Subgroups — #36545742

Order 22784.

690 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#26032600	C₂	1	1	yes	maximal, characteristic
3	#26031899	C₂	1	2	yes	maximal, complemented
4	#26019870	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019742	C₂	1	2	yes	maximal, complemented
6	#26019646	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019641	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019503	C₂	1	2	yes	maximal, complemented
9	#26019489	C₂	1	2	yes	maximal, complemented
10	#26019138	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668041	—	2	2	no	—
13	Q₈×D₃₅₆	C₂²	1	2	yes	—
14	#19152252	C₂²	1	2	yes	—
15	#19152003	—	2	1	no	—
16	#19152003	C₂²	1	1	yes	characteristic
17	#19152003	C₂²	1	1	yes	characteristic
18	#19152003	C₂²	1	2	yes	—
19	#19151978	C₂²	1	2	yes	—
20	#19151912	—	2	2	no	—
21	#19151879	—	2	2	no	—
22	#19151879	C₂²	1	2	yes	—
23	#19151879	C₂²	1	2	yes	—
24	#19151857	—	2	2	no	—
25	#19151834	—	2	1	no	—
26	#19151808	C₂²	1	2	yes	—
27	#19151803	C₂²	1	2	yes	—
28	#19151761	—	2	2	no	—
29	#19151755	—	2	2	no	—
30	C₄×D₇₁₂	—	2	1	no	—
31	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
32	#19151677	C₂²	1	2	yes	—
33	C₂×#15667946	—	2	1	no	—
34	#19151351	C₂²	1	1	yes	characteristic
35	#19151343	C₂²	1	2	yes	—
36	#19151343	C₂²	1	2	yes	—
37	#19151327	C₂²	1	2	yes	—
38	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
41	#334×C₈₉	C₂²	1	1	yes	characteristic
42	Q₈⋊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×#13083573	—	2	2	no	—
46	C₂×Q₈×D₈₉	—	2	2	no	—
47	#15668041	—	4	8	no	—
48	C₂×Dic₃₅₆	—	2	2	no	—
49	C₂×Dic₃₅₆	—	2	2	no	—
50	C₂×#13083564	—	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	—	2	2	no	—
55	#15668012	C₂³	1	2	yes	—
56	#15668012	C₂³	1	2	yes	—
57	#15667996	—	4	2	no	—
58	#15667992	—	2	2	no	—
59	#15667992	—	2	2	no	—
60	#15667992	—	4	2	no	—
61	#15667986	—	2	2	no	—
62	C₄×D₃₅₆	—	2	2	no	—
63	C₄×D₃₅₆	C₂³	1	2	yes	—
64	C₂×C₄⋊Dic₈₉	—	2	2	no	—
65	#15667946	—	2	2	no	—
66	#15667946	—	2	2	no	—
67	#15667946	—	2	1	no	—
68	#15667946	—	2	1	no	—
69	#15667946	—	2	1	no	—
70	#15667946	—	2	1	no	—
71	#15667946	—	4	2	no	—
72	#15667946	D₄	1	1	yes	characteristic
73	C₄⋊#10623494	C₂³	1	2	yes	—
74	#15667935	—	2	2	no	—
75	#15667935	—	2	2	no	—
76	#15667934	—	2	2	no	—
77	#15667934	—	2	2	no	—
78	#15667934	D₄	1	2	yes	—
79	#15667924	—	2	2	no	—
80	C₂×SD₁₆×C₈₉	D₄	1	1	yes	characteristic
81	C₂²×C₈×C₈₉	—	2	1	no	—
82	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
83	C₄×D₄×C₈₉	C₂³	1	1	yes	characteristic
84	#106×C₈₉	D₄	1	1	yes	characteristic
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	#102×C₈₉	D₄	1	1	yes	characteristic
90	C₂²⋊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	#13083573	—	4	4	no	—
96	#13083573	—	8	2	no	—
97	C₂×Dic₁₇₈	—	2	2	no	—
98	C₂×Dic₁₇₈	—	2	2	no	—
99	C₂×Dic₁₇₈	—	4	2	no	—
100	Q₈×D₈₉	—	4	4	no	—
101	Q₈×D₈₉	—	8	2	no	—
102	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
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	4	no	—
108	Dic₃₅₆	—	8	2	no	—
109	#13083564	—	4	4	no	—
110	#13083562	—	4	4	no	—
111	D₇₁₂	—	2	2	no	—
112	#13083557	—	4	4	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	—	4	2	no	—
120	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
121	#13083551	—	2	2	no	—
122	#13083551	—	4	2	no	—
123	#13083551	—	4	2	no	—
124	C₂×Q₈×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	SD₁₆×C₈₉	D₈	1	4	yes	complemented
128	C₂×C₈×C₈₉	—	2	1	no	—
129	C₂×C₈×C₈₉	—	2	1	no	—
130	C₂×C₈×C₈₉	—	2	1	no	—
131	C₂×C₈×C₈₉	—	4	2	no	—
132	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄×C₈₉	—	2	1	no	—
137	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
138	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
139	C₂²⋊C₄×C₈₉	—	2	1	no	—
140	C₄²×C₈₉	—	2	1	no	—
141	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
142	C₂³×C₈₉	—	2	1	no	—
143	C₂²×D₈₉	—	2	2	no	—
144	Q₈×C₈₉	—	2	1	no	—
145	Q₈×C₈₉	C₂×D₈	1	2	yes	—
146	D₄×C₈₉	—	2	1	no	—
147	D₄×C₈₉	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₈₉	—	2	1	no	—
149	C₂×C₄×C₈₉	—	2	1	no	—
150	C₂×C₄×C₈₉	—	2	1	no	—
151	C₂×C₄×C₈₉	—	4	1	no	—
152	C₂×C₄×C₈₉	—	4	1	no	—
153	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²⋊D₈₉	—	8	4	no	—
157	C₂×Dic₈₉	—	2	2	no	—
158	C₂×Dic₈₉	—	4	2	no	—
159	C₂×Dic₈₉	—	4	2	no	—
160	C₂×Dic₈₉	—	8	4	no	—
161	D₃₅₆	—	2	2	no	—
162	D₃₅₆	—	2	2	no	—
163	C₄×D₈₉	—	4	2	no	—
164	C₄×D₈₉	—	4	2	no	—
165	C₄×D₈₉	—	8	4	no	—
166	Dic₁₇₈	—	4	2	no	—
167	Dic₁₇₈	—	4	2	no	—
168	Dic₁₇₈	—	4	2	no	—
169	Dic₁₇₈	—	4	2	no	—
170	Dic₁₇₈	—	8	4	no	—
171	C₈×C₈₉	—	2	2	no	—
172	C₈×C₈₉	—	2	1	no	—
173	C₈×C₈₉	—	2	1	no	—
174	C₈×C₈₉	—	4	1	no	—
175	C₈×C₈₉	—	4	1	no	—
176	C₈×C₈₉	C₂×D₈	1	2	yes	—
177	C₈×C₈₉	#137	1	2	yes	complemented
178	#10623494	—	4	2	no	—
179	C₂²×C₈₉	—	2	2	no	—
180	C₂²×C₈₉	—	4	1	no	—
181	C₂²×C₈₉	—	4	1	no	—
182	C₂²×C₈₉	D₄²	1	1	yes	characteristic
183	C₂×D₈₉	—	4	2	no	—
184	C₂×D₈₉	—	4	2	no	—
185	C₂×D₈₉	—	4	2	no	—
186	C₄×C₈₉	—	2	2	no	—
187	C₄×C₈₉	—	2	1	no	—
188	C₄×C₈₉	—	4	1	no	—
189	C₄×C₈₉	—	4	1	no	—
190	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
191	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
192	C₄×C₈₉	#576	1	1	yes	characteristic
193	C₄×C₈₉	#578	1	1	yes	characteristic
194	Dic₈₉	—	4	2	no	—
195	Dic₈₉	—	8	2	no	—
196	Dic₈₉	—	8	2	no	—
197	#19887	—	89	1	no	maximal, Sylow
198	C₂×C₈₉	—	4	2	no	—
199	C₂×C₈₉	#3276	1	1	yes	characteristic
200	C₂×C₈₉	D₄×D₈	1	1	yes	characteristic
201	C₂×C₈₉	#3293	1	1	yes	characteristic
202	D₈₉	—	8	2	no	—
203	Q₈×D₈	—	89	1	no	—
204	#3334	—	89	1	no	—
205	#3276	—	89	2	no	—
206	#1713	—	89	1	no	—
207	#1689	—	89	2	no	—
208	#1674	—	89	1	no	—
209	#1673	—	89	1	no	—
210	#1649	—	89	2	no	—
211	#1647	—	89	2	no	—
212	#1589	—	89	1	no	—
213	C₈×SD₁₆	D₈₉	1	1	yes	characteristic, complemented
214	C₈₉	#19887	1	1	yes	characteristic, complemented, Sylow
215	C₂×#137	—	178	2	no	—
216	D₄×Q₈	—	89	2	no	—
217	#547	—	89	2	no	—
218	#500	—	89	2	no	—
219	#500	—	89	1	no	—
220	#500	—	89	1	no	—
221	#500	—	178	1	no	—
222	#494	—	89	2	no	—
223	#480	—	178	2	no	—
224	#474	—	89	2	no	—
225	#474	—	89	2	no	—
226	#474	—	178	2	no	—
227	#470	—	178	2	no	—
228	#466	—	178	1	no	—
229	#462	—	89	2	no	—
230	#461	—	89	2	no	—
231	#452	—	178	2	no	—
232	#451	—	178	2	no	—
233	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
234	C₄×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
235	C₄×D₈	—	89	1	no	—
236	C₄×D₈	—	178	1	no	—
237	#436	—	89	2	no	—
238	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
239	C₂×#107	—	178	1	no	—
240	#335	—	89	1	no	—
241	#334	C₂×D₈₉	1	1	yes	characteristic
242	#332	—	89	2	no	—
243	#332	—	89	2	no	—
244	Q₈⋊C₈	C₂×D₈₉	1	1	yes	characteristic
245	D₄⋊C₈	—	89	2	no	—
246	D₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
247	C₈²	C₂×D₈₉	1	1	yes	characteristic
248	C₂×#41	—	178	2	no	—
249	C₂²×Q₈	—	178	2	no	—
250	#137	—	356	8	no	—
251	C₂×Q₁₆	—	178	2	no	—
252	C₂×Q₁₆	—	178	2	no	—
253	C₂×Q₁₆	—	178	2	no	—
254	C₂×SD₁₆	—	178	2	no	—
255	C₂×SD₁₆	D₃₅₆	1	1	yes	characteristic
256	C₂×D₈	—	89	1	no	—
257	C₂×M₁₆	—	178	2	no	—
258	C₂²×C₈	—	2	1	no	—
259	C₄⋊Q₈	—	89	2	no	—
260	C₄⋊Q₈	—	89	2	no	—
261	C₄⋊Q₈	—	178	2	no	—
262	#123	—	356	2	no	—
263	C₂²⋊Q₈	—	356	2	no	—
264	C₂²⋊Q₈	—	178	2	no	—
265	C₂²⋊Q₈	—	178	2	no	—
266	#121	—	178	2	no	—
267	C₄×Q₈	C₂²×D₈₉	1	1	yes	characteristic
268	C₄×D₄	—	89	2	no	—
269	C₄×D₄	—	178	2	no	—
270	C₄×D₄	C₂²×D₈₉	1	1	yes	characteristic
271	C₂×C₄⋊C₄	—	178	2	no	—
272	#107	—	89	1	no	—
273	#107	—	356	2	no	—
274	#107	—	178	2	no	—
275	#107	—	178	2	no	—
276	#107	—	178	1	no	—
277	#107	—	178	1	no	—
278	#107	—	178	1	no	—
279	#107	—	178	1	no	—
280	#106	D₃₅₆	1	1	yes	characteristic
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	89	2	no	—
283	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
284	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
285	#103	—	178	2	no	—
286	#103	—	178	2	no	—
287	#103	D₃₅₆	1	1	yes	characteristic
288	#102	—	89	2	no	—
289	#102	—	178	2	no	—
290	#102	—	178	2	no	—
291	#102	D₃₅₆	1	1	yes	characteristic
292	C₂²⋊C₈	—	2	1	no	—
293	C₂²⋊C₈	—	178	2	no	—
294	C₄×C₈	—	2	1	no	—
295	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
296	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
297	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
298	#41	—	356	4	no	—
299	#41	—	712	2	no	—
300	C₂×Q₈	—	356	2	no	—
301	C₂×Q₈	—	356	4	no	—
302	C₂×Q₈	—	178	2	no	—
303	C₂×Q₈	—	178	2	no	—
304	C₂×Q₈	—	712	2	no	—
305	C₂×Q₈	C₂×D₃₅₆	1	1	yes	characteristic
306	C₂×D₄	—	89	2	no	—
307	C₂×D₄	—	356	2	no	—
308	C₂×D₄	C₂×D₃₅₆	1	1	yes	characteristic
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	356	2	no	—
311	C₂²×C₄	—	356	2	no	—
312	C₂²×C₄	—	178	2	no	—
313	Q₁₆	—	356	4	no	—
314	Q₁₆	—	356	4	no	—
315	Q₁₆	—	712	2	no	—
316	SD₁₆	—	356	4	no	—
317	SD₁₆	D₇₁₂	1	4	yes	complemented
318	D₈	—	178	2	no	—
319	M₁₆	—	356	4	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	4	2	no	—
324	C₂×C₈	—	178	2	no	—
325	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
326	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
327	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
328	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	89	2	no	—
331	C₄⋊C₄	—	356	2	no	—
332	C₄⋊C₄	—	356	2	no	—
333	C₄⋊C₄	—	178	2	no	—
334	C₄⋊C₄	—	178	2	no	—
335	C₄⋊C₄	—	178	2	no	—
336	C₄⋊C₄	—	178	2	no	—
337	C₄⋊C₄	C₂×D₃₅₆	1	1	yes	characteristic
338	C₄⋊C₄	C₂×D₃₅₆	1	1	yes	characteristic
339	C₂²⋊C₄	—	2	1	no	—
340	C₂²⋊C₄	—	356	2	no	—
341	C₂²⋊C₄	—	356	2	no	—
342	C₂²⋊C₄	—	178	2	no	—
343	C₄²	—	2	1	no	—
344	C₄²	C₂³×D₈₉	1	1	yes	characteristic
345	C₂³	—	2	1	no	—
346	C₂³	—	178	2	no	—
347	Q₈	—	2	1	no	—
348	Q₈	—	356	2	no	—
349	Q₈	—	356	2	no	—
350	Q₈	—	356	2	no	—
351	Q₈	—	356	2	no	—
352	Q₈	—	712	4	no	—
353	Q₈	C₂×D₇₁₂	1	2	yes	—
354	D₄	—	2	1	no	—
355	D₄	—	178	2	no	—
356	D₄	—	178	2	no	—
357	D₄	—	712	4	no	—
358	D₄	C₂×D₇₁₂	1	2	yes	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	356	2	no	—
365	C₂×C₄	—	356	2	no	—
366	C₂×C₄	—	356	2	no	—
367	C₂×C₄	—	356	2	no	—
368	C₂×C₄	—	178	2	no	—
369	C₂×C₄	—	712	4	no	—
370	C₂×C₄	—	712	4	no	—
371	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
372	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
373	C₂×C₄	#15668058	1	1	yes	characteristic
374	C₈	—	2	2	no	—
375	C₈	—	2	1	no	—
376	C₈	—	2	1	no	—
377	C₈	—	4	1	no	—
378	C₈	—	4	1	no	—
379	C₈	—	356	2	no	—
380	C₈	C₂×D₇₁₂	1	2	yes	—
381	C₈	#15668041	1	2	yes	complemented
382	C₂²	—	2	2	no	—
383	C₂²	—	4	1	no	—
384	C₂²	—	4	1	no	—
385	C₂²	—	356	2	no	—
386	C₂²	—	356	2	no	—
387	C₂²	—	356	2	no	—
388	C₂²	D₄×D₃₅₆	1	1	yes	characteristic
389	C₄	—	2	2	no	—
390	C₄	—	2	1	no	—
391	C₄	—	4	1	no	—
392	C₄	—	4	1	no	—
393	C₄	—	356	2	no	—
394	C₄	—	712	2	no	—
395	C₄	—	712	2	no	—
396	C₄	C₂²×D₇₁₂	1	1	yes	characteristic
397	C₄	C₂×#15668041	1	1	yes	characteristic
398	C₄	#19152403	1	1	yes	characteristic
399	C₄	#19152412	1	1	yes	characteristic
400	C₂	—	4	2	no	—
401	C₂	—	712	2	no	—
402	C₂	#26031899	1	1	yes	characteristic
403	C₂	D₄×D₇₁₂	1	1	yes	characteristic
404	C₂	#26032057	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₅₆
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈×SD₁₆×C₈₉	C₂
socle	C₂²×C₈₉	D₄²
prime radical	#26032600	C₂
hypercenter	C₈×SD₁₆	D₈₉
hypocenter	C₈₉	#19887
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆

Complements

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

N	H ≅ G/N	type
1. Q₈×D₇₁₂	C₂	semidirect
3. #26031899	C₂	semidirect
4. #26019870	C₂	semidirect
5. #26019742	C₂	semidirect
6. #26019646	C₂	semidirect
7. #26019641	C₂	semidirect
8. #26019503	C₂	semidirect
9. #26019489	C₂	semidirect
10. #26019138	C₂	semidirect
11. C₈×SD₁₆×C₈₉	C₂	semidirect
127. SD₁₆×C₈₉	D₈	semidirect
177. C₈×C₈₉	#137	semidirect
213. C₈×SD₁₆	D₈₉	semidirect
214. C₈₉	#19887	semidirect
317. SD₁₆	D₇₁₂	semidirect
381. C₈	#15668041	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₇₁₂	2
#26032600	2
#26031899	2
#26019870	2
#26019742	2
#26019646	2
#26019641	2
#26019503	2
#26019489	2
#26019138	2
C₈×SD₁₆×C₈₉	2
#19887	89