Subgroups — C₈₉×#35294 #36413142

Order 22784.

1722 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2868	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈₉×#2859	C₂	1	2	yes	maximal, complemented
3	C₈₉×#2859	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2629	C₂	1	2	yes	maximal
5	C₈₉×#2628	C₂	1	2	yes	maximal
6	C₈₉×#2570	C₂	1	4	yes	maximal, complemented
7	C₈₉×#2550	C₂	1	4	yes	maximal, complemented
8	C₈₉×#2441	C₂	1	4	yes	maximal
9	C₈₉×#2439	C₂	1	4	yes	maximal
10	C₈₉×#2331	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2330	C₂	1	2	yes	maximal, complemented
12	C₈₉×#2285	C₂	1	2	yes	maximal
13	C₄×#117×C₈₉	C₂	1	1	yes	maximal, characteristic
14	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₈₉	C₂²	1	4	yes	—
16	C₂×#125×C₈₉	C₂²	1	2	yes	—
17	C₂×#125×C₈₉	C₂²	1	4	yes	—
18	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#117×C₈₉	C₂²	1	4	yes	—
21	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
22	#400×C₈₉	C₂²	1	4	yes	—
23	#399×C₈₉	C₂²	1	4	yes	—
24	#398×C₈₉	C₂²	1	8	yes	—
25	#396×C₈₉	C₂²	1	8	yes	—
26	#395×C₈₉	C₂²	1	4	yes	—
27	#393×C₈₉	C₂²	1	4	yes	—
28	#391×C₈₉	C₂²	1	8	yes	—
29	#389×C₈₉	C₂²	1	8	yes	—
30	#389×C₈₉	C₂²	1	4	yes	—
31	#389×C₈₉	C₂²	1	4	yes	—
32	#389×C₈₉	C₂²	1	4	yes	—
33	#389×C₈₉	C₂²	1	4	yes	—
34	#389×C₈₉	C₂²	1	4	yes	—
35	#389×C₈₉	C₂²	1	4	yes	—
36	#387×C₈₉	C₂²	1	8	yes	—
37	#387×C₈₉	C₂²	1	8	yes	—
38	#384×C₈₉	C₂²	1	4	yes	—
39	#384×C₈₉	C₂²	1	2	yes	—
40	#384×C₈₉	C₂²	1	2	yes	—
41	#384×C₈₉	C₂²	1	2	yes	—
42	#382×C₈₉	C₂²	1	4	yes	—
43	#382×C₈₉	C₂²	1	4	yes	—
44	#382×C₈₉	C₂²	1	2	yes	—
45	#380×C₈₉	C₂²	1	4	yes	—
46	#380×C₈₉	C₂²	1	4	yes	—
47	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
48	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
49	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
50	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
51	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
52	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
53	#376×C₈₉	C₂²	1	1	yes	characteristic
54	#376×C₈₉	C₂²	1	1	yes	characteristic
55	C₄³×C₈₉	C₂²	1	1	yes	characteristic
56	C₄³×C₈₉	C₂²	1	1	yes	characteristic
57	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
58	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
59	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₈₉	—	2	8	no	—
61	C₄⋊Q₈×C₈₉	—	4	8	no	—
62	#125×C₈₉	—	2	8	no	—
63	#125×C₈₉	—	4	8	no	—
64	#117×C₈₉	—	2	4	no	—
65	#117×C₈₉	—	2	4	no	—
66	#117×C₈₉	—	4	8	no	—
67	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
71	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
72	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
73	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
74	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
75	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
76	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
77	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
78	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
79	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
80	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
81	C₂×C₂²⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
82	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
83	C₂×C₂²⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
84	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
85	C₂×C₄²×C₈₉	Q₈	1	8	yes	complemented
86	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
87	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
89	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
90	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
91	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
92	C₂×C₄²×C₈₉	C₂³	1	8	yes	—
93	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
96	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
98	#95×C₈₉	C₂³	1	2	yes	—
99	#95×C₈₉	C₂³	1	2	yes	—
100	#95×C₈₉	C₂³	1	2	yes	—
101	#95×C₈₉	C₂³	1	2	yes	—
102	#95×C₈₉	C₂³	1	8	yes	—
103	#95×C₈₉	C₂³	1	8	yes	—
104	#95×C₈₉	C₂³	1	8	yes	—
105	#95×C₈₉	C₂³	1	4	yes	—
106	#95×C₈₉	C₂³	1	8	yes	—
107	#95×C₈₉	C₂³	1	4	yes	—
108	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
109	C₂×Q₈×C₈₉	—	2	8	no	—
110	C₂×Q₈×C₈₉	—	4	8	no	—
111	C₂²×C₄×C₈₉	—	2	2	no	—
112	C₂²×C₄×C₈₉	—	2	2	no	—
113	C₂²×C₄×C₈₉	—	2	4	no	—
114	C₂²×C₄×C₈₉	—	2	4	no	—
115	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
116	C₂²×C₄×C₈₉	C₂×Q₈	1	8	yes	—
117	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
118	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
119	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
120	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
121	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
122	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
123	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
124	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
125	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
126	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
127	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
128	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
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	16	no	—
133	C₄⋊C₄×C₈₉	—	2	8	no	—
134	C₄⋊C₄×C₈₉	—	2	8	no	—
135	C₄⋊C₄×C₈₉	—	2	8	no	—
136	C₄⋊C₄×C₈₉	—	2	8	no	—
137	C₄⋊C₄×C₈₉	—	2	8	no	—
138	C₄⋊C₄×C₈₉	—	2	8	no	—
139	C₄⋊C₄×C₈₉	—	2	8	no	—
140	C₄⋊C₄×C₈₉	—	4	8	no	—
141	C₄⋊C₄×C₈₉	—	4	8	no	—
142	C₄⋊C₄×C₈₉	—	4	8	no	—
143	C₂²⋊C₄×C₈₉	—	2	4	no	—
144	C₂²⋊C₄×C₈₉	—	2	2	no	—
145	C₂²⋊C₄×C₈₉	—	2	2	no	—
146	C₂²⋊C₄×C₈₉	—	4	8	no	—
147	C₄²×C₈₉	—	2	2	no	—
148	C₄²×C₈₉	—	2	2	no	—
149	C₄²×C₈₉	—	2	4	no	—
150	C₄²×C₈₉	—	2	4	no	—
151	C₄²×C₈₉	—	2	4	no	—
152	C₄²×C₈₉	—	2	4	no	—
153	C₄²×C₈₉	—	4	8	no	—
154	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
155	C₄²×C₈₉	C₂×Q₈	1	8	yes	—
156	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
157	C₄²×C₈₉	#41	1	4	yes	—
158	C₄²×C₈₉	#41	1	4	yes	—
159	C₂³×C₈₉	—	2	1	no	—
160	C₂³×C₈₉	—	2	1	no	—
161	C₂³×C₈₉	—	2	2	no	—
162	C₂³×C₈₉	—	2	2	no	—
163	C₂³×C₈₉	C₂²×Q₈	1	2	yes	—
164	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
165	Q₈×C₈₉	—	8	8	no	—
166	C₂×C₄×C₈₉	—	2	1	no	—
167	C₂×C₄×C₈₉	—	2	1	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	—	2	2	no	—
172	C₂×C₄×C₈₉	—	2	2	no	—
173	C₂×C₄×C₈₉	—	2	2	no	—
174	C₂×C₄×C₈₉	—	2	8	no	—
175	C₂×C₄×C₈₉	—	2	16	no	—
176	C₂×C₄×C₈₉	—	2	1	no	—
177	C₂×C₄×C₈₉	—	2	1	no	—
178	C₂×C₄×C₈₉	—	2	2	no	—
179	C₂×C₄×C₈₉	—	2	8	no	—
180	C₂×C₄×C₈₉	—	2	8	no	—
181	C₂×C₄×C₈₉	—	2	2	no	—
182	C₂×C₄×C₈₉	—	2	2	no	—
183	C₂×C₄×C₈₉	—	2	8	no	—
184	C₂×C₄×C₈₉	—	2	8	no	—
185	C₂×C₄×C₈₉	—	2	8	no	—
186	C₂×C₄×C₈₉	—	2	2	no	—
187	C₂×C₄×C₈₉	—	2	2	no	—
188	C₂×C₄×C₈₉	—	2	8	no	—
189	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
190	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
191	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
192	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
193	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
194	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
195	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
196	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
197	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
198	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
199	C₂×C₄×C₈₉	2₋¹⁺⁴	1	4	yes	—
200	C₂²×C₈₉	—	2	2	no	—
201	C₂²×C₈₉	—	2	4	no	—
202	C₂²×C₈₉	—	4	1	no	—
203	C₂²×C₈₉	—	4	1	no	—
204	C₂²×C₈₉	—	4	2	no	—
205	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
206	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
207	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
208	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
209	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	2	yes	—
210	C₄×C₈₉	—	2	2	no	—
211	C₄×C₈₉	—	2	4	no	—
212	C₄×C₈₉	—	2	2	no	—
213	C₄×C₈₉	—	2	2	no	—
214	C₄×C₈₉	—	2	4	no	—
215	C₄×C₈₉	—	4	1	no	—
216	C₄×C₈₉	—	4	8	no	—
217	C₄×C₈₉	—	4	1	no	—
218	C₄×C₈₉	—	4	2	no	—
219	C₄×C₈₉	—	4	8	no	—
220	C₄×C₈₉	#533	1	4	yes	—
221	C₄×C₈₉	#544	1	8	yes	—
222	#35294	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
223	C₂×C₈₉	—	4	2	no	—
224	C₂×C₈₉	C₂×#533	1	1	yes	characteristic
225	C₂×C₈₉	C₂×#544	1	2	yes	—
226	C₂×C₈₉	#3508	1	1	yes	characteristic
227	C₂×C₈₉	#3509	1	1	yes	characteristic
228	C₂×C₈₉	#3511	1	2	yes	—
229	#2868	C₂×C₈₉	1	1	yes	characteristic, complemented
230	#2859	C₂×C₈₉	1	1	yes	characteristic, complemented
231	#2859	C₂×C₈₉	1	2	yes	complemented
232	#2629	C₂×C₈₉	1	2	yes	—
233	#2628	C₂×C₈₉	1	2	yes	—
234	#2570	C₂×C₈₉	1	4	yes	complemented
235	#2550	C₂×C₈₉	1	4	yes	complemented
236	#2441	C₂×C₈₉	1	4	yes	—
237	#2439	C₂×C₈₉	1	4	yes	—
238	#2331	C₂×C₈₉	1	2	yes	complemented
239	#2330	C₂×C₈₉	1	2	yes	complemented
240	#2285	C₂×C₈₉	1	2	yes	—
241	C₄×#117	C₂×C₈₉	1	1	yes	characteristic
242	C₈₉	#35294	1	1	yes	characteristic, direct product factor, Sylow
243	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
244	C₂×C₄⋊Q₈	C₂²×C₈₉	1	4	yes	—
245	C₂×#125	C₂²×C₈₉	1	2	yes	—
246	C₂×#125	C₂²×C₈₉	1	4	yes	—
247	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
248	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
249	C₂×#117	C₂²×C₈₉	1	4	yes	—
250	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
251	#400	C₂²×C₈₉	1	4	yes	—
252	#399	C₂²×C₈₉	1	4	yes	—
253	#398	C₂²×C₈₉	1	8	yes	—
254	#396	C₂²×C₈₉	1	8	yes	—
255	#395	C₂²×C₈₉	1	4	yes	—
256	#393	C₂²×C₈₉	1	4	yes	—
257	#391	C₂²×C₈₉	1	8	yes	—
258	#389	C₂²×C₈₉	1	4	yes	—
259	#389	C₂²×C₈₉	1	4	yes	—
260	#389	C₂²×C₈₉	1	4	yes	—
261	#389	C₂²×C₈₉	1	8	yes	—
262	#389	C₂²×C₈₉	1	4	yes	—
263	#389	C₂²×C₈₉	1	4	yes	—
264	#389	C₂²×C₈₉	1	4	yes	—
265	#387	C₂²×C₈₉	1	8	yes	—
266	#387	C₂²×C₈₉	1	8	yes	—
267	#384	C₂²×C₈₉	1	4	yes	—
268	#384	C₂²×C₈₉	1	2	yes	—
269	#384	C₂²×C₈₉	1	2	yes	—
270	#384	C₂²×C₈₉	1	2	yes	—
271	#382	C₂²×C₈₉	1	4	yes	—
272	#382	C₂²×C₈₉	1	4	yes	—
273	#382	C₂²×C₈₉	1	2	yes	—
274	#380	C₂²×C₈₉	1	4	yes	—
275	#380	C₂²×C₈₉	1	4	yes	—
276	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
277	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
278	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
279	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
280	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
281	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
282	#376	C₂²×C₈₉	1	1	yes	characteristic
283	#376	C₂²×C₈₉	1	1	yes	characteristic
284	C₄³	C₂²×C₈₉	1	1	yes	characteristic
285	C₄³	C₂²×C₈₉	1	1	yes	characteristic
286	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
287	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
288	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
289	C₄⋊Q₈	—	2	8	no	—
290	C₄⋊Q₈	—	4	8	no	—
291	#125	—	2	8	no	—
292	#125	—	4	8	no	—
293	#117	—	2	4	no	—
294	#117	—	2	4	no	—
295	#117	—	4	8	no	—
296	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
297	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
298	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
299	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
300	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
301	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
302	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
303	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
304	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
305	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
306	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
307	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
308	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
309	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
310	C₂×C₂²⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
311	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
312	C₂×C₂²⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
313	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
314	C₂×C₄²	Q₈×C₈₉	1	8	yes	complemented
315	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
316	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
317	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
318	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
319	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
320	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
321	C₂×C₄²	C₂³×C₈₉	1	8	yes	—
322	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
323	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
324	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
325	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
326	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
327	#95	C₂³×C₈₉	1	2	yes	—
328	#95	C₂³×C₈₉	1	2	yes	—
329	#95	C₂³×C₈₉	1	2	yes	—
330	#95	C₂³×C₈₉	1	2	yes	—
331	#95	C₂³×C₈₉	1	8	yes	—
332	#95	C₂³×C₈₉	1	8	yes	—
333	#95	C₂³×C₈₉	1	8	yes	—
334	#95	C₂³×C₈₉	1	4	yes	—
335	#95	C₂³×C₈₉	1	8	yes	—
336	#95	C₂³×C₈₉	1	4	yes	—
337	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
338	C₂×Q₈	—	2	8	no	—
339	C₂×Q₈	—	4	8	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	4	no	—
343	C₂²×C₄	—	2	4	no	—
344	C₂²×C₄	C₂×Q₈×C₈₉	1	4	yes	—
345	C₂²×C₄	C₂×Q₈×C₈₉	1	8	yes	—
346	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
347	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
348	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
349	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
350	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
351	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
352	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
353	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
354	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
355	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
356	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
357	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
358	C₄⋊C₄	—	2	4	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	16	no	—
362	C₄⋊C₄	—	2	8	no	—
363	C₄⋊C₄	—	2	8	no	—
364	C₄⋊C₄	—	2	8	no	—
365	C₄⋊C₄	—	2	8	no	—
366	C₄⋊C₄	—	2	8	no	—
367	C₄⋊C₄	—	2	8	no	—
368	C₄⋊C₄	—	2	8	no	—
369	C₄⋊C₄	—	4	8	no	—
370	C₄⋊C₄	—	4	8	no	—
371	C₄⋊C₄	—	4	8	no	—
372	C₂²⋊C₄	—	2	4	no	—
373	C₂²⋊C₄	—	2	2	no	—
374	C₂²⋊C₄	—	2	2	no	—
375	C₂²⋊C₄	—	4	8	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	2	2	no	—
378	C₄²	—	2	4	no	—
379	C₄²	—	2	4	no	—
380	C₄²	—	2	4	no	—
381	C₄²	—	2	4	no	—
382	C₄²	—	4	8	no	—
383	C₄²	C₂×Q₈×C₈₉	1	4	yes	—
384	C₄²	C₂×Q₈×C₈₉	1	8	yes	—
385	C₄²	C₂×Q₈×C₈₉	1	4	yes	—
386	C₄²	#41×C₈₉	1	4	yes	—
387	C₄²	#41×C₈₉	1	4	yes	—
388	C₂³	—	2	1	no	—
389	C₂³	—	2	1	no	—
390	C₂³	—	2	2	no	—
391	C₂³	—	2	2	no	—
392	C₂³	C₂²×Q₈×C₈₉	1	2	yes	—
393	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
394	Q₈	—	8	8	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	8	no	—
404	C₂×C₄	—	2	16	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	8	no	—
409	C₂×C₄	—	2	8	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	8	no	—
413	C₂×C₄	—	2	8	no	—
414	C₂×C₄	—	2	8	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	8	no	—
418	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
419	C₂×C₄	C₂²×Q₈×C₈₉	1	4	yes	—
420	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
421	C₂×C₄	C₂²×Q₈×C₈₉	1	4	yes	—
422	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
423	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
424	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
425	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
426	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
427	C₂×C₄	H(5,𝔽₂)×C₈₉	1	4	yes	—
428	C₂×C₄	2₋¹⁺⁴×C₈₉	1	4	yes	—
429	C₂²	—	2	2	no	—
430	C₂²	—	2	4	no	—
431	C₂²	—	4	1	no	—
432	C₂²	—	4	1	no	—
433	C₂²	—	4	2	no	—
434	C₂²	C₂³×Q₈×C₈₉	1	1	yes	characteristic
435	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
436	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
437	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
438	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	2	yes	—
439	C₄	—	2	2	no	—
440	C₄	—	2	4	no	—
441	C₄	—	2	2	no	—
442	C₄	—	2	2	no	—
443	C₄	—	2	4	no	—
444	C₄	—	4	8	no	—
445	C₄	—	4	1	no	—
446	C₄	—	4	1	no	—
447	C₄	—	4	2	no	—
448	C₄	—	4	8	no	—
449	C₄	#533×C₈₉	1	4	yes	—
450	C₄	#544×C₈₉	1	8	yes	—
451	C₂	—	4	2	no	—
452	C₂	C₂×#533×C₈₉	1	1	yes	characteristic
453	C₂	C₂×#544×C₈₉	1	2	yes	—
454	C₂	C₈₉×#3508	1	1	yes	characteristic
455	C₂	C₈₉×#3509	1	1	yes	characteristic
456	C₂	C₈₉×#3511	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂⁴×C₈₉	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#2868	C₂	semidirect
2. C₈₉×#2859	C₂	semidirect
3. C₈₉×#2859	C₂	semidirect
6. C₈₉×#2570	C₂	semidirect
7. C₈₉×#2550	C₂	semidirect
10. C₈₉×#2331	C₂	semidirect
11. C₈₉×#2330	C₂	semidirect
85. C₂×C₄²×C₈₉	Q₈	semidirect
222. #35294	C₈₉	direct (N × H)
229. #2868	C₂×C₈₉	semidirect
230. #2859	C₂×C₈₉	semidirect
231. #2859	C₂×C₈₉	semidirect
234. #2570	C₂×C₈₉	semidirect
235. #2550	C₂×C₈₉	semidirect
238. #2331	C₂×C₈₉	semidirect
239. #2330	C₂×C₈₉	semidirect
242. C₈₉	#35294	direct (N × H)
314. C₂×C₄²	Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#2868	2
C₈₉×#2859	2
C₈₉×#2629	2
C₈₉×#2628	2
C₈₉×#2570	2
C₈₉×#2550	2
C₈₉×#2441	2
C₈₉×#2439	2
C₈₉×#2331	2
C₈₉×#2330	2
C₈₉×#2285	2
C₄×#117×C₈₉	2
#35294	89

Subgroups — C89×#35294 #36413142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35294 #36413142