Subgroups — C₈₉×#20482 #36398330

Order 22784.

950 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2283	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1770	C₂	1	2	yes	maximal, complemented
5	C₈×C₄⋊C₄×C₈₉	C₂	1	2	yes	maximal, complemented
6	C₈₉×#1753	C₂	1	4	yes	maximal
7	C₈₉×#1746	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1726	C₂	1	2	yes	maximal, complemented
9	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
10	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
11	#431×C₈₉	C₄	1	8	yes	complemented
12	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
15	C₄×M₁₆×C₈₉	C₄	1	8	yes	complemented
16	C₂×#97×C₈₉	C₂²	1	4	yes	—
17	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₈₉	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
22	#383×C₈₉	C₄	1	2	yes	—
23	#382×C₈₉	C₄	1	2	yes	—
24	#381×C₈₉	C₄	1	2	yes	—
25	#380×C₈₉	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
27	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
28	#376×C₈₉	C₂²	1	2	yes	—
29	#336×C₈₉	C₂²	1	4	yes	—
30	#336×C₈₉	C₂²	1	4	yes	—
31	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₈₉	—	2	4	no	—
35	C₂×M₁₆×C₈₉	—	2	4	no	—
36	C₂×M₁₆×C₈₉	C₂×C₄	1	8	yes	—
37	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
38	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
39	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
40	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
42	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
48	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
51	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
52	C₂²⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
53	#97×C₈₉	—	2	8	no	—
54	#97×C₈₉	C₂×C₄	1	4	yes	—
55	#97×C₈₉	C₂×C₄	1	4	yes	—
56	C₄×C₈×C₈₉	—	2	8	no	—
57	C₄×C₈×C₈₉	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₈₉	C₂×C₄	1	8	yes	—
59	#95×C₈₉	C₂×C₄	1	4	yes	—
60	#95×C₈₉	C₂×C₄	1	4	yes	—
61	#95×C₈₉	C₂×C₄	1	4	yes	—
62	#95×C₈₉	C₂×C₄	1	4	yes	—
63	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	—	2	4	no	—
65	C₂²×C₄×C₈₉	—	2	2	no	—
66	C₂²×C₄×C₈₉	—	2	2	no	—
67	C₂²×C₄×C₈₉	—	4	2	no	—
68	C₂²×C₄×C₈₉	C₄²	1	4	yes	—
69	C₂²×C₄×C₈₉	C₄²	1	4	yes	—
70	C₂²×C₄×C₈₉	C₂²×C₄	1	8	yes	—
71	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
79	M₁₆×C₈₉	—	2	16	no	—
80	C₂×C₈×C₈₉	—	2	4	no	—
81	C₂×C₈×C₈₉	—	2	4	no	—
82	C₂×C₈×C₈₉	—	2	2	no	—
83	C₂×C₈×C₈₉	—	2	2	no	—
84	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₈₉	#41	1	4	yes	—
89	C₂×C₈×C₈₉	#41	1	4	yes	—
90	C₄⋊C₄×C₈₉	—	2	8	no	—
91	C₂²⋊C₄×C₈₉	—	4	4	no	—
92	C₄²×C₈₉	—	2	4	no	—
93	C₄²×C₈₉	—	2	4	no	—
94	C₄²×C₈₉	C₄²	1	4	yes	—
95	C₄²×C₈₉	C₄²	1	4	yes	—
96	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
97	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
98	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
99	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
100	C₂³×C₈₉	—	2	2	no	—
101	C₂³×C₈₉	—	2	2	no	—
102	C₂³×C₈₉	—	2	2	no	—
103	C₂³×C₈₉	C₂×C₄²	1	2	yes	—
104	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	—	2	2	no	—
106	C₂×C₄×C₈₉	—	2	2	no	—
107	C₂×C₄×C₈₉	—	2	2	no	—
108	C₂×C₄×C₈₉	—	2	2	no	—
109	C₂×C₄×C₈₉	—	2	4	no	—
110	C₂×C₄×C₈₉	—	2	2	no	—
111	C₂×C₄×C₈₉	—	2	2	no	—
112	C₂×C₄×C₈₉	—	2	8	no	—
113	C₂×C₄×C₈₉	—	2	8	no	—
114	C₂×C₄×C₈₉	—	2	8	no	—
115	C₂×C₄×C₈₉	—	2	8	no	—
116	C₂×C₄×C₈₉	—	4	4	no	—
117	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₈₉	#131	1	4	yes	—
123	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
126	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
127	C₈×C₈₉	—	2	4	no	—
128	C₈×C₈₉	—	2	4	no	—
129	C₈×C₈₉	—	2	4	no	—
130	C₈×C₈₉	#117	1	8	yes	—
131	C₂²×C₈₉	—	2	2	no	—
132	C₂²×C₈₉	—	2	2	no	—
133	C₂²×C₈₉	—	2	2	no	—
134	C₂²×C₈₉	—	4	4	no	—
135	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
136	C₂²×C₈₉	C₄×#41	1	2	yes	—
137	C₂²×C₈₉	C₂×#131	1	2	yes	—
138	C₂²×C₈₉	#568	1	2	yes	—
139	C₄×C₈₉	—	2	2	no	—
140	C₄×C₈₉	—	2	2	no	—
141	C₄×C₈₉	—	2	2	no	—
142	C₄×C₈₉	—	2	2	no	—
143	C₄×C₈₉	—	4	8	no	—
144	C₄×C₈₉	#405	1	4	yes	—
145	C₄×C₈₉	C₂×#117	1	2	yes	—
146	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
147	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
148	#20482	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂×C₈₉	—	4	2	no	—
150	C₂×C₈₉	C₄×#117	1	1	yes	characteristic
151	C₂×C₈₉	C₂×#405	1	1	yes	characteristic
152	C₂×C₈₉	#2873	1	1	yes	characteristic
153	C₂×C₈₉	#2969	1	4	yes	—
154	C₂×#431	C₂×C₈₉	1	1	yes	characteristic
155	C₂×C₄×M₁₆	C₂×C₈₉	1	1	yes	characteristic
156	#2283	C₂×C₈₉	1	1	yes	characteristic
157	#1770	C₂×C₈₉	1	2	yes	complemented
158	C₈×C₄⋊C₄	C₂×C₈₉	1	2	yes	complemented
159	#1753	C₂×C₈₉	1	4	yes	—
160	#1746	C₂×C₈₉	1	2	yes	complemented
161	#1726	C₂×C₈₉	1	2	yes	complemented
162	C₈₉	#20482	1	1	yes	characteristic, direct product factor, Sylow
163	C₂²×M₁₆	C₂²×C₈₉	1	2	yes	—
164	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
165	#431	C₄×C₈₉	1	8	yes	complemented
166	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
167	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
168	C₂×C₂²⋊C₈	C₂²×C₈₉	1	2	yes	—
169	C₄×M₁₆	C₄×C₈₉	1	8	yes	complemented
170	C₂×#97	C₂²×C₈₉	1	4	yes	—
171	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
172	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
173	C₂×C₄×C₈	C₂²×C₈₉	1	4	yes	—
174	C₂×C₄×C₈	C₂²×C₈₉	1	2	yes	—
175	C₂×C₄×C₈	C₂²×C₈₉	1	2	yes	—
176	#383	C₄×C₈₉	1	2	yes	—
177	#382	C₄×C₈₉	1	2	yes	—
178	#381	C₄×C₈₉	1	2	yes	—
179	#380	C₄×C₈₉	1	2	yes	—
180	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
181	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
182	#376	C₂²×C₈₉	1	2	yes	—
183	#336	C₂²×C₈₉	1	4	yes	—
184	#336	C₂²×C₈₉	1	4	yes	—
185	C₂³×C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
186	C₂³×C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
187	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
188	C₂×M₁₆	—	2	4	no	—
189	C₂×M₁₆	—	2	4	no	—
190	C₂×M₁₆	C₂×C₄×C₈₉	1	8	yes	—
191	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
192	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
193	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
194	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
195	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	4	yes	—
196	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	4	yes	—
197	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
198	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
199	C₂×C₄²	C₂×C₄×C₈₉	1	4	yes	—
200	C₂×C₄²	C₂×C₄×C₈₉	1	4	yes	—
201	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
202	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
203	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
204	C₄⋊C₈	C₂×C₄×C₈₉	1	4	yes	—
205	C₄⋊C₈	C₂×C₄×C₈₉	1	4	yes	—
206	C₂²⋊C₈	C₂×C₄×C₈₉	1	8	yes	—
207	#97	—	2	8	no	—
208	#97	C₂×C₄×C₈₉	1	4	yes	—
209	#97	C₂×C₄×C₈₉	1	4	yes	—
210	C₄×C₈	—	2	8	no	—
211	C₄×C₈	C₂×C₄×C₈₉	1	8	yes	—
212	C₄×C₈	C₂×C₄×C₈₉	1	8	yes	—
213	#95	C₂×C₄×C₈₉	1	4	yes	—
214	#95	C₂×C₄×C₈₉	1	4	yes	—
215	#95	C₂×C₄×C₈₉	1	4	yes	—
216	#95	C₂×C₄×C₈₉	1	4	yes	—
217	C₂⁴	C₂²×C₄×C₈₉	1	1	yes	characteristic
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	C₄²×C₈₉	1	4	yes	—
223	C₂²×C₄	C₄²×C₈₉	1	4	yes	—
224	C₂²×C₄	C₂²×C₄×C₈₉	1	8	yes	—
225	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
228	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
229	C₂²×C₄	C₂²×C₄×C₈₉	1	2	yes	—
230	C₂²×C₄	C₂²×C₄×C₈₉	1	2	yes	—
231	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
232	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
233	M₁₆	—	2	16	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	C₂²×C₄×C₈₉	1	4	yes	—
239	C₂×C₈	C₂²×C₄×C₈₉	1	4	yes	—
240	C₂×C₈	C₂²×C₄×C₈₉	1	4	yes	—
241	C₂×C₈	C₂²×C₄×C₈₉	1	4	yes	—
242	C₂×C₈	#41×C₈₉	1	4	yes	—
243	C₂×C₈	#41×C₈₉	1	4	yes	—
244	C₄⋊C₄	—	2	8	no	—
245	C₂²⋊C₄	—	4	4	no	—
246	C₄²	—	2	4	no	—
247	C₄²	—	2	4	no	—
248	C₄²	C₄²×C₈₉	1	4	yes	—
249	C₄²	C₄²×C₈₉	1	4	yes	—
250	C₄²	C₂²×C₄×C₈₉	1	2	yes	—
251	C₄²	C₂²×C₄×C₈₉	1	2	yes	—
252	C₄²	C₂²×C₄×C₈₉	1	2	yes	—
253	C₄²	C₂²×C₄×C₈₉	1	2	yes	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	C₂×C₄²×C₈₉	1	2	yes	—
258	C₂³	C₂³×C₄×C₈₉	1	1	yes	characteristic
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	C₂×C₄²×C₈₉	1	2	yes	—
272	C₂×C₄	C₂×C₄²×C₈₉	1	2	yes	—
273	C₂×C₄	C₂×C₄²×C₈₉	1	2	yes	—
274	C₂×C₄	C₂×C₄²×C₈₉	1	2	yes	—
275	C₂×C₄	C₂×C₄²×C₈₉	1	2	yes	—
276	C₂×C₄	#131×C₈₉	1	4	yes	—
277	C₂×C₄	C₂³×C₄×C₈₉	1	1	yes	characteristic
278	C₂×C₄	C₂³×C₄×C₈₉	1	1	yes	characteristic
279	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
280	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₈	—	2	4	no	—
284	C₈	#117×C₈₉	1	8	yes	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	4	4	no	—
289	C₂²	C₂²×C₄²×C₈₉	1	1	yes	characteristic
290	C₂²	C₄×#41×C₈₉	1	2	yes	—
291	C₂²	C₂×#131×C₈₉	1	2	yes	—
292	C₂²	#568×C₈₉	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	8	no	—
298	C₄	#405×C₈₉	1	4	yes	—
299	C₄	C₂×#117×C₈₉	1	2	yes	—
300	C₄	C₄×#41×C₈₉	1	1	yes	characteristic
301	C₄	C₄×#41×C₈₉	1	1	yes	characteristic
302	C₂	—	4	2	no	—
303	C₂	C₄×#117×C₈₉	1	1	yes	characteristic
304	C₂	C₂×#405×C₈₉	1	1	yes	characteristic
305	C₂	C₈₉×#2873	1	1	yes	characteristic
306	C₂	C₈₉×#2969	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₈₉×#1770	C₂	semidirect
5. C₈×C₄⋊C₄×C₈₉	C₂	semidirect
7. C₈₉×#1746	C₂	semidirect
8. C₈₉×#1726	C₂	semidirect
11. #431×C₈₉	C₄	semidirect
15. C₄×M₁₆×C₈₉	C₄	semidirect
148. #20482	C₈₉	direct (N × H)
157. #1770	C₂×C₈₉	semidirect
158. C₈×C₄⋊C₄	C₂×C₈₉	semidirect
160. #1746	C₂×C₈₉	semidirect
161. #1726	C₂×C₈₉	semidirect
162. C₈₉	#20482	direct (N × H)
165. #431	C₄×C₈₉	semidirect
169. C₄×M₁₆	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×#431×C₈₉	2
C₂×C₄×M₁₆×C₈₉	2
C₈₉×#2283	2
C₈₉×#1770	2
C₈×C₄⋊C₄×C₈₉	2
C₈₉×#1753	2
C₈₉×#1746	2
C₈₉×#1726	2
#20482	89

Subgroups — C89×#20482 #36398330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#20482 #36398330