Subgroups — #36545984

Order 22784.

604 conjugacy classes of subgroups.

The subgroup lattice has 390 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	#26032674	C₂	1	1	yes	maximal, characteristic
3	#26032543	C₂	1	2	yes	maximal, complemented
4	#26019861	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019830	C₂	1	2	yes	maximal, complemented
6	#26019642	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019481	C₂	1	2	yes	maximal, complemented
8	#26019415	C₂	1	2	yes	maximal, complemented
9	#26019136	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#13083561	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1676	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152314	C₂²	1	2	yes	—
13	Q₈×D₃₅₆	C₂²	1	2	yes	—
14	#19152003	—	2	1	no	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19151950	C₂²	1	2	yes	—
17	#19151948	C₂²	1	2	yes	—
18	#19151941	—	2	2	no	—
19	#19151925	—	2	2	no	—
20	#19151885	C₂²	1	2	yes	—
21	#19151879	—	2	2	no	—
22	#19151879	C₂²	1	2	yes	—
23	#19151845	—	2	2	no	—
24	#19151835	—	2	1	no	—
25	#19151804	C₂²	1	2	yes	—
26	#19151738	—	2	2	no	—
27	C₄×#13083561	C₂²	1	1	yes	characteristic
28	C₄×D₇₁₂	—	2	1	no	—
29	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
30	#19151675	C₂²	1	2	yes	—
31	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
32	#19151635	—	2	1	no	—
33	#19151605	—	2	2	no	—
34	#19151348	C₂²	1	1	yes	characteristic
35	#19151345	C₂²	1	2	yes	—
36	Q₈⋊#10623494	C₂²	1	1	yes	characteristic
37	#19151325	C₂²	1	2	yes	—
38	#19151325	C₂²	1	1	yes	characteristic
39	C₈×#10623494	C₂²	1	1	yes	characteristic
40	#500×C₈₉	C₂²	1	1	yes	characteristic
41	#477×C₈₉	C₂²	1	2	yes	—
42	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
43	#335×C₈₉	C₂²	1	1	yes	characteristic
44	#333×C₈₉	C₂²	1	2	yes	—
45	C₂×Q₈×D₈₉	—	2	2	no	—
46	C₂×#13083563	—	2	2	no	—
47	C₂×#13083561	D₄	1	1	yes	characteristic
48	C₂×D₇₁₂	D₄	1	1	yes	characteristic
49	C₂×#13083557	—	2	2	no	—
50	C₂×C₈×D₈₉	—	2	1	no	—
51	#15668012	—	2	2	no	—
52	#15668012	C₂³	1	2	yes	—
53	#15668001	—	4	2	no	—
54	#15667992	—	2	2	no	—
55	#15667992	—	4	2	no	—
56	#15667987	—	2	2	no	—
57	#15667987	—	2	2	no	—
58	C₄×D₃₅₆	—	2	2	no	—
59	C₄×D₃₅₆	C₂³	1	2	yes	—
60	C₄×D₃₅₆	C₂³	1	1	yes	characteristic
61	#15667970	—	2	2	no	—
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	#15667943	—	2	2	no	—
68	#15667943	D₄	1	1	yes	characteristic
69	C₄⋊#10623494	C₂³	1	1	yes	characteristic
70	#15667937	—	2	2	no	—
71	#15667936	—	4	2	no	—
72	#15667936	D₄	1	1	yes	characteristic
73	#15667935	—	2	2	no	—
74	#15667935	—	2	2	no	—
75	#15667934	—	2	2	no	—
76	#15667934	D₄	1	2	yes	—
77	#15667932	—	2	2	no	—
78	#15667932	D₄	1	1	yes	characteristic
79	#15667924	—	2	2	no	—
80	#15667924	—	2	1	no	—
81	C₄×#10623494	C₂³	1	1	yes	characteristic
82	C₄×#10623494	C₂³	1	1	yes	characteristic
83	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
84	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
85	#107×C₈₉	—	2	1	no	—
86	#107×C₈₉	—	2	1	no	—
87	#107×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	Q₈×D₈₉	—	4	4	no	—
97	Q₈×D₈₉	—	8	2	no	—
98	C₂×D₃₅₆	—	4	2	no	—
99	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
100	C₂×D₃₅₆	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×D₈₉	—	2	1	no	—
102	C₂×C₄×D₈₉	—	2	2	no	—
103	C₂×C₄×D₈₉	—	4	2	no	—
104	#13083563	—	8	2	no	—
105	#13083561	—	2	2	no	—
106	D₇₁₂	—	2	2	no	—
107	#13083557	—	8	2	no	—
108	C₂×#10623494	—	2	1	no	—
109	C₂×#10623494	—	2	1	no	—
110	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
111	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
112	C₈×D₈₉	—	4	2	no	—
113	C₄⋊Dic₈₉	—	2	2	no	—
114	C₄⋊Dic₈₉	—	2	2	no	—
115	C₄⋊Dic₈₉	—	4	2	no	—
116	C₄⋊Dic₈₉	C₂×D₄	1	1	yes	characteristic
117	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
118	#13083551	—	2	2	no	—
119	#13083551	—	2	1	no	—
120	#13083551	—	4	2	no	—
121	#13083551	—	4	2	no	—
122	#13083551	—	4	2	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	C₂×C₈×C₈₉	—	2	2	no	—
127	C₂×C₈×C₈₉	—	2	1	no	—
128	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₄⋊C₄×C₈₉	—	2	1	no	—
131	C₄⋊C₄×C₈₉	—	2	2	no	—
132	C₄⋊C₄×C₈₉	—	2	2	no	—
133	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₈₉	—	2	1	no	—
135	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
136	C₂²×D₈₉	—	2	2	no	—
137	C₂²×D₈₉	—	2	1	no	—
138	Q₈×C₈₉	—	2	1	no	—
139	Q₈×C₈₉	—	8	2	no	—
140	Q₈×C₈₉	#135	1	2	yes	—
141	C₂×C₄×C₈₉	—	2	1	no	—
142	C₂×C₄×C₈₉	—	2	1	no	—
143	C₂×C₄×C₈₉	—	4	2	no	—
144	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×Dic₈₉	—	2	1	no	—
148	C₂×Dic₈₉	—	2	2	no	—
149	C₂×Dic₈₉	—	4	2	no	—
150	D₃₅₆	—	2	1	no	—
151	D₃₅₆	—	2	2	no	—
152	D₃₅₆	—	2	1	no	—
153	D₃₅₆	—	2	2	no	—
154	D₃₅₆	—	16	2	no	—
155	C₄×D₈₉	—	4	2	no	—
156	C₄×D₈₉	—	4	1	no	—
157	C₄×D₈₉	—	4	2	no	—
158	C₄×D₈₉	—	4	1	no	—
159	C₄×D₈₉	—	8	4	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	1	no	—
168	#10623494	—	2	1	no	—
169	#10623494	—	4	1	no	—
170	#10623494	—	4	1	no	—
171	C₂²×C₈₉	D₄²	1	1	yes	characteristic
172	C₂×D₈₉	—	4	1	no	—
173	C₂×D₈₉	—	4	2	no	—
174	C₂×D₈₉	—	4	2	no	—
175	C₂×D₈₉	—	4	1	no	—
176	C₂×D₈₉	—	4	1	no	—
177	C₂×D₈₉	—	4	2	no	—
178	C₄×C₈₉	—	2	2	no	—
179	C₄×C₈₉	—	2	1	no	—
180	C₄×C₈₉	—	4	1	no	—
181	C₄×C₈₉	—	8	2	no	—
182	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
183	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
184	C₄×C₈₉	#576	1	1	yes	characteristic
185	C₄×C₈₉	#578	1	1	yes	characteristic
186	Dic₈₉	—	4	1	no	—
187	Dic₈₉	—	4	2	no	—
188	Dic₈₉	—	8	2	no	—
189	#19905	—	89	1	no	maximal, Sylow
190	C₂×C₈₉	#3276	1	1	yes	characteristic
191	C₂×C₈₉	#3280	1	1	yes	characteristic
192	C₂×C₈₉	#3290	1	1	yes	characteristic
193	D₈₉	—	8	1	no	—
194	D₈₉	—	8	2	no	—
195	Q₈×D₈	—	89	1	no	—
196	#3341	—	89	1	no	—
197	#3329	—	89	2	no	—
198	#1710	—	89	1	no	—
199	#1703	—	89	2	no	—
200	#1676	D₈₉	1	1	yes	characteristic, complemented
201	#1673	—	89	1	no	—
202	#1646	—	89	2	no	—
203	#1637	—	89	2	no	—
204	#1589	—	89	1	no	—
205	C₈×SD₁₆	—	89	1	no	—
206	C₈₉	#19905	1	1	yes	characteristic, complemented, Sylow
207	#555	—	89	2	no	—
208	D₄×Q₈	—	89	2	no	—
209	#500	—	89	1	no	—
210	#500	—	178	1	no	—
211	#500	C₂×D₈₉	1	1	yes	characteristic
212	#487	—	89	2	no	—
213	#487	—	89	2	no	—
214	#485	—	178	2	no	—
215	#482	—	178	2	no	—
216	#477	C₂×D₈₉	1	2	yes	—
217	#476	—	89	2	no	—
218	#474	—	89	2	no	—
219	#474	—	178	2	no	—
220	#468	—	178	2	no	—
221	#466	—	178	1	no	—
222	#461	—	89	2	no	—
223	#448	—	178	2	no	—
224	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
225	C₄×SD₁₆	—	89	1	no	—
226	C₄×D₈	—	89	1	no	—
227	C₄×D₈	—	178	1	no	—
228	#436	—	89	2	no	—
229	C₈×D₄	—	89	1	no	—
230	#427	—	178	1	no	—
231	#419	—	178	2	no	—
232	#335	C₂×D₈₉	1	1	yes	characteristic
233	#334	—	89	1	no	—
234	#333	C₂×D₈₉	1	2	yes	—
235	#332	—	89	2	no	—
236	Q₈⋊C₈	—	89	1	no	—
237	D₄⋊C₈	—	89	1	no	—
238	D₄⋊C₈	—	89	2	no	—
239	C₈²	—	89	1	no	—
240	C₂²×Q₈	—	178	2	no	—
241	C₂×SD₁₆	—	89	1	no	—
242	C₂×SD₁₆	—	178	2	no	—
243	C₂×D₈	—	89	1	no	—
244	C₂×M₁₆	—	178	2	no	—
245	C₂²×C₈	—	178	1	no	—
246	C₄⋊Q₈	—	89	2	no	—
247	C₄⋊Q₈	—	178	2	no	—
248	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
249	#124	—	356	2	no	—
250	C₂²⋊Q₈	—	356	2	no	—
251	C₂²⋊Q₈	—	178	2	no	—
252	#121	—	178	2	no	—
253	#121	—	178	2	no	—
254	C₄×Q₈	C₂²×D₈₉	1	1	yes	characteristic
255	C₄×D₄	—	89	2	no	—
256	C₄×D₄	—	89	1	no	—
257	C₄×D₄	—	178	2	no	—
258	#117	—	178	2	no	—
259	#117	—	178	2	no	—
260	#107	—	2	1	no	—
261	#107	—	2	1	no	—
262	#107	—	2	2	no	—
263	#107	—	89	1	no	—
264	#107	—	178	2	no	—
265	#107	—	178	1	no	—
266	#107	—	178	1	no	—
267	#106	—	89	1	no	—
268	#106	—	178	2	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	89	1	no	—
271	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
272	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
273	#103	—	2	2	no	—
274	#103	—	89	1	no	—
275	#103	—	356	2	no	—
276	#103	—	178	2	no	—
277	#103	—	178	2	no	—
278	#103	—	178	2	no	—
279	#102	—	89	2	no	—
280	#102	—	89	1	no	—
281	#102	—	178	2	no	—
282	#102	—	178	2	no	—
283	C₂²⋊C₈	—	178	2	no	—
284	C₂²⋊C₈	—	178	1	no	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	—	89	1	no	—
287	C₄×C₈	—	89	1	no	—
288	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	—	356	2	no	—
291	C₂×Q₈	—	356	4	no	—
292	C₂×Q₈	—	178	2	no	—
293	C₂×Q₈	—	712	2	no	—
294	C₂×Q₈	D₄×D₈₉	1	1	yes	characteristic
295	C₂×D₄	—	89	2	no	—
296	C₂×D₄	—	89	1	no	—
297	C₂×D₄	—	356	2	no	—
298	C₂²×C₄	—	356	2	no	—
299	C₂²×C₄	—	178	1	no	—
300	C₂²×C₄	—	178	2	no	—
301	SD₁₆	—	178	2	no	—
302	SD₁₆	—	712	2	no	—
303	D₈	—	178	2	no	—
304	M₁₆	—	712	2	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	89	1	no	—
308	C₂×C₈	—	89	1	no	—
309	C₂×C₈	—	356	2	no	—
310	C₂×C₈	—	178	1	no	—
311	C₂×C₈	—	178	1	no	—
312	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
313	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	89	2	no	—
318	C₄⋊C₄	—	89	1	no	—
319	C₄⋊C₄	—	356	2	no	—
320	C₄⋊C₄	—	178	2	no	—
321	C₄⋊C₄	—	178	2	no	—
322	C₄⋊C₄	D₄×D₈₉	1	1	yes	characteristic
323	C₂²⋊C₄	—	356	2	no	—
324	C₂²⋊C₄	—	356	2	no	—
325	C₂²⋊C₄	—	356	2	no	—
326	C₂²⋊C₄	—	178	2	no	—
327	C₂²⋊C₄	—	178	1	no	—
328	C₄²	—	2	1	no	—
329	C₄²	—	356	2	no	—
330	C₄²	C₂³×D₈₉	1	1	yes	characteristic
331	C₂³	—	178	2	no	—
332	C₂³	—	178	1	no	—
333	Q₈	—	2	1	no	—
334	Q₈	—	8	2	no	—
335	Q₈	—	356	2	no	—
336	Q₈	—	356	2	no	—
337	Q₈	—	712	4	no	—
338	Q₈	#15668035	1	2	yes	—
339	D₄	—	1424	2	no	—
340	D₄	—	178	2	no	—
341	D₄	—	178	1	no	—
342	D₄	—	178	2	no	—
343	D₄	—	178	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	356	1	no	—
348	C₂×C₄	—	356	2	no	—
349	C₂×C₄	—	356	1	no	—
350	C₂×C₄	—	356	2	no	—
351	C₂×C₄	—	356	2	no	—
352	C₂×C₄	—	178	1	no	—
353	C₂×C₄	—	178	2	no	—
354	C₂×C₄	—	712	4	no	—
355	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
356	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
357	C₂×C₄	#15668058	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	4	2	no	—
360	C₈	—	4	1	no	—
361	C₈	—	356	1	no	—
362	C₈	—	356	1	no	—
363	C₈	—	178	1	no	—
364	C₈	—	178	1	no	—
365	C₈	#15668044	1	2	yes	—
366	C₂²	—	356	1	no	—
367	C₂²	—	356	2	no	—
368	C₂²	—	356	2	no	—
369	C₂²	—	356	1	no	—
370	C₂²	—	356	1	no	—
371	C₂²	—	356	2	no	—
372	C₂²	#19152232	1	1	yes	characteristic
373	C₄	—	2	2	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	1	no	—
376	C₄	—	8	2	no	—
377	C₄	—	356	1	no	—
378	C₄	—	356	2	no	—
379	C₄	—	712	2	no	—
380	C₄	C₂×#15668035	1	1	yes	characteristic
381	C₄	C₂×#15668044	1	1	yes	characteristic
382	C₄	#19152400	1	1	yes	characteristic
383	C₄	#19152414	1	1	yes	characteristic
384	C₂	—	712	1	no	—
385	C₂	—	712	2	no	—
386	C₂	#26031893	1	1	yes	characteristic
387	C₂	#26031934	1	1	yes	characteristic
388	C₂	#26032033	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19152232
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1676	C₂
socle	C₂²×C₈₉	D₄²
prime radical	#26032674	C₂
hypercenter	#1676	D₈₉
hypocenter	C₈₉	#19905
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1676

Complements

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

N	H ≅ G/N	type
1. Q₈×D₇₁₂	C₂	semidirect
3. #26032543	C₂	semidirect
4. #26019861	C₂	semidirect
5. #26019830	C₂	semidirect
6. #26019642	C₂	semidirect
7. #26019481	C₂	semidirect
8. #26019415	C₂	semidirect
9. #26019136	C₂	semidirect
10. C₈×#13083561	C₂	semidirect
11. C₈₉×#1676	C₂	semidirect
200. #1676	D₈₉	semidirect
206. C₈₉	#19905	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
#26032674	2
#26032543	2
#26019861	2
#26019830	2
#26019642	2
#26019481	2
#26019415	2
#26019136	2
C₈×#13083561	2
C₈₉×#1676	2
#19905	89