Subgroups — C₈₉×#35138 #36412986

Order 22784.

2306 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#41×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2851	C₂	1	4	yes	maximal, complemented
4	C₈₉×#2634	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2633	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2611	C₂	1	4	yes	maximal, complemented
7	C₈₉×#2584	C₂	1	2	yes	maximal, complemented
8	C₈₉×#2562	C₂	1	2	yes	maximal, complemented
9	C₈₉×#2487	C₂	1	2	yes	maximal, complemented
10	C₈₉×#2486	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2484	C₂	1	2	yes	maximal, complemented
12	C₈₉×#2406	C₂	1	2	yes	maximal
13	C₈₉×#2322	C₂	1	4	yes	maximal, complemented
14	C₈₉×#2283	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#125×C₈₉	—	2	8	no	—
17	C₂×#125×C₈₉	C₂²	1	2	yes	—
18	C₂×#125×C₈₉	C₂²	1	2	yes	—
19	C₂×#125×C₈₉	C₂²	1	2	yes	—
20	C₂×#125×C₈₉	C₂²	1	2	yes	—
21	C₂×#123×C₈₉	C₂²	1	4	yes	—
22	C₄×#41×C₈₉	—	2	8	no	—
23	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
29	C₂×#117×C₈₉	C₂²	1	2	yes	—
30	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
31	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
32	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
33	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
34	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
35	C₂²×C₄²×C₈₉	C₂²	1	2	yes	—
36	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
37	#400×C₈₉	C₂²	1	4	yes	—
38	#400×C₈₉	C₂²	1	4	yes	—
39	#399×C₈₉	C₂²	1	4	yes	—
40	#398×C₈₉	C₂²	1	4	yes	—
41	#398×C₈₉	C₂²	1	4	yes	—
42	#397×C₈₉	C₂²	1	4	yes	—
43	#395×C₈₉	C₂²	1	4	yes	—
44	#394×C₈₉	C₂²	1	4	yes	—
45	#389×C₈₉	C₂²	1	8	yes	—
46	#389×C₈₉	C₂²	1	4	yes	—
47	#389×C₈₉	C₂²	1	4	yes	—
48	#388×C₈₉	C₂²	1	8	yes	—
49	#387×C₈₉	C₂²	1	8	yes	—
50	#387×C₈₉	C₂²	1	8	yes	—
51	#386×C₈₉	C₂²	1	4	yes	—
52	#385×C₈₉	C₂²	1	8	yes	—
53	#385×C₈₉	C₂²	1	4	yes	—
54	#385×C₈₉	C₂²	1	8	yes	—
55	#383×C₈₉	C₂²	1	2	yes	—
56	#382×C₈₉	C₂²	1	2	yes	—
57	#381×C₈₉	C₂²	1	4	yes	—
58	#381×C₈₉	C₂²	1	2	yes	—
59	#380×C₈₉	C₂²	1	4	yes	—
60	#380×C₈₉	C₂²	1	4	yes	—
61	#380×C₈₉	C₂²	1	2	yes	—
62	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
63	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
64	#376×C₈₉	C₂²	1	2	yes	—
65	C₂×#41×C₈₉	—	2	8	no	—
66	C₂×#41×C₈₉	—	2	4	no	—
67	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
68	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
69	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
70	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
71	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
72	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
73	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
76	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
77	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
78	#125×C₈₉	—	2	8	no	—
79	#125×C₈₉	—	2	8	no	—
80	#125×C₈₉	—	2	8	no	—
81	#125×C₈₉	—	2	8	no	—
82	#123×C₈₉	—	2	16	no	—
83	C₄×Q₈×C₈₉	—	2	4	no	—
84	C₄×Q₈×C₈₉	—	2	4	no	—
85	C₄×D₄×C₈₉	—	2	8	no	—
86	C₄×D₄×C₈₉	—	2	8	no	—
87	C₄×D₄×C₈₉	—	2	4	no	—
88	C₄×D₄×C₈₉	—	2	4	no	—
89	#117×C₈₉	—	2	4	no	—
90	#117×C₈₉	—	2	8	no	—
91	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
92	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
93	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
94	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
95	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
96	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
101	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
104	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
110	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
111	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
112	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
116	C₂×C₄²×C₈₉	—	2	8	no	—
117	C₂×C₄²×C₈₉	D₄	1	8	yes	—
118	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
119	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
120	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
121	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
122	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
126	#95×C₈₉	C₂³	1	4	yes	—
127	#95×C₈₉	C₂³	1	4	yes	—
128	#95×C₈₉	C₂³	1	4	yes	—
129	#95×C₈₉	C₂³	1	4	yes	—
130	#95×C₈₉	C₂³	1	8	yes	—
131	#95×C₈₉	C₂³	1	8	yes	—
132	#95×C₈₉	C₂³	1	8	yes	—
133	#95×C₈₉	C₂³	1	8	yes	—
134	C₂⁴×C₈₉	C₂⁴	1	2	yes	—
135	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
136	#41×C₈₉	—	2	32	no	—
137	C₂×Q₈×C₈₉	—	2	2	no	—
138	C₂×Q₈×C₈₉	—	2	4	no	—
139	C₂×D₄×C₈₉	—	2	4	no	—
140	C₂×D₄×C₈₉	—	2	4	no	—
141	C₂×D₄×C₈₉	—	2	4	no	—
142	C₂×D₄×C₈₉	—	2	2	no	—
143	C₂×D₄×C₈₉	#41	1	8	yes	—
144	C₂²×C₄×C₈₉	—	2	4	no	—
145	C₂²×C₄×C₈₉	—	2	4	no	—
146	C₂²×C₄×C₈₉	—	2	4	no	—
147	C₂²×C₄×C₈₉	—	2	4	no	—
148	C₂²×C₄×C₈₉	—	2	4	no	—
149	C₂²×C₄×C₈₉	—	2	4	no	—
150	C₂²×C₄×C₈₉	—	2	4	no	—
151	C₂²×C₄×C₈₉	—	2	8	no	—
152	C₂²×C₄×C₈₉	—	2	4	no	—
153	C₂²×C₄×C₈₉	—	2	4	no	—
154	C₂²×C₄×C₈₉	—	2	8	no	—
155	C₂²×C₄×C₈₉	—	4	2	no	—
156	C₂²×C₄×C₈₉	—	4	4	no	—
157	C₂²×C₄×C₈₉	—	4	4	no	—
158	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
159	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
160	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
161	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
163	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
167	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
170	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
171	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
173	C₄⋊C₄×C₈₉	—	2	8	no	—
174	C₄⋊C₄×C₈₉	—	2	8	no	—
175	C₄⋊C₄×C₈₉	—	2	8	no	—
176	C₄⋊C₄×C₈₉	—	2	4	no	—
177	C₄⋊C₄×C₈₉	—	2	4	no	—
178	C₄⋊C₄×C₈₉	—	2	8	no	—
179	C₄⋊C₄×C₈₉	—	2	4	no	—
180	C₄⋊C₄×C₈₉	—	2	4	no	—
181	C₄⋊C₄×C₈₉	—	2	4	no	—
182	C₄⋊C₄×C₈₉	—	2	4	no	—
183	C₄⋊C₄×C₈₉	—	2	2	no	—
184	C₄⋊C₄×C₈₉	—	2	2	no	—
185	C₄⋊C₄×C₈₉	—	2	16	no	—
186	C₄⋊C₄×C₈₉	—	2	8	no	—
187	C₄⋊C₄×C₈₉	—	4	8	no	—
188	C₂²⋊C₄×C₈₉	—	2	4	no	—
189	C₂²⋊C₄×C₈₉	—	2	8	no	—
190	C₂²⋊C₄×C₈₉	—	2	16	no	—
191	C₂²⋊C₄×C₈₉	—	2	4	no	—
192	C₂²⋊C₄×C₈₉	—	2	4	no	—
193	C₂²⋊C₄×C₈₉	—	4	8	no	—
194	C₄²×C₈₉	—	2	4	no	—
195	C₄²×C₈₉	—	2	4	no	—
196	C₄²×C₈₉	—	2	4	no	—
197	C₄²×C₈₉	—	4	4	no	—
198	C₄²×C₈₉	C₂×D₄	1	4	yes	—
199	C₄²×C₈₉	C₂×D₄	1	4	yes	—
200	C₄²×C₈₉	C₂×D₄	1	4	yes	—
201	C₄²×C₈₉	C₂×D₄	1	4	yes	—
202	C₂³×C₈₉	—	2	2	no	—
203	C₂³×C₈₉	—	2	2	no	—
204	C₂³×C₈₉	—	2	2	no	—
205	C₂³×C₈₉	—	2	2	no	—
206	C₂³×C₈₉	—	2	4	no	—
207	C₂³×C₈₉	—	2	4	no	—
208	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
209	C₂³×C₈₉	C₂×#41	1	4	yes	—
210	C₂³×C₈₉	C₂×#41	1	4	yes	—
211	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
212	Q₈×C₈₉	—	2	8	no	—
213	D₄×C₈₉	—	2	8	no	—
214	D₄×C₈₉	—	2	16	no	—
215	C₂×C₄×C₈₉	—	2	16	no	—
216	C₂×C₄×C₈₉	—	2	8	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	—	2	2	no	—
222	C₂×C₄×C₈₉	—	2	2	no	—
223	C₂×C₄×C₈₉	—	2	2	no	—
224	C₂×C₄×C₈₉	—	2	8	no	—
225	C₂×C₄×C₈₉	—	2	2	no	—
226	C₂×C₄×C₈₉	—	2	2	no	—
227	C₂×C₄×C₈₉	—	2	4	no	—
228	C₂×C₄×C₈₉	—	2	4	no	—
229	C₂×C₄×C₈₉	—	2	8	no	—
230	C₂×C₄×C₈₉	—	2	8	no	—
231	C₂×C₄×C₈₉	—	2	8	no	—
232	C₂×C₄×C₈₉	—	2	4	no	—
233	C₂×C₄×C₈₉	—	2	8	no	—
234	C₂×C₄×C₈₉	—	2	4	no	—
235	C₂×C₄×C₈₉	—	2	4	no	—
236	C₂×C₄×C₈₉	—	2	4	no	—
237	C₂×C₄×C₈₉	—	4	4	no	—
238	C₂×C₄×C₈₉	—	4	8	no	—
239	C₂×C₄×C₈₉	—	4	8	no	—
240	C₂×C₄×C₈₉	—	4	8	no	—
241	C₂×C₄×C₈₉	—	4	8	no	—
242	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
243	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
244	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
245	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
246	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
247	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
248	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
249	C₂×C₄×C₈₉	2₋¹⁺⁴	1	4	yes	—
250	C₂²×C₈₉	—	2	2	no	—
251	C₂²×C₈₉	—	2	2	no	—
252	C₂²×C₈₉	—	2	8	no	—
253	C₂²×C₈₉	—	2	4	no	—
254	C₂²×C₈₉	—	2	8	no	—
255	C₂²×C₈₉	—	4	2	no	—
256	C₂²×C₈₉	—	4	2	no	—
257	C₂²×C₈₉	—	4	4	no	—
258	C₂²×C₈₉	#525	1	4	yes	—
259	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
260	C₂²×C₈₉	C₂²×#41	1	2	yes	—
261	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	2	yes	—
262	C₂²×C₈₉	#585	1	2	yes	—
263	C₄×C₈₉	—	2	2	no	—
264	C₄×C₈₉	—	2	2	no	—
265	C₄×C₈₉	—	2	4	no	—
266	C₄×C₈₉	—	2	2	no	—
267	C₄×C₈₉	—	2	2	no	—
268	C₄×C₈₉	—	2	2	no	—
269	C₄×C₈₉	—	4	4	no	—
270	C₄×C₈₉	—	4	2	no	—
271	C₄×C₈₉	—	4	2	no	—
272	C₄×C₈₉	—	4	4	no	—
273	C₄×C₈₉	—	4	8	no	—
274	C₄×C₈₉	#536	1	4	yes	—
275	#35138	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
276	C₂×C₈₉	—	2	4	no	—
277	C₂×C₈₉	—	4	4	no	—
278	C₂×C₈₉	C₂×#525	1	1	yes	characteristic
279	C₂×C₈₉	C₂×#536	1	1	yes	characteristic
280	C₂×C₈₉	#3449	1	1	yes	characteristic
281	C₂×C₈₉	#3515	1	4	yes	—
282	C₂²×#125	C₂×C₈₉	1	1	yes	characteristic, complemented
283	C₂×C₄×#41	C₂×C₈₉	1	1	yes	characteristic
284	#2851	C₂×C₈₉	1	4	yes	complemented
285	#2634	C₂×C₈₉	1	2	yes	complemented
286	#2633	C₂×C₈₉	1	2	yes	complemented
287	#2611	C₂×C₈₉	1	4	yes	complemented
288	#2584	C₂×C₈₉	1	2	yes	complemented
289	#2562	C₂×C₈₉	1	2	yes	complemented
290	#2487	C₂×C₈₉	1	2	yes	complemented
291	#2486	C₂×C₈₉	1	2	yes	complemented
292	#2484	C₂×C₈₉	1	2	yes	complemented
293	#2406	C₂×C₈₉	1	2	yes	—
294	#2322	C₂×C₈₉	1	4	yes	complemented
295	#2283	C₂×C₈₉	1	1	yes	characteristic, complemented
296	C₈₉	#35138	1	1	yes	characteristic, direct product factor, Sylow
297	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
298	C₂×#125	—	2	8	no	—
299	C₂×#125	C₂²×C₈₉	1	2	yes	—
300	C₂×#125	C₂²×C₈₉	1	2	yes	—
301	C₂×#125	C₂²×C₈₉	1	2	yes	—
302	C₂×#125	C₂²×C₈₉	1	2	yes	—
303	C₂×#123	C₂²×C₈₉	1	4	yes	—
304	C₄×#41	—	2	8	no	—
305	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
306	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
307	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
308	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
309	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
310	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
311	C₂×#117	C₂²×C₈₉	1	2	yes	—
312	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
313	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
314	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
315	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
316	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
317	C₂²×C₄²	C₂²×C₈₉	1	2	yes	—
318	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
319	#400	C₂²×C₈₉	1	4	yes	—
320	#400	C₂²×C₈₉	1	4	yes	—
321	#399	C₂²×C₈₉	1	4	yes	—
322	#398	C₂²×C₈₉	1	4	yes	—
323	#398	C₂²×C₈₉	1	4	yes	—
324	#397	C₂²×C₈₉	1	4	yes	—
325	#395	C₂²×C₈₉	1	4	yes	—
326	#394	C₂²×C₈₉	1	4	yes	—
327	#389	C₂²×C₈₉	1	8	yes	—
328	#389	C₂²×C₈₉	1	4	yes	—
329	#389	C₂²×C₈₉	1	4	yes	—
330	#388	C₂²×C₈₉	1	8	yes	—
331	#387	C₂²×C₈₉	1	8	yes	—
332	#387	C₂²×C₈₉	1	8	yes	—
333	#386	C₂²×C₈₉	1	4	yes	—
334	#385	C₂²×C₈₉	1	8	yes	—
335	#385	C₂²×C₈₉	1	4	yes	—
336	#385	C₂²×C₈₉	1	8	yes	—
337	#383	C₂²×C₈₉	1	2	yes	—
338	#382	C₂²×C₈₉	1	2	yes	—
339	#381	C₂²×C₈₉	1	4	yes	—
340	#381	C₂²×C₈₉	1	2	yes	—
341	#380	C₂²×C₈₉	1	4	yes	—
342	#380	C₂²×C₈₉	1	4	yes	—
343	#380	C₂²×C₈₉	1	2	yes	—
344	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
345	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
346	#376	C₂²×C₈₉	1	2	yes	—
347	C₂×#41	—	2	8	no	—
348	C₂×#41	—	2	4	no	—
349	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
350	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
351	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
352	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
353	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
354	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
355	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
357	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
358	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
359	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
360	#125	—	2	8	no	—
361	#125	—	2	8	no	—
362	#125	—	2	8	no	—
363	#125	—	2	8	no	—
364	#123	—	2	16	no	—
365	C₄×Q₈	—	2	4	no	—
366	C₄×Q₈	—	2	4	no	—
367	C₄×D₄	—	2	8	no	—
368	C₄×D₄	—	2	8	no	—
369	C₄×D₄	—	2	4	no	—
370	C₄×D₄	—	2	4	no	—
371	#117	—	2	4	no	—
372	#117	—	2	8	no	—
373	C₂×C₄⋊C₄	—	2	8	no	—
374	C₂×C₄⋊C₄	—	2	4	no	—
375	C₂×C₄⋊C₄	—	2	4	no	—
376	C₂×C₄⋊C₄	—	2	8	no	—
377	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
378	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
379	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
380	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
381	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
382	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
383	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
384	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
385	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
386	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
388	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
389	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
390	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
391	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
392	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
393	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
394	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
395	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
396	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
397	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
398	C₂×C₄²	—	2	8	no	—
399	C₂×C₄²	D₄×C₈₉	1	8	yes	—
400	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
401	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
402	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
403	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
404	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
405	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
406	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
407	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
408	#95	C₂³×C₈₉	1	4	yes	—
409	#95	C₂³×C₈₉	1	4	yes	—
410	#95	C₂³×C₈₉	1	4	yes	—
411	#95	C₂³×C₈₉	1	4	yes	—
412	#95	C₂³×C₈₉	1	8	yes	—
413	#95	C₂³×C₈₉	1	8	yes	—
414	#95	C₂³×C₈₉	1	8	yes	—
415	#95	C₂³×C₈₉	1	8	yes	—
416	C₂⁴	C₂⁴×C₈₉	1	2	yes	—
417	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
418	#41	—	2	32	no	—
419	C₂×Q₈	—	2	2	no	—
420	C₂×Q₈	—	2	4	no	—
421	C₂×D₄	—	2	4	no	—
422	C₂×D₄	—	2	4	no	—
423	C₂×D₄	—	2	4	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	#41×C₈₉	1	8	yes	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	4	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	8	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	8	no	—
437	C₂²×C₄	—	4	2	no	—
438	C₂²×C₄	—	4	4	no	—
439	C₂²×C₄	—	4	4	no	—
440	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
441	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
442	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
443	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
444	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
445	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
446	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
447	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
448	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
449	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
450	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
451	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
452	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
453	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
454	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
455	C₄⋊C₄	—	2	8	no	—
456	C₄⋊C₄	—	2	8	no	—
457	C₄⋊C₄	—	2	8	no	—
458	C₄⋊C₄	—	2	4	no	—
459	C₄⋊C₄	—	2	4	no	—
460	C₄⋊C₄	—	2	8	no	—
461	C₄⋊C₄	—	2	4	no	—
462	C₄⋊C₄	—	2	4	no	—
463	C₄⋊C₄	—	2	4	no	—
464	C₄⋊C₄	—	2	4	no	—
465	C₄⋊C₄	—	2	2	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	2	16	no	—
468	C₄⋊C₄	—	2	8	no	—
469	C₄⋊C₄	—	4	8	no	—
470	C₂²⋊C₄	—	2	4	no	—
471	C₂²⋊C₄	—	2	8	no	—
472	C₂²⋊C₄	—	2	16	no	—
473	C₂²⋊C₄	—	2	4	no	—
474	C₂²⋊C₄	—	2	4	no	—
475	C₂²⋊C₄	—	4	8	no	—
476	C₄²	—	2	4	no	—
477	C₄²	—	2	4	no	—
478	C₄²	—	2	4	no	—
479	C₄²	—	4	4	no	—
480	C₄²	C₂×D₄×C₈₉	1	4	yes	—
481	C₄²	C₂×D₄×C₈₉	1	4	yes	—
482	C₄²	C₂×D₄×C₈₉	1	4	yes	—
483	C₄²	C₂×D₄×C₈₉	1	4	yes	—
484	C₂³	—	2	2	no	—
485	C₂³	—	2	2	no	—
486	C₂³	—	2	2	no	—
487	C₂³	—	2	2	no	—
488	C₂³	—	2	4	no	—
489	C₂³	—	2	4	no	—
490	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
491	C₂³	C₂×#41×C₈₉	1	4	yes	—
492	C₂³	C₂×#41×C₈₉	1	4	yes	—
493	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
494	Q₈	—	2	8	no	—
495	D₄	—	2	8	no	—
496	D₄	—	2	16	no	—
497	C₂×C₄	—	2	16	no	—
498	C₂×C₄	—	2	8	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	8	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	8	no	—
512	C₂×C₄	—	2	8	no	—
513	C₂×C₄	—	2	8	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	8	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	4	4	no	—
520	C₂×C₄	—	4	8	no	—
521	C₂×C₄	—	4	8	no	—
522	C₂×C₄	—	4	8	no	—
523	C₂×C₄	—	4	8	no	—
524	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
525	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
526	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
527	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
528	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
529	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
530	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
531	C₂×C₄	2₋¹⁺⁴×C₈₉	1	4	yes	—
532	C₂²	—	2	2	no	—
533	C₂²	—	2	2	no	—
534	C₂²	—	2	8	no	—
535	C₂²	—	2	4	no	—
536	C₂²	—	2	8	no	—
537	C₂²	—	4	2	no	—
538	C₂²	—	4	2	no	—
539	C₂²	—	4	4	no	—
540	C₂²	#525×C₈₉	1	4	yes	—
541	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
542	C₂²	C₂²×#41×C₈₉	1	2	yes	—
543	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	2	yes	—
544	C₂²	#585×C₈₉	1	2	yes	—
545	C₄	—	2	2	no	—
546	C₄	—	2	2	no	—
547	C₄	—	2	4	no	—
548	C₄	—	2	2	no	—
549	C₄	—	2	2	no	—
550	C₄	—	2	2	no	—
551	C₄	—	4	4	no	—
552	C₄	—	4	2	no	—
553	C₄	—	4	2	no	—
554	C₄	—	4	4	no	—
555	C₄	—	4	8	no	—
556	C₄	#536×C₈₉	1	4	yes	—
557	C₂	—	2	4	no	—
558	C₂	—	4	4	no	—
559	C₂	C₂×#525×C₈₉	1	1	yes	characteristic
560	C₂	C₂×#536×C₈₉	1	1	yes	characteristic
561	C₂	C₈₉×#3449	1	1	yes	characteristic
562	C₂	C₈₉×#3515	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂²×#41×C₈₉	C₂²
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. C₂²×#125×C₈₉	C₂	semidirect
3. C₈₉×#2851	C₂	semidirect
4. C₈₉×#2634	C₂	semidirect
5. C₈₉×#2633	C₂	semidirect
6. C₈₉×#2611	C₂	semidirect
7. C₈₉×#2584	C₂	semidirect
8. C₈₉×#2562	C₂	semidirect
9. C₈₉×#2487	C₂	semidirect
10. C₈₉×#2486	C₂	semidirect
11. C₈₉×#2484	C₂	semidirect
13. C₈₉×#2322	C₂	semidirect
14. C₈₉×#2283	C₂	semidirect
275. #35138	C₈₉	direct (N × H)
282. C₂²×#125	C₂×C₈₉	semidirect
284. #2851	C₂×C₈₉	semidirect
285. #2634	C₂×C₈₉	semidirect
286. #2633	C₂×C₈₉	semidirect
287. #2611	C₂×C₈₉	semidirect
288. #2584	C₂×C₈₉	semidirect
289. #2562	C₂×C₈₉	semidirect
290. #2487	C₂×C₈₉	semidirect
291. #2486	C₂×C₈₉	semidirect
292. #2484	C₂×C₈₉	semidirect
294. #2322	C₂×C₈₉	semidirect
295. #2283	C₂×C₈₉	semidirect
296. C₈₉	#35138	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#125×C₈₉	2
C₂×C₄×#41×C₈₉	2
C₈₉×#2851	2
C₈₉×#2634	2
C₈₉×#2633	2
C₈₉×#2611	2
C₈₉×#2584	2
C₈₉×#2562	2
C₈₉×#2487	2
C₈₉×#2486	2
C₈₉×#2484	2
C₈₉×#2406	2
C₈₉×#2322	2
C₈₉×#2283	2
#35138	89

Subgroups — C89×#35138 #36412986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35138 #36412986