Subgroups — #2177×D₈₉ #36707727

Order 22784.

798 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
2	C₄×D₈×D₈₉	C₂	1	2	yes	maximal, complemented
3	#26022129	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022128	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022127	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022126	C₂	1	2	yes	maximal, complemented
7	#26022125	C₂	1	2	yes	maximal, complemented
8	#367×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#357×D₈₉	C₂	1	2	yes	maximal, complemented
10	#346×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2177	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₈₉	C₂²	1	2	yes	—
13	C₄×D₄×D₈₉	—	2	2	no	—
14	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
15	C₂×#15667953	C₂²	1	2	yes	—
16	D₁₆×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×D₇₁₂	C₂²	1	2	yes	complemented
20	D₈×Dic₈₉	C₂²	1	2	yes	complemented
21	#19151680	C₂²	1	2	yes	complemented
22	C₄×#13083559	—	2	2	no	—
23	#19151678	—	2	2	no	—
24	#107×D₈₉	C₂²	1	2	yes	—
25	#102×D₈₉	—	2	2	no	—
26	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
27	#19151426	C₂²	1	1	yes	characteristic, complemented
28	#19151425	C₂²	1	2	yes	complemented
29	#19151400	C₂²	1	2	yes	complemented
30	#19151399	C₂²	1	2	yes	complemented
31	#19151398	C₂²	1	2	yes	complemented
32	#19151374	C₂²	1	1	yes	characteristic, complemented
33	#19151373	C₂²	1	1	yes	characteristic, complemented
34	#19151373	C₂²	1	1	yes	characteristic, complemented
35	C₂×D₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₈₉	C₂²	1	2	yes	complemented
37	#367×C₈₉	C₂²	1	1	yes	characteristic, complemented
38	#357×C₈₉	C₂²	1	2	yes	complemented
39	#346×C₈₉	C₂²	1	1	yes	characteristic, complemented
40	C₂×D₄×D₈₉	—	2	2	no	—
41	C₂²×C₄×D₈₉	—	4	2	no	—
42	C₂×D₇₁₂	C₂³	1	2	yes	—
43	C₂×#13083559	—	2	2	no	—
44	D₈×D₈₉	—	2	4	no	—
45	D₈×D₈₉	C₂×C₄	1	8	yes	complemented
46	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
47	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
48	C₄×D₃₅₆	—	2	2	no	—
49	D₄×Dic₈₉	—	2	2	no	—
50	#15667974	—	2	2	no	—
51	C₄×C₂²⋊D₈₉	—	4	2	no	—
52	#15667972	—	4	2	no	—
53	C₄⋊C₄×D₈₉	—	2	2	no	—
54	C₂²⋊C₄×D₈₉	—	4	2	no	—
55	C₄²×D₈₉	D₄	1	1	yes	characteristic
56	D₁₄₂₄	C₂×C₄	1	4	yes	complemented
57	#15667953	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	complemented
62	#15667946	C₂³	1	2	yes	—
63	#15667945	—	2	2	no	—
64	#15667934	—	2	2	no	—
65	#15667933	—	2	2	no	—
66	#15667932	—	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×D₈×C₈₉	C₂³	1	2	yes	—
71	C₄×D₄×C₈₉	—	2	2	no	—
72	D₁₆×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	#102×C₈₉	—	2	2	no	—
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₂³×D₈₉	—	4	2	no	—
79	C₂×D₃₅₆	—	2	2	no	—
80	C₂×C₂²⋊D₈₉	—	4	2	no	—
81	D₄×D₈₉	—	2	8	no	—
82	D₄×D₈₉	—	4	4	no	—
83	C₂²×Dic₈₉	—	4	2	no	—
84	C₂×C₄×D₈₉	—	4	2	no	—
85	C₂×C₄×D₈₉	—	4	2	no	—
86	C₂×C₄×D₈₉	—	4	2	no	—
87	C₂×C₄×D₈₉	—	8	8	no	—
88	C₂×C₄×D₈₉	D₈	1	2	yes	—
89	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
90	D₇₁₂	—	2	2	no	—
91	D₇₁₂	C₂²×C₄	1	4	yes	complemented
92	#13083559	—	2	8	no	—
93	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
94	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
95	C₈×D₈₉	—	2	2	no	—
96	C₈×D₈₉	C₂²×C₄	1	2	yes	—
97	C₈×D₈₉	#41	1	2	yes	—
98	C₄⋊Dic₈₉	—	2	2	no	—
99	#13083553	—	4	2	no	—
100	C₂²⋊Dic₈₉	—	4	2	no	—
101	#13083551	—	4	2	no	—
102	#13083551	—	4	2	no	—
103	C₄×Dic₈₉	D₈	1	2	yes	—
104	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
105	#13083549	—	2	1	no	—
106	#13083549	C₂²×C₄	1	2	yes	complemented
107	C₂×D₄×C₈₉	—	2	2	no	—
108	C₂²×C₄×C₈₉	—	4	2	no	—
109	D₈×C₈₉	—	2	2	no	—
110	D₈×C₈₉	C₂²×C₄	1	4	yes	complemented
111	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₈₉	—	2	2	no	—
114	C₂²⋊C₄×C₈₉	—	4	2	no	—
115	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₁₆×C₈₉	—	2	1	no	—
117	C₁₆×C₈₉	C₂²×C₄	1	2	yes	complemented
118	C₂³×C₈₉	—	4	2	no	—
119	C₂²×D₈₉	—	4	2	no	—
120	C₂²×D₈₉	—	4	8	no	—
121	C₂²×D₈₉	—	8	4	no	—
122	C₂²×D₈₉	—	8	4	no	—
123	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
124	D₄×C₈₉	—	2	4	no	—
125	D₄×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	—	4	2	no	—
127	C₂×C₄×C₈₉	—	8	4	no	—
128	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
130	C₂²⋊D₈₉	—	4	8	no	—
131	C₂×Dic₈₉	—	4	2	no	—
132	C₂×Dic₈₉	—	8	4	no	—
133	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
134	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
135	D₃₅₆	—	2	4	no	—
136	D₃₅₆	—	4	2	no	—
137	C₄×D₈₉	—	2	4	no	—
138	C₄×D₈₉	—	8	4	no	—
139	C₄×D₈₉	—	8	4	no	—
140	C₄×D₈₉	—	8	4	no	—
141	C₄×D₈₉	C₄×D₄	1	2	yes	—
142	C₄×D₈₉	#135	1	2	yes	—
143	C₈×C₈₉	—	2	1	no	—
144	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
145	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
146	#10623494	—	2	1	no	—
147	#10623494	C₄×D₄	1	2	yes	—
148	C₂²×C₈₉	—	4	4	no	—
149	C₂²×C₈₉	—	8	2	no	—
150	C₂²×C₈₉	—	8	2	no	—
151	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
152	C₂×D₈₉	—	4	4	no	—
153	C₂×D₈₉	—	8	2	no	—
154	C₂×D₈₉	—	8	2	no	—
155	C₂×D₈₉	—	8	16	no	—
156	C₂×D₈₉	C₄×D₈	1	2	yes	—
157	C₂×D₈₉	#509	1	4	yes	—
158	C₄×C₈₉	—	2	2	no	—
159	C₄×C₈₉	—	8	2	no	—
160	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
161	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
162	Dic₈₉	—	2	2	no	—
163	Dic₈₉	—	8	2	no	—
164	Dic₈₉	C₄×D₈	1	2	yes	—
165	C₂×#2177	—	89	1	no	maximal, Sylow
166	C₂×C₈₉	—	8	4	no	—
167	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
168	C₂×C₈₉	C₂×#509	1	2	yes	—
169	D₈₉	—	8	4	no	—
170	D₈₉	#2177	1	4	yes	direct product factor
171	C₂²×D₁₆	—	89	1	no	—
172	C₂×C₄×D₈	—	89	2	no	—
173	#2177	—	89	1	no	—
174	#2177	—	89	2	no	—
175	#2177	—	89	1	no	—
176	#2177	—	89	2	no	—
177	#2177	—	89	1	no	—
178	#2177	D₈₉	1	1	yes	characteristic, direct product factor
179	C₂×#367	—	89	1	no	—
180	C₂×#357	—	89	2	no	—
181	C₂×#346	—	89	1	no	—
182	C₈₉	C₂×#2177	1	1	yes	characteristic, complemented, Sylow
183	C₂²×D₈	—	89	2	no	—
184	C₂×C₄×D₄	—	178	2	no	—
185	C₂×D₁₆	—	89	2	no	—
186	C₂×D₁₆	—	89	8	no	—
187	C₂×D₁₆	—	89	1	no	—
188	C₂×D₁₆	C₂×D₈₉	1	1	yes	characteristic
189	C₂²×C₁₆	—	89	1	no	—
190	C₂²×C₁₆	—	89	1	no	—
191	C₄×D₈	—	89	2	no	—
192	C₄×D₈	—	89	2	no	—
193	C₄×D₈	—	89	2	no	—
194	C₄×D₈	—	178	2	no	—
195	C₄×D₈	—	178	2	no	—
196	C₄×D₈	C₂×D₈₉	1	2	yes	complemented
197	C₂×#107	—	89	2	no	—
198	C₂×#102	—	178	2	no	—
199	C₂×C₄×C₈	—	89	1	no	—
200	#367	—	89	1	no	—
201	#367	—	89	2	no	—
202	#367	C₂×D₈₉	1	1	yes	characteristic, complemented
203	#357	—	89	2	no	—
204	#357	—	89	2	no	—
205	#357	—	89	2	no	—
206	#357	C₂×D₈₉	1	2	yes	complemented
207	#346	—	89	1	no	—
208	#346	—	89	1	no	—
209	#346	—	89	1	no	—
210	#346	C₂×D₈₉	1	1	yes	characteristic, complemented
211	C₂²×D₄	—	178	2	no	—
212	C₂³×C₄	—	356	2	no	—
213	C₂×D₈	—	89	8	no	—
214	C₂×D₈	—	89	2	no	—
215	C₂×D₈	—	178	4	no	—
216	C₂×D₈	—	178	2	no	—
217	C₂×D₈	C₂²×D₈₉	1	2	yes	—
218	C₂²×C₈	—	89	1	no	—
219	C₂²×C₈	—	89	1	no	—
220	C₄×D₄	—	2	2	no	—
221	C₄×D₄	—	356	2	no	—
222	C₄×D₄	—	356	2	no	—
223	C₄×D₄	—	178	2	no	—
224	C₄×D₄	—	178	2	no	—
225	C₄×D₄	—	178	2	no	—
226	C₂×C₄⋊C₄	—	178	2	no	—
227	C₂×C₂²⋊C₄	—	356	2	no	—
228	C₂×C₄²	—	89	1	no	—
229	D₁₆	—	89	8	no	—
230	D₁₆	—	89	4	no	—
231	D₁₆	C₄×D₈₉	1	4	yes	complemented
232	C₂×C₁₆	—	89	1	no	—
233	C₂×C₁₆	—	89	4	no	—
234	C₂×C₁₆	—	89	1	no	—
235	C₂×C₁₆	—	178	2	no	—
236	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
237	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
238	#107	—	89	2	no	—
239	#107	—	178	2	no	—
240	#107	C₂²×D₈₉	1	2	yes	—
241	#102	—	2	2	no	—
242	#102	—	178	2	no	—
243	#102	—	178	2	no	—
244	#102	—	178	2	no	—
245	C₄×C₈	—	89	1	no	—
246	C₄×C₈	—	89	1	no	—
247	C₄×C₈	—	89	1	no	—
248	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
249	C₂⁴	—	356	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	356	4	no	—
252	C₂×D₄	—	356	2	no	—
253	C₂×D₄	—	178	8	no	—
254	C₂×D₄	—	178	2	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	89	1	no	—
257	C₂²×C₄	—	89	2	no	—
258	C₂²×C₄	—	356	2	no	—
259	C₂²×C₄	—	356	2	no	—
260	C₂²×C₄	—	356	2	no	—
261	C₂²×C₄	—	356	2	no	—
262	C₂²×C₄	—	712	8	no	—
263	D₈	—	2	2	no	—
264	D₈	—	89	4	no	—
265	D₈	—	178	8	no	—
266	D₈	—	178	2	no	—
267	D₈	C₂×C₄×D₈₉	1	4	yes	complemented
268	C₂×C₈	—	89	1	no	—
269	C₂×C₈	—	89	2	no	—
270	C₂×C₈	—	89	2	no	—
271	C₂×C₈	—	89	1	no	—
272	C₂×C₈	—	178	2	no	—
273	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	356	2	no	—
277	C₄⋊C₄	—	178	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	356	2	no	—
280	C₂²⋊C₄	—	356	2	no	—
281	C₂²⋊C₄	—	356	2	no	—
282	C₄²	—	89	2	no	—
283	C₄²	—	89	1	no	—
284	C₄²	D₄×D₈₉	1	1	yes	characteristic
285	C₁₆	—	2	1	no	—
286	C₁₆	—	89	2	no	—
287	C₁₆	—	178	1	no	—
288	C₁₆	C₂×C₄×D₈₉	1	2	yes	complemented
289	C₂³	—	4	2	no	—
290	C₂³	—	89	1	no	—
291	C₂³	—	356	8	no	—
292	C₂³	—	356	2	no	—
293	C₂³	—	712	4	no	—
294	C₂³	—	712	4	no	—
295	D₄	—	2	4	no	—
296	D₄	—	4	2	no	—
297	D₄	—	356	8	no	—
298	D₄	—	356	2	no	—
299	D₄	—	178	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	—	89	1	no	—
303	C₂×C₄	—	89	2	no	—
304	C₂×C₄	—	89	2	no	—
305	C₂×C₄	—	89	2	no	—
306	C₂×C₄	—	356	2	no	—
307	C₂×C₄	—	178	4	no	—
308	C₂×C₄	—	712	4	no	—
309	C₂×C₄	—	712	4	no	—
310	C₂×C₄	—	712	4	no	—
311	C₂×C₄	—	712	4	no	—
312	C₂×C₄	D₈×D₈₉	1	2	yes	—
313	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	89	2	no	—
316	C₈	—	178	1	no	—
317	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
318	C₈	#41×D₈₉	1	1	yes	characteristic
319	C₂²	—	4	4	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	89	2	no	—
323	C₂²	—	89	4	no	—
324	C₂²	—	356	4	no	—
325	C₂²	—	712	16	no	—
326	C₂²	—	712	2	no	—
327	C₂²	—	712	2	no	—
328	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	89	2	no	—
332	C₄	—	178	2	no	—
333	C₄	—	712	2	no	—
334	C₄	C₄×D₄×D₈₉	1	1	yes	characteristic
335	C₄	#135×D₈₉	1	1	yes	characteristic
336	C₂	—	8	4	no	—
337	C₂	—	89	4	no	—
338	C₂	—	712	4	no	—
339	C₂	C₄×D₈×D₈₉	1	1	yes	characteristic
340	C₂	#509×D₈₉	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2177	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₁₆×D₈₉	C₂
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2177	D₈₉
hypocenter	C₈₉	C₂×#2177
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2177

Complements

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

N	H ≅ G/N	type
2. C₄×D₈×D₈₉	C₂	semidirect
3. #26022129	C₂	semidirect
4. #26022128	C₂	semidirect
5. #26022127	C₂	semidirect
6. #26022126	C₂	semidirect
7. #26022125	C₂	semidirect
8. #367×D₈₉	C₂	semidirect
9. #357×D₈₉	C₂	semidirect
10. #346×D₈₉	C₂	semidirect
11. C₈₉×#2177	C₂	semidirect
16. D₁₆×D₈₉	C₄	semidirect
19. C₄×D₇₁₂	C₂²	semidirect
20. D₈×Dic₈₉	C₂²	semidirect
21. #19151680	C₂²	semidirect
27. #19151426	C₂²	semidirect
28. #19151425	C₂²	semidirect
29. #19151400	C₂²	semidirect
30. #19151399	C₂²	semidirect
31. #19151398	C₂²	semidirect
32. #19151374	C₂²	semidirect
33. #19151373	C₂²	semidirect
34. #19151373	C₂²	semidirect
36. C₄×D₈×C₈₉	C₂²	semidirect
37. #367×C₈₉	C₂²	semidirect
38. #357×C₈₉	C₂²	semidirect
39. #346×C₈₉	C₂²	semidirect
45. D₈×D₈₉	C₂×C₄	semidirect
56. D₁₄₂₄	C₂×C₄	semidirect
57. #15667953	C₂×C₄	semidirect
61. C₁₆×D₈₉	C₂×C₄	semidirect
72. D₁₆×C₈₉	C₂×C₄	semidirect
91. D₇₁₂	C₂²×C₄	semidirect
106. #13083549	C₂²×C₄	semidirect
110. D₈×C₈₉	C₂²×C₄	semidirect
117. C₁₆×C₈₉	C₂²×C₄	semidirect
170. D₈₉	#2177	direct (N × H)
178. #2177	D₈₉	direct (N × H)
182. C₈₉	C₂×#2177	semidirect
196. C₄×D₈	C₂×D₈₉	semidirect
202. #367	C₂×D₈₉	semidirect
206. #357	C₂×D₈₉	semidirect
210. #346	C₂×D₈₉	semidirect
231. D₁₆	C₄×D₈₉	semidirect
267. D₈	C₂×C₄×D₈₉	semidirect
288. C₁₆	C₂×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×D₁₆×D₈₉	2
C₄×D₈×D₈₉	2
#26022129	2
#26022128	2
#26022127	2
#26022126	2
#26022125	2
#367×D₈₉	2
#357×D₈₉	2
#346×D₈₉	2
C₈₉×#2177	2
C₂×#2177	89

Subgroups — #2177×D89 #36707727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2177×D₈₉ #36707727