Subgroups — #36526200

Order 22784.

586 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030835	C₂	1	1	yes	maximal, characteristic
2	#26029194	C₂	1	2	yes	maximal
3	#26019150	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019149	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019105	C₂	1	2	yes	maximal, complemented
6	#26019102	C₂	1	2	yes	maximal, complemented
7	#26019058	C₂	1	1	yes	maximal, characteristic, complemented
8	#335×D₈₉	C₂	1	1	yes	maximal, characteristic
9	#26018653	C₂	1	2	yes	maximal
10	C₈×#13083557	C₂	1	1	yes	maximal, characteristic
11	C₈×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152152	C₂²	1	2	yes	—
13	#19152002	C₄	1	1	yes	characteristic
14	#19151976	C₄	1	1	yes	characteristic
15	#19151947	C₄	1	2	yes	—
16	#19151840	C₄	1	1	yes	characteristic
17	#19151838	C₄	1	2	yes	—
18	#19151836	C₄	1	1	yes	characteristic
19	#19151729	C₂²	1	2	yes	—
20	#19151728	—	2	2	no	—
21	#19151728	C₂²	1	2	yes	—
22	#19151727	—	2	2	no	—
23	#19151722	C₂²	1	2	yes	—
24	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
25	C₄×#13083564	C₂²	1	2	yes	—
26	#19151649	C₂²	1	1	yes	characteristic
27	#19151623	—	2	2	no	—
28	C₄⋊C₈×D₈₉	C₂²	1	2	yes	—
29	#19151552	—	2	2	no	—
30	C₄×#13083557	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
32	#19151351	C₂²	1	1	yes	characteristic
33	#19151350	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	C₈×#10623494	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₈₉	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
42	#335×C₈₉	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₈₉	C₂²	1	2	yes	—
44	C₈²×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
46	C₂×#13083564	C₂×C₄	1	2	yes	—
47	C₂×#13083557	D₄	1	1	yes	characteristic
48	C₂×#13083557	D₄	1	1	yes	characteristic
49	C₂×C₈×D₈₉	—	2	2	no	—
50	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
52	#15668011	C₂×C₄	1	2	yes	—
53	#15668010	C₂×C₄	1	2	yes	—
54	#15668002	—	2	2	no	—
55	#15667992	C₂×C₄	1	2	yes	—
56	#15667990	—	2	2	no	—
57	#15667988	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₇₈	—	2	2	no	—
59	C₄×Dic₁₇₈	C₂³	1	2	yes	—
60	#15667969	—	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	#15667923	—	2	2	no	—
75	#15667922	—	2	2	no	—
76	#15667922	D₄	1	1	yes	characteristic
77	#15667922	D₄	1	1	yes	characteristic
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	—	2	2	no	—
82	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
84	C₂×Q₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
86	#107×C₈₉	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₈₉	—	2	2	no	—
88	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
89	#103×C₈₉	C₂×C₄	1	2	yes	—
90	C₄×C₈×C₈₉	—	2	2	no	—
91	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₇₈	—	2	2	no	—
95	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₈₉	#41	1	1	yes	characteristic
99	Dic₃₅₆	—	2	2	no	—
100	#13083564	—	4	2	no	—
101	#13083557	—	2	2	no	—
102	#13083557	Q₁₆	1	4	yes	complemented
103	C₂×#10623494	—	2	2	no	—
104	C₂×#10623494	—	2	1	no	—
105	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
106	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
107	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
108	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
109	C₈×D₈₉	—	2	2	no	—
110	C₈×D₈₉	—	4	1	no	—
111	C₈×D₈₉	—	4	4	no	—
112	C₄⋊Dic₈₉	—	2	2	no	—
113	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
114	#13083553	—	2	2	no	—
115	#13083553	—	2	2	no	—
116	#13083553	—	2	2	no	—
117	#13083553	—	2	2	no	—
118	#13083553	—	2	2	no	—
119	#13083551	—	2	2	no	—
120	#13083551	—	2	2	no	—
121	C₄×Dic₈₉	—	2	2	no	—
122	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₈₉	#41	1	1	yes	characteristic
125	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
126	Q₁₆×C₈₉	—	2	2	no	—
127	C₂×C₈×C₈₉	—	2	2	no	—
128	C₂×C₈×C₈₉	—	2	1	no	—
129	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
133	C₄⋊C₄×C₈₉	—	2	2	no	—
134	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
135	C₄²×C₈₉	—	2	2	no	—
136	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
137	C₂²×D₈₉	C₄×D₄	1	1	yes	characteristic
138	Q₈×C₈₉	—	2	2	no	—
139	Q₈×C₈₉	—	2	2	no	—
140	C₂×C₄×C₈₉	—	2	2	no	—
141	C₂×C₄×C₈₉	—	2	2	no	—
142	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
145	C₂×Dic₈₉	—	2	2	no	—
146	C₂×Dic₈₉	—	2	2	no	—
147	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
148	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
149	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
150	C₄×D₈₉	—	2	1	no	—
151	C₄×D₈₉	—	2	1	no	—
152	C₄×D₈₉	—	4	1	no	—
153	C₄×D₈₉	C₂×Q₁₆	1	2	yes	—
154	C₄×D₈₉	#135	1	2	yes	—
155	Dic₁₇₈	—	2	2	no	—
156	Dic₁₇₈	—	2	2	no	—
157	Dic₁₇₈	—	8	2	no	—
158	C₈×C₈₉	—	2	1	no	—
159	C₈×C₈₉	—	2	1	no	—
160	C₈×C₈₉	—	4	1	no	—
161	C₈×C₈₉	—	4	2	no	—
162	C₈×C₈₉	#131	1	2	yes	—
163	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
164	#10623494	—	2	1	no	—
165	#10623494	—	2	1	no	—
166	#10623494	—	2	1	no	—
167	#10623494	—	4	1	no	—
168	#10623494	—	4	2	no	—
169	#10623494	C₂×Q₁₆	1	2	yes	—
170	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×D₈₉	—	2	1	no	—
172	C₂×D₈₉	—	2	1	no	—
173	C₂×D₈₉	C₄×Q₁₆	1	2	yes	—
174	C₄×C₈₉	—	2	1	no	—
175	C₄×C₈₉	—	4	2	no	—
176	C₄×C₈₉	—	4	2	no	—
177	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
178	C₄×C₈₉	#568	1	1	yes	characteristic
179	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
180	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
181	Dic₈₉	—	2	1	no	—
182	Dic₈₉	—	2	1	no	—
183	Dic₈₉	—	4	2	no	—
184	Dic₈₉	—	4	2	no	—
185	Dic₈₉	C₄×Q₁₆	1	2	yes	—
186	#18068	—	89	1	no	maximal, Sylow
187	C₂×C₈₉	#2935	1	1	yes	characteristic
188	C₂×C₈₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
189	C₂×C₈₉	#2957	1	1	yes	characteristic
190	D₈₉	—	2	2	no	—
191	#3170	—	89	1	no	—
192	#2962	—	89	2	no	—
193	#1591	—	89	1	no	—
194	#1591	—	89	1	no	—
195	#1584	—	89	2	no	—
196	#1584	—	89	2	no	—
197	C₈×Q₁₆	—	89	1	no	—
198	C₈×Q₁₆	D₈₉	1	1	yes	characteristic, complemented
199	C₂×#335	—	89	1	no	—
200	#1481	—	89	2	no	—
201	C₈×M₁₆	—	89	1	no	—
202	C₈₉	#18068	1	1	yes	characteristic, complemented, Sylow
203	#533	—	89	2	no	—
204	#500	—	89	1	no	—
205	#494	—	89	1	no	—
206	#487	—	89	2	no	—
207	#467	—	89	1	no	—
208	#467	—	89	1	no	—
209	#467	—	89	2	no	—
210	#446	—	89	2	no	—
211	#446	—	89	2	no	—
212	#446	—	178	2	no	—
213	#446	—	178	2	no	—
214	C₈×Q₈	—	89	2	no	—
215	C₈×Q₈	C₂×D₈₉	1	2	yes	—
216	C₄×Q₁₆	—	89	2	no	—
217	C₄×Q₁₆	—	89	1	no	—
218	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
219	#431	—	89	1	no	—
220	#424	—	178	2	no	—
221	C₂×C₄⋊C₈	—	89	2	no	—
222	#405	—	178	2	no	—
223	C₄×M₁₆	—	89	1	no	—
224	C₂×C₄×C₈	—	89	1	no	—
225	#335	—	89	2	no	—
226	#335	—	89	1	no	—
227	#335	C₂×D₈₉	1	1	yes	characteristic
228	Q₈⋊C₈	—	89	2	no	—
229	Q₈⋊C₈	—	89	2	no	—
230	Q₈⋊C₈	—	89	2	no	—
231	Q₈⋊C₈	C₂×D₈₉	1	2	yes	—
232	#322	—	89	1	no	—
233	#322	—	89	1	no	—
234	C₈²	—	89	1	no	—
235	C₈²	C₂×D₈₉	1	1	yes	characteristic
236	C₂×Q₁₆	—	89	2	no	—
237	C₂×Q₁₆	—	89	1	no	—
238	C₂×Q₁₆	C₄×D₈₉	1	1	yes	characteristic
239	C₂×M₁₆	—	89	1	no	—
240	C₂×M₁₆	—	89	1	no	—
241	C₂²×C₈	—	89	1	no	—
242	C₂²×C₈	—	89	1	no	—
243	C₂²×C₈	—	178	2	no	—
244	C₄⋊Q₈	—	89	2	no	—
245	C₄⋊Q₈	—	89	2	no	—
246	#125	—	178	2	no	—
247	C₂²⋊Q₈	—	89	2	no	—
248	C₂²⋊Q₈	—	89	2	no	—
249	C₂²⋊Q₈	—	178	2	no	—
250	C₄×Q₈	—	89	2	no	—
251	C₄×Q₈	—	178	2	no	—
252	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
253	#117	—	178	2	no	—
254	C₂×C₄²	—	89	1	no	—
255	#107	—	89	1	no	—
256	#107	—	89	2	no	—
257	#107	C₄×D₈₉	1	1	yes	characteristic
258	C₄⋊C₈	—	2	2	no	—
259	C₄⋊C₈	—	89	2	no	—
260	C₄⋊C₈	—	178	2	no	—
261	C₄⋊C₈	—	178	2	no	—
262	C₄⋊C₈	—	178	1	no	—
263	C₄⋊C₈	—	178	2	no	—
264	C₄⋊C₈	—	178	2	no	—
265	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
266	#103	—	89	2	no	—
267	#103	—	89	2	no	—
268	#103	—	89	2	no	—
269	#103	C₄×D₈₉	1	2	yes	—
270	C₂²⋊C₈	—	178	1	no	—
271	#97	—	89	1	no	—
272	#97	—	89	1	no	—
273	#97	—	178	2	no	—
274	#97	—	178	2	no	—
275	C₄×C₈	—	2	2	no	—
276	C₄×C₈	—	89	1	no	—
277	C₄×C₈	—	89	1	no	—
278	C₄×C₈	—	89	1	no	—
279	C₄×C₈	—	89	1	no	—
280	C₄×C₈	—	89	1	no	—
281	C₄×C₈	—	178	2	no	—
282	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
283	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
285	C₂×Q₈	—	89	2	no	—
286	C₂×Q₈	—	178	2	no	—
287	C₂×Q₈	C₂×C₄×D₈₉	1	2	yes	—
288	C₂²×C₄	—	89	1	no	—
289	C₂²×C₄	—	89	1	no	—
290	C₂²×C₄	—	89	1	no	—
291	Q₁₆	—	2	2	no	—
292	Q₁₆	—	356	2	no	—
293	Q₁₆	—	178	2	no	—
294	M₁₆	—	89	4	no	—
295	M₁₆	—	178	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	89	1	no	—
299	C₂×C₈	—	89	1	no	—
300	C₂×C₈	—	89	1	no	—
301	C₂×C₈	—	89	1	no	—
302	C₂×C₈	—	356	1	no	—
303	C₂×C₈	—	356	4	no	—
304	C₂×C₈	—	178	2	no	—
305	C₂×C₈	—	178	1	no	—
306	C₂×C₈	—	178	2	no	—
307	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
308	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
309	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
310	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	89	2	no	—
313	C₄⋊C₄	—	178	2	no	—
314	C₄⋊C₄	—	178	2	no	—
315	C₄⋊C₄	—	178	2	no	—
316	C₄⋊C₄	—	178	2	no	—
317	C₄⋊C₄	—	178	2	no	—
318	C₄⋊C₄	—	178	2	no	—
319	C₄⋊C₄	C₂×C₄×D₈₉	1	2	yes	—
320	C₂²⋊C₄	—	178	2	no	—
321	C₂²⋊C₄	—	178	2	no	—
322	C₄²	—	2	2	no	—
323	C₄²	—	89	1	no	—
324	C₄²	—	89	1	no	—
325	C₄²	—	89	1	no	—
326	C₄²	—	178	2	no	—
327	C₄²	C₂³×D₈₉	1	1	yes	characteristic
328	C₂³	—	89	1	no	—
329	Q₈	—	2	2	no	—
330	Q₈	—	2	2	no	—
331	Q₈	—	178	2	no	—
332	Q₈	—	178	2	no	—
333	Q₈	—	712	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	89	1	no	—
337	C₂×C₄	—	89	2	no	—
338	C₂×C₄	—	89	2	no	—
339	C₂×C₄	—	89	1	no	—
340	C₂×C₄	—	89	1	no	—
341	C₂×C₄	—	356	1	no	—
342	C₂×C₄	—	178	2	no	—
343	C₂×C₄	—	178	2	no	—
344	C₂×C₄	—	178	1	no	—
345	C₂×C₄	—	178	1	no	—
346	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
347	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
348	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	4	1	no	—
352	C₈	—	4	2	no	—
353	C₈	—	89	2	no	—
354	C₈	—	356	1	no	—
355	C₈	—	356	2	no	—
356	C₈	—	178	1	no	—
357	C₈	—	178	1	no	—
358	C₈	—	178	1	no	—
359	C₈	#15668018	1	2	yes	—
360	C₈	Q₁₆×D₈₉	1	2	yes	—
361	C₂²	—	89	2	no	—
362	C₂²	—	178	1	no	—
363	C₂²	—	178	1	no	—
364	C₂²	C₄×D₄×D₈₉	1	1	yes	characteristic
365	C₄	—	2	1	no	—
366	C₄	—	4	2	no	—
367	C₄	—	4	2	no	—
368	C₄	—	89	2	no	—
369	C₄	—	356	2	no	—
370	C₄	—	356	2	no	—
371	C₄	—	178	1	no	—
372	C₄	—	178	1	no	—
373	C₄	C₂×#15668018	1	1	yes	characteristic
374	C₄	#19152363	1	1	yes	characteristic
375	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
376	C₄	#135×D₈₉	1	1	yes	characteristic
377	C₂	—	178	2	no	—
378	C₂	#26028991	1	1	yes	characteristic
379	C₂	C₄×Q₁₆×D₈₉	1	1	yes	characteristic
380	C₂	#26029164	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈×Q₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂²×D₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₈×Q₁₆	D₈₉
hypocenter	C₈₉	#18068
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #26019150	C₂	semidirect
4. #26019149	C₂	semidirect
5. #26019105	C₂	semidirect
6. #26019102	C₂	semidirect
7. #26019058	C₂	semidirect
11. C₈×Q₁₆×C₈₉	C₂	semidirect
102. #13083557	Q₁₆	semidirect
198. C₈×Q₁₆	D₈₉	semidirect
202. C₈₉	#18068	semidirect

Maximal subgroups

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

subgroup	index
#26030835	2
#26029194	2
#26019150	2
#26019149	2
#26019105	2
#26019102	2
#26019058	2
#335×D₈₉	2
#26018653	2
C₈×#13083557	2
C₈×Q₁₆×C₈₉	2
#18068	89