Subgroups — #36545945

Order 22784.

672 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032600	C₂	1	1	yes	maximal, characteristic, complemented
2	#26032559	C₂	1	2	yes	maximal, complemented
3	#26031974	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019849	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019800	C₂	1	2	yes	maximal, complemented
6	#26019641	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019617	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019539	C₂	1	2	yes	maximal, complemented
9	#26019538	C₂	1	2	yes	maximal, complemented
10	#26019138	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668034	—	2	1	no	—
13	#19152253	C₂²	1	2	yes	—
14	#19152252	C₂²	1	2	yes	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19151996	C₂²	1	2	yes	—
17	#19151978	C₂²	1	1	yes	characteristic
18	#19151949	C₂²	1	2	yes	—
19	#19151935	—	2	2	no	—
20	#19151912	—	2	2	no	—
21	#19151879	C₂²	1	2	yes	—
22	#19151878	C₂²	1	2	yes	—
23	#19151843	—	2	2	no	—
24	#19151839	—	2	1	no	—
25	#19151834	—	2	1	no	—
26	#19151804	C₂²	1	2	yes	—
27	#19151803	C₂²	1	2	yes	—
28	#19151762	—	2	2	no	—
29	#19151740	—	2	2	no	—
30	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
31	#19151677	C₂²	1	2	yes	—
32	C₂×#15667946	—	2	1	no	—
33	#19151596	—	2	2	no	—
34	#19151351	C₂²	1	1	yes	characteristic
35	#19151345	C₂²	1	2	yes	—
36	#19151343	C₂²	1	2	yes	—
37	#19151327	C₂²	1	2	yes	—
38	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
41	#335×C₈₉	C₂²	1	1	yes	characteristic
42	D₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
44	C₈²×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×#13083574	—	2	2	no	—
46	C₂×#13083573	—	2	2	no	—
47	#15668034	—	4	4	no	—
48	C₂×Dic₃₅₆	—	2	1	no	—
49	C₂×Dic₃₅₆	—	2	1	no	—
50	C₂×#13083563	—	2	2	no	—
51	C₂×#13083562	—	2	2	no	—
52	C₂×D₇₁₂	D₄	1	1	yes	characteristic
53	C₂×#13083557	—	2	2	no	—
54	#15668012	C₂³	1	2	yes	—
55	#15668012	C₂³	1	2	yes	—
56	#15667996	—	4	2	no	—
57	#15667994	—	4	2	no	—
58	#15667992	—	2	2	no	—
59	#15667991	—	2	2	no	—
60	#15667986	—	2	2	no	—
61	#15667983	—	2	2	no	—
62	C₄×D₃₅₆	C₂³	1	2	yes	—
63	C₂×C₄⋊Dic₈₉	—	2	2	no	—
64	C₄⋊C₄×D₈₉	—	2	2	no	—
65	#15667946	—	2	2	no	—
66	#15667946	—	2	1	no	—
67	#15667946	—	2	1	no	—
68	#15667946	D₄	1	1	yes	characteristic
69	#15667945	—	2	2	no	—
70	#15667944	—	4	2	no	—
71	C₄⋊#10623494	C₂³	1	2	yes	—
72	#15667937	—	2	2	no	—
73	#15667937	—	2	2	no	—
74	#15667937	—	2	2	no	—
75	#15667935	—	2	2	no	—
76	#15667934	—	2	2	no	—
77	#15667934	D₄	1	2	yes	—
78	#15667933	—	2	2	no	—
79	#15667924	—	2	2	no	—
80	C₂×D₈×C₈₉	D₄	1	1	yes	characteristic
81	C₂²×C₈×C₈₉	—	2	1	no	—
82	C₂²×C₈×C₈₉	—	2	1	no	—
83	C₄×D₄×C₈₉	C₂³	1	1	yes	characteristic
84	C₄×D₄×C₈₉	C₂³	1	1	yes	characteristic
85	#107×C₈₉	D₄	1	1	yes	characteristic
86	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
88	#102×C₈₉	D₄	1	1	yes	characteristic
89	#102×C₈₉	D₄	1	1	yes	characteristic
90	C₂²⋊C₈×C₈₉	—	2	1	no	—
91	C₂²⋊C₈×C₈₉	—	2	1	no	—
92	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
95	#13083574	—	4	4	no	—
96	#13083574	—	8	2	no	—
97	#13083573	—	4	4	no	—
98	#13083573	—	8	2	no	—
99	C₂×Dic₁₇₈	—	2	2	no	—
100	C₂×Dic₁₇₈	—	2	2	no	—
101	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
102	C₂×C₂²⋊D₈₉	—	4	2	no	—
103	C₂×C₂²⋊D₈₉	—	4	2	no	—
104	C₂²×Dic₈₉	—	4	2	no	—
105	C₂×C₄×D₈₉	—	2	2	no	—
106	C₂×C₄×D₈₉	—	4	2	no	—
107	Dic₃₅₆	—	4	2	no	—
108	Dic₃₅₆	—	8	1	no	—
109	#13083563	—	4	4	no	—
110	#13083562	—	8	2	no	—
111	D₇₁₂	—	2	2	no	—
112	#13083557	—	8	2	no	—
113	C₂×#10623494	—	2	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	—	4	2	no	—
119	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
120	#13083553	—	4	2	no	—
121	C₂²⋊Dic₈₉	—	4	2	no	—
122	#13083551	—	2	2	no	—
123	#13083551	—	4	2	no	—
124	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₈₉	—	2	1	no	—
127	C₂²×C₄×C₈₉	—	2	1	no	—
128	D₈×C₈₉	—	2	2	no	—
129	C₂×C₈×C₈₉	—	2	1	no	—
130	C₂×C₈×C₈₉	—	2	1	no	—
131	C₂×C₈×C₈₉	—	2	1	no	—
132	C₂×C₈×C₈₉	—	4	2	no	—
133	C₂×C₈×C₈₉	—	4	2	no	—
134	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
138	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
139	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
140	C₂²⋊C₄×C₈₉	—	2	1	no	—
141	C₂²⋊C₄×C₈₉	—	2	1	no	—
142	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
143	C₂³×C₈₉	—	2	1	no	—
144	C₂³×C₈₉	—	2	1	no	—
145	C₂²×D₈₉	—	2	2	no	—
146	D₄×C₈₉	—	2	1	no	—
147	D₄×C₈₉	—	2	1	no	—
148	D₄×C₈₉	—	2	1	no	—
149	D₄×C₈₉	#135	1	2	yes	complemented
150	C₂×C₄×C₈₉	—	2	1	no	—
151	C₂×C₄×C₈₉	—	2	1	no	—
152	C₂×C₄×C₈₉	—	4	1	no	—
153	C₂×C₄×C₈₉	—	4	1	no	—
154	C₂×C₄×C₈₉	—	4	1	no	—
155	C₂×C₄×C₈₉	—	4	1	no	—
156	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂²⋊D₈₉	—	8	4	no	—
160	C₂²⋊D₈₉	—	8	4	no	—
161	C₂×Dic₈₉	—	2	2	no	—
162	C₂×Dic₈₉	—	4	2	no	—
163	C₂×Dic₈₉	—	4	2	no	—
164	C₂×Dic₈₉	—	8	4	no	—
165	D₃₅₆	—	2	2	no	—
166	D₃₅₆	—	2	2	no	—
167	C₄×D₈₉	—	4	2	no	—
168	C₄×D₈₉	—	4	2	no	—
169	C₄×D₈₉	—	8	4	no	—
170	Dic₁₇₈	—	4	2	no	—
171	Dic₁₇₈	—	4	2	no	—
172	Dic₁₇₈	—	4	2	no	—
173	Dic₁₇₈	—	4	2	no	—
174	C₈×C₈₉	—	2	2	no	—
175	C₈×C₈₉	—	2	1	no	—
176	C₈×C₈₉	—	2	1	no	—
177	C₈×C₈₉	—	2	1	no	—
178	C₈×C₈₉	—	4	1	no	—
179	C₈×C₈₉	—	4	1	no	—
180	C₈×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
181	#10623494	—	4	2	no	—
182	C₂²×C₈₉	—	2	2	no	—
183	C₂²×C₈₉	—	4	1	no	—
184	C₂²×C₈₉	—	4	1	no	—
185	C₂²×C₈₉	—	4	1	no	—
186	C₂²×C₈₉	—	4	1	no	—
187	C₂²×C₈₉	—	4	1	no	—
188	C₂²×C₈₉	D₄²	1	1	yes	characteristic
189	C₂×D₈₉	—	4	2	no	—
190	C₂×D₈₉	—	4	2	no	—
191	C₂×D₈₉	—	4	2	no	—
192	C₄×C₈₉	—	2	1	no	—
193	C₄×C₈₉	—	4	1	no	—
194	C₄×C₈₉	—	4	1	no	—
195	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
196	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
197	C₄×C₈₉	#576	1	1	yes	characteristic
198	C₄×C₈₉	#577	1	1	yes	characteristic
199	Dic₈₉	—	4	2	no	—
200	Dic₈₉	—	8	2	no	—
201	Dic₈₉	—	8	2	no	—
202	#19902	—	89	1	no	maximal, Sylow
203	C₂×C₈₉	—	4	2	no	—
204	C₂×C₈₉	—	8	1	no	—
205	C₂×C₈₉	#3272	1	1	yes	characteristic
206	C₂×C₈₉	#3280	1	1	yes	characteristic
207	C₂×C₈₉	#3293	1	1	yes	characteristic
208	D₈₉	—	8	2	no	—
209	#3334	—	89	1	no	—
210	#3330	—	89	2	no	—
211	#3283	—	89	1	no	—
212	#1707	—	89	1	no	—
213	#1699	—	89	2	no	—
214	#1673	—	89	1	no	—
215	#1669	—	89	1	no	—
216	#1654	—	89	2	no	—
217	#1654	—	89	2	no	—
218	#1589	—	89	1	no	—
219	C₈×D₈	D₈₉	1	1	yes	characteristic, complemented
220	C₈₉	#19902	1	1	yes	characteristic, complemented, Sylow
221	C₂×#135	—	178	1	no	—
222	#547	—	89	2	no	—
223	#547	—	89	2	no	—
224	#500	—	89	1	no	—
225	#498	—	89	2	no	—
226	#494	—	89	1	no	—
227	#487	—	89	2	no	—
228	#484	—	178	2	no	—
229	#480	—	178	2	no	—
230	#474	—	89	2	no	—
231	#474	—	89	2	no	—
232	#468	—	178	2	no	—
233	#467	—	178	1	no	—
234	#466	—	178	1	no	—
235	#461	—	89	2	no	—
236	#461	—	89	2	no	—
237	#452	—	178	2	no	—
238	#449	—	178	2	no	—
239	C₄×D₈	—	89	1	no	—
240	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
241	#436	—	89	2	no	—
242	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
243	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
244	C₂×#107	—	178	1	no	—
245	#417	—	178	2	no	—
246	#335	—	89	1	no	—
247	#335	C₂×D₈₉	1	1	yes	characteristic
248	#332	—	89	2	no	—
249	#332	—	89	2	no	—
250	D₄⋊C₈	—	89	2	no	—
251	D₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
252	D₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
253	C₈²	C₂×D₈₉	1	1	yes	characteristic
254	C₂×#41	—	178	2	no	—
255	C₂×#41	—	178	2	no	—
256	#135	—	356	4	no	—
257	C₂×Q₁₆	—	178	1	no	—
258	C₂×Q₁₆	—	178	1	no	—
259	C₂×SD₁₆	—	178	2	no	—
260	C₂×SD₁₆	—	178	2	no	—
261	C₂×D₈	—	89	1	no	—
262	C₂×D₈	D₃₅₆	1	1	yes	characteristic
263	C₂×M₁₆	—	178	2	no	—
264	C₂²×C₈	—	2	1	no	—
265	C₂²×C₈	—	2	1	no	—
266	C₄⋊Q₈	—	89	2	no	—
267	C₄⋊Q₈	—	89	2	no	—
268	#123	—	356	2	no	—
269	#123	—	356	2	no	—
270	C₂²⋊Q₈	—	178	2	no	—
271	C₂²⋊Q₈	—	178	2	no	—
272	#121	—	178	2	no	—
273	#121	—	178	2	no	—
274	C₄×D₄	—	89	2	no	—
275	C₄×D₄	C₂²×D₈₉	1	1	yes	characteristic
276	C₄×D₄	C₂²×D₈₉	1	1	yes	characteristic
277	C₂×C₄⋊C₄	—	178	2	no	—
278	C₂×C₄⋊C₄	—	178	2	no	—
279	#107	—	89	1	no	—
280	#107	—	178	2	no	—
281	#107	—	178	1	no	—
282	#107	—	178	1	no	—
283	#107	—	178	2	no	—
284	#107	D₃₅₆	1	1	yes	characteristic
285	#106	—	356	2	no	—
286	C₄⋊C₈	—	89	2	no	—
287	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
288	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
289	#103	—	178	2	no	—
290	#103	—	178	2	no	—
291	#103	—	178	2	no	—
292	#103	—	178	2	no	—
293	#102	—	89	2	no	—
294	#102	—	178	2	no	—
295	#102	—	178	2	no	—
296	#102	D₃₅₆	1	1	yes	characteristic
297	#102	D₃₅₆	1	1	yes	characteristic
298	C₂²⋊C₈	—	2	1	no	—
299	C₂²⋊C₈	—	2	1	no	—
300	C₂²⋊C₈	—	178	2	no	—
301	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
302	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
303	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
304	#41	—	356	4	no	—
305	#41	—	356	4	no	—
306	#41	—	712	2	no	—
307	#41	—	712	2	no	—
308	C₂×Q₈	—	178	2	no	—
309	C₂×Q₈	—	178	2	no	—
310	C₂×D₄	—	89	2	no	—
311	C₂×D₄	—	356	2	no	—
312	C₂×D₄	—	356	2	no	—
313	C₂×D₄	C₂×D₃₅₆	1	1	yes	characteristic
314	C₂×D₄	C₂×D₃₅₆	1	1	yes	characteristic
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	356	2	no	—
318	C₂²×C₄	—	356	2	no	—
319	C₂²×C₄	—	178	2	no	—
320	Q₁₆	—	356	2	no	—
321	Q₁₆	—	712	1	no	—
322	SD₁₆	—	356	4	no	—
323	SD₁₆	—	712	2	no	—
324	D₈	—	2	2	no	—
325	D₈	—	178	2	no	—
326	M₁₆	—	712	2	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	4	2	no	—
331	C₂×C₈	—	4	2	no	—
332	C₂×C₈	—	178	2	no	—
333	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
334	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
335	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
336	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
337	C₄⋊C₄	—	89	2	no	—
338	C₄⋊C₄	—	356	2	no	—
339	C₄⋊C₄	—	356	2	no	—
340	C₄⋊C₄	—	178	2	no	—
341	C₄⋊C₄	—	178	2	no	—
342	C₄⋊C₄	—	178	2	no	—
343	C₄⋊C₄	—	178	2	no	—
344	C₄⋊C₄	C₂×D₃₅₆	1	1	yes	characteristic
345	C₄⋊C₄	C₂×D₃₅₆	1	1	yes	characteristic
346	C₂²⋊C₄	—	2	1	no	—
347	C₂²⋊C₄	—	2	1	no	—
348	C₂²⋊C₄	—	356	2	no	—
349	C₂²⋊C₄	—	356	2	no	—
350	C₂²⋊C₄	—	178	2	no	—
351	C₄²	C₂³×D₈₉	1	1	yes	characteristic
352	C₂³	—	2	1	no	—
353	C₂³	—	2	1	no	—
354	C₂³	—	178	2	no	—
355	Q₈	—	356	2	no	—
356	Q₈	—	356	2	no	—
357	Q₈	—	356	2	no	—
358	Q₈	—	356	2	no	—
359	D₄	—	2	1	no	—
360	D₄	—	2	1	no	—
361	D₄	—	2	1	no	—
362	D₄	—	178	2	no	—
363	D₄	—	178	2	no	—
364	D₄	—	712	4	no	—
365	D₄	—	712	4	no	—
366	D₄	#15668034	1	2	yes	complemented
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	356	2	no	—
374	C₂×C₄	—	356	2	no	—
375	C₂×C₄	—	356	2	no	—
376	C₂×C₄	—	356	2	no	—
377	C₂×C₄	—	178	2	no	—
378	C₂×C₄	—	712	4	no	—
379	C₂×C₄	—	712	4	no	—
380	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
381	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
382	C₂×C₄	#15668058	1	1	yes	characteristic
383	C₈	—	2	2	no	—
384	C₈	—	2	1	no	—
385	C₈	—	2	1	no	—
386	C₈	—	2	1	no	—
387	C₈	—	4	1	no	—
388	C₈	—	4	1	no	—
389	C₈	—	356	2	no	—
390	C₈	#15668036	1	2	yes	—
391	C₂²	—	2	2	no	—
392	C₂²	—	4	1	no	—
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	1	no	—
396	C₂²	—	4	1	no	—
397	C₂²	—	356	2	no	—
398	C₂²	—	356	2	no	—
399	C₂²	—	356	2	no	—
400	C₂²	D₄×D₃₅₆	1	1	yes	characteristic
401	C₄	—	2	1	no	—
402	C₄	—	4	1	no	—
403	C₄	—	4	1	no	—
404	C₄	—	356	2	no	—
405	C₄	—	712	2	no	—
406	C₄	—	712	2	no	—
407	C₄	C₂×#15668034	1	1	yes	characteristic
408	C₄	C₂×#15668036	1	1	yes	characteristic
409	C₄	#19152403	1	1	yes	characteristic
410	C₄	#19152405	1	1	yes	characteristic
411	C₂	—	4	2	no	—
412	C₂	—	8	1	no	—
413	C₂	—	712	2	no	—
414	C₂	#26031853	1	1	yes	characteristic
415	C₂	#26031939	1	1	yes	characteristic
416	C₂	#26032057	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₅₆
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈×D₈×C₈₉	C₂
socle	C₂²×C₈₉	D₄²
hypercenter	C₈×D₈	D₈₉
hypocenter	C₈₉	#19902
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈

Complements

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

N	H ≅ G/N	type
1. #26032600	C₂	semidirect
2. #26032559	C₂	semidirect
3. #26031974	C₂	semidirect
4. #26019849	C₂	semidirect
5. #26019800	C₂	semidirect
6. #26019641	C₂	semidirect
7. #26019617	C₂	semidirect
8. #26019539	C₂	semidirect
9. #26019538	C₂	semidirect
10. #26019138	C₂	semidirect
11. C₈×D₈×C₈₉	C₂	semidirect
149. D₄×C₈₉	#135	semidirect
219. C₈×D₈	D₈₉	semidirect
220. C₈₉	#19902	semidirect
366. D₄	#15668034	semidirect

Maximal subgroups

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

subgroup	index
#26032600	2
#26032559	2
#26031974	2
#26019849	2
#26019800	2
#26019641	2
#26019617	2
#26019539	2
#26019538	2
#26019138	2
C₈×D₈×C₈₉	2
#19902	89