Subgroups — #36563204

Order 22784.

750 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152002	C₂	1	1	yes	maximal, characteristic
2	#26022996	C₂	1	2	yes	maximal
3	#26021223	C₂	1	2	yes	maximal
4	#26020989	C₂	1	2	yes	maximal
5	#26020986	C₂	1	1	yes	maximal, characteristic
6	#26020983	C₂	1	1	yes	maximal, characteristic
7	#26020415	C₂	1	1	yes	maximal, characteristic
8	#26020150	C₂	1	2	yes	maximal
9	C₄×#15667946	C₂	1	1	yes	maximal, characteristic
10	#26019938	C₂	1	1	yes	maximal, characteristic
11	C₄×#107×C₈₉	C₂	1	1	yes	maximal, characteristic
12	C₂×#15668011	C₂²	1	2	yes	—
13	#19152002	C₄	1	8	yes	complemented
14	C₂×#15667946	C₂²	1	1	yes	characteristic
15	C₂×#15667946	C₂²	1	1	yes	characteristic
16	C₂×#15667945	C₂²	1	2	yes	—
17	C₂×#15667943	C₂²	1	2	yes	—
18	C₂×#15667923	C₂²	1	1	yes	characteristic
19	C₂×#15667922	C₂²	1	1	yes	characteristic
20	C₂×#15667922	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
23	#19151504	—	2	2	no	—
24	#19151502	—	2	2	no	—
25	#19151497	C₂²	1	2	yes	—
26	#19151494	C₂²	1	2	yes	—
27	#19151485	—	2	2	no	—
28	#19151482	—	2	2	no	—
29	#19151463	C₂²	1	2	yes	—
30	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
31	#19151444	C₂²	1	1	yes	characteristic
32	#19151361	C₂²	1	2	yes	—
33	#19151360	C₂²	1	2	yes	—
34	#19151360	C₂²	1	2	yes	—
35	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
39	#340×C₈₉	C₂²	1	2	yes	—
40	C₂²×Dic₁₇₈	—	2	2	no	—
41	C₂²×#10623494	C₂³	1	1	yes	characteristic
42	C₂²×#10623494	C₂³	1	1	yes	characteristic
43	#15668011	—	2	4	no	—
44	#15668011	C₂×C₄	1	8	yes	—
45	C₂×C₄⋊Dic₈₉	—	2	2	no	—
46	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
48	C₂×#13083553	—	2	2	no	—
49	C₂×#13083553	—	2	2	no	—
50	C₂×C₄×Dic₈₉	—	2	2	no	—
51	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
52	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
53	#15667946	—	2	2	no	—
54	#15667946	C₂×C₄	1	4	yes	—
55	#15667945	C₂×C₄	1	8	yes	—
56	#15667943	—	2	4	no	—
57	#15667923	Q₈	1	4	yes	complemented
58	#15667922	—	2	2	no	—
59	#15667922	—	2	2	no	—
60	C₈×Dic₈₉	—	2	2	no	—
61	C₈×Dic₈₉	C₂×C₄	1	4	yes	—
62	#15667919	—	2	2	no	—
63	#15667919	—	2	2	no	—
64	#15667919	—	2	2	no	—
65	#15667919	—	2	2	no	—
66	#15667919	—	2	2	no	—
67	#15667919	D₄	1	1	yes	characteristic
68	#15667919	D₄	1	1	yes	characteristic
69	#15667919	D₄	1	1	yes	characteristic
70	#15667919	D₄	1	1	yes	characteristic
71	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
75	C₂×C₄²×C₈₉	—	2	2	no	—
76	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
77	#107×C₈₉	—	2	2	no	—
78	#107×C₈₉	C₂×C₄	1	4	yes	—
79	C₄×C₈×C₈₉	Q₈	1	4	yes	complemented
80	#95×C₈₉	—	2	2	no	—
81	C₂×Dic₁₇₈	—	2	8	no	—
82	C₂×Dic₁₇₈	—	8	2	no	—
83	C₂×Dic₁₇₈	—	8	2	no	—
84	C₂²×Dic₈₉	—	2	2	no	—
85	C₂²×Dic₈₉	—	2	2	no	—
86	C₂²×Dic₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×Dic₈₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×Dic₈₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×Dic₈₉	C₂×D₄	1	1	yes	characteristic
90	C₂×#10623494	—	2	1	no	—
91	C₂×#10623494	—	2	1	no	—
92	C₂×#10623494	C₂²×C₄	1	2	yes	—
93	C₂×#10623494	C₂×Q₈	1	2	yes	—
94	C₂×#10623494	C₂×Q₈	1	2	yes	—
95	C₂×#10623494	#41	1	2	yes	—
96	C₄⋊Dic₈₉	—	2	2	no	—
97	C₄⋊Dic₈₉	—	2	2	no	—
98	C₄⋊Dic₈₉	—	2	2	no	—
99	C₄⋊Dic₈₉	—	8	2	no	—
100	C₄⋊Dic₈₉	C₂²×C₄	1	4	yes	—
101	#13083553	—	2	8	no	—
102	#13083553	—	8	2	no	—
103	C₄×Dic₈₉	—	2	1	no	—
104	C₄×Dic₈₉	—	2	1	no	—
105	C₄×Dic₈₉	—	8	2	no	—
106	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
107	C₄×Dic₈₉	#41	1	2	yes	—
108	C₂²×C₄×C₈₉	—	2	2	no	—
109	C₂²×C₄×C₈₉	—	2	2	no	—
110	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
113	C₂×C₈×C₈₉	—	2	1	no	—
114	C₂×C₈×C₈₉	—	2	1	no	—
115	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
116	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
117	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
118	C₂×C₈×C₈₉	#41	1	2	yes	—
119	C₄⋊C₄×C₈₉	—	2	2	no	—
120	C₄⋊C₄×C₈₉	—	2	2	no	—
121	C₄⋊C₄×C₈₉	—	2	2	no	—
122	C₄⋊C₄×C₈₉	C₂²×C₄	1	4	yes	—
123	C₄²×C₈₉	—	8	2	no	—
124	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
125	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
126	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	—	2	1	no	—
128	C₂×C₄×C₈₉	—	2	1	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	—	2	1	no	—
131	C₂×C₄×C₈₉	—	2	4	no	—
132	C₂×C₄×C₈₉	—	4	2	no	—
133	C₂×C₄×C₈₉	—	4	2	no	—
134	C₂×C₄×C₈₉	—	4	2	no	—
135	C₂×C₄×C₈₉	—	4	2	no	—
136	C₂×C₄×C₈₉	—	4	2	no	—
137	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
138	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
139	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
145	C₂×Dic₈₉	—	2	1	no	—
146	C₂×Dic₈₉	—	2	1	no	—
147	C₂×Dic₈₉	—	2	1	no	—
148	C₂×Dic₈₉	—	2	1	no	—
149	C₂×Dic₈₉	—	2	4	no	—
150	C₂×Dic₈₉	—	2	1	no	—
151	C₂×Dic₈₉	—	2	1	no	—
152	C₂×Dic₈₉	—	2	1	no	—
153	C₂×Dic₈₉	—	2	1	no	—
154	C₂×Dic₈₉	—	2	1	no	—
155	C₂×Dic₈₉	—	2	1	no	—
156	C₂×Dic₈₉	—	4	2	no	—
157	C₂×Dic₈₉	—	4	2	no	—
158	C₂×Dic₈₉	—	4	2	no	—
159	C₂×Dic₈₉	—	4	2	no	—
160	C₂×Dic₈₉	—	4	2	no	—
161	C₂×Dic₈₉	C₄×D₄	1	2	yes	—
162	C₂×Dic₈₉	C₄×D₄	1	2	yes	—
163	Dic₁₇₈	—	8	8	no	—
164	C₈×C₈₉	—	2	2	no	—
165	C₈×C₈₉	C₄×Q₈	1	4	yes	—
166	#10623494	—	2	2	no	—
167	#10623494	C₄×Q₈	1	4	yes	—
168	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂²×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
170	C₂²×C₈₉	#532	1	1	yes	characteristic
171	C₂²×C₈₉	#576	1	1	yes	characteristic
172	C₂²×C₈₉	#576	1	1	yes	characteristic
173	C₂²×C₈₉	#578	1	1	yes	characteristic
174	C₂²×C₈₉	#578	1	1	yes	characteristic
175	C₄×C₈₉	—	2	2	no	—
176	C₄×C₈₉	—	2	2	no	—
177	C₄×C₈₉	—	4	4	no	—
178	C₄×C₈₉	—	8	2	no	—
179	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
180	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
181	C₄×C₈₉	#533	1	1	yes	characteristic
182	C₄×C₈₉	#533	1	1	yes	characteristic
183	Dic₈₉	—	2	2	no	—
184	Dic₈₉	—	2	2	no	—
185	Dic₈₉	—	4	1	no	—
186	Dic₈₉	—	4	1	no	—
187	Dic₈₉	—	4	4	no	—
188	Dic₈₉	—	8	2	no	—
189	#21725	—	89	1	no	maximal, Sylow
190	C₂×C₈₉	C₄×C₄⋊Q₈	1	1	yes	characteristic
191	C₂×C₈₉	#2948	1	1	yes	characteristic
192	C₂×C₈₉	#2950	1	1	yes	characteristic
193	C₂×C₈₉	#3164	1	1	yes	characteristic
194	C₂×C₈₉	#3164	1	1	yes	characteristic
195	C₂×C₈₉	#3175	1	1	yes	characteristic
196	C₂×C₈₉	#3175	1	1	yes	characteristic
197	C₂×#500	—	89	1	no	—
198	#2331	—	89	2	no	—
199	#1993	—	89	2	no	—
200	#1943	—	89	2	no	—
201	#1942	—	89	1	no	—
202	#1942	—	89	1	no	—
203	#1846	—	89	1	no	—
204	#1776	—	89	2	no	—
205	C₄×#107	—	89	1	no	—
206	C₄×#107	D₈₉	1	1	yes	characteristic
207	#1726	—	89	1	no	—
208	C₈₉	#21725	1	1	yes	characteristic, complemented, Sylow
209	C₂×C₄⋊Q₈	—	89	2	no	—
210	#500	—	89	8	no	—
211	C₂×#107	—	89	1	no	—
212	C₂×#107	—	89	2	no	—
213	C₂×#107	—	89	1	no	—
214	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
215	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
216	C₂×#106	—	89	2	no	—
217	C₂×#97	—	89	1	no	—
218	C₂×#97	—	89	1	no	—
219	C₂×#97	—	89	1	no	—
220	C₂×C₄×C₈	—	89	1	no	—
221	C₂×C₄×C₈	—	89	1	no	—
222	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
223	#391	—	178	2	no	—
224	#391	—	178	2	no	—
225	#389	—	89	2	no	—
226	#389	—	89	2	no	—
227	#387	—	178	2	no	—
228	#387	—	178	2	no	—
229	#382	—	89	2	no	—
230	C₄×C₄⋊C₄	—	89	2	no	—
231	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
232	#376	—	89	1	no	—
233	#340	—	89	2	no	—
234	#340	—	89	2	no	—
235	#340	—	89	2	no	—
236	#340	C₂×D₈₉	1	2	yes	—
237	C₂²×Q₈	—	178	2	no	—
238	C₂²×C₈	—	89	1	no	—
239	C₂²×C₈	—	89	1	no	—
240	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
241	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
242	C₄⋊Q₈	—	89	8	no	—
243	C₄⋊Q₈	—	178	4	no	—
244	C₂×C₄⋊C₄	—	89	2	no	—
245	C₂×C₄⋊C₄	—	89	2	no	—
246	C₂×C₄⋊C₄	—	178	2	no	—
247	C₂×C₄⋊C₄	—	178	2	no	—
248	C₂×C₄⋊C₄	—	178	2	no	—
249	C₂×C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
250	C₂×C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
251	C₂×C₄²	—	2	2	no	—
252	C₂×C₄²	—	89	1	no	—
253	C₂×C₄²	—	89	1	no	—
254	C₂×C₄²	—	178	2	no	—
255	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
256	#107	—	2	2	no	—
257	#107	—	89	4	no	—
258	#107	—	89	8	no	—
259	#107	—	178	2	no	—
260	#107	C₄×D₈₉	1	4	yes	—
261	#106	—	178	4	no	—
262	#97	—	89	4	no	—
263	#97	—	178	2	no	—
264	#97	—	178	2	no	—
265	C₄×C₈	—	89	4	no	—
266	C₄×C₈	—	178	2	no	—
267	C₄×C₈	Dic₁₇₈	1	4	yes	complemented
268	#95	—	2	2	no	—
269	#95	—	89	1	no	—
270	#95	—	89	1	no	—
271	#95	—	89	1	no	—
272	#95	—	89	1	no	—
273	#95	—	178	2	no	—
274	#95	—	178	2	no	—
275	#95	—	178	2	no	—
276	#95	—	178	2	no	—
277	#95	—	178	2	no	—
278	C₂×Q₈	—	178	8	no	—
279	C₂×Q₈	—	712	2	no	—
280	C₂×Q₈	—	712	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	—	89	1	no	—
284	C₂²×C₄	—	89	1	no	—
285	C₂²×C₄	—	89	1	no	—
286	C₂²×C₄	—	89	1	no	—
287	C₂²×C₄	—	178	2	no	—
288	C₂²×C₄	—	178	2	no	—
289	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
290	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
291	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	89	2	no	—
295	C₂×C₈	—	89	2	no	—
296	C₂×C₈	—	89	2	no	—
297	C₂×C₈	—	89	2	no	—
298	C₂×C₈	—	178	1	no	—
299	C₂×C₈	—	178	1	no	—
300	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
301	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
302	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
303	C₂×C₈	#13083574	1	2	yes	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	89	4	no	—
308	C₄⋊C₄	—	178	8	no	—
309	C₄⋊C₄	—	178	2	no	—
310	C₄⋊C₄	—	178	2	no	—
311	C₄⋊C₄	—	178	2	no	—
312	C₄⋊C₄	—	712	2	no	—
313	C₄⋊C₄	—	712	2	no	—
314	C₄⋊C₄	C₂×C₄×D₈₉	1	4	yes	—
315	C₄²	—	8	2	no	—
316	C₄²	—	89	2	no	—
317	C₄²	—	89	2	no	—
318	C₄²	—	178	1	no	—
319	C₄²	—	178	1	no	—
320	C₄²	—	712	2	no	—
321	C₄²	Q₈×D₈₉	1	2	yes	—
322	C₄²	C₂×Dic₁₇₈	1	2	yes	—
323	C₂³	C₂×D₄×D₈₉	1	1	yes	characteristic
324	Q₈	—	712	8	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	89	2	no	—
336	C₂×C₄	—	89	2	no	—
337	C₂×C₄	—	356	2	no	—
338	C₂×C₄	—	356	2	no	—
339	C₂×C₄	—	356	2	no	—
340	C₂×C₄	—	356	2	no	—
341	C₂×C₄	—	356	2	no	—
342	C₂×C₄	—	178	1	no	—
343	C₂×C₄	—	178	1	no	—
344	C₂×C₄	—	178	1	no	—
345	C₂×C₄	—	178	1	no	—
346	C₂×C₄	—	178	1	no	—
347	C₂×C₄	—	178	1	no	—
348	C₂×C₄	—	178	4	no	—
349	C₂×C₄	—	178	1	no	—
350	C₂×C₄	—	178	1	no	—
351	C₂×C₄	—	178	1	no	—
352	C₂×C₄	—	178	1	no	—
353	C₂×C₄	#15668011	1	2	yes	—
354	C₂×C₄	#15668011	1	2	yes	—
355	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
356	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
357	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
358	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
359	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
360	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
361	C₈	—	2	2	no	—
362	C₈	—	89	4	no	—
363	C₈	—	178	2	no	—
364	C₈	C₄×Dic₁₇₈	1	4	yes	—
365	C₂²	C₄×D₄×D₈₉	1	1	yes	characteristic
366	C₂²	C₂×#15668011	1	1	yes	characteristic
367	C₂²	#19152145	1	1	yes	characteristic
368	C₂²	#19152399	1	1	yes	characteristic
369	C₂²	#19152400	1	1	yes	characteristic
370	C₂²	#19152411	1	1	yes	characteristic
371	C₂²	#19152412	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	4	4	no	—
375	C₄	—	8	2	no	—
376	C₄	—	356	1	no	—
377	C₄	—	356	4	no	—
378	C₄	—	356	1	no	—
379	C₄	—	178	2	no	—
380	C₄	—	178	2	no	—
381	C₄	—	712	2	no	—
382	C₄	C₄×Q₈×D₈₉	1	1	yes	characteristic
383	C₄	C₂×C₄×Dic₁₇₈	1	1	yes	characteristic
384	C₄	#19152151	1	1	yes	characteristic
385	C₄	#19152152	1	1	yes	characteristic
386	C₂	C₄×#15668011	1	1	yes	characteristic
387	C₂	#26029099	1	1	yes	characteristic
388	C₂	#26029115	1	1	yes	characteristic
389	C₂	#26030788	1	1	yes	characteristic
390	C₂	#26030792	1	1	yes	characteristic
391	C₂	#26030887	1	1	yes	characteristic
392	C₂	#26030893	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₄×#107×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄×#107	D₈₉
hypocenter	C₈₉	#21725
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
13. #19152002	C₄	semidirect
57. #15667923	Q₈	semidirect
79. C₄×C₈×C₈₉	Q₈	semidirect
208. C₈₉	#21725	semidirect
267. C₄×C₈	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152002	2
#26022996	2
#26021223	2
#26020989	2
#26020986	2
#26020983	2
#26020415	2
#26020150	2
C₄×#15667946	2
#26019938	2
C₄×#107×C₈₉	2
#21725	89