Subgroups — C₈₉×#29371 #36407219

Order 22784.

746 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2295	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2099	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2096	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2043	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#2016	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1927	C₂	1	2	yes	maximal
8	C₈₉×#1878	C₂	1	2	yes	maximal
9	C₈₉×#1878	C₂	1	2	yes	maximal
10	C₈₉×#1865	C₂	1	2	yes	maximal
11	C₈₉×#1839	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	2	yes	—
16	C₂×#106×C₈₉	C₂²	1	2	yes	—
17	C₂×#103×C₈₉	C₂²	1	2	yes	—
18	C₂×#103×C₈₉	C₂²	1	2	yes	—
19	C₂×#103×C₈₉	C₂²	1	2	yes	—
20	C₂×#103×C₈₉	C₂²	1	2	yes	—
21	#391×C₈₉	C₂²	1	1	yes	characteristic
22	#391×C₈₉	C₂²	1	1	yes	characteristic
23	#391×C₈₉	C₂²	1	2	yes	—
24	#389×C₈₉	—	2	1	no	—
25	#389×C₈₉	—	2	1	no	—
26	#389×C₈₉	C₂²	1	1	yes	characteristic
27	#389×C₈₉	C₂²	1	1	yes	characteristic
28	#387×C₈₉	—	2	1	no	—
29	#387×C₈₉	—	2	1	no	—
30	#384×C₈₉	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
32	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
33	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
34	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
35	#376×C₈₉	—	2	1	no	—
36	C₄³×C₈₉	C₂²	1	1	yes	characteristic
37	#340×C₈₉	C₂²	1	2	yes	—
38	#340×C₈₉	C₂²	1	2	yes	—
39	#336×C₈₉	C₂²	1	1	yes	characteristic
40	#336×C₈₉	C₂²	1	1	yes	characteristic
41	#336×C₈₉	C₂²	1	2	yes	—
42	#336×C₈₉	C₂²	1	2	yes	—
43	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
44	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
45	C₂×Q₁₆×C₈₉	—	2	4	no	—
46	C₂×Q₁₆×C₈₉	—	2	4	no	—
47	C₂×Q₁₆×C₈₉	—	4	2	no	—
48	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
49	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₈₉	—	2	4	no	—
51	C₄×Q₈×C₈₉	—	2	2	no	—
52	C₄×Q₈×C₈₉	—	2	2	no	—
53	C₄×Q₈×C₈₉	—	4	2	no	—
54	C₄×Q₈×C₈₉	—	4	2	no	—
55	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
57	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
58	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
59	C₂×C₄⋊C₄×C₈₉	—	2	1	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	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₈₉	—	2	1	no	—
68	C₂×C₄²×C₈₉	—	2	1	no	—
69	C₂×C₄²×C₈₉	—	2	2	no	—
70	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
77	#106×C₈₉	—	4	2	no	—
78	#103×C₈₉	—	2	4	no	—
79	#103×C₈₉	—	2	4	no	—
80	#103×C₈₉	—	4	2	no	—
81	#103×C₈₉	—	4	2	no	—
82	#95×C₈₉	—	2	1	no	—
83	#95×C₈₉	—	2	1	no	—
84	#95×C₈₉	—	2	1	no	—
85	#95×C₈₉	—	2	1	no	—
86	#95×C₈₉	—	2	1	no	—
87	#95×C₈₉	—	2	1	no	—
88	#95×C₈₉	—	2	2	no	—
89	#95×C₈₉	—	2	2	no	—
90	#95×C₈₉	—	2	1	no	—
91	C₂×Q₈×C₈₉	—	2	1	no	—
92	C₂×Q₈×C₈₉	—	2	2	no	—
93	C₂×Q₈×C₈₉	—	2	2	no	—
94	C₂×Q₈×C₈₉	—	2	2	no	—
95	C₂×Q₈×C₈₉	—	2	2	no	—
96	C₂×Q₈×C₈₉	—	2	2	no	—
97	C₂×Q₈×C₈₉	—	2	2	no	—
98	C₂×Q₈×C₈₉	—	2	2	no	—
99	C₂×Q₈×C₈₉	#41	1	4	yes	—
100	C₂×Q₈×C₈₉	#41	1	2	yes	—
101	C₂²×C₄×C₈₉	—	2	1	no	—
102	C₂²×C₄×C₈₉	—	2	1	no	—
103	C₂²×C₄×C₈₉	—	2	2	no	—
104	C₂²×C₄×C₈₉	—	2	1	no	—
105	C₂²×C₄×C₈₉	—	2	1	no	—
106	C₂²×C₄×C₈₉	—	2	2	no	—
107	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
114	Q₁₆×C₈₉	—	4	8	no	—
115	C₂×C₈×C₈₉	—	2	2	no	—
116	C₂×C₈×C₈₉	—	2	2	no	—
117	C₂×C₈×C₈₉	—	2	2	no	—
118	C₂×C₈×C₈₉	—	2	2	no	—
119	C₂×C₈×C₈₉	—	2	2	no	—
120	C₂×C₈×C₈₉	—	2	2	no	—
121	C₄⋊C₄×C₈₉	—	2	4	no	—
122	C₄⋊C₄×C₈₉	—	2	4	no	—
123	C₄⋊C₄×C₈₉	—	2	4	no	—
124	C₄⋊C₄×C₈₉	—	2	1	no	—
125	C₄⋊C₄×C₈₉	—	2	2	no	—
126	C₄⋊C₄×C₈₉	—	2	2	no	—
127	C₄⋊C₄×C₈₉	—	2	2	no	—
128	C₄⋊C₄×C₈₉	—	2	2	no	—
129	C₄⋊C₄×C₈₉	—	4	4	no	—
130	C₄⋊C₄×C₈₉	—	4	2	no	—
131	C₄⋊C₄×C₈₉	—	4	2	no	—
132	C₄⋊C₄×C₈₉	—	4	2	no	—
133	C₄⋊C₄×C₈₉	—	4	2	no	—
134	C₄⋊C₄×C₈₉	#41	1	2	yes	—
135	C₄²×C₈₉	—	2	4	no	—
136	C₄²×C₈₉	—	2	2	no	—
137	C₄²×C₈₉	—	2	2	no	—
138	C₄²×C₈₉	—	2	2	no	—
139	C₄²×C₈₉	—	2	2	no	—
140	C₄²×C₈₉	—	2	1	no	—
141	C₄²×C₈₉	—	2	1	no	—
142	C₄²×C₈₉	—	2	2	no	—
143	C₄²×C₈₉	—	2	2	no	—
144	C₄²×C₈₉	—	4	4	no	—
145	C₄²×C₈₉	—	4	2	no	—
146	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
147	Q₈×C₈₉	—	2	4	no	—
148	Q₈×C₈₉	—	2	4	no	—
149	Q₈×C₈₉	—	2	4	no	—
150	Q₈×C₈₉	—	4	2	no	—
151	Q₈×C₈₉	—	4	2	no	—
152	Q₈×C₈₉	—	4	2	no	—
153	C₂×C₄×C₈₉	—	2	1	no	—
154	C₂×C₄×C₈₉	—	2	1	no	—
155	C₂×C₄×C₈₉	—	2	1	no	—
156	C₂×C₄×C₈₉	—	2	1	no	—
157	C₂×C₄×C₈₉	—	2	1	no	—
158	C₂×C₄×C₈₉	—	2	1	no	—
159	C₂×C₄×C₈₉	—	2	1	no	—
160	C₂×C₄×C₈₉	—	2	1	no	—
161	C₂×C₄×C₈₉	—	2	1	no	—
162	C₂×C₄×C₈₉	—	2	2	no	—
163	C₂×C₄×C₈₉	—	2	2	no	—
164	C₂×C₄×C₈₉	—	2	4	no	—
165	C₂×C₄×C₈₉	—	2	1	no	—
166	C₂×C₄×C₈₉	—	2	1	no	—
167	C₂×C₄×C₈₉	—	2	1	no	—
168	C₂×C₄×C₈₉	—	2	1	no	—
169	C₂×C₄×C₈₉	—	2	1	no	—
170	C₂×C₄×C₈₉	—	4	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	1	no	—
175	C₂×C₄×C₈₉	—	4	1	no	—
176	C₂×C₄×C₈₉	—	4	1	no	—
177	C₂×C₄×C₈₉	—	4	2	no	—
178	C₂×C₄×C₈₉	—	4	1	no	—
179	C₂×C₄×C₈₉	—	4	1	no	—
180	C₂×C₄×C₈₉	—	4	2	no	—
181	C₂×C₄×C₈₉	—	4	1	no	—
182	C₂×C₄×C₈₉	—	4	1	no	—
183	C₂×C₄×C₈₉	—	4	2	no	—
184	C₂×C₄×C₈₉	—	4	1	no	—
185	C₂×C₄×C₈₉	#135	1	2	yes	—
186	C₂×C₄×C₈₉	#137	1	2	yes	—
187	C₂×C₄×C₈₉	#137	1	2	yes	—
188	C₂×C₄×C₈₉	#137	1	2	yes	—
189	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
190	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
191	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
192	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
193	C₈×C₈₉	—	4	2	no	—
194	C₈×C₈₉	—	4	2	no	—
195	C₂²×C₈₉	#525	1	1	yes	characteristic
196	C₂²×C₈₉	#525	1	1	yes	characteristic
197	C₂²×C₈₉	#535	1	1	yes	characteristic
198	C₂²×C₈₉	C₂×#135	1	1	yes	characteristic
199	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
200	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
201	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
202	C₄×C₈₉	—	2	2	no	—
203	C₄×C₈₉	—	2	2	no	—
204	C₄×C₈₉	—	2	2	no	—
205	C₄×C₈₉	—	2	2	no	—
206	C₄×C₈₉	—	4	1	no	—
207	C₄×C₈₉	—	4	2	no	—
208	C₄×C₈₉	—	4	2	no	—
209	C₄×C₈₉	—	4	4	no	—
210	C₄×C₈₉	—	4	1	no	—
211	C₄×C₈₉	—	8	1	no	—
212	C₄×C₈₉	—	8	2	no	—
213	C₄×C₈₉	—	8	1	no	—
214	C₄×C₈₉	#551	1	2	yes	—
215	C₄×C₈₉	#553	1	2	yes	—
216	#29371	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂×C₈₉	#2598	1	1	yes	characteristic
218	C₂×C₈₉	#3107	1	1	yes	characteristic
219	C₂×C₈₉	#3108	1	1	yes	characteristic
220	C₂×C₈₉	#3113	1	1	yes	characteristic
221	C₂×C₈₉	#3113	1	1	yes	characteristic
222	C₂×C₈₉	#3210	1	1	yes	characteristic
223	C₂×C₈₉	#3218	1	1	yes	characteristic
224	C₄×C₄⋊Q₈	C₂×C₈₉	1	1	yes	characteristic
225	#2295	C₂×C₈₉	1	1	yes	characteristic
226	#2099	C₂×C₈₉	1	1	yes	characteristic
227	#2096	C₂×C₈₉	1	1	yes	characteristic
228	#2043	C₂×C₈₉	1	1	yes	characteristic
229	#2016	C₂×C₈₉	1	1	yes	characteristic
230	#1927	C₂×C₈₉	1	2	yes	—
231	#1878	C₂×C₈₉	1	2	yes	—
232	#1878	C₂×C₈₉	1	2	yes	—
233	#1865	C₂×C₈₉	1	2	yes	—
234	#1839	C₂×C₈₉	1	1	yes	characteristic
235	C₈₉	#29371	1	1	yes	characteristic, direct product factor, Sylow
236	C₂²×Q₁₆	C₂²×C₈₉	1	2	yes	—
237	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
238	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
239	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
240	C₂×#106	C₂²×C₈₉	1	2	yes	—
241	C₂×#103	C₂²×C₈₉	1	2	yes	—
242	C₂×#103	C₂²×C₈₉	1	2	yes	—
243	C₂×#103	C₂²×C₈₉	1	2	yes	—
244	C₂×#103	C₂²×C₈₉	1	2	yes	—
245	#391	C₂²×C₈₉	1	2	yes	—
246	#391	C₂²×C₈₉	1	1	yes	characteristic
247	#391	C₂²×C₈₉	1	1	yes	characteristic
248	#389	—	2	1	no	—
249	#389	—	2	1	no	—
250	#389	C₂²×C₈₉	1	1	yes	characteristic
251	#389	C₂²×C₈₉	1	1	yes	characteristic
252	#387	—	2	1	no	—
253	#387	—	2	1	no	—
254	#384	C₂²×C₈₉	1	1	yes	characteristic
255	C₄×C₄⋊C₄	—	2	1	no	—
256	C₄×C₄⋊C₄	—	2	1	no	—
257	C₄×C₄⋊C₄	—	2	1	no	—
258	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
259	#376	—	2	1	no	—
260	C₄³	C₂²×C₈₉	1	1	yes	characteristic
261	#340	C₂²×C₈₉	1	2	yes	—
262	#340	C₂²×C₈₉	1	2	yes	—
263	#336	C₂²×C₈₉	1	1	yes	characteristic
264	#336	C₂²×C₈₉	1	1	yes	characteristic
265	#336	C₂²×C₈₉	1	2	yes	—
266	#336	C₂²×C₈₉	1	2	yes	—
267	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
268	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
269	C₂×Q₁₆	—	2	4	no	—
270	C₂×Q₁₆	—	2	4	no	—
271	C₂×Q₁₆	—	4	2	no	—
272	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
273	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
274	C₄⋊Q₈	—	2	4	no	—
275	C₄×Q₈	—	2	2	no	—
276	C₄×Q₈	—	2	2	no	—
277	C₄×Q₈	—	4	2	no	—
278	C₄×Q₈	—	4	2	no	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	—	2	2	no	—
283	C₂×C₄⋊C₄	—	2	1	no	—
284	C₂×C₄⋊C₄	—	2	1	no	—
285	C₂×C₄⋊C₄	—	2	1	no	—
286	C₂×C₄⋊C₄	—	2	1	no	—
287	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
288	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
289	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
291	C₂×C₄²	—	2	1	no	—
292	C₂×C₄²	—	2	2	no	—
293	C₂×C₄²	—	2	1	no	—
294	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
295	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
296	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
297	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
298	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
299	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
300	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
301	#106	—	4	2	no	—
302	#103	—	2	4	no	—
303	#103	—	2	4	no	—
304	#103	—	4	2	no	—
305	#103	—	4	2	no	—
306	#95	—	2	1	no	—
307	#95	—	2	1	no	—
308	#95	—	2	1	no	—
309	#95	—	2	1	no	—
310	#95	—	2	1	no	—
311	#95	—	2	1	no	—
312	#95	—	2	2	no	—
313	#95	—	2	2	no	—
314	#95	—	2	1	no	—
315	C₂×Q₈	—	2	1	no	—
316	C₂×Q₈	—	2	2	no	—
317	C₂×Q₈	—	2	2	no	—
318	C₂×Q₈	—	2	2	no	—
319	C₂×Q₈	—	2	2	no	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	—	2	2	no	—
322	C₂×Q₈	—	2	2	no	—
323	C₂×Q₈	#41×C₈₉	1	4	yes	—
324	C₂×Q₈	#41×C₈₉	1	2	yes	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
332	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
333	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
334	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
335	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
336	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
337	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
338	Q₁₆	—	4	8	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₄⋊C₄	—	2	4	no	—
346	C₄⋊C₄	—	2	4	no	—
347	C₄⋊C₄	—	2	4	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	4	4	no	—
354	C₄⋊C₄	—	4	2	no	—
355	C₄⋊C₄	—	4	2	no	—
356	C₄⋊C₄	—	4	2	no	—
357	C₄⋊C₄	—	4	2	no	—
358	C₄⋊C₄	#41×C₈₉	1	2	yes	—
359	C₄²	—	2	4	no	—
360	C₄²	—	2	2	no	—
361	C₄²	—	2	2	no	—
362	C₄²	—	2	2	no	—
363	C₄²	—	2	2	no	—
364	C₄²	—	2	1	no	—
365	C₄²	—	2	1	no	—
366	C₄²	—	2	2	no	—
367	C₄²	—	2	2	no	—
368	C₄²	—	4	4	no	—
369	C₄²	—	4	2	no	—
370	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
371	Q₈	—	2	4	no	—
372	Q₈	—	2	4	no	—
373	Q₈	—	2	4	no	—
374	Q₈	—	4	2	no	—
375	Q₈	—	4	2	no	—
376	Q₈	—	4	2	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	4	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	4	1	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	4	1	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	—	4	1	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	#135×C₈₉	1	2	yes	—
410	C₂×C₄	#137×C₈₉	1	2	yes	—
411	C₂×C₄	#137×C₈₉	1	2	yes	—
412	C₂×C₄	#137×C₈₉	1	2	yes	—
413	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
414	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
415	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
416	C₂×C₄	H(5,𝔽₂)×C₈₉	1	2	yes	—
417	C₈	—	4	2	no	—
418	C₈	—	4	2	no	—
419	C₂²	#525×C₈₉	1	1	yes	characteristic
420	C₂²	#525×C₈₉	1	1	yes	characteristic
421	C₂²	#535×C₈₉	1	1	yes	characteristic
422	C₂²	C₂×#135×C₈₉	1	1	yes	characteristic
423	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
424	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
425	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
426	C₄	—	2	2	no	—
427	C₄	—	2	2	no	—
428	C₄	—	2	2	no	—
429	C₄	—	2	2	no	—
430	C₄	—	4	1	no	—
431	C₄	—	4	2	no	—
432	C₄	—	4	2	no	—
433	C₄	—	4	4	no	—
434	C₄	—	4	1	no	—
435	C₄	—	8	1	no	—
436	C₄	—	8	2	no	—
437	C₄	—	8	1	no	—
438	C₄	#551×C₈₉	1	2	yes	—
439	C₄	#553×C₈₉	1	2	yes	—
440	C₂	C₈₉×#2598	1	1	yes	characteristic
441	C₂	C₈₉×#3107	1	1	yes	characteristic
442	C₂	C₈₉×#3108	1	1	yes	characteristic
443	C₂	C₈₉×#3113	1	1	yes	characteristic
444	C₂	C₈₉×#3113	1	1	yes	characteristic
445	C₂	C₈₉×#3210	1	1	yes	characteristic
446	C₂	C₈₉×#3218	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 ◁ C₄³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
216. #29371	C₈₉	direct (N × H)
235. C₈₉	#29371	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₈₉	2
C₈₉×#2295	2
C₈₉×#2099	2
C₈₉×#2096	2
C₈₉×#2043	2
C₈₉×#2016	2
C₈₉×#1927	2
C₈₉×#1878	2
C₈₉×#1865	2
C₈₉×#1839	2
#29371	89

Subgroups — C89×#29371 #36407219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#29371 #36407219