Subgroups — #1384×D₈₉ #36469335

Order 22784.

586 conjugacy classes of subgroups.

The subgroup lattice has 304 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	#427×D₈₉	C₂	1	2	yes	maximal
3	#26018354	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018353	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1384	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
7	#117×D₈₉	—	2	2	no	—
8	C₂×#15667952	C₂²	1	1	yes	characteristic
9	C₂×#15667951	C₄	1	2	yes	complemented
10	M₃₂×D₈₉	—	2	2	no	—
11	M₃₂×D₈₉	—	2	2	no	—
12	C₂×C₁₆×D₈₉	C₄	1	2	yes	complemented
13	#19151636	C₂²	1	2	yes	—
14	#19151635	—	2	2	no	—
15	#19151634	—	2	2	no	—
16	#19151633	C₄	1	4	yes	complemented
17	#107×D₈₉	—	2	2	no	—
18	#106×D₈₉	C₄	1	4	yes	complemented
19	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
20	#427×C₈₉	C₂²	1	2	yes	—
21	C₂²×C₄×D₈₉	D₄	1	1	yes	characteristic
22	C₂²×#10623494	D₄	1	1	yes	characteristic
23	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
26	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
27	C₂×C₈×D₈₉	D₄	1	2	yes	—
28	C₂×C₈×D₈₉	D₄	1	2	yes	—
29	C₂×C₈×D₈₉	Q₈	1	1	yes	characteristic
30	C₂×C₈×D₈₉	Q₈	1	1	yes	characteristic
31	#15667971	—	2	2	no	—
32	#15667970	—	2	4	no	—
33	#15667969	—	4	2	no	—
34	#15667968	—	4	2	no	—
35	C₄⋊C₄×D₈₉	—	2	4	no	—
36	C₂²⋊C₄×D₈₉	—	4	2	no	—
37	C₄²×D₈₉	—	4	2	no	—
38	#15667952	—	2	2	no	—
39	#15667951	—	2	4	no	—
40	C₂×#13083549	C₂×C₄	1	2	yes	complemented
41	C₁₆×D₈₉	—	2	2	no	—
42	C₁₆×D₈₉	—	2	2	no	—
43	#15667946	—	2	2	no	—
44	#15667945	—	2	4	no	—
45	#15667944	C₂×C₄	1	4	yes	complemented
46	#15667943	—	2	4	no	—
47	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
48	#117×C₈₉	—	2	2	no	—
49	M₃₂×C₈₉	—	2	2	no	—
50	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	complemented
51	#107×C₈₉	—	2	2	no	—
52	#106×C₈₉	C₂×C₄	1	4	yes	complemented
53	C₂³×D₈₉	D₈	1	1	yes	characteristic
54	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
55	C₂×C₄×D₈₉	—	4	4	no	—
56	C₂×C₄×D₈₉	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×D₈₉	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×D₈₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×D₈₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×D₈₉	SD₁₆	1	2	yes	—
61	C₂×C₄×D₈₉	SD₁₆	1	2	yes	—
62	C₂×C₄×D₈₉	Q₁₆	1	1	yes	characteristic
63	C₂×C₄×D₈₉	Q₁₆	1	1	yes	characteristic
64	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
65	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
66	C₂×#10623494	C₄⋊C₄	1	1	yes	characteristic
67	C₂×#10623494	C₄⋊C₄	1	1	yes	characteristic
68	C₈×D₈₉	C₄²	1	1	yes	characteristic
69	C₈×D₈₉	C₄²	1	1	yes	characteristic
70	C₈×D₈₉	C₄²	1	2	yes	—
71	C₈×D₈₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₈×D₈₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₈×D₈₉	C₂²⋊C₄	1	2	yes	—
74	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
75	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
76	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
77	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
78	C₄⋊Dic₈₉	—	2	4	no	—
79	#13083553	—	4	4	no	—
80	C₂²⋊Dic₈₉	—	4	2	no	—
81	#13083551	—	4	4	no	—
82	C₄×Dic₈₉	—	4	2	no	—
83	C₄×Dic₈₉	—	4	4	no	—
84	#13083549	—	2	2	no	—
85	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
90	C₄⋊C₄×C₈₉	—	2	4	no	—
91	C₂²⋊C₄×C₈₉	—	4	2	no	—
92	C₄²×C₈₉	—	4	2	no	—
93	C₁₆×C₈₉	—	2	2	no	—
94	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
95	C₂²×D₈₉	—	4	2	no	—
96	C₂²×D₈₉	#102	1	2	yes	—
97	C₂²×D₈₉	#102	1	2	yes	—
98	C₂²×D₈₉	#107	1	1	yes	characteristic
99	C₂²×D₈₉	#107	1	1	yes	characteristic
100	C₂×C₄×C₈₉	—	4	4	no	—
101	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
104	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
105	C₂×Dic₈₉	—	4	4	no	—
106	C₂×Dic₈₉	#102	1	2	yes	—
107	C₂×Dic₈₉	#102	1	2	yes	—
108	C₂×Dic₈₉	#107	1	1	yes	characteristic
109	C₂×Dic₈₉	#107	1	1	yes	characteristic
110	C₄×D₈₉	—	8	8	no	—
111	C₄×D₈₉	#95	1	1	yes	characteristic
112	C₄×D₈₉	#95	1	2	yes	—
113	C₄×D₈₉	#95	1	1	yes	characteristic
114	C₄×D₈₉	#103	1	2	yes	—
115	C₄×D₈₉	#103	1	2	yes	—
116	C₄×D₈₉	#103	1	2	yes	—
117	C₄×D₈₉	#103	1	2	yes	—
118	C₄×D₈₉	#106	1	1	yes	characteristic
119	C₄×D₈₉	#106	1	2	yes	—
120	C₄×D₈₉	#106	1	1	yes	characteristic
121	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
122	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
124	#10623494	#95	1	1	yes	characteristic
125	#10623494	#95	1	1	yes	characteristic
126	#10623494	#95	1	2	yes	—
127	C₂²×C₈₉	—	4	1	no	—
128	C₂²×C₈₉	C₂×#102	1	2	yes	—
129	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
130	C₂×D₈₉	—	2	4	no	—
131	C₂×D₈₉	—	2	2	no	—
132	C₂×D₈₉	—	4	2	no	—
133	C₂×D₈₉	—	4	1	no	—
134	C₂×D₈₉	#340	1	2	yes	—
135	C₂×D₈₉	#340	1	1	yes	characteristic
136	C₂×D₈₉	#340	1	1	yes	characteristic
137	C₄×C₈₉	—	8	4	no	—
138	C₄×C₈₉	C₂×#95	1	1	yes	characteristic
139	C₄×C₈₉	C₂×#103	1	2	yes	—
140	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
141	Dic₈₉	—	8	4	no	—
142	Dic₈₉	#340	1	1	yes	characteristic
143	Dic₈₉	#340	1	2	yes	—
144	Dic₈₉	#340	1	1	yes	characteristic
145	C₂×#1384	—	89	1	no	maximal, Sylow
146	C₂×C₈₉	—	2	2	no	—
147	C₂×C₈₉	—	2	1	no	—
148	C₂×C₈₉	C₂×#340	1	1	yes	characteristic
149	D₈₉	—	2	2	no	—
150	D₈₉	—	2	1	no	—
151	D₈₉	#1384	1	2	yes	direct product factor
152	C₂²×M₃₂	—	89	1	no	—
153	C₂×#427	—	89	2	no	—
154	#1384	—	89	1	no	—
155	#1384	—	89	2	no	—
156	#1384	D₈₉	1	1	yes	characteristic, direct product factor
157	C₈₉	C₂×#1384	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₈	—	89	1	no	—
159	C₂×#117	—	178	2	no	—
160	C₂×M₃₂	—	89	1	no	—
161	C₂×M₃₂	—	89	2	no	—
162	C₂×M₃₂	—	178	2	no	—
163	C₂×M₃₂	—	178	2	no	—
164	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
165	C₂²×C₁₆	—	89	2	no	—
166	#427	—	89	2	no	—
167	#427	—	89	4	no	—
168	#427	—	178	2	no	—
169	#427	—	178	2	no	—
170	#427	C₂×D₈₉	1	2	yes	—
171	C₂×#107	—	178	2	no	—
172	C₂×#106	—	89	4	no	—
173	C₂³×C₄	—	89	1	no	—
174	C₂²×C₈	—	89	1	no	—
175	C₂²×C₈	—	89	2	no	—
176	C₂²×C₈	—	89	2	no	—
177	C₂²×C₈	—	89	1	no	—
178	C₂²×C₈	—	89	1	no	—
179	C₂²×C₈	—	89	2	no	—
180	C₂²×C₈	—	89	2	no	—
181	C₂²×C₈	—	89	1	no	—
182	C₂²×C₈	—	89	1	no	—
183	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
184	#117	—	2	2	no	—
185	#117	—	356	2	no	—
186	#117	—	356	2	no	—
187	#117	—	178	4	no	—
188	#117	—	178	2	no	—
189	C₂×C₄⋊C₄	—	178	4	no	—
190	C₂×C₂²⋊C₄	—	356	2	no	—
191	C₂×C₄²	—	356	2	no	—
192	M₃₂	—	2	2	no	—
193	M₃₂	—	178	4	no	—
194	M₃₂	—	178	2	no	—
195	C₂×C₁₆	—	89	2	no	—
196	C₂×C₁₆	—	178	2	no	—
197	C₂×C₁₆	—	178	2	no	—
198	C₂×C₁₆	C₄×D₈₉	1	2	yes	complemented
199	#107	—	2	2	no	—
200	#107	—	178	4	no	—
201	#107	—	178	2	no	—
202	#106	—	89	4	no	—
203	#106	—	178	4	no	—
204	#106	C₄×D₈₉	1	4	yes	complemented
205	C₂⁴	—	89	1	no	—
206	C₂²×C₄	—	89	1	no	—
207	C₂²×C₄	—	89	2	no	—
208	C₂²×C₄	—	89	2	no	—
209	C₂²×C₄	—	89	1	no	—
210	C₂²×C₄	—	89	1	no	—
211	C₂²×C₄	—	89	2	no	—
212	C₂²×C₄	—	89	2	no	—
213	C₂²×C₄	—	89	1	no	—
214	C₂²×C₄	—	89	1	no	—
215	C₂²×C₄	—	356	4	no	—
216	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
217	C₂×C₈	—	89	2	no	—
218	C₂×C₈	—	89	2	no	—
219	C₂×C₈	—	89	1	no	—
220	C₂×C₈	—	89	1	no	—
221	C₂×C₈	—	89	1	no	—
222	C₂×C₈	—	89	2	no	—
223	C₂×C₈	—	89	1	no	—
224	C₂×C₈	—	89	1	no	—
225	C₂×C₈	—	89	1	no	—
226	C₂×C₈	—	89	2	no	—
227	C₂×C₈	—	89	2	no	—
228	C₂×C₈	—	89	2	no	—
229	C₂×C₈	—	89	2	no	—
230	C₂×C₈	—	89	2	no	—
231	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
232	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
233	C₂×C₈	D₄×D₈₉	1	2	yes	—
234	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	356	4	no	—
237	C₄⋊C₄	—	178	4	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	356	4	no	—
240	C₂²⋊C₄	—	356	2	no	—
241	C₄²	—	4	2	no	—
242	C₄²	—	356	4	no	—
243	C₄²	—	356	2	no	—
244	C₁₆	—	2	2	no	—
245	C₁₆	—	178	2	no	—
246	C₂³	—	89	1	no	—
247	C₂³	—	89	2	no	—
248	C₂³	—	89	1	no	—
249	C₂³	—	89	2	no	—
250	C₂³	—	356	2	no	—
251	C₂³	D₈×D₈₉	1	1	yes	characteristic
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	89	1	no	—
254	C₂×C₄	—	89	1	no	—
255	C₂×C₄	—	89	2	no	—
256	C₂×C₄	—	89	2	no	—
257	C₂×C₄	—	89	1	no	—
258	C₂×C₄	—	89	2	no	—
259	C₂×C₄	—	89	1	no	—
260	C₂×C₄	—	89	1	no	—
261	C₂×C₄	—	89	1	no	—
262	C₂×C₄	—	89	2	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	—	356	4	no	—
268	C₂×C₄	—	712	8	no	—
269	C₂×C₄	C₂²⋊C₄×D₈₉	1	2	yes	—
270	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
271	C₂×C₄	SD₁₆×D₈₉	1	2	yes	—
272	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
273	C₈	—	89	1	no	—
274	C₈	—	89	2	no	—
275	C₈	—	89	1	no	—
276	C₈	C₄²×D₈₉	1	1	yes	characteristic
277	C₈	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
278	C₈	C₄⋊C₄×D₈₉	1	2	yes	—
279	C₂²	—	4	1	no	—
280	C₂²	—	89	1	no	—
281	C₂²	—	89	2	no	—
282	C₂²	—	89	1	no	—
283	C₂²	—	356	2	no	—
284	C₂²	—	356	1	no	—
285	C₂²	—	178	4	no	—
286	C₂²	—	178	2	no	—
287	C₂²	#102×D₈₉	1	2	yes	—
288	C₂²	#107×D₈₉	1	1	yes	characteristic
289	C₄	—	8	4	no	—
290	C₄	—	89	1	no	—
291	C₄	—	89	2	no	—
292	C₄	—	89	1	no	—
293	C₄	—	712	4	no	—
294	C₄	#95×D₈₉	1	1	yes	characteristic
295	C₄	#103×D₈₉	1	2	yes	—
296	C₄	#106×D₈₉	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	2	1	no	—
299	C₂	—	89	2	no	—
300	C₂	—	178	2	no	—
301	C₂	—	178	1	no	—
302	C₂	#340×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#106×D₈₉
commutator	C₈×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1384	C₂
socle	C₂×C₈₉	C₂×#340
prime radical	C₂³×D₈₉	D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1384	D₈₉
hypocenter	C₈₉	C₂×#1384
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

Complements

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

N	H ≅ G/N	type
3. #26018354	C₂	semidirect
4. #26018353	C₂	semidirect
5. C₈₉×#1384	C₂	semidirect
9. C₂×#15667951	C₄	semidirect
12. C₂×C₁₆×D₈₉	C₄	semidirect
16. #19151633	C₄	semidirect
18. #106×D₈₉	C₄	semidirect
40. C₂×#13083549	C₂×C₄	semidirect
45. #15667944	C₂×C₄	semidirect
50. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
52. #106×C₈₉	C₂×C₄	semidirect
151. D₈₉	#1384	direct (N × H)
156. #1384	D₈₉	direct (N × H)
157. C₈₉	C₂×#1384	semidirect
198. C₂×C₁₆	C₄×D₈₉	semidirect
204. #106	C₄×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
#427×D₈₉	2
#26018354	2
#26018353	2
C₈₉×#1384	2
C₂×#1384	89

Subgroups — #1384×D89 #36469335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1384×D₈₉ #36469335