Subgroups — #27375406

Order 13568.

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×D₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#20287063	C₂	1	1	yes	maximal, characteristic
3	#20286362	C₂	1	2	yes	maximal, complemented
4	#20274333	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274205	C₂	1	2	yes	maximal, complemented
6	#20274109	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274104	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273966	C₂	1	2	yes	maximal, complemented
9	#20273952	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	C₂×#13133246	—	2	2	no	—
13	Q₈×D₂₁₂	C₂²	1	2	yes	—
14	#16811277	C₂²	1	2	yes	—
15	#16811028	—	2	1	no	—
16	#16811028	C₂²	1	1	yes	characteristic
17	#16811028	C₂²	1	1	yes	characteristic
18	#16811028	C₂²	1	2	yes	—
19	#16811003	C₂²	1	2	yes	—
20	#16810937	—	2	2	no	—
21	#16810904	—	2	2	no	—
22	#16810904	C₂²	1	2	yes	—
23	#16810904	C₂²	1	2	yes	—
24	#16810882	—	2	2	no	—
25	#16810859	—	2	1	no	—
26	#16810833	C₂²	1	2	yes	—
27	#16810828	C₂²	1	2	yes	—
28	#16810786	—	2	2	no	—
29	#16810780	—	2	2	no	—
30	C₄×D₄₂₄	—	2	1	no	—
31	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
32	#16810702	C₂²	1	2	yes	—
33	C₂×#13133151	—	2	1	no	—
34	#16810376	C₂²	1	1	yes	characteristic
35	#16810368	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×#11719947	—	2	2	no	—
46	C₂×Q₈×D₅₃	—	2	2	no	—
47	#13133246	—	4	8	no	—
48	C₂×Dic₂₁₂	—	2	2	no	—
49	C₂×Dic₂₁₂	—	2	2	no	—
50	C₂×#11719925	—	2	2	no	—
51	C₂×#11719923	—	2	2	no	—
52	C₂×D₄₂₄	D₄	1	1	yes	characteristic
53	C₂×#11719912	—	2	2	no	—
54	#13133217	—	2	2	no	—
55	#13133217	C₂³	1	2	yes	—
56	#13133217	C₂³	1	2	yes	—
57	#13133201	—	4	2	no	—
58	#13133197	—	2	2	no	—
59	#13133197	—	2	2	no	—
60	#13133197	—	4	2	no	—
61	#13133191	—	2	2	no	—
62	C₄×D₂₁₂	—	2	2	no	—
63	C₄×D₂₁₂	C₂³	1	2	yes	—
64	C₂×C₄⋊Dic₅₃	—	2	2	no	—
65	#13133151	—	2	1	no	—
66	#13133151	—	2	1	no	—
67	#13133151	—	2	1	no	—
68	#13133151	—	2	1	no	—
69	#13133151	—	2	2	no	—
70	#13133151	—	2	2	no	—
71	#13133151	—	4	2	no	—
72	#13133151	D₄	1	1	yes	characteristic
73	C₄⋊#88322	C₂³	1	2	yes	—
74	#13133140	—	2	2	no	—
75	#13133140	—	2	2	no	—
76	#13133139	—	2	2	no	—
77	#13133139	—	2	2	no	—
78	#13133139	D₄	1	2	yes	—
79	#13133129	—	2	2	no	—
80	C₂×SD₁₆×C₅₃	D₄	1	1	yes	characteristic
81	C₂²×C₈×C₅₃	—	2	1	no	—
82	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
83	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
84	#106×C₅₃	D₄	1	1	yes	characteristic
85	C₄⋊C₈×C₅₃	—	2	1	no	—
86	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
88	#103×C₅₃	D₄	1	1	yes	characteristic
89	#102×C₅₃	D₄	1	1	yes	characteristic
90	C₂²⋊C₈×C₅₃	—	2	1	no	—
91	C₄×C₈×C₅₃	—	2	1	no	—
92	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	—	2	1	no	—
98	C₂×C₂²⋊D₅₃	—	4	2	no	—
99	C₂²×Dic₅₃	—	4	2	no	—
100	Q₈×D₅₃	—	4	4	no	—
101	Q₈×D₅₃	—	8	2	no	—
102	#11719947	—	4	4	no	—
103	#11719947	—	8	2	no	—
104	C₂×D₂₁₂	C₂×D₄	1	2	yes	—
105	C₂×C₄×D₅₃	—	2	2	no	—
106	C₂×C₄×D₅₃	—	4	2	no	—
107	C₂×Dic₁₀₆	—	2	2	no	—
108	C₂×Dic₁₀₆	—	2	2	no	—
109	C₂×Dic₁₀₆	—	4	2	no	—
110	SD₁₆×C₅₃	D₈	1	4	yes	complemented
111	C₂×C₈×C₅₃	—	2	1	no	—
112	C₂×C₈×C₅₃	—	2	1	no	—
113	C₂×C₈×C₅₃	—	2	1	no	—
114	C₂×C₈×C₅₃	—	4	2	no	—
115	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₄⋊C₄×C₅₃	—	2	1	no	—
120	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
122	C₂²⋊C₄×C₅₃	—	2	1	no	—
123	C₄²×C₅₃	—	2	1	no	—
124	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
125	#11719925	—	4	4	no	—
126	#11719923	—	4	4	no	—
127	#11719921	—	2	2	no	—
128	#11719921	—	4	2	no	—
129	#11719921	—	4	2	no	—
130	C₄⋊Dic₅₃	—	2	2	no	—
131	C₄⋊Dic₅₃	—	2	2	no	—
132	C₄⋊Dic₅₃	—	2	2	no	—
133	C₄⋊Dic₅₃	—	2	2	no	—
134	C₄⋊Dic₅₃	—	4	2	no	—
135	C₄⋊Dic₅₃	—	4	2	no	—
136	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
137	C₂×#88322	—	2	2	no	—
138	Dic₂₁₂	—	4	4	no	—
139	Dic₂₁₂	—	8	2	no	—
140	D₄₂₄	—	2	2	no	—
141	#11719912	—	4	4	no	—
142	C₂³×C₅₃	—	2	1	no	—
143	C₂²×D₅₃	—	2	2	no	—
144	Q₈×C₅₃	—	2	1	no	—
145	Q₈×C₅₃	C₂×D₈	1	2	yes	—
146	D₄×C₅₃	—	2	1	no	—
147	D₄×C₅₃	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₅₃	—	2	1	no	—
149	C₂×C₄×C₅₃	—	2	1	no	—
150	C₂×C₄×C₅₃	—	2	1	no	—
151	C₂×C₄×C₅₃	—	4	1	no	—
152	C₂×C₄×C₅₃	—	4	1	no	—
153	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²⋊D₅₃	—	8	4	no	—
157	C₂×Dic₅₃	—	2	2	no	—
158	C₂×Dic₅₃	—	4	2	no	—
159	C₂×Dic₅₃	—	4	2	no	—
160	C₂×Dic₅₃	—	8	4	no	—
161	D₂₁₂	—	2	2	no	—
162	D₂₁₂	—	2	2	no	—
163	C₄×D₅₃	—	4	2	no	—
164	C₄×D₅₃	—	4	2	no	—
165	C₄×D₅₃	—	8	4	no	—
166	Dic₁₀₆	—	4	2	no	—
167	Dic₁₀₆	—	4	2	no	—
168	Dic₁₀₆	—	4	2	no	—
169	Dic₁₀₆	—	4	2	no	—
170	Dic₁₀₆	—	8	4	no	—
171	C₈×C₅₃	—	2	2	no	—
172	C₈×C₅₃	—	2	1	no	—
173	C₈×C₅₃	—	2	1	no	—
174	C₈×C₅₃	—	4	1	no	—
175	C₈×C₅₃	—	4	1	no	—
176	C₈×C₅₃	C₂×D₈	1	2	yes	—
177	C₈×C₅₃	#137	1	2	yes	complemented
178	#88322	—	4	2	no	—
179	#19887	—	53	1	no	maximal, Sylow
180	C₂²×C₅₃	—	2	2	no	—
181	C₂²×C₅₃	—	4	1	no	—
182	C₂²×C₅₃	—	4	1	no	—
183	C₂²×C₅₃	D₄²	1	1	yes	characteristic
184	C₂×D₅₃	—	4	2	no	—
185	C₂×D₅₃	—	4	2	no	—
186	C₂×D₅₃	—	4	2	no	—
187	C₄×C₅₃	—	2	2	no	—
188	C₄×C₅₃	—	2	1	no	—
189	C₄×C₅₃	—	4	1	no	—
190	C₄×C₅₃	—	4	1	no	—
191	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
192	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
193	C₄×C₅₃	#576	1	1	yes	characteristic
194	C₄×C₅₃	#578	1	1	yes	characteristic
195	Dic₅₃	—	4	2	no	—
196	Dic₅₃	—	8	2	no	—
197	Dic₅₃	—	8	2	no	—
198	Q₈×D₈	—	53	1	no	—
199	#3334	—	53	1	no	—
200	#3276	—	53	2	no	—
201	#1713	—	53	1	no	—
202	#1689	—	53	2	no	—
203	#1674	—	53	1	no	—
204	#1673	—	53	1	no	—
205	#1649	—	53	2	no	—
206	#1647	—	53	2	no	—
207	#1589	—	53	1	no	—
208	C₈×SD₁₆	D₅₃	1	1	yes	characteristic, complemented
209	C₂×C₅₃	—	4	2	no	—
210	C₂×C₅₃	#3276	1	1	yes	characteristic
211	C₂×C₅₃	D₄×D₈	1	1	yes	characteristic
212	C₂×C₅₃	#3293	1	1	yes	characteristic
213	D₅₃	—	8	2	no	—
214	C₂×#137	—	106	2	no	—
215	D₄×Q₈	—	53	2	no	—
216	#547	—	53	2	no	—
217	#500	—	53	1	no	—
218	#500	—	53	1	no	—
219	#500	—	53	2	no	—
220	#500	—	106	1	no	—
221	#494	—	53	2	no	—
222	#480	—	106	2	no	—
223	#474	—	53	2	no	—
224	#474	—	53	2	no	—
225	#474	—	106	2	no	—
226	#470	—	106	2	no	—
227	#466	—	106	1	no	—
228	#462	—	53	2	no	—
229	#461	—	53	2	no	—
230	#452	—	106	2	no	—
231	#451	—	106	2	no	—
232	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
233	C₄×SD₁₆	C₂×D₅₃	1	1	yes	characteristic
234	C₄×D₈	—	53	1	no	—
235	C₄×D₈	—	106	1	no	—
236	#436	—	53	2	no	—
237	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
238	C₂×#107	—	106	1	no	—
239	#335	—	53	1	no	—
240	#334	C₂×D₅₃	1	1	yes	characteristic
241	#332	—	53	2	no	—
242	#332	—	53	2	no	—
243	Q₈⋊C₈	C₂×D₅₃	1	1	yes	characteristic
244	D₄⋊C₈	—	53	2	no	—
245	D₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
246	C₈²	C₂×D₅₃	1	1	yes	characteristic
247	C₅₃	#19887	1	1	yes	characteristic, complemented, Sylow
248	C₂×#41	—	106	2	no	—
249	C₂²×Q₈	—	106	2	no	—
250	#137	—	212	8	no	—
251	C₂×Q₁₆	—	106	2	no	—
252	C₂×Q₁₆	—	106	2	no	—
253	C₂×Q₁₆	—	106	2	no	—
254	C₂×SD₁₆	—	106	2	no	—
255	C₂×SD₁₆	D₂₁₂	1	1	yes	characteristic
256	C₂×D₈	—	53	1	no	—
257	C₂×M₁₆	—	106	2	no	—
258	C₂²×C₈	—	2	1	no	—
259	C₄⋊Q₈	—	53	2	no	—
260	C₄⋊Q₈	—	53	2	no	—
261	C₄⋊Q₈	—	106	2	no	—
262	#123	—	212	2	no	—
263	C₂²⋊Q₈	—	106	2	no	—
264	C₂²⋊Q₈	—	106	2	no	—
265	C₂²⋊Q₈	—	212	2	no	—
266	#121	—	106	2	no	—
267	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
268	C₄×D₄	—	53	2	no	—
269	C₄×D₄	—	106	2	no	—
270	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
271	C₂×C₄⋊C₄	—	106	2	no	—
272	#107	—	53	1	no	—
273	#107	—	106	2	no	—
274	#107	—	106	2	no	—
275	#107	—	106	1	no	—
276	#107	—	106	1	no	—
277	#107	—	106	1	no	—
278	#107	—	106	1	no	—
279	#107	—	212	2	no	—
280	#106	D₂₁₂	1	1	yes	characteristic
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	53	2	no	—
283	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
284	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
285	#103	—	106	2	no	—
286	#103	—	106	2	no	—
287	#103	D₂₁₂	1	1	yes	characteristic
288	#102	—	53	2	no	—
289	#102	—	106	2	no	—
290	#102	—	106	2	no	—
291	#102	D₂₁₂	1	1	yes	characteristic
292	C₂²⋊C₈	—	2	1	no	—
293	C₂²⋊C₈	—	106	2	no	—
294	C₄×C₈	—	2	1	no	—
295	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
296	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
297	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
298	#41	—	424	2	no	—
299	#41	—	212	4	no	—
300	C₂×Q₈	—	106	2	no	—
301	C₂×Q₈	—	106	2	no	—
302	C₂×Q₈	—	424	2	no	—
303	C₂×Q₈	—	212	2	no	—
304	C₂×Q₈	—	212	4	no	—
305	C₂×Q₈	C₂×D₂₁₂	1	1	yes	characteristic
306	C₂×D₄	—	53	2	no	—
307	C₂×D₄	—	212	2	no	—
308	C₂×D₄	C₂×D₂₁₂	1	1	yes	characteristic
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	106	2	no	—
311	C₂²×C₄	—	212	2	no	—
312	C₂²×C₄	—	212	2	no	—
313	Q₁₆	—	424	2	no	—
314	Q₁₆	—	212	4	no	—
315	Q₁₆	—	212	4	no	—
316	SD₁₆	—	212	4	no	—
317	SD₁₆	D₄₂₄	1	4	yes	complemented
318	D₈	—	106	2	no	—
319	M₁₆	—	212	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	—	106	2	no	—
325	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
326	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
327	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
328	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	53	2	no	—
331	C₄⋊C₄	—	106	2	no	—
332	C₄⋊C₄	—	106	2	no	—
333	C₄⋊C₄	—	106	2	no	—
334	C₄⋊C₄	—	106	2	no	—
335	C₄⋊C₄	—	212	2	no	—
336	C₄⋊C₄	—	212	2	no	—
337	C₄⋊C₄	C₂×D₂₁₂	1	1	yes	characteristic
338	C₄⋊C₄	C₂×D₂₁₂	1	1	yes	characteristic
339	C₂²⋊C₄	—	2	1	no	—
340	C₂²⋊C₄	—	106	2	no	—
341	C₂²⋊C₄	—	212	2	no	—
342	C₂²⋊C₄	—	212	2	no	—
343	C₄²	—	2	1	no	—
344	C₄²	C₂³×D₅₃	1	1	yes	characteristic
345	C₂³	—	2	1	no	—
346	C₂³	—	106	2	no	—
347	Q₈	—	2	1	no	—
348	Q₈	—	424	4	no	—
349	Q₈	—	212	2	no	—
350	Q₈	—	212	2	no	—
351	Q₈	—	212	2	no	—
352	Q₈	—	212	2	no	—
353	Q₈	C₂×D₄₂₄	1	2	yes	—
354	D₄	—	2	1	no	—
355	D₄	—	106	2	no	—
356	D₄	—	106	2	no	—
357	D₄	—	424	4	no	—
358	D₄	C₂×D₄₂₄	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	—	106	2	no	—
365	C₂×C₄	—	424	4	no	—
366	C₂×C₄	—	424	4	no	—
367	C₂×C₄	—	212	2	no	—
368	C₂×C₄	—	212	2	no	—
369	C₂×C₄	—	212	2	no	—
370	C₂×C₄	—	212	2	no	—
371	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
372	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
373	C₂×C₄	#13133263	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	—	212	2	no	—
380	C₈	C₂×D₄₂₄	1	2	yes	—
381	C₈	#13133246	1	2	yes	complemented
382	C₂²	—	2	2	no	—
383	C₂²	—	4	1	no	—
384	C₂²	—	4	1	no	—
385	C₂²	—	212	2	no	—
386	C₂²	—	212	2	no	—
387	C₂²	—	212	2	no	—
388	C₂²	D₄×D₂₁₂	1	1	yes	characteristic
389	C₄	—	2	1	no	—
390	C₄	—	2	2	no	—
391	C₄	—	4	1	no	—
392	C₄	—	4	1	no	—
393	C₄	—	424	2	no	—
394	C₄	—	424	2	no	—
395	C₄	—	212	2	no	—
396	C₄	C₂²×D₄₂₄	1	1	yes	characteristic
397	C₄	C₂×#13133246	1	1	yes	characteristic
398	C₄	#16811428	1	1	yes	characteristic
399	C₄	#16811437	1	1	yes	characteristic
400	C₂	—	4	2	no	—
401	C₂	—	424	2	no	—
402	C₂	#20286362	1	1	yes	characteristic
403	C₂	D₄×D₄₂₄	1	1	yes	characteristic
404	C₂	#20286520	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	#20287063	C₂
hypercenter	C₈×SD₁₆	D₅₃
hypocenter	C₅₃	#19887
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. #20286362	C₂	semidirect
4. #20274333	C₂	semidirect
5. #20274205	C₂	semidirect
6. #20274109	C₂	semidirect
7. #20274104	C₂	semidirect
8. #20273966	C₂	semidirect
9. #20273952	C₂	semidirect
10. #20273601	C₂	semidirect
11. C₈×SD₁₆×C₅₃	C₂	semidirect
110. SD₁₆×C₅₃	D₈	semidirect
177. C₈×C₅₃	#137	semidirect
208. C₈×SD₁₆	D₅₃	semidirect
247. C₅₃	#19887	semidirect
317. SD₁₆	D₄₂₄	semidirect
381. C₈	#13133246	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
#20287063	2
#20286362	2
#20274333	2
#20274205	2
#20274109	2
#20274104	2
#20273966	2
#20273952	2
#20273601	2
C₈×SD₁₆×C₅₃	2
#19887	53