Subgroups — C₈₉×#20763 #36398611

Order 22784.

894 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#405×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2317	C₂	1	2	yes	maximal
3	C₈₉×#1836	C₂	1	4	yes	maximal, complemented
4	C₈₉×#1835	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1834	C₂	1	2	yes	maximal, complemented
6	C₈₉×#1736	C₂	1	2	yes	maximal
7	C₈₉×#1735	C₂	1	1	yes	maximal, characteristic
8	C₂×#340×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×#117×C₈₉	C₂²	1	2	yes	—
12	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
14	#405×C₈₉	C₄	1	8	yes	complemented
15	C₂×#97×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
17	#389×C₈₉	—	2	2	no	—
18	#389×C₈₉	—	2	2	no	—
19	#385×C₈₉	—	2	4	no	—
20	#384×C₈₉	C₂²	1	4	yes	—
21	#382×C₈₉	C₂²	1	4	yes	—
22	#340×C₈₉	C₂²	1	1	yes	characteristic
23	#340×C₈₉	C₂²	1	2	yes	—
24	#340×C₈₉	C₂²	1	4	yes	—
25	#340×C₈₉	C₂²	1	1	yes	characteristic
26	#340×C₈₉	C₂²	1	4	yes	—
27	#340×C₈₉	C₂²	1	2	yes	—
28	#340×C₈₉	C₂²	1	2	yes	—
29	C₂³×C₄×C₈₉	—	2	2	no	—
30	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₈₉	—	2	4	no	—
32	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₈₉	—	2	4	no	—
34	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
38	#117×C₈₉	—	2	4	no	—
39	#117×C₈₉	—	2	4	no	—
40	#117×C₈₉	C₂×C₄	1	4	yes	—
41	#117×C₈₉	Q₈	1	4	yes	—
42	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
43	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
44	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
46	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
47	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
51	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
52	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
54	C₂×C₄²×C₈₉	—	2	2	no	—
55	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
56	#97×C₈₉	C₂×C₄	1	8	yes	complemented
57	C₄×C₈×C₈₉	C₂×C₄	1	8	yes	complemented
58	#95×C₈₉	—	2	4	no	—
59	#95×C₈₉	—	2	4	no	—
60	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	—	2	2	no	—
62	C₂²×C₄×C₈₉	—	2	4	no	—
63	C₂²×C₄×C₈₉	—	2	2	no	—
64	C₂²×C₄×C₈₉	—	4	4	no	—
65	C₂²×C₄×C₈₉	—	8	2	no	—
66	C₂²×C₄×C₈₉	—	8	2	no	—
67	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₈₉	#41	1	4	yes	—
73	C₂²×C₄×C₈₉	#41	1	4	yes	—
74	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
75	M₁₆×C₈₉	—	2	8	no	—
76	C₂×C₈×C₈₉	—	2	8	no	—
77	C₂×C₈×C₈₉	—	2	2	no	—
78	C₂×C₈×C₈₉	—	2	2	no	—
79	C₂×C₈×C₈₉	—	2	4	no	—
80	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
83	C₄⋊C₄×C₈₉	—	2	4	no	—
84	C₄⋊C₄×C₈₉	—	2	4	no	—
85	C₄⋊C₄×C₈₉	—	2	2	no	—
86	C₄⋊C₄×C₈₉	—	2	2	no	—
87	C₄⋊C₄×C₈₉	—	2	2	no	—
88	C₄⋊C₄×C₈₉	—	2	2	no	—
89	C₄⋊C₄×C₈₉	—	4	4	no	—
90	C₄⋊C₄×C₈₉	—	8	4	no	—
91	C₄⋊C₄×C₈₉	—	8	4	no	—
92	C₄⋊C₄×C₈₉	C₄⋊C₄	1	4	yes	—
93	C₄⋊C₄×C₈₉	C₄⋊C₄	1	4	yes	—
94	C₂²⋊C₄×C₈₉	—	4	4	no	—
95	C₂²⋊C₄×C₈₉	C₄⋊C₄	1	8	yes	—
96	C₄²×C₈₉	—	4	4	no	—
97	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
98	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
99	C₂³×C₈₉	—	2	1	no	—
100	C₂³×C₈₉	—	2	2	no	—
101	C₂³×C₈₉	—	2	1	no	—
102	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
103	C₂³×C₈₉	#124	1	4	yes	—
104	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	—	2	2	no	—
106	C₂×C₄×C₈₉	—	2	4	no	—
107	C₂×C₄×C₈₉	—	2	2	no	—
108	C₂×C₄×C₈₉	—	4	2	no	—
109	C₂×C₄×C₈₉	—	4	2	no	—
110	C₂×C₄×C₈₉	—	4	4	no	—
111	C₂×C₄×C₈₉	—	4	2	no	—
112	C₂×C₄×C₈₉	—	4	4	no	—
113	C₂×C₄×C₈₉	—	4	2	no	—
114	C₂×C₄×C₈₉	—	4	4	no	—
115	C₂×C₄×C₈₉	—	4	2	no	—
116	C₂×C₄×C₈₉	—	4	2	no	—
117	C₂×C₄×C₈₉	—	8	8	no	—
118	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
119	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₈₉	#117	1	4	yes	—
122	C₂×C₄×C₈₉	#117	1	4	yes	—
123	C₂×C₄×C₈₉	#124	1	4	yes	—
124	C₂×C₄×C₈₉	#125	1	8	yes	—
125	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
128	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
129	C₈×C₈₉	—	2	2	no	—
130	C₈×C₈₉	—	2	2	no	—
131	C₈×C₈₉	—	2	4	no	—
132	C₂²×C₈₉	—	2	4	no	—
133	C₂²×C₈₉	—	2	4	no	—
134	C₂²×C₈₉	—	2	2	no	—
135	C₂²×C₈₉	—	2	2	no	—
136	C₂²×C₈₉	#382	1	4	yes	—
137	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₈₉	C₂×#124	1	2	yes	—
139	C₂²×C₈₉	#576	1	1	yes	characteristic
140	C₂²×C₈₉	#577	1	2	yes	—
141	C₂²×C₈₉	#578	1	1	yes	characteristic
142	C₄×C₈₉	—	2	4	no	—
143	C₄×C₈₉	—	2	4	no	—
144	C₄×C₈₉	—	8	2	no	—
145	C₄×C₈₉	—	8	4	no	—
146	C₄×C₈₉	—	8	2	no	—
147	C₄×C₈₉	#382	1	4	yes	—
148	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
149	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
150	C₄×C₈₉	C₂×#125	1	2	yes	—
151	#20763	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂×C₈₉	—	2	4	no	—
153	C₂×C₈₉	C₂×#382	1	1	yes	characteristic
154	C₂×C₈₉	#2912	1	1	yes	characteristic
155	C₂×C₈₉	#2913	1	1	yes	characteristic
156	C₂×C₈₉	#3136	1	2	yes	—
157	C₂×C₈₉	#3137	1	2	yes	—
158	C₂×#405	C₂×C₈₉	1	1	yes	characteristic
159	#2317	C₂×C₈₉	1	2	yes	—
160	#1836	C₂×C₈₉	1	4	yes	complemented
161	#1835	C₂×C₈₉	1	2	yes	complemented
162	#1834	C₂×C₈₉	1	2	yes	complemented
163	#1736	C₂×C₈₉	1	2	yes	—
164	#1735	C₂×C₈₉	1	1	yes	characteristic
165	C₂×#340	C₂×C₈₉	1	1	yes	characteristic
166	C₈₉	#20763	1	1	yes	characteristic, direct product factor, Sylow
167	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
168	C₂³×C₈	C₂²×C₈₉	1	1	yes	characteristic
169	C₂×#117	C₂²×C₈₉	1	2	yes	—
170	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
171	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
172	#405	C₄×C₈₉	1	8	yes	complemented
173	C₂×#97	C₂²×C₈₉	1	2	yes	—
174	C₂×C₄×C₈	C₂²×C₈₉	1	2	yes	—
175	#389	—	2	2	no	—
176	#389	—	2	2	no	—
177	#385	—	2	4	no	—
178	#384	C₂²×C₈₉	1	4	yes	—
179	#382	C₂²×C₈₉	1	4	yes	—
180	#340	C₂²×C₈₉	1	1	yes	characteristic
181	#340	C₂²×C₈₉	1	2	yes	—
182	#340	C₂²×C₈₉	1	4	yes	—
183	#340	C₂²×C₈₉	1	1	yes	characteristic
184	#340	C₂²×C₈₉	1	4	yes	—
185	#340	C₂²×C₈₉	1	2	yes	—
186	#340	C₂²×C₈₉	1	2	yes	—
187	C₂³×C₄	—	2	2	no	—
188	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
189	C₂×M₁₆	—	2	4	no	—
190	C₂×M₁₆	C₂×C₄×C₈₉	1	4	yes	—
191	C₂²×C₈	—	2	4	no	—
192	C₂²×C₈	C₂×C₄×C₈₉	1	4	yes	—
193	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
196	#117	—	2	4	no	—
197	#117	—	2	4	no	—
198	#117	C₂×C₄×C₈₉	1	4	yes	—
199	#117	Q₈×C₈₉	1	4	yes	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	4	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
206	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
207	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
208	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
209	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
210	C₂×C₂²⋊C₄	—	2	2	no	—
211	C₂×C₂²⋊C₄	D₄×C₈₉	1	2	yes	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
214	#97	C₂×C₄×C₈₉	1	8	yes	complemented
215	C₄×C₈	C₂×C₄×C₈₉	1	8	yes	complemented
216	#95	—	2	4	no	—
217	#95	—	2	4	no	—
218	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	8	2	no	—
224	C₂²×C₄	—	8	2	no	—
225	C₂²×C₄	C₂²×C₄×C₈₉	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
227	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
228	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
229	C₂²×C₄	C₂×Q₈×C₈₉	1	2	yes	—
230	C₂²×C₄	#41×C₈₉	1	4	yes	—
231	C₂²×C₄	#41×C₈₉	1	4	yes	—
232	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
233	M₁₆	—	2	8	no	—
234	C₂×C₈	—	2	8	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	C₂²×C₄×C₈₉	1	2	yes	—
239	C₂×C₈	C₂²×C₄×C₈₉	1	4	yes	—
240	C₂×C₈	C₂²×C₄×C₈₉	1	2	yes	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	4	4	no	—
248	C₄⋊C₄	—	8	4	no	—
249	C₄⋊C₄	—	8	4	no	—
250	C₄⋊C₄	C₄⋊C₄×C₈₉	1	4	yes	—
251	C₄⋊C₄	C₄⋊C₄×C₈₉	1	4	yes	—
252	C₂²⋊C₄	—	4	4	no	—
253	C₂²⋊C₄	C₄⋊C₄×C₈₉	1	8	yes	—
254	C₄²	—	4	4	no	—
255	C₄²	C₂²×C₄×C₈₉	1	4	yes	—
256	C₄²	C₂×Q₈×C₈₉	1	4	yes	—
257	C₂³	—	2	1	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	1	no	—
260	C₂³	C₂×C₄⋊C₄×C₈₉	1	2	yes	—
261	C₂³	#124×C₈₉	1	4	yes	—
262	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	8	8	no	—
276	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	4	yes	—
277	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	4	yes	—
278	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	2	yes	—
279	C₂×C₄	#117×C₈₉	1	4	yes	—
280	C₂×C₄	#117×C₈₉	1	4	yes	—
281	C₂×C₄	#124×C₈₉	1	4	yes	—
282	C₂×C₄	#125×C₈₉	1	8	yes	—
283	C₂×C₄	C₂³×C₄×C₈₉	1	1	yes	characteristic
284	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
285	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
286	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
287	C₈	—	2	2	no	—
288	C₈	—	2	2	no	—
289	C₈	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	#382×C₈₉	1	4	yes	—
295	C₂²	C₂²×C₄⋊C₄×C₈₉	1	1	yes	characteristic
296	C₂²	C₂×#124×C₈₉	1	2	yes	—
297	C₂²	#576×C₈₉	1	1	yes	characteristic
298	C₂²	#577×C₈₉	1	2	yes	—
299	C₂²	#578×C₈₉	1	1	yes	characteristic
300	C₄	—	2	4	no	—
301	C₄	—	2	4	no	—
302	C₄	—	8	4	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	#382×C₈₉	1	4	yes	—
306	C₄	C₂×#117×C₈₉	1	1	yes	characteristic
307	C₄	C₂×#117×C₈₉	1	1	yes	characteristic
308	C₄	C₂×#125×C₈₉	1	2	yes	—
309	C₂	—	2	4	no	—
310	C₂	C₂×#382×C₈₉	1	1	yes	characteristic
311	C₂	C₈₉×#2912	1	1	yes	characteristic
312	C₂	C₈₉×#2913	1	1	yes	characteristic
313	C₂	C₈₉×#3136	1	2	yes	—
314	C₂	C₈₉×#3137	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₈₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂⁴×C₈₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂×#117×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₈₉×#1836	C₂	semidirect
4. C₈₉×#1835	C₂	semidirect
5. C₈₉×#1834	C₂	semidirect
14. #405×C₈₉	C₄	semidirect
56. #97×C₈₉	C₂×C₄	semidirect
57. C₄×C₈×C₈₉	C₂×C₄	semidirect
151. #20763	C₈₉	direct (N × H)
160. #1836	C₂×C₈₉	semidirect
161. #1835	C₂×C₈₉	semidirect
162. #1834	C₂×C₈₉	semidirect
166. C₈₉	#20763	direct (N × H)
172. #405	C₄×C₈₉	semidirect
214. #97	C₂×C₄×C₈₉	semidirect
215. C₄×C₈	C₂×C₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#405×C₈₉	2
C₈₉×#2317	2
C₈₉×#1836	2
C₈₉×#1835	2
C₈₉×#1834	2
C₈₉×#1736	2
C₈₉×#1735	2
C₂×#340×C₈₉	2
#20763	89

Subgroups — C89×#20763 #36398611

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#20763 #36398611