Subgroups — C₈₉×#29223 #36407071

Order 22784.

678 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2859	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2100	C₂	1	2	yes	maximal
4	C₈₉×#2099	C₂	1	2	yes	maximal
5	C₈₉×#1993	C₂	1	2	yes	maximal
6	C₈₉×#1989	C₂	1	4	yes	maximal
7	C₈₉×#1836	C₂	1	2	yes	maximal
8	C₈₉×#1767	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×#125×C₈₉	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
12	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
13	C₂×#97×C₈₉	C₂²	1	2	yes	—
14	#391×C₈₉	—	2	1	no	—
15	#389×C₈₉	—	2	1	no	—
16	#389×C₈₉	—	2	1	no	—
17	#389×C₈₉	—	2	1	no	—
18	#389×C₈₉	—	2	1	no	—
19	#389×C₈₉	—	2	1	no	—
20	#389×C₈₉	C₂²	1	2	yes	—
21	#384×C₈₉	C₂²	1	1	yes	characteristic
22	#384×C₈₉	C₂²	1	1	yes	characteristic
23	#384×C₈₉	C₂²	1	1	yes	characteristic
24	#382×C₈₉	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
26	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
27	C₄³×C₈₉	C₂²	1	1	yes	characteristic
28	#340×C₈₉	C₂²	1	4	yes	—
29	#340×C₈₉	C₂²	1	4	yes	—
30	#340×C₈₉	C₂²	1	4	yes	—
31	#340×C₈₉	C₂²	1	4	yes	—
32	#336×C₈₉	C₂²	1	2	yes	—
33	C₂²×Q₈×C₈₉	—	2	1	no	—
34	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
35	C₄⋊Q₈×C₈₉	—	2	2	no	—
36	C₄⋊Q₈×C₈₉	Q₈	1	4	yes	—
37	#125×C₈₉	—	2	8	no	—
38	C₄×Q₈×C₈₉	—	4	4	no	—
39	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
41	C₂×C₄⋊C₄×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	—	2	1	no	—
45	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
46	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
48	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
49	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
53	C₂×C₄²×C₈₉	—	2	2	no	—
54	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₈₉	—	2	4	no	—
62	#97×C₈₉	—	2	4	no	—
63	#95×C₈₉	—	2	2	no	—
64	#95×C₈₉	—	2	2	no	—
65	#95×C₈₉	—	2	2	no	—
66	#95×C₈₉	—	2	2	no	—
67	C₂×Q₈×C₈₉	—	2	4	no	—
68	C₂×Q₈×C₈₉	—	4	2	no	—
69	C₂×Q₈×C₈₉	—	4	2	no	—
70	C₂²×C₄×C₈₉	—	2	2	no	—
71	C₂²×C₄×C₈₉	—	2	2	no	—
72	C₂²×C₄×C₈₉	—	2	2	no	—
73	C₂²×C₄×C₈₉	—	2	2	no	—
74	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₈₉	—	2	4	no	—
82	C₂×C₈×C₈₉	—	2	4	no	—
83	C₂×C₈×C₈₉	—	2	4	no	—
84	C₄⋊C₄×C₈₉	—	2	4	no	—
85	C₄⋊C₄×C₈₉	—	2	2	no	—
86	C₄⋊C₄×C₈₉	—	2	4	no	—
87	C₄⋊C₄×C₈₉	—	2	4	no	—
88	C₄⋊C₄×C₈₉	—	2	4	no	—
89	C₄⋊C₄×C₈₉	—	4	2	no	—
90	C₄⋊C₄×C₈₉	—	4	2	no	—
91	C₄⋊C₄×C₈₉	—	4	2	no	—
92	C₄⋊C₄×C₈₉	—	4	2	no	—
93	C₄⋊C₄×C₈₉	—	4	4	no	—
94	C₄⋊C₄×C₈₉	—	4	4	no	—
95	C₄⋊C₄×C₈₉	—	4	2	no	—
96	C₄⋊C₄×C₈₉	—	4	4	no	—
97	C₄⋊C₄×C₈₉	—	4	2	no	—
98	C₄⋊C₄×C₈₉	C₂×Q₈	1	4	yes	—
99	C₄²×C₈₉	—	2	2	no	—
100	C₄²×C₈₉	—	2	2	no	—
101	C₄²×C₈₉	—	2	2	no	—
102	C₄²×C₈₉	—	4	1	no	—
103	C₄²×C₈₉	—	4	1	no	—
104	C₄²×C₈₉	—	4	4	no	—
105	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
106	C₄²×C₈₉	#41	1	2	yes	—
107	C₄²×C₈₉	#41	1	4	yes	—
108	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₈₉	—	4	8	no	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	4	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	4	2	no	—
125	C₂×C₄×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	—	4	2	no	—
127	C₂×C₄×C₈₉	—	4	2	no	—
128	C₂×C₄×C₈₉	—	4	2	no	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	—	4	2	no	—
131	C₂×C₄×C₈₉	—	4	2	no	—
132	C₂×C₄×C₈₉	—	4	2	no	—
133	C₂×C₄×C₈₉	—	4	2	no	—
134	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
136	C₂×C₄×C₈₉	#124	1	2	yes	—
137	C₂×C₄×C₈₉	#124	1	2	yes	—
138	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
139	C₂×C₄×C₈₉	#137	1	2	yes	—
140	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
143	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
144	C₈×C₈₉	—	4	4	no	—
145	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
146	C₂²×C₈₉	C₂×#124	1	1	yes	characteristic
147	C₂²×C₈₉	#536	1	1	yes	characteristic
148	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
150	C₂²×C₈₉	#575	1	1	yes	characteristic
151	C₂²×C₈₉	#575	1	1	yes	characteristic
152	C₄×C₈₉	—	2	2	no	—
153	C₄×C₈₉	—	2	2	no	—
154	C₄×C₈₉	—	2	2	no	—
155	C₄×C₈₉	—	2	2	no	—
156	C₄×C₈₉	—	2	2	no	—
157	C₄×C₈₉	—	2	2	no	—
158	C₄×C₈₉	—	4	4	no	—
159	C₄×C₈₉	—	8	2	no	—
160	C₄×C₈₉	—	8	2	no	—
161	C₄×C₈₉	—	8	2	no	—
162	C₄×C₈₉	#556	1	2	yes	—
163	C₄×C₈₉	#557	1	2	yes	—
164	#29223	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂×C₈₉	#2568	1	1	yes	characteristic
166	C₂×C₈₉	#3077	1	1	yes	characteristic
167	C₂×C₈₉	#3078	1	1	yes	characteristic
168	C₂×C₈₉	#3135	1	1	yes	characteristic
169	C₂×C₈₉	#3141	1	1	yes	characteristic
170	C₂×C₈₉	#3234	1	1	yes	characteristic
171	C₂×C₈₉	#3236	1	1	yes	characteristic
172	#2859	C₂×C₈₉	1	1	yes	characteristic
173	C₄×C₄⋊Q₈	C₂×C₈₉	1	1	yes	characteristic
174	#2100	C₂×C₈₉	1	2	yes	—
175	#2099	C₂×C₈₉	1	2	yes	—
176	#1993	C₂×C₈₉	1	2	yes	—
177	#1989	C₂×C₈₉	1	4	yes	—
178	#1836	C₂×C₈₉	1	2	yes	—
179	#1767	C₂×C₈₉	1	1	yes	characteristic
180	C₈₉	#29223	1	1	yes	characteristic, direct product factor, Sylow
181	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
182	C₂×#125	C₂²×C₈₉	1	2	yes	—
183	C₂×C₄×Q₈	—	2	1	no	—
184	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
185	C₂×#97	C₂²×C₈₉	1	2	yes	—
186	#391	—	2	1	no	—
187	#389	—	2	1	no	—
188	#389	—	2	1	no	—
189	#389	—	2	1	no	—
190	#389	—	2	1	no	—
191	#389	—	2	1	no	—
192	#389	C₂²×C₈₉	1	2	yes	—
193	#384	C₂²×C₈₉	1	1	yes	characteristic
194	#384	C₂²×C₈₉	1	1	yes	characteristic
195	#384	C₂²×C₈₉	1	1	yes	characteristic
196	#382	C₂²×C₈₉	1	2	yes	—
197	C₄×C₄⋊C₄	—	2	1	no	—
198	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
199	C₄³	C₂²×C₈₉	1	1	yes	characteristic
200	#340	C₂²×C₈₉	1	4	yes	—
201	#340	C₂²×C₈₉	1	4	yes	—
202	#340	C₂²×C₈₉	1	4	yes	—
203	#340	C₂²×C₈₉	1	4	yes	—
204	#336	C₂²×C₈₉	1	2	yes	—
205	C₂²×Q₈	—	2	1	no	—
206	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
207	C₄⋊Q₈	—	2	2	no	—
208	C₄⋊Q₈	Q₈×C₈₉	1	4	yes	—
209	#125	—	2	8	no	—
210	C₄×Q₈	—	4	4	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
222	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
223	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
224	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
225	C₂×C₄²	—	2	2	no	—
226	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
227	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
228	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
229	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
230	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
231	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
232	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
233	C₄⋊C₈	—	2	4	no	—
234	#97	—	2	4	no	—
235	#95	—	2	2	no	—
236	#95	—	2	2	no	—
237	#95	—	2	2	no	—
238	#95	—	2	2	no	—
239	C₂×Q₈	—	2	4	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
252	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	4	no	—
266	C₄⋊C₄	—	4	4	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	C₂×Q₈×C₈₉	1	4	yes	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	4	4	no	—
277	C₄²	C₂×Q₈×C₈₉	1	2	yes	—
278	C₄²	#41×C₈₉	1	2	yes	—
279	C₄²	#41×C₈₉	1	4	yes	—
280	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
281	Q₈	—	4	8	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	C₂²⋊Q₈×C₈₉	1	2	yes	—
307	C₂×C₄	C₂²⋊Q₈×C₈₉	1	2	yes	—
308	C₂×C₄	#124×C₈₉	1	2	yes	—
309	C₂×C₄	#124×C₈₉	1	2	yes	—
310	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
311	C₂×C₄	#137×C₈₉	1	2	yes	—
312	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
313	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
314	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
315	C₂×C₄	2₋¹⁺⁴×C₈₉	1	2	yes	—
316	C₈	—	4	4	no	—
317	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
318	C₂²	C₂×#124×C₈₉	1	1	yes	characteristic
319	C₂²	#536×C₈₉	1	1	yes	characteristic
320	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
321	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
322	C₂²	#575×C₈₉	1	1	yes	characteristic
323	C₂²	#575×C₈₉	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	4	4	no	—
331	C₄	—	8	2	no	—
332	C₄	—	8	2	no	—
333	C₄	—	8	2	no	—
334	C₄	#556×C₈₉	1	2	yes	—
335	C₄	#557×C₈₉	1	2	yes	—
336	C₂	C₈₉×#2568	1	1	yes	characteristic
337	C₂	C₈₉×#3077	1	1	yes	characteristic
338	C₂	C₈₉×#3078	1	1	yes	characteristic
339	C₂	C₈₉×#3135	1	1	yes	characteristic
340	C₂	C₈₉×#3141	1	1	yes	characteristic
341	C₂	C₈₉×#3234	1	1	yes	characteristic
342	C₂	C₈₉×#3236	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₄³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
164. #29223	C₈₉	direct (N × H)
180. C₈₉	#29223	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×#2859	2
C₄×C₄⋊Q₈×C₈₉	2
C₈₉×#2100	2
C₈₉×#2099	2
C₈₉×#1993	2
C₈₉×#1989	2
C₈₉×#1836	2
C₈₉×#1767	2
#29223	89

Subgroups — C89×#29223 #36407071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#29223 #36407071