Subgroups — #36525921

Order 22784.

630 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#13083557	C₂	1	2	yes	maximal
2	C₄×Q₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#26019162	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019160	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019123	C₂	1	2	yes	maximal, complemented
6	#26019120	C₂	1	2	yes	maximal, complemented
7	#26019071	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019006	C₂	1	1	yes	maximal, characteristic
9	#26018679	C₂	1	2	yes	maximal
10	#26018538	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#1579	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₈₉	C₄	1	1	yes	characteristic
13	C₄×Q₈×D₈₉	C₂²	1	2	yes	—
14	#19151728	—	2	2	no	—
15	#19151728	C₂²	1	2	yes	—
16	#19151725	—	2	2	no	—
17	#19151725	C₂²	1	2	yes	—
18	Q₈×#10623494	C₂²	1	2	yes	—
19	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
20	Q₁₆×Dic₈₉	C₄	1	1	yes	characteristic
21	#19151692	C₄	1	1	yes	characteristic
22	C₄×#13083564	C₂²	1	2	yes	—
23	#19151690	C₄	1	2	yes	—
24	#19151649	C₂²	1	1	yes	characteristic
25	#107×D₈₉	C₄	1	1	yes	characteristic
26	#19151620	—	2	2	no	—
27	#19151620	C₂²	1	2	yes	—
28	#103×D₈₉	C₄	1	2	yes	—
29	C₄×#13083557	—	2	2	no	—
30	C₄×#13083557	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
32	#19151349	C₂²	1	1	yes	characteristic
33	#19151348	C₂²	1	2	yes	—
34	#19151330	C₂²	1	2	yes	—
35	Q₈⋊#10623494	C₂²	1	2	yes	—
36	#19151328	C₂²	1	2	yes	—
37	#19151320	C₂²	1	1	yes	characteristic
38	#19151319	C₂²	1	1	yes	characteristic
39	#19151319	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₈₉	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
42	#334×C₈₉	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₈₉	C₂²	1	2	yes	—
44	#322×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₈₉	C₂×C₄	1	2	yes	—
46	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
47	C₂×#13083564	C₂×C₄	1	2	yes	—
48	Q₁₆×D₈₉	—	2	4	no	—
49	C₂×#13083557	—	2	2	no	—
50	C₂×#13083557	D₄	1	2	yes	—
51	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
53	Q₈×Dic₈₉	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₇₈	—	2	2	no	—
55	C₄×Dic₁₇₈	C₂³	1	2	yes	—
56	#15667977	—	2	2	no	—
57	#15667977	C₂×C₄	1	2	yes	—
58	C₄⋊C₄×D₈₉	—	2	2	no	—
59	C₄⋊C₄×D₈₉	C₂×C₄	1	2	yes	—
60	C₄²×D₈₉	—	2	2	no	—
61	C₄²×D₈₉	C₂³	1	1	yes	characteristic
62	#15667946	C₂×C₄	1	1	yes	characteristic
63	#15667945	C₂×C₄	1	2	yes	—
64	C₄⋊#10623494	—	2	2	no	—
65	C₄⋊#10623494	C₂³	1	2	yes	—
66	#15667941	—	2	2	no	—
67	#15667941	—	2	2	no	—
68	#15667941	—	2	1	no	—
69	#15667941	—	2	2	no	—
70	#15667937	C₂×C₄	1	2	yes	—
71	#15667936	C₂×C₄	1	2	yes	—
72	#15667935	C₂×C₄	1	2	yes	—
73	#15667924	—	2	1	no	—
74	#15667922	—	2	2	no	—
75	#15667922	—	2	2	no	—
76	#15667922	D₄	1	2	yes	—
77	C₄×#10623494	—	2	2	no	—
78	C₄×#10623494	C₂³	1	1	yes	characteristic
79	C₄×#10623494	C₂³	1	1	yes	characteristic
80	C₄×#10623494	C₂³	1	1	yes	characteristic
81	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
83	C₂×Q₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
84	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
85	#107×C₈₉	C₂×C₄	1	1	yes	characteristic
86	C₄⋊C₈×C₈₉	—	2	2	no	—
87	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
88	#103×C₈₉	C₂×C₄	1	2	yes	—
89	C₄×C₈×C₈₉	—	2	2	no	—
90	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
93	C₂×Dic₁₇₈	—	2	2	no	—
94	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
95	Q₈×D₈₉	—	2	4	no	—
96	Q₈×D₈₉	—	4	2	no	—
97	C₂×C₄×D₈₉	—	2	2	no	—
98	C₂×C₄×D₈₉	—	2	2	no	—
99	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×D₈₉	#41	1	1	yes	characteristic
102	Dic₃₅₆	—	2	2	no	—
103	#13083564	—	2	4	no	—
104	#13083557	—	2	4	no	—
105	#13083557	—	4	4	no	—
106	C₂×#10623494	—	2	2	no	—
107	C₂×#10623494	—	2	1	no	—
108	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
109	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
110	C₂×#10623494	C₂×D₄	1	2	yes	—
111	C₈×D₈₉	—	2	2	no	—
112	C₈×D₈₉	—	4	1	no	—
113	C₄⋊Dic₈₉	—	2	2	no	—
114	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
115	#13083553	—	2	2	no	—
116	#13083553	—	2	2	no	—
117	#13083553	—	2	2	no	—
118	C₄×Dic₈₉	—	2	2	no	—
119	C₄×Dic₈₉	—	2	2	no	—
120	C₄×Dic₈₉	—	2	2	no	—
121	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
122	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
123	C₄×Dic₈₉	#41	1	1	yes	characteristic
124	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
125	Q₁₆×C₈₉	M₁₆	1	4	yes	complemented
126	C₂×C₈×C₈₉	—	2	2	no	—
127	C₂×C₈×C₈₉	—	2	1	no	—
128	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
131	C₄⋊C₄×C₈₉	—	2	2	no	—
132	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
133	C₄²×C₈₉	—	2	2	no	—
134	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
135	C₂²×D₈₉	C₄×D₄	1	1	yes	characteristic
136	Q₈×C₈₉	—	2	2	no	—
137	Q₈×C₈₉	C₂×M₁₆	1	4	yes	—
138	C₂×C₄×C₈₉	—	2	2	no	—
139	C₂×C₄×C₈₉	—	2	2	no	—
140	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
143	C₂×Dic₈₉	—	2	2	no	—
144	C₂×Dic₈₉	—	2	2	no	—
145	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
146	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
147	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
148	C₄×D₈₉	—	2	1	no	—
149	C₄×D₈₉	—	2	1	no	—
150	C₄×D₈₉	—	4	1	no	—
151	C₄×D₈₉	—	4	4	no	—
152	C₄×D₈₉	—	8	2	no	—
153	C₄×D₈₉	#137	1	4	yes	—
154	Dic₁₇₈	—	2	2	no	—
155	Dic₁₇₈	—	2	2	no	—
156	Dic₁₇₈	—	4	4	no	—
157	C₈×C₈₉	—	2	1	no	—
158	C₈×C₈₉	—	2	2	no	—
159	C₈×C₈₉	—	4	1	no	—
160	C₈×C₈₉	—	4	2	no	—
161	C₈×C₈₉	C₂×M₁₆	1	2	yes	—
162	#10623494	—	2	1	no	—
163	#10623494	—	2	1	no	—
164	#10623494	—	2	2	no	—
165	#10623494	—	4	1	no	—
166	#10623494	—	4	2	no	—
167	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂×D₈₉	—	2	1	no	—
169	C₂×D₈₉	—	2	1	no	—
170	C₂×D₈₉	#441	1	2	yes	—
171	C₄×C₈₉	—	2	4	no	—
172	C₄×C₈₉	—	2	1	no	—
173	C₄×C₈₉	—	4	2	no	—
174	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
175	C₄×C₈₉	#568	1	1	yes	characteristic
176	C₄×C₈₉	C₂×#137	1	2	yes	—
177	Dic₈₉	—	2	1	no	—
178	Dic₈₉	—	2	1	no	—
179	Dic₈₉	—	4	2	no	—
180	Dic₈₉	—	4	2	no	—
181	Dic₈₉	#441	1	2	yes	—
182	#18042	—	89	1	no	maximal, Sylow
183	C₂×C₈₉	D₄×M₁₆	1	1	yes	characteristic
184	C₂×C₈₉	C₂×#441	1	1	yes	characteristic
185	C₂×C₈₉	#2957	1	1	yes	characteristic
186	D₈₉	—	2	2	no	—
187	Q₈×M₁₆	—	89	2	no	—
188	C₂×C₄×Q₁₆	—	89	1	no	—
189	#1593	—	89	1	no	—
190	#1593	—	89	1	no	—
191	#1587	—	89	2	no	—
192	#1587	—	89	2	no	—
193	#1579	—	89	1	no	—
194	#1579	D₈₉	1	1	yes	characteristic, complemented
195	#1566	—	89	1	no	—
196	#1487	—	89	2	no	—
197	#1451	—	89	1	no	—
198	C₈₉	#18042	1	1	yes	characteristic, complemented, Sylow
199	C₂²×Q₁₆	—	89	1	no	—
200	C₂×C₄×Q₈	—	89	2	no	—
201	#446	—	89	2	no	—
202	#446	—	89	2	no	—
203	#446	—	178	2	no	—
204	#446	—	178	2	no	—
205	C₈×Q₈	—	89	2	no	—
206	C₈×Q₈	C₂×D₈₉	1	2	yes	—
207	C₄×Q₁₆	—	89	1	no	—
208	C₄×Q₁₆	—	89	2	no	—
209	C₄×Q₁₆	—	89	1	no	—
210	C₄×Q₁₆	—	89	2	no	—
211	C₄×Q₁₆	—	89	1	no	—
212	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
213	#431	—	89	1	no	—
214	C₂×#107	—	89	1	no	—
215	#423	—	89	2	no	—
216	#423	—	178	2	no	—
217	C₂×#103	—	89	2	no	—
218	C₄×M₁₆	—	89	1	no	—
219	C₄×M₁₆	—	178	2	no	—
220	C₂×C₄×C₈	—	89	1	no	—
221	#334	—	89	2	no	—
222	#334	—	89	1	no	—
223	#334	C₂×D₈₉	1	1	yes	characteristic
224	Q₈⋊C₈	—	89	2	no	—
225	Q₈⋊C₈	—	89	2	no	—
226	Q₈⋊C₈	—	89	2	no	—
227	Q₈⋊C₈	C₂×D₈₉	1	2	yes	—
228	#322	—	89	1	no	—
229	#322	—	89	1	no	—
230	#322	—	89	1	no	—
231	#322	C₂×D₈₉	1	1	yes	characteristic
232	C₂²×Q₈	—	89	2	no	—
233	C₂×Q₁₆	—	89	2	no	—
234	C₂×Q₁₆	—	89	1	no	—
235	C₂×Q₁₆	—	178	4	no	—
236	C₂×Q₁₆	C₄×D₈₉	1	1	yes	characteristic
237	C₂×M₁₆	—	89	2	no	—
238	C₂×M₁₆	—	178	2	no	—
239	C₂²×C₈	—	89	1	no	—
240	C₂²×C₈	—	89	1	no	—
241	C₄×Q₈	—	89	2	no	—
242	C₄×Q₈	—	89	2	no	—
243	C₄×Q₈	—	89	2	no	—
244	C₄×Q₈	—	178	2	no	—
245	C₄×Q₈	—	178	2	no	—
246	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
247	C₂×C₄⋊C₄	—	89	2	no	—
248	C₂×C₄⋊C₄	—	178	2	no	—
249	C₂×C₄²	—	89	1	no	—
250	C₂×C₄²	—	178	2	no	—
251	#107	—	89	1	no	—
252	#107	—	89	2	no	—
253	#107	C₄×D₈₉	1	1	yes	characteristic
254	C₄⋊C₈	—	2	2	no	—
255	C₄⋊C₈	—	89	2	no	—
256	C₄⋊C₈	—	178	2	no	—
257	C₄⋊C₈	—	178	2	no	—
258	C₄⋊C₈	—	178	1	no	—
259	C₄⋊C₈	—	178	2	no	—
260	C₄⋊C₈	—	178	2	no	—
261	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
262	#103	—	89	2	no	—
263	#103	—	89	2	no	—
264	#103	—	89	2	no	—
265	#103	C₄×D₈₉	1	2	yes	—
266	C₂²⋊C₈	—	178	1	no	—
267	#97	—	89	2	no	—
268	#97	—	178	2	no	—
269	#97	—	178	2	no	—
270	C₄×C₈	—	2	2	no	—
271	C₄×C₈	—	89	1	no	—
272	C₄×C₈	—	89	1	no	—
273	C₄×C₈	—	89	1	no	—
274	C₄×C₈	—	89	1	no	—
275	C₄×C₈	—	89	1	no	—
276	C₄×C₈	—	178	2	no	—
277	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
280	C₂×Q₈	—	89	2	no	—
281	C₂×Q₈	—	356	2	no	—
282	C₂×Q₈	—	178	2	no	—
283	C₂×Q₈	—	178	4	no	—
284	C₂×Q₈	C₂×C₄×D₈₉	1	2	yes	—
285	C₂²×C₄	—	89	1	no	—
286	C₂²×C₄	—	89	1	no	—
287	C₂²×C₄	—	89	1	no	—
288	C₂²×C₄	—	178	2	no	—
289	C₂²×C₄	—	178	2	no	—
290	Q₁₆	—	178	4	no	—
291	Q₁₆	—	178	2	no	—
292	Q₁₆	#13083557	1	4	yes	complemented
293	M₁₆	—	356	4	no	—
294	M₁₆	—	178	4	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	89	1	no	—
298	C₂×C₈	—	89	1	no	—
299	C₂×C₈	—	89	2	no	—
300	C₂×C₈	—	356	1	no	—
301	C₂×C₈	—	178	2	no	—
302	C₂×C₈	—	178	1	no	—
303	C₂×C₈	—	178	2	no	—
304	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
305	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
306	C₂×C₈	D₄×D₈₉	1	2	yes	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	89	2	no	—
309	C₄⋊C₄	—	178	2	no	—
310	C₄⋊C₄	—	178	2	no	—
311	C₄⋊C₄	—	178	2	no	—
312	C₄⋊C₄	—	178	2	no	—
313	C₄⋊C₄	C₂×C₄×D₈₉	1	2	yes	—
314	C₄²	—	2	2	no	—
315	C₄²	—	89	1	no	—
316	C₄²	—	89	1	no	—
317	C₄²	—	89	1	no	—
318	C₄²	—	178	2	no	—
319	C₄²	—	178	2	no	—
320	C₄²	—	178	2	no	—
321	C₄²	C₂³×D₈₉	1	1	yes	characteristic
322	C₂³	—	89	1	no	—
323	Q₈	—	2	2	no	—
324	Q₈	—	356	4	no	—
325	Q₈	—	178	2	no	—
326	Q₈	—	178	2	no	—
327	Q₈	C₂×#13083557	1	4	yes	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	89	1	no	—
331	C₂×C₄	—	89	4	no	—
332	C₂×C₄	—	89	1	no	—
333	C₂×C₄	—	89	1	no	—
334	C₂×C₄	—	356	4	no	—
335	C₂×C₄	—	356	1	no	—
336	C₂×C₄	—	178	2	no	—
337	C₂×C₄	—	178	2	no	—
338	C₂×C₄	—	178	1	no	—
339	C₂×C₄	—	178	1	no	—
340	C₂×C₄	—	712	2	no	—
341	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
342	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
343	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
344	C₈	—	2	2	no	—
345	C₈	—	2	1	no	—
346	C₈	—	4	1	no	—
347	C₈	—	4	2	no	—
348	C₈	—	356	1	no	—
349	C₈	—	356	2	no	—
350	C₈	—	178	2	no	—
351	C₈	—	178	1	no	—
352	C₈	—	178	1	no	—
353	C₈	C₂×#13083557	1	2	yes	—
354	C₂²	—	89	2	no	—
355	C₂²	—	178	1	no	—
356	C₂²	—	178	1	no	—
357	C₂²	C₄×D₄×D₈₉	1	1	yes	characteristic
358	C₄	—	2	1	no	—
359	C₄	—	2	4	no	—
360	C₄	—	4	2	no	—
361	C₄	—	89	2	no	—
362	C₄	—	356	2	no	—
363	C₄	—	356	2	no	—
364	C₄	—	178	1	no	—
365	C₄	—	178	1	no	—
366	C₄	C₂²×#13083557	1	1	yes	characteristic
367	C₄	#19152363	1	1	yes	characteristic
368	C₄	#137×D₈₉	1	2	yes	—
369	C₂	—	178	2	no	—
370	C₂	D₄×#13083557	1	1	yes	characteristic
371	C₂	#441×D₈₉	1	1	yes	characteristic
372	C₂	#26029166	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1579	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂²×D₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1579	D₈₉
hypocenter	C₈₉	#18042
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1579

Complements

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

N	H ≅ G/N	type
3. #26019162	C₂	semidirect
4. #26019160	C₂	semidirect
5. #26019123	C₂	semidirect
6. #26019120	C₂	semidirect
7. #26019071	C₂	semidirect
11. C₈₉×#1579	C₂	semidirect
125. Q₁₆×C₈₉	M₁₆	semidirect
194. #1579	D₈₉	semidirect
198. C₈₉	#18042	semidirect
292. Q₁₆	#13083557	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#13083557	2
C₄×Q₁₆×D₈₉	2
#26019162	2
#26019160	2
#26019123	2
#26019120	2
#26019071	2
#26019006	2
#26018679	2
#26018538	2
C₈₉×#1579	2
#18042	89