Subgroups — #36537865

Order 22784.

562 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26031433	C₂	1	2	yes	maximal
2	#26030838	C₂	1	1	yes	maximal, characteristic, complemented
3	#26029227	C₂	1	1	yes	maximal, characteristic
4	#26019781	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019776	C₂	1	1	yes	maximal, characteristic, complemented
6	#26019747	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019742	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019711	C₂	1	2	yes	maximal, complemented
9	#26019704	C₂	1	2	yes	maximal, complemented
10	#26018960	C₂	1	2	yes	maximal, complemented
11	C₈₉×#1564	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152155	C₂²	1	2	yes	—
13	#19152132	C₂²	1	1	yes	characteristic
14	#19152003	C₂²	1	1	yes	characteristic
15	#19152000	—	2	2	no	—
16	#19151978	C₂²	1	1	yes	characteristic
17	#19151953	—	2	2	no	—
18	#19151949	C₂²	1	2	yes	—
19	#19151940	—	2	2	no	—
20	#19151923	—	2	2	no	—
21	#19151894	C₂²	1	2	yes	—
22	#19151889	C₂²	1	2	yes	—
23	#19151884	C₂²	1	2	yes	—
24	#19151879	C₂²	1	2	yes	—
25	#19151839	—	2	1	no	—
26	#19151839	—	2	1	no	—
27	#19151728	C₂²	1	1	yes	characteristic
28	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
29	C₄×Dic₃₅₆	—	2	1	no	—
30	C₄×Dic₃₅₆	—	2	1	no	—
31	#19151677	C₂²	1	1	yes	characteristic
32	#19151669	—	2	1	no	—
33	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
34	C₈×C₂²⋊D₈₉	—	2	1	no	—
35	#19151657	—	2	1	no	—
36	#19151655	C₂²	1	1	yes	characteristic
37	#19151552	—	2	1	no	—
38	#19151346	C₂²	1	2	yes	—
39	#19151346	C₂²	1	2	yes	—
40	#19151343	C₂²	1	2	yes	—
41	#19151343	C₂²	1	2	yes	—
42	#431×C₈₉	C₂²	1	2	yes	—
43	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
44	#335×C₈₉	C₂²	1	2	yes	—
45	#334×C₈₉	C₂²	1	1	yes	characteristic
46	#334×C₈₉	C₂²	1	1	yes	characteristic
47	C₂×Dic₃₅₆	—	2	1	no	—
48	C₂×Dic₃₅₆	—	2	1	no	—
49	C₂×#13083557	—	2	1	no	—
50	C₂×C₈×D₈₉	—	2	1	no	—
51	#15668012	C₂³	1	2	yes	—
52	#15668012	C₂³	1	2	yes	—
53	#15668006	—	2	1	no	—
54	#15668004	—	2	2	no	—
55	#15668004	D₄	1	1	yes	characteristic
56	#15668000	D₄	1	1	yes	characteristic
57	#15667995	—	2	1	no	—
58	#15667991	—	2	2	no	—
59	#15667991	—	2	2	no	—
60	#15667991	D₄	1	1	yes	characteristic
61	#15667986	D₄	1	1	yes	characteristic
62	C₄×Dic₁₇₈	—	2	2	no	—
63	C₄×Dic₁₇₈	—	2	2	no	—
64	C₄×Dic₁₇₈	C₂³	1	1	yes	characteristic
65	C₄×D₃₅₆	C₂³	1	1	yes	characteristic
66	C₄×C₂²⋊D₈₉	—	2	1	no	—
67	#15667971	—	2	2	no	—
68	#15667969	—	2	1	no	—
69	#15667946	—	2	2	no	—
70	#15667946	—	2	2	no	—
71	#15667946	—	2	1	no	—
72	#15667946	—	2	1	no	—
73	#15667944	—	2	2	no	—
74	#15667944	—	2	2	no	—
75	C₄⋊#10623494	C₂³	1	1	yes	characteristic
76	C₄⋊#10623494	C₂³	1	1	yes	characteristic
77	#15667941	—	2	1	no	—
78	#15667941	—	2	1	no	—
79	#15667937	—	2	2	no	—
80	#15667937	—	2	2	no	—
81	#15667937	—	2	2	no	—
82	#15667937	—	2	2	no	—
83	#15667934	—	2	2	no	—
84	#15667934	—	2	2	no	—
85	C₂²⋊#10623494	—	2	1	no	—
86	#15667924	—	2	1	no	—
87	#15667924	—	2	1	no	—
88	#15667923	—	2	1	no	—
89	#15667922	—	2	1	no	—
90	C₈×Dic₈₉	—	2	1	no	—
91	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
92	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
93	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
95	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
96	C₂²⋊C₈×C₈₉	—	2	2	no	—
97	C₄×C₈×C₈₉	—	2	2	no	—
98	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
99	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
100	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×Dic₁₇₈	—	2	2	no	—
103	C₂×Dic₁₇₈	—	2	2	no	—
104	C₂×Dic₁₇₈	C₂×D₄	1	1	yes	characteristic
105	C₂×D₃₅₆	C₂×D₄	1	1	yes	characteristic
106	C₂×C₂²⋊D₈₉	—	2	1	no	—
107	C₂×C₄×D₈₉	—	2	1	no	—
108	Dic₃₅₆	—	4	2	no	—
109	Dic₃₅₆	—	4	2	no	—
110	#13083557	—	4	2	no	—
111	C₂×#10623494	—	2	1	no	—
112	C₂×#10623494	—	2	1	no	—
113	C₈×D₈₉	—	4	2	no	—
114	C₄⋊Dic₈₉	—	2	2	no	—
115	C₄⋊Dic₈₉	—	2	2	no	—
116	C₄⋊Dic₈₉	—	2	2	no	—
117	C₄⋊Dic₈₉	—	2	2	no	—
118	C₄⋊Dic₈₉	C₂×D₄	1	1	yes	characteristic
119	C₄⋊Dic₈₉	C₂×D₄	1	1	yes	characteristic
120	#13083553	—	2	1	no	—
121	#13083553	—	2	1	no	—
122	#13083553	—	2	1	no	—
123	#13083553	—	2	1	no	—
124	#13083553	—	4	2	no	—
125	#13083553	—	4	2	no	—
126	C₂²⋊Dic₈₉	—	2	1	no	—
127	C₂²⋊Dic₈₉	—	4	2	no	—
128	#13083551	—	2	1	no	—
129	#13083551	—	2	1	no	—
130	#13083551	—	2	1	no	—
131	#13083551	—	2	1	no	—
132	C₄×Dic₈₉	—	2	1	no	—
133	C₄×Dic₈₉	—	4	2	no	—
134	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
137	C₂×C₈×C₈₉	—	2	2	no	—
138	C₂×C₈×C₈₉	—	2	2	no	—
139	C₂×C₈×C₈₉	—	2	2	no	—
140	C₂×C₈×C₈₉	—	4	1	no	—
141	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
145	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
146	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₈₉	#41	1	1	yes	characteristic
148	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
149	C₂³×C₈₉	#121	1	1	yes	characteristic
150	C₂²×D₈₉	—	2	1	no	—
151	C₂×C₄×C₈₉	—	2	1	no	—
152	C₂×C₄×C₈₉	—	2	1	no	—
153	C₂×C₄×C₈₉	—	2	2	no	—
154	C₂×C₄×C₈₉	—	4	1	no	—
155	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
156	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
157	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
158	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
161	C₂²⋊D₈₉	—	4	2	no	—
162	C₂×Dic₈₉	—	2	1	no	—
163	C₂×Dic₈₉	—	2	1	no	—
164	C₂×Dic₈₉	—	2	1	no	—
165	C₂×Dic₈₉	—	4	2	no	—
166	C₂×Dic₈₉	—	4	2	no	—
167	D₃₅₆	—	2	2	no	—
168	C₄×D₈₉	—	4	1	no	—
169	C₄×D₈₉	—	4	1	no	—
170	Dic₁₇₈	—	2	2	no	—
171	Dic₁₇₈	—	4	2	no	—
172	Dic₁₇₈	—	4	2	no	—
173	Dic₁₇₈	—	8	2	no	—
174	C₈×C₈₉	—	2	1	no	—
175	C₈×C₈₉	—	2	1	no	—
176	C₈×C₈₉	—	2	1	no	—
177	C₈×C₈₉	—	4	2	no	—
178	C₈×C₈₉	—	4	2	no	—
179	C₈×C₈₉	#135	1	2	yes	—
180	#10623494	—	4	1	no	—
181	#10623494	—	4	1	no	—
182	C₂²×C₈₉	—	2	1	no	—
183	C₂²×C₈₉	—	2	1	no	—
184	C₂²×C₈₉	—	2	1	no	—
185	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
186	C₂×D₈₉	—	4	1	no	—
187	C₂×D₈₉	—	4	1	no	—
188	C₂×D₈₉	—	4	1	no	—
189	C₄×C₈₉	—	2	1	no	—
190	C₄×C₈₉	—	2	1	no	—
191	C₄×C₈₉	—	2	1	no	—
192	C₄×C₈₉	—	2	1	no	—
193	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
194	C₄×C₈₉	#575	1	1	yes	characteristic
195	C₄×C₈₉	#578	1	2	yes	—
196	Dic₈₉	—	4	1	no	—
197	Dic₈₉	—	4	1	no	—
198	Dic₈₉	—	4	1	no	—
199	Dic₈₉	—	8	2	no	—
200	Dic₈₉	—	8	2	no	—
201	#19226	—	89	1	no	maximal, Sylow
202	C₂×C₈₉	—	4	1	no	—
203	C₂×C₈₉	#3035	1	1	yes	characteristic
204	C₂×C₈₉	#3059	1	1	yes	characteristic
205	C₂×C₈₉	#3067	1	1	yes	characteristic
206	D₈₉	—	8	1	no	—
207	#3233	—	89	2	no	—
208	#3170	—	89	1	no	—
209	#2966	—	89	1	no	—
210	#1696	—	89	1	no	—
211	#1695	—	89	1	no	—
212	#1690	—	89	1	no	—
213	#1689	—	89	1	no	—
214	#1684	—	89	2	no	—
215	#1683	—	89	2	no	—
216	#1564	D₈₉	1	1	yes	characteristic, complemented
217	#1554	—	89	2	no	—
218	C₈₉	#19226	1	1	yes	characteristic, complemented, Sylow
219	#533	—	89	2	no	—
220	#529	—	89	1	no	—
221	#500	—	89	1	no	—
222	#499	—	178	2	no	—
223	#494	—	89	1	no	—
224	#488	—	178	2	no	—
225	#487	—	89	2	no	—
226	#485	—	178	2	no	—
227	#482	—	178	2	no	—
228	#477	—	89	2	no	—
229	#476	—	89	2	no	—
230	#475	—	89	2	no	—
231	#474	—	89	2	no	—
232	#467	—	178	1	no	—
233	#467	—	178	1	no	—
234	#446	—	89	1	no	—
235	C₈×Q₈	—	89	1	no	—
236	C₄×Q₁₆	—	178	1	no	—
237	C₄×Q₁₆	—	178	1	no	—
238	#436	—	89	1	no	—
239	#435	—	178	1	no	—
240	C₈×D₄	—	89	1	no	—
241	C₈×D₄	—	178	1	no	—
242	#433	—	178	1	no	—
243	#432	—	89	1	no	—
244	#431	C₂×D₈₉	1	2	yes	—
245	#405	—	178	1	no	—
246	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
247	#335	C₂×D₈₉	1	2	yes	—
248	#334	C₂×D₈₉	1	1	yes	characteristic
249	#334	C₂×D₈₉	1	1	yes	characteristic
250	#333	—	89	2	no	—
251	#333	—	89	2	no	—
252	#332	—	89	2	no	—
253	#332	—	89	2	no	—
254	C₂×Q₁₆	—	178	1	no	—
255	C₂×Q₁₆	—	178	1	no	—
256	C₂×M₁₆	—	178	1	no	—
257	C₂²×C₈	—	178	1	no	—
258	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
259	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
260	C₄⋊Q₈	—	89	2	no	—
261	C₄⋊Q₈	—	89	2	no	—
262	#126	—	178	1	no	—
263	#125	—	89	1	no	—
264	#125	—	178	2	no	—
265	#124	—	89	1	no	—
266	#123	—	178	1	no	—
267	C₂²⋊Q₈	—	89	1	no	—
268	C₂²⋊Q₈	—	178	2	no	—
269	C₂²⋊Q₈	—	178	2	no	—
270	#121	—	89	1	no	—
271	C₄×Q₈	—	89	1	no	—
272	C₄×Q₈	—	178	2	no	—
273	C₄×Q₈	—	178	2	no	—
274	C₄×D₄	—	89	1	no	—
275	C₄×D₄	—	178	1	no	—
276	#117	—	178	2	no	—
277	#117	—	178	1	no	—
278	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
279	#107	—	178	2	no	—
280	#107	—	178	2	no	—
281	#107	—	178	1	no	—
282	#107	—	178	1	no	—
283	#106	—	178	2	no	—
284	#106	—	178	2	no	—
285	C₄⋊C₈	—	89	1	no	—
286	C₄⋊C₈	—	89	1	no	—
287	C₄⋊C₈	—	178	1	no	—
288	C₄⋊C₈	—	178	1	no	—
289	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
290	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
291	#103	—	178	2	no	—
292	#103	—	178	2	no	—
293	#103	—	178	2	no	—
294	#103	—	178	2	no	—
295	#102	—	178	2	no	—
296	#102	—	178	2	no	—
297	C₂²⋊C₈	—	2	2	no	—
298	C₂²⋊C₈	—	178	1	no	—
299	C₂²⋊C₈	—	178	1	no	—
300	C₂²⋊C₈	—	178	1	no	—
301	#97	—	178	1	no	—
302	#97	—	178	1	no	—
303	C₄×C₈	—	2	2	no	—
304	C₄×C₈	—	178	1	no	—
305	C₄×C₈	D₃₅₆	1	1	yes	characteristic
306	C₄×C₈	D₃₅₆	1	1	yes	characteristic
307	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
308	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
309	C₂×Q₈	—	89	1	no	—
310	C₂×Q₈	—	178	2	no	—
311	C₂×Q₈	—	178	2	no	—
312	C₂×D₄	—	89	1	no	—
313	C₂×D₄	—	178	1	no	—
314	C₂²×C₄	—	178	1	no	—
315	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
316	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
317	C₂²×C₄	#13083575	1	1	yes	characteristic
318	Q₁₆	—	356	2	no	—
319	Q₁₆	—	356	2	no	—
320	M₁₆	—	356	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	4	1	no	—
325	C₂×C₈	—	356	2	no	—
326	C₂×C₈	—	178	1	no	—
327	C₂×C₈	—	178	1	no	—
328	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
329	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
330	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
331	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
332	C₄⋊C₄	—	89	1	no	—
333	C₄⋊C₄	—	89	1	no	—
334	C₄⋊C₄	—	356	2	no	—
335	C₄⋊C₄	—	356	2	no	—
336	C₄⋊C₄	—	178	2	no	—
337	C₄⋊C₄	—	178	2	no	—
338	C₄⋊C₄	—	178	2	no	—
339	C₄⋊C₄	—	178	2	no	—
340	C₄⋊C₄	—	178	1	no	—
341	C₄⋊C₄	—	178	1	no	—
342	C₄⋊C₄	—	178	1	no	—
343	C₄⋊C₄	—	178	1	no	—
344	C₂²⋊C₄	—	356	2	no	—
345	C₂²⋊C₄	—	178	1	no	—
346	C₂²⋊C₄	—	178	1	no	—
347	C₂²⋊C₄	—	178	1	no	—
348	C₂²⋊C₄	—	178	1	no	—
349	C₂²⋊C₄	—	178	1	no	—
350	C₄²	—	356	2	no	—
351	C₄²	—	178	1	no	—
352	C₄²	D₄×D₈₉	1	1	yes	characteristic
353	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
354	C₄²	#13083575	1	1	yes	characteristic
355	C₄²	C₂³×D₈₉	1	1	yes	characteristic
356	C₂³	—	178	1	no	—
357	C₂³	#15667987	1	1	yes	characteristic
358	Q₈	—	356	2	no	—
359	Q₈	—	356	2	no	—
360	Q₈	—	178	2	no	—
361	Q₈	—	712	2	no	—
362	D₄	—	356	2	no	—
363	D₄	—	178	2	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	4	1	no	—
368	C₂×C₄	—	356	1	no	—
369	C₂×C₄	—	356	1	no	—
370	C₂×C₄	—	356	2	no	—
371	C₂×C₄	—	356	2	no	—
372	C₂×C₄	—	178	1	no	—
373	C₂×C₄	—	178	1	no	—
374	C₂×C₄	—	178	1	no	—
375	C₂×C₄	#15667987	1	1	yes	characteristic
376	C₂×C₄	#15667987	1	1	yes	characteristic
377	C₂×C₄	#15667987	1	1	yes	characteristic
378	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
379	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
380	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
381	C₈	—	2	1	no	—
382	C₈	—	2	1	no	—
383	C₈	—	2	1	no	—
384	C₈	—	4	2	no	—
385	C₈	—	4	2	no	—
386	C₈	—	356	1	no	—
387	C₈	—	356	1	no	—
388	C₈	#15668034	1	2	yes	—
389	C₂²	—	2	1	no	—
390	C₂²	—	2	1	no	—
391	C₂²	—	2	1	no	—
392	C₂²	—	356	1	no	—
393	C₂²	—	356	1	no	—
394	C₂²	—	356	1	no	—
395	C₂²	C₂×#15667987	1	1	yes	characteristic
396	C₄	—	2	1	no	—
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	356	1	no	—
401	C₄	—	356	1	no	—
402	C₄	—	356	1	no	—
403	C₄	—	712	2	no	—
404	C₄	—	712	2	no	—
405	C₄	C₂×#15668034	1	1	yes	characteristic
406	C₄	#19152398	1	1	yes	characteristic
407	C₄	#19152412	1	2	yes	—
408	C₂	—	4	1	no	—
409	C₂	—	712	1	no	—
410	C₂	#26029709	1	1	yes	characteristic
411	C₂	#26029908	1	1	yes	characteristic
412	C₂	#26029973	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667987
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1564	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	#15667986	D₄
hypercenter	#1564	D₈₉
hypocenter	C₈₉	#19226
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
2. #26030838	C₂	semidirect
4. #26019781	C₂	semidirect
5. #26019776	C₂	semidirect
6. #26019747	C₂	semidirect
7. #26019742	C₂	semidirect
8. #26019711	C₂	semidirect
9. #26019704	C₂	semidirect
10. #26018960	C₂	semidirect
11. C₈₉×#1564	C₂	semidirect
216. #1564	D₈₉	semidirect
218. C₈₉	#19226	semidirect

Maximal subgroups

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

subgroup	index
#26031433	2
#26030838	2
#26029227	2
#26019781	2
#26019776	2
#26019747	2
#26019742	2
#26019711	2
#26019704	2
#26018960	2
C₈₉×#1564	2
#19226	89