Subgroups — C₈₉×#8426 #36386274

Order 22784.

836 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3508	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1982	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#1806	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#1550	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1545	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#1503	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1477	C₂	1	1	yes	maximal, characteristic, complemented
8	#559×C₈₉	—	2	1	no	—
9	#559×C₈₉	C₄	1	2	yes	complemented
10	#550×C₈₉	—	2	1	no	—
11	#550×C₈₉	C₄	1	2	yes	complemented
12	#546×C₈₉	—	2	2	no	—
13	#546×C₈₉	—	2	2	no	—
14	#540×C₈₉	—	2	2	no	—
15	#535×C₈₉	—	2	1	no	—
16	#533×C₈₉	C₂²	1	1	yes	characteristic
17	#532×C₈₉	—	2	1	no	—
18	#532×C₈₉	—	2	1	no	—
19	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
20	#518×C₈₉	—	2	1	no	—
21	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
22	#431×C₈₉	C₂²	1	1	yes	characteristic
23	#423×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×#104×C₈₉	—	2	1	no	—
26	C₂×#102×C₈₉	—	2	2	no	—
27	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
28	#339×C₈₉	—	2	1	no	—
29	#333×C₈₉	—	2	1	no	—
30	#332×C₈₉	—	2	1	no	—
31	#331×C₈₉	—	2	2	no	—
32	#328×C₈₉	—	2	2	no	—
33	Q₈⋊C₈×C₈₉	—	2	1	no	—
34	D₄⋊C₈×C₈₉	—	2	1	no	—
35	C₂×#41×C₈₉	—	2	2	no	—
36	C₂×#41×C₈₉	—	2	2	no	—
37	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
38	C₂²×D₄×C₈₉	—	2	1	no	—
39	C₂²×D₄×C₈₉	—	2	1	no	—
40	C₂²×D₄×C₈₉	D₄	1	2	yes	—
41	C₂×M₁₆×C₈₉	—	2	1	no	—
42	C₂²×C₈×C₈₉	—	2	1	no	—
43	C₄⋊Q₈×C₈₉	D₄	1	1	yes	characteristic
44	C₄⋊Q₈×C₈₉	D₄	1	1	yes	characteristic
45	#127×C₈₉	—	2	2	no	—
46	#127×C₈₉	—	2	1	no	—
47	#127×C₈₉	—	2	1	no	—
48	#127×C₈₉	C₂×C₄	1	2	yes	—
49	#127×C₈₉	D₄	1	2	yes	—
50	#125×C₈₉	—	2	1	no	—
51	#124×C₈₉	—	2	2	no	—
52	#124×C₈₉	—	2	2	no	—
53	#124×C₈₉	—	2	2	no	—
54	#123×C₈₉	—	2	4	no	—
55	#123×C₈₉	—	2	4	no	—
56	C₂²⋊Q₈×C₈₉	—	2	1	no	—
57	C₂²⋊Q₈×C₈₉	D₄	1	2	yes	—
58	#121×C₈₉	—	2	2	no	—
59	#121×C₈₉	—	2	2	no	—
60	#121×C₈₉	—	2	2	no	—
61	#121×C₈₉	—	2	4	no	—
62	#121×C₈₉	—	2	4	no	—
63	#121×C₈₉	—	2	2	no	—
64	#120×C₈₉	—	2	4	no	—
65	C₄×Q₈×C₈₉	—	2	1	no	—
66	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
67	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
68	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
69	C₄×D₄×C₈₉	—	2	2	no	—
70	C₄×D₄×C₈₉	—	2	2	no	—
71	C₄×D₄×C₈₉	—	2	2	no	—
72	C₄×D₄×C₈₉	—	2	1	no	—
73	C₄×D₄×C₈₉	—	2	1	no	—
74	C₄×D₄×C₈₉	—	2	2	no	—
75	C₄×D₄×C₈₉	—	2	2	no	—
76	C₄×D₄×C₈₉	—	2	2	no	—
77	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
78	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
79	#117×C₈₉	—	2	1	no	—
80	#117×C₈₉	—	2	1	no	—
81	#117×C₈₉	D₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
83	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
84	C₂×C₄²×C₈₉	—	2	1	no	—
85	C₂×C₄²×C₈₉	—	2	1	no	—
86	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₈₉	—	2	1	no	—
88	C₄⋊C₈×C₈₉	—	2	1	no	—
89	C₄⋊C₈×C₈₉	—	2	1	no	—
90	C₄⋊C₈×C₈₉	—	2	1	no	—
91	C₄⋊C₈×C₈₉	—	2	1	no	—
92	#104×C₈₉	—	4	2	no	—
93	#102×C₈₉	—	4	4	no	—
94	C₂²⋊C₈×C₈₉	—	2	1	no	—
95	C₄×C₈×C₈₉	—	2	1	no	—
96	#95×C₈₉	—	2	1	no	—
97	C₂⁴×C₈₉	—	2	2	no	—
98	#41×C₈₉	—	2	1	no	—
99	#41×C₈₉	—	2	1	no	—
100	#41×C₈₉	—	2	1	no	—
101	#41×C₈₉	—	2	1	no	—
102	#41×C₈₉	—	8	4	no	—
103	#41×C₈₉	—	8	2	no	—
104	#41×C₈₉	—	8	2	no	—
105	C₂×Q₈×C₈₉	—	2	1	no	—
106	C₂×Q₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×Q₈×C₈₉	C₂²⋊C₄	1	2	yes	—
108	C₂×D₄×C₈₉	—	2	1	no	—
109	C₂×D₄×C₈₉	—	2	4	no	—
110	C₂×D₄×C₈₉	—	2	4	no	—
111	C₂×D₄×C₈₉	—	2	2	no	—
112	C₂×D₄×C₈₉	—	2	2	no	—
113	C₂×D₄×C₈₉	—	2	2	no	—
114	C₂×D₄×C₈₉	—	2	2	no	—
115	C₂×D₄×C₈₉	—	2	2	no	—
116	C₂×D₄×C₈₉	—	4	2	no	—
117	C₂×D₄×C₈₉	—	4	2	no	—
118	C₂×D₄×C₈₉	—	4	1	no	—
119	C₂×D₄×C₈₉	—	4	1	no	—
120	C₂×D₄×C₈₉	—	4	1	no	—
121	C₂×D₄×C₈₉	—	4	1	no	—
122	C₂×D₄×C₈₉	—	8	2	no	—
123	C₂×D₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
124	C₂×D₄×C₈₉	C₂²⋊C₄	1	4	yes	—
125	C₂²×C₄×C₈₉	—	2	4	no	—
126	C₂²×C₄×C₈₉	—	2	4	no	—
127	C₂²×C₄×C₈₉	—	2	1	no	—
128	C₂²×C₄×C₈₉	—	2	1	no	—
129	C₂²×C₄×C₈₉	—	2	1	no	—
130	C₂²×C₄×C₈₉	—	4	1	no	—
131	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
134	M₁₆×C₈₉	—	4	1	no	—
135	M₁₆×C₈₉	—	4	1	no	—
136	C₂×C₈×C₈₉	—	4	2	no	—
137	C₂×C₈×C₈₉	—	4	1	no	—
138	C₂×C₈×C₈₉	—	4	1	no	—
139	C₂×C₈×C₈₉	—	4	1	no	—
140	C₄⋊C₄×C₈₉	—	2	1	no	—
141	C₄⋊C₄×C₈₉	—	2	2	no	—
142	C₄⋊C₄×C₈₉	—	2	1	no	—
143	C₄⋊C₄×C₈₉	—	2	1	no	—
144	C₄⋊C₄×C₈₉	—	2	1	no	—
145	C₄⋊C₄×C₈₉	—	2	1	no	—
146	C₄⋊C₄×C₈₉	—	2	2	no	—
147	C₄⋊C₄×C₈₉	—	2	2	no	—
148	C₄⋊C₄×C₈₉	—	4	1	no	—
149	C₄⋊C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
150	C₄⋊C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
151	C₄⋊C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
152	C₂²⋊C₄×C₈₉	—	2	4	no	—
153	C₂²⋊C₄×C₈₉	—	2	4	no	—
154	C₂²⋊C₄×C₈₉	—	2	4	no	—
155	C₂²⋊C₄×C₈₉	—	2	4	no	—
156	C₂²⋊C₄×C₈₉	—	2	1	no	—
157	C₂²⋊C₄×C₈₉	—	2	2	no	—
158	C₂²⋊C₄×C₈₉	—	2	1	no	—
159	C₂²⋊C₄×C₈₉	—	2	1	no	—
160	C₄²×C₈₉	—	2	1	no	—
161	C₄²×C₈₉	—	2	2	no	—
162	C₄²×C₈₉	—	2	1	no	—
163	C₄²×C₈₉	—	2	1	no	—
164	C₄²×C₈₉	—	4	1	no	—
165	C₄²×C₈₉	—	4	1	no	—
166	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
167	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
168	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
169	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
170	C₂³×C₈₉	—	2	4	no	—
171	C₂³×C₈₉	—	2	1	no	—
172	C₂³×C₈₉	—	4	2	no	—
173	C₂³×C₈₉	—	4	2	no	—
174	C₂³×C₈₉	—	8	2	no	—
175	C₂³×C₈₉	—	8	2	no	—
176	C₂³×C₈₉	#120	1	1	yes	characteristic
177	Q₈×C₈₉	—	2	1	no	—
178	Q₈×C₈₉	—	2	1	no	—
179	Q₈×C₈₉	—	4	2	no	—
180	Q₈×C₈₉	—	4	1	no	—
181	Q₈×C₈₉	—	4	1	no	—
182	D₄×C₈₉	—	2	1	no	—
183	D₄×C₈₉	—	2	1	no	—
184	D₄×C₈₉	—	4	4	no	—
185	D₄×C₈₉	—	4	4	no	—
186	D₄×C₈₉	—	4	1	no	—
187	D₄×C₈₉	—	4	1	no	—
188	D₄×C₈₉	—	4	2	no	—
189	D₄×C₈₉	—	4	2	no	—
190	D₄×C₈₉	—	4	2	no	—
191	D₄×C₈₉	—	4	2	no	—
192	D₄×C₈₉	—	4	2	no	—
193	D₄×C₈₉	—	4	2	no	—
194	D₄×C₈₉	—	8	2	no	—
195	D₄×C₈₉	—	8	4	no	—
196	D₄×C₈₉	—	8	4	no	—
197	C₂×C₄×C₈₉	—	2	1	no	—
198	C₂×C₄×C₈₉	—	2	1	no	—
199	C₂×C₄×C₈₉	—	2	1	no	—
200	C₂×C₄×C₈₉	—	2	1	no	—
201	C₂×C₄×C₈₉	—	2	1	no	—
202	C₂×C₄×C₈₉	—	2	2	no	—
203	C₂×C₄×C₈₉	—	2	2	no	—
204	C₂×C₄×C₈₉	—	4	1	no	—
205	C₂×C₄×C₈₉	—	4	2	no	—
206	C₂×C₄×C₈₉	—	4	1	no	—
207	C₂×C₄×C₈₉	—	4	1	no	—
208	C₂×C₄×C₈₉	—	4	1	no	—
209	C₂×C₄×C₈₉	—	4	1	no	—
210	C₂×C₄×C₈₉	—	4	2	no	—
211	C₂×C₄×C₈₉	—	8	4	no	—
212	C₂×C₄×C₈₉	—	8	4	no	—
213	C₂×C₄×C₈₉	—	8	1	no	—
214	C₂×C₄×C₈₉	—	8	1	no	—
215	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
216	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
218	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
219	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
220	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
221	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
222	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
223	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
224	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
225	C₈×C₈₉	—	8	1	no	—
226	C₈×C₈₉	—	8	1	no	—
227	C₈×C₈₉	—	8	1	no	—
228	C₂²×C₈₉	—	2	1	no	—
229	C₂²×C₈₉	—	2	1	no	—
230	C₂²×C₈₉	—	4	1	no	—
231	C₂²×C₈₉	—	4	4	no	—
232	C₂²×C₈₉	—	4	1	no	—
233	C₂²×C₈₉	—	4	4	no	—
234	C₂²×C₈₉	—	4	1	no	—
235	C₂²×C₈₉	—	4	4	no	—
236	C₂²×C₈₉	—	16	2	no	—
237	C₂²×C₈₉	—	16	2	no	—
238	C₂²×C₈₉	#379	1	1	yes	characteristic
239	C₂²×C₈₉	#449	1	1	yes	characteristic
240	C₂²×C₈₉	#452	1	1	yes	characteristic
241	C₄×C₈₉	—	2	1	no	—
242	C₄×C₈₉	—	2	1	no	—
243	C₄×C₈₉	—	2	1	no	—
244	C₄×C₈₉	—	4	1	no	—
245	C₄×C₈₉	—	4	2	no	—
246	C₄×C₈₉	—	4	1	no	—
247	C₄×C₈₉	—	4	2	no	—
248	C₄×C₈₉	—	4	1	no	—
249	C₄×C₈₉	—	8	2	no	—
250	C₄×C₈₉	#416	1	1	yes	characteristic
251	C₄×C₈₉	#417	1	1	yes	characteristic
252	C₄×C₈₉	#449	1	1	yes	characteristic
253	C₄×C₈₉	#452	1	1	yes	characteristic
254	C₄×C₈₉	#453	1	1	yes	characteristic
255	C₄×C₈₉	#455	1	1	yes	characteristic
256	#8426	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
257	C₂×C₈₉	—	2	1	no	—
258	C₂×C₈₉	—	2	1	no	—
259	C₂×C₈₉	—	8	1	no	—
260	C₂×C₈₉	—	8	4	no	—
261	C₂×C₈₉	#1788	1	1	yes	characteristic
262	C₂×C₈₉	#1792	1	1	yes	characteristic
263	C₂×C₈₉	#1795	1	1	yes	characteristic
264	#3508	C₂×C₈₉	1	1	yes	characteristic
265	#1982	C₂×C₈₉	1	1	yes	characteristic, complemented
266	#1806	C₂×C₈₉	1	1	yes	characteristic, complemented
267	#1550	C₂×C₈₉	1	1	yes	characteristic, complemented
268	#1545	C₂×C₈₉	1	1	yes	characteristic, complemented
269	#1503	C₂×C₈₉	1	1	yes	characteristic, complemented
270	#1477	C₂×C₈₉	1	1	yes	characteristic, complemented
271	C₈₉	#8426	1	1	yes	characteristic, direct product factor, Sylow
272	#559	—	2	1	no	—
273	#559	C₄×C₈₉	1	2	yes	complemented
274	#550	—	2	1	no	—
275	#550	C₄×C₈₉	1	2	yes	complemented
276	#546	—	2	2	no	—
277	#546	—	2	2	no	—
278	#540	—	2	2	no	—
279	#535	—	2	1	no	—
280	#533	C₂²×C₈₉	1	1	yes	characteristic
281	#532	—	2	1	no	—
282	#532	—	2	1	no	—
283	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
284	#518	—	2	1	no	—
285	C₄×#41	C₂²×C₈₉	1	1	yes	characteristic
286	#431	C₂²×C₈₉	1	1	yes	characteristic
287	#423	C₂²×C₈₉	1	1	yes	characteristic
288	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
289	C₂×#104	—	2	1	no	—
290	C₂×#102	—	2	2	no	—
291	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
292	#339	—	2	1	no	—
293	#333	—	2	1	no	—
294	#332	—	2	1	no	—
295	#331	—	2	2	no	—
296	#328	—	2	2	no	—
297	Q₈⋊C₈	—	2	1	no	—
298	D₄⋊C₈	—	2	1	no	—
299	C₂×#41	—	2	2	no	—
300	C₂×#41	—	2	2	no	—
301	C₂×#41	D₄×C₈₉	1	1	yes	characteristic
302	C₂²×D₄	—	2	1	no	—
303	C₂²×D₄	—	2	1	no	—
304	C₂²×D₄	D₄×C₈₉	1	2	yes	—
305	C₂×M₁₆	—	2	1	no	—
306	C₂²×C₈	—	2	1	no	—
307	C₄⋊Q₈	D₄×C₈₉	1	1	yes	characteristic
308	C₄⋊Q₈	D₄×C₈₉	1	1	yes	characteristic
309	#127	—	2	2	no	—
310	#127	—	2	1	no	—
311	#127	—	2	1	no	—
312	#127	C₂×C₄×C₈₉	1	2	yes	—
313	#127	D₄×C₈₉	1	2	yes	—
314	#125	—	2	1	no	—
315	#124	—	2	2	no	—
316	#124	—	2	2	no	—
317	#124	—	2	2	no	—
318	#123	—	2	4	no	—
319	#123	—	2	4	no	—
320	C₂²⋊Q₈	—	2	1	no	—
321	C₂²⋊Q₈	D₄×C₈₉	1	2	yes	—
322	#121	—	2	2	no	—
323	#121	—	2	2	no	—
324	#121	—	2	2	no	—
325	#121	—	2	4	no	—
326	#121	—	2	4	no	—
327	#121	—	2	2	no	—
328	#120	—	2	4	no	—
329	C₄×Q₈	—	2	1	no	—
330	C₄×Q₈	C₂×C₄×C₈₉	1	1	yes	characteristic
331	C₄×Q₈	C₂×C₄×C₈₉	1	2	yes	—
332	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
333	C₄×D₄	—	2	2	no	—
334	C₄×D₄	—	2	2	no	—
335	C₄×D₄	—	2	2	no	—
336	C₄×D₄	—	2	1	no	—
337	C₄×D₄	—	2	1	no	—
338	C₄×D₄	—	2	2	no	—
339	C₄×D₄	—	2	2	no	—
340	C₄×D₄	—	2	2	no	—
341	C₄×D₄	C₂×C₄×C₈₉	1	1	yes	characteristic
342	C₄×D₄	D₄×C₈₉	1	1	yes	characteristic
343	#117	—	2	1	no	—
344	#117	—	2	1	no	—
345	#117	D₄×C₈₉	1	1	yes	characteristic
346	C₂×C₄⋊C₄	—	2	1	no	—
347	C₂×C₂²⋊C₄	—	2	4	no	—
348	C₂×C₄²	—	2	1	no	—
349	C₂×C₄²	—	2	1	no	—
350	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
351	C₄⋊C₈	—	2	1	no	—
352	C₄⋊C₈	—	2	1	no	—
353	C₄⋊C₈	—	2	1	no	—
354	C₄⋊C₈	—	2	1	no	—
355	C₄⋊C₈	—	2	1	no	—
356	#104	—	4	2	no	—
357	#102	—	4	4	no	—
358	C₂²⋊C₈	—	2	1	no	—
359	C₄×C₈	—	2	1	no	—
360	#95	—	2	1	no	—
361	C₂⁴	—	2	2	no	—
362	#41	—	2	1	no	—
363	#41	—	2	1	no	—
364	#41	—	2	1	no	—
365	#41	—	2	1	no	—
366	#41	—	8	4	no	—
367	#41	—	8	2	no	—
368	#41	—	8	2	no	—
369	C₂×Q₈	—	2	1	no	—
370	C₂×Q₈	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
371	C₂×Q₈	C₂²⋊C₄×C₈₉	1	2	yes	—
372	C₂×D₄	—	2	1	no	—
373	C₂×D₄	—	2	4	no	—
374	C₂×D₄	—	2	4	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	4	2	no	—
381	C₂×D₄	—	4	2	no	—
382	C₂×D₄	—	4	1	no	—
383	C₂×D₄	—	4	1	no	—
384	C₂×D₄	—	4	1	no	—
385	C₂×D₄	—	4	1	no	—
386	C₂×D₄	—	8	2	no	—
387	C₂×D₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
388	C₂×D₄	C₂²⋊C₄×C₈₉	1	4	yes	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	4	1	no	—
395	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
397	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
398	M₁₆	—	4	1	no	—
399	M₁₆	—	4	1	no	—
400	C₂×C₈	—	4	2	no	—
401	C₂×C₈	—	4	1	no	—
402	C₂×C₈	—	4	1	no	—
403	C₂×C₈	—	4	1	no	—
404	C₄⋊C₄	—	2	1	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	4	1	no	—
413	C₄⋊C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
414	C₄⋊C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
415	C₄⋊C₄	C₂²⋊C₄×C₈₉	1	2	yes	—
416	C₂²⋊C₄	—	2	4	no	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₂²⋊C₄	—	2	4	no	—
419	C₂²⋊C₄	—	2	4	no	—
420	C₂²⋊C₄	—	2	1	no	—
421	C₂²⋊C₄	—	2	2	no	—
422	C₂²⋊C₄	—	2	1	no	—
423	C₂²⋊C₄	—	2	1	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	2	2	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	2	1	no	—
428	C₄²	—	4	1	no	—
429	C₄²	—	4	1	no	—
430	C₄²	C₂²×C₄×C₈₉	1	1	yes	characteristic
431	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
432	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
433	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
434	C₂³	—	2	4	no	—
435	C₂³	—	2	1	no	—
436	C₂³	—	4	2	no	—
437	C₂³	—	4	2	no	—
438	C₂³	—	8	2	no	—
439	C₂³	—	8	2	no	—
440	C₂³	#120×C₈₉	1	1	yes	characteristic
441	Q₈	—	2	1	no	—
442	Q₈	—	2	1	no	—
443	Q₈	—	4	2	no	—
444	Q₈	—	4	1	no	—
445	Q₈	—	4	1	no	—
446	D₄	—	2	1	no	—
447	D₄	—	2	1	no	—
448	D₄	—	4	4	no	—
449	D₄	—	4	4	no	—
450	D₄	—	4	1	no	—
451	D₄	—	4	1	no	—
452	D₄	—	4	2	no	—
453	D₄	—	4	2	no	—
454	D₄	—	4	2	no	—
455	D₄	—	4	2	no	—
456	D₄	—	4	2	no	—
457	D₄	—	4	2	no	—
458	D₄	—	8	2	no	—
459	D₄	—	8	4	no	—
460	D₄	—	8	4	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	1	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	8	4	no	—
476	C₂×C₄	—	8	4	no	—
477	C₂×C₄	—	8	1	no	—
478	C₂×C₄	—	8	1	no	—
479	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
480	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
481	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
482	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
483	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
484	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
485	C₂×C₄	#135×C₈₉	1	1	yes	characteristic
486	C₂×C₄	#135×C₈₉	1	1	yes	characteristic
487	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
488	C₂×C₄	#137×C₈₉	1	1	yes	characteristic
489	C₈	—	8	1	no	—
490	C₈	—	8	1	no	—
491	C₈	—	8	1	no	—
492	C₂²	—	2	1	no	—
493	C₂²	—	2	1	no	—
494	C₂²	—	4	1	no	—
495	C₂²	—	4	4	no	—
496	C₂²	—	4	1	no	—
497	C₂²	—	4	4	no	—
498	C₂²	—	4	1	no	—
499	C₂²	—	4	4	no	—
500	C₂²	—	16	2	no	—
501	C₂²	—	16	2	no	—
502	C₂²	#379×C₈₉	1	1	yes	characteristic
503	C₂²	#449×C₈₉	1	1	yes	characteristic
504	C₂²	#452×C₈₉	1	1	yes	characteristic
505	C₄	—	2	1	no	—
506	C₄	—	2	1	no	—
507	C₄	—	2	1	no	—
508	C₄	—	4	1	no	—
509	C₄	—	4	1	no	—
510	C₄	—	4	2	no	—
511	C₄	—	4	1	no	—
512	C₄	—	4	2	no	—
513	C₄	—	8	2	no	—
514	C₄	#416×C₈₉	1	1	yes	characteristic
515	C₄	#417×C₈₉	1	1	yes	characteristic
516	C₄	#449×C₈₉	1	1	yes	characteristic
517	C₄	#452×C₈₉	1	1	yes	characteristic
518	C₄	#453×C₈₉	1	1	yes	characteristic
519	C₄	#455×C₈₉	1	1	yes	characteristic
520	C₂	—	2	1	no	—
521	C₂	—	2	1	no	—
522	C₂	—	8	1	no	—
523	C₂	—	8	4	no	—
524	C₂	C₈₉×#1788	1	1	yes	characteristic
525	C₂	C₈₉×#1792	1	1	yes	characteristic
526	C₂	C₈₉×#1795	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	#379
commutator	C₄²	C₂²×C₄×C₈₉
Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
prime radical	C₈₉×#3508	C₂
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×#1982	C₂	semidirect
3. C₈₉×#1806	C₂	semidirect
4. C₈₉×#1550	C₂	semidirect
5. C₈₉×#1545	C₂	semidirect
6. C₈₉×#1503	C₂	semidirect
7. C₈₉×#1477	C₂	semidirect
9. #559×C₈₉	C₄	semidirect
11. #550×C₈₉	C₄	semidirect
256. #8426	C₈₉	direct (N × H)
265. #1982	C₂×C₈₉	semidirect
266. #1806	C₂×C₈₉	semidirect
267. #1550	C₂×C₈₉	semidirect
268. #1545	C₂×C₈₉	semidirect
269. #1503	C₂×C₈₉	semidirect
270. #1477	C₂×C₈₉	semidirect
271. C₈₉	#8426	direct (N × H)
273. #559	C₄×C₈₉	semidirect
275. #550	C₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3508	2
C₈₉×#1982	2
C₈₉×#1806	2
C₈₉×#1550	2
C₈₉×#1545	2
C₈₉×#1503	2
C₈₉×#1477	2
#8426	89

Subgroups — C89×#8426 #36386274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#8426 #36386274