Subgroups — #36546010

Order 22784.

602 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033242	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₇₁₂	C₂	1	1	yes	maximal, characteristic
3	#26032532	C₂	1	2	yes	maximal, complemented
4	#26019846	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019802	C₂	1	2	yes	maximal, complemented
6	#26019647	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019523	C₂	1	2	yes	maximal, complemented
8	#26019514	C₂	1	2	yes	maximal, complemented
9	#26019136	C₂	1	1	yes	maximal, characteristic
10	C₈×#13083564	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1670	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152322	C₂²	1	2	yes	—
13	Q₈×D₃₅₆	C₂²	1	2	yes	—
14	#19152003	—	2	1	no	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19151994	C₂²	1	2	yes	—
17	#19151983	—	2	1	no	—
18	#19151950	C₂²	1	2	yes	—
19	#19151942	—	2	2	no	—
20	#19151879	—	2	2	no	—
21	#19151879	C₂²	1	2	yes	—
22	#19151857	—	2	2	no	—
23	#19151822	—	2	2	no	—
24	#19151809	C₂²	1	2	yes	—
25	#19151800	C₂²	1	2	yes	—
26	#19151755	—	2	2	no	—
27	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
28	Q₁₆×Dic₈₉	—	2	1	no	—
29	C₄×#13083564	C₂²	1	1	yes	characteristic
30	C₄×D₇₁₂	—	2	1	no	—
31	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
32	#19151675	C₂²	1	2	yes	—
33	#19151605	—	2	2	no	—
34	#19151350	C₂²	1	1	yes	characteristic
35	#19151345	C₂²	1	2	yes	—
36	Q₈⋊#10623494	C₂²	1	1	yes	characteristic
37	#19151328	C₂²	1	1	yes	characteristic
38	#19151325	C₂²	1	2	yes	—
39	C₈×#10623494	C₂²	1	1	yes	characteristic
40	#494×C₈₉	C₂²	1	1	yes	characteristic
41	#475×C₈₉	C₂²	1	2	yes	—
42	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
43	#335×C₈₉	C₂²	1	1	yes	characteristic
44	#332×C₈₉	C₂²	1	2	yes	—
45	C₂×Q₈×D₈₉	—	2	2	no	—
46	C₂×Dic₃₅₆	—	2	2	no	—
47	C₂×#13083564	D₄	1	1	yes	characteristic
48	C₂×#13083561	—	2	2	no	—
49	C₂×D₇₁₂	D₄	1	1	yes	characteristic
50	C₂×#13083557	—	2	2	no	—
51	#15668012	—	2	2	no	—
52	#15668012	C₂³	1	2	yes	—
53	#15668007	—	4	2	no	—
54	#15668001	—	2	2	no	—
55	#15667992	—	2	2	no	—
56	#15667992	—	2	2	no	—
57	#15667992	—	4	2	no	—
58	Q₈×Dic₈₉	—	2	2	no	—
59	C₄×Dic₁₇₈	C₂³	1	1	yes	characteristic
60	C₄×D₃₅₆	—	2	2	no	—
61	C₄×D₃₅₆	C₂³	1	2	yes	—
62	#15667970	—	2	2	no	—
63	#15667946	—	2	2	no	—
64	#15667946	—	2	1	no	—
65	#15667946	—	2	1	no	—
66	#15667946	D₄	1	1	yes	characteristic
67	#15667945	D₄	1	1	yes	characteristic
68	#15667943	—	4	2	no	—
69	C₄⋊#10623494	C₂³	1	1	yes	characteristic
70	#15667941	—	2	1	no	—
71	#15667937	—	2	2	no	—
72	#15667936	—	2	2	no	—
73	#15667936	D₄	1	1	yes	characteristic
74	#15667935	—	2	2	no	—
75	#15667935	—	2	2	no	—
76	#15667935	D₄	1	1	yes	characteristic
77	#15667934	—	2	2	no	—
78	#15667934	D₄	1	2	yes	—
79	#15667924	—	2	2	no	—
80	C₄×#10623494	C₂³	1	1	yes	characteristic
81	C₄×#10623494	C₂³	1	1	yes	characteristic
82	C₈×Dic₈₉	—	2	1	no	—
83	C₂×Q₁₆×C₈₉	—	2	1	no	—
84	C₂×Q₁₆×C₈₉	—	2	1	no	—
85	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
86	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
87	#106×C₈₉	—	2	2	no	—
88	C₄⋊C₈×C₈₉	—	2	1	no	—
89	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
90	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
91	#103×C₈₉	—	2	2	no	—
92	C₄×C₈×C₈₉	—	2	1	no	—
93	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₇₈	—	2	2	no	—
95	C₂×Dic₁₇₈	—	4	2	no	—
96	C₂×Dic₁₇₈	C₂×D₄	1	1	yes	characteristic
97	Q₈×D₈₉	—	4	4	no	—
98	Q₈×D₈₉	—	8	2	no	—
99	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
100	C₂×C₄×D₈₉	—	2	2	no	—
101	C₂×C₄×D₈₉	—	4	2	no	—
102	Dic₃₅₆	—	8	2	no	—
103	#13083564	—	2	2	no	—
104	#13083561	—	8	2	no	—
105	D₇₁₂	—	2	2	no	—
106	#13083557	—	8	2	no	—
107	C₂×#10623494	—	2	1	no	—
108	C₂×#10623494	—	2	1	no	—
109	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
110	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
111	C₄⋊Dic₈₉	—	2	2	no	—
112	C₄⋊Dic₈₉	—	2	2	no	—
113	C₄⋊Dic₈₉	—	4	2	no	—
114	C₄⋊Dic₈₉	—	4	2	no	—
115	C₄⋊Dic₈₉	C₂×D₄	1	1	yes	characteristic
116	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
117	#13083553	—	2	1	no	—
118	#13083551	—	2	2	no	—
119	#13083551	—	4	2	no	—
120	#13083551	—	4	2	no	—
121	#13083551	—	4	2	no	—
122	C₄×Dic₈₉	—	2	1	no	—
123	C₄×Dic₈₉	—	4	2	no	—
124	C₂×Q₈×C₈₉	—	2	2	no	—
125	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
126	Q₁₆×C₈₉	—	4	2	no	—
127	Q₁₆×C₈₉	—	8	1	no	—
128	C₂×C₈×C₈₉	—	2	2	no	—
129	C₂×C₈×C₈₉	—	2	1	no	—
130	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄×C₈₉	—	2	1	no	—
133	C₄⋊C₄×C₈₉	—	2	2	no	—
134	C₄⋊C₄×C₈₉	—	2	2	no	—
135	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₈₉	—	2	1	no	—
137	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
138	C₂²×D₈₉	—	2	2	no	—
139	Q₈×C₈₉	—	2	1	no	—
140	Q₈×C₈₉	—	4	2	no	—
141	Q₈×C₈₉	—	4	2	no	—
142	Q₈×C₈₉	#135	1	2	yes	—
143	C₂×C₄×C₈₉	—	2	1	no	—
144	C₂×C₄×C₈₉	—	2	1	no	—
145	C₂×C₄×C₈₉	—	4	2	no	—
146	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂×Dic₈₉	—	2	1	no	—
150	C₂×Dic₈₉	—	2	1	no	—
151	C₂×Dic₈₉	—	2	2	no	—
152	C₂×Dic₈₉	—	4	2	no	—
153	D₃₅₆	—	2	2	no	—
154	D₃₅₆	—	2	2	no	—
155	C₄×D₈₉	—	4	2	no	—
156	C₄×D₈₉	—	4	2	no	—
157	C₄×D₈₉	—	8	4	no	—
158	Dic₁₇₈	—	2	1	no	—
159	Dic₁₇₈	—	2	1	no	—
160	Dic₁₇₈	—	4	2	no	—
161	Dic₁₇₈	—	4	2	no	—
162	Dic₁₇₈	—	8	4	no	—
163	C₈×C₈₉	—	2	1	no	—
164	C₈×C₈₉	—	4	2	no	—
165	C₈×C₈₉	—	4	1	no	—
166	C₈×C₈₉	#137	1	2	yes	—
167	#10623494	—	2	2	no	—
168	#10623494	—	2	1	no	—
169	#10623494	—	2	1	no	—
170	#10623494	—	4	1	no	—
171	C₂²×C₈₉	D₄²	1	1	yes	characteristic
172	C₂×D₈₉	—	4	2	no	—
173	C₂×D₈₉	—	4	2	no	—
174	C₂×D₈₉	—	4	2	no	—
175	C₄×C₈₉	—	2	2	no	—
176	C₄×C₈₉	—	2	1	no	—
177	C₄×C₈₉	—	4	1	no	—
178	C₄×C₈₉	—	8	2	no	—
179	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
180	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
181	C₄×C₈₉	#576	1	1	yes	characteristic
182	C₄×C₈₉	#577	1	1	yes	characteristic
183	Dic₈₉	—	4	1	no	—
184	Dic₈₉	—	4	2	no	—
185	Dic₈₉	—	4	1	no	—
186	Dic₈₉	—	8	2	no	—
187	#19907	—	89	1	no	maximal, Sylow
188	C₂×C₈₉	#3278	1	1	yes	characteristic
189	C₂×C₈₉	#3280	1	1	yes	characteristic
190	C₂×C₈₉	#3293	1	1	yes	characteristic
191	D₈₉	—	8	2	no	—
192	#3393	—	89	1	no	—
193	Q₈×D₈	—	89	1	no	—
194	#3328	—	89	2	no	—
195	#1707	—	89	1	no	—
196	#1699	—	89	2	no	—
197	#1674	—	89	1	no	—
198	#1670	D₈₉	1	1	yes	characteristic, complemented
199	#1652	—	89	2	no	—
200	#1651	—	89	2	no	—
201	#1589	—	89	1	no	—
202	C₈×Q₁₆	—	89	1	no	—
203	C₈₉	#19907	1	1	yes	characteristic, complemented, Sylow
204	#556	—	89	2	no	—
205	D₄×Q₈	—	89	2	no	—
206	#500	—	89	1	no	—
207	#500	—	178	1	no	—
208	#498	—	89	2	no	—
209	#495	—	178	1	no	—
210	#494	C₂×D₈₉	1	1	yes	characteristic
211	#487	—	89	2	no	—
212	#485	—	178	2	no	—
213	#475	C₂×D₈₉	1	2	yes	—
214	#474	—	89	2	no	—
215	#474	—	178	2	no	—
216	#470	—	178	2	no	—
217	#464	—	178	2	no	—
218	#462	—	89	2	no	—
219	#461	—	89	2	no	—
220	#451	—	178	2	no	—
221	C₈×Q₈	—	89	1	no	—
222	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
223	C₄×Q₁₆	—	89	1	no	—
224	C₄×Q₁₆	—	178	1	no	—
225	C₄×D₈	—	89	1	no	—
226	C₄×D₈	—	178	1	no	—
227	#436	—	89	2	no	—
228	#419	—	178	2	no	—
229	#335	—	89	1	no	—
230	#335	C₂×D₈₉	1	1	yes	characteristic
231	#332	—	89	2	no	—
232	#332	C₂×D₈₉	1	2	yes	—
233	Q₈⋊C₈	—	89	1	no	—
234	Q₈⋊C₈	—	89	1	no	—
235	D₄⋊C₈	—	89	2	no	—
236	C₈²	—	89	1	no	—
237	C₂²×Q₈	—	178	2	no	—
238	C₂×Q₁₆	—	2	1	no	—
239	C₂×Q₁₆	—	2	1	no	—
240	C₂×Q₁₆	—	89	1	no	—
241	C₂×Q₁₆	—	178	2	no	—
242	C₂×SD₁₆	—	178	2	no	—
243	C₂×D₈	—	89	1	no	—
244	C₂×M₁₆	—	178	2	no	—
245	C₄⋊Q₈	—	89	2	no	—
246	C₄⋊Q₈	—	178	2	no	—
247	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
248	#126	—	356	2	no	—
249	#124	—	178	2	no	—
250	C₂²⋊Q₈	—	356	2	no	—
251	C₂²⋊Q₈	—	178	2	no	—
252	C₂²⋊Q₈	—	178	2	no	—
253	C₄×Q₈	—	89	1	no	—
254	C₄×Q₈	—	178	2	no	—
255	C₄×Q₈	C₂²×D₈₉	1	1	yes	characteristic
256	C₄×D₄	—	89	2	no	—
257	C₄×D₄	—	178	2	no	—
258	#117	—	178	2	no	—
259	#107	—	89	1	no	—
260	#107	—	89	1	no	—
261	#107	—	178	2	no	—
262	#107	—	178	1	no	—
263	#107	—	178	1	no	—
264	#106	—	2	2	no	—
265	#106	—	356	2	no	—
266	C₄⋊C₈	—	2	1	no	—
267	C₄⋊C₈	—	89	1	no	—
268	C₄⋊C₈	—	178	1	no	—
269	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
270	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
271	#103	—	2	2	no	—
272	#103	—	89	1	no	—
273	#103	—	89	1	no	—
274	#103	—	178	2	no	—
275	#103	—	178	2	no	—
276	#103	—	178	2	no	—
277	#103	—	178	2	no	—
278	#102	—	89	2	no	—
279	#102	—	178	2	no	—
280	C₂²⋊C₈	—	178	2	no	—
281	C₄×C₈	—	2	1	no	—
282	C₄×C₈	—	89	1	no	—
283	C₄×C₈	—	89	1	no	—
284	C₄×C₈	—	178	1	no	—
285	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
286	C₂×Q₈	—	2	2	no	—
287	C₂×Q₈	—	89	1	no	—
288	C₂×Q₈	—	356	2	no	—
289	C₂×Q₈	—	356	4	no	—
290	C₂×Q₈	—	178	2	no	—
291	C₂×Q₈	—	712	2	no	—
292	C₂×Q₈	D₄×D₈₉	1	1	yes	characteristic
293	C₂×D₄	—	89	2	no	—
294	C₂²×C₄	—	356	2	no	—
295	C₂²×C₄	—	178	2	no	—
296	Q₁₆	—	4	2	no	—
297	Q₁₆	—	8	1	no	—
298	Q₁₆	—	178	2	no	—
299	Q₁₆	—	712	2	no	—
300	SD₁₆	—	712	2	no	—
301	D₈	—	178	2	no	—
302	M₁₆	—	712	2	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	89	1	no	—
306	C₂×C₈	—	89	1	no	—
307	C₂×C₈	—	178	1	no	—
308	C₂×C₈	—	178	1	no	—
309	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
310	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	89	2	no	—
315	C₄⋊C₄	—	89	1	no	—
316	C₄⋊C₄	—	356	2	no	—
317	C₄⋊C₄	—	356	2	no	—
318	C₄⋊C₄	—	178	2	no	—
319	C₄⋊C₄	—	178	2	no	—
320	C₄⋊C₄	—	178	1	no	—
321	C₄⋊C₄	D₄×D₈₉	1	1	yes	characteristic
322	C₂²⋊C₄	—	356	2	no	—
323	C₂²⋊C₄	—	356	2	no	—
324	C₂²⋊C₄	—	356	2	no	—
325	C₂²⋊C₄	—	178	2	no	—
326	C₄²	—	2	1	no	—
327	C₄²	—	356	2	no	—
328	C₄²	—	178	1	no	—
329	C₄²	C₂³×D₈₉	1	1	yes	characteristic
330	C₂³	—	178	2	no	—
331	Q₈	—	2	1	no	—
332	Q₈	—	4	2	no	—
333	Q₈	—	4	2	no	—
334	Q₈	—	356	2	no	—
335	Q₈	—	356	2	no	—
336	Q₈	—	178	1	no	—
337	Q₈	—	178	1	no	—
338	Q₈	—	712	4	no	—
339	Q₈	#15668035	1	2	yes	—
340	D₄	—	178	2	no	—
341	D₄	—	178	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	356	2	no	—
346	C₂×C₄	—	356	2	no	—
347	C₂×C₄	—	356	2	no	—
348	C₂×C₄	—	178	1	no	—
349	C₂×C₄	—	178	2	no	—
350	C₂×C₄	—	178	1	no	—
351	C₂×C₄	—	712	4	no	—
352	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
353	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
354	C₂×C₄	#15668058	1	1	yes	characteristic
355	C₈	—	2	1	no	—
356	C₈	—	4	2	no	—
357	C₈	—	4	1	no	—
358	C₈	—	356	1	no	—
359	C₈	—	178	2	no	—
360	C₈	—	178	1	no	—
361	C₈	—	178	1	no	—
362	C₈	#15668041	1	2	yes	—
363	C₂²	—	356	2	no	—
364	C₂²	—	356	2	no	—
365	C₂²	—	356	2	no	—
366	C₂²	#19152232	1	1	yes	characteristic
367	C₄	—	2	2	no	—
368	C₄	—	2	1	no	—
369	C₄	—	4	1	no	—
370	C₄	—	8	2	no	—
371	C₄	—	356	2	no	—
372	C₄	—	356	1	no	—
373	C₄	—	356	1	no	—
374	C₄	—	712	2	no	—
375	C₄	C₂×#15668035	1	1	yes	characteristic
376	C₄	C₂×#15668041	1	1	yes	characteristic
377	C₄	#19152400	1	1	yes	characteristic
378	C₄	#19152408	1	1	yes	characteristic
379	C₂	—	712	2	no	—
380	C₂	#26031921	1	1	yes	characteristic
381	C₂	#26031934	1	1	yes	characteristic
382	C₂	#26032055	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19152232
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1670	C₂
socle	C₂²×C₈₉	D₄²
prime radical	C₂×D₇₁₂	D₄
hypercenter	#1670	D₈₉
hypocenter	C₈₉	#19907
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1670

Complements

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

N	H ≅ G/N	type
3. #26032532	C₂	semidirect
4. #26019846	C₂	semidirect
5. #26019802	C₂	semidirect
6. #26019647	C₂	semidirect
7. #26019523	C₂	semidirect
8. #26019514	C₂	semidirect
10. C₈×#13083564	C₂	semidirect
11. C₈₉×#1670	C₂	semidirect
198. #1670	D₈₉	semidirect
203. C₈₉	#19907	semidirect

Maximal subgroups

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

subgroup	index
#26033242	2
Q₈×D₇₁₂	2
#26032532	2
#26019846	2
#26019802	2
#26019647	2
#26019523	2
#26019514	2
#26019136	2
C₈×#13083564	2
C₈₉×#1670	2
#19907	89