Subgroups — #36524566

Order 22784.

638 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030256	C₂	1	1	yes	maximal, characteristic
2	#26029275	C₂	1	1	yes	maximal, characteristic
3	#26019107	C₂	1	2	yes	maximal, complemented
4	#26019086	C₂	1	2	yes	maximal, complemented
5	#26019062	C₂	1	2	yes	maximal, complemented
6	C₈×#13083561	C₂	1	2	yes	maximal, complemented
7	#26019033	C₂	1	1	yes	maximal, characteristic, complemented
8	#26018694	C₂	1	1	yes	maximal, characteristic
9	#26018673	C₂	1	1	yes	maximal, characteristic, complemented
10	#26018547	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2985	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
13	#19151926	C₄	1	2	yes	—
14	#19151885	C₄	1	2	yes	—
15	#19151800	C₄	1	2	yes	—
16	#19151748	C₄	1	2	yes	—
17	C₄×#13083561	C₂²	1	2	yes	—
18	C₄×#13083561	C₂²	1	2	yes	—
19	C₈×D₃₅₆	C₂²	1	2	yes	—
20	C₈×D₃₅₆	C₂²	1	2	yes	—
21	#19151663	—	2	2	no	—
22	#19151651	C₂²	1	1	yes	characteristic
23	#19151651	C₂²	1	1	yes	characteristic
24	C₂²⋊C₈×D₈₉	—	2	2	no	—
25	C₄×#13083558	C₂²	1	1	yes	characteristic
26	#19151350	C₂²	1	2	yes	complemented
27	#19151348	C₂²	1	2	yes	complemented
28	Q₈⋊#10623494	C₂²	1	2	yes	complemented
29	Q₈⋊#10623494	C₂²	1	2	yes	complemented
30	#19151325	C₂²	1	2	yes	—
31	#19151325	C₂²	1	2	yes	—
32	#19151320	C₂²	1	1	yes	characteristic, complemented
33	#19151320	C₂²	1	1	yes	characteristic, complemented
34	#19151319	C₂²	1	1	yes	characteristic, complemented
35	C₈×#10623494	C₂²	1	1	yes	characteristic, complemented
36	#533×C₈₉	C₂²	1	1	yes	characteristic
37	#446×C₈₉	—	2	2	no	—
38	#446×C₈₉	C₂²	1	2	yes	complemented
39	C₈×Q₈×C₈₉	C₂²	1	2	yes	complemented
40	#433×C₈₉	—	2	2	no	—
41	#432×C₈₉	C₂²	1	1	yes	characteristic
42	#431×C₈₉	C₂²	1	1	yes	characteristic
43	C₂²×D₃₅₆	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×D₈₉	—	2	1	no	—
45	C₂×#13083561	C₂×C₄	1	2	yes	—
46	C₂×#13083561	C₂×C₄	1	2	yes	—
47	C₂×#13083558	—	2	1	no	—
48	C₂×C₈×D₈₉	—	2	2	no	—
49	C₂×C₈×D₈₉	—	2	2	no	—
50	C₄×D₃₅₆	—	2	1	no	—
51	C₄×D₃₅₆	—	2	1	no	—
52	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
53	C₄×D₃₅₆	C₂³	1	2	yes	—
54	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
55	C₂×#13083551	—	2	1	no	—
56	#15667943	C₂×C₄	1	2	yes	—
57	#15667943	C₂×C₄	1	2	yes	—
58	C₄⋊#10623494	C₂×C₄	1	2	yes	—
59	C₄⋊#10623494	C₂³	1	2	yes	—
60	#15667936	C₂×C₄	1	2	yes	—
61	#15667936	C₂×C₄	1	2	yes	—
62	#15667932	C₂×C₄	1	2	yes	—
63	#15667932	C₂×C₄	1	2	yes	—
64	C₂²⋊#10623494	—	2	1	no	—
65	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
66	#15667924	—	2	2	no	—
67	#15667924	—	2	2	no	—
68	#15667923	—	2	1	no	—
69	C₄×#10623494	—	2	1	no	—
70	C₄×#10623494	C₂³	1	1	yes	characteristic
71	C₄×#10623494	C₂³	1	1	yes	characteristic
72	C₄×#10623494	C₂³	1	1	yes	characteristic
73	C₄×#10623494	C₂³	1	1	yes	characteristic
74	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
75	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
76	#125×C₈₉	—	2	1	no	—
77	C₂²⋊Q₈×C₈₉	—	2	1	no	—
78	C₂²⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
79	C₄×Q₈×C₈₉	—	2	1	no	—
80	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
81	#117×C₈₉	—	2	1	no	—
82	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₈₉	—	2	2	no	—
84	C₄⋊C₈×C₈₉	—	2	2	no	—
85	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
86	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
87	C₂²⋊C₈×C₈₉	—	2	1	no	—
88	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
89	#97×C₈₉	—	2	2	no	—
90	#97×C₈₉	—	2	1	no	—
91	C₄×C₈×C₈₉	—	2	2	no	—
92	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
94	C₂³×D₈₉	—	2	1	no	—
95	C₂×D₃₅₆	—	2	1	no	—
96	C₂×D₃₅₆	—	2	4	no	—
97	C₂×D₃₅₆	C₂²×C₄	1	2	yes	—
98	C₂²×Dic₈₉	—	2	1	no	—
99	C₂×C₄×D₈₉	—	2	2	no	—
100	C₂×C₄×D₈₉	—	2	2	no	—
101	C₂×C₄×D₈₉	—	8	2	no	—
102	#13083561	—	2	4	no	—
103	#13083561	—	2	4	no	—
104	#13083558	—	4	2	no	—
105	C₂×#10623494	—	2	1	no	—
106	C₂×#10623494	—	2	2	no	—
107	C₂×#10623494	—	2	1	no	—
108	C₂×#10623494	C₂²×C₄	1	2	yes	—
109	C₂×#10623494	C₂²×C₄	1	2	yes	—
110	C₈×D₈₉	—	8	2	no	—
111	C₈×D₈₉	—	8	2	no	—
112	C₄⋊Dic₈₉	—	2	1	no	—
113	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
114	#13083551	—	2	2	no	—
115	#13083551	—	2	2	no	—
116	C₂×Q₈×C₈₉	—	2	1	no	—
117	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
118	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
121	C₂×C₈×C₈₉	—	2	2	no	—
122	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
123	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
124	C₄⋊C₄×C₈₉	—	2	1	no	—
125	C₄⋊C₄×C₈₉	—	2	1	no	—
126	C₄⋊C₄×C₈₉	—	2	1	no	—
127	C₄⋊C₄×C₈₉	—	2	2	no	—
128	C₄⋊C₄×C₈₉	—	2	2	no	—
129	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
130	C₂²⋊C₄×C₈₉	—	2	2	no	—
131	C₄²×C₈₉	—	2	2	no	—
132	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₈₉	#41	1	1	yes	characteristic
135	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
136	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
137	C₂²×D₈₉	—	2	2	no	—
138	C₂²×D₈₉	—	2	4	no	—
139	C₂²×D₈₉	—	8	1	no	—
140	C₂²×D₈₉	—	8	1	no	—
141	Q₈×C₈₉	—	2	2	no	—
142	Q₈×C₈₉	—	2	2	no	—
143	Q₈×C₈₉	—	8	1	no	—
144	C₂×C₄×C₈₉	—	2	2	no	—
145	C₂×C₄×C₈₉	—	2	2	no	—
146	C₂×C₄×C₈₉	—	2	2	no	—
147	C₂×C₄×C₈₉	—	2	2	no	—
148	C₂×C₄×C₈₉	—	4	1	no	—
149	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
155	C₂×Dic₈₉	—	2	2	no	—
156	C₂×Dic₈₉	—	8	1	no	—
157	D₃₅₆	—	2	4	no	—
158	D₃₅₆	—	2	2	no	—
159	D₃₅₆	#131	1	4	yes	—
160	C₄×D₈₉	—	4	2	no	—
161	C₄×D₈₉	—	4	2	no	—
162	C₄×D₈₉	—	8	2	no	—
163	C₈×C₈₉	—	2	2	no	—
164	C₈×C₈₉	—	2	2	no	—
165	C₈×C₈₉	—	4	2	no	—
166	#10623494	—	2	2	no	—
167	#10623494	—	2	2	no	—
168	#10623494	—	2	2	no	—
169	#10623494	—	4	1	no	—
170	#10623494	—	4	2	no	—
171	C₂²×C₈₉	—	2	1	no	—
172	C₂²×C₈₉	—	2	1	no	—
173	C₂²×C₈₉	—	2	1	no	—
174	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
175	C₂×D₈₉	—	2	4	no	—
176	C₂×D₈₉	—	4	2	no	—
177	C₂×D₈₉	—	4	2	no	—
178	C₂×D₈₉	—	8	4	no	—
179	C₄×C₈₉	—	2	1	no	—
180	C₄×C₈₉	—	2	1	no	—
181	C₄×C₈₉	—	2	1	no	—
182	C₄×C₈₉	—	2	1	no	—
183	C₄×C₈₉	—	4	2	no	—
184	C₄×C₈₉	—	4	2	no	—
185	C₄×C₈₉	C₂×#131	1	2	yes	—
186	C₄×C₈₉	#577	1	2	yes	—
187	Dic₈₉	—	4	2	no	—
188	#17924	—	89	1	no	maximal, Sylow
189	C₂×C₈₉	—	4	1	no	—
190	C₂×C₈₉	#2929	1	1	yes	characteristic
191	C₂×C₈₉	#2949	1	1	yes	characteristic
192	C₂×C₈₉	#2956	1	1	yes	characteristic
193	D₈₉	—	4	4	no	—
194	#3101	—	89	1	no	—
195	#2985	D₈₉	1	1	yes	characteristic, complemented
196	#2972	—	89	1	no	—
197	#1585	—	89	2	no	—
198	#1582	—	89	2	no	—
199	SD₁₆⋊C₈	—	89	2	no	—
200	C₈×SD₁₆	—	89	2	no	—
201	#1572	—	89	1	no	—
202	#1490	—	89	1	no	—
203	#1485	—	89	1	no	—
204	#1453	—	89	1	no	—
205	C₈₉	#17924	1	1	yes	characteristic, complemented, Sylow
206	#533	C₂×D₈₉	1	1	yes	characteristic
207	C₂×C₄×D₄	—	89	1	no	—
208	#483	—	89	2	no	—
209	#476	—	89	2	no	—
210	#461	—	89	2	no	—
211	#450	—	89	2	no	—
212	#446	—	2	2	no	—
213	#446	C₂×D₈₉	1	2	yes	complemented
214	C₈×Q₈	C₂×D₈₉	1	2	yes	complemented
215	C₄×SD₁₆	—	89	2	no	—
216	C₄×SD₁₆	—	89	2	no	—
217	C₈×D₄	—	89	2	no	—
218	C₈×D₄	—	89	2	no	—
219	C₈×D₄	—	178	2	no	—
220	#433	—	2	2	no	—
221	#432	C₂×D₈₉	1	1	yes	characteristic
222	#431	—	89	1	no	—
223	#431	—	89	1	no	—
224	#431	C₂×D₈₉	1	1	yes	characteristic
225	C₂×C₂²⋊C₈	—	178	2	no	—
226	C₄×M₁₆	—	89	1	no	—
227	#335	—	89	2	no	—
228	#334	—	89	2	no	—
229	Q₈⋊C₈	—	89	2	no	—
230	Q₈⋊C₈	—	89	2	no	—
231	D₄⋊C₈	—	89	2	no	—
232	D₄⋊C₈	—	89	2	no	—
233	#322	—	89	1	no	—
234	#322	—	89	1	no	—
235	#322	—	89	1	no	—
236	C₈²	—	89	1	no	—
237	C₂²×D₄	—	89	1	no	—
238	C₂³×C₄	—	178	1	no	—
239	C₂×SD₁₆	—	89	2	no	—
240	C₂×SD₁₆	—	89	2	no	—
241	C₂×M₁₆	—	178	1	no	—
242	C₂²×C₈	—	178	2	no	—
243	C₂²×C₈	—	178	2	no	—
244	C₄⋊Q₈	C₄×D₈₉	1	1	yes	characteristic
245	C₄⋊Q₈	C₄×D₈₉	1	1	yes	characteristic
246	#125	—	2	1	no	—
247	C₂²⋊Q₈	—	2	1	no	—
248	C₂²⋊Q₈	C₄×D₈₉	1	2	yes	—
249	C₄×Q₈	—	2	1	no	—
250	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
251	C₄×D₄	—	89	2	no	—
252	C₄×D₄	—	89	2	no	—
253	C₄×D₄	—	178	1	no	—
254	C₄×D₄	—	178	1	no	—
255	#117	—	2	1	no	—
256	C₂×C₄⋊C₄	—	89	1	no	—
257	C₂×C₂²⋊C₄	—	178	1	no	—
258	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
259	#106	—	89	2	no	—
260	#106	—	89	2	no	—
261	C₄⋊C₈	—	2	2	no	—
262	C₄⋊C₈	—	2	2	no	—
263	C₄⋊C₈	—	89	2	no	—
264	C₄⋊C₈	—	89	2	no	—
265	C₄⋊C₈	C₂²⋊D₈₉	1	2	yes	—
266	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
267	#103	—	89	2	no	—
268	#103	—	89	2	no	—
269	#102	—	89	2	no	—
270	#102	—	89	2	no	—
271	C₂²⋊C₈	—	2	1	no	—
272	C₂²⋊C₈	—	89	2	no	—
273	C₂²⋊C₈	—	178	1	no	—
274	C₂²⋊C₈	—	178	2	no	—
275	C₂²⋊C₈	—	178	2	no	—
276	C₂²⋊C₈	C₂²⋊D₈₉	1	2	yes	—
277	#97	—	2	2	no	—
278	#97	—	2	1	no	—
279	#97	—	178	1	no	—
280	C₄×C₈	—	2	2	no	—
281	C₄×C₈	—	89	1	no	—
282	C₄×C₈	—	89	1	no	—
283	C₄×C₈	—	89	1	no	—
284	C₄×C₈	—	89	1	no	—
285	C₄×C₈	—	178	1	no	—
286	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
288	C₂⁴	—	178	1	no	—
289	C₂×Q₈	—	2	1	no	—
290	C₂×Q₈	C₂×C₄×D₈₉	1	2	yes	—
291	C₂×D₄	—	89	2	no	—
292	C₂×D₄	—	178	4	no	—
293	C₂×D₄	—	178	1	no	—
294	C₂²×C₄	—	178	2	no	—
295	C₂²×C₄	—	178	1	no	—
296	C₂²×C₄	—	178	2	no	—
297	C₂²×C₄	—	712	2	no	—
298	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
299	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
300	C₂²×C₄	#13083574	1	1	yes	characteristic
301	SD₁₆	—	178	4	no	—
302	SD₁₆	—	178	4	no	—
303	M₁₆	—	356	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	89	2	no	—
306	C₂×C₈	—	89	2	no	—
307	C₂×C₈	—	178	2	no	—
308	C₂×C₈	—	178	1	no	—
309	C₂×C₈	—	178	1	no	—
310	C₂×C₈	—	712	2	no	—
311	C₂×C₈	—	712	2	no	—
312	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
313	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	2	1	no	—
316	C₄⋊C₄	—	2	1	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	89	2	no	—
320	C₄⋊C₄	—	178	1	no	—
321	C₄⋊C₄	C₂×C₄×D₈₉	1	2	yes	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₂²⋊C₄	—	178	2	no	—
324	C₂²⋊C₄	—	178	2	no	—
325	C₄²	—	2	2	no	—
326	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
327	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
328	C₄²	#13083574	1	1	yes	characteristic
329	C₄²	C₂³×D₈₉	1	1	yes	characteristic
330	C₂³	—	178	4	no	—
331	C₂³	—	178	2	no	—
332	C₂³	—	712	1	no	—
333	C₂³	—	712	1	no	—
334	C₂³	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
335	Q₈	—	2	2	no	—
336	Q₈	—	2	2	no	—
337	Q₈	—	8	1	no	—
338	D₄	—	89	4	no	—
339	D₄	—	178	4	no	—
340	D₄	—	178	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	356	2	no	—
347	C₂×C₄	—	356	2	no	—
348	C₂×C₄	—	178	2	no	—
349	C₂×C₄	—	712	2	no	—
350	C₂×C₄	—	712	1	no	—
351	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
352	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
353	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
354	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
356	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
357	C₈	—	2	2	no	—
358	C₈	—	2	2	no	—
359	C₈	—	4	2	no	—
360	C₈	—	356	1	no	—
361	C₈	—	356	2	no	—
362	C₈	—	178	2	no	—
363	C₈	—	178	2	no	—
364	C₈	—	178	2	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	356	2	no	—
369	C₂²	—	356	2	no	—
370	C₂²	—	178	4	no	—
371	C₂²	—	712	4	no	—
372	C₂²	C₂×C₄×C₂²⋊D₈₉	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	2	no	—
379	C₄	—	356	2	no	—
380	C₄	#131×D₈₉	1	2	yes	—
381	C₄	#19152406	1	2	yes	—
382	C₂	—	4	1	no	—
383	C₂	—	356	4	no	—
384	C₂	#26028944	1	1	yes	characteristic
385	C₂	#26029111	1	1	yes	characteristic
386	C₂	#26029159	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#2985	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂²×D₃₅₆	C₂×C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#2985	D₈₉
hypocenter	C₈₉	#17924
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #2985

Complements

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

N	H ≅ G/N	type
3. #26019107	C₂	semidirect
4. #26019086	C₂	semidirect
5. #26019062	C₂	semidirect
6. C₈×#13083561	C₂	semidirect
7. #26019033	C₂	semidirect
9. #26018673	C₂	semidirect
10. #26018547	C₂	semidirect
11. C₈₉×#2985	C₂	semidirect
26. #19151350	C₂²	semidirect
27. #19151348	C₂²	semidirect
28. Q₈⋊#10623494	C₂²	semidirect
29. Q₈⋊#10623494	C₂²	semidirect
32. #19151320	C₂²	semidirect
33. #19151320	C₂²	semidirect
34. #19151319	C₂²	semidirect
35. C₈×#10623494	C₂²	semidirect
38. #446×C₈₉	C₂²	semidirect
39. C₈×Q₈×C₈₉	C₂²	semidirect
195. #2985	D₈₉	semidirect
205. C₈₉	#17924	semidirect
213. #446	C₂×D₈₉	semidirect
214. C₈×Q₈	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26030256	2
#26029275	2
#26019107	2
#26019086	2
#26019062	2
C₈×#13083561	2
#26019033	2
#26018694	2
#26018673	2
#26018547	2
C₈₉×#2985	2
#17924	89