Subgroups — C₈₉×#35177 #36413025

Order 22784.

1796 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2857	C₂	1	2	yes	maximal, complemented
2	C₈₉×#2855	C₂	1	2	yes	maximal, complemented
3	C₈₉×#2665	C₂	1	4	yes	maximal, complemented
4	C₈₉×#2567	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2563	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#2551	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2538	C₂	1	1	yes	maximal, characteristic
8	C₈₉×#2509	C₂	1	2	yes	maximal, complemented
9	C₈₉×#2497	C₂	1	2	yes	maximal, complemented
10	C₈₉×#2492	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2465	C₂	1	4	yes	maximal, complemented
12	C₈₉×#2452	C₂	1	2	yes	maximal
13	C₈₉×#2331	C₂	1	2	yes	maximal, complemented
14	C₈₉×#2319	C₂	1	2	yes	maximal
15	C₄×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₈₉×#2295	C₂	1	1	yes	maximal, characteristic, complemented
17	C₄²×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic
18	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
19	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
21	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	2	yes	—
22	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	4	yes	—
23	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₈₉	C₂²	1	2	yes	—
29	#400×C₈₉	C₂²	1	4	yes	—
30	#400×C₈₉	C₂²	1	4	yes	—
31	#396×C₈₉	C₂²	1	4	yes	—
32	#396×C₈₉	C₂²	1	4	yes	—
33	#395×C₈₉	C₂²	1	4	yes	—
34	#395×C₈₉	C₂²	1	4	yes	—
35	#393×C₈₉	C₂²	1	4	yes	—
36	#393×C₈₉	C₂²	1	4	yes	—
37	#391×C₈₉	C₂²	1	1	yes	characteristic
38	#391×C₈₉	C₂²	1	1	yes	characteristic
39	#391×C₈₉	C₂²	1	2	yes	—
40	#391×C₈₉	C₂²	1	2	yes	—
41	#391×C₈₉	C₂²	1	4	yes	—
42	#391×C₈₉	C₂²	1	4	yes	—
43	#390×C₈₉	C₂²	1	2	yes	—
44	#390×C₈₉	C₂²	1	2	yes	—
45	#389×C₈₉	C₂²	1	1	yes	characteristic
46	#389×C₈₉	C₂²	1	1	yes	characteristic
47	#389×C₈₉	C₂²	1	2	yes	—
48	#389×C₈₉	C₂²	1	2	yes	—
49	#389×C₈₉	C₂²	1	2	yes	—
50	#389×C₈₉	C₂²	1	2	yes	—
51	#389×C₈₉	C₂²	1	2	yes	—
52	#389×C₈₉	C₂²	1	2	yes	—
53	#389×C₈₉	C₂²	1	2	yes	—
54	#389×C₈₉	C₂²	1	4	yes	—
55	#388×C₈₉	C₂²	1	4	yes	—
56	#388×C₈₉	C₂²	1	4	yes	—
57	#387×C₈₉	C₂²	1	2	yes	—
58	#387×C₈₉	C₂²	1	2	yes	—
59	#387×C₈₉	C₂²	1	4	yes	—
60	#387×C₈₉	C₂²	1	4	yes	—
61	#387×C₈₉	C₂²	1	4	yes	—
62	#385×C₈₉	C₂²	1	4	yes	—
63	#384×C₈₉	C₂²	1	1	yes	characteristic
64	#384×C₈₉	C₂²	1	2	yes	—
65	#382×C₈₉	C₂²	1	2	yes	—
66	#381×C₈₉	C₂²	1	4	yes	—
67	#380×C₈₉	C₂²	1	2	yes	—
68	#380×C₈₉	C₂²	1	2	yes	—
69	#380×C₈₉	C₂²	1	4	yes	—
70	#380×C₈₉	C₂²	1	2	yes	—
71	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
72	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
73	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
74	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
75	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
76	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
77	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
78	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
79	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
80	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
81	#376×C₈₉	C₂²	1	2	yes	—
82	C₄³×C₈₉	C₂²	1	1	yes	characteristic
83	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
84	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
85	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
86	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
87	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
88	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
89	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
90	C₄⋊Q₈×C₈₉	—	2	8	no	—
91	C₄⋊Q₈×C₈₉	—	2	4	no	—
92	C₄⋊Q₈×C₈₉	—	4	4	no	—
93	C₂²⋊Q₈×C₈₉	—	4	8	no	—
94	C₂²⋊Q₈×C₈₉	—	4	4	no	—
95	C₄×Q₈×C₈₉	—	2	4	no	—
96	C₄×Q₈×C₈₉	—	4	4	no	—
97	C₄×D₄×C₈₉	—	2	4	no	—
98	C₄×D₄×C₈₉	—	2	4	no	—
99	C₄×D₄×C₈₉	Q₈	1	8	yes	complemented
100	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
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	4	yes	—
110	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
111	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
112	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
114	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
115	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
116	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
117	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
118	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
120	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
121	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
123	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
124	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
125	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
126	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
127	C₂×C₄²×C₈₉	—	4	4	no	—
128	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
131	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
133	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
134	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
135	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
136	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
137	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
138	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
139	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
140	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
141	#95×C₈₉	C₂³	1	2	yes	—
142	#95×C₈₉	C₂³	1	2	yes	—
143	#95×C₈₉	C₂³	1	2	yes	—
144	#95×C₈₉	C₂³	1	2	yes	—
145	#95×C₈₉	C₂³	1	2	yes	—
146	#95×C₈₉	C₂³	1	2	yes	—
147	#95×C₈₉	C₂³	1	2	yes	—
148	#95×C₈₉	C₂³	1	4	yes	—
149	#95×C₈₉	C₂³	1	2	yes	—
150	#95×C₈₉	C₂³	1	2	yes	—
151	#95×C₈₉	C₂³	1	2	yes	—
152	#95×C₈₉	C₂³	1	4	yes	—
153	#95×C₈₉	C₂³	1	4	yes	—
154	#95×C₈₉	C₂³	1	4	yes	—
155	#95×C₈₉	C₂³	1	4	yes	—
156	#95×C₈₉	C₂³	1	4	yes	—
157	#95×C₈₉	C₂³	1	4	yes	—
158	#95×C₈₉	C₂³	1	4	yes	—
159	C₂⁴×C₈₉	C₂⁴	1	2	yes	—
160	C₂×Q₈×C₈₉	—	2	2	no	—
161	C₂×Q₈×C₈₉	—	2	4	no	—
162	C₂×Q₈×C₈₉	—	2	8	no	—
163	C₂×Q₈×C₈₉	—	2	2	no	—
164	C₂×Q₈×C₈₉	—	2	2	no	—
165	C₂×Q₈×C₈₉	—	4	2	no	—
166	C₂×Q₈×C₈₉	—	4	2	no	—
167	C₂×Q₈×C₈₉	—	4	4	no	—
168	C₂×Q₈×C₈₉	—	4	4	no	—
169	C₂×D₄×C₈₉	—	2	2	no	—
170	C₂×D₄×C₈₉	—	2	2	no	—
171	C₂×D₄×C₈₉	C₂×Q₈	1	4	yes	—
172	C₂²×C₄×C₈₉	—	2	4	no	—
173	C₂²×C₄×C₈₉	—	2	4	no	—
174	C₂²×C₄×C₈₉	—	2	4	no	—
175	C₂²×C₄×C₈₉	—	2	4	no	—
176	C₂²×C₄×C₈₉	—	2	4	no	—
177	C₂²×C₄×C₈₉	—	4	2	no	—
178	C₂²×C₄×C₈₉	—	4	2	no	—
179	C₂²×C₄×C₈₉	—	4	2	no	—
180	C₂²×C₄×C₈₉	C₂×Q₈	1	8	yes	—
181	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
186	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
187	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
188	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
189	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
190	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
191	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
192	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
193	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
194	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
195	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
196	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
197	C₄⋊C₄×C₈₉	—	2	2	no	—
198	C₄⋊C₄×C₈₉	—	2	2	no	—
199	C₄⋊C₄×C₈₉	—	2	4	no	—
200	C₄⋊C₄×C₈₉	—	2	8	no	—
201	C₄⋊C₄×C₈₉	—	2	4	no	—
202	C₄⋊C₄×C₈₉	—	2	4	no	—
203	C₄⋊C₄×C₈₉	—	2	4	no	—
204	C₄⋊C₄×C₈₉	—	2	8	no	—
205	C₄⋊C₄×C₈₉	—	2	4	no	—
206	C₄⋊C₄×C₈₉	—	2	4	no	—
207	C₄⋊C₄×C₈₉	—	2	2	no	—
208	C₄⋊C₄×C₈₉	—	2	2	no	—
209	C₄⋊C₄×C₈₉	—	4	4	no	—
210	C₄⋊C₄×C₈₉	—	4	2	no	—
211	C₄⋊C₄×C₈₉	—	4	2	no	—
212	C₄⋊C₄×C₈₉	—	4	4	no	—
213	C₄⋊C₄×C₈₉	—	4	4	no	—
214	C₄⋊C₄×C₈₉	—	4	4	no	—
215	C₄⋊C₄×C₈₉	C₂×Q₈	1	4	yes	—
216	C₄⋊C₄×C₈₉	#41	1	4	yes	—
217	C₂²⋊C₄×C₈₉	—	2	4	no	—
218	C₂²⋊C₄×C₈₉	—	2	4	no	—
219	C₂²⋊C₄×C₈₉	—	4	4	no	—
220	C₂²⋊C₄×C₈₉	—	4	4	no	—
221	C₂²⋊C₄×C₈₉	—	4	4	no	—
222	C₂²⋊C₄×C₈₉	—	4	4	no	—
223	C₂²⋊C₄×C₈₉	C₂×Q₈	1	8	yes	—
224	C₄²×C₈₉	—	2	2	no	—
225	C₄²×C₈₉	—	2	4	no	—
226	C₄²×C₈₉	—	2	4	no	—
227	C₄²×C₈₉	—	2	2	no	—
228	C₄²×C₈₉	—	2	4	no	—
229	C₄²×C₈₉	—	2	2	no	—
230	C₄²×C₈₉	—	2	2	no	—
231	C₄²×C₈₉	—	4	2	no	—
232	C₄²×C₈₉	—	4	2	no	—
233	C₄²×C₈₉	—	4	1	no	—
234	C₄²×C₈₉	—	4	4	no	—
235	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
236	C₄²×C₈₉	#41	1	4	yes	—
237	C₂³×C₈₉	—	2	2	no	—
238	C₂³×C₈₉	—	2	2	no	—
239	C₂³×C₈₉	—	2	2	no	—
240	C₂³×C₈₉	—	2	2	no	—
241	C₂³×C₈₉	—	2	2	no	—
242	C₂³×C₈₉	—	2	2	no	—
243	C₂³×C₈₉	C₂²×Q₈	1	4	yes	—
244	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
245	Q₈×C₈₉	—	4	8	no	—
246	Q₈×C₈₉	—	4	4	no	—
247	Q₈×C₈₉	—	8	8	no	—
248	D₄×C₈₉	—	2	8	no	—
249	C₂×C₄×C₈₉	—	2	1	no	—
250	C₂×C₄×C₈₉	—	2	1	no	—
251	C₂×C₄×C₈₉	—	2	1	no	—
252	C₂×C₄×C₈₉	—	2	1	no	—
253	C₂×C₄×C₈₉	—	2	2	no	—
254	C₂×C₄×C₈₉	—	2	2	no	—
255	C₂×C₄×C₈₉	—	2	1	no	—
256	C₂×C₄×C₈₉	—	2	2	no	—
257	C₂×C₄×C₈₉	—	2	1	no	—
258	C₂×C₄×C₈₉	—	2	1	no	—
259	C₂×C₄×C₈₉	—	2	1	no	—
260	C₂×C₄×C₈₉	—	2	2	no	—
261	C₂×C₄×C₈₉	—	2	1	no	—
262	C₂×C₄×C₈₉	—	2	2	no	—
263	C₂×C₄×C₈₉	—	2	2	no	—
264	C₂×C₄×C₈₉	—	2	2	no	—
265	C₂×C₄×C₈₉	—	2	2	no	—
266	C₂×C₄×C₈₉	—	2	2	no	—
267	C₂×C₄×C₈₉	—	2	4	no	—
268	C₂×C₄×C₈₉	—	2	4	no	—
269	C₂×C₄×C₈₉	—	2	2	no	—
270	C₂×C₄×C₈₉	—	2	4	no	—
271	C₂×C₄×C₈₉	—	2	4	no	—
272	C₂×C₄×C₈₉	—	2	2	no	—
273	C₂×C₄×C₈₉	—	2	2	no	—
274	C₂×C₄×C₈₉	—	2	2	no	—
275	C₂×C₄×C₈₉	—	2	4	no	—
276	C₂×C₄×C₈₉	—	2	2	no	—
277	C₂×C₄×C₈₉	—	2	1	no	—
278	C₂×C₄×C₈₉	—	2	2	no	—
279	C₂×C₄×C₈₉	—	2	2	no	—
280	C₂×C₄×C₈₉	—	2	2	no	—
281	C₂×C₄×C₈₉	—	2	1	no	—
282	C₂×C₄×C₈₉	—	2	2	no	—
283	C₂×C₄×C₈₉	—	2	4	no	—
284	C₂×C₄×C₈₉	—	2	2	no	—
285	C₂×C₄×C₈₉	—	4	8	no	—
286	C₂×C₄×C₈₉	—	4	8	no	—
287	C₂×C₄×C₈₉	—	4	8	no	—
288	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
289	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
290	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
291	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
292	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
293	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
294	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
295	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
296	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
297	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
298	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
299	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
300	C₂²×C₈₉	—	2	4	no	—
301	C₂²×C₈₉	—	2	4	no	—
302	C₂²×C₈₉	—	2	4	no	—
303	C₂²×C₈₉	—	4	2	no	—
304	C₂²×C₈₉	—	4	2	no	—
305	C₂²×C₈₉	—	4	2	no	—
306	C₂²×C₈₉	—	4	2	no	—
307	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
308	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
309	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
310	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
311	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
312	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
313	C₂²×C₈₉	#585	1	1	yes	characteristic
314	C₄×C₈₉	—	2	2	no	—
315	C₄×C₈₉	—	2	2	no	—
316	C₄×C₈₉	—	2	4	no	—
317	C₄×C₈₉	—	2	2	no	—
318	C₄×C₈₉	—	2	2	no	—
319	C₄×C₈₉	—	2	4	no	—
320	C₄×C₈₉	—	4	2	no	—
321	C₄×C₈₉	—	4	2	no	—
322	C₄×C₈₉	—	4	2	no	—
323	C₄×C₈₉	—	4	4	no	—
324	C₄×C₈₉	—	4	1	no	—
325	C₄×C₈₉	—	4	2	no	—
326	C₄×C₈₉	—	4	4	no	—
327	C₄×C₈₉	—	4	2	no	—
328	C₄×C₈₉	#533	1	4	yes	—
329	C₄×C₈₉	#542	1	4	yes	—
330	#35177	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
331	C₂×C₈₉	—	4	4	no	—
332	C₂×C₈₉	C₂×#533	1	1	yes	characteristic
333	C₂×C₈₉	C₂×#542	1	1	yes	characteristic
334	C₂×C₈₉	#3460	1	1	yes	characteristic
335	C₂×C₈₉	#3501	1	1	yes	characteristic
336	C₂×C₈₉	#3503	1	1	yes	characteristic
337	C₂×C₈₉	#3518	1	1	yes	characteristic
338	C₂×C₈₉	#3519	1	1	yes	characteristic
339	#2857	C₂×C₈₉	1	2	yes	complemented
340	#2855	C₂×C₈₉	1	2	yes	complemented
341	#2665	C₂×C₈₉	1	4	yes	complemented
342	#2567	C₂×C₈₉	1	1	yes	characteristic, complemented
343	#2563	C₂×C₈₉	1	1	yes	characteristic
344	#2551	C₂×C₈₉	1	1	yes	characteristic, complemented
345	#2538	C₂×C₈₉	1	1	yes	characteristic
346	#2509	C₂×C₈₉	1	2	yes	complemented
347	#2497	C₂×C₈₉	1	2	yes	complemented
348	#2492	C₂×C₈₉	1	2	yes	complemented
349	#2465	C₂×C₈₉	1	4	yes	complemented
350	#2452	C₂×C₈₉	1	2	yes	—
351	#2331	C₂×C₈₉	1	2	yes	complemented
352	#2319	C₂×C₈₉	1	2	yes	—
353	C₄×C₄⋊Q₈	C₂×C₈₉	1	1	yes	characteristic, complemented
354	#2295	C₂×C₈₉	1	1	yes	characteristic, complemented
355	C₄²×D₄	C₂×C₈₉	1	1	yes	characteristic
356	C₈₉	#35177	1	1	yes	characteristic, direct product factor, Sylow
357	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
358	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
359	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
360	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	2	yes	—
361	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	4	yes	—
362	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
363	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
364	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
365	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
366	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
367	C₂²×C₄²	C₂²×C₈₉	1	2	yes	—
368	#400	C₂²×C₈₉	1	4	yes	—
369	#400	C₂²×C₈₉	1	4	yes	—
370	#396	C₂²×C₈₉	1	4	yes	—
371	#396	C₂²×C₈₉	1	4	yes	—
372	#395	C₂²×C₈₉	1	4	yes	—
373	#395	C₂²×C₈₉	1	4	yes	—
374	#393	C₂²×C₈₉	1	4	yes	—
375	#393	C₂²×C₈₉	1	4	yes	—
376	#391	C₂²×C₈₉	1	2	yes	—
377	#391	C₂²×C₈₉	1	2	yes	—
378	#391	C₂²×C₈₉	1	4	yes	—
379	#391	C₂²×C₈₉	1	4	yes	—
380	#391	C₂²×C₈₉	1	1	yes	characteristic
381	#391	C₂²×C₈₉	1	1	yes	characteristic
382	#390	C₂²×C₈₉	1	2	yes	—
383	#390	C₂²×C₈₉	1	2	yes	—
384	#389	C₂²×C₈₉	1	2	yes	—
385	#389	C₂²×C₈₉	1	2	yes	—
386	#389	C₂²×C₈₉	1	2	yes	—
387	#389	C₂²×C₈₉	1	2	yes	—
388	#389	C₂²×C₈₉	1	2	yes	—
389	#389	C₂²×C₈₉	1	2	yes	—
390	#389	C₂²×C₈₉	1	2	yes	—
391	#389	C₂²×C₈₉	1	1	yes	characteristic
392	#389	C₂²×C₈₉	1	4	yes	—
393	#389	C₂²×C₈₉	1	1	yes	characteristic
394	#388	C₂²×C₈₉	1	4	yes	—
395	#388	C₂²×C₈₉	1	4	yes	—
396	#387	C₂²×C₈₉	1	4	yes	—
397	#387	C₂²×C₈₉	1	4	yes	—
398	#387	C₂²×C₈₉	1	4	yes	—
399	#387	C₂²×C₈₉	1	2	yes	—
400	#387	C₂²×C₈₉	1	2	yes	—
401	#385	C₂²×C₈₉	1	4	yes	—
402	#384	C₂²×C₈₉	1	1	yes	characteristic
403	#384	C₂²×C₈₉	1	2	yes	—
404	#382	C₂²×C₈₉	1	2	yes	—
405	#381	C₂²×C₈₉	1	4	yes	—
406	#380	C₂²×C₈₉	1	2	yes	—
407	#380	C₂²×C₈₉	1	2	yes	—
408	#380	C₂²×C₈₉	1	4	yes	—
409	#380	C₂²×C₈₉	1	2	yes	—
410	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
411	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
412	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
413	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
414	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
415	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
416	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
417	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
418	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
419	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
420	#376	C₂²×C₈₉	1	2	yes	—
421	C₄³	C₂²×C₈₉	1	1	yes	characteristic
422	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
423	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
424	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
425	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
426	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
427	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
428	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
429	C₄⋊Q₈	—	2	8	no	—
430	C₄⋊Q₈	—	2	4	no	—
431	C₄⋊Q₈	—	4	4	no	—
432	C₂²⋊Q₈	—	4	8	no	—
433	C₂²⋊Q₈	—	4	4	no	—
434	C₄×Q₈	—	2	4	no	—
435	C₄×Q₈	—	4	4	no	—
436	C₄×D₄	—	2	4	no	—
437	C₄×D₄	—	2	4	no	—
438	C₄×D₄	Q₈×C₈₉	1	8	yes	complemented
439	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
440	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
441	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
442	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
443	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
444	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
445	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
446	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
447	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
448	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
449	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
450	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
451	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
452	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
453	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
454	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
455	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
456	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
457	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
458	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
459	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
460	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
461	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
462	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
463	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
464	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
465	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
466	C₂×C₄²	—	4	4	no	—
467	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
468	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
469	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
470	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
471	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
472	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
473	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
474	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
475	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
476	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
477	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
478	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
479	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
480	#95	C₂³×C₈₉	1	2	yes	—
481	#95	C₂³×C₈₉	1	2	yes	—
482	#95	C₂³×C₈₉	1	2	yes	—
483	#95	C₂³×C₈₉	1	2	yes	—
484	#95	C₂³×C₈₉	1	2	yes	—
485	#95	C₂³×C₈₉	1	2	yes	—
486	#95	C₂³×C₈₉	1	2	yes	—
487	#95	C₂³×C₈₉	1	4	yes	—
488	#95	C₂³×C₈₉	1	2	yes	—
489	#95	C₂³×C₈₉	1	2	yes	—
490	#95	C₂³×C₈₉	1	2	yes	—
491	#95	C₂³×C₈₉	1	4	yes	—
492	#95	C₂³×C₈₉	1	4	yes	—
493	#95	C₂³×C₈₉	1	4	yes	—
494	#95	C₂³×C₈₉	1	4	yes	—
495	#95	C₂³×C₈₉	1	4	yes	—
496	#95	C₂³×C₈₉	1	4	yes	—
497	#95	C₂³×C₈₉	1	4	yes	—
498	C₂⁴	C₂⁴×C₈₉	1	2	yes	—
499	C₂×Q₈	—	2	2	no	—
500	C₂×Q₈	—	2	4	no	—
501	C₂×Q₈	—	2	8	no	—
502	C₂×Q₈	—	2	2	no	—
503	C₂×Q₈	—	2	2	no	—
504	C₂×Q₈	—	4	2	no	—
505	C₂×Q₈	—	4	2	no	—
506	C₂×Q₈	—	4	4	no	—
507	C₂×Q₈	—	4	4	no	—
508	C₂×D₄	—	2	2	no	—
509	C₂×D₄	—	2	2	no	—
510	C₂×D₄	C₂×Q₈×C₈₉	1	4	yes	—
511	C₂²×C₄	—	2	4	no	—
512	C₂²×C₄	—	2	4	no	—
513	C₂²×C₄	—	2	4	no	—
514	C₂²×C₄	—	2	4	no	—
515	C₂²×C₄	—	2	4	no	—
516	C₂²×C₄	—	4	2	no	—
517	C₂²×C₄	—	4	2	no	—
518	C₂²×C₄	—	4	2	no	—
519	C₂²×C₄	C₂×Q₈×C₈₉	1	8	yes	—
520	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
521	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
522	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
523	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
524	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
525	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
526	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
527	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
528	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
529	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
530	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
531	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
532	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
533	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
534	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
535	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
536	C₄⋊C₄	—	2	2	no	—
537	C₄⋊C₄	—	2	2	no	—
538	C₄⋊C₄	—	2	4	no	—
539	C₄⋊C₄	—	2	8	no	—
540	C₄⋊C₄	—	2	4	no	—
541	C₄⋊C₄	—	2	4	no	—
542	C₄⋊C₄	—	2	4	no	—
543	C₄⋊C₄	—	2	8	no	—
544	C₄⋊C₄	—	2	4	no	—
545	C₄⋊C₄	—	2	4	no	—
546	C₄⋊C₄	—	2	2	no	—
547	C₄⋊C₄	—	2	2	no	—
548	C₄⋊C₄	—	4	2	no	—
549	C₄⋊C₄	—	4	4	no	—
550	C₄⋊C₄	—	4	4	no	—
551	C₄⋊C₄	—	4	4	no	—
552	C₄⋊C₄	—	4	4	no	—
553	C₄⋊C₄	—	4	2	no	—
554	C₄⋊C₄	C₂×Q₈×C₈₉	1	4	yes	—
555	C₄⋊C₄	#41×C₈₉	1	4	yes	—
556	C₂²⋊C₄	—	2	4	no	—
557	C₂²⋊C₄	—	2	4	no	—
558	C₂²⋊C₄	—	4	4	no	—
559	C₂²⋊C₄	—	4	4	no	—
560	C₂²⋊C₄	—	4	4	no	—
561	C₂²⋊C₄	—	4	4	no	—
562	C₂²⋊C₄	C₂×Q₈×C₈₉	1	8	yes	—
563	C₄²	—	2	2	no	—
564	C₄²	—	2	4	no	—
565	C₄²	—	2	4	no	—
566	C₄²	—	2	2	no	—
567	C₄²	—	2	4	no	—
568	C₄²	—	2	2	no	—
569	C₄²	—	2	2	no	—
570	C₄²	—	4	2	no	—
571	C₄²	—	4	2	no	—
572	C₄²	—	4	4	no	—
573	C₄²	—	4	1	no	—
574	C₄²	C₂×Q₈×C₈₉	1	4	yes	—
575	C₄²	#41×C₈₉	1	4	yes	—
576	C₂³	—	2	2	no	—
577	C₂³	—	2	2	no	—
578	C₂³	—	2	2	no	—
579	C₂³	—	2	2	no	—
580	C₂³	—	2	2	no	—
581	C₂³	—	2	2	no	—
582	C₂³	C₂²×Q₈×C₈₉	1	4	yes	—
583	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
584	Q₈	—	4	8	no	—
585	Q₈	—	4	4	no	—
586	Q₈	—	8	8	no	—
587	D₄	—	2	8	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	4	no	—
607	C₂×C₄	—	2	4	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	4	no	—
610	C₂×C₄	—	2	4	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	2	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	4	no	—
615	C₂×C₄	—	2	2	no	—
616	C₂×C₄	—	2	1	no	—
617	C₂×C₄	—	2	2	no	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	4	no	—
623	C₂×C₄	—	2	2	no	—
624	C₂×C₄	—	4	8	no	—
625	C₂×C₄	—	4	8	no	—
626	C₂×C₄	—	4	8	no	—
627	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
628	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
629	C₂×C₄	C₂²×Q₈×C₈₉	1	4	yes	—
630	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
631	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
632	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
633	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
634	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
635	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
636	C₂×C₄	H(5,𝔽₂)×C₈₉	1	2	yes	—
637	C₂×C₄	H(5,𝔽₂)×C₈₉	1	2	yes	—
638	C₂×C₄	2₋¹⁺⁴×C₈₉	1	2	yes	—
639	C₂²	—	2	4	no	—
640	C₂²	—	2	4	no	—
641	C₂²	—	2	4	no	—
642	C₂²	—	4	2	no	—
643	C₂²	—	4	2	no	—
644	C₂²	—	4	2	no	—
645	C₂²	—	4	2	no	—
646	C₂²	C₂³×Q₈×C₈₉	1	1	yes	characteristic
647	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
648	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
649	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
650	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
651	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
652	C₂²	#585×C₈₉	1	1	yes	characteristic
653	C₄	—	2	2	no	—
654	C₄	—	2	2	no	—
655	C₄	—	2	4	no	—
656	C₄	—	2	2	no	—
657	C₄	—	2	2	no	—
658	C₄	—	2	4	no	—
659	C₄	—	4	2	no	—
660	C₄	—	4	2	no	—
661	C₄	—	4	4	no	—
662	C₄	—	4	2	no	—
663	C₄	—	4	1	no	—
664	C₄	—	4	2	no	—
665	C₄	—	4	2	no	—
666	C₄	—	4	4	no	—
667	C₄	#533×C₈₉	1	4	yes	—
668	C₄	#542×C₈₉	1	4	yes	—
669	C₂	—	4	4	no	—
670	C₂	C₂×#533×C₈₉	1	1	yes	characteristic
671	C₂	C₂×#542×C₈₉	1	1	yes	characteristic
672	C₂	C₈₉×#3460	1	1	yes	characteristic
673	C₂	C₈₉×#3501	1	1	yes	characteristic
674	C₂	C₈₉×#3503	1	1	yes	characteristic
675	C₂	C₈₉×#3518	1	1	yes	characteristic
676	C₂	C₈₉×#3519	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂²×D₄×C₈₉	C₂³
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×#2857	C₂	semidirect
2. C₈₉×#2855	C₂	semidirect
3. C₈₉×#2665	C₂	semidirect
4. C₈₉×#2567	C₂	semidirect
6. C₈₉×#2551	C₂	semidirect
8. C₈₉×#2509	C₂	semidirect
9. C₈₉×#2497	C₂	semidirect
10. C₈₉×#2492	C₂	semidirect
11. C₈₉×#2465	C₂	semidirect
13. C₈₉×#2331	C₂	semidirect
15. C₄×C₄⋊Q₈×C₈₉	C₂	semidirect
16. C₈₉×#2295	C₂	semidirect
99. C₄×D₄×C₈₉	Q₈	semidirect
330. #35177	C₈₉	direct (N × H)
339. #2857	C₂×C₈₉	semidirect
340. #2855	C₂×C₈₉	semidirect
341. #2665	C₂×C₈₉	semidirect
342. #2567	C₂×C₈₉	semidirect
344. #2551	C₂×C₈₉	semidirect
346. #2509	C₂×C₈₉	semidirect
347. #2497	C₂×C₈₉	semidirect
348. #2492	C₂×C₈₉	semidirect
349. #2465	C₂×C₈₉	semidirect
351. #2331	C₂×C₈₉	semidirect
353. C₄×C₄⋊Q₈	C₂×C₈₉	semidirect
354. #2295	C₂×C₈₉	semidirect
356. C₈₉	#35177	direct (N × H)
438. C₄×D₄	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×#2857	2
C₈₉×#2855	2
C₈₉×#2665	2
C₈₉×#2567	2
C₈₉×#2563	2
C₈₉×#2551	2
C₈₉×#2538	2
C₈₉×#2509	2
C₈₉×#2497	2
C₈₉×#2492	2
C₈₉×#2465	2
C₈₉×#2452	2
C₈₉×#2331	2
C₈₉×#2319	2
C₄×C₄⋊Q₈×C₈₉	2
C₈₉×#2295	2
C₄²×D₄×C₈₉	2
#35177	89

Subgroups — C89×#35177 #36413025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35177 #36413025