Subgroups — #27402621

Order 13568.

808 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283633	C₂	1	1	yes	maximal, characteristic
2	#20283631	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810744	C₂	1	1	yes	maximal, characteristic
4	#20283325	C₂	1	2	yes	maximal, complemented
5	#20275506	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274958	C₂	1	2	yes	maximal, complemented
7	#20274952	C₂	1	2	yes	maximal, complemented
8	#20274888	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274768	C₂	1	2	yes	maximal, complemented
10	#20274600	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1953	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133239	C₂²	1	1	yes	characteristic
13	#16811389	C₂²	1	2	yes	—
14	#16811389	C₂²	1	2	yes	—
15	C₂×#13133225	C₂²	1	2	yes	—
16	#16810746	—	2	2	no	—
17	#16810744	—	2	2	no	—
18	#16810744	C₄	1	4	yes	complemented
19	#16810744	C₄	1	4	yes	complemented
20	#16810741	—	2	2	no	—
21	#16810739	—	2	2	no	—
22	#16810672	C₂²	1	2	yes	—
23	#16810670	—	2	4	no	—
24	C₂×#13133153	C₂²	1	1	yes	characteristic
25	C₂×#13133152	C₂²	1	2	yes	—
26	C₂×#13133152	C₂²	1	1	yes	characteristic
27	#108×D₅₃	—	2	4	no	—
28	#16810659	C₂²	1	1	yes	characteristic
29	#16810646	C₂²	1	2	yes	—
30	#16810641	C₂²	1	2	yes	—
31	#16810639	C₂²	1	2	yes	—
32	C₂×#13133143	C₂²	1	2	yes	—
33	#16810578	C₂²	1	1	yes	characteristic
34	C₂×#13133127	C₂²	1	1	yes	characteristic
35	#16810395	C₂²	1	2	yes	—
36	#16810384	C₂²	1	1	yes	characteristic
37	#16810382	C₂²	1	1	yes	characteristic
38	C₅₃×#430	C₂²	1	2	yes	—
39	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#405	C₂²	1	1	yes	characteristic
42	C₅₃×#344	C₂²	1	2	yes	—
43	C₂×#11719945	C₂³	1	2	yes	—
44	#13133239	C₂×C₄	1	2	yes	—
45	#13133239	C₂×C₄	1	2	yes	—
46	#13133239	D₄	1	2	yes	—
47	#13133239	D₄	1	2	yes	—
48	C₂×Dic₂₁₂	Q₈	1	1	yes	characteristic
49	C₂×#11719913	Q₈	1	2	yes	—
50	C₂×D₄₂₄	Q₈	1	1	yes	characteristic
51	#13133225	—	2	2	no	—
52	#13133225	—	2	2	no	—
53	#13133225	—	2	4	no	—
54	#13133225	—	4	2	no	—
55	#13133225	C₂×C₄	1	4	yes	—
56	#13133225	C₂×C₄	1	4	yes	—
57	#13133223	—	2	4	no	—
58	#13133223	—	4	2	no	—
59	C₂×#11719917	C₂³	1	2	yes	—
60	C₂×#11719917	C₂³	1	1	yes	characteristic
61	C₂×#11719912	—	2	2	no	—
62	M₁₆×D₅₃	—	2	4	no	—
63	M₁₆×D₅₃	—	2	4	no	—
64	C₂²×#88322	C₂³	1	1	yes	characteristic
65	C₂×C₈×D₅₃	—	2	2	no	—
66	#13133176	C₂³	1	1	yes	characteristic
67	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
68	#13133153	—	2	1	no	—
69	#13133153	—	2	1	no	—
70	#13133153	—	2	4	no	—
71	#13133152	—	2	2	no	—
72	#13133152	—	2	2	no	—
73	#13133152	D₄	1	4	yes	—
74	#13133150	—	2	1	no	—
75	#13133148	—	2	1	no	—
76	#13133146	—	4	2	no	—
77	#13133145	—	2	4	no	—
78	#13133143	—	2	4	no	—
79	#13133143	C₂×C₄	1	8	yes	—
80	#13133127	—	2	1	no	—
81	#13133127	C₂×C₄	1	2	yes	—
82	#13133127	C₂×C₄	1	2	yes	—
83	C₈×Dic₅₃	—	2	1	no	—
84	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
85	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
86	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
87	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
88	#117×C₅₃	C₂³	1	1	yes	characteristic
89	#108×C₅₃	—	2	4	no	—
90	#108×C₅₃	—	2	1	no	—
91	#108×C₅₃	—	2	1	no	—
92	#108×C₅₃	C₂×C₄	1	2	yes	—
93	#108×C₅₃	C₂×C₄	1	2	yes	—
94	#97×C₅₃	—	2	1	no	—
95	C₄×C₈×C₅₃	—	2	1	no	—
96	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
97	C₂×C₂²⋊D₅₃	—	2	2	no	—
98	C₂²×Dic₅₃	—	2	1	no	—
99	#11719945	—	2	2	no	—
100	#11719945	—	2	2	no	—
101	#11719945	C₂²×C₄	1	4	yes	—
102	#11719945	C₂×D₄	1	4	yes	—
103	C₂×D₂₁₂	C₂×Q₈	1	2	yes	—
104	C₂×C₄×D₅₃	—	2	2	no	—
105	C₂×Dic₁₀₆	C₂×Q₈	1	2	yes	—
106	M₁₆×C₅₃	—	2	2	no	—
107	M₁₆×C₅₃	—	2	1	no	—
108	M₁₆×C₅₃	—	2	2	no	—
109	M₁₆×C₅₃	—	2	2	no	—
110	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
111	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₅₃	—	2	1	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	—	2	2	no	—
115	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₅₃	C₂²×C₄	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	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
120	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
121	C₄⋊C₄×C₅₃	—	2	1	no	—
122	C₂²⋊C₄×C₅₃	—	2	1	no	—
123	C₄²×C₅₃	—	2	1	no	—
124	C₂²⋊Dic₅₃	—	2	1	no	—
125	C₄⋊Dic₅₃	—	2	1	no	—
126	C₄×Dic₅₃	—	2	1	no	—
127	C₄×Dic₅₃	—	2	1	no	—
128	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
129	#11719917	—	2	1	no	—
130	#11719917	—	2	2	no	—
131	#11719917	—	2	2	no	—
132	#11719917	C₂×D₄	1	2	yes	—
133	C₂×#88322	—	2	1	no	—
134	C₂×#88322	—	2	2	no	—
135	C₂×#88322	C₂²×C₄	1	2	yes	—
136	C₂×#88322	C₂×D₄	1	2	yes	—
137	C₂×#88322	#41	1	1	yes	characteristic
138	C₂×#88322	#41	1	1	yes	characteristic
139	Dic₂₁₂	C₄⋊C₄	1	2	yes	—
140	Dic₂₁₂	C₄⋊C₄	1	2	yes	—
141	D₄₂₄	C₄⋊C₄	1	2	yes	—
142	D₄₂₄	C₄⋊C₄	1	2	yes	—
143	#11719913	C₄⋊C₄	1	4	yes	—
144	#11719913	C₄⋊C₄	1	4	yes	—
145	#11719912	—	2	4	no	—
146	#11719912	—	2	4	no	—
147	#11719912	—	4	4	no	—
148	#11719912	—	4	4	no	—
149	C₈×D₅₃	—	2	4	no	—
150	C₈×D₅₃	—	2	4	no	—
151	C₈×D₅₃	—	4	4	no	—
152	C₈×D₅₃	—	4	4	no	—
153	C₂³×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
154	C₂²×D₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	4	1	no	—
156	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
161	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂²⋊D₅₃	—	2	4	no	—
163	C₂²⋊D₅₃	—	2	4	no	—
164	C₂×Dic₅₃	—	2	2	no	—
165	C₂×Dic₅₃	—	2	2	no	—
166	C₂×Dic₅₃	—	4	1	no	—
167	C₂×Dic₅₃	—	8	1	no	—
168	D₂₁₂	—	2	2	no	—
169	D₂₁₂	C₂×C₄⋊C₄	1	4	yes	—
170	C₄×D₅₃	—	2	4	no	—
171	C₄×D₅₃	—	2	4	no	—
172	Dic₁₀₆	—	2	2	no	—
173	Dic₁₀₆	C₂×C₄⋊C₄	1	4	yes	—
174	C₈×C₅₃	—	2	4	no	—
175	C₈×C₅₃	—	2	2	no	—
176	C₈×C₅₃	—	4	2	no	—
177	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
181	#88322	—	2	2	no	—
182	#88322	—	4	2	no	—
183	#88322	C₄×D₄	1	2	yes	—
184	#88322	C₄×D₄	1	2	yes	—
185	#22695	—	53	1	no	maximal, Sylow
186	C₂²×C₅₃	—	4	1	no	—
187	C₂²×C₅₃	#520	1	1	yes	characteristic
188	C₂²×C₅₃	#547	1	1	yes	characteristic
189	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
190	C₂×D₅₃	—	2	4	no	—
191	C₂×D₅₃	—	8	2	no	—
192	C₄×C₅₃	—	8	1	no	—
193	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
194	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
195	C₄×C₅₃	D₄²	1	1	yes	characteristic
196	C₄×C₅₃	#554	1	1	yes	characteristic
197	Dic₅₃	—	2	4	no	—
198	Dic₅₃	—	4	2	no	—
199	Dic₅₃	—	8	1	no	—
200	#2957	—	53	1	no	—
201	#2957	—	53	1	no	—
202	C₂×#444	—	53	1	no	—
203	#2916	—	53	2	no	—
204	#1953	—	53	1	no	—
205	#1953	D₅₃	1	1	yes	characteristic, complemented
206	#1861	—	53	2	no	—
207	#1861	—	53	2	no	—
208	#1849	—	53	1	no	—
209	#1814	—	53	2	no	—
210	#1773	—	53	1	no	—
211	C₂×C₅₃	—	2	1	no	—
212	C₂×C₅₃	—	2	1	no	—
213	C₂×C₅₃	—	2	1	no	—
214	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
215	D₅₃	—	4	4	no	—
216	C₂×#135	—	53	1	no	—
217	#568	—	53	2	no	—
218	#568	—	53	2	no	—
219	C₂×#131	—	53	2	no	—
220	#444	—	53	4	no	—
221	#444	—	53	4	no	—
222	#444	—	106	2	no	—
223	#444	—	106	2	no	—
224	#443	—	106	2	no	—
225	#443	—	106	2	no	—
226	#430	—	53	2	no	—
227	#430	—	106	4	no	—
228	#430	C₂×D₅₃	1	2	yes	—
229	C₂×#108	—	53	1	no	—
230	C₂×#108	—	53	2	no	—
231	C₂×#108	—	53	1	no	—
232	C₂×#108	—	106	4	no	—
233	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
234	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
235	#427	—	53	1	no	—
236	#423	—	53	2	no	—
237	#421	—	53	2	no	—
238	#421	—	53	2	no	—
239	C₂×#104	—	53	2	no	—
240	#405	—	53	1	no	—
241	#405	C₂×D₅₃	1	1	yes	characteristic
242	C₂×#97	—	53	1	no	—
243	#344	—	53	2	no	—
244	#344	C₂×D₅₃	1	2	yes	—
245	#339	—	53	1	no	—
246	#339	—	53	1	no	—
247	C₅₃	#22695	1	1	yes	characteristic, complemented, Sylow
248	C₂×#41	—	53	2	no	—
249	#135	—	53	2	no	—
250	#135	—	53	2	no	—
251	#135	—	53	2	no	—
252	#135	—	53	2	no	—
253	C₂×Q₁₆	—	53	1	no	—
254	C₂×SD₁₆	—	53	2	no	—
255	C₂×D₈	—	53	1	no	—
256	#131	—	53	4	no	—
257	#131	—	53	4	no	—
258	#131	—	106	4	no	—
259	#131	—	106	2	no	—
260	#131	—	106	2	no	—
261	#131	—	106	4	no	—
262	#131	—	212	2	no	—
263	#131	—	212	2	no	—
264	C₂×M₁₆	—	53	1	no	—
265	C₂×M₁₆	—	53	2	no	—
266	C₂×M₁₆	—	106	2	no	—
267	C₂×M₁₆	—	106	4	no	—
268	C₂×M₁₆	—	106	4	no	—
269	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
270	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
271	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
272	C₂²×C₈	—	53	1	no	—
273	C₂²×C₈	—	106	2	no	—
274	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
275	#117	—	53	1	no	—
276	#117	C₂²×D₅₃	1	1	yes	characteristic
277	C₂×C₄²	—	53	1	no	—
278	#108	—	2	4	no	—
279	#108	—	2	1	no	—
280	#108	—	2	1	no	—
281	#108	—	53	4	no	—
282	#108	—	106	4	no	—
283	#108	—	106	2	no	—
284	#108	—	106	2	no	—
285	#108	—	106	1	no	—
286	#108	—	106	1	no	—
287	#108	C₄×D₅₃	1	2	yes	—
288	#108	C₄×D₅₃	1	2	yes	—
289	#107	—	106	1	no	—
290	#106	—	106	1	no	—
291	C₄⋊C₈	—	212	2	no	—
292	#104	—	53	8	no	—
293	#104	—	106	4	no	—
294	#104	—	106	4	no	—
295	#97	—	2	1	no	—
296	#97	—	53	2	no	—
297	#97	—	53	2	no	—
298	#97	—	106	1	no	—
299	C₄×C₈	—	2	1	no	—
300	C₄×C₈	—	106	1	no	—
301	#41	—	53	4	no	—
302	#41	—	53	4	no	—
303	#41	—	106	2	no	—
304	#41	—	106	2	no	—
305	C₂×Q₈	—	53	2	no	—
306	C₂×D₄	—	53	2	no	—
307	C₂×D₄	—	106	2	no	—
308	C₂²×C₄	—	106	2	no	—
309	C₂²×C₄	—	106	1	no	—
310	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
311	Q₁₆	—	53	2	no	—
312	Q₁₆	—	53	2	no	—
313	SD₁₆	—	53	4	no	—
314	SD₁₆	—	53	4	no	—
315	D₈	—	53	2	no	—
316	D₈	—	53	2	no	—
317	M₁₆	—	2	2	no	—
318	M₁₆	—	2	1	no	—
319	M₁₆	—	2	2	no	—
320	M₁₆	—	2	2	no	—
321	M₁₆	—	53	2	no	—
322	M₁₆	—	106	4	no	—
323	M₁₆	—	106	4	no	—
324	M₁₆	—	106	1	no	—
325	M₁₆	—	106	2	no	—
326	M₁₆	—	106	2	no	—
327	M₁₆	—	212	4	no	—
328	M₁₆	—	212	4	no	—
329	M₁₆	C₂×C₄×D₅₃	1	4	yes	—
330	M₁₆	D₄×D₅₃	1	2	yes	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	53	2	no	—
335	C₂×C₈	—	53	2	no	—
336	C₂×C₈	—	53	1	no	—
337	C₂×C₈	—	53	1	no	—
338	C₂×C₈	—	106	4	no	—
339	C₂×C₈	—	106	4	no	—
340	C₂×C₈	—	106	1	no	—
341	C₂×C₈	—	106	2	no	—
342	C₂×C₈	—	212	4	no	—
343	C₂×C₈	—	212	4	no	—
344	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
345	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
346	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
347	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
348	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
349	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	106	1	no	—
352	C₂²⋊C₄	—	2	1	no	—
353	C₂²⋊C₄	—	106	1	no	—
354	C₄²	—	2	1	no	—
355	C₄²	—	53	2	no	—
356	C₄²	—	106	1	no	—
357	C₄²	—	106	1	no	—
358	C₂³	—	106	2	no	—
359	C₂³	#13133265	1	1	yes	characteristic
360	Q₈	—	53	4	no	—
361	Q₈	—	106	2	no	—
362	D₄	—	53	4	no	—
363	D₄	—	106	4	no	—
364	D₄	—	106	4	no	—
365	D₄	—	106	2	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	106	2	no	—
368	C₂×C₄	—	106	4	no	—
369	C₂×C₄	—	106	4	no	—
370	C₂×C₄	—	106	2	no	—
371	C₂×C₄	—	424	1	no	—
372	C₂×C₄	—	212	1	no	—
373	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
374	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
375	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
376	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
377	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
378	C₂×C₄	#13133263	1	1	yes	characteristic
379	C₈	—	2	4	no	—
380	C₈	—	2	2	no	—
381	C₈	—	4	2	no	—
382	C₈	—	53	2	no	—
383	C₈	—	53	2	no	—
384	C₈	—	106	2	no	—
385	C₈	—	212	2	no	—
386	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
387	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
388	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
389	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
390	C₂²	—	4	1	no	—
391	C₂²	—	106	4	no	—
392	C₂²	—	424	2	no	—
393	C₂²	#16811105	1	1	yes	characteristic
394	C₂²	#16811275	1	1	yes	characteristic
395	C₂²	#16811288	1	1	yes	characteristic
396	C₄	—	8	1	no	—
397	C₄	—	106	4	no	—
398	C₄	—	424	1	no	—
399	C₄	—	212	2	no	—
400	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
401	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
402	C₄	#16811257	1	1	yes	characteristic
403	C₄	#16811329	1	1	yes	characteristic
404	C₂	—	2	1	no	—
405	C₂	—	2	1	no	—
406	C₂	—	2	1	no	—
407	C₂	—	212	4	no	—
408	C₂	#20277616	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16811257
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1953	C₂
socle	C₂×C₅₃	D₄×C₄⋊C₄
prime radical	C₂×#13133239	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1953	D₅₃
hypocenter	C₅₃	#22695
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1953

Complements

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

N	H ≅ G/N	type
4. #20283325	C₂	semidirect
5. #20275506	C₂	semidirect
6. #20274958	C₂	semidirect
7. #20274952	C₂	semidirect
8. #20274888	C₂	semidirect
9. #20274768	C₂	semidirect
10. #20274600	C₂	semidirect
11. C₅₃×#1953	C₂	semidirect
18. #16810744	C₄	semidirect
19. #16810744	C₄	semidirect
205. #1953	D₅₃	semidirect
247. C₅₃	#22695	semidirect

Maximal subgroups

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

subgroup	index
#20283633	2
#20283631	2
C₂×#16810744	2
#20283325	2
#20275506	2
#20274958	2
#20274952	2
#20274888	2
#20274768	2
#20274600	2
C₅₃×#1953	2
#22695	53