Subgroups — C₈₉×#19512 #36397360

Order 22784.

588 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3235	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#3229	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3206	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2981	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1699	C₂	1	2	yes	maximal, complemented
6	C₈₉×#1691	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1679	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1671	C₂	1	2	yes	maximal, complemented
9	C₈₉×#1572	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1556	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1554	C₂	1	1	yes	maximal, characteristic, complemented
12	#533×C₈₉	C₂²	1	1	yes	characteristic
13	#532×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	#501×C₈₉	—	2	1	no	—
16	#498×C₈₉	C₂²	1	1	yes	characteristic
17	#498×C₈₉	C₂²	1	1	yes	characteristic
18	#492×C₈₉	—	2	1	no	—
19	#490×C₈₉	—	2	1	no	—
20	#487×C₈₉	—	2	1	no	—
21	#487×C₈₉	C₂²	1	1	yes	characteristic
22	#487×C₈₉	C₂²	1	1	yes	characteristic
23	#483×C₈₉	—	2	2	no	—
24	#482×C₈₉	—	2	2	no	—
25	#476×C₈₉	C₂²	1	2	yes	—
26	#475×C₈₉	—	2	2	no	—
27	#474×C₈₉	C₂²	1	2	yes	—
28	#461×C₈₉	C₂²	1	2	yes	—
29	#461×C₈₉	C₂²	1	2	yes	—
30	#448×C₈₉	—	2	2	no	—
31	#436×C₈₉	C₂²	1	2	yes	—
32	#435×C₈₉	—	2	2	no	—
33	C₈×D₄×C₈₉	C₂²	1	2	yes	—
34	#432×C₈₉	C₂²	1	1	yes	characteristic
35	#432×C₈₉	C₂²	1	1	yes	characteristic
36	#431×C₈₉	C₂²	1	1	yes	characteristic
37	#431×C₈₉	C₂²	1	1	yes	characteristic
38	#408×C₈₉	—	2	2	no	—
39	#335×C₈₉	C₂²	1	2	yes	—
40	#334×C₈₉	C₂²	1	2	yes	—
41	#333×C₈₉	C₂²	1	2	yes	—
42	#332×C₈₉	C₂²	1	2	yes	—
43	#332×C₈₉	C₂²	1	2	yes	—
44	#332×C₈₉	C₂²	1	2	yes	—
45	C₂×#41×C₈₉	—	2	1	no	—
46	C₂²×Q₈×C₈₉	—	2	1	no	—
47	C₂×SD₁₆×C₈₉	—	2	2	no	—
48	C₂×SD₁₆×C₈₉	—	2	2	no	—
49	C₂×M₁₆×C₈₉	—	2	2	no	—
50	C₂²×C₈×C₈₉	—	2	2	no	—
51	C₄⋊Q₈×C₈₉	—	2	1	no	—
52	C₄⋊Q₈×C₈₉	—	2	1	no	—
53	C₄⋊Q₈×C₈₉	—	2	1	no	—
54	C₄⋊Q₈×C₈₉	D₄	1	1	yes	characteristic
55	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
58	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
59	#127×C₈₉	D₄	1	1	yes	characteristic
60	#125×C₈₉	—	2	1	no	—
61	#124×C₈₉	—	2	1	no	—
62	C₂²⋊Q₈×C₈₉	—	2	1	no	—
63	C₂²⋊Q₈×C₈₉	—	2	1	no	—
64	#121×C₈₉	—	2	1	no	—
65	#121×C₈₉	D₄	1	2	yes	—
66	C₄×Q₈×C₈₉	—	2	1	no	—
67	C₄×Q₈×C₈₉	—	2	1	no	—
68	C₄×D₄×C₈₉	—	2	1	no	—
69	C₄×D₄×C₈₉	C₂³	1	2	yes	—
70	#117×C₈₉	—	2	1	no	—
71	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
72	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
73	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
74	#107×C₈₉	—	2	2	no	—
75	#106×C₈₉	—	2	2	no	—
76	#106×C₈₉	—	2	2	no	—
77	#106×C₈₉	—	2	2	no	—
78	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
79	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
80	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
81	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
82	#103×C₈₉	—	2	2	no	—
83	#103×C₈₉	—	2	2	no	—
84	#103×C₈₉	—	2	2	no	—
85	#103×C₈₉	—	2	2	no	—
86	#102×C₈₉	—	2	2	no	—
87	#102×C₈₉	—	2	2	no	—
88	C₂²⋊C₈×C₈₉	—	2	1	no	—
89	C₂²⋊C₈×C₈₉	—	2	1	no	—
90	C₂²⋊C₈×C₈₉	—	2	1	no	—
91	C₂²⋊C₈×C₈₉	—	2	2	no	—
92	C₂²⋊C₈×C₈₉	—	2	2	no	—
93	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
94	#97×C₈₉	—	2	1	no	—
95	#97×C₈₉	—	2	1	no	—
96	C₄×C₈×C₈₉	—	2	1	no	—
97	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
99	#41×C₈₉	—	8	2	no	—
100	C₂×Q₈×C₈₉	—	2	1	no	—
101	C₂×Q₈×C₈₉	—	2	1	no	—
102	C₂×Q₈×C₈₉	—	2	1	no	—
103	C₂×Q₈×C₈₉	—	2	1	no	—
104	C₂×Q₈×C₈₉	—	2	1	no	—
105	C₂×Q₈×C₈₉	—	4	1	no	—
106	C₂×Q₈×C₈₉	—	8	2	no	—
107	C₂×D₄×C₈₉	—	2	2	no	—
108	C₂×D₄×C₈₉	—	2	1	no	—
109	C₂×D₄×C₈₉	—	2	1	no	—
110	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₈₉	—	2	2	no	—
112	C₂²×C₄×C₈₉	—	4	1	no	—
113	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
116	SD₁₆×C₈₉	—	8	2	no	—
117	SD₁₆×C₈₉	—	8	2	no	—
118	M₁₆×C₈₉	—	8	2	no	—
119	C₂×C₈×C₈₉	—	2	2	no	—
120	C₂×C₈×C₈₉	—	2	2	no	—
121	C₂×C₈×C₈₉	—	2	2	no	—
122	C₂×C₈×C₈₉	—	8	2	no	—
123	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
124	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
125	C₄⋊C₄×C₈₉	—	2	1	no	—
126	C₄⋊C₄×C₈₉	—	2	1	no	—
127	C₄⋊C₄×C₈₉	—	2	1	no	—
128	C₄⋊C₄×C₈₉	—	2	1	no	—
129	C₄⋊C₄×C₈₉	—	2	1	no	—
130	C₄⋊C₄×C₈₉	—	2	1	no	—
131	C₄⋊C₄×C₈₉	—	2	1	no	—
132	C₄⋊C₄×C₈₉	—	2	1	no	—
133	C₄⋊C₄×C₈₉	—	2	1	no	—
134	C₄⋊C₄×C₈₉	—	4	1	no	—
135	C₄⋊C₄×C₈₉	—	4	1	no	—
136	C₄⋊C₄×C₈₉	—	4	1	no	—
137	C₄⋊C₄×C₈₉	—	4	1	no	—
138	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
139	C₂²⋊C₄×C₈₉	—	2	2	no	—
140	C₂²⋊C₄×C₈₉	—	2	2	no	—
141	C₂²⋊C₄×C₈₉	—	4	1	no	—
142	C₄²×C₈₉	—	4	1	no	—
143	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
144	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
145	C₄²×C₈₉	#41	1	1	yes	characteristic
146	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
147	C₂³×C₈₉	—	2	2	no	—
148	C₂³×C₈₉	#121	1	1	yes	characteristic
149	Q₈×C₈₉	—	4	2	no	—
150	Q₈×C₈₉	—	4	2	no	—
151	Q₈×C₈₉	—	4	2	no	—
152	Q₈×C₈₉	—	8	1	no	—
153	Q₈×C₈₉	—	8	1	no	—
154	Q₈×C₈₉	—	8	2	no	—
155	D₄×C₈₉	—	2	2	no	—
156	D₄×C₈₉	—	2	2	no	—
157	D₄×C₈₉	—	8	1	no	—
158	D₄×C₈₉	—	8	1	no	—
159	D₄×C₈₉	—	8	2	no	—
160	C₂×C₄×C₈₉	—	2	2	no	—
161	C₂×C₄×C₈₉	—	2	2	no	—
162	C₂×C₄×C₈₉	—	2	2	no	—
163	C₂×C₄×C₈₉	—	4	1	no	—
164	C₂×C₄×C₈₉	—	4	2	no	—
165	C₂×C₄×C₈₉	—	4	2	no	—
166	C₂×C₄×C₈₉	—	4	1	no	—
167	C₂×C₄×C₈₉	—	4	1	no	—
168	C₂×C₄×C₈₉	—	4	1	no	—
169	C₂×C₄×C₈₉	—	4	1	no	—
170	C₂×C₄×C₈₉	—	8	2	no	—
171	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
172	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
173	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
174	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
177	C₈×C₈₉	—	2	2	no	—
178	C₈×C₈₉	—	2	2	no	—
179	C₈×C₈₉	—	4	2	no	—
180	C₈×C₈₉	—	4	2	no	—
181	C₈×C₈₉	—	4	2	no	—
182	C₂²×C₈₉	—	2	1	no	—
183	C₂²×C₈₉	—	2	1	no	—
184	C₂²×C₈₉	—	2	1	no	—
185	C₂²×C₈₉	—	4	2	no	—
186	C₂²×C₈₉	—	4	2	no	—
187	C₂²×C₈₉	—	4	2	no	—
188	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
189	C₄×C₈₉	—	2	1	no	—
190	C₄×C₈₉	—	2	1	no	—
191	C₄×C₈₉	—	2	1	no	—
192	C₄×C₈₉	—	2	1	no	—
193	C₄×C₈₉	—	4	2	no	—
194	C₄×C₈₉	—	8	1	no	—
195	C₄×C₈₉	—	8	1	no	—
196	C₄×C₈₉	—	8	1	no	—
197	C₄×C₈₉	—	8	1	no	—
198	C₄×C₈₉	#577	1	2	yes	—
199	C₄×C₈₉	#577	1	2	yes	—
200	#19512	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂×C₈₉	—	4	1	no	—
202	C₂×C₈₉	—	8	2	no	—
203	C₂×C₈₉	#3042	1	1	yes	characteristic
204	C₂×C₈₉	#3057	1	1	yes	characteristic
205	C₂×C₈₉	#3069	1	1	yes	characteristic
206	#3235	C₂×C₈₉	1	1	yes	characteristic
207	#3229	C₂×C₈₉	1	1	yes	characteristic, complemented
208	#3206	C₂×C₈₉	1	1	yes	characteristic
209	#2981	C₂×C₈₉	1	1	yes	characteristic
210	#1699	C₂×C₈₉	1	2	yes	complemented
211	#1691	C₂×C₈₉	1	2	yes	complemented
212	#1679	C₂×C₈₉	1	2	yes	complemented
213	#1671	C₂×C₈₉	1	2	yes	complemented
214	#1572	C₂×C₈₉	1	1	yes	characteristic, complemented
215	#1556	C₂×C₈₉	1	1	yes	characteristic, complemented
216	#1554	C₂×C₈₉	1	1	yes	characteristic, complemented
217	C₈₉	#19512	1	1	yes	characteristic, direct product factor, Sylow
218	#533	C₂²×C₈₉	1	1	yes	characteristic
219	#532	C₂²×C₈₉	1	1	yes	characteristic
220	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
221	#501	—	2	1	no	—
222	#498	C₂²×C₈₉	1	1	yes	characteristic
223	#498	C₂²×C₈₉	1	1	yes	characteristic
224	#492	—	2	1	no	—
225	#490	—	2	1	no	—
226	#487	—	2	1	no	—
227	#487	C₂²×C₈₉	1	1	yes	characteristic
228	#487	C₂²×C₈₉	1	1	yes	characteristic
229	#483	—	2	2	no	—
230	#482	—	2	2	no	—
231	#476	C₂²×C₈₉	1	2	yes	—
232	#475	—	2	2	no	—
233	#474	C₂²×C₈₉	1	2	yes	—
234	#461	C₂²×C₈₉	1	2	yes	—
235	#461	C₂²×C₈₉	1	2	yes	—
236	#448	—	2	2	no	—
237	#436	C₂²×C₈₉	1	2	yes	—
238	#435	—	2	2	no	—
239	C₈×D₄	C₂²×C₈₉	1	2	yes	—
240	#432	C₂²×C₈₉	1	1	yes	characteristic
241	#432	C₂²×C₈₉	1	1	yes	characteristic
242	#431	C₂²×C₈₉	1	1	yes	characteristic
243	#431	C₂²×C₈₉	1	1	yes	characteristic
244	#408	—	2	2	no	—
245	#335	C₂²×C₈₉	1	2	yes	—
246	#334	C₂²×C₈₉	1	2	yes	—
247	#333	C₂²×C₈₉	1	2	yes	—
248	#332	C₂²×C₈₉	1	2	yes	—
249	#332	C₂²×C₈₉	1	2	yes	—
250	#332	C₂²×C₈₉	1	2	yes	—
251	C₂×#41	—	2	1	no	—
252	C₂²×Q₈	—	2	1	no	—
253	C₂×SD₁₆	—	2	2	no	—
254	C₂×SD₁₆	—	2	2	no	—
255	C₂×M₁₆	—	2	2	no	—
256	C₂²×C₈	—	2	2	no	—
257	C₄⋊Q₈	—	2	1	no	—
258	C₄⋊Q₈	—	2	1	no	—
259	C₄⋊Q₈	—	2	1	no	—
260	C₄⋊Q₈	D₄×C₈₉	1	1	yes	characteristic
261	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
262	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
263	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
264	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
265	#127	D₄×C₈₉	1	1	yes	characteristic
266	#125	—	2	1	no	—
267	#124	—	2	1	no	—
268	C₂²⋊Q₈	—	2	1	no	—
269	C₂²⋊Q₈	—	2	1	no	—
270	#121	—	2	1	no	—
271	#121	D₄×C₈₉	1	2	yes	—
272	C₄×Q₈	—	2	1	no	—
273	C₄×Q₈	—	2	1	no	—
274	C₄×D₄	—	2	1	no	—
275	C₄×D₄	C₂³×C₈₉	1	2	yes	—
276	#117	—	2	1	no	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	—	2	1	no	—
279	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
280	#107	—	2	2	no	—
281	#106	—	2	2	no	—
282	#106	—	2	2	no	—
283	#106	—	2	2	no	—
284	C₄⋊C₈	D₄×C₈₉	1	2	yes	—
285	C₄⋊C₈	C₂³×C₈₉	1	2	yes	—
286	C₄⋊C₈	C₂³×C₈₉	1	2	yes	—
287	C₄⋊C₈	C₂³×C₈₉	1	2	yes	—
288	#103	—	2	2	no	—
289	#103	—	2	2	no	—
290	#103	—	2	2	no	—
291	#103	—	2	2	no	—
292	#102	—	2	2	no	—
293	#102	—	2	2	no	—
294	C₂²⋊C₈	—	2	1	no	—
295	C₂²⋊C₈	—	2	1	no	—
296	C₂²⋊C₈	—	2	1	no	—
297	C₂²⋊C₈	—	2	2	no	—
298	C₂²⋊C₈	—	2	2	no	—
299	C₂²⋊C₈	D₄×C₈₉	1	2	yes	—
300	#97	—	2	1	no	—
301	#97	—	2	1	no	—
302	C₄×C₈	—	2	1	no	—
303	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
304	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
305	#41	—	8	2	no	—
306	C₂×Q₈	—	2	1	no	—
307	C₂×Q₈	—	2	1	no	—
308	C₂×Q₈	—	2	1	no	—
309	C₂×Q₈	—	2	1	no	—
310	C₂×Q₈	—	2	1	no	—
311	C₂×Q₈	—	4	1	no	—
312	C₂×Q₈	—	8	2	no	—
313	C₂×D₄	—	2	2	no	—
314	C₂×D₄	—	2	1	no	—
315	C₂×D₄	—	2	1	no	—
316	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	4	1	no	—
319	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
321	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
322	SD₁₆	—	8	2	no	—
323	SD₁₆	—	8	2	no	—
324	M₁₆	—	8	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	8	2	no	—
329	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
330	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
331	C₄⋊C₄	—	2	1	no	—
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	2	1	no	—
335	C₄⋊C₄	—	2	1	no	—
336	C₄⋊C₄	—	2	1	no	—
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	2	1	no	—
340	C₄⋊C₄	—	4	1	no	—
341	C₄⋊C₄	—	4	1	no	—
342	C₄⋊C₄	—	4	1	no	—
343	C₄⋊C₄	—	4	1	no	—
344	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
345	C₂²⋊C₄	—	2	2	no	—
346	C₂²⋊C₄	—	2	2	no	—
347	C₂²⋊C₄	—	4	1	no	—
348	C₄²	—	4	1	no	—
349	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
350	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
351	C₄²	#41×C₈₉	1	1	yes	characteristic
352	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
353	C₂³	—	2	2	no	—
354	C₂³	#121×C₈₉	1	1	yes	characteristic
355	Q₈	—	4	2	no	—
356	Q₈	—	4	2	no	—
357	Q₈	—	4	2	no	—
358	Q₈	—	8	1	no	—
359	Q₈	—	8	1	no	—
360	Q₈	—	8	2	no	—
361	D₄	—	2	2	no	—
362	D₄	—	2	2	no	—
363	D₄	—	8	1	no	—
364	D₄	—	8	1	no	—
365	D₄	—	8	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	8	2	no	—
377	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
378	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
379	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
380	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
381	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
382	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
383	C₈	—	2	2	no	—
384	C₈	—	2	2	no	—
385	C₈	—	4	2	no	—
386	C₈	—	4	2	no	—
387	C₈	—	4	2	no	—
388	C₂²	—	2	1	no	—
389	C₂²	—	2	1	no	—
390	C₂²	—	2	1	no	—
391	C₂²	—	4	2	no	—
392	C₂²	—	4	2	no	—
393	C₂²	—	4	2	no	—
394	C₂²	C₂×#121×C₈₉	1	1	yes	characteristic
395	C₄	—	2	1	no	—
396	C₄	—	2	1	no	—
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	—	4	2	no	—
400	C₄	—	8	1	no	—
401	C₄	—	8	1	no	—
402	C₄	—	8	1	no	—
403	C₄	—	8	1	no	—
404	C₄	#577×C₈₉	1	2	yes	—
405	C₄	#577×C₈₉	1	2	yes	—
406	C₂	—	4	1	no	—
407	C₂	—	8	2	no	—
408	C₂	C₈₉×#3042	1	1	yes	characteristic
409	C₂	C₈₉×#3057	1	1	yes	characteristic
410	C₂	C₈₉×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₈₉
prime radical	#532×C₈₉	C₂²
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
2. C₈₉×#3229	C₂	semidirect
5. C₈₉×#1699	C₂	semidirect
6. C₈₉×#1691	C₂	semidirect
7. C₈₉×#1679	C₂	semidirect
8. C₈₉×#1671	C₂	semidirect
9. C₈₉×#1572	C₂	semidirect
10. C₈₉×#1556	C₂	semidirect
11. C₈₉×#1554	C₂	semidirect
200. #19512	C₈₉	direct (N × H)
207. #3229	C₂×C₈₉	semidirect
210. #1699	C₂×C₈₉	semidirect
211. #1691	C₂×C₈₉	semidirect
212. #1679	C₂×C₈₉	semidirect
213. #1671	C₂×C₈₉	semidirect
214. #1572	C₂×C₈₉	semidirect
215. #1556	C₂×C₈₉	semidirect
216. #1554	C₂×C₈₉	semidirect
217. C₈₉	#19512	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×#3235	2
C₈₉×#3229	2
C₈₉×#3206	2
C₈₉×#2981	2
C₈₉×#1699	2
C₈₉×#1691	2
C₈₉×#1679	2
C₈₉×#1671	2
C₈₉×#1572	2
C₈₉×#1556	2
C₈₉×#1554	2
#19512	89

Subgroups — C89×#19512 #36397360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#19512 #36397360