Subgroups — C₈₉×#60719 #36438567

Order 22784.

1966 conjugacy classes of subgroups.

The subgroup lattice has 614 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, complemented
2	C₈₉×#3508	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3180	C₂	1	4	yes	maximal, complemented
4	C₈₉×#3179	C₂	1	8	yes	maximal, complemented
5	C₈₉×#3171	C₂	1	4	yes	maximal, complemented
6	C₈₉×#3162	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#496×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#3140	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#3138	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#3058	C₂	1	2	yes	maximal, complemented
11	C₈₉×#3057	C₂	1	2	yes	maximal, complemented
12	C₈₉×#2949	C₂	1	2	yes	maximal, complemented
13	C₈₉×#2948	C₂	1	2	yes	maximal, complemented
14	C₈₉×#2875	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×SD₁₆×C₈₉	C₂²	1	2	yes	—
16	C₂²×D₈×C₈₉	C₂²	1	2	yes	—
17	C₂×#131×C₈₉	C₂²	1	2	yes	—
18	#559×C₈₉	—	2	2	no	—
19	#559×C₈₉	—	2	2	no	—
20	#556×C₈₉	—	2	2	no	—
21	#551×C₈₉	—	2	4	no	—
22	#550×C₈₉	—	2	2	no	—
23	#546×C₈₉	—	2	4	no	—
24	#542×C₈₉	C₂²	1	4	yes	complemented
25	#540×C₈₉	C₂²	1	4	yes	complemented
26	#536×C₈₉	C₂²	1	1	yes	characteristic
27	#535×C₈₉	C₂²	1	2	yes	—
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	#532×C₈₉	C₂²	1	4	yes	complemented
32	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
33	#529×C₈₉	C₂²	1	4	yes	complemented
34	C₂×#124×C₈₉	C₂²	1	1	yes	characteristic
35	#519×C₈₉	C₂²	1	1	yes	characteristic
36	#518×C₈₉	C₂²	1	1	yes	characteristic
37	#518×C₈₉	C₂²	1	2	yes	—
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	#496×C₈₉	C₂²	1	2	yes	—
43	#496×C₈₉	C₂²	1	4	yes	—
44	#496×C₈₉	C₂²	1	2	yes	—
45	#490×C₈₉	C₂²	1	4	yes	complemented
46	#489×C₈₉	C₂²	1	4	yes	complemented
47	#488×C₈₉	C₂²	1	4	yes	complemented
48	#486×C₈₉	C₂²	1	4	yes	complemented
49	#469×C₈₉	C₂²	1	8	yes	complemented
50	#468×C₈₉	C₂²	1	8	yes	complemented
51	#466×C₈₉	C₂²	1	8	yes	complemented
52	#465×C₈₉	C₂²	1	8	yes	complemented
53	#442×C₈₉	C₂²	1	8	yes	complemented
54	#440×C₈₉	C₂²	1	8	yes	complemented
55	C₄×SD₁₆×C₈₉	C₂²	1	8	yes	complemented
56	C₄×D₈×C₈₉	C₂²	1	8	yes	complemented
57	#427×C₈₉	C₂²	1	2	yes	—
58	#427×C₈₉	C₂²	1	2	yes	—
59	#419×C₈₉	C₂²	1	2	yes	—
60	#418×C₈₉	C₂²	1	2	yes	—
61	#418×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	—	2	4	no	—
70	C₂×#41×C₈₉	C₂³	1	1	yes	characteristic
71	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₂²×D₄×C₈₉	—	2	2	no	—
73	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
74	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
75	C₂×SD₁₆×C₈₉	—	2	8	no	—
76	C₂×SD₁₆×C₈₉	C₂³	1	4	yes	—
77	C₂×SD₁₆×C₈₉	C₂³	1	4	yes	—
78	C₂×D₈×C₈₉	—	2	8	no	—
79	C₂×D₈×C₈₉	C₂³	1	4	yes	—
80	C₂×D₈×C₈₉	C₂³	1	4	yes	—
81	#131×C₈₉	—	2	8	no	—
82	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
83	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
84	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
85	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
86	C₄⋊Q₈×C₈₉	—	2	1	no	—
87	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
88	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
89	#127×C₈₉	—	2	2	no	—
90	#127×C₈₉	—	2	1	no	—
91	#127×C₈₉	—	2	2	no	—
92	#127×C₈₉	C₂³	1	2	yes	—
93	#127×C₈₉	C₂³	1	2	yes	—
94	#126×C₈₉	—	2	4	no	—
95	#126×C₈₉	—	2	4	no	—
96	#125×C₈₉	C₂³	1	2	yes	—
97	#125×C₈₉	C₂³	1	2	yes	—
98	#124×C₈₉	—	2	1	no	—
99	#124×C₈₉	—	2	2	no	—
100	#124×C₈₉	—	2	1	no	—
101	#124×C₈₉	—	2	4	no	—
102	#124×C₈₉	—	2	2	no	—
103	#124×C₈₉	—	2	4	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	8	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	C₂³	1	4	yes	—
114	#121×C₈₉	—	2	4	no	—
115	#121×C₈₉	—	2	8	no	—
116	#121×C₈₉	C₂³	1	8	yes	—
117	#121×C₈₉	C₂³	1	4	yes	—
118	#120×C₈₉	—	2	4	no	—
119	#120×C₈₉	—	2	2	no	—
120	C₄×Q₈×C₈₉	—	2	2	no	—
121	C₄×Q₈×C₈₉	—	2	2	no	—
122	C₄×Q₈×C₈₉	—	2	2	no	—
123	C₄×Q₈×C₈₉	D₄	1	2	yes	—
124	C₄×Q₈×C₈₉	C₂³	1	4	yes	—
125	C₄×D₄×C₈₉	—	2	4	no	—
126	C₄×D₄×C₈₉	—	2	4	no	—
127	C₄×D₄×C₈₉	—	2	2	no	—
128	C₄×D₄×C₈₉	—	2	4	no	—
129	C₄×D₄×C₈₉	D₄	1	4	yes	—
130	C₄×D₄×C₈₉	D₄	1	2	yes	—
131	C₄×D₄×C₈₉	C₂³	1	4	yes	—
132	C₄×D₄×C₈₉	C₂³	1	8	yes	—
133	#117×C₈₉	—	2	2	no	—
134	#117×C₈₉	—	2	4	no	—
135	#117×C₈₉	C₂³	1	1	yes	characteristic
136	#117×C₈₉	C₂³	1	1	yes	characteristic
137	#117×C₈₉	C₂³	1	2	yes	—
138	#117×C₈₉	C₂³	1	1	yes	characteristic
139	#117×C₈₉	C₂³	1	2	yes	—
140	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
141	C₂×C₂²⋊C₄×C₈₉	—	2	4	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	#102×C₈₉	C₂³	1	8	yes	—
151	#102×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	C₂⁴×C₈₉	—	2	2	no	—
162	#41×C₈₉	—	2	4	no	—
163	#41×C₈₉	—	8	8	no	—
164	C₂×Q₈×C₈₉	—	2	1	no	—
165	C₂×Q₈×C₈₉	—	2	2	no	—
166	C₂×Q₈×C₈₉	—	2	2	no	—
167	C₂×Q₈×C₈₉	—	2	4	no	—
168	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
169	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
170	C₂×D₄×C₈₉	—	2	4	no	—
171	C₂×D₄×C₈₉	—	2	8	no	—
172	C₂×D₄×C₈₉	—	2	1	no	—
173	C₂×D₄×C₈₉	—	2	2	no	—
174	C₂×D₄×C₈₉	—	2	4	no	—
175	C₂×D₄×C₈₉	—	2	4	no	—
176	C₂×D₄×C₈₉	—	2	4	no	—
177	C₂×D₄×C₈₉	—	2	4	no	—
178	C₂×D₄×C₈₉	—	2	4	no	—
179	C₂×D₄×C₈₉	—	8	4	no	—
180	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
181	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
182	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
183	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
184	C₂²×C₄×C₈₉	—	2	1	no	—
185	C₂²×C₄×C₈₉	—	2	4	no	—
186	C₂²×C₄×C₈₉	—	2	8	no	—
187	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
188	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
189	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
190	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
191	SD₁₆×C₈₉	—	2	8	no	—
192	SD₁₆×C₈₉	—	2	8	no	—
193	D₈×C₈₉	—	2	8	no	—
194	D₈×C₈₉	—	2	8	no	—
195	M₁₆×C₈₉	—	2	4	no	—
196	M₁₆×C₈₉	—	2	8	no	—
197	C₂×C₈×C₈₉	—	2	8	no	—
198	C₂×C₈×C₈₉	#41	1	8	yes	complemented
199	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
200	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
201	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
202	C₄⋊C₄×C₈₉	—	2	2	no	—
203	C₄⋊C₄×C₈₉	—	2	4	no	—
204	C₄⋊C₄×C₈₉	—	2	4	no	—
205	C₄⋊C₄×C₈₉	—	2	2	no	—
206	C₄⋊C₄×C₈₉	—	2	2	no	—
207	C₄⋊C₄×C₈₉	—	2	4	no	—
208	C₄⋊C₄×C₈₉	—	2	2	no	—
209	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
210	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
211	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
212	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
213	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
214	C₂²⋊C₄×C₈₉	—	2	2	no	—
215	C₂²⋊C₄×C₈₉	—	2	1	no	—
216	C₂²⋊C₄×C₈₉	—	2	1	no	—
217	C₂²⋊C₄×C₈₉	—	2	4	no	—
218	C₂²⋊C₄×C₈₉	—	2	4	no	—
219	C₂²⋊C₄×C₈₉	—	2	8	no	—
220	C₂²⋊C₄×C₈₉	—	2	4	no	—
221	C₂²⋊C₄×C₈₉	—	2	4	no	—
222	C₂²⋊C₄×C₈₉	—	2	4	no	—
223	C₂²⋊C₄×C₈₉	—	2	8	no	—
224	C₂²⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
225	C₂²⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
226	C₄²×C₈₉	—	2	4	no	—
227	C₄²×C₈₉	—	2	2	no	—
228	C₄²×C₈₉	—	2	1	no	—
229	C₄²×C₈₉	—	2	1	no	—
230	C₄²×C₈₉	C₂×D₄	1	4	yes	—
231	C₄²×C₈₉	C₂×D₄	1	2	yes	—
232	C₄²×C₈₉	C₂⁴	1	2	yes	—
233	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
234	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
235	C₂³×C₈₉	—	2	2	no	—
236	C₂³×C₈₉	—	2	4	no	—
237	C₂³×C₈₉	—	4	2	no	—
238	C₂³×C₈₉	—	4	4	no	—
239	C₂³×C₈₉	—	8	4	no	—
240	C₂³×C₈₉	—	8	2	no	—
241	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
242	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
243	Q₈×C₈₉	—	2	2	no	—
244	Q₈×C₈₉	—	2	4	no	—
245	Q₈×C₈₉	—	4	4	no	—
246	Q₈×C₈₉	—	4	2	no	—
247	Q₈×C₈₉	—	8	2	no	—
248	D₄×C₈₉	—	2	4	no	—
249	D₄×C₈₉	—	2	2	no	—
250	D₄×C₈₉	—	2	4	no	—
251	D₄×C₈₉	—	2	4	no	—
252	D₄×C₈₉	—	2	4	no	—
253	D₄×C₈₉	—	4	8	no	—
254	D₄×C₈₉	—	4	2	no	—
255	D₄×C₈₉	—	4	4	no	—
256	D₄×C₈₉	—	8	4	no	—
257	D₄×C₈₉	—	8	4	no	—
258	D₄×C₈₉	—	8	8	no	—
259	C₂×C₄×C₈₉	—	2	4	no	—
260	C₂×C₄×C₈₉	—	2	2	no	—
261	C₂×C₄×C₈₉	—	2	2	no	—
262	C₂×C₄×C₈₉	—	2	2	no	—
263	C₂×C₄×C₈₉	—	2	4	no	—
264	C₂×C₄×C₈₉	—	4	2	no	—
265	C₂×C₄×C₈₉	—	4	4	no	—
266	C₂×C₄×C₈₉	—	4	2	no	—
267	C₂×C₄×C₈₉	—	8	4	no	—
268	C₂×C₄×C₈₉	—	8	8	no	—
269	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
270	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
271	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
272	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
273	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
274	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
275	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
276	C₈×C₈₉	—	2	4	no	—
277	C₈×C₈₉	C₂×#41	1	4	yes	—
278	C₈×C₈₉	C₂×#41	1	4	yes	—
279	C₂²×C₈₉	—	2	2	no	—
280	C₂²×C₈₉	—	2	4	no	—
281	C₂²×C₈₉	—	2	2	no	—
282	C₂²×C₈₉	—	4	8	no	—
283	C₂²×C₈₉	—	4	4	no	—
284	C₂²×C₈₉	—	4	2	no	—
285	C₂²×C₈₉	—	4	4	no	—
286	C₂²×C₈₉	—	16	4	no	—
287	C₂²×C₈₉	—	16	2	no	—
288	C₂²×C₈₉	#532	1	2	yes	—
289	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
290	C₄×C₈₉	—	2	2	no	—
291	C₄×C₈₉	—	2	4	no	—
292	C₄×C₈₉	—	2	2	no	—
293	C₄×C₈₉	—	2	2	no	—
294	C₄×C₈₉	—	4	4	no	—
295	C₄×C₈₉	—	4	2	no	—
296	C₄×C₈₉	—	4	2	no	—
297	C₄×C₈₉	—	4	2	no	—
298	C₄×C₈₉	#532	1	2	yes	—
299	C₄×C₈₉	C₂²×#41	1	2	yes	—
300	#60719	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
301	C₂×C₈₉	—	2	2	no	—
302	C₂×C₈₉	—	4	2	no	—
303	C₂×C₈₉	—	8	4	no	—
304	C₂×C₈₉	—	8	2	no	—
305	C₂×C₈₉	C₂×#532	1	1	yes	characteristic
306	C₂×C₈₉	#3585	1	2	yes	—
307	#3510	C₂×C₈₉	1	1	yes	characteristic, complemented
308	#3508	C₂×C₈₉	1	1	yes	characteristic, complemented
309	#3180	C₂×C₈₉	1	4	yes	complemented
310	#3179	C₂×C₈₉	1	8	yes	complemented
311	#3171	C₂×C₈₉	1	4	yes	complemented
312	#3162	C₂×C₈₉	1	1	yes	characteristic, complemented
313	C₂×#496	C₂×C₈₉	1	1	yes	characteristic, complemented
314	#3140	C₂×C₈₉	1	1	yes	characteristic, complemented
315	#3138	C₂×C₈₉	1	1	yes	characteristic, complemented
316	#3058	C₂×C₈₉	1	2	yes	complemented
317	#3057	C₂×C₈₉	1	2	yes	complemented
318	#2949	C₂×C₈₉	1	2	yes	complemented
319	#2948	C₂×C₈₉	1	2	yes	complemented
320	#2875	C₂×C₈₉	1	1	yes	characteristic, complemented
321	C₈₉	#60719	1	1	yes	characteristic, direct product factor, Sylow
322	C₂²×SD₁₆	C₂²×C₈₉	1	2	yes	—
323	C₂²×D₈	C₂²×C₈₉	1	2	yes	—
324	C₂×#131	C₂²×C₈₉	1	2	yes	—
325	#559	—	2	2	no	—
326	#559	—	2	2	no	—
327	#556	—	2	2	no	—
328	#551	—	2	4	no	—
329	#550	—	2	2	no	—
330	#546	—	2	4	no	—
331	#542	C₂²×C₈₉	1	4	yes	complemented
332	#540	C₂²×C₈₉	1	4	yes	complemented
333	#536	C₂²×C₈₉	1	1	yes	characteristic
334	#535	C₂²×C₈₉	1	2	yes	—
335	#535	C₂²×C₈₉	1	1	yes	characteristic
336	#533	C₂²×C₈₉	1	1	yes	characteristic
337	#533	C₂²×C₈₉	1	1	yes	characteristic
338	#532	C₂²×C₈₉	1	4	yes	complemented
339	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
340	#529	C₂²×C₈₉	1	4	yes	complemented
341	C₂×#124	C₂²×C₈₉	1	1	yes	characteristic
342	#519	C₂²×C₈₉	1	1	yes	characteristic
343	#518	C₂²×C₈₉	1	1	yes	characteristic
344	#518	C₂²×C₈₉	1	2	yes	—
345	C₄×#41	C₂²×C₈₉	1	1	yes	characteristic
346	#501	C₂²×C₈₉	1	2	yes	complemented
347	#501	C₂²×C₈₉	1	4	yes	complemented
348	#501	C₂²×C₈₉	1	2	yes	complemented
349	#496	C₂²×C₈₉	1	2	yes	—
350	#496	C₂²×C₈₉	1	4	yes	—
351	#496	C₂²×C₈₉	1	2	yes	—
352	#490	C₂²×C₈₉	1	4	yes	complemented
353	#489	C₂²×C₈₉	1	4	yes	complemented
354	#488	C₂²×C₈₉	1	4	yes	complemented
355	#486	C₂²×C₈₉	1	4	yes	complemented
356	#469	C₂²×C₈₉	1	8	yes	complemented
357	#468	C₂²×C₈₉	1	8	yes	complemented
358	#466	C₂²×C₈₉	1	8	yes	complemented
359	#465	C₂²×C₈₉	1	8	yes	complemented
360	#442	C₂²×C₈₉	1	8	yes	complemented
361	#440	C₂²×C₈₉	1	8	yes	complemented
362	C₄×SD₁₆	C₂²×C₈₉	1	8	yes	complemented
363	C₄×D₈	C₂²×C₈₉	1	8	yes	complemented
364	#427	C₂²×C₈₉	1	2	yes	—
365	#427	C₂²×C₈₉	1	2	yes	—
366	#419	C₂²×C₈₉	1	2	yes	—
367	#418	C₂²×C₈₉	1	2	yes	—
368	#418	C₂²×C₈₉	1	2	yes	—
369	#418	C₂²×C₈₉	1	2	yes	—
370	#405	C₂²×C₈₉	1	2	yes	—
371	#405	C₂²×C₈₉	1	4	yes	complemented
372	C₄×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
373	C₄×M₁₆	C₂²×C₈₉	1	2	yes	complemented
374	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
375	C₂×#97	C₂²×C₈₉	1	2	yes	complemented
376	C₂×#41	—	2	4	no	—
377	C₂×#41	C₂³×C₈₉	1	1	yes	characteristic
378	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
379	C₂²×D₄	—	2	2	no	—
380	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
381	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
382	C₂×SD₁₆	—	2	8	no	—
383	C₂×SD₁₆	C₂³×C₈₉	1	4	yes	—
384	C₂×SD₁₆	C₂³×C₈₉	1	4	yes	—
385	C₂×D₈	—	2	8	no	—
386	C₂×D₈	C₂³×C₈₉	1	4	yes	—
387	C₂×D₈	C₂³×C₈₉	1	4	yes	—
388	#131	—	2	8	no	—
389	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
390	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
391	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
392	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
393	C₄⋊Q₈	—	2	1	no	—
394	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
395	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
396	#127	—	2	2	no	—
397	#127	—	2	1	no	—
398	#127	—	2	2	no	—
399	#127	C₂³×C₈₉	1	2	yes	—
400	#127	C₂³×C₈₉	1	2	yes	—
401	#126	—	2	4	no	—
402	#126	—	2	4	no	—
403	#125	C₂³×C₈₉	1	2	yes	—
404	#125	C₂³×C₈₉	1	2	yes	—
405	#124	—	2	1	no	—
406	#124	—	2	2	no	—
407	#124	—	2	1	no	—
408	#124	—	2	4	no	—
409	#124	—	2	2	no	—
410	#124	—	2	4	no	—
411	#124	C₂³×C₈₉	1	2	yes	—
412	#124	C₂³×C₈₉	1	2	yes	—
413	#123	—	2	2	no	—
414	#123	—	2	4	no	—
415	#123	—	2	8	no	—
416	#123	—	2	4	no	—
417	C₂²⋊Q₈	—	2	2	no	—
418	C₂²⋊Q₈	—	2	2	no	—
419	C₂²⋊Q₈	—	2	4	no	—
420	C₂²⋊Q₈	C₂³×C₈₉	1	4	yes	—
421	#121	—	2	4	no	—
422	#121	—	2	8	no	—
423	#121	C₂³×C₈₉	1	4	yes	—
424	#121	C₂³×C₈₉	1	8	yes	—
425	#120	—	2	4	no	—
426	#120	—	2	2	no	—
427	C₄×Q₈	—	2	2	no	—
428	C₄×Q₈	—	2	2	no	—
429	C₄×Q₈	—	2	2	no	—
430	C₄×Q₈	D₄×C₈₉	1	2	yes	—
431	C₄×Q₈	C₂³×C₈₉	1	4	yes	—
432	C₄×D₄	—	2	4	no	—
433	C₄×D₄	—	2	4	no	—
434	C₄×D₄	—	2	2	no	—
435	C₄×D₄	—	2	4	no	—
436	C₄×D₄	D₄×C₈₉	1	4	yes	—
437	C₄×D₄	D₄×C₈₉	1	2	yes	—
438	C₄×D₄	C₂³×C₈₉	1	4	yes	—
439	C₄×D₄	C₂³×C₈₉	1	8	yes	—
440	#117	—	2	2	no	—
441	#117	—	2	4	no	—
442	#117	C₂³×C₈₉	1	2	yes	—
443	#117	C₂³×C₈₉	1	1	yes	characteristic
444	#117	C₂³×C₈₉	1	1	yes	characteristic
445	#117	C₂³×C₈₉	1	1	yes	characteristic
446	#117	C₂³×C₈₉	1	2	yes	—
447	C₂×C₂²⋊C₄	—	2	2	no	—
448	C₂×C₂²⋊C₄	—	2	4	no	—
449	C₂×C₂²⋊C₄	—	2	2	no	—
450	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
451	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
452	#107	C₂³×C₈₉	1	4	yes	—
453	#107	C₂³×C₈₉	1	4	yes	—
454	#106	C₂³×C₈₉	1	4	yes	—
455	#106	C₂³×C₈₉	1	4	yes	—
456	#103	C₂³×C₈₉	1	8	yes	—
457	#102	C₂³×C₈₉	1	8	yes	—
458	#102	C₂³×C₈₉	1	8	yes	—
459	#102	C₂³×C₈₉	1	8	yes	—
460	#97	C₂³×C₈₉	1	1	yes	characteristic
461	#97	C₂³×C₈₉	1	1	yes	characteristic
462	#97	C₂³×C₈₉	1	2	yes	—
463	#97	C₂³×C₈₉	1	2	yes	—
464	#97	C₂³×C₈₉	1	4	yes	—
465	C₄×C₈	C₂³×C₈₉	1	2	yes	—
466	C₄×C₈	C₂³×C₈₉	1	4	yes	—
467	C₂⁴	—	2	1	no	—
468	C₂⁴	—	2	2	no	—
469	#41	—	2	4	no	—
470	#41	—	8	8	no	—
471	C₂×Q₈	—	2	1	no	—
472	C₂×Q₈	—	2	2	no	—
473	C₂×Q₈	—	2	2	no	—
474	C₂×Q₈	—	2	4	no	—
475	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
476	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
477	C₂×D₄	—	2	4	no	—
478	C₂×D₄	—	2	8	no	—
479	C₂×D₄	—	2	1	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	—	2	4	no	—
482	C₂×D₄	—	2	4	no	—
483	C₂×D₄	—	2	4	no	—
484	C₂×D₄	—	2	4	no	—
485	C₂×D₄	—	2	4	no	—
486	C₂×D₄	—	8	4	no	—
487	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
488	C₂×D₄	C₂×D₄×C₈₉	1	1	yes	characteristic
489	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
490	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	4	no	—
493	C₂²×C₄	—	2	8	no	—
494	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
495	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
496	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
497	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
498	SD₁₆	—	2	8	no	—
499	SD₁₆	—	2	8	no	—
500	D₈	—	2	8	no	—
501	D₈	—	2	8	no	—
502	M₁₆	—	2	4	no	—
503	M₁₆	—	2	8	no	—
504	C₂×C₈	—	2	8	no	—
505	C₂×C₈	#41×C₈₉	1	8	yes	complemented
506	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
507	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
508	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	2	4	no	—
511	C₄⋊C₄	—	2	4	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	2	no	—
514	C₄⋊C₄	—	2	4	no	—
515	C₄⋊C₄	—	2	2	no	—
516	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
517	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
518	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
519	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
520	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
521	C₂²⋊C₄	—	2	2	no	—
522	C₂²⋊C₄	—	2	1	no	—
523	C₂²⋊C₄	—	2	1	no	—
524	C₂²⋊C₄	—	2	4	no	—
525	C₂²⋊C₄	—	2	4	no	—
526	C₂²⋊C₄	—	2	8	no	—
527	C₂²⋊C₄	—	2	4	no	—
528	C₂²⋊C₄	—	2	4	no	—
529	C₂²⋊C₄	—	2	4	no	—
530	C₂²⋊C₄	—	2	8	no	—
531	C₂²⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
532	C₂²⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
533	C₄²	—	2	4	no	—
534	C₄²	—	2	2	no	—
535	C₄²	—	2	1	no	—
536	C₄²	—	2	1	no	—
537	C₄²	C₂×D₄×C₈₉	1	4	yes	—
538	C₄²	C₂×D₄×C₈₉	1	2	yes	—
539	C₄²	C₂⁴×C₈₉	1	2	yes	—
540	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
541	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
542	C₂³	—	2	2	no	—
543	C₂³	—	2	4	no	—
544	C₂³	—	4	2	no	—
545	C₂³	—	4	4	no	—
546	C₂³	—	8	4	no	—
547	C₂³	—	8	2	no	—
548	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
549	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
550	Q₈	—	2	2	no	—
551	Q₈	—	2	4	no	—
552	Q₈	—	4	4	no	—
553	Q₈	—	4	2	no	—
554	Q₈	—	8	2	no	—
555	D₄	—	2	4	no	—
556	D₄	—	2	2	no	—
557	D₄	—	2	4	no	—
558	D₄	—	2	4	no	—
559	D₄	—	2	4	no	—
560	D₄	—	4	8	no	—
561	D₄	—	4	2	no	—
562	D₄	—	4	4	no	—
563	D₄	—	8	4	no	—
564	D₄	—	8	4	no	—
565	D₄	—	8	8	no	—
566	C₂×C₄	—	2	4	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	4	2	no	—
572	C₂×C₄	—	4	4	no	—
573	C₂×C₄	—	4	2	no	—
574	C₂×C₄	—	8	4	no	—
575	C₂×C₄	—	8	8	no	—
576	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
577	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
578	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
579	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
580	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
581	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
582	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
583	C₈	—	2	4	no	—
584	C₈	C₂×#41×C₈₉	1	4	yes	—
585	C₈	C₂×#41×C₈₉	1	4	yes	—
586	C₂²	—	2	2	no	—
587	C₂²	—	2	4	no	—
588	C₂²	—	2	2	no	—
589	C₂²	—	4	8	no	—
590	C₂²	—	4	4	no	—
591	C₂²	—	4	2	no	—
592	C₂²	—	4	4	no	—
593	C₂²	—	16	4	no	—
594	C₂²	—	16	2	no	—
595	C₂²	#532×C₈₉	1	2	yes	—
596	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
597	C₄	—	2	2	no	—
598	C₄	—	2	4	no	—
599	C₄	—	2	2	no	—
600	C₄	—	2	2	no	—
601	C₄	—	4	4	no	—
602	C₄	—	4	2	no	—
603	C₄	—	4	2	no	—
604	C₄	—	4	2	no	—
605	C₄	#532×C₈₉	1	2	yes	—
606	C₄	C₂²×#41×C₈₉	1	2	yes	—
607	C₂	—	2	2	no	—
608	C₂	—	4	2	no	—
609	C₂	—	8	4	no	—
610	C₂	—	8	2	no	—
611	C₂	C₂×#532×C₈₉	1	1	yes	characteristic
612	C₂	C₈₉×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
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
1. C₈₉×#3510	C₂	semidirect
2. C₈₉×#3508	C₂	semidirect
3. C₈₉×#3180	C₂	semidirect
4. C₈₉×#3179	C₂	semidirect
5. C₈₉×#3171	C₂	semidirect
6. C₈₉×#3162	C₂	semidirect
7. C₂×#496×C₈₉	C₂	semidirect
8. C₈₉×#3140	C₂	semidirect
9. C₈₉×#3138	C₂	semidirect
10. C₈₉×#3058	C₂	semidirect
11. C₈₉×#3057	C₂	semidirect
12. C₈₉×#2949	C₂	semidirect
13. C₈₉×#2948	C₂	semidirect
14. C₈₉×#2875	C₂	semidirect
24. #542×C₈₉	C₂²	semidirect
25. #540×C₈₉	C₂²	semidirect
31. #532×C₈₉	C₂²	semidirect
33. #529×C₈₉	C₂²	semidirect
39. #501×C₈₉	C₂²	semidirect
40. #501×C₈₉	C₂²	semidirect
41. #501×C₈₉	C₂²	semidirect
45. #490×C₈₉	C₂²	semidirect
46. #489×C₈₉	C₂²	semidirect
47. #488×C₈₉	C₂²	semidirect
48. #486×C₈₉	C₂²	semidirect
49. #469×C₈₉	C₂²	semidirect
50. #468×C₈₉	C₂²	semidirect
51. #466×C₈₉	C₂²	semidirect
52. #465×C₈₉	C₂²	semidirect
53. #442×C₈₉	C₂²	semidirect
54. #440×C₈₉	C₂²	semidirect
55. C₄×SD₁₆×C₈₉	C₂²	semidirect
56. C₄×D₈×C₈₉	C₂²	semidirect
64. #405×C₈₉	C₂²	semidirect
66. C₄×M₁₆×C₈₉	C₂²	semidirect
68. C₂×#97×C₈₉	C₂²	semidirect
198. C₂×C₈×C₈₉	#41	semidirect
300. #60719	C₈₉	direct (N × H)
307. #3510	C₂×C₈₉	semidirect
308. #3508	C₂×C₈₉	semidirect
309. #3180	C₂×C₈₉	semidirect
310. #3179	C₂×C₈₉	semidirect
311. #3171	C₂×C₈₉	semidirect
312. #3162	C₂×C₈₉	semidirect
313. C₂×#496	C₂×C₈₉	semidirect
314. #3140	C₂×C₈₉	semidirect
315. #3138	C₂×C₈₉	semidirect
316. #3058	C₂×C₈₉	semidirect
317. #3057	C₂×C₈₉	semidirect
318. #2949	C₂×C₈₉	semidirect
319. #2948	C₂×C₈₉	semidirect
320. #2875	C₂×C₈₉	semidirect
321. C₈₉	#60719	direct (N × H)
331. #542	C₂²×C₈₉	semidirect
332. #540	C₂²×C₈₉	semidirect
338. #532	C₂²×C₈₉	semidirect
340. #529	C₂²×C₈₉	semidirect
346. #501	C₂²×C₈₉	semidirect
347. #501	C₂²×C₈₉	semidirect
348. #501	C₂²×C₈₉	semidirect
352. #490	C₂²×C₈₉	semidirect
353. #489	C₂²×C₈₉	semidirect
354. #488	C₂²×C₈₉	semidirect
355. #486	C₂²×C₈₉	semidirect
356. #469	C₂²×C₈₉	semidirect
357. #468	C₂²×C₈₉	semidirect
358. #466	C₂²×C₈₉	semidirect
359. #465	C₂²×C₈₉	semidirect
360. #442	C₂²×C₈₉	semidirect
361. #440	C₂²×C₈₉	semidirect
362. C₄×SD₁₆	C₂²×C₈₉	semidirect
363. C₄×D₈	C₂²×C₈₉	semidirect
371. #405	C₂²×C₈₉	semidirect
373. C₄×M₁₆	C₂²×C₈₉	semidirect
375. C₂×#97	C₂²×C₈₉	semidirect
505. C₂×C₈	#41×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×#3508	2
C₈₉×#3180	2
C₈₉×#3179	2
C₈₉×#3171	2
C₈₉×#3162	2
C₂×#496×C₈₉	2
C₈₉×#3140	2
C₈₉×#3138	2
C₈₉×#3058	2
C₈₉×#3057	2
C₈₉×#2949	2
C₈₉×#2948	2
C₈₉×#2875	2
#60719	89

Subgroups — C89×#60719 #36438567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60719 #36438567