Subgroups — C₈₉×#29151 #36406999

Order 22784.

734 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2660	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈₉×#2326	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2026	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2025	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2024	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#2023	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1959	C₂	1	1	yes	maximal, characteristic
8	C₈₉×#1958	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#1836	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1834	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1805	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈₉×#1804	C₂	1	1	yes	maximal, characteristic
13	C₈₉×#1803	C₂	1	1	yes	maximal, characteristic, complemented
14	C₈₉×#1802	C₂	1	1	yes	maximal, characteristic
15	C₈₉×#1768	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#125×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×#124×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#103×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×#102×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
33	#398×C₈₉	—	2	1	no	—
34	#398×C₈₉	—	2	1	no	—
35	#395×C₈₉	—	2	1	no	—
36	#395×C₈₉	—	2	1	no	—
37	#391×C₈₉	C₂²	1	1	yes	characteristic
38	#390×C₈₉	C₂²	1	1	yes	characteristic
39	#388×C₈₉	—	2	1	no	—
40	#388×C₈₉	—	2	1	no	—
41	#385×C₈₉	—	2	1	no	—
42	#385×C₈₉	—	2	1	no	—
43	#384×C₈₉	C₂²	1	1	yes	characteristic
44	#384×C₈₉	C₂²	1	1	yes	characteristic
45	#384×C₈₉	C₂²	1	1	yes	characteristic
46	#382×C₈₉	C₂²	1	1	yes	characteristic
47	#382×C₈₉	C₂²	1	1	yes	characteristic
48	#376×C₈₉	C₂²	1	1	yes	characteristic
49	#340×C₈₉	C₂²	1	1	yes	characteristic
50	#340×C₈₉	C₂²	1	1	yes	characteristic
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	—	2	1	no	—
60	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
61	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₈₉	—	2	1	no	—
63	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₈₉	—	4	1	no	—
68	C₄⋊Q₈×C₈₉	—	4	1	no	—
69	#125×C₈₉	—	4	1	no	—
70	#125×C₈₉	—	4	1	no	—
71	#124×C₈₉	—	2	2	no	—
72	#124×C₈₉	Q₈	1	4	yes	complemented
73	C₄×Q₈×C₈₉	—	2	2	no	—
74	C₄×Q₈×C₈₉	—	2	2	no	—
75	C₄×D₄×C₈₉	—	2	2	no	—
76	C₄×D₄×C₈₉	—	2	2	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	—	2	1	no	—
84	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
85	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
86	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
87	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
94	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
96	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
97	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
98	C₂×C₄²×C₈₉	—	2	1	no	—
99	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₈×C₈₉	—	2	2	no	—
103	C₄⋊C₈×C₈₉	—	2	2	no	—
104	#103×C₈₉	—	4	1	no	—
105	#103×C₈₉	—	4	1	no	—
106	#103×C₈₉	—	4	1	no	—
107	#103×C₈₉	—	4	1	no	—
108	#102×C₈₉	—	4	1	no	—
109	#102×C₈₉	—	4	1	no	—
110	#102×C₈₉	—	4	1	no	—
111	#102×C₈₉	—	4	1	no	—
112	#97×C₈₉	—	2	2	no	—
113	C₄×C₈×C₈₉	—	2	2	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	D₄	1	1	yes	characteristic
121	#95×C₈₉	D₄	1	1	yes	characteristic
122	#95×C₈₉	D₄	1	1	yes	characteristic
123	#95×C₈₉	D₄	1	1	yes	characteristic
124	C₂⁴×C₈₉	—	2	1	no	—
125	C₂×Q₈×C₈₉	—	2	1	no	—
126	C₂×Q₈×C₈₉	—	2	1	no	—
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	—	4	2	no	—
131	C₂×Q₈×C₈₉	—	8	1	no	—
132	C₂×Q₈×C₈₉	—	8	1	no	—
133	C₂×Q₈×C₈₉	C₂×Q₈	1	2	yes	—
134	C₂×D₄×C₈₉	—	2	1	no	—
135	C₂×D₄×C₈₉	—	2	1	no	—
136	C₂×D₄×C₈₉	—	2	1	no	—
137	C₂×D₄×C₈₉	—	2	1	no	—
138	C₂×D₄×C₈₉	—	2	1	no	—
139	C₂×D₄×C₈₉	C₂×Q₈	1	2	yes	—
140	C₂²×C₄×C₈₉	—	2	1	no	—
141	C₂²×C₄×C₈₉	—	2	1	no	—
142	C₂²×C₄×C₈₉	—	2	1	no	—
143	C₂²×C₄×C₈₉	—	2	1	no	—
144	C₂²×C₄×C₈₉	—	2	1	no	—
145	C₂²×C₄×C₈₉	—	2	1	no	—
146	C₂²×C₄×C₈₉	—	2	1	no	—
147	C₂²×C₄×C₈₉	—	4	2	no	—
148	C₂²×C₄×C₈₉	—	8	1	no	—
149	C₂²×C₄×C₈₉	—	8	1	no	—
150	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
157	C₂×C₈×C₈₉	—	2	1	no	—
158	C₂×C₈×C₈₉	—	2	1	no	—
159	C₂×C₈×C₈₉	—	2	1	no	—
160	C₂×C₈×C₈₉	—	2	1	no	—
161	C₂×C₈×C₈₉	—	2	1	no	—
162	C₂×C₈×C₈₉	—	2	1	no	—
163	C₂×C₈×C₈₉	—	2	1	no	—
164	C₂×C₈×C₈₉	—	2	1	no	—
165	C₂×C₈×C₈₉	—	2	1	no	—
166	C₂×C₈×C₈₉	—	2	1	no	—
167	C₂×C₈×C₈₉	—	2	1	no	—
168	C₂×C₈×C₈₉	—	2	1	no	—
169	C₄⋊C₄×C₈₉	—	2	1	no	—
170	C₄⋊C₄×C₈₉	—	2	1	no	—
171	C₄⋊C₄×C₈₉	—	2	1	no	—
172	C₄⋊C₄×C₈₉	—	2	1	no	—
173	C₄⋊C₄×C₈₉	—	2	1	no	—
174	C₄⋊C₄×C₈₉	—	2	1	no	—
175	C₄⋊C₄×C₈₉	—	2	1	no	—
176	C₄⋊C₄×C₈₉	—	2	1	no	—
177	C₄⋊C₄×C₈₉	—	2	1	no	—
178	C₄⋊C₄×C₈₉	—	2	1	no	—
179	C₄⋊C₄×C₈₉	—	2	1	no	—
180	C₄⋊C₄×C₈₉	—	2	1	no	—
181	C₄⋊C₄×C₈₉	—	4	2	no	—
182	C₄⋊C₄×C₈₉	—	4	2	no	—
183	C₄⋊C₄×C₈₉	—	4	2	no	—
184	C₄⋊C₄×C₈₉	—	4	2	no	—
185	C₄⋊C₄×C₈₉	—	8	1	no	—
186	C₄⋊C₄×C₈₉	—	8	1	no	—
187	C₄⋊C₄×C₈₉	—	8	1	no	—
188	C₄⋊C₄×C₈₉	—	8	1	no	—
189	C₄⋊C₄×C₈₉	—	8	1	no	—
190	C₄⋊C₄×C₈₉	—	8	1	no	—
191	C₂²⋊C₄×C₈₉	—	2	4	no	—
192	C₂²⋊C₄×C₈₉	—	2	4	no	—
193	C₂²⋊C₄×C₈₉	—	4	2	no	—
194	C₂²⋊C₄×C₈₉	—	8	1	no	—
195	C₂²⋊C₄×C₈₉	—	8	1	no	—
196	C₄²×C₈₉	—	2	1	no	—
197	C₄²×C₈₉	—	2	1	no	—
198	C₄²×C₈₉	—	4	2	no	—
199	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
200	C₄²×C₈₉	#41	1	2	yes	—
201	C₄²×C₈₉	#41	1	2	yes	—
202	C₄²×C₈₉	#41	1	2	yes	—
203	C₂³×C₈₉	—	2	2	no	—
204	C₂³×C₈₉	—	4	1	no	—
205	C₂³×C₈₉	—	4	1	no	—
206	C₂³×C₈₉	—	4	1	no	—
207	C₂³×C₈₉	—	4	1	no	—
208	C₂³×C₈₉	—	4	1	no	—
209	C₂³×C₈₉	—	4	1	no	—
210	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
211	Q₈×C₈₉	—	2	2	no	—
212	Q₈×C₈₉	—	2	2	no	—
213	Q₈×C₈₉	—	4	1	no	—
214	Q₈×C₈₉	—	4	1	no	—
215	Q₈×C₈₉	—	8	4	no	—
216	D₄×C₈₉	—	2	2	no	—
217	D₄×C₈₉	—	2	2	no	—
218	D₄×C₈₉	—	4	1	no	—
219	D₄×C₈₉	—	4	1	no	—
220	C₂×C₄×C₈₉	—	2	1	no	—
221	C₂×C₄×C₈₉	—	2	1	no	—
222	C₂×C₄×C₈₉	—	2	1	no	—
223	C₂×C₄×C₈₉	—	2	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	2	no	—
229	C₂×C₄×C₈₉	—	2	1	no	—
230	C₂×C₄×C₈₉	—	2	1	no	—
231	C₂×C₄×C₈₉	—	2	1	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	—	4	1	no	—
236	C₂×C₄×C₈₉	—	4	1	no	—
237	C₂×C₄×C₈₉	—	4	1	no	—
238	C₂×C₄×C₈₉	—	4	1	no	—
239	C₂×C₄×C₈₉	—	4	1	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	—	8	4	no	—
256	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
257	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
258	C₂×C₄×C₈₉	#124	1	2	yes	—
259	C₂×C₄×C₈₉	#124	1	2	yes	—
260	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
261	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
262	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
263	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
264	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
265	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
266	C₈×C₈₉	—	4	1	no	—
267	C₈×C₈₉	—	4	1	no	—
268	C₈×C₈₉	—	4	1	no	—
269	C₈×C₈₉	—	4	1	no	—
270	C₂²×C₈₉	—	4	2	no	—
271	C₂²×C₈₉	—	4	2	no	—
272	C₂²×C₈₉	—	4	2	no	—
273	C₂²×C₈₉	—	8	1	no	—
274	C₂²×C₈₉	—	8	1	no	—
275	C₂²×C₈₉	—	8	1	no	—
276	C₂²×C₈₉	—	8	1	no	—
277	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
278	C₂²×C₈₉	C₂×#124	1	1	yes	characteristic
279	C₂²×C₈₉	#535	1	1	yes	characteristic
280	C₂²×C₈₉	#576	1	1	yes	characteristic
281	C₂²×C₈₉	#577	1	1	yes	characteristic
282	C₂²×C₈₉	#577	1	1	yes	characteristic
283	C₂²×C₈₉	#578	1	1	yes	characteristic
284	C₄×C₈₉	—	2	2	no	—
285	C₄×C₈₉	—	2	2	no	—
286	C₄×C₈₉	—	2	2	no	—
287	C₄×C₈₉	—	2	2	no	—
288	C₄×C₈₉	—	4	2	no	—
289	C₄×C₈₉	—	4	1	no	—
290	C₄×C₈₉	—	4	1	no	—
291	C₄×C₈₉	—	8	1	no	—
292	C₄×C₈₉	—	8	1	no	—
293	C₄×C₈₉	—	8	1	no	—
294	C₄×C₈₉	—	8	1	no	—
295	C₄×C₈₉	—	8	1	no	—
296	C₄×C₈₉	—	8	1	no	—
297	C₄×C₈₉	#554	1	1	yes	characteristic
298	C₄×C₈₉	#554	1	1	yes	characteristic
299	C₄×C₈₉	#555	1	1	yes	characteristic
300	C₄×C₈₉	#555	1	1	yes	characteristic
301	#29151	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
302	C₂×C₈₉	—	8	2	no	—
303	C₂×C₈₉	#2563	1	1	yes	characteristic
304	C₂×C₈₉	#3083	1	1	yes	characteristic
305	C₂×C₈₉	#3084	1	1	yes	characteristic
306	C₂×C₈₉	#3136	1	1	yes	characteristic
307	C₂×C₈₉	#3137	1	1	yes	characteristic
308	C₂×C₈₉	#3224	1	1	yes	characteristic
309	C₂×C₈₉	#3225	1	1	yes	characteristic
310	#2660	C₂×C₈₉	1	1	yes	characteristic, complemented
311	#2326	C₂×C₈₉	1	1	yes	characteristic
312	#2026	C₂×C₈₉	1	1	yes	characteristic, complemented
313	#2025	C₂×C₈₉	1	1	yes	characteristic
314	#2024	C₂×C₈₉	1	1	yes	characteristic, complemented
315	#2023	C₂×C₈₉	1	1	yes	characteristic
316	#1959	C₂×C₈₉	1	1	yes	characteristic
317	#1958	C₂×C₈₉	1	1	yes	characteristic
318	#1836	C₂×C₈₉	1	1	yes	characteristic, complemented
319	#1834	C₂×C₈₉	1	1	yes	characteristic, complemented
320	#1805	C₂×C₈₉	1	1	yes	characteristic, complemented
321	#1804	C₂×C₈₉	1	1	yes	characteristic
322	#1803	C₂×C₈₉	1	1	yes	characteristic, complemented
323	#1802	C₂×C₈₉	1	1	yes	characteristic
324	#1768	C₂×C₈₉	1	1	yes	characteristic, complemented
325	C₈₉	#29151	1	1	yes	characteristic, direct product factor, Sylow
326	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
327	C₂×#125	C₂²×C₈₉	1	1	yes	characteristic
328	C₂×#124	C₂²×C₈₉	1	1	yes	characteristic
329	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
330	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
331	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
332	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
333	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
334	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
335	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
336	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
337	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
338	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
339	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
340	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
341	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
342	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
343	#398	—	2	1	no	—
344	#398	—	2	1	no	—
345	#395	—	2	1	no	—
346	#395	—	2	1	no	—
347	#391	C₂²×C₈₉	1	1	yes	characteristic
348	#390	C₂²×C₈₉	1	1	yes	characteristic
349	#388	—	2	1	no	—
350	#388	—	2	1	no	—
351	#385	—	2	1	no	—
352	#385	—	2	1	no	—
353	#384	C₂²×C₈₉	1	1	yes	characteristic
354	#384	C₂²×C₈₉	1	1	yes	characteristic
355	#384	C₂²×C₈₉	1	1	yes	characteristic
356	#382	C₂²×C₈₉	1	1	yes	characteristic
357	#382	C₂²×C₈₉	1	1	yes	characteristic
358	#376	C₂²×C₈₉	1	1	yes	characteristic
359	#340	C₂²×C₈₉	1	1	yes	characteristic
360	#340	C₂²×C₈₉	1	1	yes	characteristic
361	#340	C₂²×C₈₉	1	1	yes	characteristic
362	#340	C₂²×C₈₉	1	1	yes	characteristic
363	#340	C₂²×C₈₉	1	1	yes	characteristic
364	#340	C₂²×C₈₉	1	1	yes	characteristic
365	#340	C₂²×C₈₉	1	1	yes	characteristic
366	#340	C₂²×C₈₉	1	1	yes	characteristic
367	#336	C₂²×C₈₉	1	1	yes	characteristic
368	#336	C₂²×C₈₉	1	1	yes	characteristic
369	C₂²×Q₈	—	2	1	no	—
370	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
371	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
372	C₂³×C₄	—	2	1	no	—
373	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
374	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
375	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
376	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
377	C₄⋊Q₈	—	4	1	no	—
378	C₄⋊Q₈	—	4	1	no	—
379	#125	—	4	1	no	—
380	#125	—	4	1	no	—
381	#124	—	2	2	no	—
382	#124	Q₈×C₈₉	1	4	yes	complemented
383	C₄×Q₈	—	2	2	no	—
384	C₄×Q₈	—	2	2	no	—
385	C₄×D₄	—	2	2	no	—
386	C₄×D₄	—	2	2	no	—
387	C₂×C₄⋊C₄	—	2	1	no	—
388	C₂×C₄⋊C₄	—	2	1	no	—
389	C₂×C₄⋊C₄	—	2	1	no	—
390	C₂×C₄⋊C₄	—	2	1	no	—
391	C₂×C₄⋊C₄	—	2	1	no	—
392	C₂×C₄⋊C₄	—	2	1	no	—
393	C₂×C₄⋊C₄	—	2	1	no	—
394	C₂×C₄⋊C₄	—	2	1	no	—
395	C₂×C₄⋊C₄	—	2	1	no	—
396	C₂×C₄⋊C₄	—	2	1	no	—
397	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
398	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
399	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
400	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
401	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
402	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
403	C₂×C₂²⋊C₄	—	2	1	no	—
404	C₂×C₂²⋊C₄	—	2	1	no	—
405	C₂×C₂²⋊C₄	—	2	1	no	—
406	C₂×C₂²⋊C₄	—	2	1	no	—
407	C₂×C₂²⋊C₄	—	2	1	no	—
408	C₂×C₄²	—	2	1	no	—
409	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
410	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
411	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
412	C₄⋊C₈	—	2	2	no	—
413	C₄⋊C₈	—	2	2	no	—
414	#103	—	4	1	no	—
415	#103	—	4	1	no	—
416	#103	—	4	1	no	—
417	#103	—	4	1	no	—
418	#102	—	4	1	no	—
419	#102	—	4	1	no	—
420	#102	—	4	1	no	—
421	#102	—	4	1	no	—
422	#97	—	2	2	no	—
423	C₄×C₈	—	2	2	no	—
424	#95	—	2	1	no	—
425	#95	—	2	1	no	—
426	#95	—	2	1	no	—
427	#95	—	2	1	no	—
428	#95	—	2	1	no	—
429	#95	—	2	1	no	—
430	#95	D₄×C₈₉	1	1	yes	characteristic
431	#95	D₄×C₈₉	1	1	yes	characteristic
432	#95	D₄×C₈₉	1	1	yes	characteristic
433	#95	D₄×C₈₉	1	1	yes	characteristic
434	C₂⁴	—	2	1	no	—
435	C₂×Q₈	—	2	1	no	—
436	C₂×Q₈	—	2	1	no	—
437	C₂×Q₈	—	2	1	no	—
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	4	2	no	—
441	C₂×Q₈	—	8	1	no	—
442	C₂×Q₈	—	8	1	no	—
443	C₂×Q₈	C₂×Q₈×C₈₉	1	2	yes	—
444	C₂×D₄	—	2	1	no	—
445	C₂×D₄	—	2	1	no	—
446	C₂×D₄	—	2	1	no	—
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	C₂×Q₈×C₈₉	1	2	yes	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	4	2	no	—
458	C₂²×C₄	—	8	1	no	—
459	C₂²×C₄	—	8	1	no	—
460	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
461	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
462	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
464	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
465	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
466	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	4	2	no	—
492	C₄⋊C₄	—	4	2	no	—
493	C₄⋊C₄	—	4	2	no	—
494	C₄⋊C₄	—	4	2	no	—
495	C₄⋊C₄	—	8	1	no	—
496	C₄⋊C₄	—	8	1	no	—
497	C₄⋊C₄	—	8	1	no	—
498	C₄⋊C₄	—	8	1	no	—
499	C₄⋊C₄	—	8	1	no	—
500	C₄⋊C₄	—	8	1	no	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	4	no	—
503	C₂²⋊C₄	—	4	2	no	—
504	C₂²⋊C₄	—	8	1	no	—
505	C₂²⋊C₄	—	8	1	no	—
506	C₄²	—	2	1	no	—
507	C₄²	—	2	1	no	—
508	C₄²	—	4	2	no	—
509	C₄²	C₂×Q₈×C₈₉	1	2	yes	—
510	C₄²	#41×C₈₉	1	2	yes	—
511	C₄²	#41×C₈₉	1	2	yes	—
512	C₄²	#41×C₈₉	1	2	yes	—
513	C₂³	—	2	2	no	—
514	C₂³	—	4	1	no	—
515	C₂³	—	4	1	no	—
516	C₂³	—	4	1	no	—
517	C₂³	—	4	1	no	—
518	C₂³	—	4	1	no	—
519	C₂³	—	4	1	no	—
520	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
521	Q₈	—	2	2	no	—
522	Q₈	—	2	2	no	—
523	Q₈	—	4	1	no	—
524	Q₈	—	4	1	no	—
525	Q₈	—	8	4	no	—
526	D₄	—	2	2	no	—
527	D₄	—	2	2	no	—
528	D₄	—	4	1	no	—
529	D₄	—	4	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	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	—	8	4	no	—
566	C₂×C₄	C₂²⋊Q₈×C₈₉	1	2	yes	—
567	C₂×C₄	C₂²⋊Q₈×C₈₉	1	2	yes	—
568	C₂×C₄	#124×C₈₉	1	2	yes	—
569	C₂×C₄	#124×C₈₉	1	2	yes	—
570	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
571	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
572	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
573	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
574	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
575	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
576	C₈	—	4	1	no	—
577	C₈	—	4	1	no	—
578	C₈	—	4	1	no	—
579	C₈	—	4	1	no	—
580	C₂²	—	4	2	no	—
581	C₂²	—	4	2	no	—
582	C₂²	—	4	2	no	—
583	C₂²	—	8	1	no	—
584	C₂²	—	8	1	no	—
585	C₂²	—	8	1	no	—
586	C₂²	—	8	1	no	—
587	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
588	C₂²	C₂×#124×C₈₉	1	1	yes	characteristic
589	C₂²	#535×C₈₉	1	1	yes	characteristic
590	C₂²	#576×C₈₉	1	1	yes	characteristic
591	C₂²	#577×C₈₉	1	1	yes	characteristic
592	C₂²	#577×C₈₉	1	1	yes	characteristic
593	C₂²	#578×C₈₉	1	1	yes	characteristic
594	C₄	—	2	2	no	—
595	C₄	—	2	2	no	—
596	C₄	—	2	2	no	—
597	C₄	—	2	2	no	—
598	C₄	—	4	2	no	—
599	C₄	—	4	1	no	—
600	C₄	—	4	1	no	—
601	C₄	—	8	1	no	—
602	C₄	—	8	1	no	—
603	C₄	—	8	1	no	—
604	C₄	—	8	1	no	—
605	C₄	—	8	1	no	—
606	C₄	—	8	1	no	—
607	C₄	#554×C₈₉	1	1	yes	characteristic
608	C₄	#554×C₈₉	1	1	yes	characteristic
609	C₄	#555×C₈₉	1	1	yes	characteristic
610	C₄	#555×C₈₉	1	1	yes	characteristic
611	C₂	—	8	2	no	—
612	C₂	C₈₉×#2563	1	1	yes	characteristic
613	C₂	C₈₉×#3083	1	1	yes	characteristic
614	C₂	C₈₉×#3084	1	1	yes	characteristic
615	C₂	C₈₉×#3136	1	1	yes	characteristic
616	C₂	C₈₉×#3137	1	1	yes	characteristic
617	C₂	C₈₉×#3224	1	1	yes	characteristic
618	C₂	C₈₉×#3225	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂²×D₄×C₈₉	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ #376×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#2660	C₂	semidirect
3. C₈₉×#2026	C₂	semidirect
5. C₈₉×#2024	C₂	semidirect
9. C₈₉×#1836	C₂	semidirect
10. C₈₉×#1834	C₂	semidirect
11. C₈₉×#1805	C₂	semidirect
13. C₈₉×#1803	C₂	semidirect
15. C₈₉×#1768	C₂	semidirect
72. #124×C₈₉	Q₈	semidirect
301. #29151	C₈₉	direct (N × H)
310. #2660	C₂×C₈₉	semidirect
312. #2026	C₂×C₈₉	semidirect
314. #2024	C₂×C₈₉	semidirect
318. #1836	C₂×C₈₉	semidirect
319. #1834	C₂×C₈₉	semidirect
320. #1805	C₂×C₈₉	semidirect
322. #1803	C₂×C₈₉	semidirect
324. #1768	C₂×C₈₉	semidirect
325. C₈₉	#29151	direct (N × H)
382. #124	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×#2660	2
C₈₉×#2326	2
C₈₉×#2026	2
C₈₉×#2025	2
C₈₉×#2024	2
C₈₉×#2023	2
C₈₉×#1959	2
C₈₉×#1958	2
C₈₉×#1836	2
C₈₉×#1834	2
C₈₉×#1805	2
C₈₉×#1804	2
C₈₉×#1803	2
C₈₉×#1802	2
C₈₉×#1768	2
#29151	89

Subgroups — C89×#29151 #36406999

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#29151 #36406999