic
250	#1695	C₂×C₈₉	1	2	yes	complemented
251	#1691	C₂×C₈₉	1	2	yes	complemented
252	#1675	C₂×C₈₉	1	2	yes	complemented
253	#1671	C₂×C₈₉	1	2	yes	complemented
254	#1565	C₂×C₈₉	1	1	yes	characteristic, complemented
255	#1550	C₂×C₈₉	1	1	yes	characteristic, complemented
256	#1548	C₂×C₈₉	1	1	yes	characteristic, complemented
257	C₈₉	#19165	1	1	yes	characteristic, direct product factor, Sylow
258	C₂²×M₁₆	—	2	1	no	—
259	#533	C₂²×C₈₉	1	1	yes	characteristic
260	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
261	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
262	#501	—	2	2	no	—
263	#500	C₂²×C₈₉	1	2	yes	complemented
264	#498	C₂²×C₈₉	1	2	yes	complemented
265	#483	—	2	1	no	—
266	#476	—	2	1	no	—
267	#476	—	2	1	no	—
268	#476	C₂²×C₈₉	1	1	yes	characteristic
269	#476	C₂²×C₈₉	1	1	yes	characteristic
270	#476	C₂²×C₈₉	1	2	yes	—
271	#476	C₂²×C₈₉	1	2	yes	—
272	#461	C₂²×C₈₉	1	1	yes	characteristic
273	#461	C₂²×C₈₉	1	1	yes	characteristic
274	#450	—	2	1	no	—
275	C₄×SD₁₆	—	2	1	no	—
276	C₄×SD₁₆	—	2	1	no	—
277	#436	C₂²×C₈₉	1	2	yes	—
278	#435	—	2	2	no	—
279	C₈×D₄	C₂²×C₈₉	1	2	yes	—
280	#431	C₂²×C₈₉	1	1	yes	characteristic
281	#428	—	2	1	no	—
282	#428	—	2	1	no	—
283	C₂×#106	—	2	1	no	—
284	#423	C₂²×C₈₉	1	1	yes	characteristic
285	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
286	C₂×#102	—	2	1	no	—
287	#407	—	2	1	no	—
288	C₄×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
289	#335	C₂²×C₈₉	1	2	yes	complemented
290	#334	C₂²×C₈₉	1	2	yes	complemented
291	#333	C₂²×C₈₉	1	2	yes	complemented
292	#333	C₂²×C₈₉	1	2	yes	complemented
293	#333	C₂²×C₈₉	1	2	yes	complemented
294	#332	C₂²×C₈₉	1	2	yes	complemented
295	C₂²×Q₈	—	2	1	no	—
296	C₂²×D₄	D₄×C₈₉	1	1	yes	characteristic
297	C₂³×C₄	—	2	1	no	—
298	C₂×SD₁₆	—	2	1	no	—
299	C₂×SD₁₆	—	2	1	no	—
300	C₂×M₁₆	—	2	1	no	—
301	C₂×M₁₆	—	2	2	no	—
302	C₂×M₁₆	—	4	4	no	—
303	C₂×M₁₆	D₄×C₈₉	1	1	yes	characteristic
304	C₂×M₁₆	D₄×C₈₉	1	1	yes	characteristic
305	C₂²×C₈	—	2	1	no	—
306	C₂²×C₈	—	2	2	no	—
307	C₄⋊Q₈	—	2	1	no	—
308	C₄⋊Q₈	—	2	1	no	—
309	C₄⋊Q₈	—	2	1	no	—
310	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
311	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
312	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
313	#125	—	2	1	no	—
314	C₂²⋊Q₈	—	2	1	no	—
315	C₂²⋊Q₈	—	2	1	no	—
316	C₄×Q₈	—	2	1	no	—
317	C₄×Q₈	—	2	1	no	—
318	C₄×D₄	—	2	1	no	—
319	C₄×D₄	—	2	1	no	—
320	C₄×D₄	D₄×C₈₉	1	1	yes	characteristic
321	C₄×D₄	D₄×C₈₉	1	1	yes	characteristic
322	C₄×D₄	C₂³×C₈₉	1	1	yes	characteristic
323	C₄×D₄	C₂³×C₈₉	1	1	yes	characteristic
324	#117	—	2	1	no	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄⋊C₄	—	2	1	no	—
327	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
328	C₂×C₂²⋊C₄	—	2	1	no	—
329	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
330	#107	—	2	2	no	—
331	#107	—	2	2	no	—
332	#106	—	2	2	no	—
333	#106	—	2	2	no	—
334	#106	—	2	2	no	—
335	#106	—	2	2	no	—
336	#106	—	2	1	no	—
337	#106	—	2	1	no	—
338	C₄⋊C₈	—	2	1	no	—
339	C₄⋊C₈	—	2	1	no	—
340	C₄⋊C₈	C₂³×C₈₉	1	2	yes	—
341	C₄⋊C₈	C₂³×C₈₉	1	2	yes	—
342	C₄⋊C₈	C₂³×C₈₉	1	1	yes	characteristic
343	C₄⋊C₈	C₂³×C₈₉	1	1	yes	characteristic
344	#103	—	2	1	no	—
345	#103	—	2	1	no	—
346	#102	—	2	2	no	—
347	#102	—	2	2	no	—
348	#102	—	2	1	no	—
349	#102	—	2	1	no	—
350	C₂²⋊C₈	—	2	1	no	—
351	C₂²⋊C₈	—	2	2	no	—
352	C₂²⋊C₈	—	2	2	no	—
353	#97	D₄×C₈₉	1	2	yes	—
354	C₄×C₈	—	2	1	no	—
355	C₄×C₈	C₂³×C₈₉	1	2	yes	—
356	C₂⁴	—	2	1	no	—
357	C₂×Q₈	—	2	1	no	—
358	C₂×Q₈	—	2	1	no	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	—	4	1	no	—
361	C₂×Q₈	—	8	2	no	—
362	C₂×D₄	—	2	2	no	—
363	C₂×D₄	—	2	1	no	—
364	C₂×D₄	—	4	1	no	—
365	C₂×D₄	C₂×D₄×C₈₉	1	1	yes	characteristic
366	C₂×D₄	C₂×D₄×C₈₉	1	1	yes	characteristic
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	4	2	no	—
373	C₂²×C₄	—	4	2	no	—
374	C₂²×C₄	—	4	1	no	—
375	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
376	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
377	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
378	SD₁₆	—	4	2	no	—
379	SD₁₆	—	4	2	no	—
380	M₁₆	—	2	2	no	—
381	M₁₆	—	4	2	no	—
382	M₁₆	—	4	4	no	—
383	M₁₆	SD₁₆×C₈₉	1	4	yes	complemented
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	4	2	no	—
389	C₂×C₈	—	4	4	no	—
390	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
391	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	—	2	1	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	2	1	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	4	1	no	—
400	C₄⋊C₄	—	4	1	no	—
401	C₄⋊C₄	—	4	1	no	—
402	C₄⋊C₄	—	4	1	no	—
403	C₄⋊C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
404	C₄⋊C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
405	C₂²⋊C₄	—	2	1	no	—
406	C₂²⋊C₄	—	2	1	no	—
407	C₂²⋊C₄	—	2	1	no	—
408	C₂²⋊C₄	—	2	1	no	—
409	C₂²⋊C₄	—	4	1	no	—
410	C₄²	—	4	1	no	—
411	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
412	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
413	C₄²	#41×C₈₉	1	1	yes	characteristic
414	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
415	C₂³	—	2	1	no	—
416	C₂³	—	2	1	no	—
417	C₂³	—	4	2	no	—
418	C₂³	—	4	2	no	—
419	C₂³	—	8	1	no	—
420	C₂³	#121×C₈₉	1	1	yes	characteristic
421	Q₈	—	4	1	no	—
422	Q₈	—	4	1	no	—
423	Q₈	—	8	1	no	—
424	Q₈	—	8	2	no	—
425	Q₈	—	8	2	no	—
426	D₄	—	2	1	no	—
427	D₄	—	2	1	no	—
428	D₄	—	2	1	no	—
429	D₄	—	4	2	no	—
430	D₄	#137×C₈₉	1	2	yes	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	8	2	no	—
447	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
448	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
449	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
450	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
451	C₂×C₄	#137×C₈₉	1	2	yes	—
452	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
453	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
454	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
455	C₈	—	2	2	no	—
456	C₈	—	4	2	no	—
457	C₈	—	4	1	no	—
458	C₈	—	4	2	no	—
459	C₈	—	4	1	no	—
460	C₈	C₂×SD₁₆×C₈₉	1	4	yes	—
461	C₂²	—	2	1	no	—
462	C₂²	—	2	1	no	—
463	C₂²	—	2	2	no	—
464	C₂²	—	4	1	no	—
465	C₂²	—	4	1	no	—
466	C₂²	—	4	1	no	—
467	C₂²	—	4	1	no	—
468	C₂²	—	4	1	no	—
469	C₂²	—	8	4	no	—
470	C₂²	#461×C₈₉	1	2	yes	—
471	C₂²	C₂×#121×C₈₉	1	1	yes	characteristic
472	C₄	—	2	1	no	—
473	C₄	—	2	1	no	—
474	C₄	—	2	1	no	—
475	C₄	—	4	1	no	—
476	C₄	—	4	1	no	—
477	C₄	—	8	2	no	—
478	C₄	—	8	1	no	—
479	C₄	—	8	1	no	—
480	C₄	#461×C₈₉	1	2	yes	—
481	C₄	C₂²×SD₁₆×C₈₉	1	1	yes	characteristic
482	C₄	C₂×#137×C₈₉	1	1	yes	characteristic
483	C₄	C₂×#137×C₈₉	1	1	yes	characteristic
484	C₄	#575×C₈₉	1	1	yes	characteristic
485	C₂	—	2	2	no	—
486	C₂	—	4	2	no	—
487	C₂	—	8	1	no	—
488	C₂	C₂×#461×C₈₉	1	1	yes	characteristic
489	C₂	C₈₉×#3056	1	1	yes	characteristic
490	C₂	C₈₉×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₈₉
prime radical	C₂²×D₄×C₈₉	D₄
LCS	G ▷ 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
1. C₈₉×#3165	C₂	semidirect
5. C₈₉×#1695	C₂	semidirect
6. C₈₉×#1691	C₂	semidirect
7. C₈₉×#1675	C₂	semidirect
8. C₈₉×#1671	C₂	semidirect
9. C₈₉×#1565	C₂	semidirect
10. C₈₉×#1550	C₂	semidirect
11. C₈₉×#1548	C₂	semidirect
17. #500×C₈₉	C₂²	semidirect
18. #498×C₈₉	C₂²	semidirect
43. #335×C₈₉	C₂²	semidirect
44. #334×C₈₉	C₂²	semidirect
45. #333×C₈₉	C₂²	semidirect
46. #333×C₈₉	C₂²	semidirect
47. #333×C₈₉	C₂²	semidirect
48. #332×C₈₉	C₂²	semidirect
137. M₁₆×C₈₉	SD₁₆	semidirect
239. #19165	C₈₉	direct (N × H)
246. #3165	C₂×C₈₉	semidirect
250. #1695	C₂×C₈₉	semidirect
251. #1691	C₂×C₈₉	semidirect
252. #1675	C₂×C₈₉	semidirect
253. #1671	C₂×C₈₉	semidirect
254. #1565	C₂×C₈₉	semidirect
255. #1550	C₂×C₈₉	semidirect
256. #1548	C₂×C₈₉	semidirect
257. C₈₉	#19165	direct (N × H)
263. #500	C₂²×C₈₉	semidirect
264. #498	C₂²×C₈₉	semidirect
289. #335	C₂²×C₈₉	semidirect
290. #334	C₂²×C₈₉	semidirect
291. #333	C₂²×C₈₉	semidirect
292. #333	C₂²×C₈₉	semidirect
293. #333	C₂²×C₈₉	semidirect
294. #332	C₂²×C₈₉	semidirect
383. M₁₆	SD₁₆×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3165	2
C₈₉×#3101	2
C₂×#476×C₈₉	2
D₄×M₁₆×C₈₉	2
C₈₉×#1695	2
C₈₉×#1691	2
C₈₉×#1675	2
C₈₉×#1671	2
C₈₉×#1565	2
C₈₉×#1550	2
C₈₉×#1548	2
#19165	89

Subgroups — C89×#19165 #36397013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#19165 #36397013