Subgroups — #27375646

Order 13568.

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×D₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#20287138	C₂	1	1	yes	maximal, characteristic
3	#20287011	C₂	1	2	yes	maximal, complemented
4	#20274325	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274291	C₂	1	2	yes	maximal, complemented
6	#20274119	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274104	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273945	C₂	1	2	yes	maximal, complemented
9	#20273882	C₂	1	2	yes	maximal, complemented
10	#20273601	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811338	C₂²	1	2	yes	—
13	Q₈×D₂₁₂	C₂²	1	2	yes	—
14	#16811028	—	2	1	no	—
15	#16811028	C₂²	1	1	yes	characteristic
16	#16811028	C₂²	1	1	yes	characteristic
17	#16810974	C₂²	1	2	yes	—
18	#16810974	C₂²	1	2	yes	—
19	#16810967	—	2	2	no	—
20	#16810948	—	2	2	no	—
21	#16810918	C₂²	1	2	yes	—
22	#16810914	C₂²	1	2	yes	—
23	#16810904	—	2	2	no	—
24	#16810904	C₂²	1	2	yes	—
25	#16810868	—	2	2	no	—
26	#16810859	—	2	1	no	—
27	#16810828	C₂²	1	2	yes	—
28	#16810760	—	2	2	no	—
29	C₄×D₄₂₄	—	2	1	no	—
30	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
31	#16810702	C₂²	1	2	yes	—
32	#16810661	—	2	1	no	—
33	#16810630	—	2	2	no	—
34	#16810376	C₂²	1	1	yes	characteristic
35	#16810371	C₂²	1	2	yes	—
36	#16810368	C₂²	1	2	yes	—
37	#16810352	C₂²	1	2	yes	—
38	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#334	C₂²	1	1	yes	characteristic
42	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
43	C₅₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₅₃	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₅₃	—	2	2	no	—
46	C₂×#11719923	—	2	2	no	—
47	C₂×D₄₂₄	D₄	1	1	yes	characteristic
48	C₂×#11719912	—	2	2	no	—
49	#13133217	—	2	2	no	—
50	#13133217	C₂³	1	2	yes	—
51	#13133217	C₂³	1	2	yes	—
52	#13133203	—	4	2	no	—
53	#13133197	—	2	2	no	—
54	#13133197	—	4	2	no	—
55	#13133191	—	2	2	no	—
56	#13133188	—	2	2	no	—
57	C₄×D₂₁₂	—	2	2	no	—
58	C₄×D₂₁₂	C₂³	1	2	yes	—
59	#13133176	—	2	2	no	—
60	#13133175	—	2	2	no	—
61	#13133151	—	2	1	no	—
62	#13133151	—	2	1	no	—
63	#13133151	—	2	1	no	—
64	#13133151	—	2	1	no	—
65	#13133151	—	2	2	no	—
66	#13133151	D₄	1	1	yes	characteristic
67	#13133150	—	2	2	no	—
68	#13133149	—	2	2	no	—
69	C₄⋊#88322	C₂³	1	2	yes	—
70	#13133142	—	2	2	no	—
71	#13133142	—	2	2	no	—
72	#13133142	—	4	2	no	—
73	#13133141	—	2	2	no	—
74	#13133140	—	2	2	no	—
75	#13133139	—	2	2	no	—
76	#13133139	—	2	2	no	—
77	#13133139	D₄	1	2	yes	—
78	#13133129	—	2	2	no	—
79	C₂×SD₁₆×C₅₃	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₅₃	—	2	1	no	—
81	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
82	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
83	#106×C₅₃	D₄	1	1	yes	characteristic
84	C₄⋊C₈×C₅₃	—	2	1	no	—
85	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
87	#103×C₅₃	D₄	1	1	yes	characteristic
88	#102×C₅₃	D₄	1	1	yes	characteristic
89	C₂²⋊C₈×C₅₃	—	2	1	no	—
90	C₄×C₈×C₅₃	—	2	1	no	—
91	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	—	2	1	no	—
97	C₂×C₂²⋊D₅₃	—	4	2	no	—
98	Q₈×D₅₃	—	4	4	no	—
99	Q₈×D₅₃	—	8	2	no	—
100	C₂×D₂₁₂	C₂×D₄	1	2	yes	—
101	C₂×C₄×D₅₃	—	2	2	no	—
102	C₂×C₄×D₅₃	—	4	2	no	—
103	C₂×Dic₁₀₆	—	2	2	no	—
104	C₂×Dic₁₀₆	—	2	2	no	—
105	C₂×Dic₁₀₆	—	4	2	no	—
106	SD₁₆×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	—	2	1	no	—
108	C₂×C₈×C₅₃	—	2	1	no	—
109	C₂×C₈×C₅₃	—	2	1	no	—
110	C₂×C₈×C₅₃	—	4	2	no	—
111	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₄⋊C₄×C₅₃	—	2	1	no	—
116	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂²⋊C₄×C₅₃	—	2	1	no	—
119	C₄²×C₅₃	—	2	1	no	—
120	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
121	C₂²⋊Dic₅₃	—	4	2	no	—
122	#11719923	—	8	2	no	—
123	#11719921	—	2	2	no	—
124	#11719921	—	4	2	no	—
125	#11719921	—	4	2	no	—
126	C₄⋊Dic₅₃	—	2	2	no	—
127	C₄⋊Dic₅₃	—	2	2	no	—
128	C₄⋊Dic₅₃	—	2	2	no	—
129	C₄⋊Dic₅₃	—	2	2	no	—
130	C₄⋊Dic₅₃	—	4	2	no	—
131	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
132	C₄×Dic₅₃	—	4	2	no	—
133	C₂×#88322	—	2	2	no	—
134	D₄₂₄	—	2	2	no	—
135	#11719912	—	8	2	no	—
136	C₂³×C₅₃	—	2	1	no	—
137	C₂²×D₅₃	—	2	2	no	—
138	Q₈×C₅₃	—	2	1	no	—
139	Q₈×C₅₃	#135	1	2	yes	—
140	D₄×C₅₃	—	2	1	no	—
141	D₄×C₅₃	—	2	1	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	1	no	—
144	C₂×C₄×C₅₃	—	2	1	no	—
145	C₂×C₄×C₅₃	—	4	1	no	—
146	C₂×C₄×C₅₃	—	4	1	no	—
147	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂²⋊D₅₃	—	16	2	no	—
151	C₂×Dic₅₃	—	2	2	no	—
152	C₂×Dic₅₃	—	4	2	no	—
153	C₂×Dic₅₃	—	4	2	no	—
154	D₂₁₂	—	2	2	no	—
155	D₂₁₂	—	2	2	no	—
156	C₄×D₅₃	—	4	2	no	—
157	C₄×D₅₃	—	4	2	no	—
158	C₄×D₅₃	—	8	4	no	—
159	Dic₁₀₆	—	4	2	no	—
160	Dic₁₀₆	—	4	2	no	—
161	Dic₁₀₆	—	8	4	no	—
162	Dic₁₀₆	—	8	2	no	—
163	C₈×C₅₃	—	2	1	no	—
164	C₈×C₅₃	—	2	1	no	—
165	C₈×C₅₃	—	2	1	no	—
166	C₈×C₅₃	—	4	1	no	—
167	C₈×C₅₃	—	4	1	no	—
168	C₈×C₅₃	—	4	1	no	—
169	C₈×C₅₃	#137	1	2	yes	—
170	#88322	—	4	2	no	—
171	#19905	—	53	1	no	maximal, Sylow
172	C₂²×C₅₃	—	4	1	no	—
173	C₂²×C₅₃	—	4	1	no	—
174	C₂²×C₅₃	—	4	1	no	—
175	C₂²×C₅₃	D₄²	1	1	yes	characteristic
176	C₂×D₅₃	—	4	2	no	—
177	C₂×D₅₃	—	4	2	no	—
178	C₂×D₅₃	—	4	2	no	—
179	C₄×C₅₃	—	2	2	no	—
180	C₄×C₅₃	—	2	1	no	—
181	C₄×C₅₃	—	4	1	no	—
182	C₄×C₅₃	—	4	1	no	—
183	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
184	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
185	C₄×C₅₃	#576	1	1	yes	characteristic
186	C₄×C₅₃	#578	1	1	yes	characteristic
187	Dic₅₃	—	4	2	no	—
188	Dic₅₃	—	8	2	no	—
189	Dic₅₃	—	8	2	no	—
190	Q₈×D₈	—	53	1	no	—
191	#3341	—	53	1	no	—
192	#3329	—	53	2	no	—
193	#1710	—	53	1	no	—
194	#1703	—	53	2	no	—
195	#1676	—	53	1	no	—
196	#1673	—	53	1	no	—
197	#1646	—	53	2	no	—
198	#1637	—	53	2	no	—
199	#1589	—	53	1	no	—
200	C₈×SD₁₆	D₅₃	1	1	yes	characteristic, complemented
201	C₂×C₅₃	—	8	1	no	—
202	C₂×C₅₃	#3276	1	1	yes	characteristic
203	C₂×C₅₃	#3280	1	1	yes	characteristic
204	C₂×C₅₃	#3290	1	1	yes	characteristic
205	D₅₃	—	8	2	no	—
206	#555	—	53	2	no	—
207	D₄×Q₈	—	53	2	no	—
208	#500	—	53	1	no	—
209	#500	—	53	1	no	—
210	#500	—	106	1	no	—
211	#487	—	53	2	no	—
212	#487	—	53	2	no	—
213	#485	—	106	2	no	—
214	#482	—	106	2	no	—
215	#477	—	53	2	no	—
216	#476	—	53	2	no	—
217	#474	—	53	2	no	—
218	#474	—	106	2	no	—
219	#468	—	106	2	no	—
220	#466	—	106	1	no	—
221	#461	—	53	2	no	—
222	#448	—	106	2	no	—
223	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
224	C₄×SD₁₆	C₂×D₅₃	1	1	yes	characteristic
225	C₄×D₈	—	53	1	no	—
226	C₄×D₈	—	106	1	no	—
227	#436	—	53	2	no	—
228	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
229	#427	—	106	1	no	—
230	#419	—	106	2	no	—
231	#335	—	53	1	no	—
232	#334	C₂×D₅₃	1	1	yes	characteristic
233	#333	—	53	2	no	—
234	#332	—	53	2	no	—
235	Q₈⋊C₈	C₂×D₅₃	1	1	yes	characteristic
236	D₄⋊C₈	—	53	2	no	—
237	D₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
238	C₈²	C₂×D₅₃	1	1	yes	characteristic
239	C₅₃	#19905	1	1	yes	characteristic, complemented, Sylow
240	C₂²×Q₈	—	106	2	no	—
241	C₂×SD₁₆	—	106	2	no	—
242	C₂×SD₁₆	D₂₁₂	1	1	yes	characteristic
243	C₂×D₈	—	53	1	no	—
244	C₂×M₁₆	—	106	2	no	—
245	C₂²×C₈	—	2	1	no	—
246	C₄⋊Q₈	—	53	2	no	—
247	C₄⋊Q₈	—	53	2	no	—
248	C₄⋊Q₈	—	106	2	no	—
249	#124	—	212	2	no	—
250	C₂²⋊Q₈	—	106	2	no	—
251	C₂²⋊Q₈	—	212	2	no	—
252	#121	—	106	2	no	—
253	#121	—	106	2	no	—
254	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
255	C₄×D₄	—	53	2	no	—
256	C₄×D₄	—	106	2	no	—
257	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
258	#117	—	106	2	no	—
259	#117	—	106	2	no	—
260	#107	—	53	1	no	—
261	#107	—	106	2	no	—
262	#107	—	106	1	no	—
263	#107	—	106	1	no	—
264	#107	—	106	1	no	—
265	#107	—	106	1	no	—
266	#107	—	106	2	no	—
267	#106	—	106	2	no	—
268	#106	D₂₁₂	1	1	yes	characteristic
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	53	2	no	—
271	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
272	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
273	#103	—	106	2	no	—
274	#103	—	106	2	no	—
275	#103	—	106	2	no	—
276	#103	—	106	2	no	—
277	#103	—	212	2	no	—
278	#103	D₂₁₂	1	1	yes	characteristic
279	#102	—	53	2	no	—
280	#102	—	106	2	no	—
281	#102	—	106	2	no	—
282	#102	D₂₁₂	1	1	yes	characteristic
283	C₂²⋊C₈	—	2	1	no	—
284	C₂²⋊C₈	—	106	2	no	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
289	C₂×Q₈	—	106	2	no	—
290	C₂×Q₈	—	106	2	no	—
291	C₂×Q₈	—	424	2	no	—
292	C₂×Q₈	—	212	2	no	—
293	C₂×Q₈	—	212	4	no	—
294	C₂×Q₈	C₂×D₂₁₂	1	1	yes	characteristic
295	C₂×D₄	—	53	2	no	—
296	C₂×D₄	—	212	2	no	—
297	C₂×D₄	C₂×D₂₁₂	1	1	yes	characteristic
298	C₂²×C₄	—	2	1	no	—
299	C₂²×C₄	—	106	2	no	—
300	C₂²×C₄	—	212	2	no	—
301	SD₁₆	—	2	2	no	—
302	SD₁₆	—	424	2	no	—
303	D₈	—	106	2	no	—
304	M₁₆	—	424	2	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	4	2	no	—
309	C₂×C₈	—	106	2	no	—
310	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
311	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
312	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
313	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	53	2	no	—
316	C₄⋊C₄	—	106	2	no	—
317	C₄⋊C₄	—	106	2	no	—
318	C₄⋊C₄	—	106	2	no	—
319	C₄⋊C₄	—	106	2	no	—
320	C₄⋊C₄	—	212	2	no	—
321	C₄⋊C₄	C₂×D₂₁₂	1	1	yes	characteristic
322	C₄⋊C₄	C₂×D₂₁₂	1	1	yes	characteristic
323	C₂²⋊C₄	—	2	1	no	—
324	C₂²⋊C₄	—	106	2	no	—
325	C₂²⋊C₄	—	212	2	no	—
326	C₂²⋊C₄	—	212	2	no	—
327	C₂²⋊C₄	—	212	2	no	—
328	C₄²	—	2	1	no	—
329	C₄²	—	212	2	no	—
330	C₄²	C₂³×D₅₃	1	1	yes	characteristic
331	C₂³	—	2	1	no	—
332	C₂³	—	106	2	no	—
333	Q₈	—	2	1	no	—
334	Q₈	—	424	2	no	—
335	Q₈	—	424	4	no	—
336	Q₈	—	212	2	no	—
337	Q₈	—	212	2	no	—
338	Q₈	#13133239	1	2	yes	—
339	D₄	—	2	1	no	—
340	D₄	—	2	1	no	—
341	D₄	—	848	2	no	—
342	D₄	—	106	2	no	—
343	D₄	—	106	2	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	106	2	no	—
350	C₂×C₄	—	424	4	no	—
351	C₂×C₄	—	212	2	no	—
352	C₂×C₄	—	212	2	no	—
353	C₂×C₄	—	212	2	no	—
354	C₂×C₄	—	212	2	no	—
355	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
356	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
357	C₂×C₄	#13133263	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	4	1	no	—
362	C₈	—	4	1	no	—
363	C₈	—	4	1	no	—
364	C₈	—	212	2	no	—
365	C₈	#13133246	1	2	yes	—
366	C₂²	—	4	1	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	4	1	no	—
369	C₂²	—	212	2	no	—
370	C₂²	—	212	2	no	—
371	C₂²	—	212	2	no	—
372	C₂²	D₄×D₂₁₂	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	2	no	—
375	C₄	—	4	1	no	—
376	C₄	—	4	1	no	—
377	C₄	—	424	2	no	—
378	C₄	—	424	2	no	—
379	C₄	—	212	2	no	—
380	C₄	C₂×#13133239	1	1	yes	characteristic
381	C₄	C₂×#13133246	1	1	yes	characteristic
382	C₄	#16811428	1	1	yes	characteristic
383	C₄	#16811437	1	1	yes	characteristic
384	C₂	—	8	1	no	—
385	C₂	—	424	2	no	—
386	C₂	#20286362	1	1	yes	characteristic
387	C₂	#20286402	1	1	yes	characteristic
388	C₂	#20286502	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₁₂
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×SD₁₆×C₅₃	C₂
socle	C₂²×C₅₃	D₄²
prime radical	#20287138	C₂
hypercenter	C₈×SD₁₆	D₅₃
hypocenter	C₅₃	#19905
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆

Complements

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

N	H ≅ G/N	type
1. Q₈×D₄₂₄	C₂	semidirect
3. #20287011	C₂	semidirect
4. #20274325	C₂	semidirect
5. #20274291	C₂	semidirect
6. #20274119	C₂	semidirect
7. #20274104	C₂	semidirect
8. #20273945	C₂	semidirect
9. #20273882	C₂	semidirect
10. #20273601	C₂	semidirect
11. C₈×SD₁₆×C₅₃	C₂	semidirect
200. C₈×SD₁₆	D₅₃	semidirect
239. C₅₃	#19905	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₄₂₄	2
#20287138	2
#20287011	2
#20274325	2
#20274291	2
#20274119	2
#20274104	2
#20273945	2
#20273882	2
#20273601	2
C₈×SD₁₆×C₅₃	2
#19905	53