Subgroups — #36546006

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	#26033104	C₂	1	1	yes	maximal, characteristic
3	#26032534	C₂	1	2	yes	maximal, complemented
4	#26019796	C₂	1	2	yes	maximal, complemented
5	#26019643	C₂	1	1	yes	maximal, characteristic, complemented
6	#26019619	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019519	C₂	1	2	yes	maximal, complemented
8	#26019512	C₂	1	2	yes	maximal, complemented
9	#26019138	C₂	1	1	yes	maximal, characteristic
10	C₈×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1707	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152322	C₂²	1	2	yes	—
13	#19152263	C₂²	1	2	yes	—
14	#19152002	—	2	1	no	—
15	#19151983	—	2	1	no	—
16	#19151938	—	2	2	no	—
17	#19151880	C₂²	1	2	yes	—
18	#19151877	—	2	2	no	—
19	#19151875	C₂²	1	2	yes	—
20	#19151855	—	2	2	no	—
21	#19151822	—	2	2	no	—
22	#19151805	C₂²	1	2	yes	—
23	#19151800	C₂²	1	2	yes	—
24	#19151755	—	2	2	no	—
25	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
26	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
27	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
28	Q₁₆×Dic₈₉	—	2	1	no	—
29	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
30	#19151680	—	2	1	no	—
31	#19151677	C₂²	1	2	yes	—
32	#19151605	—	2	2	no	—
33	#19151351	C₂²	1	1	yes	characteristic
34	#19151351	C₂²	1	1	yes	characteristic
35	#19151344	C₂²	1	2	yes	—
36	#19151344	C₂²	1	2	yes	—
37	#19151330	C₂²	1	1	yes	characteristic
38	#19151330	C₂²	1	1	yes	characteristic
39	#19151327	C₂²	1	2	yes	—
40	#500×C₈₉	C₂²	1	1	yes	characteristic
41	#498×C₈₉	C₂²	1	2	yes	—
42	#494×C₈₉	C₂²	1	1	yes	characteristic
43	#487×C₈₉	C₂²	1	2	yes	—
44	C₈²×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₈₉	—	2	2	no	—
46	C₂×Dic₃₅₆	D₄	1	1	yes	characteristic
47	C₂×#13083564	—	2	2	no	—
48	C₂×#13083561	—	2	2	no	—
49	C₂×D₇₁₂	D₄	1	1	yes	characteristic
50	C₂×#13083557	—	2	2	no	—
51	#15668011	—	2	2	no	—
52	#15668007	—	4	2	no	—
53	#15668001	—	2	2	no	—
54	#15667992	—	2	2	no	—
55	#15667990	—	4	2	no	—
56	#15667988	—	2	2	no	—
57	Q₈×Dic₈₉	—	2	2	no	—
58	C₄×Dic₁₇₈	C₂³	1	1	yes	characteristic
59	C₄×Dic₁₇₈	C₂³	1	1	yes	characteristic
60	C₄×D₃₅₆	C₂³	1	2	yes	—
61	#15667974	—	2	2	no	—
62	#15667970	—	2	2	no	—
63	#15667946	D₄	1	1	yes	characteristic
64	#15667946	D₄	1	1	yes	characteristic
65	#15667945	—	2	2	no	—
66	#15667943	—	2	2	no	—
67	C₄⋊#10623494	C₂³	1	1	yes	characteristic
68	C₄⋊#10623494	C₂³	1	2	yes	—
69	C₄⋊#10623494	C₂³	1	1	yes	characteristic
70	#15667941	—	2	1	no	—
71	#15667941	—	2	1	no	—
72	#15667937	D₄	1	1	yes	characteristic
73	#15667937	D₄	1	1	yes	characteristic
74	#15667936	—	2	2	no	—
75	#15667936	—	2	2	no	—
76	#15667935	—	2	2	no	—
77	#15667934	D₄	1	2	yes	—
78	#15667932	—	2	2	no	—
79	#15667924	—	2	2	no	—
80	C₈×Dic₈₉	—	2	1	no	—
81	C₈×Dic₈₉	—	2	1	no	—
82	C₂×Q₁₆×C₈₉	—	2	1	no	—
83	C₂×Q₁₆×C₈₉	—	2	1	no	—
84	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
85	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
86	#107×C₈₉	—	2	1	no	—
87	#107×C₈₉	—	2	1	no	—
88	#106×C₈₉	—	4	2	no	—
89	#103×C₈₉	—	2	2	no	—
90	#103×C₈₉	—	2	2	no	—
91	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₇₈	—	4	2	no	—
95	C₂×Dic₁₇₈	C₂×D₄	1	1	yes	characteristic
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	—	2	2	no	—
103	#13083564	—	8	2	no	—
104	#13083561	—	8	2	no	—
105	D₇₁₂	—	2	2	no	—
106	#13083557	—	8	2	no	—
107	C₂×#10623494	—	2	2	no	—
108	C₂×#10623494	—	2	1	no	—
109	C₂×#10623494	—	2	1	no	—
110	C₄⋊Dic₈₉	—	4	2	no	—
111	C₄⋊Dic₈₉	C₂×D₄	1	1	yes	characteristic
112	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
113	C₄⋊Dic₈₉	C₂×D₄	1	1	yes	characteristic
114	#13083553	—	2	1	no	—
115	#13083553	—	2	1	no	—
116	#13083553	—	4	2	no	—
117	#13083551	—	2	2	no	—
118	#13083551	—	4	2	no	—
119	#13083551	—	4	2	no	—
120	#13083551	—	4	2	no	—
121	C₄×Dic₈₉	—	2	1	no	—
122	C₄×Dic₈₉	—	2	1	no	—
123	C₄×Dic₈₉	—	4	2	no	—
124	C₂×Q₈×C₈₉	—	2	2	no	—
125	C₂×Q₈×C₈₉	—	2	2	no	—
126	Q₁₆×C₈₉	—	4	2	no	—
127	Q₁₆×C₈₉	—	8	1	no	—
128	C₂×C₈×C₈₉	—	2	1	no	—
129	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
133	C₄⋊C₄×C₈₉	—	2	2	no	—
134	C₄⋊C₄×C₈₉	—	2	2	no	—
135	C₄⋊C₄×C₈₉	—	2	2	no	—
136	C₄⋊C₄×C₈₉	—	2	2	no	—
137	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
138	C₂²×D₈₉	—	2	2	no	—
139	Q₈×C₈₉	—	4	2	no	—
140	Q₈×C₈₉	—	4	2	no	—
141	Q₈×C₈₉	—	4	2	no	—
142	Q₈×C₈₉	—	4	2	no	—
143	C₂×C₄×C₈₉	—	4	2	no	—
144	C₂×C₄×C₈₉	—	4	2	no	—
145	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×Dic₈₉	—	2	1	no	—
149	C₂×Dic₈₉	—	2	1	no	—
150	C₂×Dic₈₉	—	2	1	no	—
151	C₂×Dic₈₉	—	2	1	no	—
152	C₂×Dic₈₉	—	2	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	—	2	1	no	—
161	Dic₁₇₈	—	8	4	no	—
162	Dic₁₇₈	#135	1	2	yes	—
163	C₈×C₈₉	—	2	2	no	—
164	C₈×C₈₉	—	2	1	no	—
165	C₈×C₈₉	—	2	1	no	—
166	C₈×C₈₉	—	2	1	no	—
167	C₈×C₈₉	#137	1	2	yes	—
168	#10623494	—	4	2	no	—
169	#10623494	—	4	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	1	no	—
176	C₄×C₈₉	—	8	2	no	—
177	C₄×C₈₉	—	8	2	no	—
178	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
179	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
180	C₄×C₈₉	#576	1	1	yes	characteristic
181	C₄×C₈₉	#577	1	1	yes	characteristic
182	Dic₈₉	—	2	2	no	—
183	Dic₈₉	—	4	1	no	—
184	Dic₈₉	—	4	2	no	—
185	Dic₈₉	—	4	1	no	—
186	Dic₈₉	—	4	1	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	D₈₉	1	1	yes	characteristic, complemented
196	#1699	—	89	2	no	—
197	#1674	—	89	1	no	—
198	#1670	—	89	1	no	—
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	—	178	1	no	—
207	#500	C₂×D₈₉	1	1	yes	characteristic
208	#498	C₂×D₈₉	1	2	yes	—
209	#495	—	178	1	no	—
210	#494	C₂×D₈₉	1	1	yes	characteristic
211	#487	C₂×D₈₉	1	2	yes	—
212	#485	—	178	2	no	—
213	#475	—	89	2	no	—
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	—	89	1	no	—
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	—	89	1	no	—
231	#332	—	89	2	no	—
232	#332	—	89	2	no	—
233	Q₈⋊C₈	—	89	1	no	—
234	Q₈⋊C₈	—	89	1	no	—
235	D₄⋊C₈	—	89	2	no	—
236	C₈²	C₂×D₈₉	1	1	yes	characteristic
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	—	178	2	no	—
246	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
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	—	89	1	no	—
255	C₄×Q₈	—	178	2	no	—
256	C₄×D₄	—	89	2	no	—
257	C₄×D₄	—	178	2	no	—
258	#117	—	178	2	no	—
259	#107	—	2	1	no	—
260	#107	—	2	1	no	—
261	#107	—	89	1	no	—
262	#107	—	89	1	no	—
263	#107	—	178	2	no	—
264	#106	—	4	2	no	—
265	#106	—	178	2	no	—
266	C₄⋊C₈	—	89	2	no	—
267	C₄⋊C₈	—	89	1	no	—
268	C₄⋊C₈	—	89	1	no	—
269	C₄⋊C₈	—	178	1	no	—
270	C₄⋊C₈	—	178	1	no	—
271	#103	—	2	2	no	—
272	#103	—	2	2	no	—
273	#103	—	89	1	no	—
274	#103	—	89	1	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	—	178	1	no	—
282	C₄×C₈	—	178	1	no	—
283	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
286	C₂×Q₈	—	2	2	no	—
287	C₂×Q₈	—	2	2	no	—
288	C₂×Q₈	—	89	1	no	—
289	C₂×Q₈	—	89	1	no	—
290	C₂×Q₈	—	356	2	no	—
291	C₂×Q₈	—	356	4	no	—
292	C₂×Q₈	—	712	2	no	—
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	—	178	1	no	—
305	C₂×C₈	—	178	1	no	—
306	C₂×C₈	—	178	2	no	—
307	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
308	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
309	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
310	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	89	2	no	—
316	C₄⋊C₄	—	89	1	no	—
317	C₄⋊C₄	—	89	1	no	—
318	C₄⋊C₄	—	356	2	no	—
319	C₄⋊C₄	—	356	2	no	—
320	C₄⋊C₄	—	178	1	no	—
321	C₄⋊C₄	—	178	1	no	—
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	—	356	2	no	—
327	C₄²	—	178	1	no	—
328	C₄²	—	178	1	no	—
329	C₄²	C₂³×D₈₉	1	1	yes	characteristic
330	C₂³	—	178	2	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	4	2	no	—
333	Q₈	—	4	2	no	—
334	Q₈	—	4	2	no	—
335	Q₈	—	89	2	no	—
336	Q₈	—	178	1	no	—
337	Q₈	—	178	1	no	—
338	Q₈	—	178	1	no	—
339	Q₈	—	712	4	no	—
340	D₄	—	178	2	no	—
341	D₄	—	178	2	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	356	2	no	—
345	C₂×C₄	—	356	2	no	—
346	C₂×C₄	—	178	1	no	—
347	C₂×C₄	—	178	2	no	—
348	C₂×C₄	—	178	1	no	—
349	C₂×C₄	—	178	1	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	#15668057	1	1	yes	characteristic
355	C₈	—	2	2	no	—
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	356	2	no	—
360	C₈	—	356	1	no	—
361	C₈	—	356	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	#19152233	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	8	2	no	—
369	C₄	—	8	2	no	—
370	C₄	—	356	2	no	—
371	C₄	—	356	1	no	—
372	C₄	—	356	1	no	—
373	C₄	—	356	1	no	—
374	C₄	—	178	2	no	—
375	C₄	#135×D₈₉	1	1	yes	characteristic
376	C₄	C₂×#15668041	1	1	yes	characteristic
377	C₄	#19152400	1	1	yes	characteristic
378	C₄	#19152404	1	1	yes	characteristic
379	C₂	—	712	2	no	—
380	C₂	#26031917	1	1	yes	characteristic
381	C₂	#26031936	1	1	yes	characteristic
382	C₂	#26032056	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19152233
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1707	C₂
socle	C₂²×C₈₉	D₄²
prime radical	C₂×D₇₁₂	D₄
hypercenter	#1707	D₈₉
hypocenter	C₈₉	#19907
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1707

Complements

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

N	H ≅ G/N	type
3. #26032534	C₂	semidirect
4. #26019796	C₂	semidirect
5. #26019643	C₂	semidirect
6. #26019619	C₂	semidirect
7. #26019519	C₂	semidirect
8. #26019512	C₂	semidirect
10. C₈×Dic₃₅₆	C₂	semidirect
11. C₈₉×#1707	C₂	semidirect
195. #1707	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
#26033104	2
#26032534	2
#26019796	2
#26019643	2
#26019619	2
#26019519	2
#26019512	2
#26019138	2
C₈×Dic₃₅₆	2
C₈₉×#1707	2
#19907	89