Subgroups — C₈₉×#18665 #36396513

Order 22784.

926 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3501	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#3249	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#3077	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1649	C₂	1	2	yes	maximal, complemented
6	C₈₉×#1644	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1637	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1632	C₂	1	2	yes	maximal, complemented
9	C₈₉×#1550	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1493	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×D₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
12	#554×C₈₉	—	2	1	no	—
13	#554×C₈₉	—	2	2	no	—
14	#554×C₈₉	—	2	2	no	—
15	#554×C₈₉	C₂²	1	2	yes	—
16	D₄×Q₈×C₈₉	—	2	1	no	—
17	D₄×Q₈×C₈₉	C₂²	1	2	yes	—
18	#544×C₈₉	—	2	1	no	—
19	#543×C₈₉	—	2	2	no	—
20	#533×C₈₉	C₂²	1	1	yes	characteristic
21	#533×C₈₉	C₂²	1	1	yes	characteristic
22	#533×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Q₈×C₈₉	—	2	2	no	—
24	#518×C₈₉	—	2	1	no	—
25	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
26	#500×C₈₉	C₂²	1	1	yes	characteristic, complemented
27	#494×C₈₉	C₂²	1	1	yes	characteristic, complemented
28	#487×C₈₉	C₂²	1	1	yes	characteristic, complemented
29	#487×C₈₉	C₂²	1	1	yes	characteristic, complemented
30	#478×C₈₉	—	2	2	no	—
31	#477×C₈₉	—	2	1	no	—
32	#477×C₈₉	C₂²	1	2	yes	complemented
33	#476×C₈₉	C₂²	1	2	yes	—
34	#474×C₈₉	C₂²	1	2	yes	—
35	#467×C₈₉	—	2	2	no	—
36	#467×C₈₉	—	2	2	no	—
37	#462×C₈₉	—	2	1	no	—
38	#462×C₈₉	C₂²	1	2	yes	complemented
39	C₄×Q₁₆×C₈₉	—	2	2	no	—
40	#428×C₈₉	—	2	1	no	—
41	#423×C₈₉	C₂²	1	1	yes	characteristic
42	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
43	#418×C₈₉	—	2	1	no	—
44	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
45	#333×C₈₉	C₂²	1	2	yes	complemented
46	#332×C₈₉	C₂²	1	2	yes	complemented
47	Q₈⋊C₈×C₈₉	C₂²	1	2	yes	complemented
48	Q₈⋊C₈×C₈₉	C₂²	1	2	yes	complemented
49	D₄⋊C₈×C₈₉	C₂²	1	2	yes	—
50	D₄⋊C₈×C₈₉	C₂²	1	2	yes	—
51	C₂²×Q₈×C₈₉	—	2	2	no	—
52	C₂²×D₄×C₈₉	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₈₉	—	2	1	no	—
54	C₂×Q₁₆×C₈₉	—	2	2	no	—
55	C₂×Q₁₆×C₈₉	—	2	2	no	—
56	C₂×M₁₆×C₈₉	—	2	1	no	—
57	C₂²×C₈×C₈₉	—	2	1	no	—
58	C₂²×C₈×C₈₉	—	2	1	no	—
59	C₄⋊Q₈×C₈₉	—	2	2	no	—
60	C₄⋊Q₈×C₈₉	—	2	2	no	—
61	C₄⋊Q₈×C₈₉	D₄	1	1	yes	characteristic
62	C₄⋊Q₈×C₈₉	D₄	1	1	yes	characteristic
63	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
66	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
67	#125×C₈₉	—	2	2	no	—
68	#125×C₈₉	—	2	2	no	—
69	#125×C₈₉	—	2	1	no	—
70	#125×C₈₉	—	2	1	no	—
71	#125×C₈₉	—	2	1	no	—
72	C₂²⋊Q₈×C₈₉	—	2	4	no	—
73	C₂²⋊Q₈×C₈₉	—	2	4	no	—
74	C₂²⋊Q₈×C₈₉	—	2	4	no	—
75	C₂²⋊Q₈×C₈₉	—	2	4	no	—
76	C₂²⋊Q₈×C₈₉	—	2	2	no	—
77	C₂²⋊Q₈×C₈₉	—	2	4	no	—
78	C₂²⋊Q₈×C₈₉	—	2	1	no	—
79	C₂²⋊Q₈×C₈₉	—	2	1	no	—
80	C₂²⋊Q₈×C₈₉	—	2	2	no	—
81	C₂²⋊Q₈×C₈₉	—	2	4	no	—
82	C₂²⋊Q₈×C₈₉	—	2	1	no	—
83	C₂²⋊Q₈×C₈₉	—	2	1	no	—
84	C₂²⋊Q₈×C₈₉	D₄	1	2	yes	—
85	C₂²⋊Q₈×C₈₉	D₄	1	2	yes	—
86	C₄×Q₈×C₈₉	—	2	1	no	—
87	C₄×Q₈×C₈₉	—	2	1	no	—
88	C₄×Q₈×C₈₉	—	2	1	no	—
89	C₄×Q₈×C₈₉	—	2	1	no	—
90	C₄×Q₈×C₈₉	D₄	1	2	yes	—
91	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
92	C₄×D₄×C₈₉	—	2	2	no	—
93	C₄×D₄×C₈₉	—	2	2	no	—
94	C₄×D₄×C₈₉	—	2	2	no	—
95	C₄×D₄×C₈₉	—	2	2	no	—
96	C₄×D₄×C₈₉	—	2	1	no	—
97	C₄×D₄×C₈₉	—	2	1	no	—
98	C₄×D₄×C₈₉	D₄	1	2	yes	—
99	C₄×D₄×C₈₉	C₂³	1	2	yes	—
100	#117×C₈₉	—	2	1	no	—
101	#117×C₈₉	—	2	1	no	—
102	#117×C₈₉	—	2	1	no	—
103	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
104	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
105	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
106	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
107	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
108	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
109	#107×C₈₉	—	2	2	no	—
110	#107×C₈₉	—	2	2	no	—
111	#107×C₈₉	—	2	2	no	—
112	#106×C₈₉	—	2	2	no	—
113	C₄⋊C₈×C₈₉	—	2	1	no	—
114	C₄⋊C₈×C₈₉	—	2	1	no	—
115	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
116	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
117	#103×C₈₉	—	2	2	no	—
118	#103×C₈₉	—	2	2	no	—
119	#103×C₈₉	—	2	2	no	—
120	#103×C₈₉	—	2	2	no	—
121	#102×C₈₉	—	2	2	no	—
122	#102×C₈₉	—	2	2	no	—
123	C₄×C₈×C₈₉	—	2	1	no	—
124	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
126	C₂⁴×C₈₉	—	2	1	no	—
127	C₂×Q₈×C₈₉	—	2	4	no	—
128	C₂×Q₈×C₈₉	—	2	1	no	—
129	C₂×Q₈×C₈₉	—	2	1	no	—
130	C₂×Q₈×C₈₉	—	2	1	no	—
131	C₂×Q₈×C₈₉	—	2	1	no	—
132	C₂×Q₈×C₈₉	—	8	2	no	—
133	C₂×Q₈×C₈₉	—	8	2	no	—
134	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
135	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
136	C₂×D₄×C₈₉	—	2	1	no	—
137	C₂×D₄×C₈₉	—	2	2	no	—
138	C₂×D₄×C₈₉	Q₁₆	1	4	yes	complemented
139	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
140	C₂²×C₄×C₈₉	—	2	4	no	—
141	C₂²×C₄×C₈₉	—	2	4	no	—
142	C₂²×C₄×C₈₉	—	2	1	no	—
143	C₂²×C₄×C₈₉	—	2	2	no	—
144	C₂²×C₄×C₈₉	—	2	2	no	—
145	C₂²×C₄×C₈₉	—	8	1	no	—
146	C₂²×C₄×C₈₉	—	8	1	no	—
147	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
150	Q₁₆×C₈₉	—	4	4	no	—
151	Q₁₆×C₈₉	—	8	2	no	—
152	M₁₆×C₈₉	—	4	2	no	—
153	C₂×C₈×C₈₉	—	2	1	no	—
154	C₂×C₈×C₈₉	—	2	1	no	—
155	C₂×C₈×C₈₉	—	2	2	no	—
156	C₂×C₈×C₈₉	—	2	2	no	—
157	C₂×C₈×C₈₉	—	4	2	no	—
158	C₂×C₈×C₈₉	—	8	1	no	—
159	C₄⋊C₄×C₈₉	—	2	4	no	—
160	C₄⋊C₄×C₈₉	—	2	4	no	—
161	C₄⋊C₄×C₈₉	—	2	4	no	—
162	C₄⋊C₄×C₈₉	—	2	2	no	—
163	C₄⋊C₄×C₈₉	—	2	2	no	—
164	C₄⋊C₄×C₈₉	—	2	1	no	—
165	C₄⋊C₄×C₈₉	—	2	1	no	—
166	C₄⋊C₄×C₈₉	—	2	1	no	—
167	C₄⋊C₄×C₈₉	—	2	1	no	—
168	C₄⋊C₄×C₈₉	—	2	2	no	—
169	C₄⋊C₄×C₈₉	—	2	1	no	—
170	C₄⋊C₄×C₈₉	—	2	1	no	—
171	C₄⋊C₄×C₈₉	—	2	1	no	—
172	C₄⋊C₄×C₈₉	—	2	2	no	—
173	C₄⋊C₄×C₈₉	—	2	1	no	—
174	C₄⋊C₄×C₈₉	—	2	1	no	—
175	C₄⋊C₄×C₈₉	—	2	1	no	—
176	C₄⋊C₄×C₈₉	—	2	1	no	—
177	C₄⋊C₄×C₈₉	—	4	1	no	—
178	C₄⋊C₄×C₈₉	—	4	1	no	—
179	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
180	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
181	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
182	C₂²⋊C₄×C₈₉	—	2	4	no	—
183	C₂²⋊C₄×C₈₉	—	2	4	no	—
184	C₂²⋊C₄×C₈₉	—	2	2	no	—
185	C₂²⋊C₄×C₈₉	—	2	2	no	—
186	C₂²⋊C₄×C₈₉	—	2	2	no	—
187	C₂²⋊C₄×C₈₉	—	2	2	no	—
188	C₂²⋊C₄×C₈₉	—	4	1	no	—
189	C₄²×C₈₉	—	2	2	no	—
190	C₄²×C₈₉	—	2	2	no	—
191	C₄²×C₈₉	—	4	1	no	—
192	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
196	C₂³×C₈₉	—	2	2	no	—
197	C₂³×C₈₉	—	2	4	no	—
198	C₂³×C₈₉	—	8	1	no	—
199	C₂³×C₈₉	—	8	1	no	—
200	C₂³×C₈₉	#120	1	1	yes	characteristic
201	Q₈×C₈₉	—	2	2	no	—
202	Q₈×C₈₉	—	2	2	no	—
203	Q₈×C₈₉	—	4	2	no	—
204	Q₈×C₈₉	—	4	2	no	—
205	Q₈×C₈₉	—	4	1	no	—
206	Q₈×C₈₉	—	4	1	no	—
207	Q₈×C₈₉	—	4	1	no	—
208	Q₈×C₈₉	—	4	1	no	—
209	Q₈×C₈₉	—	8	1	no	—
210	Q₈×C₈₉	—	8	4	no	—
211	D₄×C₈₉	—	2	2	no	—
212	D₄×C₈₉	—	2	4	no	—
213	D₄×C₈₉	C₂×Q₁₆	1	4	yes	—
214	C₂×C₄×C₈₉	—	2	1	no	—
215	C₂×C₄×C₈₉	—	2	1	no	—
216	C₂×C₄×C₈₉	—	2	2	no	—
217	C₂×C₄×C₈₉	—	2	2	no	—
218	C₂×C₄×C₈₉	—	2	2	no	—
219	C₂×C₄×C₈₉	—	2	2	no	—
220	C₂×C₄×C₈₉	—	2	2	no	—
221	C₂×C₄×C₈₉	—	4	1	no	—
222	C₂×C₄×C₈₉	—	4	2	no	—
223	C₂×C₄×C₈₉	—	4	2	no	—
224	C₂×C₄×C₈₉	—	4	1	no	—
225	C₂×C₄×C₈₉	—	4	1	no	—
226	C₂×C₄×C₈₉	—	8	2	no	—
227	C₂×C₄×C₈₉	—	8	1	no	—
228	C₂×C₄×C₈₉	—	8	4	no	—
229	C₂×C₄×C₈₉	—	8	4	no	—
230	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
231	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
232	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
233	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
234	C₂×C₄×C₈₉	#137	1	2	yes	—
235	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
238	C₈×C₈₉	—	2	2	no	—
239	C₈×C₈₉	—	4	1	no	—
240	C₈×C₈₉	—	4	2	no	—
241	C₈×C₈₉	—	4	2	no	—
242	C₂²×C₈₉	—	2	1	no	—
243	C₂²×C₈₉	—	2	1	no	—
244	C₂²×C₈₉	—	2	4	no	—
245	C₂²×C₈₉	—	4	2	no	—
246	C₂²×C₈₉	—	4	2	no	—
247	C₂²×C₈₉	—	8	4	no	—
248	C₂²×C₈₉	#451	1	2	yes	—
249	C₂²×C₈₉	C₂×#120	1	1	yes	characteristic
250	C₄×C₈₉	—	2	1	no	—
251	C₄×C₈₉	—	2	1	no	—
252	C₄×C₈₉	—	2	1	no	—
253	C₄×C₈₉	—	4	2	no	—
254	C₄×C₈₉	—	4	2	no	—
255	C₄×C₈₉	—	4	2	no	—
256	C₄×C₈₉	—	4	2	no	—
257	C₄×C₈₉	—	4	2	no	—
258	C₄×C₈₉	—	8	1	no	—
259	C₄×C₈₉	—	8	1	no	—
260	C₄×C₈₉	#451	1	2	yes	—
261	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
262	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
263	C₄×C₈₉	#576	1	1	yes	characteristic
264	C₄×C₈₉	#577	1	1	yes	characteristic
265	#18665	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
266	C₂×C₈₉	—	2	2	no	—
267	C₂×C₈₉	—	4	4	no	—
268	C₂×C₈₉	C₂×#451	1	1	yes	characteristic
269	C₂×C₈₉	#3012	1	1	yes	characteristic
270	C₂×C₈₉	#3018	1	1	yes	characteristic
271	#3501	C₂×C₈₉	1	1	yes	characteristic
272	#3249	C₂×C₈₉	1	1	yes	characteristic, complemented
273	#3170	C₂×C₈₉	1	1	yes	characteristic, complemented
274	#3077	C₂×C₈₉	1	1	yes	characteristic
275	#1649	C₂×C₈₉	1	2	yes	complemented
276	#1644	C₂×C₈₉	1	2	yes	complemented
277	#1637	C₂×C₈₉	1	2	yes	complemented
278	#1632	C₂×C₈₉	1	2	yes	complemented
279	#1550	C₂×C₈₉	1	1	yes	characteristic, complemented
280	#1493	C₂×C₈₉	1	1	yes	characteristic, complemented
281	C₂×D₄⋊C₈	C₂×C₈₉	1	1	yes	characteristic
282	C₈₉	#18665	1	1	yes	characteristic, direct product factor, Sylow
283	#554	—	2	1	no	—
284	#554	—	2	2	no	—
285	#554	—	2	2	no	—
286	#554	C₂²×C₈₉	1	2	yes	—
287	D₄×Q₈	—	2	1	no	—
288	D₄×Q₈	C₂²×C₈₉	1	2	yes	—
289	#544	—	2	1	no	—
290	#543	—	2	2	no	—
291	#533	C₂²×C₈₉	1	1	yes	characteristic
292	#533	C₂²×C₈₉	1	1	yes	characteristic
293	#533	C₂²×C₈₉	1	1	yes	characteristic
294	C₂×C₂²⋊Q₈	—	2	2	no	—
295	#518	—	2	1	no	—
296	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
297	#500	C₂²×C₈₉	1	1	yes	characteristic, complemented
298	#494	C₂²×C₈₉	1	1	yes	characteristic, complemented
299	#487	C₂²×C₈₉	1	1	yes	characteristic, complemented
300	#487	C₂²×C₈₉	1	1	yes	characteristic, complemented
301	#478	—	2	2	no	—
302	#477	—	2	1	no	—
303	#477	C₂²×C₈₉	1	2	yes	complemented
304	#476	C₂²×C₈₉	1	2	yes	—
305	#474	C₂²×C₈₉	1	2	yes	—
306	#467	—	2	2	no	—
307	#467	—	2	2	no	—
308	#462	—	2	1	no	—
309	#462	C₂²×C₈₉	1	2	yes	complemented
310	C₄×Q₁₆	—	2	2	no	—
311	#428	—	2	1	no	—
312	#423	C₂²×C₈₉	1	1	yes	characteristic
313	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
314	#418	—	2	1	no	—
315	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
316	#333	C₂²×C₈₉	1	2	yes	complemented
317	#332	C₂²×C₈₉	1	2	yes	complemented
318	Q₈⋊C₈	C₂²×C₈₉	1	2	yes	complemented
319	Q₈⋊C₈	C₂²×C₈₉	1	2	yes	complemented
320	D₄⋊C₈	C₂²×C₈₉	1	2	yes	—
321	D₄⋊C₈	C₂²×C₈₉	1	2	yes	—
322	C₂²×Q₈	—	2	2	no	—
323	C₂²×D₄	D₄×C₈₉	1	1	yes	characteristic
324	C₂³×C₄	—	2	1	no	—
325	C₂×Q₁₆	—	2	2	no	—
326	C₂×Q₁₆	—	2	2	no	—
327	C₂×M₁₆	—	2	1	no	—
328	C₂²×C₈	—	2	1	no	—
329	C₂²×C₈	—	2	1	no	—
330	C₄⋊Q₈	—	2	2	no	—
331	C₄⋊Q₈	—	2	2	no	—
332	C₄⋊Q₈	D₄×C₈₉	1	1	yes	characteristic
333	C₄⋊Q₈	D₄×C₈₉	1	1	yes	characteristic
334	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
335	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
336	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
337	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
338	#125	—	2	2	no	—
339	#125	—	2	2	no	—
340	#125	—	2	1	no	—
341	#125	—	2	1	no	—
342	#125	—	2	1	no	—
343	C₂²⋊Q₈	—	2	4	no	—
344	C₂²⋊Q₈	—	2	4	no	—
345	C₂²⋊Q₈	—	2	4	no	—
346	C₂²⋊Q₈	—	2	4	no	—
347	C₂²⋊Q₈	—	2	2	no	—
348	C₂²⋊Q₈	—	2	4	no	—
349	C₂²⋊Q₈	—	2	1	no	—
350	C₂²⋊Q₈	—	2	1	no	—
351	C₂²⋊Q₈	—	2	2	no	—
352	C₂²⋊Q₈	—	2	4	no	—
353	C₂²⋊Q₈	—	2	1	no	—
354	C₂²⋊Q₈	—	2	1	no	—
355	C₂²⋊Q₈	D₄×C₈₉	1	2	yes	—
356	C₂²⋊Q₈	D₄×C₈₉	1	2	yes	—
357	C₄×Q₈	—	2	1	no	—
358	C₄×Q₈	—	2	1	no	—
359	C₄×Q₈	—	2	1	no	—
360	C₄×Q₈	—	2	1	no	—
361	C₄×Q₈	D₄×C₈₉	1	2	yes	—
362	C₄×Q₈	C₂³×C₈₉	1	2	yes	—
363	C₄×D₄	—	2	2	no	—
364	C₄×D₄	—	2	2	no	—
365	C₄×D₄	—	2	2	no	—
366	C₄×D₄	—	2	2	no	—
367	C₄×D₄	—	2	1	no	—
368	C₄×D₄	—	2	1	no	—
369	C₄×D₄	D₄×C₈₉	1	2	yes	—
370	C₄×D₄	C₂³×C₈₉	1	2	yes	—
371	#117	—	2	1	no	—
372	#117	—	2	1	no	—
373	#117	—	2	1	no	—
374	C₂×C₄⋊C₄	—	2	4	no	—
375	C₂×C₄⋊C₄	—	2	2	no	—
376	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
377	C₂×C₂²⋊C₄	—	2	4	no	—
378	C₂×C₂²⋊C₄	—	2	1	no	—
379	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
380	#107	—	2	2	no	—
381	#107	—	2	2	no	—
382	#107	—	2	2	no	—
383	#106	—	2	2	no	—
384	C₄⋊C₈	—	2	1	no	—
385	C₄⋊C₈	—	2	1	no	—
386	C₄⋊C₈	C₂³×C₈₉	1	2	yes	—
387	C₄⋊C₈	C₂³×C₈₉	1	2	yes	—
388	#103	—	2	2	no	—
389	#103	—	2	2	no	—
390	#103	—	2	2	no	—
391	#103	—	2	2	no	—
392	#102	—	2	2	no	—
393	#102	—	2	2	no	—
394	C₄×C₈	—	2	1	no	—
395	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
396	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
397	C₂⁴	—	2	1	no	—
398	C₂×Q₈	—	2	4	no	—
399	C₂×Q₈	—	2	1	no	—
400	C₂×Q₈	—	2	1	no	—
401	C₂×Q₈	—	2	1	no	—
402	C₂×Q₈	—	2	1	no	—
403	C₂×Q₈	—	8	2	no	—
404	C₂×Q₈	—	8	2	no	—
405	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
406	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
407	C₂×D₄	—	2	1	no	—
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	Q₁₆×C₈₉	1	4	yes	complemented
410	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	8	1	no	—
417	C₂²×C₄	—	8	1	no	—
418	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
419	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
420	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
421	Q₁₆	—	4	4	no	—
422	Q₁₆	—	8	2	no	—
423	M₁₆	—	4	2	no	—
424	C₂×C₈	—	2	1	no	—
425	C₂×C₈	—	2	1	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	4	2	no	—
429	C₂×C₈	—	8	1	no	—
430	C₄⋊C₄	—	2	4	no	—
431	C₄⋊C₄	—	2	4	no	—
432	C₄⋊C₄	—	2	4	no	—
433	C₄⋊C₄	—	2	2	no	—
434	C₄⋊C₄	—	2	2	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	—	2	1	no	—
441	C₄⋊C₄	—	2	1	no	—
442	C₄⋊C₄	—	2	1	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	—	2	1	no	—
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	2	1	no	—
448	C₄⋊C₄	—	4	1	no	—
449	C₄⋊C₄	—	4	1	no	—
450	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
451	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
452	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
453	C₂²⋊C₄	—	2	4	no	—
454	C₂²⋊C₄	—	2	4	no	—
455	C₂²⋊C₄	—	2	2	no	—
456	C₂²⋊C₄	—	2	2	no	—
457	C₂²⋊C₄	—	2	2	no	—
458	C₂²⋊C₄	—	2	2	no	—
459	C₂²⋊C₄	—	4	1	no	—
460	C₄²	—	2	2	no	—
461	C₄²	—	2	2	no	—
462	C₄²	—	4	1	no	—
463	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
464	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
465	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
466	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
467	C₂³	—	2	2	no	—
468	C₂³	—	2	4	no	—
469	C₂³	—	8	1	no	—
470	C₂³	—	8	1	no	—
471	C₂³	#120×C₈₉	1	1	yes	characteristic
472	Q₈	—	2	2	no	—
473	Q₈	—	2	2	no	—
474	Q₈	—	4	2	no	—
475	Q₈	—	4	2	no	—
476	Q₈	—	4	1	no	—
477	Q₈	—	4	1	no	—
478	Q₈	—	4	1	no	—
479	Q₈	—	4	1	no	—
480	Q₈	—	8	1	no	—
481	Q₈	—	8	4	no	—
482	D₄	—	2	4	no	—
483	D₄	—	2	2	no	—
484	D₄	C₂×Q₁₆×C₈₉	1	4	yes	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	4	1	no	—
493	C₂×C₄	—	4	2	no	—
494	C₂×C₄	—	4	2	no	—
495	C₂×C₄	—	4	1	no	—
496	C₂×C₄	—	4	1	no	—
497	C₂×C₄	—	8	2	no	—
498	C₂×C₄	—	8	1	no	—
499	C₂×C₄	—	8	4	no	—
500	C₂×C₄	—	8	4	no	—
501	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
502	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
503	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
504	C₂×C₄	C₂×Q₁₆×C₈₉	1	2	yes	—
505	C₂×C₄	#137×C₈₉	1	2	yes	—
506	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
507	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
508	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
509	C₈	—	2	2	no	—
510	C₈	—	4	1	no	—
511	C₈	—	4	2	no	—
512	C₈	—	4	2	no	—
513	C₂²	—	2	1	no	—
514	C₂²	—	2	1	no	—
515	C₂²	—	2	4	no	—
516	C₂²	—	4	2	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	8	4	no	—
519	C₂²	#451×C₈₉	1	2	yes	—
520	C₂²	C₂×#120×C₈₉	1	1	yes	characteristic
521	C₄	—	2	1	no	—
522	C₄	—	2	1	no	—
523	C₄	—	2	1	no	—
524	C₄	—	4	2	no	—
525	C₄	—	4	2	no	—
526	C₄	—	4	2	no	—
527	C₄	—	4	2	no	—
528	C₄	—	4	2	no	—
529	C₄	—	8	1	no	—
530	C₄	—	8	1	no	—
531	C₄	#451×C₈₉	1	2	yes	—
532	C₄	C₂²×Q₁₆×C₈₉	1	1	yes	characteristic
533	C₄	C₂×#137×C₈₉	1	1	yes	characteristic
534	C₄	#576×C₈₉	1	1	yes	characteristic
535	C₄	#577×C₈₉	1	1	yes	characteristic
536	C₂	—	2	2	no	—
537	C₂	—	4	4	no	—
538	C₂	C₂×#451×C₈₉	1	1	yes	characteristic
539	C₂	C₈₉×#3012	1	1	yes	characteristic
540	C₂	C₈₉×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂×#120
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
2. C₈₉×#3249	C₂	semidirect
3. C₈₉×#3170	C₂	semidirect
5. C₈₉×#1649	C₂	semidirect
6. C₈₉×#1644	C₂	semidirect
7. C₈₉×#1637	C₂	semidirect
8. C₈₉×#1632	C₂	semidirect
9. C₈₉×#1550	C₂	semidirect
10. C₈₉×#1493	C₂	semidirect
26. #500×C₈₉	C₂²	semidirect
27. #494×C₈₉	C₂²	semidirect
28. #487×C₈₉	C₂²	semidirect
29. #487×C₈₉	C₂²	semidirect
32. #477×C₈₉	C₂²	semidirect
38. #462×C₈₉	C₂²	semidirect
45. #333×C₈₉	C₂²	semidirect
46. #332×C₈₉	C₂²	semidirect
47. Q₈⋊C₈×C₈₉	C₂²	semidirect
48. Q₈⋊C₈×C₈₉	C₂²	semidirect
138. C₂×D₄×C₈₉	Q₁₆	semidirect
265. #18665	C₈₉	direct (N × H)
272. #3249	C₂×C₈₉	semidirect
273. #3170	C₂×C₈₉	semidirect
275. #1649	C₂×C₈₉	semidirect
276. #1644	C₂×C₈₉	semidirect
277. #1637	C₂×C₈₉	semidirect
278. #1632	C₂×C₈₉	semidirect
279. #1550	C₂×C₈₉	semidirect
280. #1493	C₂×C₈₉	semidirect
282. C₈₉	#18665	direct (N × H)
297. #500	C₂²×C₈₉	semidirect
298. #494	C₂²×C₈₉	semidirect
299. #487	C₂²×C₈₉	semidirect
300. #487	C₂²×C₈₉	semidirect
303. #477	C₂²×C₈₉	semidirect
309. #462	C₂²×C₈₉	semidirect
316. #333	C₂²×C₈₉	semidirect
317. #332	C₂²×C₈₉	semidirect
318. Q₈⋊C₈	C₂²×C₈₉	semidirect
319. Q₈⋊C₈	C₂²×C₈₉	semidirect
409. C₂×D₄	Q₁₆×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3501	2
C₈₉×#3249	2
C₈₉×#3170	2
C₈₉×#3077	2
C₈₉×#1649	2
C₈₉×#1644	2
C₈₉×#1637	2
C₈₉×#1632	2
C₈₉×#1550	2
C₈₉×#1493	2
C₂×D₄⋊C₈×C₈₉	2
#18665	89

Subgroups — C89×#18665 #36396513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#18665 #36396513