Subgroups — #485×D₈₉ #26030140

Order 11392.

416 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊Q₈×D₈₉	C₂	1	1	yes	maximal, characteristic
2	#117×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151942	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151941	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151940	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151939	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151938	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151937	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151936	C₂	1	1	yes	maximal, characteristic, complemented
10	#107×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#106×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#103×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
13	#103×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²⋊C₈×D₈₉	C₂	1	1	yes	maximal, characteristic
15	#485×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
20	#15667992	C₂²	1	1	yes	characteristic
21	#15667991	C₂²	1	1	yes	characteristic
22	#15667990	—	2	1	no	—
23	#15667989	—	2	1	no	—
24	#15667988	C₂²	1	1	yes	characteristic
25	#15667971	C₂²	1	1	yes	characteristic
26	#15667970	C₂²	1	1	yes	characteristic
27	#15667970	C₂²	1	1	yes	characteristic
28	#15667969	—	2	1	no	—
29	#15667968	—	2	1	no	—
30	C₄⋊C₄×D₈₉	—	2	1	no	—
31	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
34	C₂²⋊C₄×D₈₉	—	2	1	no	—
35	C₂²⋊C₄×D₈₉	—	2	1	no	—
36	C₄²×D₈₉	—	2	1	no	—
37	#15667946	C₂²	1	1	yes	characteristic, complemented
38	#15667945	C₂²	1	1	yes	characteristic, complemented
39	#15667945	C₂²	1	1	yes	characteristic, complemented
40	#15667944	C₂²	1	1	yes	characteristic, complemented
41	#15667943	C₂²	1	1	yes	characteristic, complemented
42	#15667943	C₂²	1	1	yes	characteristic, complemented
43	#15667937	C₂²	1	1	yes	characteristic, complemented
44	#15667937	C₂²	1	1	yes	characteristic, complemented
45	#15667936	C₂²	1	1	yes	characteristic, complemented
46	#15667936	C₂²	1	1	yes	characteristic, complemented
47	#15667935	C₂²	1	1	yes	characteristic, complemented
48	#15667935	C₂²	1	1	yes	characteristic, complemented
49	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	#15667924	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
53	#117×C₈₉	C₂²	1	1	yes	characteristic
54	#107×C₈₉	C₂²	1	1	yes	characteristic, complemented
55	#106×C₈₉	C₂²	1	1	yes	characteristic, complemented
56	#103×C₈₉	C₂²	1	1	yes	characteristic, complemented
57	#103×C₈₉	C₂²	1	1	yes	characteristic, complemented
58	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
59	C₂³×D₈₉	D₄	1	1	yes	characteristic
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
62	Q₈×D₈₉	—	2	2	no	—
63	Q₈×D₈₉	—	2	2	no	—
64	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
65	C₂×C₄×D₈₉	—	2	2	no	—
66	C₂×C₄×D₈₉	—	2	2	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	—	2	1	no	—
69	C₂×C₄×D₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
72	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
73	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
75	C₂×#10623494	C₂³	1	1	yes	characteristic
76	C₂×#10623494	C₂³	1	1	yes	characteristic
77	C₈×D₈₉	—	2	2	no	—
78	C₈×D₈₉	—	2	2	no	—
79	C₄⋊Dic₈₉	—	2	1	no	—
80	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
83	#13083553	—	2	1	no	—
84	#13083553	—	2	1	no	—
85	#13083553	—	2	1	no	—
86	#13083553	—	2	1	no	—
87	#13083553	—	2	1	no	—
88	C₂²⋊Dic₈₉	—	2	1	no	—
89	C₂²⋊Dic₈₉	—	2	1	no	—
90	#13083551	—	2	1	no	—
91	#13083551	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	#13083551	—	2	1	no	—
94	C₄×Dic₈₉	—	2	1	no	—
95	C₄×Dic₈₉	—	2	1	no	—
96	C₄×Dic₈₉	—	2	1	no	—
97	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₉	—	2	1	no	—
102	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₈₉	—	2	1	no	—
106	C₂²⋊C₄×C₈₉	—	2	1	no	—
107	C₄²×C₈₉	—	2	1	no	—
108	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₈₉	—	2	2	no	—
110	C₂²×D₈₉	—	2	2	no	—
111	C₂²×D₈₉	—	2	2	no	—
112	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₈₉	—	2	1	no	—
115	Q₈×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
124	C₂×Dic₈₉	—	2	2	no	—
125	C₂×Dic₈₉	—	2	1	no	—
126	C₂×Dic₈₉	—	2	1	no	—
127	C₂×Dic₈₉	—	2	1	no	—
128	C₂×Dic₈₉	—	2	1	no	—
129	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
131	C₄×D₈₉	—	2	2	no	—
132	C₄×D₈₉	—	2	1	no	—
133	C₄×D₈₉	—	2	1	no	—
134	C₄×D₈₉	—	2	2	no	—
135	C₄×D₈₉	—	4	2	no	—
136	C₄×D₈₉	—	4	2	no	—
137	C₄×D₈₉	—	4	2	no	—
138	C₄×D₈₉	—	4	2	no	—
139	C₄×D₈₉	#41	1	2	yes	—
140	C₄×D₈₉	#41	1	2	yes	—
141	Dic₁₇₈	—	2	1	no	—
142	Dic₁₇₈	—	2	1	no	—
143	Dic₁₇₈	—	4	2	no	—
144	C₈×C₈₉	—	2	1	no	—
145	C₈×C₈₉	—	2	1	no	—
146	#10623494	—	2	1	no	—
147	#10623494	—	2	1	no	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	—	2	1	no	—
151	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₈₉	—	2	1	no	—
153	C₂×D₈₉	—	2	1	no	—
154	C₂×D₈₉	—	2	1	no	—
155	C₂×D₈₉	—	4	4	no	—
156	C₂×D₈₉	#123	1	2	yes	—
157	C₂×D₈₉	#135	1	2	yes	—
158	C₂×D₈₉	#137	1	2	yes	—
159	C₄×C₈₉	—	2	1	no	—
160	C₄×C₈₉	—	4	1	no	—
161	C₄×C₈₉	—	4	1	no	—
162	C₄×C₈₉	—	4	1	no	—
163	C₄×C₈₉	—	4	1	no	—
164	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
165	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
166	Dic₈₉	—	2	1	no	—
167	Dic₈₉	—	4	1	no	—
168	Dic₈₉	—	4	1	no	—
169	Dic₈₉	—	4	1	no	—
170	Dic₈₉	—	4	1	no	—
171	Dic₈₉	#123	1	2	yes	—
172	C₂×C₈₉	—	4	1	no	—
173	C₂×C₈₉	C₂×#123	1	1	yes	characteristic
174	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
175	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
176	D₈₉	—	4	1	no	—
177	D₈₉	#485	1	4	yes	direct product factor
178	C₂×#485	—	89	1	no	maximal, Sylow
179	C₈₉	C₂×#485	1	1	yes	characteristic, complemented, Sylow
180	C₂×C₂²⋊Q₈	—	89	1	no	—
181	C₂×#117	—	89	1	no	—
182	#485	—	89	1	no	—
183	#485	—	89	1	no	—
184	#485	—	89	1	no	—
185	#485	—	89	1	no	—
186	#485	—	89	1	no	—
187	#485	—	89	1	no	—
188	#485	—	89	1	no	—
189	#485	D₈₉	1	1	yes	characteristic, direct product factor
190	C₂×#107	—	89	1	no	—
191	C₂×#106	—	89	1	no	—
192	C₂×#103	—	89	1	no	—
193	C₂×#103	—	89	1	no	—
194	C₂×C₂²⋊C₈	—	89	1	no	—
195	C₂²×Q₈	—	89	1	no	—
196	C₂³×C₄	—	89	1	no	—
197	C₂²×C₈	—	89	1	no	—
198	C₂²×C₈	—	89	1	no	—
199	C₂²⋊Q₈	—	89	1	no	—
200	C₂²⋊Q₈	—	89	1	no	—
201	C₂²⋊Q₈	—	89	1	no	—
202	C₂²⋊Q₈	—	178	1	no	—
203	C₂²⋊Q₈	—	178	1	no	—
204	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
205	#117	—	89	1	no	—
206	#117	—	89	1	no	—
207	#117	—	89	1	no	—
208	#117	—	178	1	no	—
209	#117	—	178	1	no	—
210	#117	C₂×D₈₉	1	1	yes	characteristic
211	C₂×C₄⋊C₄	—	89	1	no	—
212	C₂×C₄⋊C₄	—	89	1	no	—
213	C₂×C₄⋊C₄	—	89	1	no	—
214	C₂×C₄⋊C₄	—	178	1	no	—
215	C₂×C₂²⋊C₄	—	178	1	no	—
216	C₂×C₂²⋊C₄	—	178	1	no	—
217	C₂×C₄²	—	178	1	no	—
218	#107	—	89	1	no	—
219	#107	—	89	1	no	—
220	#107	—	89	1	no	—
221	#107	C₂×D₈₉	1	1	yes	characteristic, complemented
222	#106	—	89	1	no	—
223	#106	—	89	1	no	—
224	#106	—	89	1	no	—
225	#106	C₂×D₈₉	1	1	yes	characteristic, complemented
226	#103	—	89	1	no	—
227	#103	—	89	1	no	—
228	#103	—	89	1	no	—
229	#103	—	89	1	no	—
230	#103	—	89	1	no	—
231	#103	—	89	1	no	—
232	#103	C₂×D₈₉	1	1	yes	characteristic, complemented
233	#103	C₂×D₈₉	1	1	yes	characteristic, complemented
234	C₂²⋊C₈	—	89	1	no	—
235	C₂²⋊C₈	—	89	1	no	—
236	C₂²⋊C₈	—	89	1	no	—
237	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
238	C₂⁴	—	89	1	no	—
239	C₂×Q₈	—	89	1	no	—
240	C₂×Q₈	—	178	1	no	—
241	C₂×Q₈	—	178	2	no	—
242	C₂×Q₈	—	178	2	no	—
243	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
244	C₂²×C₄	—	89	1	no	—
245	C₂²×C₄	—	89	1	no	—
246	C₂²×C₄	—	89	1	no	—
247	C₂²×C₄	—	89	1	no	—
248	C₂²×C₄	—	89	1	no	—
249	C₂²×C₄	—	178	2	no	—
250	C₂²×C₄	—	178	2	no	—
251	C₂²×C₄	—	178	1	no	—
252	C₂²×C₄	—	178	1	no	—
253	C₂²×C₄	—	178	1	no	—
254	C₂²×C₄	—	178	1	no	—
255	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
256	C₂×C₈	—	89	1	no	—
257	C₂×C₈	—	89	1	no	—
258	C₂×C₈	—	178	2	no	—
259	C₂×C₈	—	178	2	no	—
260	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	89	1	no	—
264	C₄⋊C₄	—	89	1	no	—
265	C₄⋊C₄	—	89	1	no	—
266	C₄⋊C₄	—	178	1	no	—
267	C₄⋊C₄	—	178	1	no	—
268	C₄⋊C₄	—	178	1	no	—
269	C₄⋊C₄	—	178	1	no	—
270	C₄⋊C₄	—	178	1	no	—
271	C₄⋊C₄	—	178	1	no	—
272	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
273	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	178	1	no	—
278	C₂²⋊C₄	—	178	1	no	—
279	C₂²⋊C₄	—	178	1	no	—
280	C₂²⋊C₄	—	178	1	no	—
281	C₂²⋊C₄	—	178	1	no	—
282	C₂²⋊C₄	—	178	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	178	1	no	—
285	C₄²	—	178	1	no	—
286	C₄²	—	178	1	no	—
287	C₂³	—	89	1	no	—
288	C₂³	—	89	1	no	—
289	C₂³	—	178	2	no	—
290	C₂³	—	178	2	no	—
291	C₂³	—	178	2	no	—
292	C₂³	D₄×D₈₉	1	1	yes	characteristic
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	356	2	no	—
296	Q₈	—	178	1	no	—
297	Q₈	—	178	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	89	1	no	—
305	C₂×C₄	—	89	2	no	—
306	C₂×C₄	—	89	2	no	—
307	C₂×C₄	—	89	1	no	—
308	C₂×C₄	—	356	2	no	—
309	C₂×C₄	—	356	2	no	—
310	C₂×C₄	—	356	2	no	—
311	C₂×C₄	—	356	2	no	—
312	C₂×C₄	—	178	2	no	—
313	C₂×C₄	—	178	2	no	—
314	C₂×C₄	—	178	2	no	—
315	C₂×C₄	—	178	1	no	—
316	C₂×C₄	—	178	1	no	—
317	C₂×C₄	—	178	1	no	—
318	C₂×C₄	—	178	1	no	—
319	C₂×C₄	—	178	1	no	—
320	C₂×C₄	—	178	1	no	—
321	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
322	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	178	1	no	—
326	C₈	—	178	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	89	2	no	—
331	C₂²	—	89	2	no	—
332	C₂²	—	89	2	no	—
333	C₂²	—	356	4	no	—
334	C₂²	—	178	1	no	—
335	C₂²	—	178	1	no	—
336	C₂²	—	178	1	no	—
337	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	89	2	no	—
344	C₄	—	356	1	no	—
345	C₄	—	356	1	no	—
346	C₄	—	356	1	no	—
347	C₄	—	356	1	no	—
348	C₄	—	178	1	no	—
349	C₄	#41×D₈₉	1	1	yes	characteristic
350	C₄	#41×D₈₉	1	1	yes	characteristic
351	C₂	—	4	1	no	—
352	C₂	—	89	4	no	—
353	C₂	—	356	1	no	—
354	C₂	#123×D₈₉	1	1	yes	characteristic
355	C₂	#135×D₈₉	1	1	yes	characteristic
356	C₂	#137×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#485×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂³×D₈₉	D₄
hypercenter	#485	D₈₉
hypocenter	C₈₉	C₂×#485
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
3. #19151942	C₂	semidirect
4. #19151941	C₂	semidirect
5. #19151940	C₂	semidirect
6. #19151939	C₂	semidirect
7. #19151938	C₂	semidirect
8. #19151937	C₂	semidirect
9. #19151936	C₂	semidirect
10. #107×D₈₉	C₂	semidirect
11. #106×D₈₉	C₂	semidirect
12. #103×D₈₉	C₂	semidirect
13. #103×D₈₉	C₂	semidirect
15. #485×C₈₉	C₂	semidirect
37. #15667946	C₂²	semidirect
38. #15667945	C₂²	semidirect
39. #15667945	C₂²	semidirect
40. #15667944	C₂²	semidirect
41. #15667943	C₂²	semidirect
42. #15667943	C₂²	semidirect
43. #15667937	C₂²	semidirect
44. #15667937	C₂²	semidirect
45. #15667936	C₂²	semidirect
46. #15667936	C₂²	semidirect
47. #15667935	C₂²	semidirect
48. #15667935	C₂²	semidirect
54. #107×C₈₉	C₂²	semidirect
55. #106×C₈₉	C₂²	semidirect
56. #103×C₈₉	C₂²	semidirect
57. #103×C₈₉	C₂²	semidirect
177. D₈₉	#485	direct (N × H)
179. C₈₉	C₂×#485	semidirect
189. #485	D₈₉	direct (N × H)
221. #107	C₂×D₈₉	semidirect
225. #106	C₂×D₈₉	semidirect
232. #103	C₂×D₈₉	semidirect
233. #103	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊Q₈×D₈₉	2
#117×D₈₉	2
#19151942	2
#19151941	2
#19151940	2
#19151939	2
#19151938	2
#19151937	2
#19151936	2
#107×D₈₉	2
#106×D₈₉	2
#103×D₈₉	2
C₂²⋊C₈×D₈₉	2
#485×C₈₉	2
C₂×#485	89

Subgroups — #485×D89 #26030140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #485×D₈₉ #26030140