Subgroups — #2161×D₈₉ #36707481

Order 22784.

642 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×M₃₂×D₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×D₈₉	C₂	1	2	yes	maximal
3	#26022047	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022046	C₂	1	2	yes	maximal, complemented
5	#26022045	C₂	1	4	yes	maximal, complemented
6	#368×D₈₉	C₂	1	4	yes	maximal, complemented
7	C₈₉×#2161	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₈₉	—	2	2	no	—
9	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
10	C₂×#15667952	C₂²	1	1	yes	characteristic
11	C₂×#15667951	C₂²	1	2	yes	—
12	M₃₂×D₈₉	C₄	1	4	yes	—
13	M₃₂×D₈₉	C₄	1	4	yes	—
14	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
15	C₂×#15667948	C₂²	1	2	yes	—
16	C₂×#15667947	—	2	2	no	—
17	#108×D₈₉	—	2	2	no	—
18	#108×D₈₉	—	2	2	no	—
19	#108×D₈₉	C₂²	1	4	yes	—
20	#108×D₈₉	C₂²	1	4	yes	—
21	#19151428	C₂²	1	4	yes	complemented
22	#19151427	C₂²	1	8	yes	complemented
23	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×#108×C₈₉	C₂²	1	2	yes	—
25	#368×C₈₉	C₂²	1	4	yes	complemented
26	C₂²×C₄×D₈₉	D₄	1	1	yes	characteristic
27	C₂×#13083558	—	2	2	no	—
28	C₂×#13083557	—	4	2	no	—
29	M₁₆×D₈₉	—	2	4	no	—
30	M₁₆×D₈₉	—	2	4	no	—
31	M₁₆×D₈₉	—	4	2	no	—
32	M₁₆×D₈₉	—	4	2	no	—
33	C₂²×#10623494	D₄	1	1	yes	characteristic
34	C₂×C₈×D₈₉	—	4	2	no	—
35	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
38	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
39	C₂×C₈×D₈₉	Q₈	1	2	yes	—
40	C₂×C₈×D₈₉	Q₈	1	2	yes	—
41	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
42	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
43	#15667952	C₂×C₄	1	4	yes	—
44	#15667951	C₂×C₄	1	8	yes	—
45	C₂×#13083549	C₂³	1	2	yes	—
46	C₁₆×D₈₉	C₂×C₄	1	4	yes	—
47	C₁₆×D₈₉	C₂×C₄	1	4	yes	—
48	#15667948	—	2	2	no	—
49	#15667948	C₂³	1	4	yes	—
50	#15667947	—	2	4	no	—
51	#15667947	—	2	4	no	—
52	C₂×M₁₆×C₈₉	—	2	2	no	—
53	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
54	M₃₂×C₈₉	C₂×C₄	1	4	yes	—
55	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
56	#108×C₈₉	—	2	2	no	—
57	#108×C₈₉	C₂³	1	4	yes	—
58	C₂³×D₈₉	Q₁₆	1	1	yes	characteristic
59	C₂²×Dic₈₉	Q₁₆	1	1	yes	characteristic
60	C₂×C₄×D₈₉	C₄⋊C₄	1	2	yes	—
61	C₂×C₄×D₈₉	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×D₈₉	D₈	1	2	yes	—
63	C₂×C₄×D₈₉	D₈	1	2	yes	—
64	C₂×C₄×D₈₉	Q₁₆	1	1	yes	characteristic
65	C₂×C₄×D₈₉	Q₁₆	1	1	yes	characteristic
66	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
68	#13083558	—	2	4	no	—
69	#13083558	—	4	2	no	—
70	#13083557	—	4	4	no	—
71	#13083557	—	4	4	no	—
72	C₂×#10623494	—	4	2	no	—
73	C₂×#10623494	C₄⋊C₄	1	2	yes	—
74	C₂×#10623494	C₄⋊C₄	1	2	yes	—
75	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
76	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
77	C₈×D₈₉	—	4	4	no	—
78	C₈×D₈₉	—	4	4	no	—
79	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
80	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
81	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
82	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
83	C₈×D₈₉	C₂²×C₄	1	1	yes	characteristic
84	C₈×D₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₈×D₈₉	C₂²×C₄	1	2	yes	—
86	C₈×D₈₉	C₂×Q₈	1	1	yes	characteristic
87	C₈×D₈₉	C₂×Q₈	1	1	yes	characteristic
88	C₈×D₈₉	C₂×Q₈	1	2	yes	—
89	#13083549	C₂²×C₄	1	4	yes	—
90	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
91	M₁₆×C₈₉	—	2	4	no	—
92	M₁₆×C₈₉	—	4	2	no	—
93	C₂×C₈×C₈₉	—	4	2	no	—
94	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
97	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
98	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
99	C₂³×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
100	C₂²×D₈₉	—	4	2	no	—
101	C₂²×D₈₉	#107	1	2	yes	—
102	C₂²×D₈₉	#107	1	2	yes	—
103	C₂²×D₈₉	C₂×Q₁₆	1	1	yes	characteristic
104	C₂²×D₈₉	C₂×Q₁₆	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×Dic₈₉	#107	1	2	yes	—
110	C₂×Dic₈₉	#107	1	2	yes	—
111	C₂×Dic₈₉	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×Dic₈₉	C₂×Q₁₆	1	1	yes	characteristic
113	C₄×D₈₉	#107	1	2	yes	—
114	C₄×D₈₉	#107	1	2	yes	—
115	C₄×D₈₉	#107	1	2	yes	—
116	C₄×D₈₉	#107	1	2	yes	—
117	C₄×D₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₄×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
119	C₄×D₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₄×D₈₉	C₂×D₈	1	1	yes	characteristic
121	C₄×D₈₉	C₂×D₈	1	2	yes	—
122	C₄×D₈₉	C₂×D₈	1	1	yes	characteristic
123	C₈×C₈₉	—	4	4	no	—
124	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
125	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
126	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
127	#10623494	—	4	4	no	—
128	#10623494	C₂×C₄⋊C₄	1	1	yes	characteristic
129	#10623494	C₂×C₄⋊C₄	1	1	yes	characteristic
130	#10623494	C₂×C₄⋊C₄	1	2	yes	—
131	C₂²×C₈₉	—	4	1	no	—
132	C₂²×C₈₉	C₂×#107	1	2	yes	—
133	C₂²×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
134	C₂×D₈₉	—	2	4	no	—
135	C₂×D₈₉	—	2	2	no	—
136	C₂×D₈₉	—	4	2	no	—
137	C₂×D₈₉	—	4	1	no	—
138	C₂×D₈₉	C₂×#107	1	2	yes	—
139	C₂×D₈₉	C₂×#107	1	1	yes	characteristic
140	C₂×D₈₉	C₂×#107	1	1	yes	characteristic
141	C₄×C₈₉	C₂×#107	1	2	yes	—
142	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
144	Dic₈₉	C₂×#107	1	1	yes	characteristic
145	Dic₈₉	C₂×#107	1	2	yes	—
146	Dic₈₉	C₂×#107	1	1	yes	characteristic
147	C₂×#2161	—	89	1	no	maximal, Sylow
148	C₂×C₈₉	—	2	1	no	—
149	C₂×C₈₉	—	2	2	no	—
150	C₂×C₈₉	C₂²×#107	1	1	yes	characteristic
151	D₈₉	—	2	2	no	—
152	D₈₉	—	2	1	no	—
153	D₈₉	#2161	1	2	yes	direct product factor
154	C₂²×M₃₂	—	89	1	no	—
155	C₂²×#108	—	89	2	no	—
156	#2161	—	89	1	no	—
157	#2161	—	89	2	no	—
158	#2161	—	89	4	no	—
159	#2161	D₈₉	1	1	yes	characteristic, direct product factor
160	C₂×#368	—	89	4	no	—
161	C₈₉	C₂×#2161	1	1	yes	characteristic, complemented, Sylow
162	C₂²×M₁₆	—	178	2	no	—
163	C₂³×C₈	—	89	1	no	—
164	C₂×M₃₂	—	89	1	no	—
165	C₂×M₃₂	—	89	4	no	—
166	C₂×M₃₂	—	89	4	no	—
167	C₂×M₃₂	—	89	2	no	—
168	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
169	C₂²×C₁₆	—	89	2	no	—
170	C₂×#108	—	89	2	no	—
171	C₂×#108	—	89	4	no	—
172	C₂×#108	—	89	4	no	—
173	C₂×#108	—	178	2	no	—
174	C₂×#108	—	178	2	no	—
175	C₂×#108	—	178	2	no	—
176	C₂×#108	C₂×D₈₉	1	2	yes	—
177	#368	—	89	4	no	—
178	#368	—	89	8	no	—
179	#368	C₂×D₈₉	1	4	yes	complemented
180	C₂³×C₄	—	89	1	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	356	2	no	—
183	C₂×M₁₆	—	356	2	no	—
184	C₂×M₁₆	—	356	2	no	—
185	C₂×M₁₆	—	178	4	no	—
186	C₂×M₁₆	—	178	4	no	—
187	C₂×M₁₆	—	178	2	no	—
188	C₂²×C₈	—	89	1	no	—
189	C₂²×C₈	—	89	2	no	—
190	C₂²×C₈	—	89	2	no	—
191	C₂²×C₈	—	89	1	no	—
192	C₂²×C₈	—	89	1	no	—
193	C₂²×C₈	—	89	2	no	—
194	C₂²×C₈	—	89	2	no	—
195	C₂²×C₈	—	89	1	no	—
196	C₂²×C₈	—	89	1	no	—
197	C₂²×C₈	—	356	2	no	—
198	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
199	M₃₂	—	89	4	no	—
200	M₃₂	—	89	8	no	—
201	M₃₂	C₄×D₈₉	1	4	yes	—
202	C₂×C₁₆	—	89	2	no	—
203	C₂×C₁₆	—	89	4	no	—
204	C₂×C₁₆	—	89	4	no	—
205	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
206	#108	—	2	2	no	—
207	#108	—	89	4	no	—
208	#108	—	178	4	no	—
209	#108	—	178	4	no	—
210	#108	—	178	2	no	—
211	#108	C₂²×D₈₉	1	4	yes	—
212	C₂⁴	—	89	1	no	—
213	C₂²×C₄	—	89	1	no	—
214	C₂²×C₄	—	89	2	no	—
215	C₂²×C₄	—	89	2	no	—
216	C₂²×C₄	—	89	1	no	—
217	C₂²×C₄	—	89	1	no	—
218	C₂²×C₄	—	89	1	no	—
219	C₂²×C₄	—	89	1	no	—
220	C₂²×C₄	—	89	2	no	—
221	C₂²×C₄	—	89	2	no	—
222	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
223	M₁₆	—	2	4	no	—
224	M₁₆	—	4	2	no	—
225	M₁₆	—	356	4	no	—
226	M₁₆	—	356	4	no	—
227	M₁₆	—	356	2	no	—
228	M₁₆	—	178	4	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	89	1	no	—
231	C₂×C₈	—	89	1	no	—
232	C₂×C₈	—	89	2	no	—
233	C₂×C₈	—	89	2	no	—
234	C₂×C₈	—	89	1	no	—
235	C₂×C₈	—	89	1	no	—
236	C₂×C₈	—	89	2	no	—
237	C₂×C₈	—	89	1	no	—
238	C₂×C₈	—	89	1	no	—
239	C₂×C₈	—	89	2	no	—
240	C₂×C₈	—	89	2	no	—
241	C₂×C₈	—	89	2	no	—
242	C₂×C₈	—	89	2	no	—
243	C₂×C₈	—	89	2	no	—
244	C₂×C₈	—	356	4	no	—
245	C₂×C₈	—	356	4	no	—
246	C₂×C₈	—	356	2	no	—
247	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
248	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
249	C₂×C₈	Q₈×D₈₉	1	2	yes	—
250	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
251	C₁₆	—	89	4	no	—
252	C₁₆	C₂×C₄×D₈₉	1	4	yes	—
253	C₂³	—	89	2	no	—
254	C₂³	—	89	1	no	—
255	C₂³	—	89	2	no	—
256	C₂³	—	89	1	no	—
257	C₂³	—	356	2	no	—
258	C₂³	Q₁₆×D₈₉	1	1	yes	characteristic
259	C₂×C₄	—	89	2	no	—
260	C₂×C₄	—	89	2	no	—
261	C₂×C₄	—	89	1	no	—
262	C₂×C₄	—	89	1	no	—
263	C₂×C₄	—	89	2	no	—
264	C₂×C₄	—	89	2	no	—
265	C₂×C₄	—	89	2	no	—
266	C₂×C₄	—	89	2	no	—
267	C₂×C₄	—	89	1	no	—
268	C₂×C₄	—	89	1	no	—
269	C₂×C₄	—	89	2	no	—
270	C₂×C₄	—	89	1	no	—
271	C₂×C₄	—	89	2	no	—
272	C₂×C₄	—	89	1	no	—
273	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
274	C₂×C₄	D₈×D₈₉	1	2	yes	—
275	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
276	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
277	C₈	—	4	4	no	—
278	C₈	—	89	1	no	—
279	C₈	—	89	1	no	—
280	C₈	—	89	2	no	—
281	C₈	—	356	4	no	—
282	C₈	C₄⋊C₄×D₈₉	1	2	yes	—
283	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
284	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
285	C₂²	—	4	1	no	—
286	C₂²	—	89	1	no	—
287	C₂²	—	89	2	no	—
288	C₂²	—	89	1	no	—
289	C₂²	—	356	2	no	—
290	C₂²	—	356	1	no	—
291	C₂²	—	178	4	no	—
292	C₂²	—	178	2	no	—
293	C₂²	#107×D₈₉	1	2	yes	—
294	C₂²	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
295	C₄	—	89	1	no	—
296	C₄	—	89	2	no	—
297	C₄	—	89	1	no	—
298	C₄	#107×D₈₉	1	2	yes	—
299	C₄	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
300	C₄	C₂×D₈×D₈₉	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	2	1	no	—
303	C₂	—	89	2	no	—
304	C₂	—	178	2	no	—
305	C₂	—	178	1	no	—
306	C₂	C₂×#107×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2161	C₂
socle	C₂×C₈₉	C₂²×#107
prime radical	C₂³×D₈₉	Q₁₆
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2161	D₈₉
hypocenter	C₈₉	C₂×#2161
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

Complements

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

N	H ≅ G/N	type
3. #26022047	C₂	semidirect
4. #26022046	C₂	semidirect
5. #26022045	C₂	semidirect
6. #368×D₈₉	C₂	semidirect
7. C₈₉×#2161	C₂	semidirect
21. #19151428	C₂²	semidirect
22. #19151427	C₂²	semidirect
25. #368×C₈₉	C₂²	semidirect
153. D₈₉	#2161	direct (N × H)
159. #2161	D₈₉	direct (N × H)
161. C₈₉	C₂×#2161	semidirect
179. #368	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₈₉	2
C₂×#108×D₈₉	2
#26022047	2
#26022046	2
#26022045	2
#368×D₈₉	2
C₈₉×#2161	2
C₂×#2161	89

Subgroups — #2161×D89 #36707481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2161×D₈₉ #36707481