Subgroups — #36545762

Order 22784.

710 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	#26033227	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#26031919	C₂	1	2	yes	maximal, complemented
4	#26019844	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019718	C₂	1	2	yes	maximal, complemented
6	#26019646	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019612	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019447	C₂	1	2	yes	maximal, complemented
9	#26019357	C₂	1	2	yes	maximal, complemented
10	#26019138	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668044	—	2	2	no	—
13	#19152270	C₂²	1	2	yes	—
14	Q₈×D₃₅₆	C₂²	1	2	yes	—
15	#19152003	—	2	1	no	—
16	#19152003	C₂²	1	1	yes	characteristic
17	#19151975	—	2	1	no	—
18	#19151975	C₂²	1	1	yes	characteristic
19	#19151971	C₂²	1	2	yes	—
20	#19151945	C₂²	1	2	yes	—
21	#19151879	—	2	2	no	—
22	#19151879	C₂²	1	2	yes	—
23	#19151845	—	2	2	no	—
24	#19151808	C₂²	1	2	yes	—
25	#19151798	C₂²	1	2	yes	—
26	#19151794	—	2	2	no	—
27	#19151794	C₂²	1	2	yes	—
28	#19151763	—	2	2	no	—
29	#19151738	—	2	2	no	—
30	C₄×D₇₁₂	—	2	1	no	—
31	C₄×D₇₁₂	—	2	1	no	—
32	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
33	#19151677	C₂²	1	2	yes	—
34	#19151351	C₂²	1	1	yes	characteristic
35	#19151343	C₂²	1	2	yes	—
36	#19151341	C₂²	1	2	yes	—
37	#19151327	C₂²	1	2	yes	—
38	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	#335×C₈₉	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
44	C₈²×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×#13083575	—	2	2	no	—
46	C₂×Q₈×D₈₉	—	2	2	no	—
47	#15668044	—	4	8	no	—
48	C₂×#13083564	—	2	2	no	—
49	C₂×#13083563	—	2	2	no	—
50	C₂×#13083563	—	2	2	no	—
51	C₂×#13083561	—	2	2	no	—
52	C₂×D₇₁₂	—	2	2	no	—
53	C₂×D₇₁₂	—	2	1	no	—
54	C₂×D₇₁₂	—	2	1	no	—
55	C₂×D₇₁₂	D₄	1	1	yes	characteristic
56	C₂×#13083557	—	2	2	no	—
57	#15668012	—	2	2	no	—
58	#15668012	C₂³	1	2	yes	—
59	#15668009	—	2	2	no	—
60	#15668009	C₂³	1	2	yes	—
61	#15667992	—	2	2	no	—
62	#15667992	—	4	2	no	—
63	#15667987	—	2	2	no	—
64	#15667987	—	4	2	no	—
65	C₄×D₃₅₆	—	2	2	no	—
66	C₄×D₃₅₆	—	2	2	no	—
67	C₄×D₃₅₆	C₂³	1	2	yes	—
68	#15667946	—	2	2	no	—
69	#15667946	—	2	1	no	—
70	#15667946	—	2	1	no	—
71	#15667946	D₄	1	1	yes	characteristic
72	C₄⋊#10623494	C₂³	1	2	yes	—
73	#15667935	—	2	2	no	—
74	#15667934	—	2	2	no	—
75	#15667934	—	2	2	no	—
76	#15667934	—	4	2	no	—
77	#15667934	D₄	1	2	yes	—
78	#15667932	—	2	2	no	—
79	#15667924	—	2	2	no	—
80	C₂×Q₁₆×C₈₉	D₄	1	1	yes	characteristic
81	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
82	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
83	#107×C₈₉	D₄	1	1	yes	characteristic
84	C₄⋊C₈×C₈₉	—	2	1	no	—
85	C₄⋊C₈×C₈₉	—	2	1	no	—
86	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
88	#103×C₈₉	D₄	1	1	yes	characteristic
89	#103×C₈₉	D₄	1	1	yes	characteristic
90	C₄×C₈×C₈₉	—	2	1	no	—
91	C₄×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	#13083575	—	4	4	no	—
96	#13083575	—	8	2	no	—
97	C₂×Dic₁₇₈	—	2	2	no	—
98	C₂×Dic₁₇₈	—	4	2	no	—
99	Q₈×D₈₉	—	4	4	no	—
100	Q₈×D₈₉	—	8	2	no	—
101	C₂×D₃₅₆	—	2	2	no	—
102	C₂×D₃₅₆	—	2	2	no	—
103	C₂×D₃₅₆	—	2	2	no	—
104	C₂×D₃₅₆	—	4	2	no	—
105	C₂×D₃₅₆	—	4	2	no	—
106	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
107	C₂×C₄×D₈₉	—	2	2	no	—
108	C₂×C₄×D₈₉	—	4	2	no	—
109	C₂×C₄×D₈₉	—	4	2	no	—
110	#13083564	—	4	4	no	—
111	#13083563	—	4	4	no	—
112	#13083563	—	8	2	no	—
113	#13083561	—	4	4	no	—
114	D₇₁₂	—	2	2	no	—
115	D₇₁₂	—	4	2	no	—
116	D₇₁₂	—	8	1	no	—
117	D₇₁₂	—	8	2	no	—
118	#13083557	—	4	4	no	—
119	C₂×#10623494	—	2	2	no	—
120	C₄⋊Dic₈₉	—	2	2	no	—
121	C₄⋊Dic₈₉	—	2	2	no	—
122	C₄⋊Dic₈₉	—	4	2	no	—
123	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
124	#13083551	—	2	2	no	—
125	#13083551	—	4	2	no	—
126	#13083551	—	4	2	no	—
127	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
129	Q₁₆×C₈₉	D₈	1	4	yes	complemented
130	C₂×C₈×C₈₉	—	2	1	no	—
131	C₂×C₈×C₈₉	—	2	1	no	—
132	C₂×C₈×C₈₉	—	2	1	no	—
133	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄×C₈₉	—	2	1	no	—
138	C₄⋊C₄×C₈₉	—	2	1	no	—
139	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
140	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
141	C₄²×C₈₉	—	2	1	no	—
142	C₄²×C₈₉	—	2	1	no	—
143	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
144	C₂²×D₈₉	—	2	2	no	—
145	C₂²×D₈₉	—	4	2	no	—
146	Q₈×C₈₉	—	2	1	no	—
147	Q₈×C₈₉	—	2	1	no	—
148	Q₈×C₈₉	C₂×D₈	1	2	yes	—
149	Q₈×C₈₉	C₂×D₈	1	2	yes	—
150	C₂×C₄×C₈₉	—	2	1	no	—
151	C₂×C₄×C₈₉	—	2	1	no	—
152	C₂×C₄×C₈₉	—	2	1	no	—
153	C₂×C₄×C₈₉	—	2	1	no	—
154	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂×Dic₈₉	—	2	2	no	—
158	C₂×Dic₈₉	—	4	2	no	—
159	D₃₅₆	—	2	2	no	—
160	D₃₅₆	—	2	2	no	—
161	D₃₅₆	—	4	2	no	—
162	D₃₅₆	—	4	2	no	—
163	D₃₅₆	—	4	2	no	—
164	D₃₅₆	—	4	2	no	—
165	D₃₅₆	—	8	4	no	—
166	D₃₅₆	—	8	2	no	—
167	D₃₅₆	—	16	2	no	—
168	C₄×D₈₉	—	4	2	no	—
169	C₄×D₈₉	—	4	2	no	—
170	C₄×D₈₉	—	8	4	no	—
171	C₄×D₈₉	—	8	4	no	—
172	Dic₁₇₈	—	4	2	no	—
173	Dic₁₇₈	—	4	2	no	—
174	Dic₁₇₈	—	8	4	no	—
175	C₈×C₈₉	—	2	1	no	—
176	C₈×C₈₉	—	2	1	no	—
177	C₈×C₈₉	—	4	1	no	—
178	C₈×C₈₉	—	4	1	no	—
179	C₈×C₈₉	—	4	1	no	—
180	C₈×C₈₉	C₂×D₈	1	2	yes	—
181	C₈×C₈₉	#137	1	2	yes	complemented
182	#10623494	—	4	2	no	—
183	C₂²×C₈₉	D₄²	1	1	yes	characteristic
184	C₂×D₈₉	—	4	2	no	—
185	C₂×D₈₉	—	4	2	no	—
186	C₂×D₈₉	—	4	2	no	—
187	C₂×D₈₉	—	8	2	no	—
188	C₂×D₈₉	—	8	2	no	—
189	C₂×D₈₉	—	8	2	no	—
190	C₄×C₈₉	—	2	2	no	—
191	C₄×C₈₉	—	2	2	no	—
192	C₄×C₈₉	—	2	1	no	—
193	C₄×C₈₉	—	4	1	no	—
194	C₄×C₈₉	—	4	1	no	—
195	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
196	C₄×C₈₉	C₂×#137	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	#19889	—	89	1	no	maximal, Sylow
202	C₂×C₈₉	#3278	1	1	yes	characteristic
203	C₂×C₈₉	D₄×D₈	1	1	yes	characteristic
204	C₂×C₈₉	#3290	1	1	yes	characteristic
205	D₈₉	—	8	2	no	—
206	D₈₉	—	16	2	no	—
207	#3391	—	89	1	no	—
208	Q₈×D₈	—	89	1	no	—
209	#3278	—	89	2	no	—
210	#1706	—	89	1	no	—
211	#1685	—	89	2	no	—
212	#1674	—	89	1	no	—
213	#1668	—	89	1	no	—
214	#1641	—	89	2	no	—
215	#1627	—	89	2	no	—
216	#1589	—	89	1	no	—
217	C₈×Q₁₆	D₈₉	1	1	yes	characteristic, complemented
218	C₈₉	#19889	1	1	yes	characteristic, complemented, Sylow
219	C₂×#137	—	178	2	no	—
220	#550	—	89	2	no	—
221	D₄×Q₈	—	89	2	no	—
222	#500	—	89	1	no	—
223	#500	—	178	1	no	—
224	#493	—	89	1	no	—
225	#493	—	178	1	no	—
226	#492	—	89	2	no	—
227	#486	—	89	2	no	—
228	#474	—	89	2	no	—
229	#474	—	178	2	no	—
230	#468	—	178	2	no	—
231	#462	—	89	2	no	—
232	#460	—	89	2	no	—
233	#459	—	89	2	no	—
234	#459	—	178	2	no	—
235	#452	—	178	2	no	—
236	#448	—	178	2	no	—
237	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
238	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
239	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
240	C₄×D₈	—	89	1	no	—
241	C₄×D₈	—	178	1	no	—
242	C₄×D₈	—	178	1	no	—
243	#436	—	89	2	no	—
244	#335	—	89	1	no	—
245	#335	C₂×D₈₉	1	1	yes	characteristic
246	#332	—	89	2	no	—
247	#331	—	89	2	no	—
248	Q₈⋊C₈	C₂×D₈₉	1	1	yes	characteristic
249	Q₈⋊C₈	C₂×D₈₉	1	1	yes	characteristic
250	D₄⋊C₈	—	89	2	no	—
251	C₈²	C₂×D₈₉	1	1	yes	characteristic
252	C₂×#41	—	178	2	no	—
253	C₂²×Q₈	—	178	2	no	—
254	#137	—	356	8	no	—
255	C₂×Q₁₆	—	178	2	no	—
256	C₂×Q₁₆	D₃₅₆	1	1	yes	characteristic
257	C₂×SD₁₆	—	178	2	no	—
258	C₂×SD₁₆	—	178	2	no	—
259	C₂×SD₁₆	—	178	2	no	—
260	C₂×D₈	—	89	1	no	—
261	C₂×D₈	—	178	2	no	—
262	C₂×D₈	—	178	1	no	—
263	C₂×D₈	—	178	1	no	—
264	C₂×M₁₆	—	178	2	no	—
265	C₄⋊Q₈	—	89	2	no	—
266	C₄⋊Q₈	—	178	2	no	—
267	#127	—	89	2	no	—
268	#127	—	178	2	no	—
269	C₂²⋊Q₈	—	356	2	no	—
270	C₂²⋊Q₈	—	178	2	no	—
271	#121	—	356	2	no	—
272	#121	—	178	2	no	—
273	C₄×Q₈	C₂²×D₈₉	1	1	yes	characteristic
274	C₄×Q₈	C₂²×D₈₉	1	1	yes	characteristic
275	C₄×D₄	—	89	2	no	—
276	C₄×D₄	—	178	2	no	—
277	C₄×D₄	—	178	2	no	—
278	#107	—	89	1	no	—
279	#107	—	178	2	no	—
280	#107	—	178	1	no	—
281	#107	—	178	1	no	—
282	#107	D₃₅₆	1	1	yes	characteristic
283	C₄⋊C₈	—	2	1	no	—
284	C₄⋊C₈	—	2	1	no	—
285	C₄⋊C₈	—	89	2	no	—
286	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
287	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
288	#103	—	178	2	no	—
289	#103	D₃₅₆	1	1	yes	characteristic
290	#103	D₃₅₆	1	1	yes	characteristic
291	#102	—	89	2	no	—
292	#102	—	356	2	no	—
293	#102	—	178	2	no	—
294	#102	—	178	2	no	—
295	#102	—	178	2	no	—
296	C₂²⋊C₈	—	178	2	no	—
297	C₄×C₈	—	2	1	no	—
298	C₄×C₈	—	2	1	no	—
299	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
300	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
301	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
302	#41	—	356	4	no	—
303	#41	—	712	2	no	—
304	C₂×Q₈	—	356	2	no	—
305	C₂×Q₈	—	356	4	no	—
306	C₂×Q₈	—	178	2	no	—
307	C₂×Q₈	—	712	2	no	—
308	C₂×Q₈	C₂×D₃₅₆	1	1	yes	characteristic
309	C₂×Q₈	C₂×D₃₅₆	1	1	yes	characteristic
310	C₂×D₄	—	89	2	no	—
311	C₂×D₄	—	356	2	no	—
312	C₂×D₄	—	356	2	no	—
313	C₂×D₄	—	178	2	no	—
314	C₂×D₄	—	178	2	no	—
315	C₂×D₄	—	178	2	no	—
316	C₂²×C₄	—	356	2	no	—
317	C₂²×C₄	—	356	2	no	—
318	C₂²×C₄	—	178	2	no	—
319	Q₁₆	—	356	4	no	—
320	Q₁₆	D₇₁₂	1	4	yes	complemented
321	SD₁₆	—	356	4	no	—
322	SD₁₆	—	356	4	no	—
323	SD₁₆	—	712	2	no	—
324	D₈	—	356	2	no	—
325	D₈	—	178	2	no	—
326	D₈	—	712	1	no	—
327	D₈	—	712	2	no	—
328	M₁₆	—	356	4	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	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	—	2	1	no	—
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	89	2	no	—
340	C₄⋊C₄	—	356	2	no	—
341	C₄⋊C₄	—	178	2	no	—
342	C₄⋊C₄	—	178	2	no	—
343	C₄⋊C₄	C₂×D₃₅₆	1	1	yes	characteristic
344	C₄⋊C₄	C₂×D₃₅₆	1	1	yes	characteristic
345	C₂²⋊C₄	—	356	2	no	—
346	C₂²⋊C₄	—	356	2	no	—
347	C₂²⋊C₄	—	178	2	no	—
348	C₄²	—	2	1	no	—
349	C₄²	—	2	1	no	—
350	C₄²	C₂³×D₈₉	1	1	yes	characteristic
351	C₂³	—	356	2	no	—
352	C₂³	—	178	2	no	—
353	Q₈	—	2	1	no	—
354	Q₈	—	2	1	no	—
355	Q₈	—	356	2	no	—
356	Q₈	—	356	2	no	—
357	Q₈	—	712	4	no	—
358	Q₈	C₂×D₇₁₂	1	2	yes	—
359	Q₈	C₂×D₇₁₂	1	2	yes	—
360	D₄	—	356	2	no	—
361	D₄	—	356	2	no	—
362	D₄	—	356	2	no	—
363	D₄	—	356	2	no	—
364	D₄	—	1424	2	no	—
365	D₄	—	178	2	no	—
366	D₄	—	178	2	no	—
367	D₄	—	712	4	no	—
368	D₄	—	712	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	356	2	no	—
374	C₂×C₄	—	356	2	no	—
375	C₂×C₄	—	356	2	no	—
376	C₂×C₄	—	178	2	no	—
377	C₂×C₄	—	712	4	no	—
378	C₂×C₄	—	712	4	no	—
379	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
380	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
381	C₂×C₄	#15668058	1	1	yes	characteristic
382	C₈	—	2	1	no	—
383	C₈	—	2	1	no	—
384	C₈	—	4	1	no	—
385	C₈	—	4	1	no	—
386	C₈	—	4	1	no	—
387	C₈	—	356	2	no	—
388	C₈	C₂×D₇₁₂	1	2	yes	—
389	C₈	#15668044	1	2	yes	complemented
390	C₂²	—	356	2	no	—
391	C₂²	—	356	2	no	—
392	C₂²	—	356	2	no	—
393	C₂²	—	712	2	no	—
394	C₂²	—	712	2	no	—
395	C₂²	—	712	2	no	—
396	C₂²	D₄×D₃₅₆	1	1	yes	characteristic
397	C₄	—	2	2	no	—
398	C₄	—	2	2	no	—
399	C₄	—	2	1	no	—
400	C₄	—	4	1	no	—
401	C₄	—	4	1	no	—
402	C₄	—	356	2	no	—
403	C₄	—	712	2	no	—
404	C₄	C₂²×D₇₁₂	1	1	yes	characteristic
405	C₄	C₂×#15668044	1	1	yes	characteristic
406	C₄	#19152403	1	1	yes	characteristic
407	C₄	#19152405	1	1	yes	characteristic
408	C₂	—	1424	2	no	—
409	C₂	—	712	2	no	—
410	C₂	#26031919	1	1	yes	characteristic
411	C₂	D₄×D₇₁₂	1	1	yes	characteristic
412	C₂	#26032039	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₅₆
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈×Q₁₆×C₈₉	C₂
socle	C₂²×C₈₉	D₄²
prime radical	#26033227	C₂
hypercenter	C₈×Q₁₆	D₈₉
hypocenter	C₈₉	#19889
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₇₁₂	C₂	semidirect
3. #26031919	C₂	semidirect
4. #26019844	C₂	semidirect
5. #26019718	C₂	semidirect
6. #26019646	C₂	semidirect
7. #26019612	C₂	semidirect
8. #26019447	C₂	semidirect
9. #26019357	C₂	semidirect
10. #26019138	C₂	semidirect
11. C₈×Q₁₆×C₈₉	C₂	semidirect
129. Q₁₆×C₈₉	D₈	semidirect
181. C₈×C₈₉	#137	semidirect
217. C₈×Q₁₆	D₈₉	semidirect
218. C₈₉	#19889	semidirect
320. Q₁₆	D₇₁₂	semidirect
389. C₈	#15668044	semidirect

Maximal subgroups

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

subgroup	index
#26033227	2
Q₈×D₇₁₂	2
#26031919	2
#26019844	2
#26019718	2
#26019646	2
#26019612	2
#26019447	2
#26019357	2
#26019138	2
C₈×Q₁₆×C₈₉	2
#19889	89