Subgroups — C₈₉×#33651 #36411499

Order 22784.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3059	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2913	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2237	C₂	1	2	yes	maximal, complemented
4	C₈₉×#2237	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2236	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2235	C₂	1	2	yes	maximal, complemented
7	C₈₉×#2159	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#2158	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2141	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#358×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2113	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
14	#536×C₈₉	—	2	1	no	—
15	#520×C₈₉	—	2	1	no	—
16	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	#470×C₈₉	—	2	2	no	—
19	#467×C₈₉	C₂²	1	2	yes	—
20	#467×C₈₉	C₂²	1	2	yes	—
21	#428×C₈₉	—	2	1	no	—
22	#428×C₈₉	—	2	2	no	—
23	#427×C₈₉	C₂²	1	1	yes	characteristic
24	#427×C₈₉	C₂²	1	2	yes	—
25	#427×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#107×C₈₉	C₂²	1	2	yes	—
28	#419×C₈₉	—	2	1	no	—
29	#367×C₈₉	C₂²	1	1	yes	characteristic
30	#367×C₈₉	C₂²	1	1	yes	characteristic
31	#367×C₈₉	C₂²	1	2	yes	—
32	#366×C₈₉	C₂²	1	1	yes	characteristic
33	#366×C₈₉	C₂²	1	1	yes	characteristic
34	#366×C₈₉	C₂²	1	2	yes	—
35	#358×C₈₉	C₂²	1	2	yes	—
36	#358×C₈₉	C₂²	1	2	yes	—
37	#358×C₈₉	C₂²	1	2	yes	—
38	#358×C₈₉	C₂²	1	2	yes	—
39	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
40	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
41	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
42	C₂²×Q₈×C₈₉	—	2	1	no	—
43	C₂×Q₁₆×C₈₉	—	2	1	no	—
44	C₂×Q₁₆×C₈₉	—	2	2	no	—
45	C₂×Q₁₆×C₈₉	—	2	2	no	—
46	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
47	#131×C₈₉	—	2	2	no	—
48	#131×C₈₉	—	2	2	no	—
49	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
50	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
51	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
52	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₈₉	—	4	1	no	—
54	#124×C₈₉	—	4	1	no	—
55	#123×C₈₉	—	4	2	no	—
56	C₂²⋊Q₈×C₈₉	—	2	2	no	—
57	C₂²⋊Q₈×C₈₉	—	2	2	no	—
58	C₄×Q₈×C₈₉	—	4	1	no	—
59	C₄×D₄×C₈₉	—	4	2	no	—
60	C₄×D₄×C₈₉	—	4	1	no	—
61	#117×C₈₉	—	2	1	no	—
62	#117×C₈₉	—	2	2	no	—
63	#117×C₈₉	—	2	1	no	—
64	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
65	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
66	M₃₂×C₈₉	—	2	2	no	—
67	C₂×C₁₆×C₈₉	—	2	2	no	—
68	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
69	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
71	#107×C₈₉	—	2	2	no	—
72	#107×C₈₉	—	2	1	no	—
73	#107×C₈₉	C₂³	1	2	yes	—
74	#107×C₈₉	C₂³	1	1	yes	characteristic
75	#107×C₈₉	C₂³	1	1	yes	characteristic
76	#107×C₈₉	C₂³	1	1	yes	characteristic
77	#107×C₈₉	C₂³	1	1	yes	characteristic
78	#106×C₈₉	—	2	2	no	—
79	#106×C₈₉	—	2	1	no	—
80	#106×C₈₉	—	2	1	no	—
81	#103×C₈₉	—	2	2	no	—
82	#103×C₈₉	—	2	2	no	—
83	#41×C₈₉	—	2	2	no	—
84	#41×C₈₉	—	2	2	no	—
85	C₂×Q₈×C₈₉	—	2	2	no	—
86	C₂×Q₈×C₈₉	—	4	1	no	—
87	C₂×Q₈×C₈₉	—	4	2	no	—
88	C₂×Q₈×C₈₉	—	4	2	no	—
89	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
90	C₂×D₄×C₈₉	D₈	1	2	yes	—
91	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	—	4	1	no	—
93	C₂²×C₄×C₈₉	—	4	2	no	—
94	C₂²×C₄×C₈₉	—	4	1	no	—
95	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
97	Q₁₆×C₈₉	—	2	2	no	—
98	Q₁₆×C₈₉	—	2	2	no	—
99	Q₁₆×C₈₉	—	4	2	no	—
100	M₁₆×C₈₉	—	2	2	no	—
101	M₁₆×C₈₉	—	2	4	no	—
102	C₂×C₈×C₈₉	—	2	2	no	—
103	C₂×C₈×C₈₉	—	2	2	no	—
104	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	#41	1	2	yes	—
107	C₂×C₈×C₈₉	#41	1	2	yes	—
108	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₈₉	—	2	1	no	—
110	C₄⋊C₄×C₈₉	—	2	1	no	—
111	C₄⋊C₄×C₈₉	—	2	1	no	—
112	C₄⋊C₄×C₈₉	—	2	2	no	—
113	C₄⋊C₄×C₈₉	—	2	1	no	—
114	C₄⋊C₄×C₈₉	—	4	2	no	—
115	C₄⋊C₄×C₈₉	—	4	1	no	—
116	C₄⋊C₄×C₈₉	—	4	2	no	—
117	C₄⋊C₄×C₈₉	—	4	2	no	—
118	C₂²⋊C₄×C₈₉	—	4	2	no	—
119	C₂²⋊C₄×C₈₉	—	4	1	no	—
120	C₂²⋊C₄×C₈₉	—	4	1	no	—
121	C₂²⋊C₄×C₈₉	—	4	2	no	—
122	C₂²⋊C₄×C₈₉	—	4	2	no	—
123	C₄²×C₈₉	—	4	2	no	—
124	C₄²×C₈₉	—	4	1	no	—
125	C₄²×C₈₉	—	4	1	no	—
126	C₁₆×C₈₉	—	2	1	no	—
127	C₁₆×C₈₉	—	2	1	no	—
128	C₁₆×C₈₉	—	2	2	no	—
129	C₂³×C₈₉	C₂×D₈	1	2	yes	—
130	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
131	Q₈×C₈₉	—	2	2	no	—
132	Q₈×C₈₉	—	4	2	no	—
133	Q₈×C₈₉	—	4	2	no	—
134	Q₈×C₈₉	—	8	2	no	—
135	D₄×C₈₉	—	2	4	no	—
136	D₄×C₈₉	—	2	2	no	—
137	C₂×C₄×C₈₉	—	2	2	no	—
138	C₂×C₄×C₈₉	—	2	2	no	—
139	C₂×C₄×C₈₉	—	4	1	no	—
140	C₂×C₄×C₈₉	—	4	1	no	—
141	C₂×C₄×C₈₉	—	4	1	no	—
142	C₂×C₄×C₈₉	—	4	2	no	—
143	C₂×C₄×C₈₉	—	4	1	no	—
144	C₂×C₄×C₈₉	—	4	2	no	—
145	C₂×C₄×C₈₉	—	8	2	no	—
146	C₂×C₄×C₈₉	—	16	1	no	—
147	C₂×C₄×C₈₉	—	16	2	no	—
148	C₂×C₄×C₈₉	#123	1	2	yes	—
149	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
150	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₈₉	#137	1	2	yes	—
152	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
153	C₈×C₈₉	—	2	2	no	—
154	C₈×C₈₉	#123	1	2	yes	—
155	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
156	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
157	C₂²×C₈₉	—	2	1	no	—
158	C₂²×C₈₉	—	2	1	no	—
159	C₂²×C₈₉	—	2	2	no	—
160	C₂²×C₈₉	—	2	2	no	—
161	C₂²×C₈₉	—	2	2	no	—
162	C₂²×C₈₉	#480	1	2	yes	—
163	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
164	C₄×C₈₉	—	2	2	no	—
165	C₄×C₈₉	—	8	2	no	—
166	C₄×C₈₉	—	8	1	no	—
167	C₄×C₈₉	—	8	2	no	—
168	C₄×C₈₉	—	8	1	no	—
169	C₄×C₈₉	—	8	1	no	—
170	C₄×C₈₉	—	8	1	no	—
171	C₄×C₈₉	#480	1	2	yes	—
172	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
173	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
174	#33651	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
175	C₂×C₈₉	—	2	2	no	—
176	C₂×C₈₉	—	4	2	no	—
177	C₂×C₈₉	C₂×#480	1	1	yes	characteristic
178	C₂×C₈₉	#3416	1	1	yes	characteristic
179	C₂×C₈₉	#3417	1	1	yes	characteristic
180	#3059	C₂×C₈₉	1	1	yes	characteristic
181	#2913	C₂×C₈₉	1	1	yes	characteristic
182	#2237	C₂×C₈₉	1	2	yes	complemented
183	#2237	C₂×C₈₉	1	2	yes	complemented
184	#2236	C₂×C₈₉	1	2	yes	complemented
185	#2235	C₂×C₈₉	1	2	yes	complemented
186	#2159	C₂×C₈₉	1	1	yes	characteristic, complemented
187	#2158	C₂×C₈₉	1	1	yes	characteristic, complemented
188	#2141	C₂×C₈₉	1	1	yes	characteristic, complemented
189	C₂×#358	C₂×C₈₉	1	1	yes	characteristic, complemented
190	#2113	C₂×C₈₉	1	1	yes	characteristic
191	C₈₉	#33651	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²×Q₁₆	C₂²×C₈₉	1	1	yes	characteristic
193	C₂×#131	C₂²×C₈₉	1	1	yes	characteristic
194	#536	—	2	1	no	—
195	#520	—	2	1	no	—
196	C₂×M₃₂	C₂²×C₈₉	1	1	yes	characteristic
197	C₂²×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
198	#470	—	2	2	no	—
199	#467	C₂²×C₈₉	1	2	yes	—
200	#467	C₂²×C₈₉	1	2	yes	—
201	#428	—	2	1	no	—
202	#428	—	2	2	no	—
203	#427	C₂²×C₈₉	1	1	yes	characteristic
204	#427	C₂²×C₈₉	1	2	yes	—
205	#427	C₂²×C₈₉	1	1	yes	characteristic
206	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
207	C₂×#107	C₂²×C₈₉	1	2	yes	—
208	#419	—	2	1	no	—
209	#367	C₂²×C₈₉	1	1	yes	characteristic
210	#367	C₂²×C₈₉	1	1	yes	characteristic
211	#367	C₂²×C₈₉	1	2	yes	—
212	#366	C₂²×C₈₉	1	1	yes	characteristic
213	#366	C₂²×C₈₉	1	1	yes	characteristic
214	#366	C₂²×C₈₉	1	2	yes	—
215	#358	C₂²×C₈₉	1	2	yes	—
216	#358	C₂²×C₈₉	1	2	yes	—
217	#358	C₂²×C₈₉	1	2	yes	—
218	#358	C₂²×C₈₉	1	2	yes	—
219	C₂²⋊C₁₆	C₂²×C₈₉	1	2	yes	—
220	C₂²⋊C₁₆	C₂²×C₈₉	1	2	yes	—
221	C₂×#41	D₄×C₈₉	1	1	yes	characteristic
222	C₂²×Q₈	—	2	1	no	—
223	C₂×Q₁₆	—	2	1	no	—
224	C₂×Q₁₆	—	2	2	no	—
225	C₂×Q₁₆	—	2	2	no	—
226	C₂×Q₁₆	C₂³×C₈₉	1	2	yes	—
227	#131	—	2	2	no	—
228	#131	—	2	2	no	—
229	C₂×M₁₆	D₄×C₈₉	1	1	yes	characteristic
230	C₂×M₁₆	D₄×C₈₉	1	2	yes	—
231	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
232	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
233	C₄⋊Q₈	—	4	1	no	—
234	#124	—	4	1	no	—
235	#123	—	4	2	no	—
236	C₂²⋊Q₈	—	2	2	no	—
237	C₂²⋊Q₈	—	2	2	no	—
238	C₄×Q₈	—	4	1	no	—
239	C₄×D₄	—	4	2	no	—
240	C₄×D₄	—	4	1	no	—
241	#117	—	2	2	no	—
242	#117	—	2	1	no	—
243	#117	—	2	1	no	—
244	C₂×C₄⋊C₄	—	2	2	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	M₃₂	—	2	2	no	—
247	C₂×C₁₆	—	2	2	no	—
248	C₂×C₁₆	C₂³×C₈₉	1	1	yes	characteristic
249	C₂×C₁₆	C₂³×C₈₉	1	2	yes	—
250	C₂×C₁₆	C₂³×C₈₉	1	1	yes	characteristic
251	#107	—	2	2	no	—
252	#107	—	2	1	no	—
253	#107	C₂³×C₈₉	1	2	yes	—
254	#107	C₂³×C₈₉	1	1	yes	characteristic
255	#107	C₂³×C₈₉	1	1	yes	characteristic
256	#107	C₂³×C₈₉	1	1	yes	characteristic
257	#107	C₂³×C₈₉	1	1	yes	characteristic
258	#106	—	2	2	no	—
259	#106	—	2	1	no	—
260	#106	—	2	1	no	—
261	#103	—	2	2	no	—
262	#103	—	2	2	no	—
263	#41	—	2	2	no	—
264	#41	—	2	2	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×Q₈	—	4	2	no	—
268	C₂×Q₈	—	4	2	no	—
269	C₂×Q₈	D₈×C₈₉	1	1	yes	characteristic
270	C₂×D₄	D₈×C₈₉	1	2	yes	—
271	C₂×D₄	D₈×C₈₉	1	1	yes	characteristic
272	C₂²×C₄	—	4	1	no	—
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	—	4	1	no	—
275	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
276	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
277	Q₁₆	—	2	2	no	—
278	Q₁₆	—	2	2	no	—
279	Q₁₆	—	4	2	no	—
280	M₁₆	—	2	2	no	—
281	M₁₆	—	2	4	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
285	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
286	C₂×C₈	#41×C₈₉	1	2	yes	—
287	C₂×C₈	#41×C₈₉	1	2	yes	—
288	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	4	2	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	4	2	no	—
297	C₄⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	4	2	no	—
299	C₂²⋊C₄	—	4	1	no	—
300	C₂²⋊C₄	—	4	1	no	—
301	C₂²⋊C₄	—	4	2	no	—
302	C₂²⋊C₄	—	4	2	no	—
303	C₄²	—	4	2	no	—
304	C₄²	—	4	1	no	—
305	C₄²	—	4	1	no	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	2	2	no	—
309	C₂³	C₂×D₈×C₈₉	1	2	yes	—
310	C₂³	C₂×D₈×C₈₉	1	1	yes	characteristic
311	Q₈	—	2	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	8	2	no	—
315	D₄	—	2	4	no	—
316	D₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	8	2	no	—
326	C₂×C₄	—	16	1	no	—
327	C₂×C₄	—	16	2	no	—
328	C₂×C₄	#123×C₈₉	1	2	yes	—
329	C₂×C₄	C₂×D₈×C₈₉	1	2	yes	—
330	C₂×C₄	C₂×D₈×C₈₉	1	1	yes	characteristic
331	C₂×C₄	#137×C₈₉	1	2	yes	—
332	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	#123×C₈₉	1	2	yes	—
335	C₈	C₂×#41×C₈₉	1	1	yes	characteristic
336	C₈	C₂×#41×C₈₉	1	1	yes	characteristic
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	#480×C₈₉	1	2	yes	—
343	C₂²	C₂²×D₈×C₈₉	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	2	no	—
347	C₄	—	8	1	no	—
348	C₄	—	8	1	no	—
349	C₄	—	8	1	no	—
350	C₄	—	8	1	no	—
351	C₄	#480×C₈₉	1	2	yes	—
352	C₄	C₂×#123×C₈₉	1	1	yes	characteristic
353	C₄	C₂×#137×C₈₉	1	1	yes	characteristic
354	C₂	—	2	2	no	—
355	C₂	—	4	2	no	—
356	C₂	C₂×#480×C₈₉	1	1	yes	characteristic
357	C₂	C₈₉×#3416	1	1	yes	characteristic
358	C₂	C₈₉×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₈₉
prime radical	C₂×#41×C₈₉	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×#41×C₈₉ ◁ C₂×#131×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₈₉×#2237	C₂	semidirect
4. C₈₉×#2237	C₂	semidirect
5. C₈₉×#2236	C₂	semidirect
6. C₈₉×#2235	C₂	semidirect
7. C₈₉×#2159	C₂	semidirect
8. C₈₉×#2158	C₂	semidirect
9. C₈₉×#2141	C₂	semidirect
10. C₂×#358×C₈₉	C₂	semidirect
174. #33651	C₈₉	direct (N × H)
182. #2237	C₂×C₈₉	semidirect
183. #2237	C₂×C₈₉	semidirect
184. #2236	C₂×C₈₉	semidirect
185. #2235	C₂×C₈₉	semidirect
186. #2159	C₂×C₈₉	semidirect
187. #2158	C₂×C₈₉	semidirect
188. #2141	C₂×C₈₉	semidirect
189. C₂×#358	C₂×C₈₉	semidirect
191. C₈₉	#33651	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈₉×#3059	2
C₈₉×#2913	2
C₈₉×#2237	2
C₈₉×#2236	2
C₈₉×#2235	2
C₈₉×#2159	2
C₈₉×#2158	2
C₈₉×#2141	2
C₂×#358×C₈₉	2
C₈₉×#2113	2
#33651	89

Subgroups — C89×#33651 #36411499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#33651 #36411499