Subgroups — C₈₉×#29463 #36407311

Order 22784.

766 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2855	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2075	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2035	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2035	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#2001	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1969	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#1941	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1933	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1933	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1927	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈₉×#1918	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈₉×#1878	C₂	1	1	yes	maximal, characteristic, complemented
14	C₈₉×#1865	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₂²⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
16	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
34	#391×C₈₉	C₂²	1	1	yes	characteristic
35	#391×C₈₉	C₂²	1	1	yes	characteristic
36	#391×C₈₉	C₂²	1	1	yes	characteristic
37	#389×C₈₉	C₂²	1	1	yes	characteristic
38	#389×C₈₉	C₂²	1	1	yes	characteristic
39	#389×C₈₉	C₂²	1	1	yes	characteristic
40	#387×C₈₉	—	2	1	no	—
41	#387×C₈₉	—	2	1	no	—
42	#387×C₈₉	—	2	1	no	—
43	#385×C₈₉	—	2	1	no	—
44	#385×C₈₉	—	2	1	no	—
45	#381×C₈₉	—	2	1	no	—
46	#380×C₈₉	C₂²	1	1	yes	characteristic
47	#380×C₈₉	C₂²	1	1	yes	characteristic
48	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
49	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
50	C₄×C₂²⋊C₄×C₈₉	—	2	1	no	—
51	#340×C₈₉	C₂²	1	1	yes	characteristic
52	#340×C₈₉	C₂²	1	1	yes	characteristic
53	#340×C₈₉	C₂²	1	1	yes	characteristic
54	#340×C₈₉	C₂²	1	1	yes	characteristic
55	#340×C₈₉	C₂²	1	1	yes	characteristic
56	#340×C₈₉	C₂²	1	1	yes	characteristic
57	#336×C₈₉	C₂²	1	1	yes	characteristic
58	#336×C₈₉	C₂²	1	1	yes	characteristic
59	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
62	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
63	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
64	C₂×Q₁₆×C₈₉	—	2	2	no	—
65	C₂×Q₁₆×C₈₉	—	2	2	no	—
66	C₂×Q₁₆×C₈₉	—	4	1	no	—
67	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₂²⋊Q₈×C₈₉	—	2	2	no	—
72	C₂²⋊Q₈×C₈₉	—	2	2	no	—
73	C₂²⋊Q₈×C₈₉	—	4	1	no	—
74	C₂²⋊Q₈×C₈₉	—	4	1	no	—
75	C₄×Q₈×C₈₉	—	4	1	no	—
76	C₄×Q₈×C₈₉	—	4	1	no	—
77	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
78	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
79	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
80	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
81	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
82	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
83	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
93	C₂×C₄²×C₈₉	—	2	1	no	—
94	C₂×C₄²×C₈₉	—	2	1	no	—
95	C₂×C₄²×C₈₉	—	4	1	no	—
96	C₂×C₄²×C₈₉	—	4	1	no	—
97	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
101	#107×C₈₉	—	2	2	no	—
102	#106×C₈₉	—	2	2	no	—
103	#106×C₈₉	—	4	1	no	—
104	#103×C₈₉	—	2	2	no	—
105	#103×C₈₉	—	2	2	no	—
106	#103×C₈₉	—	2	2	no	—
107	#103×C₈₉	—	2	2	no	—
108	#103×C₈₉	—	4	1	no	—
109	#103×C₈₉	—	4	1	no	—
110	C₂²⋊C₈×C₈₉	—	2	2	no	—
111	C₂²⋊C₈×C₈₉	—	2	2	no	—
112	C₄×C₈×C₈₉	—	4	1	no	—
113	C₄×C₈×C₈₉	—	4	1	no	—
114	#95×C₈₉	—	2	1	no	—
115	#95×C₈₉	—	2	1	no	—
116	#95×C₈₉	—	2	1	no	—
117	#95×C₈₉	—	2	1	no	—
118	#95×C₈₉	—	2	1	no	—
119	#95×C₈₉	—	2	1	no	—
120	#95×C₈₉	—	2	1	no	—
121	#95×C₈₉	—	2	1	no	—
122	#95×C₈₉	—	2	1	no	—
123	#95×C₈₉	—	2	1	no	—
124	#95×C₈₉	—	2	1	no	—
125	#95×C₈₉	—	2	1	no	—
126	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Q₈×C₈₉	—	2	1	no	—
128	C₂×Q₈×C₈₉	—	2	1	no	—
129	C₂×Q₈×C₈₉	—	2	1	no	—
130	C₂×Q₈×C₈₉	—	2	1	no	—
131	C₂×Q₈×C₈₉	—	2	1	no	—
132	C₂×Q₈×C₈₉	—	2	1	no	—
133	C₂×Q₈×C₈₉	—	2	1	no	—
134	C₂×Q₈×C₈₉	—	2	1	no	—
135	C₂×Q₈×C₈₉	—	2	1	no	—
136	C₂×Q₈×C₈₉	—	2	1	no	—
137	C₂×Q₈×C₈₉	—	2	1	no	—
138	C₂×Q₈×C₈₉	#41	1	2	yes	—
139	C₂²×C₄×C₈₉	—	2	2	no	—
140	C₂²×C₄×C₈₉	—	2	1	no	—
141	C₂²×C₄×C₈₉	—	2	1	no	—
142	C₂²×C₄×C₈₉	—	2	2	no	—
143	C₂²×C₄×C₈₉	—	2	2	no	—
144	C₂²×C₄×C₈₉	—	2	2	no	—
145	C₂²×C₄×C₈₉	—	2	1	no	—
146	C₂²×C₄×C₈₉	—	2	1	no	—
147	C₂²×C₄×C₈₉	—	2	1	no	—
148	C₂²×C₄×C₈₉	—	2	1	no	—
149	C₂²×C₄×C₈₉	—	2	1	no	—
150	C₂²×C₄×C₈₉	—	2	1	no	—
151	C₂²×C₄×C₈₉	—	2	1	no	—
152	C₂²×C₄×C₈₉	—	2	1	no	—
153	C₂²×C₄×C₈₉	—	2	1	no	—
154	C₂²×C₄×C₈₉	—	2	1	no	—
155	C₂²×C₄×C₈₉	—	2	1	no	—
156	C₂²×C₄×C₈₉	—	2	1	no	—
157	C₂²×C₄×C₈₉	—	4	1	no	—
158	C₂²×C₄×C₈₉	—	4	1	no	—
159	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
165	Q₁₆×C₈₉	—	4	4	no	—
166	C₂×C₈×C₈₉	—	2	1	no	—
167	C₂×C₈×C₈₉	—	2	1	no	—
168	C₂×C₈×C₈₉	—	2	1	no	—
169	C₂×C₈×C₈₉	—	2	1	no	—
170	C₂×C₈×C₈₉	—	2	1	no	—
171	C₂×C₈×C₈₉	—	2	1	no	—
172	C₂×C₈×C₈₉	—	2	1	no	—
173	C₂×C₈×C₈₉	—	2	1	no	—
174	C₂×C₈×C₈₉	—	2	1	no	—
175	C₂×C₈×C₈₉	—	2	1	no	—
176	C₂×C₈×C₈₉	#41	1	2	yes	—
177	C₂×C₈×C₈₉	#41	1	2	yes	—
178	C₄⋊C₄×C₈₉	—	2	4	no	—
179	C₄⋊C₄×C₈₉	—	2	1	no	—
180	C₄⋊C₄×C₈₉	—	2	1	no	—
181	C₄⋊C₄×C₈₉	—	2	1	no	—
182	C₄⋊C₄×C₈₉	—	2	1	no	—
183	C₄⋊C₄×C₈₉	—	2	1	no	—
184	C₄⋊C₄×C₈₉	—	2	1	no	—
185	C₄⋊C₄×C₈₉	—	2	1	no	—
186	C₄⋊C₄×C₈₉	—	2	1	no	—
187	C₄⋊C₄×C₈₉	—	2	1	no	—
188	C₄⋊C₄×C₈₉	—	2	1	no	—
189	C₄⋊C₄×C₈₉	—	2	1	no	—
190	C₄⋊C₄×C₈₉	—	4	2	no	—
191	C₄⋊C₄×C₈₉	—	4	2	no	—
192	C₄⋊C₄×C₈₉	—	4	2	no	—
193	C₄⋊C₄×C₈₉	—	4	2	no	—
194	C₄⋊C₄×C₈₉	—	4	2	no	—
195	C₄⋊C₄×C₈₉	#41	1	2	yes	—
196	C₂²⋊C₄×C₈₉	—	2	4	no	—
197	C₂²⋊C₄×C₈₉	—	4	2	no	—
198	C₂²⋊C₄×C₈₉	—	4	2	no	—
199	C₂²⋊C₄×C₈₉	—	4	2	no	—
200	C₄²×C₈₉	—	2	1	no	—
201	C₄²×C₈₉	—	2	1	no	—
202	C₄²×C₈₉	—	2	1	no	—
203	C₄²×C₈₉	—	2	1	no	—
204	C₄²×C₈₉	—	4	1	no	—
205	C₄²×C₈₉	—	4	1	no	—
206	C₄²×C₈₉	—	4	2	no	—
207	C₄²×C₈₉	—	4	2	no	—
208	C₂³×C₈₉	—	2	1	no	—
209	C₂³×C₈₉	—	2	1	no	—
210	C₂³×C₈₉	—	2	1	no	—
211	C₂³×C₈₉	—	2	1	no	—
212	C₂³×C₈₉	—	2	1	no	—
213	C₂³×C₈₉	—	2	1	no	—
214	C₂³×C₈₉	#135	1	2	yes	—
215	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
216	Q₈×C₈₉	—	2	2	no	—
217	Q₈×C₈₉	—	2	2	no	—
218	Q₈×C₈₉	—	4	1	no	—
219	Q₈×C₈₉	—	4	1	no	—
220	Q₈×C₈₉	—	4	1	no	—
221	Q₈×C₈₉	—	4	1	no	—
222	Q₈×C₈₉	—	4	1	no	—
223	Q₈×C₈₉	—	4	1	no	—
224	C₂×C₄×C₈₉	—	2	1	no	—
225	C₂×C₄×C₈₉	—	2	1	no	—
226	C₂×C₄×C₈₉	—	2	1	no	—
227	C₂×C₄×C₈₉	—	2	1	no	—
228	C₂×C₄×C₈₉	—	2	1	no	—
229	C₂×C₄×C₈₉	—	2	1	no	—
230	C₂×C₄×C₈₉	—	2	2	no	—
231	C₂×C₄×C₈₉	—	2	2	no	—
232	C₂×C₄×C₈₉	—	2	1	no	—
233	C₂×C₄×C₈₉	—	2	1	no	—
234	C₂×C₄×C₈₉	—	2	1	no	—
235	C₂×C₄×C₈₉	—	2	1	no	—
236	C₂×C₄×C₈₉	—	2	1	no	—
237	C₂×C₄×C₈₉	—	2	1	no	—
238	C₂×C₄×C₈₉	—	4	4	no	—
239	C₂×C₄×C₈₉	—	4	4	no	—
240	C₂×C₄×C₈₉	—	4	1	no	—
241	C₂×C₄×C₈₉	—	4	1	no	—
242	C₂×C₄×C₈₉	—	4	1	no	—
243	C₂×C₄×C₈₉	—	4	1	no	—
244	C₂×C₄×C₈₉	—	4	1	no	—
245	C₂×C₄×C₈₉	—	4	1	no	—
246	C₂×C₄×C₈₉	—	4	1	no	—
247	C₂×C₄×C₈₉	—	4	1	no	—
248	C₂×C₄×C₈₉	—	4	1	no	—
249	C₂×C₄×C₈₉	—	4	1	no	—
250	C₂×C₄×C₈₉	—	4	1	no	—
251	C₂×C₄×C₈₉	—	4	1	no	—
252	C₂×C₄×C₈₉	—	4	1	no	—
253	C₂×C₄×C₈₉	—	4	1	no	—
254	C₂×C₄×C₈₉	—	4	1	no	—
255	C₂×C₄×C₈₉	—	4	1	no	—
256	C₂×C₄×C₈₉	—	4	1	no	—
257	C₂×C₄×C₈₉	—	4	1	no	—
258	C₂×C₄×C₈₉	—	4	1	no	—
259	C₂×C₄×C₈₉	—	4	1	no	—
260	C₂×C₄×C₈₉	—	4	1	no	—
261	C₂×C₄×C₈₉	—	4	1	no	—
262	C₂×C₄×C₈₉	—	4	1	no	—
263	C₂×C₄×C₈₉	—	4	1	no	—
264	C₂×C₄×C₈₉	—	4	1	no	—
265	C₂×C₄×C₈₉	—	4	1	no	—
266	C₂×C₄×C₈₉	#135	1	2	yes	—
267	C₂×C₄×C₈₉	#137	1	2	yes	—
268	C₂×C₄×C₈₉	#137	1	2	yes	—
269	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
270	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
271	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
272	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
273	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
274	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
275	C₈×C₈₉	—	2	2	no	—
276	C₈×C₈₉	—	2	2	no	—
277	C₈×C₈₉	—	4	1	no	—
278	C₈×C₈₉	—	4	1	no	—
279	C₂²×C₈₉	—	2	2	no	—
280	C₂²×C₈₉	—	2	2	no	—
281	C₂²×C₈₉	—	2	2	no	—
282	C₂²×C₈₉	—	4	1	no	—
283	C₂²×C₈₉	—	4	1	no	—
284	C₂²×C₈₉	—	4	1	no	—
285	C₂²×C₈₉	—	4	1	no	—
286	C₂²×C₈₉	#525	1	1	yes	characteristic
287	C₂²×C₈₉	#532	1	1	yes	characteristic
288	C₂²×C₈₉	#534	1	1	yes	characteristic
289	C₂²×C₈₉	C₂×#135	1	1	yes	characteristic
290	C₂²×C₈₉	C₂×#135	1	1	yes	characteristic
291	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
292	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
293	C₄×C₈₉	—	2	2	no	—
294	C₄×C₈₉	—	2	2	no	—
295	C₄×C₈₉	—	2	2	no	—
296	C₄×C₈₉	—	4	2	no	—
297	C₄×C₈₉	—	4	1	no	—
298	C₄×C₈₉	—	4	2	no	—
299	C₄×C₈₉	—	4	1	no	—
300	C₄×C₈₉	—	8	1	no	—
301	C₄×C₈₉	—	8	1	no	—
302	C₄×C₈₉	—	8	1	no	—
303	C₄×C₈₉	—	8	1	no	—
304	C₄×C₈₉	—	8	1	no	—
305	C₄×C₈₉	—	8	1	no	—
306	C₄×C₈₉	#551	1	1	yes	characteristic
307	C₄×C₈₉	#553	1	1	yes	characteristic
308	C₄×C₈₉	#553	1	1	yes	characteristic
309	C₄×C₈₉	#559	1	1	yes	characteristic
310	#29463	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
311	C₂×C₈₉	—	4	2	no	—
312	C₂×C₈₉	#2602	1	1	yes	characteristic
313	C₂×C₈₉	#3107	1	1	yes	characteristic
314	C₂×C₈₉	#3108	1	1	yes	characteristic
315	C₂×C₈₉	#3168	1	1	yes	characteristic
316	C₂×C₈₉	#3185	1	1	yes	characteristic
317	C₂×C₈₉	#3190	1	1	yes	characteristic
318	C₂×C₈₉	#3190	1	1	yes	characteristic
319	#2855	C₂×C₈₉	1	1	yes	characteristic
320	C₄×C₂²⋊Q₈	C₂×C₈₉	1	1	yes	characteristic
321	#2075	C₂×C₈₉	1	1	yes	characteristic
322	#2035	C₂×C₈₉	1	1	yes	characteristic
323	#2035	C₂×C₈₉	1	1	yes	characteristic
324	#2001	C₂×C₈₉	1	1	yes	characteristic
325	#1969	C₂×C₈₉	1	1	yes	characteristic, complemented
326	#1941	C₂×C₈₉	1	1	yes	characteristic, complemented
327	#1933	C₂×C₈₉	1	1	yes	characteristic, complemented
328	#1933	C₂×C₈₉	1	1	yes	characteristic, complemented
329	#1927	C₂×C₈₉	1	1	yes	characteristic, complemented
330	#1918	C₂×C₈₉	1	1	yes	characteristic, complemented
331	#1878	C₂×C₈₉	1	1	yes	characteristic, complemented
332	#1865	C₂×C₈₉	1	1	yes	characteristic, complemented
333	C₄×C₂²⋊C₈	C₂×C₈₉	1	1	yes	characteristic
334	C₈₉	#29463	1	1	yes	characteristic, direct product factor, Sylow
335	C₂²×Q₁₆	C₂²×C₈₉	1	1	yes	characteristic
336	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
337	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
338	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
339	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
340	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
341	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
342	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
343	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
344	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
345	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
346	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
347	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
348	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
349	C₂×C₂²⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
350	C₂×C₂²⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
351	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
352	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
353	#391	C₂²×C₈₉	1	1	yes	characteristic
354	#391	C₂²×C₈₉	1	1	yes	characteristic
355	#391	C₂²×C₈₉	1	1	yes	characteristic
356	#389	C₂²×C₈₉	1	1	yes	characteristic
357	#389	C₂²×C₈₉	1	1	yes	characteristic
358	#389	C₂²×C₈₉	1	1	yes	characteristic
359	#387	—	2	1	no	—
360	#387	—	2	1	no	—
361	#387	—	2	1	no	—
362	#385	—	2	1	no	—
363	#385	—	2	1	no	—
364	#381	—	2	1	no	—
365	#380	C₂²×C₈₉	1	1	yes	characteristic
366	#380	C₂²×C₈₉	1	1	yes	characteristic
367	C₄×C₄⋊C₄	—	2	1	no	—
368	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
369	C₄×C₂²⋊C₄	—	2	1	no	—
370	#340	C₂²×C₈₉	1	1	yes	characteristic
371	#340	C₂²×C₈₉	1	1	yes	characteristic
372	#340	C₂²×C₈₉	1	1	yes	characteristic
373	#340	C₂²×C₈₉	1	1	yes	characteristic
374	#340	C₂²×C₈₉	1	1	yes	characteristic
375	#340	C₂²×C₈₉	1	1	yes	characteristic
376	#336	C₂²×C₈₉	1	1	yes	characteristic
377	#336	C₂²×C₈₉	1	1	yes	characteristic
378	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
379	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
380	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
381	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
382	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
383	C₂×Q₁₆	—	2	2	no	—
384	C₂×Q₁₆	—	2	2	no	—
385	C₂×Q₁₆	—	4	1	no	—
386	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
387	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
388	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
389	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
390	C₂²⋊Q₈	—	2	2	no	—
391	C₂²⋊Q₈	—	2	2	no	—
392	C₂²⋊Q₈	—	4	1	no	—
393	C₂²⋊Q₈	—	4	1	no	—
394	C₄×Q₈	—	4	1	no	—
395	C₄×Q₈	—	4	1	no	—
396	C₂×C₄⋊C₄	—	2	1	no	—
397	C₂×C₄⋊C₄	—	2	1	no	—
398	C₂×C₄⋊C₄	—	2	1	no	—
399	C₂×C₄⋊C₄	—	2	1	no	—
400	C₂×C₄⋊C₄	—	2	1	no	—
401	C₂×C₄⋊C₄	—	2	1	no	—
402	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
403	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
404	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
405	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
406	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
408	C₂×C₂²⋊C₄	—	2	1	no	—
409	C₂×C₂²⋊C₄	—	2	1	no	—
410	C₂×C₂²⋊C₄	—	2	1	no	—
411	C₂×C₂²⋊C₄	—	2	1	no	—
412	C₂×C₄²	—	2	1	no	—
413	C₂×C₄²	—	2	1	no	—
414	C₂×C₄²	—	4	1	no	—
415	C₂×C₄²	—	4	1	no	—
416	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
417	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
418	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
419	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
420	#107	—	2	2	no	—
421	#106	—	2	2	no	—
422	#106	—	4	1	no	—
423	#103	—	2	2	no	—
424	#103	—	2	2	no	—
425	#103	—	2	2	no	—
426	#103	—	2	2	no	—
427	#103	—	4	1	no	—
428	#103	—	4	1	no	—
429	C₂²⋊C₈	—	2	2	no	—
430	C₂²⋊C₈	—	2	2	no	—
431	C₄×C₈	—	4	1	no	—
432	C₄×C₈	—	4	1	no	—
433	#95	—	2	1	no	—
434	#95	—	2	1	no	—
435	#95	—	2	1	no	—
436	#95	—	2	1	no	—
437	#95	—	2	1	no	—
438	#95	—	2	1	no	—
439	#95	—	2	1	no	—
440	#95	—	2	1	no	—
441	#95	—	2	1	no	—
442	#95	—	2	1	no	—
443	#95	—	2	1	no	—
444	#95	—	2	1	no	—
445	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	—	2	1	no	—
448	C₂×Q₈	—	2	1	no	—
449	C₂×Q₈	—	2	1	no	—
450	C₂×Q₈	—	2	1	no	—
451	C₂×Q₈	—	2	1	no	—
452	C₂×Q₈	—	2	1	no	—
453	C₂×Q₈	—	2	1	no	—
454	C₂×Q₈	—	2	1	no	—
455	C₂×Q₈	—	2	1	no	—
456	C₂×Q₈	—	2	1	no	—
457	C₂×Q₈	#41×C₈₉	1	2	yes	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	1	no	—
472	C₂²×C₄	—	2	1	no	—
473	C₂²×C₄	—	2	1	no	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	4	1	no	—
477	C₂²×C₄	—	4	1	no	—
478	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
479	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
480	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
481	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
482	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
483	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
484	Q₁₆	—	4	4	no	—
485	C₂×C₈	—	2	1	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	2	1	no	—
488	C₂×C₈	—	2	1	no	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	—	2	1	no	—
491	C₂×C₈	—	2	1	no	—
492	C₂×C₈	—	2	1	no	—
493	C₂×C₈	—	2	1	no	—
494	C₂×C₈	—	2	1	no	—
495	C₂×C₈	#41×C₈₉	1	2	yes	—
496	C₂×C₈	#41×C₈₉	1	2	yes	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	2	1	no	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	1	no	—
506	C₄⋊C₄	—	2	1	no	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	—	4	2	no	—
510	C₄⋊C₄	—	4	2	no	—
511	C₄⋊C₄	—	4	2	no	—
512	C₄⋊C₄	—	4	2	no	—
513	C₄⋊C₄	—	4	2	no	—
514	C₄⋊C₄	#41×C₈₉	1	2	yes	—
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	C₄²	—	2	1	no	—
520	C₄²	—	2	1	no	—
521	C₄²	—	2	1	no	—
522	C₄²	—	2	1	no	—
523	C₄²	—	4	1	no	—
524	C₄²	—	4	1	no	—
525	C₄²	—	4	2	no	—
526	C₄²	—	4	2	no	—
527	C₂³	—	2	1	no	—
528	C₂³	—	2	1	no	—
529	C₂³	—	2	1	no	—
530	C₂³	—	2	1	no	—
531	C₂³	—	2	1	no	—
532	C₂³	—	2	1	no	—
533	C₂³	#135×C₈₉	1	2	yes	—
534	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
535	Q₈	—	2	2	no	—
536	Q₈	—	2	2	no	—
537	Q₈	—	4	1	no	—
538	Q₈	—	4	1	no	—
539	Q₈	—	4	1	no	—
540	Q₈	—	4	1	no	—
541	Q₈	—	4	1	no	—
542	Q₈	—	4	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	4	4	no	—
558	C₂×C₄	—	4	4	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	1	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	—	4	1	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	#135×C₈₉	1	2	yes	—
586	C₂×C₄	#137×C₈₉	1	2	yes	—
587	C₂×C₄	#137×C₈₉	1	2	yes	—
588	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
589	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
590	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
591	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
592	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
593	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
594	C₈	—	2	2	no	—
595	C₈	—	2	2	no	—
596	C₈	—	4	1	no	—
597	C₈	—	4	1	no	—
598	C₂²	—	2	2	no	—
599	C₂²	—	2	2	no	—
600	C₂²	—	2	2	no	—
601	C₂²	—	4	1	no	—
602	C₂²	—	4	1	no	—
603	C₂²	—	4	1	no	—
604	C₂²	—	4	1	no	—
605	C₂²	#525×C₈₉	1	1	yes	characteristic
606	C₂²	#532×C₈₉	1	1	yes	characteristic
607	C₂²	#534×C₈₉	1	1	yes	characteristic
608	C₂²	C₂×#135×C₈₉	1	1	yes	characteristic
609	C₂²	C₂×#135×C₈₉	1	1	yes	characteristic
610	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
611	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
612	C₄	—	2	2	no	—
613	C₄	—	2	2	no	—
614	C₄	—	2	2	no	—
615	C₄	—	4	2	no	—
616	C₄	—	4	1	no	—
617	C₄	—	4	2	no	—
618	C₄	—	4	1	no	—
619	C₄	—	8	1	no	—
620	C₄	—	8	1	no	—
621	C₄	—	8	1	no	—
622	C₄	—	8	1	no	—
623	C₄	—	8	1	no	—
624	C₄	—	8	1	no	—
625	C₄	#551×C₈₉	1	1	yes	characteristic
626	C₄	#553×C₈₉	1	1	yes	characteristic
627	C₄	#553×C₈₉	1	1	yes	characteristic
628	C₄	#559×C₈₉	1	1	yes	characteristic
629	C₂	—	4	2	no	—
630	C₂	C₈₉×#2602	1	1	yes	characteristic
631	C₂	C₈₉×#3107	1	1	yes	characteristic
632	C₂	C₈₉×#3108	1	1	yes	characteristic
633	C₂	C₈₉×#3168	1	1	yes	characteristic
634	C₂	C₈₉×#3185	1	1	yes	characteristic
635	C₂	C₈₉×#3190	1	1	yes	characteristic
636	C₂	C₈₉×#3190	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂⁴×C₈₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂²×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₈₉×#1969	C₂	semidirect
8. C₈₉×#1941	C₂	semidirect
9. C₈₉×#1933	C₂	semidirect
10. C₈₉×#1933	C₂	semidirect
11. C₈₉×#1927	C₂	semidirect
12. C₈₉×#1918	C₂	semidirect
13. C₈₉×#1878	C₂	semidirect
14. C₈₉×#1865	C₂	semidirect
310. #29463	C₈₉	direct (N × H)
325. #1969	C₂×C₈₉	semidirect
326. #1941	C₂×C₈₉	semidirect
327. #1933	C₂×C₈₉	semidirect
328. #1933	C₂×C₈₉	semidirect
329. #1927	C₂×C₈₉	semidirect
330. #1918	C₂×C₈₉	semidirect
331. #1878	C₂×C₈₉	semidirect
332. #1865	C₂×C₈₉	semidirect
334. C₈₉	#29463	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈₉×#2855	2
C₄×C₂²⋊Q₈×C₈₉	2
C₈₉×#2075	2
C₈₉×#2035	2
C₈₉×#2001	2
C₈₉×#1969	2
C₈₉×#1941	2
C₈₉×#1933	2
C₈₉×#1927	2
C₈₉×#1918	2
C₈₉×#1878	2
C₈₉×#1865	2
C₄×C₂²⋊C₈×C₈₉	2
#29463	89

Subgroups — C89×#29463 #36407311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#29463 #36407311