Subgroups — C₈₉×#8706 #36386554

Order 22784.

680 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1797	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂²⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1279	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄⋊C₄×C₈₉	C₄	1	1	yes	characteristic
6	C₂²×C₄⋊C₄×C₈₉	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₈₉	—	2	1	no	—
8	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
10	C₂×C₂²⋊C₈×C₈₉	—	2	1	no	—
11	C₂×C₂²⋊C₈×C₈₉	—	2	1	no	—
12	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
13	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
14	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
15	C₄×C₄⋊C₄×C₈₉	C₄	1	2	yes	—
16	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
17	C₂×#95×C₈₉	—	2	1	no	—
18	#336×C₈₉	—	2	1	no	—
19	#336×C₈₉	—	2	1	no	—
20	C₂³×C₄×C₈₉	—	2	1	no	—
21	C₂³×C₄×C₈₉	—	2	1	no	—
22	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₈₉	—	2	2	no	—
26	C₂²×C₈×C₈₉	—	2	2	no	—
27	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
31	C₂×C₄⋊C₄×C₈₉	C₈	1	4	yes	complemented
32	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₈₉	—	2	4	no	—
35	C₂×C₄²×C₈₉	—	2	2	no	—
36	C₂×C₄²×C₈₉	—	2	2	no	—
37	C₂×C₄²×C₈₉	—	4	2	no	—
38	C₂×C₄²×C₈₉	—	4	2	no	—
39	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₈₉	—	2	4	no	—
44	C₂²⋊C₈×C₈₉	—	2	4	no	—
45	C₂²⋊C₈×C₈₉	—	4	2	no	—
46	C₄×C₈×C₈₉	—	2	4	no	—
47	#95×C₈₉	—	2	2	no	—
48	#95×C₈₉	—	2	2	no	—
49	C₂⁴×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	—	2	2	no	—
51	C₂²×C₄×C₈₉	—	2	2	no	—
52	C₂²×C₄×C₈₉	—	2	2	no	—
53	C₂²×C₄×C₈₉	—	2	2	no	—
54	C₂²×C₄×C₈₉	—	2	4	no	—
55	C₂²×C₄×C₈₉	—	2	4	no	—
56	C₂²×C₄×C₈₉	—	2	2	no	—
57	C₂²×C₄×C₈₉	—	2	2	no	—
58	C₂²×C₄×C₈₉	—	2	2	no	—
59	C₂²×C₄×C₈₉	—	2	2	no	—
60	C₂²×C₄×C₈₉	—	4	1	no	—
61	C₂²×C₄×C₈₉	—	4	2	no	—
62	C₂²×C₄×C₈₉	—	4	2	no	—
63	C₂²×C₄×C₈₉	—	4	2	no	—
64	C₂²×C₄×C₈₉	—	4	2	no	—
65	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
69	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
70	C₂²×C₄×C₈₉	D₈	1	2	yes	—
71	C₂²×C₄×C₈₉	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₈₉	—	2	4	no	—
76	C₂×C₈×C₈₉	—	4	2	no	—
77	C₂×C₈×C₈₉	—	4	2	no	—
78	C₂×C₈×C₈₉	—	4	2	no	—
79	C₂×C₈×C₈₉	—	4	2	no	—
80	C₂×C₈×C₈₉	—	4	2	no	—
81	C₄⋊C₄×C₈₉	—	2	4	no	—
82	C₄⋊C₄×C₈₉	—	2	4	no	—
83	C₄⋊C₄×C₈₉	—	2	2	no	—
84	C₄⋊C₄×C₈₉	—	2	2	no	—
85	C₄⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
86	C₄⋊C₄×C₈₉	M₁₆	1	4	yes	—
87	C₄²×C₈₉	—	2	2	no	—
88	C₄²×C₈₉	—	2	2	no	—
89	C₄²×C₈₉	—	2	2	no	—
90	C₄²×C₈₉	—	4	4	no	—
91	C₄²×C₈₉	—	4	2	no	—
92	C₄²×C₈₉	—	4	2	no	—
93	C₄²×C₈₉	#41	1	4	yes	—
94	C₂³×C₈₉	—	2	1	no	—
95	C₂³×C₈₉	—	2	1	no	—
96	C₂³×C₈₉	—	2	1	no	—
97	C₂³×C₈₉	—	2	1	no	—
98	C₂³×C₈₉	C₂²⋊C₈	1	2	yes	—
99	C₂³×C₈₉	#102	1	2	yes	—
100	C₂³×C₈₉	#104	1	2	yes	—
101	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₈₉	—	2	1	no	—
103	C₂×C₄×C₈₉	—	2	1	no	—
104	C₂×C₄×C₈₉	—	2	1	no	—
105	C₂×C₄×C₈₉	—	2	1	no	—
106	C₂×C₄×C₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	—	2	4	no	—
109	C₂×C₄×C₈₉	—	2	4	no	—
110	C₂×C₄×C₈₉	—	2	4	no	—
111	C₂×C₄×C₈₉	—	2	4	no	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	4	no	—
114	C₂×C₄×C₈₉	—	2	4	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	4	4	no	—
118	C₂×C₄×C₈₉	—	4	8	no	—
119	C₂×C₄×C₈₉	—	4	8	no	—
120	C₂×C₄×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	—	4	2	no	—
122	C₂×C₄×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
125	C₂×C₄×C₈₉	#102	1	2	yes	—
126	C₂×C₄×C₈₉	#117	1	2	yes	—
127	C₂×C₄×C₈₉	#123	1	2	yes	—
128	C₂×C₄×C₈₉	#123	1	2	yes	—
129	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
131	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₈₉	#137	1	2	yes	—
135	C₈×C₈₉	—	4	4	no	—
136	C₈×C₈₉	—	8	2	no	—
137	C₂²×C₈₉	—	2	2	no	—
138	C₂²×C₈₉	—	2	2	no	—
139	C₂²×C₈₉	—	2	2	no	—
140	C₂²×C₈₉	—	2	2	no	—
141	C₂²×C₈₉	—	2	2	no	—
142	C₂²×C₈₉	—	2	2	no	—
143	C₂²×C₈₉	D₄⋊C₈	1	4	yes	—
144	C₂²×C₈₉	#381	1	1	yes	characteristic
145	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
146	C₂²×C₈₉	#407	1	1	yes	characteristic
147	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
148	C₂²×C₈₉	#419	1	1	yes	characteristic
149	C₂²×C₈₉	C₂×#104	1	1	yes	characteristic
150	C₂²×C₈₉	#421	1	1	yes	characteristic
151	C₄×C₈₉	—	2	2	no	—
152	C₄×C₈₉	—	2	2	no	—
153	C₄×C₈₉	—	2	2	no	—
154	C₄×C₈₉	—	4	1	no	—
155	C₄×C₈₉	—	4	4	no	—
156	C₄×C₈₉	—	4	4	no	—
157	C₄×C₈₉	#431	1	2	yes	—
158	C₄×C₈₉	#432	1	2	yes	—
159	C₄×C₈₉	#480	1	2	yes	—
160	C₄×C₈₉	#483	1	2	yes	—
161	#8706	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂×C₈₉	—	2	4	no	—
163	C₂×C₈₉	C₂×D₄⋊C₈	1	1	yes	characteristic
164	C₂×C₈₉	#1479	1	1	yes	characteristic
165	C₂×C₈₉	#1486	1	1	yes	characteristic
166	C₂×C₈₉	#1491	1	1	yes	characteristic
167	C₂×C₈₉	#1817	1	1	yes	characteristic
168	C₂×C₈₉	#1823	1	1	yes	characteristic
169	C₂×C₈₉	#1826	1	1	yes	characteristic
170	C₂×C₄×C₄⋊C₄	C₂×C₈₉	1	1	yes	characteristic
171	#1797	C₂×C₈₉	1	1	yes	characteristic
172	C₄×C₂²⋊C₈	C₂×C₈₉	1	1	yes	characteristic
173	#1279	C₂×C₈₉	1	4	yes	complemented
174	C₈₉	#8706	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²×C₄⋊C₄	C₄×C₈₉	1	1	yes	characteristic
176	C₂²×C₄⋊C₄	C₄×C₈₉	1	1	yes	characteristic
177	C₂²×C₄²	—	2	1	no	—
178	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
179	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
180	C₂×C₂²⋊C₈	—	2	1	no	—
181	C₂×C₂²⋊C₈	—	2	1	no	—
182	C₂×C₄×C₈	C₂²×C₈₉	1	2	yes	—
183	C₄×C₄⋊C₄	—	2	1	no	—
184	C₄×C₄⋊C₄	—	2	1	no	—
185	C₄×C₄⋊C₄	C₄×C₈₉	1	2	yes	—
186	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
187	C₂×#95	—	2	1	no	—
188	#336	—	2	1	no	—
189	#336	—	2	1	no	—
190	C₂³×C₄	—	2	1	no	—
191	C₂³×C₄	—	2	1	no	—
192	C₂³×C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
193	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
194	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	C₈×C₈₉	1	4	yes	complemented
202	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	2	yes	—
203	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	2	yes	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	—	4	2	no	—
208	C₂×C₄²	—	4	2	no	—
209	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
210	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
211	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
212	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
213	C₄⋊C₈	—	2	4	no	—
214	C₂²⋊C₈	—	2	4	no	—
215	C₂²⋊C₈	—	4	2	no	—
216	C₄×C₈	—	2	4	no	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	C₂⁴	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
236	C₂²×C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
237	C₂²×C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₈×C₈₉	1	2	yes	—
239	C₂²×C₄	M₁₆×C₈₉	1	2	yes	—
240	C₂²×C₄	D₈×C₈₉	1	2	yes	—
241	C₂²×C₄	SD₁₆×C₈₉	1	2	yes	—
242	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	C₂×C₈×C₈₉	1	4	yes	—
256	C₄⋊C₄	M₁₆×C₈₉	1	4	yes	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	4	4	no	—
261	C₄²	—	4	2	no	—
262	C₄²	—	4	2	no	—
263	C₄²	#41×C₈₉	1	4	yes	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	C₂²⋊C₈×C₈₉	1	2	yes	—
269	C₂³	#102×C₈₉	1	2	yes	—
270	C₂³	#104×C₈₉	1	2	yes	—
271	C₂³	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	8	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	8	no	—
294	C₂×C₄	C₂²⋊C₈×C₈₉	1	2	yes	—
295	C₂×C₄	#102×C₈₉	1	2	yes	—
296	C₂×C₄	#117×C₈₉	1	2	yes	—
297	C₂×C₄	#123×C₈₉	1	2	yes	—
298	C₂×C₄	#123×C₈₉	1	2	yes	—
299	C₂×C₄	C₂²×C₈×C₈₉	1	1	yes	characteristic
300	C₂×C₄	C₂×M₁₆×C₈₉	1	2	yes	—
301	C₂×C₄	C₂×M₁₆×C₈₉	1	1	yes	characteristic
302	C₂×C₄	C₂×D₈×C₈₉	1	1	yes	characteristic
303	C₂×C₄	C₂×SD₁₆×C₈₉	1	1	yes	characteristic
304	C₂×C₄	#137×C₈₉	1	2	yes	—
305	C₈	—	4	4	no	—
306	C₈	—	8	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	D₄⋊C₈×C₈₉	1	4	yes	—
314	C₂²	#381×C₈₉	1	1	yes	characteristic
315	C₂²	C₂×C₂²⋊C₈×C₈₉	1	1	yes	characteristic
316	C₂²	#407×C₈₉	1	1	yes	characteristic
317	C₂²	C₂×#102×C₈₉	1	1	yes	characteristic
318	C₂²	#419×C₈₉	1	1	yes	characteristic
319	C₂²	C₂×#104×C₈₉	1	1	yes	characteristic
320	C₂²	#421×C₈₉	1	1	yes	characteristic
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	4	no	—
326	C₄	—	4	4	no	—
327	C₄	#431×C₈₉	1	2	yes	—
328	C₄	#432×C₈₉	1	2	yes	—
329	C₄	#480×C₈₉	1	2	yes	—
330	C₄	#483×C₈₉	1	2	yes	—
331	C₂	—	2	4	no	—
332	C₂	C₂×D₄⋊C₈×C₈₉	1	1	yes	characteristic
333	C₂	C₈₉×#1479	1	1	yes	characteristic
334	C₂	C₈₉×#1486	1	1	yes	characteristic
335	C₂	C₈₉×#1491	1	1	yes	characteristic
336	C₂	C₈₉×#1817	1	1	yes	characteristic
337	C₂	C₈₉×#1823	1	1	yes	characteristic
338	C₂	C₈₉×#1826	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₈₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₈₉
Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
socle	C₂³×C₈₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₈₉	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₈₉ ◁ C₂²×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₈₉×#1279	C₂	semidirect
31. C₂×C₄⋊C₄×C₈₉	C₈	semidirect
161. #8706	C₈₉	direct (N × H)
173. #1279	C₂×C₈₉	semidirect
174. C₈₉	#8706	direct (N × H)
201. C₂×C₄⋊C₄	C₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄⋊C₄×C₈₉	2
C₈₉×#1797	2
C₄×C₂²⋊C₈×C₈₉	2
C₈₉×#1279	2
#8706	89

Subgroups — C89×#8706 #36386554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#8706 #36386554