Subgroups — #2176×D₈₉ #36707721

Order 22784.

670 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×D₈₉	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆×D₈₉	C₂	1	2	yes	maximal
3	#26022124	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022123	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022122	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022121	C₂	1	2	yes	maximal, complemented
7	#26022120	C₂	1	2	yes	maximal, complemented
8	#367×D₈₉	C₂	1	1	yes	maximal, characteristic
9	#358×D₈₉	C₂	1	2	yes	maximal
10	#346×D₈₉	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2176	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₈₉	C₂²	1	2	yes	—
13	C₄×Q₈×D₈₉	—	2	2	no	—
14	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
15	C₂×#15667958	C₂²	1	2	yes	—
16	Q₃₂×D₈₉	C₄	1	8	yes	complemented
17	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
19	C₄×Dic₃₅₆	C₂²	1	2	yes	—
20	Q₁₆×Dic₈₉	C₂²	1	2	yes	—
21	#19151692	C₂²	1	2	yes	—
22	C₄×#13083564	—	2	2	no	—
23	#19151690	—	2	2	no	—
24	#107×D₈₉	C₂²	1	2	yes	—
25	#103×D₈₉	—	2	2	no	—
26	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
27	#19151426	C₂²	1	1	yes	characteristic
28	#19151425	C₂²	1	2	yes	—
29	#19151403	C₂²	1	2	yes	—
30	#19151402	C₂²	1	2	yes	—
31	#19151401	C₂²	1	2	yes	—
32	#19151374	C₂²	1	1	yes	characteristic
33	#19151373	C₂²	1	1	yes	characteristic
34	#19151373	C₂²	1	1	yes	characteristic
35	C₂×Q₃₂×C₈₉	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
37	#367×C₈₉	C₂²	1	1	yes	characteristic
38	#358×C₈₉	C₂²	1	2	yes	—
39	#346×C₈₉	C₂²	1	1	yes	characteristic
40	C₂×Q₈×D₈₉	—	2	2	no	—
41	C₂×Dic₃₅₆	C₂³	1	2	yes	—
42	C₂×#13083564	—	2	2	no	—
43	Q₁₆×D₈₉	—	2	4	no	—
44	Q₁₆×D₈₉	C₂×C₄	1	8	yes	—
45	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
46	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
47	Q₈×Dic₈₉	—	2	2	no	—
48	C₄×Dic₁₇₈	—	2	2	no	—
49	C₄×Dic₁₇₈	—	4	2	no	—
50	#15667977	—	2	2	no	—
51	#15667977	—	4	2	no	—
52	C₄⋊C₄×D₈₉	—	2	2	no	—
53	C₄⋊C₄×D₈₉	—	4	2	no	—
54	C₄²×D₈₉	—	4	2	no	—
55	C₄²×D₈₉	D₄	1	1	yes	characteristic
56	Dic₇₁₂	C₂×C₄	1	4	yes	complemented
57	#15667958	C₂×C₄	1	8	yes	complemented
58	C₂×#13083549	C₂³	1	1	yes	characteristic
59	C₂×#13083549	C₂³	1	1	yes	characteristic
60	C₁₆×D₈₉	—	2	2	no	—
61	C₁₆×D₈₉	C₂×C₄	1	4	yes	—
62	#15667946	C₂³	1	2	yes	—
63	#15667945	—	2	2	no	—
64	#15667937	—	2	2	no	—
65	#15667936	—	2	2	no	—
66	#15667935	—	2	2	no	—
67	C₄×#10623494	D₄	1	1	yes	characteristic
68	C₈×Dic₈₉	D₄	1	1	yes	characteristic
69	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
71	C₄×Q₈×C₈₉	—	2	2	no	—
72	Q₃₂×C₈₉	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
75	#107×C₈₉	C₂³	1	2	yes	—
76	#103×C₈₉	—	2	2	no	—
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₂×Dic₁₇₈	—	2	2	no	—
79	C₂×Dic₁₇₈	—	4	2	no	—
80	Q₈×D₈₉	—	2	8	no	—
81	Q₈×D₈₉	—	4	4	no	—
82	C₂×C₄×D₈₉	—	4	2	no	—
83	C₂×C₄×D₈₉	—	4	2	no	—
84	C₂×C₄×D₈₉	D₈	1	2	yes	—
85	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
86	Dic₃₅₆	—	2	2	no	—
87	Dic₃₅₆	C₂²×C₄	1	4	yes	—
88	#13083564	—	2	8	no	—
89	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
90	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
91	C₈×D₈₉	—	2	2	no	—
92	C₈×D₈₉	C₂²×C₄	1	2	yes	—
93	C₈×D₈₉	#41	1	2	yes	—
94	C₄⋊Dic₈₉	—	2	2	no	—
95	C₄⋊Dic₈₉	—	4	2	no	—
96	#13083553	—	4	2	no	—
97	#13083553	—	4	2	no	—
98	#13083553	—	4	2	no	—
99	C₄×Dic₈₉	—	4	2	no	—
100	C₄×Dic₈₉	—	4	2	no	—
101	C₄×Dic₈₉	—	4	2	no	—
102	C₄×Dic₈₉	D₈	1	2	yes	—
103	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
104	#13083549	—	2	1	no	—
105	#13083549	C₂²×C₄	1	2	yes	—
106	C₂×Q₈×C₈₉	—	2	2	no	—
107	Q₁₆×C₈₉	—	2	2	no	—
108	Q₁₆×C₈₉	C₂²×C₄	1	4	yes	—
109	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₈₉	—	2	2	no	—
112	C₄⋊C₄×C₈₉	—	4	2	no	—
113	C₄²×C₈₉	—	4	2	no	—
114	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₈₉	—	2	1	no	—
116	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
117	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
118	Q₈×C₈₉	—	2	4	no	—
119	Q₈×C₈₉	—	4	2	no	—
120	C₂×C₄×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	—	4	2	no	—
122	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×Dic₈₉	—	4	2	no	—
125	C₂×Dic₈₉	—	4	2	no	—
126	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
127	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
128	C₄×D₈₉	—	2	4	no	—
129	C₄×D₈₉	—	4	8	no	—
130	C₄×D₈₉	—	8	4	no	—
131	C₄×D₈₉	C₄×D₄	1	2	yes	—
132	C₄×D₈₉	#135	1	2	yes	—
133	Dic₁₇₈	—	2	4	no	—
134	Dic₁₇₈	—	4	2	no	—
135	Dic₁₇₈	—	4	8	no	—
136	C₈×C₈₉	—	2	1	no	—
137	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
138	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
139	#10623494	—	2	1	no	—
140	#10623494	C₄×D₄	1	2	yes	—
141	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
142	C₂×D₈₉	C₄×D₈	1	2	yes	—
143	C₂×D₈₉	#510	1	4	yes	—
144	C₄×C₈₉	—	2	2	no	—
145	C₄×C₈₉	—	4	4	no	—
146	C₄×C₈₉	—	8	2	no	—
147	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
149	Dic₈₉	—	2	2	no	—
150	Dic₈₉	—	4	4	no	—
151	Dic₈₉	—	8	2	no	—
152	Dic₈₉	C₄×D₈	1	2	yes	—
153	C₂×#2176	—	89	1	no	maximal, Sylow
154	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
155	C₂×C₈₉	C₂×#510	1	2	yes	—
156	D₈₉	#2176	1	4	yes	direct product factor
157	C₂²×Q₃₂	—	89	1	no	—
158	C₂×C₄×Q₁₆	—	89	2	no	—
159	#2176	—	89	1	no	—
160	#2176	—	89	2	no	—
161	#2176	—	89	1	no	—
162	#2176	—	89	2	no	—
163	#2176	—	89	1	no	—
164	#2176	D₈₉	1	1	yes	characteristic, direct product factor
165	C₂×#367	—	89	1	no	—
166	C₂×#358	—	89	2	no	—
167	C₂×#346	—	89	1	no	—
168	C₈₉	C₂×#2176	1	1	yes	characteristic, complemented, Sylow
169	C₂²×Q₁₆	—	89	2	no	—
170	C₂×C₄×Q₈	—	178	2	no	—
171	C₂×Q₃₂	—	89	2	no	—
172	C₂×Q₃₂	—	89	8	no	—
173	C₂×Q₃₂	—	89	1	no	—
174	C₂×Q₃₂	C₂×D₈₉	1	1	yes	characteristic
175	C₂²×C₁₆	—	89	1	no	—
176	C₂²×C₁₆	—	89	1	no	—
177	C₄×Q₁₆	—	89	2	no	—
178	C₄×Q₁₆	—	89	2	no	—
179	C₄×Q₁₆	—	89	2	no	—
180	C₄×Q₁₆	—	178	2	no	—
181	C₄×Q₁₆	—	178	2	no	—
182	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
183	C₂×#107	—	89	2	no	—
184	C₂×#103	—	178	2	no	—
185	C₂×C₄×C₈	—	89	1	no	—
186	#367	—	89	1	no	—
187	#367	—	89	2	no	—
188	#367	C₂×D₈₉	1	1	yes	characteristic
189	#358	—	89	2	no	—
190	#358	—	89	2	no	—
191	#358	—	89	2	no	—
192	#358	C₂×D₈₉	1	2	yes	—
193	#346	—	89	1	no	—
194	#346	—	89	1	no	—
195	#346	—	89	1	no	—
196	#346	C₂×D₈₉	1	1	yes	characteristic
197	C₂²×Q₈	—	178	2	no	—
198	C₂×Q₁₆	—	89	8	no	—
199	C₂×Q₁₆	—	89	2	no	—
200	C₂×Q₁₆	—	178	4	no	—
201	C₂×Q₁₆	—	178	2	no	—
202	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
203	C₂²×C₈	—	89	1	no	—
204	C₂²×C₈	—	89	1	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×Q₈	—	356	2	no	—
207	C₄×Q₈	—	356	2	no	—
208	C₄×Q₈	—	178	2	no	—
209	C₄×Q₈	—	178	2	no	—
210	C₄×Q₈	—	178	2	no	—
211	C₂×C₄⋊C₄	—	356	2	no	—
212	C₂×C₄⋊C₄	—	178	2	no	—
213	C₂×C₄²	—	89	1	no	—
214	C₂×C₄²	—	356	2	no	—
215	Q₃₂	—	89	8	no	—
216	Q₃₂	—	89	4	no	—
217	Q₃₂	C₄×D₈₉	1	4	yes	complemented
218	C₂×C₁₆	—	89	1	no	—
219	C₂×C₁₆	—	89	4	no	—
220	C₂×C₁₆	—	89	1	no	—
221	C₂×C₁₆	—	178	2	no	—
222	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
223	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
224	#107	—	89	2	no	—
225	#107	—	178	2	no	—
226	#107	C₂²×D₈₉	1	2	yes	—
227	#103	—	2	2	no	—
228	#103	—	178	2	no	—
229	#103	—	178	2	no	—
230	#103	—	178	2	no	—
231	C₄×C₈	—	89	1	no	—
232	C₄×C₈	—	89	1	no	—
233	C₄×C₈	—	89	1	no	—
234	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	356	2	no	—
237	C₂×Q₈	—	356	4	no	—
238	C₂×Q₈	—	178	8	no	—
239	C₂×Q₈	—	178	2	no	—
240	C₂²×C₄	—	89	1	no	—
241	C₂²×C₄	—	89	2	no	—
242	C₂²×C₄	—	356	2	no	—
243	C₂²×C₄	—	356	2	no	—
244	Q₁₆	—	2	2	no	—
245	Q₁₆	—	89	4	no	—
246	Q₁₆	—	178	8	no	—
247	Q₁₆	—	178	2	no	—
248	Q₁₆	C₂×C₄×D₈₉	1	4	yes	—
249	C₂×C₈	—	89	1	no	—
250	C₂×C₈	—	89	2	no	—
251	C₂×C₈	—	89	2	no	—
252	C₂×C₈	—	89	1	no	—
253	C₂×C₈	—	178	2	no	—
254	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
255	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	356	2	no	—
259	C₄⋊C₄	—	356	2	no	—
260	C₄⋊C₄	—	356	2	no	—
261	C₄⋊C₄	—	356	2	no	—
262	C₄⋊C₄	—	178	2	no	—
263	C₄²	—	4	2	no	—
264	C₄²	—	89	2	no	—
265	C₄²	—	89	1	no	—
266	C₄²	—	356	2	no	—
267	C₄²	—	356	2	no	—
268	C₄²	—	356	2	no	—
269	C₄²	D₄×D₈₉	1	1	yes	characteristic
270	C₁₆	—	2	1	no	—
271	C₁₆	—	89	2	no	—
272	C₁₆	—	178	1	no	—
273	C₁₆	C₂×C₄×D₈₉	1	2	yes	—
274	C₂³	—	89	1	no	—
275	Q₈	—	2	4	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	356	8	no	—
278	Q₈	—	356	2	no	—
279	Q₈	—	178	4	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	89	1	no	—
283	C₂×C₄	—	89	2	no	—
284	C₂×C₄	—	89	2	no	—
285	C₂×C₄	—	89	2	no	—
286	C₂×C₄	—	356	8	no	—
287	C₂×C₄	—	356	2	no	—
288	C₂×C₄	—	356	2	no	—
289	C₂×C₄	—	178	4	no	—
290	C₂×C₄	—	712	4	no	—
291	C₂×C₄	D₈×D₈₉	1	2	yes	—
292	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	89	2	no	—
295	C₈	—	178	1	no	—
296	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
297	C₈	#41×D₈₉	1	1	yes	characteristic
298	C₂²	—	89	2	no	—
299	C₂²	—	89	4	no	—
300	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
301	C₄	—	2	2	no	—
302	C₄	—	4	4	no	—
303	C₄	—	8	2	no	—
304	C₄	—	89	2	no	—
305	C₄	—	356	4	no	—
306	C₄	—	178	2	no	—
307	C₄	—	712	2	no	—
308	C₄	C₄×D₄×D₈₉	1	1	yes	characteristic
309	C₄	#135×D₈₉	1	1	yes	characteristic
310	C₂	—	89	4	no	—
311	C₂	C₄×D₈×D₈₉	1	1	yes	characteristic
312	C₂	#510×D₈₉	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2176	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂²×D₈₉	C₂×D₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2176	D₈₉
hypocenter	C₈₉	C₂×#2176
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

Complements

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

N	H ≅ G/N	type
3. #26022124	C₂	semidirect
4. #26022123	C₂	semidirect
5. #26022122	C₂	semidirect
6. #26022121	C₂	semidirect
7. #26022120	C₂	semidirect
11. C₈₉×#2176	C₂	semidirect
16. Q₃₂×D₈₉	C₄	semidirect
56. Dic₇₁₂	C₂×C₄	semidirect
57. #15667958	C₂×C₄	semidirect
72. Q₃₂×C₈₉	C₂×C₄	semidirect
156. D₈₉	#2176	direct (N × H)
164. #2176	D₈₉	direct (N × H)
168. C₈₉	C₂×#2176	semidirect
217. Q₃₂	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×Q₃₂×D₈₉	2
C₄×Q₁₆×D₈₉	2
#26022124	2
#26022123	2
#26022122	2
#26022121	2
#26022120	2
#367×D₈₉	2
#358×D₈₉	2
#346×D₈₉	2
C₈₉×#2176	2
C₂×#2176	89

Subgroups — #2176×D89 #36707721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2176×D₈₉ #36707721