Subgroups — #27368036

Order 13568.

608 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285226	C₂	1	1	yes	maximal, characteristic, complemented
2	#20284876	C₂	1	2	yes	maximal
3	C₈×#11719949	C₂	1	1	yes	maximal, characteristic
4	#20274119	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274115	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274101	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274095	C₂	1	2	yes	maximal, complemented
8	#20274089	C₂	1	2	yes	maximal, complemented
9	#20273477	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273422	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1674	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811178	C₂²	1	2	yes	—
13	C₄×#11719949	C₂²	1	1	yes	characteristic
14	#16811028	C₂²	1	1	yes	characteristic
15	#16811024	—	2	1	no	—
16	#16811024	—	2	1	no	—
17	#16811019	C₂²	1	2	yes	—
18	#16810962	—	2	2	no	—
19	#16810918	C₂²	1	2	yes	—
20	#16810914	C₂²	1	2	yes	—
21	#16810825	C₂²	1	2	yes	—
22	#16810784	—	2	2	no	—
23	Q₈×#88322	C₂²	1	1	yes	characteristic
24	#16810730	—	2	2	no	—
25	#16810726	—	2	2	no	—
26	C₈×D₂₁₂	—	2	1	no	—
27	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
28	#16810688	—	2	1	no	—
29	#16810688	C₂²	1	1	yes	characteristic
30	#16810679	C₂²	1	2	yes	—
31	#16810675	—	2	1	no	—
32	#16810675	C₂²	1	1	yes	characteristic
33	#16810660	—	2	1	no	—
34	#16810660	—	2	1	no	—
35	C₄×C₈×D₅₃	—	2	1	no	—
36	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
37	#16810376	C₂²	1	1	yes	characteristic
38	#16810373	C₂²	1	2	yes	—
39	#16810371	C₂²	1	2	yes	—
40	#16810371	C₂²	1	2	yes	—
41	#16810369	C₂²	1	2	yes	—
42	C₅₃×#500	C₂²	1	1	yes	characteristic
43	C₅₃×#462	C₂²	1	2	yes	—
44	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#335	C₂²	1	1	yes	characteristic
46	C₅₃×#332	C₂²	1	2	yes	—
47	C₂×#11719949	D₄	1	1	yes	characteristic
48	C₂×#11719924	—	2	2	no	—
49	C₂×C₈×D₅₃	—	2	1	no	—
50	C₂×C₈×D₅₃	—	2	1	no	—
51	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
52	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
53	#13133217	C₂³	1	2	yes	—
54	#13133208	—	2	2	no	—
55	#13133197	—	2	2	no	—
56	#13133193	—	2	2	no	—
57	Q₈×Dic₅₃	—	2	2	no	—
58	Q₈×Dic₅₃	D₄	1	1	yes	characteristic
59	#13133182	—	2	2	no	—
60	C₄×D₂₁₂	—	2	1	no	—
61	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
62	#13133179	—	2	1	no	—
63	#13133179	D₄	1	1	yes	characteristic
64	#13133175	—	2	1	no	—
65	#13133175	—	2	2	no	—
66	#13133175	D₄	1	1	yes	characteristic
67	C₄²×D₅₃	—	2	1	no	—
68	C₄²×D₅₃	C₂³	1	1	yes	characteristic
69	#13133151	—	2	1	no	—
70	#13133151	—	2	1	no	—
71	#13133150	—	2	2	no	—
72	#13133149	—	2	2	no	—
73	#13133148	—	2	2	no	—
74	#13133148	—	2	2	no	—
75	C₄⋊#88322	—	2	1	no	—
76	C₄⋊#88322	C₂³	1	2	yes	—
77	C₄⋊#88322	C₂³	1	1	yes	characteristic
78	#13133141	—	2	2	no	—
79	#13133141	—	2	2	no	—
80	#13133139	—	2	2	no	—
81	#13133129	—	2	1	no	—
82	#13133129	—	2	1	no	—
83	#13133129	—	2	2	no	—
84	#13133129	—	2	1	no	—
85	#13133127	—	2	2	no	—
86	C₄×#88322	—	2	1	no	—
87	C₄×#88322	C₂³	1	1	yes	characteristic
88	C₈×Dic₅₃	—	2	1	no	—
89	C₈×Dic₅₃	D₄	1	1	yes	characteristic
90	C₈×Dic₅₃	D₄	1	1	yes	characteristic
91	C₂×Q₁₆×C₅₃	—	2	2	no	—
92	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
93	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
94	#107×C₅₃	—	2	1	no	—
95	#107×C₅₃	—	2	1	no	—
96	C₄⋊C₈×C₅₃	—	2	1	no	—
97	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
99	#103×C₅₃	—	2	2	no	—
100	C₄×C₈×C₅₃	—	2	1	no	—
101	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₅₃	—	2	2	no	—
103	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
104	#11719949	—	2	2	no	—
105	#11719949	—	2	2	no	—
106	C₂×D₂₁₂	—	2	1	no	—
107	C₂×D₂₁₂	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×D₅₃	—	2	1	no	—
109	C₂×C₄×D₅₃	—	2	1	no	—
110	C₂×C₄×D₅₃	—	2	1	no	—
111	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
114	C₂×Dic₁₀₆	—	2	2	no	—
115	Q₁₆×C₅₃	—	4	4	no	—
116	C₂×C₈×C₅₃	—	2	1	no	—
117	C₂×C₈×C₅₃	—	2	2	no	—
118	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₄⋊C₄×C₅₃	—	2	1	no	—
121	C₄⋊C₄×C₅₃	—	2	2	no	—
122	C₄⋊C₄×C₅₃	—	2	2	no	—
123	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₅₃	—	2	1	no	—
125	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
126	#11719924	—	4	4	no	—
127	#11719921	—	2	1	no	—
128	#11719921	—	2	1	no	—
129	#11719921	—	2	1	no	—
130	#11719921	—	4	2	no	—
131	C₄⋊Dic₅₃	—	2	1	no	—
132	C₄⋊Dic₅₃	—	2	2	no	—
133	C₄⋊Dic₅₃	—	2	2	no	—
134	C₄⋊Dic₅₃	—	4	2	no	—
135	C₄⋊Dic₅₃	C₂×D₄	1	1	yes	characteristic
136	#11719919	—	4	2	no	—
137	C₄×Dic₅₃	—	2	1	no	—
138	C₄×Dic₅₃	—	2	1	no	—
139	C₄×Dic₅₃	—	4	2	no	—
140	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
141	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
142	C₄×Dic₅₃	#41	1	1	yes	characteristic
143	C₂×#88322	—	2	1	no	—
144	C₂×#88322	—	2	2	no	—
145	C₂×#88322	C₂×D₄	1	1	yes	characteristic
146	C₂×#88322	C₂×D₄	1	1	yes	characteristic
147	C₈×D₅₃	—	2	2	no	—
148	C₈×D₅₃	—	2	2	no	—
149	C₈×D₅₃	—	4	2	no	—
150	C₈×D₅₃	—	4	2	no	—
151	C₂²×D₅₃	—	2	1	no	—
152	C₂²×D₅₃	#121	1	1	yes	characteristic
153	Q₈×C₅₃	—	2	1	no	—
154	Q₈×C₅₃	—	4	2	no	—
155	Q₈×C₅₃	—	4	2	no	—
156	Q₈×C₅₃	#135	1	2	yes	—
157	C₂×C₄×C₅₃	—	2	1	no	—
158	C₂×C₄×C₅₃	—	2	1	no	—
159	C₂×C₄×C₅₃	—	4	2	no	—
160	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
163	C₂×Dic₅₃	—	2	1	no	—
164	C₂×Dic₅₃	—	4	2	no	—
165	C₂×Dic₅₃	#121	1	1	yes	characteristic
166	C₂×Dic₅₃	#121	1	1	yes	characteristic
167	C₂×Dic₅₃	#121	1	1	yes	characteristic
168	D₂₁₂	—	2	4	no	—
169	D₂₁₂	—	2	1	no	—
170	D₂₁₂	#135	1	2	yes	—
171	C₄×D₅₃	—	2	4	no	—
172	C₄×D₅₃	—	2	1	no	—
173	C₄×D₅₃	—	2	1	no	—
174	C₄×D₅₃	—	2	1	no	—
175	C₄×D₅₃	—	2	1	no	—
176	C₄×D₅₃	—	4	1	no	—
177	C₄×D₅₃	—	4	2	no	—
178	C₄×D₅₃	—	4	1	no	—
179	C₄×D₅₃	—	4	1	no	—
180	Dic₁₀₆	—	8	2	no	—
181	C₈×C₅₃	—	2	1	no	—
182	C₈×C₅₃	—	4	2	no	—
183	C₈×C₅₃	—	4	1	no	—
184	C₈×C₅₃	#137	1	2	yes	—
185	#88322	—	2	1	no	—
186	#88322	—	4	2	no	—
187	#88322	—	4	1	no	—
188	#88322	#137	1	2	yes	—
189	#19273	—	53	1	no	maximal, Sylow
190	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
191	C₂×D₅₃	—	2	1	no	—
192	C₂×D₅₃	—	2	1	no	—
193	C₂×D₅₃	—	2	2	no	—
194	C₂×D₅₃	—	2	1	no	—
195	C₂×D₅₃	—	4	1	no	—
196	C₂×D₅₃	—	4	1	no	—
197	C₄×C₅₃	—	2	2	no	—
198	C₄×C₅₃	—	2	1	no	—
199	C₄×C₅₃	—	4	1	no	—
200	C₄×C₅₃	—	8	2	no	—
201	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
202	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
203	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
204	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
205	Dic₅₃	—	2	1	no	—
206	Dic₅₃	—	2	1	no	—
207	Dic₅₃	—	2	1	no	—
208	Dic₅₃	—	4	1	no	—
209	Dic₅₃	—	8	2	no	—
210	#3160	—	53	1	no	—
211	#3116	—	53	2	no	—
212	C₈×#41	—	53	1	no	—
213	#1676	—	53	1	no	—
214	#1675	—	53	1	no	—
215	#1674	D₅₃	1	1	yes	characteristic, complemented
216	#1673	—	53	1	no	—
217	#1672	—	53	2	no	—
218	#1671	—	53	2	no	—
219	#1568	—	53	1	no	—
220	#1554	—	53	2	no	—
221	C₂×C₅₃	#3039	1	1	yes	characteristic
222	C₂×C₅₃	#3050	1	1	yes	characteristic
223	C₂×C₅₃	#3067	1	1	yes	characteristic
224	D₅₃	—	4	1	no	—
225	D₅₃	—	4	2	no	—
226	#533	—	53	2	no	—
227	C₄×#41	—	53	1	no	—
228	#500	—	53	1	no	—
229	#500	C₂×D₅₃	1	1	yes	characteristic
230	#499	—	106	1	no	—
231	#499	—	106	1	no	—
232	#498	—	53	2	no	—
233	#485	—	106	2	no	—
234	#477	—	53	2	no	—
235	#476	—	53	2	no	—
236	#462	C₂×D₅₃	1	2	yes	—
237	#461	—	53	2	no	—
238	#452	—	106	2	no	—
239	C₈×Q₈	—	53	1	no	—
240	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
241	#441	—	106	2	no	—
242	#440	—	106	2	no	—
243	C₈×D₄	—	53	1	no	—
244	C₈×D₄	—	53	1	no	—
245	C₈×D₄	—	106	1	no	—
246	C₈×D₄	—	106	1	no	—
247	#432	—	53	2	no	—
248	#431	—	53	1	no	—
249	#431	—	106	1	no	—
250	#427	—	106	1	no	—
251	#427	—	106	1	no	—
252	C₂×C₄×C₈	—	53	1	no	—
253	C₂×C₄×C₈	—	106	1	no	—
254	#335	—	53	1	no	—
255	#335	C₂×D₅₃	1	1	yes	characteristic
256	#334	—	53	2	no	—
257	#333	—	53	2	no	—
258	#333	—	53	2	no	—
259	#332	—	53	2	no	—
260	#332	C₂×D₅₃	1	2	yes	—
261	C₅₃	#19273	1	1	yes	characteristic, complemented, Sylow
262	C₂×#41	—	53	1	no	—
263	C₂×Q₁₆	—	2	2	no	—
264	C₂×SD₁₆	—	106	2	no	—
265	C₂²×C₈	—	53	1	no	—
266	C₂²×C₈	—	53	1	no	—
267	C₂²×C₈	—	106	1	no	—
268	C₂²×C₈	—	106	1	no	—
269	C₄⋊Q₈	—	53	2	no	—
270	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
271	#125	—	106	2	no	—
272	C₂²⋊Q₈	—	106	2	no	—
273	C₂²⋊Q₈	—	106	2	no	—
274	C₄×Q₈	—	53	1	no	—
275	C₄×Q₈	—	106	2	no	—
276	C₄×Q₈	—	106	2	no	—
277	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
278	C₄×D₄	—	53	1	no	—
279	C₄×D₄	—	53	1	no	—
280	C₄×D₄	—	106	1	no	—
281	C₄×D₄	—	106	1	no	—
282	#117	—	53	1	no	—
283	#117	—	106	1	no	—
284	#117	—	106	2	no	—
285	C₂×C₄²	—	53	1	no	—
286	C₂×C₄²	—	106	1	no	—
287	#107	—	2	1	no	—
288	#107	—	2	1	no	—
289	#107	—	106	1	no	—
290	#107	—	106	1	no	—
291	#107	—	106	2	no	—
292	#106	—	106	2	no	—
293	#106	—	106	2	no	—
294	#106	—	106	2	no	—
295	C₄⋊C₈	—	2	1	no	—
296	C₄⋊C₈	—	53	1	no	—
297	C₄⋊C₈	—	53	2	no	—
298	C₄⋊C₈	—	106	1	no	—
299	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
300	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
301	#103	—	2	2	no	—
302	#103	—	106	2	no	—
303	#103	—	106	2	no	—
304	#102	—	106	2	no	—
305	C₂²⋊C₈	—	106	1	no	—
306	C₂²⋊C₈	—	106	1	no	—
307	C₂²⋊C₈	—	106	2	no	—
308	C₂²⋊C₈	—	106	1	no	—
309	#97	—	106	2	no	—
310	C₄×C₈	—	2	1	no	—
311	C₄×C₈	—	53	1	no	—
312	C₄×C₈	—	53	1	no	—
313	C₄×C₈	—	53	1	no	—
314	C₄×C₈	—	106	1	no	—
315	C₄×C₈	—	106	1	no	—
316	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
317	#41	—	106	2	no	—
318	#41	—	106	2	no	—
319	C₂×Q₈	—	2	2	no	—
320	C₂×Q₈	—	106	2	no	—
321	C₂×Q₈	D₄×D₅₃	1	1	yes	characteristic
322	C₂×D₄	—	53	1	no	—
323	C₂×D₄	—	106	1	no	—
324	C₂²×C₄	—	53	1	no	—
325	C₂²×C₄	—	53	1	no	—
326	C₂²×C₄	—	53	1	no	—
327	C₂²×C₄	—	106	1	no	—
328	C₂²×C₄	—	106	1	no	—
329	C₂²×C₄	—	106	1	no	—
330	Q₁₆	—	4	4	no	—
331	SD₁₆	—	212	4	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	53	1	no	—
335	C₂×C₈	—	53	1	no	—
336	C₂×C₈	—	106	2	no	—
337	C₂×C₈	—	106	2	no	—
338	C₂×C₈	—	106	1	no	—
339	C₂×C₈	—	106	2	no	—
340	C₂×C₈	—	212	2	no	—
341	C₂×C₈	—	212	2	no	—
342	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
343	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	53	1	no	—
348	C₄⋊C₄	—	106	1	no	—
349	C₄⋊C₄	—	106	2	no	—
350	C₄⋊C₄	—	106	2	no	—
351	C₄⋊C₄	—	212	2	no	—
352	C₄⋊C₄	—	212	2	no	—
353	C₄⋊C₄	D₄×D₅₃	1	1	yes	characteristic
354	C₂²⋊C₄	—	106	1	no	—
355	C₂²⋊C₄	—	106	1	no	—
356	C₂²⋊C₄	—	106	1	no	—
357	C₂²⋊C₄	—	212	2	no	—
358	C₄²	—	2	1	no	—
359	C₄²	—	53	1	no	—
360	C₄²	—	53	1	no	—
361	C₄²	—	53	1	no	—
362	C₄²	—	106	1	no	—
363	C₄²	—	106	1	no	—
364	C₄²	—	212	2	no	—
365	C₄²	C₂³×D₅₃	1	1	yes	characteristic
366	C₂³	—	53	1	no	—
367	C₂³	—	106	1	no	—
368	Q₈	—	2	1	no	—
369	Q₈	—	4	2	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	424	2	no	—
372	Q₈	#13133236	1	2	yes	—
373	D₄	—	53	2	no	—
374	D₄	—	106	4	no	—
375	D₄	—	106	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	53	1	no	—
380	C₂×C₄	—	53	1	no	—
381	C₂×C₄	—	53	1	no	—
382	C₂×C₄	—	106	4	no	—
383	C₂×C₄	—	106	1	no	—
384	C₂×C₄	—	106	1	no	—
385	C₂×C₄	—	106	1	no	—
386	C₂×C₄	—	106	1	no	—
387	C₂×C₄	—	106	1	no	—
388	C₂×C₄	—	212	1	no	—
389	C₂×C₄	—	212	2	no	—
390	C₂×C₄	—	212	2	no	—
391	C₂×C₄	—	212	1	no	—
392	C₂×C₄	—	212	1	no	—
393	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
394	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
395	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
396	C₈	—	2	1	no	—
397	C₈	—	4	2	no	—
398	C₈	—	4	1	no	—
399	C₈	—	53	2	no	—
400	C₈	—	106	1	no	—
401	C₈	—	212	2	no	—
402	C₈	—	212	1	no	—
403	C₈	#13133249	1	2	yes	—
404	C₂²	—	106	1	no	—
405	C₂²	—	106	1	no	—
406	C₂²	—	106	2	no	—
407	C₂²	—	106	1	no	—
408	C₂²	—	212	1	no	—
409	C₂²	—	212	1	no	—
410	C₂²	#121×D₅₃	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	2	2	no	—
413	C₄	—	4	1	no	—
414	C₄	—	8	2	no	—
415	C₄	—	106	1	no	—
416	C₄	—	106	1	no	—
417	C₄	—	106	1	no	—
418	C₄	—	424	2	no	—
419	C₄	—	212	1	no	—
420	C₄	#135×D₅₃	1	1	yes	characteristic
421	C₄	C₂×#13133236	1	1	yes	characteristic
422	C₄	#137×D₅₃	1	1	yes	characteristic
423	C₄	C₂×#13133249	1	1	yes	characteristic
424	C₂	—	212	1	no	—
425	C₂	—	212	2	no	—
426	C₂	#20284199	1	1	yes	characteristic
427	C₂	#20284294	1	1	yes	characteristic
428	C₂	#20284437	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1674	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂×#11719949	D₄
hypercenter	#1674	D₅₃
hypocenter	C₅₃	#19273
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1674

Complements

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

N	H ≅ G/N	type
1. #20285226	C₂	semidirect
4. #20274119	C₂	semidirect
5. #20274115	C₂	semidirect
6. #20274101	C₂	semidirect
7. #20274095	C₂	semidirect
8. #20274089	C₂	semidirect
9. #20273477	C₂	semidirect
10. #20273422	C₂	semidirect
11. C₅₃×#1674	C₂	semidirect
215. #1674	D₅₃	semidirect
261. C₅₃	#19273	semidirect

Maximal subgroups

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

subgroup	index
#20285226	2
#20284876	2
C₈×#11719949	2
#20274119	2
#20274115	2
#20274101	2
#20274095	2
#20274089	2
#20273477	2
#20273422	2
C₅₃×#1674	2
#19273	53