Subgroups — #36524104

Order 22784.

674 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	#26029333	C₂	1	1	yes	maximal, characteristic, complemented
2	#26029292	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#13083561	C₂	1	1	yes	maximal, characteristic
4	#26019127	C₂	1	4	yes	maximal, complemented
5	#26019092	C₂	1	4	yes	maximal, complemented
6	#26018998	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018644	C₂	1	1	yes	maximal, characteristic
8	#26018545	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1479	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#13083561	C₄	1	1	yes	characteristic
11	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
12	Q₈×#10623494	—	2	2	no	—
13	Q₈×#10623494	C₂²	1	4	yes	complemented
14	C₄×#13083561	C₄	1	2	yes	—
15	C₄×#13083561	C₄	1	2	yes	—
16	C₄×#13083561	C₂²	1	2	yes	—
17	C₄×#13083561	C₂²	1	2	yes	—
18	#19151677	—	2	2	no	—
19	#19151675	C₂²	1	4	yes	—
20	#19151651	—	2	2	no	—
21	#19151651	C₂²	1	1	yes	characteristic
22	#19151651	C₂²	1	1	yes	characteristic
23	C₂×#15667943	C₄	1	1	yes	characteristic
24	C₂×#15667936	C₄	1	1	yes	characteristic
25	C₂×#15667932	C₄	1	1	yes	characteristic
26	#19151560	—	2	2	no	—
27	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
28	#19151350	C₂²	1	2	yes	complemented
29	#19151348	C₂²	1	2	yes	complemented
30	#19151327	C₂²	1	4	yes	—
31	#19151320	C₂²	1	2	yes	complemented
32	#19151319	C₂²	1	2	yes	complemented
33	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
34	#431×C₈₉	C₂²	1	1	yes	characteristic
35	#431×C₈₉	C₂²	1	1	yes	characteristic
36	Q₈⋊C₈×C₈₉	C₂²	1	4	yes	complemented
37	C₂²×D₃₅₆	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×D₈₉	—	2	1	no	—
39	C₂×#13083561	—	2	4	no	—
40	C₂×#13083561	C₂×C₄	1	2	yes	—
41	C₂×#13083561	C₂×C₄	1	2	yes	—
42	C₂×#13083557	—	2	4	no	—
43	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
44	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
45	C₄×D₃₅₆	—	2	1	no	—
46	C₄×D₃₅₆	—	2	1	no	—
47	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
48	C₄×D₃₅₆	C₂³	1	2	yes	—
49	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂×#13083551	—	2	1	no	—
51	#15667943	C₂×C₄	1	2	yes	—
52	#15667943	C₂×C₄	1	2	yes	—
53	C₄⋊#10623494	—	2	4	no	—
54	C₄⋊#10623494	D₄	1	2	yes	—
55	C₄⋊#10623494	C₂³	1	2	yes	—
56	#15667936	C₂×C₄	1	2	yes	—
57	#15667936	C₂×C₄	1	2	yes	—
58	#15667932	C₂×C₄	1	2	yes	—
59	#15667932	C₂×C₄	1	2	yes	—
60	C₂²⋊#10623494	—	2	1	no	—
61	C₂²⋊#10623494	D₄	1	2	yes	—
62	#15667924	—	2	4	no	—
63	C₄×#10623494	—	2	4	no	—
64	C₄×#10623494	—	2	1	no	—
65	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
66	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
67	C₄×#10623494	C₂³	1	1	yes	characteristic
68	C₄×#10623494	C₂³	1	1	yes	characteristic
69	C₄×#10623494	C₂³	1	2	yes	—
70	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
71	C₄×Q₈×C₈₉	—	2	1	no	—
72	C₄×Q₈×C₈₉	—	2	1	no	—
73	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
74	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
76	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₈₉	—	2	1	no	—
78	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₈₉	—	2	1	no	—
80	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
81	C₂²⋊C₈×C₈₉	—	2	1	no	—
82	C₂²⋊C₈×C₈₉	—	2	1	no	—
83	C₄×C₈×C₈₉	—	2	1	no	—
84	C₄×C₈×C₈₉	C₂³	1	2	yes	—
85	C₂³×D₈₉	—	2	1	no	—
86	C₂×D₃₅₆	—	2	2	no	—
87	C₂×D₃₅₆	—	2	2	no	—
88	C₂×D₃₅₆	—	2	1	no	—
89	C₂×D₃₅₆	C₂²×C₄	1	2	yes	—
90	C₂²×Dic₈₉	—	2	1	no	—
91	C₂×C₄×D₈₉	—	2	2	no	—
92	C₂×C₄×D₈₉	—	2	2	no	—
93	C₂×C₄×D₈₉	—	8	2	no	—
94	#13083561	—	2	4	no	—
95	#13083561	—	2	4	no	—
96	#13083557	—	8	4	no	—
97	C₂×#10623494	—	2	2	no	—
98	C₂×#10623494	—	2	2	no	—
99	C₂×#10623494	—	2	2	no	—
100	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
101	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
102	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
103	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
104	C₂×#10623494	C₂×D₄	1	4	yes	—
105	C₄⋊Dic₈₉	—	2	1	no	—
106	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
107	#13083551	—	2	2	no	—
108	#13083551	—	2	2	no	—
109	C₂×Q₈×C₈₉	—	2	1	no	—
110	C₂×Q₈×C₈₉	—	2	2	no	—
111	C₂×Q₈×C₈₉	M₁₆	1	4	yes	complemented
112	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
113	C₂²×C₄×C₈₉	—	2	1	no	—
114	C₂²×C₄×C₈₉	—	2	1	no	—
115	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
118	C₂×C₈×C₈₉	—	2	2	no	—
119	C₂×C₈×C₈₉	—	2	2	no	—
120	C₄⋊C₄×C₈₉	—	2	2	no	—
121	C₄⋊C₄×C₈₉	—	2	1	no	—
122	C₄⋊C₄×C₈₉	—	2	2	no	—
123	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
124	C₄²×C₈₉	—	2	2	no	—
125	C₄²×C₈₉	—	2	2	no	—
126	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₈₉	#41	1	1	yes	characteristic
129	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
130	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
131	C₂²×D₈₉	—	2	2	no	—
132	C₂²×D₈₉	—	4	2	no	—
133	C₂²×D₈₉	—	8	1	no	—
134	C₂²×D₈₉	—	8	1	no	—
135	Q₈×C₈₉	—	2	2	no	—
136	Q₈×C₈₉	—	2	4	no	—
137	Q₈×C₈₉	C₂×M₁₆	1	4	yes	—
138	C₂×C₄×C₈₉	—	2	2	no	—
139	C₂×C₄×C₈₉	—	2	4	no	—
140	C₂×C₄×C₈₉	—	2	2	no	—
141	C₂×C₄×C₈₉	—	2	2	no	—
142	C₂×C₄×C₈₉	—	4	1	no	—
143	C₂×C₄×C₈₉	—	8	1	no	—
144	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
148	C₂×C₄×C₈₉	#131	1	2	yes	—
149	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
152	C₂×Dic₈₉	—	2	2	no	—
153	C₂×Dic₈₉	—	8	1	no	—
154	D₃₅₆	—	2	2	no	—
155	D₃₅₆	—	2	2	no	—
156	D₃₅₆	—	4	2	no	—
157	C₄×D₈₉	—	4	4	no	—
158	C₄×D₈₉	—	8	2	no	—
159	C₈×C₈₉	—	4	2	no	—
160	C₈×C₈₉	—	4	2	no	—
161	#10623494	—	2	1	no	—
162	#10623494	—	2	1	no	—
163	#10623494	—	2	1	no	—
164	#10623494	—	2	1	no	—
165	#10623494	—	2	4	no	—
166	#10623494	—	4	2	no	—
167	C₂²×C₈₉	—	2	1	no	—
168	C₂²×C₈₉	—	2	1	no	—
169	C₂²×C₈₉	#435	1	2	yes	—
170	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×D₈₉	—	4	2	no	—
172	C₂×D₈₉	—	4	2	no	—
173	C₂×D₈₉	—	4	2	no	—
174	C₂×D₈₉	—	8	4	no	—
175	C₄×C₈₉	—	2	4	no	—
176	C₄×C₈₉	—	2	1	no	—
177	C₄×C₈₉	—	2	1	no	—
178	C₄×C₈₉	—	2	1	no	—
179	C₄×C₈₉	—	4	2	no	—
180	C₄×C₈₉	#435	1	2	yes	—
181	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
182	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
183	C₄×C₈₉	#577	1	2	yes	—
184	Dic₈₉	—	4	2	no	—
185	#17878	—	89	1	no	maximal, Sylow
186	C₂×C₈₉	—	2	2	no	—
187	C₂×C₈₉	C₂×#435	1	1	yes	characteristic
188	C₂×C₈₉	#2949	1	1	yes	characteristic
189	C₂×C₈₉	#2957	1	1	yes	characteristic
190	D₈₉	—	8	2	no	—
191	#2982	—	89	1	no	—
192	#2975	—	89	1	no	—
193	C₂×C₄×SD₁₆	—	89	1	no	—
194	#1588	—	89	4	no	—
195	#1583	—	89	4	no	—
196	#1564	—	89	1	no	—
197	#1479	D₈₉	1	1	yes	characteristic, complemented
198	#1478	—	89	1	no	—
199	#1452	—	89	1	no	—
200	C₈₉	#17878	1	1	yes	characteristic, complemented, Sylow
201	C₂²×SD₁₆	—	89	1	no	—
202	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
203	C₂×C₄×D₄	—	89	1	no	—
204	C₈×Q₈	—	89	4	no	—
205	C₈×Q₈	—	178	2	no	—
206	C₄×SD₁₆	—	89	2	no	—
207	C₄×SD₁₆	—	89	2	no	—
208	C₄×SD₁₆	—	89	2	no	—
209	C₄×SD₁₆	—	89	2	no	—
210	#436	—	89	4	no	—
211	#436	—	178	2	no	—
212	#431	—	89	1	no	—
213	#431	—	89	1	no	—
214	#431	—	178	2	no	—
215	#431	C₂×D₈₉	1	1	yes	characteristic
216	#431	C₂×D₈₉	1	1	yes	characteristic
217	C₂×#106	—	89	1	no	—
218	C₂×#103	—	89	1	no	—
219	C₂×#102	—	89	1	no	—
220	#407	—	178	2	no	—
221	C₂×C₄×C₈	—	89	1	no	—
222	#335	—	89	2	no	—
223	#334	—	89	2	no	—
224	Q₈⋊C₈	C₂×D₈₉	1	4	yes	complemented
225	D₄⋊C₈	—	89	4	no	—
226	#322	—	89	2	no	—
227	#322	—	89	2	no	—
228	C₂²×Q₈	C₄×D₈₉	1	1	yes	characteristic
229	C₂²×D₄	—	89	1	no	—
230	C₂³×C₄	—	178	1	no	—
231	C₂×SD₁₆	—	89	2	no	—
232	C₂×SD₁₆	—	89	2	no	—
233	C₂×SD₁₆	—	178	4	no	—
234	C₂×M₁₆	—	178	4	no	—
235	C₂²×C₈	—	89	1	no	—
236	C₂²×C₈	—	89	1	no	—
237	C₄×Q₈	—	2	1	no	—
238	C₄×Q₈	—	2	1	no	—
239	C₄×Q₈	C₄×D₈₉	1	2	yes	—
240	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
241	C₄×D₄	—	89	2	no	—
242	C₄×D₄	—	89	2	no	—
243	C₄×D₄	—	178	1	no	—
244	C₄×D₄	—	178	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	89	1	no	—
247	C₂×C₄⋊C₄	C₄×D₈₉	1	1	yes	characteristic
248	C₂×C₂²⋊C₄	—	178	1	no	—
249	C₂×C₄²	—	2	1	no	—
250	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
251	#106	—	89	2	no	—
252	#106	—	89	2	no	—
253	C₄⋊C₈	—	2	1	no	—
254	C₄⋊C₈	—	89	2	no	—
255	C₄⋊C₈	—	89	2	no	—
256	C₄⋊C₈	—	178	4	no	—
257	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
258	#103	—	89	2	no	—
259	#103	—	89	2	no	—
260	#102	—	89	2	no	—
261	#102	—	89	2	no	—
262	C₂²⋊C₈	—	2	1	no	—
263	C₂²⋊C₈	—	2	1	no	—
264	C₂²⋊C₈	—	89	2	no	—
265	C₂²⋊C₈	—	178	4	no	—
266	C₂²⋊C₈	—	178	1	no	—
267	C₄×C₈	—	2	1	no	—
268	C₄×C₈	—	89	1	no	—
269	C₄×C₈	—	89	1	no	—
270	C₄×C₈	—	89	1	no	—
271	C₄×C₈	—	89	1	no	—
272	C₄×C₈	—	89	2	no	—
273	C₄×C₈	—	178	4	no	—
274	C₄×C₈	—	178	1	no	—
275	C₄×C₈	C₂²×D₈₉	1	2	yes	—
276	C₂⁴	—	178	1	no	—
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	2	2	no	—
279	C₂×Q₈	#13083557	1	4	yes	complemented
280	C₂×Q₈	C₂×C₄×D₈₉	1	2	yes	—
281	C₂×D₄	—	89	2	no	—
282	C₂×D₄	—	178	1	no	—
283	C₂×D₄	—	178	2	no	—
284	C₂×D₄	—	178	2	no	—
285	C₂²×C₄	—	2	1	no	—
286	C₂²×C₄	—	2	1	no	—
287	C₂²×C₄	—	178	2	no	—
288	C₂²×C₄	—	178	1	no	—
289	C₂²×C₄	—	178	2	no	—
290	C₂²×C₄	—	712	2	no	—
291	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
292	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
293	C₂²×C₄	#13083573	1	1	yes	characteristic
294	SD₁₆	—	178	4	no	—
295	SD₁₆	—	178	4	no	—
296	M₁₆	—	712	4	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	89	1	no	—
300	C₂×C₈	—	89	1	no	—
301	C₂×C₈	—	89	1	no	—
302	C₂×C₈	—	89	1	no	—
303	C₂×C₈	—	89	4	no	—
304	C₂×C₈	—	178	2	no	—
305	C₂×C₈	—	178	2	no	—
306	C₂×C₈	—	178	2	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	89	2	no	—
311	C₄⋊C₄	—	178	1	no	—
312	C₄⋊C₄	C₂×C₄×D₈₉	1	2	yes	—
313	C₂²⋊C₄	—	178	2	no	—
314	C₂²⋊C₄	—	178	2	no	—
315	C₄²	—	2	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
318	C₄²	D₄×D₈₉	1	1	yes	characteristic
319	C₄²	#13083573	1	1	yes	characteristic
320	C₄²	C₂³×D₈₉	1	1	yes	characteristic
321	C₂³	—	356	2	no	—
322	C₂³	—	178	2	no	—
323	C₂³	—	712	1	no	—
324	C₂³	—	712	1	no	—
325	C₂³	#15667972	1	1	yes	characteristic
326	Q₈	—	2	4	no	—
327	Q₈	—	2	2	no	—
328	Q₈	C₂×#13083557	1	4	yes	—
329	D₄	—	356	2	no	—
330	D₄	—	178	2	no	—
331	D₄	—	178	2	no	—
332	C₂×C₄	—	2	4	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	8	1	no	—
338	C₂×C₄	—	356	4	no	—
339	C₂×C₄	—	178	2	no	—
340	C₂×C₄	—	712	2	no	—
341	C₂×C₄	—	712	1	no	—
342	C₂×C₄	#15667972	1	1	yes	characteristic
343	C₂×C₄	#15667972	1	1	yes	characteristic
344	C₂×C₄	#15667972	1	1	yes	characteristic
345	C₂×C₄	C₂×#13083557	1	2	yes	—
346	C₂×C₄	#15668020	1	2	yes	—
347	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
348	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
349	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
350	C₈	—	4	2	no	—
351	C₈	—	4	2	no	—
352	C₈	—	356	2	no	—
353	C₈	—	178	4	no	—
354	C₈	—	178	1	no	—
355	C₈	—	178	1	no	—
356	C₈	—	178	1	no	—
357	C₈	—	178	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	356	2	no	—
361	C₂²	—	356	2	no	—
362	C₂²	—	356	2	no	—
363	C₂²	—	712	4	no	—
364	C₂²	#19151666	1	2	yes	—
365	C₂²	C₂×#15667972	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	2	4	no	—
370	C₄	—	4	2	no	—
371	C₄	—	356	2	no	—
372	C₄	#19151666	1	2	yes	—
373	C₄	C₂²×#13083557	1	1	yes	characteristic
374	C₄	C₂×#15668020	1	1	yes	characteristic
375	C₄	#19152407	1	2	yes	—
376	C₂	—	2	2	no	—
377	C₂	—	712	2	no	—
378	C₂	C₂×#19151666	1	1	yes	characteristic
379	C₂	#26029107	1	1	yes	characteristic
380	C₂	#26029165	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×#1479	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂²×D₃₅₆	C₂×C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1479	D₈₉
hypocenter	C₈₉	#17878
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1479

Complements

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

N	H ≅ G/N	type
1. #26029333	C₂	semidirect
4. #26019127	C₂	semidirect
5. #26019092	C₂	semidirect
6. #26018998	C₂	semidirect
8. #26018545	C₂	semidirect
9. C₈₉×#1479	C₂	semidirect
13. Q₈×#10623494	C₂²	semidirect
28. #19151350	C₂²	semidirect
29. #19151348	C₂²	semidirect
31. #19151320	C₂²	semidirect
32. #19151319	C₂²	semidirect
36. Q₈⋊C₈×C₈₉	C₂²	semidirect
111. C₂×Q₈×C₈₉	M₁₆	semidirect
197. #1479	D₈₉	semidirect
200. C₈₉	#17878	semidirect
224. Q₈⋊C₈	C₂×D₈₉	semidirect
279. C₂×Q₈	#13083557	semidirect

Maximal subgroups

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

subgroup	index
#26029333	2
#26029292	2
C₂×C₄×#13083561	2
#26019127	2
#26019092	2
#26018998	2
#26018644	2
#26018545	2
C₈₉×#1479	2
#17878	89