Subgroups — #2158×D₈₉ #36707453

Order 22784.

694 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
2	#427×D₈₉	C₂	1	2	yes	maximal
3	#26022036	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022035	C₂	1	2	yes	maximal, complemented
5	#26022034	C₂	1	2	yes	maximal, complemented
6	#26022033	C₂	1	2	yes	maximal, complemented
7	#367×D₈₉	C₂	1	2	yes	maximal, complemented
8	#366×D₈₉	C₂	1	2	yes	maximal, complemented
9	C₈₉×#2158	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
11	#117×D₈₉	—	2	2	no	—
12	C₂²×#13083549	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×D₈₉	C₄	1	8	yes	complemented
14	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
18	#19151636	C₂²	1	2	yes	—
19	#19151635	C₂²	1	4	yes	—
20	#19151634	—	2	2	no	—
21	#19151633	—	2	2	no	—
22	#107×D₈₉	C₂²	1	4	yes	—
23	#106×D₈₉	—	2	2	no	—
24	#19151426	C₂²	1	2	yes	complemented
25	#19151425	C₂²	1	4	yes	complemented
26	#19151424	C₂²	1	2	yes	complemented
27	#19151423	C₂²	1	4	yes	complemented
28	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
29	#427×C₈₉	C₂²	1	2	yes	—
30	#367×C₈₉	C₂²	1	2	yes	complemented
31	#366×C₈₉	C₂²	1	2	yes	complemented
32	C₂²×C₄×D₈₉	D₄	1	1	yes	characteristic
33	C₂²×#10623494	D₄	1	1	yes	characteristic
34	C₂×C₈×D₈₉	C₂×C₄	1	4	yes	—
35	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
36	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
37	C₂×C₈×D₈₉	Q₈	1	4	yes	—
38	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
39	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
40	#15667971	—	2	2	no	—
41	#15667970	—	2	4	no	—
42	#15667969	—	4	2	no	—
43	#15667968	—	4	2	no	—
44	C₄⋊C₄×D₈₉	—	2	4	no	—
45	C₂²⋊C₄×D₈₉	—	4	2	no	—
46	C₄²×D₈₉	—	4	2	no	—
47	C₂×#13083549	C₂×C₄	1	4	yes	complemented
48	C₂×#13083549	C₂³	1	1	yes	characteristic
49	C₂×#13083549	C₂³	1	1	yes	characteristic
50	C₁₆×D₈₉	C₂×C₄	1	4	yes	—
51	C₁₆×D₈₉	C₂×C₄	1	4	yes	—
52	C₁₆×D₈₉	C₂×C₄	1	4	yes	complemented
53	C₁₆×D₈₉	C₂×C₄	1	4	yes	complemented
54	#15667946	C₂³	1	4	yes	—
55	#15667945	—	2	4	no	—
56	#15667944	—	2	2	no	—
57	#15667943	—	2	4	no	—
58	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
59	#117×C₈₉	—	2	2	no	—
60	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
61	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
63	#107×C₈₉	C₂³	1	4	yes	—
64	#106×C₈₉	—	2	2	no	—
65	C₂³×D₈₉	D₈	1	1	yes	characteristic
66	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
67	C₂×C₄×D₈₉	—	4	4	no	—
68	C₂×C₄×D₈₉	C₄⋊C₄	1	4	yes	—
69	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
70	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
71	C₂×C₄×D₈₉	Q₁₆	1	4	yes	—
72	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
74	C₂×#10623494	C₄⋊C₄	1	4	yes	—
75	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
76	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
77	C₈×D₈₉	C₄⋊C₄	1	4	yes	—
78	C₈×D₈₉	C₄⋊C₄	1	4	yes	—
79	C₈×D₈₉	C₂²×C₄	1	2	yes	—
80	C₈×D₈₉	C₂²×C₄	1	2	yes	—
81	C₈×D₈₉	C₂×Q₈	1	2	yes	—
82	C₈×D₈₉	C₂×Q₈	1	2	yes	—
83	C₄⋊Dic₈₉	—	2	4	no	—
84	#13083553	—	4	4	no	—
85	C₂²⋊Dic₈₉	—	4	2	no	—
86	#13083551	—	4	4	no	—
87	C₄×Dic₈₉	—	4	2	no	—
88	C₄×Dic₈₉	—	4	4	no	—
89	#13083549	C₂²×C₄	1	2	yes	—
90	#13083549	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
95	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₈₉	—	2	4	no	—
97	C₂²⋊C₄×C₈₉	—	4	2	no	—
98	C₄²×C₈₉	—	4	2	no	—
99	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
100	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
101	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
102	C₂²×D₈₉	—	2	4	no	—
103	C₂²×D₈₉	#107	1	4	yes	—
104	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
105	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₈₉	—	4	4	no	—
107	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
110	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
111	C₂×Dic₈₉	—	4	4	no	—
112	C₂×Dic₈₉	#107	1	4	yes	—
113	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
114	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
115	C₄×D₈₉	—	8	8	no	—
116	C₄×D₈₉	#107	1	4	yes	—
117	C₄×D₈₉	#107	1	4	yes	—
118	C₄×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
119	C₄×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
120	C₄×D₈₉	C₂×Q₁₆	1	2	yes	—
121	C₄×D₈₉	C₂×Q₁₆	1	2	yes	—
122	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
123	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
124	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
125	#10623494	C₂×C₄⋊C₄	1	2	yes	—
126	#10623494	C₂×C₄⋊C₄	1	2	yes	—
127	C₂²×C₈₉	—	2	2	no	—
128	C₂²×C₈₉	C₂×#107	1	2	yes	—
129	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
130	C₂×D₈₉	—	2	2	no	—
131	C₂×D₈₉	—	2	8	no	—
132	C₂×D₈₉	C₂×#107	1	2	yes	—
133	C₂×D₈₉	C₂×#107	1	2	yes	—
134	C₂×D₈₉	#508	1	4	yes	—
135	C₄×C₈₉	—	8	4	no	—
136	C₄×C₈₉	C₂×#107	1	2	yes	—
137	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
139	Dic₈₉	—	8	4	no	—
140	Dic₈₉	C₂×#107	1	2	yes	—
141	Dic₈₉	C₂×#107	1	2	yes	—
142	C₂×#2158	—	89	1	no	maximal, Sylow
143	C₂×C₈₉	—	2	2	no	—
144	C₂×C₈₉	C₂²×#107	1	1	yes	characteristic
145	C₂×C₈₉	C₂×#508	1	2	yes	—
146	D₈₉	—	2	2	no	—
147	D₈₉	#2158	1	4	yes	direct product factor
148	C₂³×C₁₆	—	89	1	no	—
149	C₂×#427	—	89	2	no	—
150	#2158	—	89	1	no	—
151	#2158	—	89	2	no	—
152	#2158	—	89	2	no	—
153	#2158	—	89	2	no	—
154	#2158	D₈₉	1	1	yes	characteristic, direct product factor
155	C₂×#367	—	89	2	no	—
156	C₂×#366	—	89	2	no	—
157	C₈₉	C₂×#2158	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₈	—	89	1	no	—
159	C₂×#117	—	178	2	no	—
160	C₂²×C₁₆	—	89	1	no	—
161	C₂²×C₁₆	—	89	8	no	—
162	C₂²×C₁₆	—	89	1	no	—
163	C₂²×C₁₆	—	89	1	no	—
164	C₂²×C₁₆	—	89	1	no	—
165	C₂²×C₁₆	—	89	1	no	—
166	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
167	#427	—	89	2	no	—
168	#427	—	89	4	no	—
169	#427	—	178	2	no	—
170	#427	—	178	2	no	—
171	#427	C₂×D₈₉	1	2	yes	—
172	C₂×#107	—	89	4	no	—
173	C₂×#106	—	178	2	no	—
174	#367	—	89	2	no	—
175	#367	—	89	4	no	—
176	#367	C₂×D₈₉	1	2	yes	complemented
177	#366	—	89	2	no	—
178	#366	—	89	4	no	—
179	#366	C₂×D₈₉	1	2	yes	complemented
180	C₂³×C₄	—	89	1	no	—
181	C₂²×C₈	—	89	1	no	—
182	C₂²×C₈	—	89	4	no	—
183	C₂²×C₈	—	89	1	no	—
184	C₂²×C₈	—	89	1	no	—
185	C₂²×C₈	—	89	4	no	—
186	C₂²×C₈	—	89	1	no	—
187	C₂²×C₈	—	89	1	no	—
188	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
189	#117	—	2	2	no	—
190	#117	—	356	2	no	—
191	#117	—	356	2	no	—
192	#117	—	178	4	no	—
193	#117	—	178	2	no	—
194	C₂×C₄⋊C₄	—	178	4	no	—
195	C₂×C₂²⋊C₄	—	356	2	no	—
196	C₂×C₄²	—	356	2	no	—
197	C₂×C₁₆	—	89	4	no	—
198	C₂×C₁₆	—	89	1	no	—
199	C₂×C₁₆	—	89	1	no	—
200	C₂×C₁₆	—	89	4	no	—
201	C₂×C₁₆	—	89	4	no	—
202	C₂×C₁₆	—	89	4	no	—
203	C₂×C₁₆	—	89	4	no	—
204	C₂×C₁₆	C₄×D₈₉	1	4	yes	complemented
205	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
206	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
207	#107	—	89	4	no	—
208	#107	—	178	4	no	—
209	#107	C₂²×D₈₉	1	4	yes	—
210	#106	—	2	2	no	—
211	#106	—	178	4	no	—
212	#106	—	178	2	no	—
213	C₂⁴	—	89	1	no	—
214	C₂²×C₄	—	89	1	no	—
215	C₂²×C₄	—	89	1	no	—
216	C₂²×C₄	—	89	1	no	—
217	C₂²×C₄	—	89	4	no	—
218	C₂²×C₄	—	89	1	no	—
219	C₂²×C₄	—	89	1	no	—
220	C₂²×C₄	—	89	4	no	—
221	C₂²×C₄	—	356	4	no	—
222	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
223	C₂×C₈	—	89	1	no	—
224	C₂×C₈	—	89	1	no	—
225	C₂×C₈	—	89	4	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	4	no	—
231	C₂×C₈	—	89	4	no	—
232	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
233	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
234	C₂×C₈	Q₈×D₈₉	1	2	yes	—
235	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	356	4	no	—
238	C₄⋊C₄	—	178	4	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	356	4	no	—
241	C₂²⋊C₄	—	356	2	no	—
242	C₄²	—	4	2	no	—
243	C₄²	—	356	4	no	—
244	C₄²	—	356	2	no	—
245	C₁₆	—	89	2	no	—
246	C₁₆	—	89	2	no	—
247	C₁₆	C₂×C₄×D₈₉	1	2	yes	—
248	C₁₆	C₂×C₄×D₈₉	1	2	yes	—
249	C₂³	—	89	1	no	—
250	C₂³	—	89	4	no	—
251	C₂³	—	89	1	no	—
252	C₂³	—	178	4	no	—
253	C₂³	D₈×D₈₉	1	1	yes	characteristic
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	89	4	no	—
256	C₂×C₄	—	89	1	no	—
257	C₂×C₄	—	89	1	no	—
258	C₂×C₄	—	89	4	no	—
259	C₂×C₄	—	89	4	no	—
260	C₂×C₄	—	89	2	no	—
261	C₂×C₄	—	89	2	no	—
262	C₂×C₄	—	89	2	no	—
263	C₂×C₄	—	89	2	no	—
264	C₂×C₄	—	356	4	no	—
265	C₂×C₄	—	712	8	no	—
266	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
267	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
268	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
269	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
270	C₈	—	89	2	no	—
271	C₈	—	89	2	no	—
272	C₈	C₄⋊C₄×D₈₉	1	2	yes	—
273	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
274	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	89	2	no	—
277	C₂²	—	89	4	no	—
278	C₂²	—	89	2	no	—
279	C₂²	—	178	8	no	—
280	C₂²	—	178	2	no	—
281	C₂²	#107×D₈₉	1	2	yes	—
282	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
283	C₄	—	8	4	no	—
284	C₄	—	89	2	no	—
285	C₄	—	89	2	no	—
286	C₄	—	712	4	no	—
287	C₄	#107×D₈₉	1	2	yes	—
288	C₄	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
289	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	89	4	no	—
292	C₂	—	178	2	no	—
293	C₂	C₂×#107×D₈₉	1	1	yes	characteristic
294	C₂	#508×D₈₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2158	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂³×D₈₉	D₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2158	D₈₉
hypocenter	C₈₉	C₂×#2158
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
3. #26022036	C₂	semidirect
4. #26022035	C₂	semidirect
5. #26022034	C₂	semidirect
6. #26022033	C₂	semidirect
7. #367×D₈₉	C₂	semidirect
8. #366×D₈₉	C₂	semidirect
9. C₈₉×#2158	C₂	semidirect
13. C₂×C₁₆×D₈₉	C₄	semidirect
24. #19151426	C₂²	semidirect
25. #19151425	C₂²	semidirect
26. #19151424	C₂²	semidirect
27. #19151423	C₂²	semidirect
30. #367×C₈₉	C₂²	semidirect
31. #366×C₈₉	C₂²	semidirect
47. C₂×#13083549	C₂×C₄	semidirect
52. C₁₆×D₈₉	C₂×C₄	semidirect
53. C₁₆×D₈₉	C₂×C₄	semidirect
60. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
147. D₈₉	#2158	direct (N × H)
154. #2158	D₈₉	direct (N × H)
157. C₈₉	C₂×#2158	semidirect
176. #367	C₂×D₈₉	semidirect
179. #366	C₂×D₈₉	semidirect
204. C₂×C₁₆	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₈₉	2
#427×D₈₉	2
#26022036	2
#26022035	2
#26022034	2
#26022033	2
#367×D₈₉	2
#366×D₈₉	2
C₈₉×#2158	2
C₂×#2158	89

Subgroups — #2158×D89 #36707453

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2158×D₈₉ #36707453