Subgroups — C₈₉×#60720 #36438568

Order 22784.

1846 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3510	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#3509	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3181	C₂	1	4	yes	maximal, complemented
4	C₈₉×#3178	C₂	1	8	yes	maximal, complemented
5	C₈₉×#3172	C₂	1	4	yes	maximal, complemented
6	C₈₉×#3162	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#497×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#3140	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#3139	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#3059	C₂	1	2	yes	maximal, complemented
11	C₈₉×#3057	C₂	1	2	yes	maximal
12	C₈₉×#2950	C₂	1	2	yes	maximal, complemented
13	C₈₉×#2949	C₂	1	2	yes	maximal, complemented
14	C₈₉×#2875	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×Q₁₆×C₈₉	C₂²	1	2	yes	—
16	C₂²×SD₁₆×C₈₉	C₂²	1	2	yes	—
17	C₂×#131×C₈₉	C₂²	1	2	yes	—
18	#559×C₈₉	—	2	2	no	—
19	#557×C₈₉	—	2	2	no	—
20	#556×C₈₉	—	2	2	no	—
21	#556×C₈₉	—	2	2	no	—
22	#552×C₈₉	—	2	4	no	—
23	#551×C₈₉	—	2	4	no	—
24	#542×C₈₉	C₂²	1	4	yes	—
25	#541×C₈₉	C₂²	1	4	yes	complemented
26	#536×C₈₉	C₂²	1	2	yes	—
27	#536×C₈₉	C₂²	1	1	yes	characteristic
28	#535×C₈₉	C₂²	1	1	yes	characteristic
29	#533×C₈₉	C₂²	1	1	yes	characteristic
30	#533×C₈₉	C₂²	1	1	yes	characteristic
31	#533×C₈₉	C₂²	1	4	yes	complemented
32	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
33	#529×C₈₉	C₂²	1	4	yes	—
34	C₂×#124×C₈₉	C₂²	1	1	yes	characteristic
35	#519×C₈₉	C₂²	1	2	yes	—
36	#519×C₈₉	C₂²	1	1	yes	characteristic
37	#518×C₈₉	C₂²	1	1	yes	characteristic
38	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
39	#501×C₈₉	C₂²	1	2	yes	complemented
40	#501×C₈₉	C₂²	1	4	yes	complemented
41	#501×C₈₉	C₂²	1	2	yes	complemented
42	#497×C₈₉	C₂²	1	2	yes	—
43	#497×C₈₉	C₂²	1	4	yes	—
44	#497×C₈₉	C₂²	1	2	yes	—
45	#491×C₈₉	C₂²	1	4	yes	complemented
46	#489×C₈₉	C₂²	1	4	yes	—
47	#488×C₈₉	C₂²	1	4	yes	—
48	#487×C₈₉	C₂²	1	4	yes	complemented
49	#470×C₈₉	C₂²	1	8	yes	complemented
50	#468×C₈₉	C₂²	1	8	yes	—
51	#467×C₈₉	C₂²	1	8	yes	complemented
52	#465×C₈₉	C₂²	1	8	yes	—
53	#441×C₈₉	C₂²	1	8	yes	complemented
54	#440×C₈₉	C₂²	1	8	yes	—
55	C₄×Q₁₆×C₈₉	C₂²	1	8	yes	complemented
56	C₄×SD₁₆×C₈₉	C₂²	1	8	yes	—
57	#427×C₈₉	C₂²	1	2	yes	—
58	#427×C₈₉	C₂²	1	2	yes	—
59	#419×C₈₉	C₂²	1	2	yes	—
60	#419×C₈₉	C₂²	1	2	yes	—
61	#419×C₈₉	C₂²	1	2	yes	—
62	#418×C₈₉	C₂²	1	2	yes	—
63	#405×C₈₉	C₂²	1	2	yes	—
64	#405×C₈₉	C₂²	1	4	yes	complemented
65	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
66	C₄×M₁₆×C₈₉	C₂²	1	2	yes	complemented
67	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
68	C₂×#97×C₈₉	C₂²	1	2	yes	complemented
69	C₂×#41×C₈₉	C₂³	1	1	yes	characteristic
70	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
72	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×Q₁₆×C₈₉	—	2	8	no	—
74	C₂×Q₁₆×C₈₉	C₂³	1	4	yes	—
75	C₂×Q₁₆×C₈₉	C₂³	1	4	yes	—
76	C₂×SD₁₆×C₈₉	—	2	8	no	—
77	C₂×SD₁₆×C₈₉	C₂³	1	4	yes	—
78	C₂×SD₁₆×C₈₉	C₂³	1	4	yes	—
79	#131×C₈₉	—	2	8	no	—
80	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
81	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
82	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
83	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
84	C₄⋊Q₈×C₈₉	—	2	1	no	—
85	C₄⋊Q₈×C₈₉	—	2	2	no	—
86	C₄⋊Q₈×C₈₉	—	2	2	no	—
87	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
88	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
89	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
90	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
91	#127×C₈₉	—	2	1	no	—
92	#126×C₈₉	—	2	4	no	—
93	#126×C₈₉	—	2	8	no	—
94	#126×C₈₉	—	2	4	no	—
95	#125×C₈₉	—	2	8	no	—
96	#125×C₈₉	—	2	4	no	—
97	#125×C₈₉	C₂³	1	2	yes	—
98	#125×C₈₉	C₂³	1	2	yes	—
99	#124×C₈₉	—	2	1	no	—
100	#124×C₈₉	—	2	2	no	—
101	#124×C₈₉	—	2	1	no	—
102	#124×C₈₉	—	2	4	no	—
103	#124×C₈₉	—	2	2	no	—
104	#124×C₈₉	C₂³	1	2	yes	—
105	#124×C₈₉	C₂³	1	2	yes	—
106	#123×C₈₉	—	2	2	no	—
107	#123×C₈₉	—	2	4	no	—
108	#123×C₈₉	—	2	4	no	—
109	#123×C₈₉	—	2	4	no	—
110	C₂²⋊Q₈×C₈₉	—	2	2	no	—
111	C₂²⋊Q₈×C₈₉	—	2	2	no	—
112	C₂²⋊Q₈×C₈₉	—	2	4	no	—
113	C₂²⋊Q₈×C₈₉	—	2	4	no	—
114	C₂²⋊Q₈×C₈₉	C₂³	1	8	yes	—
115	C₂²⋊Q₈×C₈₉	C₂³	1	4	yes	—
116	#121×C₈₉	C₂³	1	4	yes	—
117	#120×C₈₉	—	2	2	no	—
118	C₄×Q₈×C₈₉	—	2	4	no	—
119	C₄×Q₈×C₈₉	—	2	2	no	—
120	C₄×Q₈×C₈₉	—	2	2	no	—
121	C₄×Q₈×C₈₉	—	2	2	no	—
122	C₄×Q₈×C₈₉	—	2	4	no	—
123	C₄×Q₈×C₈₉	D₄	1	2	yes	—
124	C₄×Q₈×C₈₉	C₂³	1	8	yes	—
125	C₄×Q₈×C₈₉	C₂³	1	4	yes	—
126	C₄×D₄×C₈₉	—	2	4	no	—
127	C₄×D₄×C₈₉	—	2	2	no	—
128	C₄×D₄×C₈₉	D₄	1	4	yes	—
129	C₄×D₄×C₈₉	D₄	1	2	yes	—
130	C₄×D₄×C₈₉	C₂³	1	4	yes	—
131	#117×C₈₉	—	2	4	no	—
132	#117×C₈₉	—	2	2	no	—
133	#117×C₈₉	—	2	4	no	—
134	#117×C₈₉	—	2	4	no	—
135	#117×C₈₉	C₂³	1	2	yes	—
136	#117×C₈₉	C₂³	1	1	yes	characteristic
137	#117×C₈₉	C₂³	1	1	yes	characteristic
138	#117×C₈₉	C₂³	1	1	yes	characteristic
139	#117×C₈₉	C₂³	1	2	yes	—
140	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
141	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
142	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
143	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
144	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
145	#107×C₈₉	C₂³	1	4	yes	—
146	#107×C₈₉	C₂³	1	4	yes	—
147	#106×C₈₉	C₂³	1	4	yes	—
148	#106×C₈₉	C₂³	1	4	yes	—
149	#103×C₈₉	C₂³	1	8	yes	—
150	#103×C₈₉	C₂³	1	8	yes	—
151	#103×C₈₉	C₂³	1	8	yes	—
152	#102×C₈₉	C₂³	1	8	yes	—
153	#97×C₈₉	C₂³	1	1	yes	characteristic
154	#97×C₈₉	C₂³	1	1	yes	characteristic
155	#97×C₈₉	C₂³	1	2	yes	—
156	#97×C₈₉	C₂³	1	2	yes	—
157	#97×C₈₉	C₂³	1	4	yes	—
158	C₄×C₈×C₈₉	C₂³	1	2	yes	—
159	C₄×C₈×C₈₉	C₂³	1	4	yes	—
160	C₂⁴×C₈₉	—	2	1	no	—
161	#41×C₈₉	—	2	4	no	—
162	C₂×Q₈×C₈₉	—	2	2	no	—
163	C₂×Q₈×C₈₉	—	2	1	no	—
164	C₂×Q₈×C₈₉	—	2	2	no	—
165	C₂×Q₈×C₈₉	—	2	2	no	—
166	C₂×Q₈×C₈₉	—	2	4	no	—
167	C₂×Q₈×C₈₉	—	2	4	no	—
168	C₂×Q₈×C₈₉	—	2	4	no	—
169	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
170	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
171	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
172	C₂×D₄×C₈₉	—	2	4	no	—
173	C₂×D₄×C₈₉	—	2	1	no	—
174	C₂×D₄×C₈₉	—	2	4	no	—
175	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
176	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
177	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₈₉	—	2	1	no	—
179	C₂²×C₄×C₈₉	—	2	2	no	—
180	C₂²×C₄×C₈₉	—	2	4	no	—
181	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
182	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
183	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
185	Q₁₆×C₈₉	—	2	8	no	—
186	Q₁₆×C₈₉	—	2	8	no	—
187	SD₁₆×C₈₉	—	2	8	no	—
188	SD₁₆×C₈₉	—	2	8	no	—
189	M₁₆×C₈₉	—	2	4	no	—
190	M₁₆×C₈₉	—	2	8	no	—
191	C₂×C₈×C₈₉	—	2	8	no	—
192	C₂×C₈×C₈₉	#41	1	8	yes	—
193	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
194	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
195	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
196	C₄⋊C₄×C₈₉	—	2	4	no	—
197	C₄⋊C₄×C₈₉	—	2	4	no	—
198	C₄⋊C₄×C₈₉	—	2	2	no	—
199	C₄⋊C₄×C₈₉	—	2	2	no	—
200	C₄⋊C₄×C₈₉	—	2	4	no	—
201	C₄⋊C₄×C₈₉	—	2	8	no	—
202	C₄⋊C₄×C₈₉	—	2	4	no	—
203	C₄⋊C₄×C₈₉	—	2	2	no	—
204	C₄⋊C₄×C₈₉	—	2	2	no	—
205	C₄⋊C₄×C₈₉	—	2	4	no	—
206	C₄⋊C₄×C₈₉	—	2	8	no	—
207	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
208	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
209	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
210	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
211	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
212	C₂²⋊C₄×C₈₉	—	2	1	no	—
213	C₂²⋊C₄×C₈₉	—	2	1	no	—
214	C₂²⋊C₄×C₈₉	—	2	2	no	—
215	C₂²⋊C₄×C₈₉	—	2	8	no	—
216	C₂²⋊C₄×C₈₉	—	2	4	no	—
217	C₂²⋊C₄×C₈₉	—	2	4	no	—
218	C₂²⋊C₄×C₈₉	—	2	4	no	—
219	C₂²⋊C₄×C₈₉	—	2	4	no	—
220	C₂²⋊C₄×C₈₉	—	2	4	no	—
221	C₂²⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
222	C₂²⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
223	C₄²×C₈₉	—	2	4	no	—
224	C₄²×C₈₉	—	2	8	no	—
225	C₄²×C₈₉	—	2	1	no	—
226	C₄²×C₈₉	—	2	1	no	—
227	C₄²×C₈₉	—	2	2	no	—
228	C₄²×C₈₉	C₂×D₄	1	4	yes	—
229	C₄²×C₈₉	C₂×D₄	1	2	yes	—
230	C₄²×C₈₉	C₂⁴	1	2	yes	—
231	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
232	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
233	C₂³×C₈₉	—	2	2	no	—
234	C₂³×C₈₉	—	4	2	no	—
235	C₂³×C₈₉	—	8	2	no	—
236	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
237	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
238	Q₈×C₈₉	—	2	2	no	—
239	Q₈×C₈₉	—	2	4	no	—
240	Q₈×C₈₉	—	2	4	no	—
241	Q₈×C₈₉	—	2	4	no	—
242	Q₈×C₈₉	—	4	4	no	—
243	Q₈×C₈₉	—	4	2	no	—
244	Q₈×C₈₉	—	4	4	no	—
245	Q₈×C₈₉	—	8	2	no	—
246	D₄×C₈₉	—	2	4	no	—
247	D₄×C₈₉	—	2	2	no	—
248	D₄×C₈₉	—	2	4	no	—
249	D₄×C₈₉	—	4	2	no	—
250	D₄×C₈₉	—	8	4	no	—
251	C₂×C₄×C₈₉	—	2	4	no	—
252	C₂×C₄×C₈₉	—	2	4	no	—
253	C₂×C₄×C₈₉	—	2	2	no	—
254	C₂×C₄×C₈₉	—	2	2	no	—
255	C₂×C₄×C₈₉	—	2	4	no	—
256	C₂×C₄×C₈₉	—	2	2	no	—
257	C₂×C₄×C₈₉	—	4	4	no	—
258	C₂×C₄×C₈₉	—	4	2	no	—
259	C₂×C₄×C₈₉	—	4	2	no	—
260	C₂×C₄×C₈₉	—	4	4	no	—
261	C₂×C₄×C₈₉	—	8	4	no	—
262	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
263	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
264	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
265	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
266	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
267	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
268	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
269	C₈×C₈₉	—	2	4	no	—
270	C₈×C₈₉	C₂×#41	1	4	yes	—
271	C₈×C₈₉	C₂×#41	1	4	yes	—
272	C₂²×C₈₉	—	2	2	no	—
273	C₂²×C₈₉	—	2	4	no	—
274	C₂²×C₈₉	—	2	2	no	—
275	C₂²×C₈₉	—	4	4	no	—
276	C₂²×C₈₉	—	4	2	no	—
277	C₂²×C₈₉	—	16	2	no	—
278	C₂²×C₈₉	#532	1	2	yes	—
279	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
280	C₄×C₈₉	—	2	2	no	—
281	C₄×C₈₉	—	2	4	no	—
282	C₄×C₈₉	—	2	2	no	—
283	C₄×C₈₉	—	2	2	no	—
284	C₄×C₈₉	—	4	2	no	—
285	C₄×C₈₉	—	4	2	no	—
286	C₄×C₈₉	—	4	4	no	—
287	C₄×C₈₉	—	4	2	no	—
288	C₄×C₈₉	—	4	4	no	—
289	C₄×C₈₉	#532	1	2	yes	—
290	C₄×C₈₉	C₂²×#41	1	2	yes	—
291	#60720	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
292	C₂×C₈₉	—	2	2	no	—
293	C₂×C₈₉	—	4	2	no	—
294	C₂×C₈₉	—	8	2	no	—
295	C₂×C₈₉	C₂×#532	1	1	yes	characteristic
296	C₂×C₈₉	#3586	1	2	yes	—
297	#3510	C₂×C₈₉	1	1	yes	characteristic
298	#3509	C₂×C₈₉	1	1	yes	characteristic, complemented
299	#3181	C₂×C₈₉	1	4	yes	complemented
300	#3178	C₂×C₈₉	1	8	yes	complemented
301	#3172	C₂×C₈₉	1	4	yes	complemented
302	#3162	C₂×C₈₉	1	1	yes	characteristic, complemented
303	C₂×#497	C₂×C₈₉	1	1	yes	characteristic, complemented
304	#3140	C₂×C₈₉	1	1	yes	characteristic, complemented
305	#3139	C₂×C₈₉	1	1	yes	characteristic, complemented
306	#3059	C₂×C₈₉	1	2	yes	complemented
307	#3057	C₂×C₈₉	1	2	yes	—
308	#2950	C₂×C₈₉	1	2	yes	complemented
309	#2949	C₂×C₈₉	1	2	yes	complemented
310	#2875	C₂×C₈₉	1	1	yes	characteristic, complemented
311	C₈₉	#60720	1	1	yes	characteristic, direct product factor, Sylow
312	C₂²×Q₁₆	C₂²×C₈₉	1	2	yes	—
313	C₂²×SD₁₆	C₂²×C₈₉	1	2	yes	—
314	C₂×#131	C₂²×C₈₉	1	2	yes	—
315	#559	—	2	2	no	—
316	#557	—	2	2	no	—
317	#556	—	2	2	no	—
318	#556	—	2	2	no	—
319	#552	—	2	4	no	—
320	#551	—	2	4	no	—
321	#542	C₂²×C₈₉	1	4	yes	—
322	#541	C₂²×C₈₉	1	4	yes	complemented
323	#536	C₂²×C₈₉	1	2	yes	—
324	#536	C₂²×C₈₉	1	1	yes	characteristic
325	#535	C₂²×C₈₉	1	1	yes	characteristic
326	#533	C₂²×C₈₉	1	1	yes	characteristic
327	#533	C₂²×C₈₉	1	1	yes	characteristic
328	#533	C₂²×C₈₉	1	4	yes	complemented
329	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
330	#529	C₂²×C₈₉	1	4	yes	—
331	C₂×#124	C₂²×C₈₉	1	1	yes	characteristic
332	#519	C₂²×C₈₉	1	2	yes	—
333	#519	C₂²×C₈₉	1	1	yes	characteristic
334	#518	C₂²×C₈₉	1	1	yes	characteristic
335	C₄×#41	C₂²×C₈₉	1	1	yes	characteristic
336	#501	C₂²×C₈₉	1	2	yes	complemented
337	#501	C₂²×C₈₉	1	4	yes	complemented
338	#501	C₂²×C₈₉	1	2	yes	complemented
339	#497	C₂²×C₈₉	1	2	yes	—
340	#497	C₂²×C₈₉	1	4	yes	—
341	#497	C₂²×C₈₉	1	2	yes	—
342	#491	C₂²×C₈₉	1	4	yes	complemented
343	#489	C₂²×C₈₉	1	4	yes	—
344	#488	C₂²×C₈₉	1	4	yes	—
345	#487	C₂²×C₈₉	1	4	yes	complemented
346	#470	C₂²×C₈₉	1	8	yes	complemented
347	#468	C₂²×C₈₉	1	8	yes	—
348	#467	C₂²×C₈₉	1	8	yes	complemented
349	#465	C₂²×C₈₉	1	8	yes	—
350	#441	C₂²×C₈₉	1	8	yes	complemented
351	#440	C₂²×C₈₉	1	8	yes	—
352	C₄×Q₁₆	C₂²×C₈₉	1	8	yes	complemented
353	C₄×SD₁₆	C₂²×C₈₉	1	8	yes	—
354	#427	C₂²×C₈₉	1	2	yes	—
355	#427	C₂²×C₈₉	1	2	yes	—
356	#419	C₂²×C₈₉	1	2	yes	—
357	#419	C₂²×C₈₉	1	2	yes	—
358	#419	C₂²×C₈₉	1	2	yes	—
359	#418	C₂²×C₈₉	1	2	yes	—
360	#405	C₂²×C₈₉	1	2	yes	—
361	#405	C₂²×C₈₉	1	4	yes	complemented
362	C₄×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
363	C₄×M₁₆	C₂²×C₈₉	1	2	yes	complemented
364	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
365	C₂×#97	C₂²×C₈₉	1	2	yes	complemented
366	C₂×#41	C₂³×C₈₉	1	1	yes	characteristic
367	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
368	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
369	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
370	C₂×Q₁₆	—	2	8	no	—
371	C₂×Q₁₆	C₂³×C₈₉	1	4	yes	—
372	C₂×Q₁₆	C₂³×C₈₉	1	4	yes	—
373	C₂×SD₁₆	—	2	8	no	—
374	C₂×SD₁₆	C₂³×C₈₉	1	4	yes	—
375	C₂×SD₁₆	C₂³×C₈₉	1	4	yes	—
376	#131	—	2	8	no	—
377	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
378	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
379	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
380	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
381	C₄⋊Q₈	—	2	1	no	—
382	C₄⋊Q₈	—	2	2	no	—
383	C₄⋊Q₈	—	2	2	no	—
384	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
385	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
386	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
387	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
388	#127	—	2	1	no	—
389	#126	—	2	4	no	—
390	#126	—	2	8	no	—
391	#126	—	2	4	no	—
392	#125	—	2	8	no	—
393	#125	—	2	4	no	—
394	#125	C₂³×C₈₉	1	2	yes	—
395	#125	C₂³×C₈₉	1	2	yes	—
396	#124	—	2	1	no	—
397	#124	—	2	2	no	—
398	#124	—	2	1	no	—
399	#124	—	2	4	no	—
400	#124	—	2	2	no	—
401	#124	C₂³×C₈₉	1	2	yes	—
402	#124	C₂³×C₈₉	1	2	yes	—
403	#123	—	2	2	no	—
404	#123	—	2	4	no	—
405	#123	—	2	4	no	—
406	#123	—	2	4	no	—
407	C₂²⋊Q₈	—	2	2	no	—
408	C₂²⋊Q₈	—	2	2	no	—
409	C₂²⋊Q₈	—	2	4	no	—
410	C₂²⋊Q₈	—	2	4	no	—
411	C₂²⋊Q₈	C₂³×C₈₉	1	4	yes	—
412	C₂²⋊Q₈	C₂³×C₈₉	1	8	yes	—
413	#121	C₂³×C₈₉	1	4	yes	—
414	#120	—	2	2	no	—
415	C₄×Q₈	—	2	4	no	—
416	C₄×Q₈	—	2	2	no	—
417	C₄×Q₈	—	2	2	no	—
418	C₄×Q₈	—	2	2	no	—
419	C₄×Q₈	—	2	4	no	—
420	C₄×Q₈	D₄×C₈₉	1	2	yes	—
421	C₄×Q₈	C₂³×C₈₉	1	4	yes	—
422	C₄×Q₈	C₂³×C₈₉	1	8	yes	—
423	C₄×D₄	—	2	4	no	—
424	C₄×D₄	—	2	2	no	—
425	C₄×D₄	D₄×C₈₉	1	4	yes	—
426	C₄×D₄	D₄×C₈₉	1	2	yes	—
427	C₄×D₄	C₂³×C₈₉	1	4	yes	—
428	#117	—	2	4	no	—
429	#117	—	2	2	no	—
430	#117	—	2	4	no	—
431	#117	—	2	4	no	—
432	#117	C₂³×C₈₉	1	1	yes	characteristic
433	#117	C₂³×C₈₉	1	1	yes	characteristic
434	#117	C₂³×C₈₉	1	2	yes	—
435	#117	C₂³×C₈₉	1	1	yes	characteristic
436	#117	C₂³×C₈₉	1	2	yes	—
437	C₂×C₄⋊C₄	—	2	2	no	—
438	C₂×C₂²⋊C₄	—	2	2	no	—
439	C₂×C₂²⋊C₄	—	2	2	no	—
440	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
441	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
442	#107	C₂³×C₈₉	1	4	yes	—
443	#107	C₂³×C₈₉	1	4	yes	—
444	#106	C₂³×C₈₉	1	4	yes	—
445	#106	C₂³×C₈₉	1	4	yes	—
446	#103	C₂³×C₈₉	1	8	yes	—
447	#103	C₂³×C₈₉	1	8	yes	—
448	#103	C₂³×C₈₉	1	8	yes	—
449	#102	C₂³×C₈₉	1	8	yes	—
450	#97	C₂³×C₈₉	1	1	yes	characteristic
451	#97	C₂³×C₈₉	1	1	yes	characteristic
452	#97	C₂³×C₈₉	1	2	yes	—
453	#97	C₂³×C₈₉	1	2	yes	—
454	#97	C₂³×C₈₉	1	4	yes	—
455	C₄×C₈	C₂³×C₈₉	1	2	yes	—
456	C₄×C₈	C₂³×C₈₉	1	4	yes	—
457	C₂⁴	—	2	1	no	—
458	#41	—	2	4	no	—
459	C₂×Q₈	—	2	2	no	—
460	C₂×Q₈	—	2	1	no	—
461	C₂×Q₈	—	2	2	no	—
462	C₂×Q₈	—	2	2	no	—
463	C₂×Q₈	—	2	4	no	—
464	C₂×Q₈	—	2	4	no	—
465	C₂×Q₈	—	2	4	no	—
466	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
467	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
468	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
469	C₂×D₄	—	2	4	no	—
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	—	2	4	no	—
472	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
473	C₂×D₄	C₂×D₄×C₈₉	1	1	yes	characteristic
474	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	2	2	no	—
477	C₂²×C₄	—	2	4	no	—
478	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
479	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
480	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
481	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
482	Q₁₆	—	2	8	no	—
483	Q₁₆	—	2	8	no	—
484	SD₁₆	—	2	8	no	—
485	SD₁₆	—	2	8	no	—
486	M₁₆	—	2	4	no	—
487	M₁₆	—	2	8	no	—
488	C₂×C₈	—	2	8	no	—
489	C₂×C₈	#41×C₈₉	1	8	yes	—
490	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
491	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
492	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	4	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	8	no	—
499	C₄⋊C₄	—	2	4	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	—	2	8	no	—
504	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
505	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
506	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
507	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
508	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
509	C₂²⋊C₄	—	2	1	no	—
510	C₂²⋊C₄	—	2	1	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	2	8	no	—
513	C₂²⋊C₄	—	2	4	no	—
514	C₂²⋊C₄	—	2	4	no	—
515	C₂²⋊C₄	—	2	4	no	—
516	C₂²⋊C₄	—	2	4	no	—
517	C₂²⋊C₄	—	2	4	no	—
518	C₂²⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
519	C₂²⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
520	C₄²	—	2	4	no	—
521	C₄²	—	2	8	no	—
522	C₄²	—	2	1	no	—
523	C₄²	—	2	1	no	—
524	C₄²	—	2	2	no	—
525	C₄²	C₂×D₄×C₈₉	1	4	yes	—
526	C₄²	C₂×D₄×C₈₉	1	2	yes	—
527	C₄²	C₂⁴×C₈₉	1	2	yes	—
528	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
529	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
530	C₂³	—	2	2	no	—
531	C₂³	—	4	2	no	—
532	C₂³	—	8	2	no	—
533	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
534	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
535	Q₈	—	2	2	no	—
536	Q₈	—	2	4	no	—
537	Q₈	—	2	4	no	—
538	Q₈	—	2	4	no	—
539	Q₈	—	4	4	no	—
540	Q₈	—	4	2	no	—
541	Q₈	—	4	4	no	—
542	Q₈	—	8	2	no	—
543	D₄	—	2	4	no	—
544	D₄	—	2	2	no	—
545	D₄	—	2	4	no	—
546	D₄	—	4	2	no	—
547	D₄	—	8	4	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	4	4	no	—
555	C₂×C₄	—	4	2	no	—
556	C₂×C₄	—	4	2	no	—
557	C₂×C₄	—	4	4	no	—
558	C₂×C₄	—	8	4	no	—
559	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
560	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
561	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
562	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
563	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
564	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
565	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
566	C₈	—	2	4	no	—
567	C₈	C₂×#41×C₈₉	1	4	yes	—
568	C₈	C₂×#41×C₈₉	1	4	yes	—
569	C₂²	—	2	2	no	—
570	C₂²	—	2	4	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	4	4	no	—
573	C₂²	—	4	2	no	—
574	C₂²	—	16	2	no	—
575	C₂²	#532×C₈₉	1	2	yes	—
576	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
577	C₄	—	2	2	no	—
578	C₄	—	2	4	no	—
579	C₄	—	2	2	no	—
580	C₄	—	2	2	no	—
581	C₄	—	4	2	no	—
582	C₄	—	4	2	no	—
583	C₄	—	4	4	no	—
584	C₄	—	4	2	no	—
585	C₄	—	4	4	no	—
586	C₄	#532×C₈₉	1	2	yes	—
587	C₄	C₂²×#41×C₈₉	1	2	yes	—
588	C₂	—	2	2	no	—
589	C₂	—	4	2	no	—
590	C₂	—	8	2	no	—
591	C₂	C₂×#532×C₈₉	1	1	yes	characteristic
592	C₂	C₈₉×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
prime radical	#535×C₈₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₄×#41×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#3509	C₂	semidirect
3. C₈₉×#3181	C₂	semidirect
4. C₈₉×#3178	C₂	semidirect
5. C₈₉×#3172	C₂	semidirect
6. C₈₉×#3162	C₂	semidirect
7. C₂×#497×C₈₉	C₂	semidirect
8. C₈₉×#3140	C₂	semidirect
9. C₈₉×#3139	C₂	semidirect
10. C₈₉×#3059	C₂	semidirect
12. C₈₉×#2950	C₂	semidirect
13. C₈₉×#2949	C₂	semidirect
14. C₈₉×#2875	C₂	semidirect
25. #541×C₈₉	C₂²	semidirect
31. #533×C₈₉	C₂²	semidirect
39. #501×C₈₉	C₂²	semidirect
40. #501×C₈₉	C₂²	semidirect
41. #501×C₈₉	C₂²	semidirect
45. #491×C₈₉	C₂²	semidirect
48. #487×C₈₉	C₂²	semidirect
49. #470×C₈₉	C₂²	semidirect
51. #467×C₈₉	C₂²	semidirect
53. #441×C₈₉	C₂²	semidirect
55. C₄×Q₁₆×C₈₉	C₂²	semidirect
64. #405×C₈₉	C₂²	semidirect
66. C₄×M₁₆×C₈₉	C₂²	semidirect
68. C₂×#97×C₈₉	C₂²	semidirect
291. #60720	C₈₉	direct (N × H)
298. #3509	C₂×C₈₉	semidirect
299. #3181	C₂×C₈₉	semidirect
300. #3178	C₂×C₈₉	semidirect
301. #3172	C₂×C₈₉	semidirect
302. #3162	C₂×C₈₉	semidirect
303. C₂×#497	C₂×C₈₉	semidirect
304. #3140	C₂×C₈₉	semidirect
305. #3139	C₂×C₈₉	semidirect
306. #3059	C₂×C₈₉	semidirect
308. #2950	C₂×C₈₉	semidirect
309. #2949	C₂×C₈₉	semidirect
310. #2875	C₂×C₈₉	semidirect
311. C₈₉	#60720	direct (N × H)
322. #541	C₂²×C₈₉	semidirect
328. #533	C₂²×C₈₉	semidirect
336. #501	C₂²×C₈₉	semidirect
337. #501	C₂²×C₈₉	semidirect
338. #501	C₂²×C₈₉	semidirect
342. #491	C₂²×C₈₉	semidirect
345. #487	C₂²×C₈₉	semidirect
346. #470	C₂²×C₈₉	semidirect
348. #467	C₂²×C₈₉	semidirect
350. #441	C₂²×C₈₉	semidirect
352. C₄×Q₁₆	C₂²×C₈₉	semidirect
361. #405	C₂²×C₈₉	semidirect
363. C₄×M₁₆	C₂²×C₈₉	semidirect
365. C₂×#97	C₂²×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3510	2
C₈₉×#3509	2
C₈₉×#3181	2
C₈₉×#3178	2
C₈₉×#3172	2
C₈₉×#3162	2
C₂×#497×C₈₉	2
C₈₉×#3140	2
C₈₉×#3139	2
C₈₉×#3059	2
C₈₉×#3057	2
C₈₉×#2950	2
C₈₉×#2949	2
C₈₉×#2875	2
#60720	89

Subgroups — C89×#60720 #36438568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60720 #36438568