Subgroups — #28328358

Order 13568.

648 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291683	C₂	1	2	yes	maximal, complemented
2	#20291107	C₂	1	2	yes	maximal, complemented
3	#20291105	C₂	1	2	yes	maximal, complemented
4	#20291103	C₂	1	4	yes	maximal, complemented
5	D₅₃⋊#364	C₂	1	1	yes	maximal, characteristic, complemented
6	#20290958	C₂	1	2	yes	maximal, complemented
7	C₄×#13133292	C₂	1	1	yes	maximal, characteristic, complemented
8	#443×D₅₃	C₂	1	1	yes	maximal, characteristic
9	#16811682	—	2	4	no	—
10	#16811682	C₂²	1	4	yes	complemented
11	#16811681	C₂²	1	4	yes	complemented
12	C₂×#13133292	—	2	2	no	—
13	C₂×#13133292	C₂²	1	1	yes	characteristic
14	C₂×#13133292	C₂²	1	1	yes	characteristic
15	D₅₃⋊M₃₂	—	2	2	no	—
16	D₅₃⋊M₃₂	C₂²	1	2	yes	—
17	#16811546	C₂²	1	2	yes	complemented
18	#16811545	C₂²	1	2	yes	complemented
19	#16811522	C₂²	1	4	yes	complemented
20	#16811521	C₂²	1	4	yes	complemented
21	#16811512	C₂²	1	2	yes	complemented
22	C₄×#11719910	C₂²	1	2	yes	complemented
23	#135×D₅₃	C₄	1	1	yes	characteristic
24	#131×D₅₃	C₂²	1	2	yes	—
25	#16810741	C₄	1	1	yes	characteristic
26	#16810740	C₂²	1	1	yes	characteristic
27	#16810739	C₂²	1	1	yes	characteristic
28	#16810738	C₄	1	2	yes	—
29	#16810737	C₂²	1	2	yes	—
30	#108×D₅₃	C₄	1	1	yes	characteristic
31	#104×D₅₃	C₄	1	2	yes	—
32	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#443	C₄	1	1	yes	characteristic
34	#13133292	—	2	2	no	—
35	#13133292	—	2	1	no	—
36	#13133292	—	2	1	no	—
37	#13133292	—	2	4	no	—
38	#13133292	D₄	1	2	yes	—
39	#13133292	D₄	1	2	yes	—
40	#13133291	—	2	4	no	—
41	#13133291	C₂³	1	4	yes	—
42	D₅₃⋊C₁₆	—	2	2	no	—
43	D₅₃⋊C₁₆	—	2	4	no	—
44	C₂×#11719910	—	2	4	no	—
45	C₂×#11719910	C₂³	1	2	yes	—
46	C₂×#11719910	C₂³	1	2	yes	—
47	#41×D₅₃	C₂×C₄	1	2	yes	—
48	#13133240	C₈	1	1	yes	characteristic
49	#13133239	C₂×C₄	1	1	yes	characteristic
50	#13133238	C₈	1	2	yes	—
51	#13133237	C₂×C₄	1	2	yes	—
52	#13133236	C₈	1	1	yes	characteristic
53	Q₁₆×D₅₃	C₈	1	1	yes	characteristic
54	SD₁₆×D₅₃	C₈	1	2	yes	—
55	D₈×D₅₃	C₈	1	1	yes	characteristic
56	#13133225	—	2	2	no	—
57	#13133225	C₂³	1	2	yes	—
58	#13133224	C₂³	1	2	yes	—
59	#13133223	—	2	2	no	—
60	#13133223	C₂×C₄	1	2	yes	—
61	M₁₆×D₅₃	—	2	2	no	—
62	M₁₆×D₅₃	C₂×C₄	1	2	yes	—
63	C₂×C₈×D₅₃	—	2	2	no	—
64	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
66	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
67	#13133153	C₂×C₄	1	1	yes	characteristic
68	#13133152	C₂×C₄	1	2	yes	—
69	#13133145	C₂×C₄	1	2	yes	—
70	#13133144	C₂×C₄	1	2	yes	—
71	#13133143	C₂×C₄	1	2	yes	—
72	C₄×#88322	C₂³	1	1	yes	characteristic
73	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
74	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
75	#135×C₅₃	C₂×C₄	1	1	yes	characteristic
76	#131×C₅₃	C₂×C₄	1	2	yes	—
77	#108×C₅₃	C₂×C₄	1	1	yes	characteristic
78	#104×C₅₃	C₂×C₄	1	2	yes	—
79	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
80	#41×C₅₃	C₂²×C₄	1	2	yes	—
81	#11719949	C₂×C₈	1	2	yes	—
82	Q₈×D₅₃	C₂×C₈	1	2	yes	—
83	#11719947	—	2	2	no	—
84	#11719947	C₂×C₈	1	2	yes	—
85	D₄×D₅₃	—	2	2	no	—
86	D₄×D₅₃	C₂×C₈	1	2	yes	—
87	#11719945	—	2	2	no	—
88	#11719945	C₂²×C₄	1	2	yes	—
89	C₂×C₄×D₅₃	—	2	2	no	—
90	C₂×C₄×D₅₃	—	2	1	no	—
91	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
92	Q₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
93	SD₁₆×C₅₃	C₂×C₈	1	2	yes	—
94	D₈×C₅₃	C₂×C₈	1	1	yes	characteristic
95	M₁₆×C₅₃	—	2	2	no	—
96	M₁₆×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₅₃	—	2	2	no	—
98	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	#11719925	C₂×C₈	1	2	yes	—
102	#11719924	C₂×C₈	1	2	yes	—
103	#11719923	C₂×C₈	1	2	yes	—
104	#11719922	C₂×C₈	1	2	yes	—
105	C₄×Dic₅₃	—	2	1	no	—
106	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
107	#11719917	—	2	2	no	—
108	#11719917	C₂²×C₄	1	2	yes	—
109	C₂×#88322	—	2	2	no	—
110	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
111	C₂×#88322	C₂⁴	1	1	yes	characteristic
112	Dic₂₁₂	C₂×C₈	1	1	yes	characteristic
113	D₄₂₄	C₂×C₈	1	1	yes	characteristic
114	#11719913	C₂×C₈	1	2	yes	—
115	#11719912	—	2	2	no	—
116	#11719912	—	2	2	no	—
117	#11719912	—	2	2	no	—
118	C₈×D₅₃	—	2	2	no	—
119	C₈×D₅₃	—	2	2	no	—
120	C₈×D₅₃	—	2	2	no	—
121	C₈×D₅₃	C₂×C₈	1	1	yes	characteristic
122	C₈×D₅₃	C₂×C₈	1	1	yes	characteristic
123	C₈×D₅₃	C₂×C₈	1	1	yes	characteristic
124	C₈×D₅₃	C₂×C₈	1	1	yes	characteristic
125	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
126	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
127	C₈×D₅₃	#41	1	1	yes	characteristic
128	C₈×D₅₃	#41	1	1	yes	characteristic
129	#11719910	—	2	4	no	—
130	#11719910	—	2	2	no	—
131	#11719910	C₂×D₄	1	4	yes	—
132	C₂²×D₅₃	—	2	2	no	—
133	C₂²×D₅₃	#131	1	1	yes	characteristic
134	Q₈×C₅₃	C₂²×C₈	1	2	yes	—
135	D₄×C₅₃	—	2	2	no	—
136	D₄×C₅₃	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₅₃	—	2	1	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
140	C₂²⋊D₅₃	—	2	2	no	—
141	C₂²⋊D₅₃	—	2	2	no	—
142	C₂²⋊D₅₃	—	2	2	no	—
143	C₂×Dic₅₃	—	2	2	no	—
144	C₂×Dic₅₃	—	2	1	no	—
145	C₂×Dic₅₃	#131	1	1	yes	characteristic
146	D₂₁₂	—	2	2	no	—
147	D₂₁₂	C₂²×C₈	1	2	yes	—
148	C₄×D₅₃	—	2	2	no	—
149	C₄×D₅₃	—	2	2	no	—
150	C₄×D₅₃	—	2	2	no	—
151	C₄×D₅₃	—	4	2	no	—
152	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
153	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
154	C₄×D₅₃	C₂²×C₈	1	1	yes	characteristic
155	C₄×D₅₃	C₂²×C₈	1	1	yes	characteristic
156	Dic₁₀₆	—	2	2	no	—
157	Dic₁₀₆	C₂²×C₈	1	2	yes	—
158	C₈×C₅₃	—	2	2	no	—
159	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
160	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
161	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
162	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
163	#88322	—	2	2	no	—
164	#88322	C₂²×C₈	1	1	yes	characteristic
165	#88322	C₂²×C₈	1	1	yes	characteristic
166	#88322	C₂²×D₄	1	1	yes	characteristic
167	#88322	C₂×#41	1	1	yes	characteristic
168	#33443	—	53	1	no	maximal, Sylow
169	C₂²×C₅₃	—	2	2	no	—
170	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
171	C₂×D₅₃	—	2	2	no	—
172	C₂×D₅₃	—	4	1	no	—
173	C₂×D₅₃	—	8	2	no	—
174	C₂×D₅₃	C₈×D₄	1	1	yes	characteristic
175	C₂×D₅₃	C₈×D₄	1	1	yes	characteristic
176	C₄×C₅₃	—	2	2	no	—
177	C₄×C₅₃	—	2	2	no	—
178	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
179	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
180	Dic₅₃	—	2	2	no	—
181	Dic₅₃	—	4	1	no	—
182	Dic₅₃	C₈×D₄	1	1	yes	characteristic
183	Dic₅₃	C₈×D₄	1	1	yes	characteristic
184	#3407	—	53	2	no	—
185	C₂×#443	—	53	1	no	—
186	#2171	—	53	2	no	—
187	#2171	—	53	2	no	—
188	#2170	—	53	4	no	—
189	C₂×#364	—	53	1	no	—
190	#2117	—	53	2	no	—
191	C₄×M₃₂	—	53	1	no	—
192	C₂×C₅₃	—	2	1	no	—
193	C₂×C₅₃	—	4	2	no	—
194	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
195	D₅₃	—	2	1	no	—
196	D₅₃	—	2	1	no	—
197	D₅₃	—	4	2	no	—
198	C₂×#135	—	53	1	no	—
199	C₂×#131	—	53	2	no	—
200	#504	—	53	4	no	—
201	#504	—	53	4	no	—
202	#504	—	106	4	no	—
203	C₂×M₃₂	—	53	1	no	—
204	C₂×M₃₂	—	53	1	no	—
205	C₂×M₃₂	—	53	2	no	—
206	C₂×M₃₂	—	106	2	no	—
207	C₂×M₃₂	—	106	2	no	—
208	#443	—	53	1	no	—
209	#443	—	53	2	no	—
210	#443	—	53	1	no	—
211	#443	—	53	2	no	—
212	#443	—	53	1	no	—
213	#443	#6157	1	1	yes	characteristic
214	C₂×#108	—	53	1	no	—
215	C₂×#104	—	53	2	no	—
216	C₂×C₄×C₈	—	53	1	no	—
217	#364	—	53	2	no	—
218	#364	—	53	2	no	—
219	#350	—	53	4	no	—
220	#350	—	53	4	no	—
221	#346	—	53	2	no	—
222	C₄×C₁₆	—	53	2	no	—
223	C₅₃	#33443	1	1	yes	characteristic, complemented, Sylow
224	C₂×#41	—	53	2	no	—
225	#135	—	53	2	no	—
226	#135	—	53	1	no	—
227	#135	—	53	2	no	—
228	#135	—	53	1	no	—
229	#135	—	53	1	no	—
230	#135	C₂×#6157	1	1	yes	characteristic
231	C₂×Q₁₆	—	53	1	no	—
232	C₂×SD₁₆	—	53	2	no	—
233	C₂×D₈	—	53	1	no	—
234	#131	—	53	2	no	—
235	#131	—	53	2	no	—
236	#131	—	53	2	no	—
237	#131	—	106	2	no	—
238	#131	—	106	2	no	—
239	#131	C₂×#6157	1	2	yes	—
240	C₂×M₁₆	—	53	2	no	—
241	C₂×M₁₆	—	106	2	no	—
242	C₂²×C₈	—	53	1	no	—
243	C₂²×C₈	—	53	1	no	—
244	C₂²×C₈	—	106	2	no	—
245	C₂×C₄²	—	53	1	no	—
246	M₃₂	—	53	2	no	—
247	M₃₂	—	53	2	no	—
248	M₃₂	—	53	4	no	—
249	M₃₂	—	106	4	no	—
250	M₃₂	—	106	4	no	—
251	M₃₂	—	106	1	no	—
252	M₃₂	—	106	1	no	—
253	M₃₂	—	106	2	no	—
254	C₂×C₁₆	—	53	2	no	—
255	C₂×C₁₆	—	53	2	no	—
256	C₂×C₁₆	—	106	4	no	—
257	C₂×C₁₆	—	106	4	no	—
258	C₂×C₁₆	—	106	2	no	—
259	#108	—	53	2	no	—
260	#108	—	53	1	no	—
261	#108	C₂×#6157	1	1	yes	characteristic
262	#104	—	53	2	no	—
263	#104	—	53	2	no	—
264	#104	—	53	2	no	—
265	#104	C₂×#6157	1	2	yes	—
266	C₄×C₈	—	53	1	no	—
267	C₄×C₈	—	53	1	no	—
268	C₄×C₈	—	53	1	no	—
269	C₄×C₈	C₂×#6157	1	1	yes	characteristic
270	#41	—	53	2	no	—
271	#41	—	53	2	no	—
272	#41	—	53	2	no	—
273	#41	—	106	2	no	—
274	#41	—	106	2	no	—
275	#41	C₂²×#6157	1	2	yes	—
276	C₂×Q₈	—	53	2	no	—
277	C₂×D₄	—	53	2	no	—
278	C₂×D₄	—	106	2	no	—
279	C₂²×C₄	—	53	1	no	—
280	C₂²×C₄	—	106	2	no	—
281	C₂²×C₄	—	106	1	no	—
282	Q₁₆	—	53	2	no	—
283	Q₁₆	—	53	1	no	—
284	Q₁₆	D₅₃⋊C₈	1	1	yes	characteristic
285	SD₁₆	—	53	2	no	—
286	SD₁₆	—	53	2	no	—
287	SD₁₆	—	53	2	no	—
288	SD₁₆	D₅₃⋊C₈	1	2	yes	—
289	D₈	—	53	2	no	—
290	D₈	—	53	1	no	—
291	D₈	D₅₃⋊C₈	1	1	yes	characteristic
292	M₁₆	—	2	2	no	—
293	M₁₆	—	53	2	no	—
294	M₁₆	—	106	2	no	—
295	M₁₆	—	106	2	no	—
296	M₁₆	—	106	2	no	—
297	M₁₆	—	106	2	no	—
298	M₁₆	C₂²×#6157	1	2	yes	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	53	1	no	—
301	C₂×C₈	—	53	1	no	—
302	C₂×C₈	—	53	1	no	—
303	C₂×C₈	—	53	1	no	—
304	C₂×C₈	—	53	1	no	—
305	C₂×C₈	—	53	1	no	—
306	C₂×C₈	—	53	1	no	—
307	C₂×C₈	—	53	1	no	—
308	C₂×C₈	—	53	1	no	—
309	C₂×C₈	—	53	1	no	—
310	C₂×C₈	—	106	2	no	—
311	C₂×C₈	—	106	2	no	—
312	C₂×C₈	—	106	2	no	—
313	C₂×C₈	—	106	2	no	—
314	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
315	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
316	C₄²	—	53	1	no	—
317	C₄²	—	106	1	no	—
318	C₄²	C₂²×#6157	1	1	yes	characteristic
319	C₁₆	—	53	4	no	—
320	C₁₆	—	106	2	no	—
321	C₁₆	—	106	4	no	—
322	C₂³	—	53	1	no	—
323	C₂³	—	106	2	no	—
324	Q₈	—	53	2	no	—
325	Q₈	—	106	2	no	—
326	Q₈	C₂×D₅₃⋊C₈	1	2	yes	—
327	D₄	—	2	2	no	—
328	D₄	—	53	2	no	—
329	D₄	—	106	2	no	—
330	D₄	—	106	2	no	—
331	D₄	—	106	2	no	—
332	D₄	—	106	2	no	—
333	D₄	C₂×D₅₃⋊C₈	1	2	yes	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	53	1	no	—
337	C₂×C₄	—	53	1	no	—
338	C₂×C₄	—	53	1	no	—
339	C₂×C₄	—	53	1	no	—
340	C₂×C₄	—	53	1	no	—
341	C₂×C₄	—	106	2	no	—
342	C₂×C₄	—	106	2	no	—
343	C₂×C₄	—	106	2	no	—
344	C₂×C₄	—	106	2	no	—
345	C₂×C₄	—	106	1	no	—
346	C₂×C₄	—	212	2	no	—
347	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
348	C₈	—	2	2	no	—
349	C₈	—	53	1	no	—
350	C₈	—	53	1	no	—
351	C₈	—	53	1	no	—
352	C₈	—	53	1	no	—
353	C₈	—	106	2	no	—
354	C₈	D₄×#6157	1	1	yes	characteristic
355	C₈	D₄×#6157	1	1	yes	characteristic
356	C₈	C₂×D₅₃⋊C₈	1	1	yes	characteristic
357	C₈	C₂×D₅₃⋊C₈	1	1	yes	characteristic
358	C₂²	—	2	2	no	—
359	C₂²	—	53	1	no	—
360	C₂²	—	53	1	no	—
361	C₂²	—	106	2	no	—
362	C₂²	—	424	2	no	—
363	C₂²	—	212	1	no	—
364	C₂²	D₅₃⋊#131	1	1	yes	characteristic
365	C₄	—	2	2	no	—
366	C₄	—	2	2	no	—
367	C₄	—	53	1	no	—
368	C₄	—	53	1	no	—
369	C₄	—	106	2	no	—
370	C₄	—	212	1	no	—
371	C₄	C₂×D₄×#6157	1	1	yes	characteristic
372	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
373	C₂	—	2	1	no	—
374	C₂	—	4	2	no	—
375	C₂	—	106	1	no	—
376	C₂	—	106	1	no	—
377	C₂	—	212	2	no	—
378	C₂	D₄×D₅₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×#6157
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#443	C₄
socle	C₂×C₅₃	C₂×C₈×D₄
prime radical	#135×D₅₃	C₄
cosocle	C₂×#88322	C₂⁴
hypercenter	#443	#6157
hypocenter	C₅₃	#33443
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
1. #20291683	C₂	semidirect
2. #20291107	C₂	semidirect
3. #20291105	C₂	semidirect
4. #20291103	C₂	semidirect
5. D₅₃⋊#364	C₂	semidirect
6. #20290958	C₂	semidirect
7. C₄×#13133292	C₂	semidirect
10. #16811682	C₂²	semidirect
11. #16811681	C₂²	semidirect
17. #16811546	C₂²	semidirect
18. #16811545	C₂²	semidirect
19. #16811522	C₂²	semidirect
20. #16811521	C₂²	semidirect
21. #16811512	C₂²	semidirect
22. C₄×#11719910	C₂²	semidirect
223. C₅₃	#33443	semidirect

Maximal subgroups

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

subgroup	index
#20291683	2
#20291107	2
#20291105	2
#20291103	2
D₅₃⋊#364	2
#20290958	2
C₄×#13133292	2
#443×D₅₃	2
#33443	53