Subgroups — C₈₉×#10444 #36388292

Order 22784.

440 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#461×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1800	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1571	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#1546	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1546	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#1478	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₈₉	—	2	1	no	—
9	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
11	#461×C₈₉	—	2	2	no	—
12	#461×C₈₉	—	2	2	no	—
13	#431×C₈₉	C₂²	1	1	yes	characteristic
14	#431×C₈₉	C₂²	1	1	yes	characteristic
15	#423×C₈₉	C₂²	1	1	yes	characteristic
16	#423×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×#103×C₈₉	—	2	1	no	—
19	C₂×#100×C₈₉	—	2	1	no	—
20	#343×C₈₉	—	2	1	no	—
21	#335×C₈₉	—	2	2	no	—
22	#332×C₈₉	—	2	2	no	—
23	#332×C₈₉	—	2	2	no	—
24	D₄⋊C₈×C₈₉	—	2	2	no	—
25	C₂²×Q₈×C₈₉	—	2	1	no	—
26	C₂²×D₄×C₈₉	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₈₉	—	2	1	no	—
28	C₂×SD₁₆×C₈₉	—	2	4	no	—
29	C₂×SD₁₆×C₈₉	—	4	2	no	—
30	C₂×SD₁₆×C₈₉	—	4	2	no	—
31	C₂×M₁₆×C₈₉	—	2	1	no	—
32	C₂×M₁₆×C₈₉	—	2	1	no	—
33	C₂²×C₈×C₈₉	—	2	1	no	—
34	C₄⋊Q₈×C₈₉	—	2	1	no	—
35	C₄⋊Q₈×C₈₉	—	2	1	no	—
36	C₄⋊Q₈×C₈₉	—	2	1	no	—
37	C₄⋊Q₈×C₈₉	D₄	1	2	yes	—
38	C₄×D₄×C₈₉	—	2	1	no	—
39	C₄×D₄×C₈₉	—	2	1	no	—
40	C₄×D₄×C₈₉	—	2	1	no	—
41	C₄×D₄×C₈₉	—	2	1	no	—
42	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
43	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
44	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
45	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
46	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₈₉	—	2	1	no	—
48	C₄⋊C₈×C₈₉	—	2	1	no	—
49	C₄⋊C₈×C₈₉	—	2	1	no	—
50	C₄⋊C₈×C₈₉	—	2	1	no	—
51	C₄⋊C₈×C₈₉	—	2	1	no	—
52	C₄⋊C₈×C₈₉	—	2	1	no	—
53	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
54	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
55	#103×C₈₉	—	2	4	no	—
56	#100×C₈₉	—	4	2	no	—
57	C₂²⋊C₈×C₈₉	—	2	1	no	—
58	C₂²⋊C₈×C₈₉	—	2	1	no	—
59	C₄×C₈×C₈₉	—	2	1	no	—
60	C₄×C₈×C₈₉	—	2	1	no	—
61	C₂⁴×C₈₉	—	2	1	no	—
62	C₂×Q₈×C₈₉	—	2	2	no	—
63	C₂×Q₈×C₈₉	—	4	1	no	—
64	C₂×Q₈×C₈₉	—	4	1	no	—
65	C₂×Q₈×C₈₉	—	4	1	no	—
66	C₂×Q₈×C₈₉	—	4	1	no	—
67	C₂×Q₈×C₈₉	—	4	1	no	—
68	C₂×D₄×C₈₉	—	2	1	no	—
69	C₂×D₄×C₈₉	—	2	1	no	—
70	C₂×D₄×C₈₉	—	2	1	no	—
71	C₂×D₄×C₈₉	—	2	1	no	—
72	C₂×D₄×C₈₉	—	2	1	no	—
73	C₂×D₄×C₈₉	D₈	1	2	yes	—
74	C₂²×C₄×C₈₉	—	2	1	no	—
75	C₂²×C₄×C₈₉	—	4	1	no	—
76	C₂²×C₄×C₈₉	—	4	1	no	—
77	C₂²×C₄×C₈₉	—	4	1	no	—
78	C₂²×C₄×C₈₉	—	8	2	no	—
79	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
82	SD₁₆×C₈₉	—	4	8	no	—
83	M₁₆×C₈₉	—	4	1	no	—
84	M₁₆×C₈₉	—	4	1	no	—
85	M₁₆×C₈₉	—	4	1	no	—
86	M₁₆×C₈₉	—	4	1	no	—
87	C₂×C₈×C₈₉	—	2	2	no	—
88	C₂×C₈×C₈₉	—	2	2	no	—
89	C₂×C₈×C₈₉	—	4	1	no	—
90	C₂×C₈×C₈₉	—	4	1	no	—
91	C₂×C₈×C₈₉	—	4	1	no	—
92	C₂×C₈×C₈₉	—	4	1	no	—
93	C₂×C₈×C₈₉	—	4	1	no	—
94	C₄⋊C₄×C₈₉	—	2	1	no	—
95	C₄⋊C₄×C₈₉	—	2	2	no	—
96	C₄⋊C₄×C₈₉	—	2	1	no	—
97	C₄⋊C₄×C₈₉	—	2	2	no	—
98	C₄⋊C₄×C₈₉	—	4	1	no	—
99	C₄⋊C₄×C₈₉	—	4	1	no	—
100	C₂²⋊C₄×C₈₉	—	4	1	no	—
101	C₂²⋊C₄×C₈₉	—	4	1	no	—
102	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₈₉	#41	1	1	yes	characteristic
106	C₂³×C₈₉	—	2	2	no	—
107	C₂³×C₈₉	—	4	1	no	—
108	C₂³×C₈₉	—	4	1	no	—
109	C₂³×C₈₉	—	8	1	no	—
110	C₂³×C₈₉	—	8	1	no	—
111	C₂³×C₈₉	#123	1	1	yes	characteristic
112	Q₈×C₈₉	—	4	2	no	—
113	Q₈×C₈₉	—	4	2	no	—
114	Q₈×C₈₉	—	8	1	no	—
115	Q₈×C₈₉	—	8	1	no	—
116	D₄×C₈₉	—	2	2	no	—
117	D₄×C₈₉	—	2	2	no	—
118	D₄×C₈₉	—	4	1	no	—
119	D₄×C₈₉	—	4	1	no	—
120	C₂×C₄×C₈₉	—	2	2	no	—
121	C₂×C₄×C₈₉	—	4	1	no	—
122	C₂×C₄×C₈₉	—	4	1	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	8	2	no	—
125	C₂×C₄×C₈₉	—	8	1	no	—
126	C₂×C₄×C₈₉	—	8	1	no	—
127	C₂×C₄×C₈₉	—	8	2	no	—
128	C₂×C₄×C₈₉	—	8	1	no	—
129	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
130	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
131	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
132	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
133	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
134	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
137	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
139	C₈×C₈₉	—	4	2	no	—
140	C₈×C₈₉	—	4	2	no	—
141	C₈×C₈₉	—	8	1	no	—
142	C₈×C₈₉	—	8	1	no	—
143	C₈×C₈₉	—	8	1	no	—
144	C₂²×C₈₉	—	2	1	no	—
145	C₂²×C₈₉	—	2	1	no	—
146	C₂²×C₈₉	—	4	2	no	—
147	C₂²×C₈₉	—	8	2	no	—
148	C₂²×C₈₉	—	8	2	no	—
149	C₂²×C₈₉	—	8	1	no	—
150	C₂²×C₈₉	—	8	1	no	—
151	C₂²×C₈₉	#394	1	1	yes	characteristic
152	C₂²×C₈₉	#480	1	1	yes	characteristic
153	C₂²×C₈₉	#482	1	1	yes	characteristic
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	8	2	no	—
158	C₄×C₈₉	—	8	1	no	—
159	C₄×C₈₉	#447	1	1	yes	characteristic
160	C₄×C₈₉	#452	1	1	yes	characteristic
161	C₄×C₈₉	#466	1	1	yes	characteristic
162	C₄×C₈₉	#468	1	1	yes	characteristic
163	C₄×C₈₉	#471	1	2	yes	—
164	#10444	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂×C₈₉	—	2	1	no	—
166	C₂×C₈₉	—	2	1	no	—
167	C₂×C₈₉	—	8	2	no	—
168	C₂×C₈₉	#2033	1	1	yes	characteristic
169	C₂×C₈₉	#2046	1	1	yes	characteristic
170	C₂×C₈₉	#2049	1	1	yes	characteristic
171	C₂×#461	C₂×C₈₉	1	1	yes	characteristic
172	#1800	C₂×C₈₉	1	1	yes	characteristic
173	#1571	C₂×C₈₉	1	1	yes	characteristic, complemented
174	#1546	C₂×C₈₉	1	1	yes	characteristic, complemented
175	#1546	C₂×C₈₉	1	1	yes	characteristic, complemented
176	#1478	C₂×C₈₉	1	1	yes	characteristic
177	#1301	C₂×C₈₉	1	1	yes	characteristic, complemented
178	C₈₉	#10444	1	1	yes	characteristic, direct product factor, Sylow
179	C₂²×SD₁₆	—	2	1	no	—
180	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
181	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
182	#461	—	2	2	no	—
183	#461	—	2	2	no	—
184	#431	C₂²×C₈₉	1	1	yes	characteristic
185	#431	C₂²×C₈₉	1	1	yes	characteristic
186	#423	C₂²×C₈₉	1	1	yes	characteristic
187	#423	C₂²×C₈₉	1	1	yes	characteristic
188	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
189	C₂×#103	—	2	1	no	—
190	C₂×#100	—	2	1	no	—
191	#343	—	2	1	no	—
192	#335	—	2	2	no	—
193	#332	—	2	2	no	—
194	#332	—	2	2	no	—
195	D₄⋊C₈	—	2	2	no	—
196	C₂²×Q₈	—	2	1	no	—
197	C₂²×D₄	D₄×C₈₉	1	1	yes	characteristic
198	C₂³×C₄	—	2	1	no	—
199	C₂×SD₁₆	—	2	4	no	—
200	C₂×SD₁₆	—	4	2	no	—
201	C₂×SD₁₆	—	4	2	no	—
202	C₂×M₁₆	—	2	1	no	—
203	C₂×M₁₆	—	2	1	no	—
204	C₂²×C₈	—	2	1	no	—
205	C₄⋊Q₈	—	2	1	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	D₄×C₈₉	1	2	yes	—
209	C₄×D₄	—	2	1	no	—
210	C₄×D₄	—	2	1	no	—
211	C₄×D₄	—	2	1	no	—
212	C₄×D₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	—	2	1	no	—
217	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	2	1	no	—
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	D₄×C₈₉	1	2	yes	—
225	C₄⋊C₈	D₄×C₈₉	1	2	yes	—
226	#103	—	2	4	no	—
227	#100	—	4	2	no	—
228	C₂²⋊C₈	—	2	1	no	—
229	C₂²⋊C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₂⁴	—	2	1	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	D₈×C₈₉	1	2	yes	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	8	2	no	—
250	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
251	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
252	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
253	SD₁₆	—	4	8	no	—
254	M₁₆	—	4	1	no	—
255	M₁₆	—	4	1	no	—
256	M₁₆	—	4	1	no	—
257	M₁₆	—	4	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	4	1	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₂²⋊C₄	—	4	1	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
275	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
276	C₄²	#41×C₈₉	1	1	yes	characteristic
277	C₂³	—	2	2	no	—
278	C₂³	—	4	1	no	—
279	C₂³	—	4	1	no	—
280	C₂³	—	8	1	no	—
281	C₂³	—	8	1	no	—
282	C₂³	#123×C₈₉	1	1	yes	characteristic
283	Q₈	—	4	2	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	8	1	no	—
286	Q₈	—	8	1	no	—
287	D₄	—	2	2	no	—
288	D₄	—	2	2	no	—
289	D₄	—	4	1	no	—
290	D₄	—	4	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	8	2	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
301	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
302	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
303	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
304	C₂×C₄	#123×C₈₉	1	1	yes	characteristic
305	C₂×C₄	#124×C₈₉	1	1	yes	characteristic
306	C₂×C₄	C₂×D₈×C₈₉	1	1	yes	characteristic
307	C₂×C₄	#135×C₈₉	1	1	yes	characteristic
308	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
309	C₂×C₄	#137×C₈₉	1	1	yes	characteristic
310	C₈	—	4	2	no	—
311	C₈	—	4	2	no	—
312	C₈	—	8	1	no	—
313	C₈	—	8	1	no	—
314	C₈	—	8	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	#394×C₈₉	1	1	yes	characteristic
323	C₂²	#480×C₈₉	1	1	yes	characteristic
324	C₂²	#482×C₈₉	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	8	1	no	—
329	C₄	—	8	2	no	—
330	C₄	#447×C₈₉	1	1	yes	characteristic
331	C₄	#452×C₈₉	1	1	yes	characteristic
332	C₄	#466×C₈₉	1	1	yes	characteristic
333	C₄	#468×C₈₉	1	1	yes	characteristic
334	C₄	#471×C₈₉	1	2	yes	—
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	8	2	no	—
338	C₂	C₈₉×#2033	1	1	yes	characteristic
339	C₂	C₈₉×#2046	1	1	yes	characteristic
340	C₂	C₈₉×#2049	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
prime radical	C₂²×D₄×C₈₉	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₈₉×#1571	C₂	semidirect
4. C₈₉×#1546	C₂	semidirect
5. C₈₉×#1546	C₂	semidirect
7. C₈₉×#1301	C₂	semidirect
164. #10444	C₈₉	direct (N × H)
173. #1571	C₂×C₈₉	semidirect
174. #1546	C₂×C₈₉	semidirect
175. #1546	C₂×C₈₉	semidirect
177. #1301	C₂×C₈₉	semidirect
178. C₈₉	#10444	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#461×C₈₉	2
C₈₉×#1800	2
C₈₉×#1571	2
C₈₉×#1546	2
C₈₉×#1478	2
C₈₉×#1301	2
#10444	89

Subgroups — C89×#10444 #36388292

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#10444 #36388292