Subgroups — C₈₉×#25323 #36403171

Order 22784.

790 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#487×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#462×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2568	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2365	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2074	C₂	1	2	yes	maximal
6	C₈₉×#2056	C₂	1	2	yes	maximal
7	C₈₉×#1988	C₂	1	1	yes	maximal, characteristic
8	C₈₉×#1939	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#1878	C₂	1	2	yes	maximal
10	C₈₉×#1805	C₂	1	2	yes	maximal
11	C₄×#103×C₈₉	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₈₉	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
17	#487×C₈₉	—	2	4	no	—
18	#462×C₈₉	—	2	4	no	—
19	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
21	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×#103×C₈₉	C₂²	1	2	yes	—
23	C₂×#103×C₈₉	C₂²	1	2	yes	—
24	C₂×#103×C₈₉	C₂²	1	2	yes	—
25	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
27	#400×C₈₉	—	2	2	no	—
28	#391×C₈₉	—	2	1	no	—
29	#391×C₈₉	—	2	1	no	—
30	#391×C₈₉	C₂²	1	2	yes	—
31	#391×C₈₉	C₂²	1	2	yes	—
32	#389×C₈₉	—	2	1	no	—
33	#387×C₈₉	—	2	2	no	—
34	#382×C₈₉	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
36	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
38	#340×C₈₉	C₂²	1	2	yes	—
39	#340×C₈₉	C₂²	1	1	yes	characteristic
40	#340×C₈₉	C₂²	1	2	yes	—
41	#340×C₈₉	C₂²	1	1	yes	characteristic
42	#336×C₈₉	C₂²	1	2	yes	—
43	C₂²×Q₈×C₈₉	—	2	1	no	—
44	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
45	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
46	C₂×Q₁₆×C₈₉	—	2	4	no	—
47	C₂×Q₁₆×C₈₉	—	2	4	no	—
48	C₂×Q₁₆×C₈₉	—	4	2	no	—
49	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
52	C₄⋊Q₈×C₈₉	—	2	2	no	—
53	C₄⋊Q₈×C₈₉	—	2	2	no	—
54	C₄⋊Q₈×C₈₉	—	2	2	no	—
55	C₄⋊Q₈×C₈₉	D₄	1	4	yes	—
56	C₄×Q₈×C₈₉	—	2	4	no	—
57	C₄×Q₈×C₈₉	—	2	2	no	—
58	C₄×Q₈×C₈₉	—	2	2	no	—
59	C₄×Q₈×C₈₉	—	4	2	no	—
60	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
61	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
62	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
63	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
64	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
65	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₈₉	—	2	1	no	—
72	C₂×C₄²×C₈₉	—	2	2	no	—
73	C₂×C₄²×C₈₉	—	2	1	no	—
74	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
75	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₈₉	—	2	2	no	—
77	C₄⋊C₈×C₈₉	—	4	2	no	—
78	#103×C₈₉	—	2	4	no	—
79	#103×C₈₉	—	2	4	no	—
80	#103×C₈₉	—	2	4	no	—
81	#103×C₈₉	—	2	2	no	—
82	#103×C₈₉	—	2	2	no	—
83	C₄×C₈×C₈₉	—	2	2	no	—
84	#95×C₈₉	—	2	2	no	—
85	#95×C₈₉	—	2	1	no	—
86	#95×C₈₉	—	2	2	no	—
87	#95×C₈₉	—	2	2	no	—
88	#95×C₈₉	—	2	2	no	—
89	#95×C₈₉	—	2	2	no	—
90	#95×C₈₉	—	2	2	no	—
91	#95×C₈₉	D₄	1	2	yes	—
92	C₂×Q₈×C₈₉	—	2	2	no	—
93	C₂×Q₈×C₈₉	—	2	1	no	—
94	C₂×Q₈×C₈₉	—	2	1	no	—
95	C₂×Q₈×C₈₉	—	2	1	no	—
96	C₂×Q₈×C₈₉	—	2	1	no	—
97	C₂×Q₈×C₈₉	—	2	2	no	—
98	C₂×Q₈×C₈₉	—	2	2	no	—
99	C₂×Q₈×C₈₉	—	2	2	no	—
100	C₂×Q₈×C₈₉	—	2	1	no	—
101	C₂×Q₈×C₈₉	—	2	2	no	—
102	C₂×Q₈×C₈₉	—	4	2	no	—
103	C₂×Q₈×C₈₉	—	4	2	no	—
104	C₂×Q₈×C₈₉	—	4	2	no	—
105	C₂×Q₈×C₈₉	C₂×D₄	1	4	yes	—
106	C₂×Q₈×C₈₉	#41	1	2	yes	—
107	C₂²×C₄×C₈₉	—	2	1	no	—
108	C₂²×C₄×C₈₉	—	2	2	no	—
109	C₂²×C₄×C₈₉	—	2	1	no	—
110	C₂²×C₄×C₈₉	—	2	1	no	—
111	C₂²×C₄×C₈₉	—	2	1	no	—
112	C₂²×C₄×C₈₉	—	2	2	no	—
113	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
118	Q₁₆×C₈₉	—	4	8	no	—
119	C₂×C₈×C₈₉	—	2	1	no	—
120	C₂×C₈×C₈₉	—	2	1	no	—
121	C₂×C₈×C₈₉	—	2	2	no	—
122	C₂×C₈×C₈₉	—	2	1	no	—
123	C₂×C₈×C₈₉	—	2	1	no	—
124	C₂×C₈×C₈₉	—	2	2	no	—
125	C₂×C₈×C₈₉	—	2	1	no	—
126	C₂×C₈×C₈₉	—	2	2	no	—
127	C₂×C₈×C₈₉	—	2	1	no	—
128	C₄⋊C₄×C₈₉	—	2	4	no	—
129	C₄⋊C₄×C₈₉	—	2	4	no	—
130	C₄⋊C₄×C₈₉	—	2	2	no	—
131	C₄⋊C₄×C₈₉	—	2	2	no	—
132	C₄⋊C₄×C₈₉	—	2	1	no	—
133	C₄⋊C₄×C₈₉	—	2	1	no	—
134	C₄⋊C₄×C₈₉	—	2	2	no	—
135	C₄⋊C₄×C₈₉	—	2	1	no	—
136	C₄⋊C₄×C₈₉	—	2	1	no	—
137	C₄⋊C₄×C₈₉	—	2	2	no	—
138	C₄⋊C₄×C₈₉	—	2	1	no	—
139	C₄⋊C₄×C₈₉	—	4	4	no	—
140	C₄⋊C₄×C₈₉	—	4	2	no	—
141	C₄⋊C₄×C₈₉	—	4	2	no	—
142	C₄⋊C₄×C₈₉	#41	1	2	yes	—
143	C₄²×C₈₉	—	2	2	no	—
144	C₄²×C₈₉	—	2	2	no	—
145	C₄²×C₈₉	—	4	4	no	—
146	C₄²×C₈₉	—	4	2	no	—
147	C₄²×C₈₉	—	4	2	no	—
148	C₄²×C₈₉	C₂×D₄	1	2	yes	—
149	C₄²×C₈₉	#41	1	2	yes	—
150	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
151	Q₈×C₈₉	—	2	2	no	—
152	Q₈×C₈₉	—	2	2	no	—
153	Q₈×C₈₉	—	2	4	no	—
154	Q₈×C₈₉	—	2	4	no	—
155	Q₈×C₈₉	—	4	2	no	—
156	Q₈×C₈₉	—	4	2	no	—
157	Q₈×C₈₉	—	4	1	no	—
158	Q₈×C₈₉	—	4	1	no	—
159	Q₈×C₈₉	—	8	4	no	—
160	C₂×C₄×C₈₉	—	2	1	no	—
161	C₂×C₄×C₈₉	—	2	1	no	—
162	C₂×C₄×C₈₉	—	2	4	no	—
163	C₂×C₄×C₈₉	—	2	2	no	—
164	C₂×C₄×C₈₉	—	2	2	no	—
165	C₂×C₄×C₈₉	—	2	2	no	—
166	C₂×C₄×C₈₉	—	2	2	no	—
167	C₂×C₄×C₈₉	—	2	2	no	—
168	C₂×C₄×C₈₉	—	2	2	no	—
169	C₂×C₄×C₈₉	—	2	2	no	—
170	C₂×C₄×C₈₉	—	2	2	no	—
171	C₂×C₄×C₈₉	—	4	1	no	—
172	C₂×C₄×C₈₉	—	4	2	no	—
173	C₂×C₄×C₈₉	—	4	1	no	—
174	C₂×C₄×C₈₉	—	4	2	no	—
175	C₂×C₄×C₈₉	—	4	1	no	—
176	C₂×C₄×C₈₉	—	4	1	no	—
177	C₂×C₄×C₈₉	—	4	1	no	—
178	C₂×C₄×C₈₉	—	4	2	no	—
179	C₂×C₄×C₈₉	—	4	1	no	—
180	C₂×C₄×C₈₉	—	4	1	no	—
181	C₂×C₄×C₈₉	—	4	2	no	—
182	C₂×C₄×C₈₉	—	4	2	no	—
183	C₂×C₄×C₈₉	—	4	1	no	—
184	C₂×C₄×C₈₉	—	4	1	no	—
185	C₂×C₄×C₈₉	—	4	1	no	—
186	C₂×C₄×C₈₉	#121	1	2	yes	—
187	C₂×C₄×C₈₉	#121	1	2	yes	—
188	C₂×C₄×C₈₉	#135	1	2	yes	—
189	C₂×C₄×C₈₉	#137	1	2	yes	—
190	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
192	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
195	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
196	C₈×C₈₉	—	4	1	no	—
197	C₈×C₈₉	—	4	2	no	—
198	C₈×C₈₉	—	4	1	no	—
199	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
200	C₂²×C₈₉	#546	1	1	yes	characteristic
201	C₂²×C₈₉	#548	1	1	yes	characteristic
202	C₂²×C₈₉	C₂×#135	1	1	yes	characteristic
203	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
204	C₂²×C₈₉	#578	1	1	yes	characteristic
205	C₂²×C₈₉	#578	1	1	yes	characteristic
206	C₄×C₈₉	—	2	2	no	—
207	C₄×C₈₉	—	4	2	no	—
208	C₄×C₈₉	—	4	4	no	—
209	C₄×C₈₉	—	4	2	no	—
210	C₄×C₈₉	—	4	2	no	—
211	C₄×C₈₉	—	4	2	no	—
212	C₄×C₈₉	—	8	1	no	—
213	C₄×C₈₉	—	8	2	no	—
214	C₄×C₈₉	—	8	1	no	—
215	C₄×C₈₉	#464	1	4	yes	—
216	C₄×C₈₉	#529	1	1	yes	characteristic
217	C₄×C₈₉	#542	1	1	yes	characteristic
218	C₄×C₈₉	#546	1	1	yes	characteristic
219	C₄×C₈₉	#547	1	1	yes	characteristic
220	#25323	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂×C₈₉	#2427	1	1	yes	characteristic
222	C₂×C₈₉	C₂×#464	1	1	yes	characteristic
223	C₂×C₈₉	#3044	1	1	yes	characteristic
224	C₂×C₈₉	#3302	1	1	yes	characteristic
225	C₂×C₈₉	#3328	1	1	yes	characteristic
226	C₂×C₈₉	#3364	1	1	yes	characteristic
227	C₂×C₈₉	#3374	1	1	yes	characteristic
228	C₂×#487	C₂×C₈₉	1	1	yes	characteristic
229	C₂×#462	C₂×C₈₉	1	1	yes	characteristic
230	#2568	C₂×C₈₉	1	1	yes	characteristic
231	#2365	C₂×C₈₉	1	1	yes	characteristic
232	#2074	C₂×C₈₉	1	2	yes	—
233	#2056	C₂×C₈₉	1	2	yes	—
234	#1988	C₂×C₈₉	1	1	yes	characteristic
235	#1939	C₂×C₈₉	1	1	yes	characteristic
236	#1878	C₂×C₈₉	1	2	yes	—
237	#1805	C₂×C₈₉	1	2	yes	—
238	C₄×#103	C₂×C₈₉	1	1	yes	characteristic
239	C₈₉	#25323	1	1	yes	characteristic, direct product factor, Sylow
240	C₂²×Q₁₆	C₂²×C₈₉	1	2	yes	—
241	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
242	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
243	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
244	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
245	#487	—	2	4	no	—
246	#462	—	2	4	no	—
247	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
248	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
249	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
250	C₂×#103	C₂²×C₈₉	1	2	yes	—
251	C₂×#103	C₂²×C₈₉	1	2	yes	—
252	C₂×#103	C₂²×C₈₉	1	2	yes	—
253	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
254	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
255	#400	—	2	2	no	—
256	#391	—	2	1	no	—
257	#391	—	2	1	no	—
258	#391	C₂²×C₈₉	1	2	yes	—
259	#391	C₂²×C₈₉	1	2	yes	—
260	#389	—	2	1	no	—
261	#387	—	2	2	no	—
262	#382	C₂²×C₈₉	1	2	yes	—
263	C₄×C₄⋊C₄	—	2	1	no	—
264	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
265	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
266	#340	C₂²×C₈₉	1	2	yes	—
267	#340	C₂²×C₈₉	1	1	yes	characteristic
268	#340	C₂²×C₈₉	1	2	yes	—
269	#340	C₂²×C₈₉	1	1	yes	characteristic
270	#336	C₂²×C₈₉	1	2	yes	—
271	C₂²×Q₈	—	2	1	no	—
272	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
273	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
274	C₂×Q₁₆	—	2	4	no	—
275	C₂×Q₁₆	—	2	4	no	—
276	C₂×Q₁₆	—	4	2	no	—
277	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
278	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
279	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
280	C₄⋊Q₈	—	2	2	no	—
281	C₄⋊Q₈	—	2	2	no	—
282	C₄⋊Q₈	—	2	2	no	—
283	C₄⋊Q₈	D₄×C₈₉	1	4	yes	—
284	C₄×Q₈	—	2	4	no	—
285	C₄×Q₈	—	2	2	no	—
286	C₄×Q₈	—	2	2	no	—
287	C₄×Q₈	—	4	2	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	—	2	2	no	—
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	—	2	1	no	—
293	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
294	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
295	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
296	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
297	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
298	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
299	C₂×C₄²	—	2	2	no	—
300	C₂×C₄²	—	2	1	no	—
301	C₂×C₄²	—	2	1	no	—
302	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
303	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
304	C₄⋊C₈	—	2	2	no	—
305	C₄⋊C₈	—	4	2	no	—
306	#103	—	2	4	no	—
307	#103	—	2	4	no	—
308	#103	—	2	4	no	—
309	#103	—	2	2	no	—
310	#103	—	2	2	no	—
311	C₄×C₈	—	2	2	no	—
312	#95	—	2	2	no	—
313	#95	—	2	1	no	—
314	#95	—	2	2	no	—
315	#95	—	2	2	no	—
316	#95	—	2	2	no	—
317	#95	—	2	2	no	—
318	#95	—	2	2	no	—
319	#95	D₄×C₈₉	1	2	yes	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	—	2	1	no	—
322	C₂×Q₈	—	2	1	no	—
323	C₂×Q₈	—	2	1	no	—
324	C₂×Q₈	—	2	1	no	—
325	C₂×Q₈	—	2	2	no	—
326	C₂×Q₈	—	2	2	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	2	1	no	—
329	C₂×Q₈	—	2	2	no	—
330	C₂×Q₈	—	4	2	no	—
331	C₂×Q₈	—	4	2	no	—
332	C₂×Q₈	—	4	2	no	—
333	C₂×Q₈	C₂×D₄×C₈₉	1	4	yes	—
334	C₂×Q₈	#41×C₈₉	1	2	yes	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
342	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
343	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
344	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
345	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
346	Q₁₆	—	4	8	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	1	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	1	no	—
356	C₄⋊C₄	—	2	4	no	—
357	C₄⋊C₄	—	2	4	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	1	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	4	4	no	—
368	C₄⋊C₄	—	4	2	no	—
369	C₄⋊C₄	—	4	2	no	—
370	C₄⋊C₄	#41×C₈₉	1	2	yes	—
371	C₄²	—	2	2	no	—
372	C₄²	—	2	2	no	—
373	C₄²	—	4	4	no	—
374	C₄²	—	4	2	no	—
375	C₄²	—	4	2	no	—
376	C₄²	C₂×D₄×C₈₉	1	2	yes	—
377	C₄²	#41×C₈₉	1	2	yes	—
378	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
379	Q₈	—	2	2	no	—
380	Q₈	—	2	2	no	—
381	Q₈	—	2	4	no	—
382	Q₈	—	2	4	no	—
383	Q₈	—	4	2	no	—
384	Q₈	—	4	2	no	—
385	Q₈	—	4	1	no	—
386	Q₈	—	4	1	no	—
387	Q₈	—	8	4	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	4	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	1	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	#121×C₈₉	1	2	yes	—
415	C₂×C₄	#121×C₈₉	1	2	yes	—
416	C₂×C₄	#135×C₈₉	1	2	yes	—
417	C₂×C₄	#137×C₈₉	1	2	yes	—
418	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
419	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
420	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
421	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
422	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
423	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
424	C₈	—	4	1	no	—
425	C₈	—	4	2	no	—
426	C₈	—	4	1	no	—
427	C₂²	C₂×#121×C₈₉	1	1	yes	characteristic
428	C₂²	#546×C₈₉	1	1	yes	characteristic
429	C₂²	#548×C₈₉	1	1	yes	characteristic
430	C₂²	C₂×#135×C₈₉	1	1	yes	characteristic
431	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
432	C₂²	#578×C₈₉	1	1	yes	characteristic
433	C₂²	#578×C₈₉	1	1	yes	characteristic
434	C₄	—	2	2	no	—
435	C₄	—	4	2	no	—
436	C₄	—	4	4	no	—
437	C₄	—	4	2	no	—
438	C₄	—	4	2	no	—
439	C₄	—	4	2	no	—
440	C₄	—	8	1	no	—
441	C₄	—	8	2	no	—
442	C₄	—	8	1	no	—
443	C₄	#464×C₈₉	1	4	yes	—
444	C₄	#529×C₈₉	1	1	yes	characteristic
445	C₄	#542×C₈₉	1	1	yes	characteristic
446	C₄	#546×C₈₉	1	1	yes	characteristic
447	C₄	#547×C₈₉	1	1	yes	characteristic
448	C₂	C₈₉×#2427	1	1	yes	characteristic
449	C₂	C₂×#464×C₈₉	1	1	yes	characteristic
450	C₂	C₈₉×#3044	1	1	yes	characteristic
451	C₂	C₈₉×#3302	1	1	yes	characteristic
452	C₂	C₈₉×#3328	1	1	yes	characteristic
453	C₂	C₈₉×#3364	1	1	yes	characteristic
454	C₂	C₈₉×#3374	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₄×C₄⋊C₄×C₈₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
220. #25323	C₈₉	direct (N × H)
239. C₈₉	#25323	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#487×C₈₉	2
C₂×#462×C₈₉	2
C₈₉×#2568	2
C₈₉×#2365	2
C₈₉×#2074	2
C₈₉×#2056	2
C₈₉×#1988	2
C₈₉×#1939	2
C₈₉×#1878	2
C₈₉×#1805	2
C₄×#103×C₈₉	2
#25323	89

Subgroups — C89×#25323 #36403171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#25323 #36403171