Subgroups — #28328356

Order 13568.

648 conjugacy classes of subgroups.

The subgroup lattice has 384 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	#20291106	C₂	1	4	yes	maximal, complemented
3	#20291104	C₂	1	2	yes	maximal, complemented
4	#20291102	C₂	1	2	yes	maximal, complemented
5	D₅₃⋊#364	C₂	1	1	yes	maximal, characteristic, complemented
6	#20290958	C₂	1	2	yes	maximal, complemented
7	#20290922	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	#16811511	C₂²	1	2	yes	complemented
22	#16811509	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	1	yes	characteristic
39	#13133292	D₄	1	1	yes	characteristic
40	#13133292	D₄	1	2	yes	—
41	#13133291	—	2	4	no	—
42	#13133291	C₂³	1	4	yes	—
43	D₅₃⋊C₁₆	—	2	2	no	—
44	D₅₃⋊C₁₆	—	2	4	no	—
45	C₂×#11719910	—	2	4	no	—
46	C₂×#11719910	C₂³	1	2	yes	—
47	C₂×#11719910	C₂³	1	2	yes	—
48	#41×D₅₃	C₂×C₄	1	2	yes	—
49	#13133240	C₈	1	1	yes	characteristic
50	#13133239	C₂×C₄	1	1	yes	characteristic
51	#13133238	C₈	1	2	yes	—
52	#13133237	C₂×C₄	1	2	yes	—
53	#13133236	C₈	1	1	yes	characteristic
54	Q₁₆×D₅₃	C₈	1	1	yes	characteristic
55	SD₁₆×D₅₃	C₈	1	2	yes	—
56	D₈×D₅₃	C₈	1	1	yes	characteristic
57	#13133225	—	2	2	no	—
58	#13133225	C₂³	1	2	yes	—
59	#13133224	C₂³	1	2	yes	—
60	#13133223	—	2	2	no	—
61	#13133223	C₂×C₄	1	2	yes	—
62	M₁₆×D₅₃	—	2	2	no	—
63	M₁₆×D₅₃	C₂×C₄	1	2	yes	—
64	C₂×C₈×D₅₃	—	2	2	no	—
65	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
67	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
68	#13133153	C₂×C₄	1	1	yes	characteristic
69	#13133152	C₂×C₄	1	2	yes	—
70	#13133145	C₂×C₄	1	2	yes	—
71	#13133144	C₂×C₄	1	2	yes	—
72	#13133143	C₂×C₄	1	2	yes	—
73	C₄×#88322	C₂³	1	1	yes	characteristic
74	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
75	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
76	#135×C₅₃	C₂×C₄	1	1	yes	characteristic
77	#131×C₅₃	C₂×C₄	1	2	yes	—
78	#108×C₅₃	C₂×C₄	1	1	yes	characteristic
79	#104×C₅₃	C₂×C₄	1	2	yes	—
80	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
81	#41×C₅₃	C₂²×C₄	1	2	yes	—
82	#11719949	C₂×C₈	1	2	yes	—
83	Q₈×D₅₃	C₂×C₈	1	2	yes	—
84	#11719947	—	2	2	no	—
85	#11719947	C₂×C₈	1	2	yes	—
86	D₄×D₅₃	—	2	2	no	—
87	D₄×D₅₃	C₂×C₈	1	2	yes	—
88	#11719945	—	2	2	no	—
89	#11719945	C₂²×C₄	1	2	yes	—
90	C₂×C₄×D₅₃	—	2	2	no	—
91	C₂×C₄×D₅₃	—	2	1	no	—
92	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
93	Q₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
94	SD₁₆×C₅₃	C₂×C₈	1	2	yes	—
95	D₈×C₅₃	C₂×C₈	1	1	yes	characteristic
96	M₁₆×C₅₃	—	2	2	no	—
97	M₁₆×C₅₃	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₅₃	—	2	2	no	—
99	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	#11719925	C₂×C₈	1	2	yes	—
103	#11719924	C₂×C₈	1	2	yes	—
104	#11719923	C₂×C₈	1	2	yes	—
105	#11719922	C₂×C₈	1	2	yes	—
106	C₄×Dic₅₃	—	2	1	no	—
107	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
108	#11719917	—	2	2	no	—
109	#11719917	C₂²×C₄	1	2	yes	—
110	C₂×#88322	—	2	2	no	—
111	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
112	C₂×#88322	C₂⁴	1	1	yes	characteristic
113	Dic₂₁₂	C₂×C₈	1	1	yes	characteristic
114	D₄₂₄	C₂×C₈	1	1	yes	characteristic
115	#11719913	C₂×C₈	1	2	yes	—
116	#11719912	—	2	2	no	—
117	#11719912	—	2	2	no	—
118	#11719912	—	2	2	no	—
119	C₈×D₅₃	—	2	2	no	—
120	C₈×D₅₃	—	2	2	no	—
121	C₈×D₅₃	—	2	2	no	—
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×C₈	1	1	yes	characteristic
126	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
127	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
128	C₈×D₅₃	#41	1	1	yes	characteristic
129	C₈×D₅₃	#41	1	1	yes	characteristic
130	#11719910	—	2	4	no	—
131	#11719910	—	2	2	no	—
132	#11719910	C₂×D₄	1	2	yes	—
133	#11719910	C₂×D₄	1	2	yes	—
134	C₂²×D₅₃	—	2	2	no	—
135	C₂²×D₅₃	#131	1	1	yes	characteristic
136	Q₈×C₅₃	C₂²×C₈	1	2	yes	—
137	D₄×C₅₃	—	2	2	no	—
138	D₄×C₅₃	C₂²×C₈	1	2	yes	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	1	no	—
141	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
142	C₂²⋊D₅₃	—	2	2	no	—
143	C₂²⋊D₅₃	—	2	2	no	—
144	C₂²⋊D₅₃	—	2	2	no	—
145	C₂×Dic₅₃	—	2	1	no	—
146	C₂×Dic₅₃	—	2	2	no	—
147	C₂×Dic₅₃	#131	1	1	yes	characteristic
148	D₂₁₂	—	2	2	no	—
149	D₂₁₂	C₂²×C₈	1	2	yes	—
150	C₄×D₅₃	—	2	2	no	—
151	C₄×D₅₃	—	2	2	no	—
152	C₄×D₅₃	—	2	2	no	—
153	C₄×D₅₃	—	4	2	no	—
154	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
155	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
156	C₄×D₅₃	C₂²×C₈	1	1	yes	characteristic
157	C₄×D₅₃	C₂²×C₈	1	1	yes	characteristic
158	Dic₁₀₆	—	2	2	no	—
159	Dic₁₀₆	C₂²×C₈	1	2	yes	—
160	C₈×C₅₃	—	2	2	no	—
161	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
162	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
163	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
164	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
165	#88322	—	2	2	no	—
166	#88322	C₂²×C₈	1	1	yes	characteristic
167	#88322	C₂²×C₈	1	1	yes	characteristic
168	#88322	C₂²×D₄	1	1	yes	characteristic
169	#88322	C₂×#41	1	1	yes	characteristic
170	#33443	—	53	1	no	maximal, Sylow
171	C₂²×C₅₃	—	2	2	no	—
172	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
173	C₂×D₅₃	—	2	2	no	—
174	C₂×D₅₃	—	4	1	no	—
175	C₂×D₅₃	—	8	2	no	—
176	C₂×D₅₃	C₈×D₄	1	1	yes	characteristic
177	C₂×D₅₃	C₈×D₄	1	1	yes	characteristic
178	C₄×C₅₃	—	2	2	no	—
179	C₄×C₅₃	—	4	1	no	—
180	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
181	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
182	Dic₅₃	—	2	2	no	—
183	Dic₅₃	—	2	2	no	—
184	Dic₅₃	C₈×D₄	1	1	yes	characteristic
185	Dic₅₃	C₈×D₄	1	1	yes	characteristic
186	#3407	—	53	2	no	—
187	C₂×#443	—	53	1	no	—
188	#2171	—	53	4	no	—
189	#2170	—	53	2	no	—
190	#2170	—	53	2	no	—
191	C₂×#364	—	53	1	no	—
192	#2117	—	53	2	no	—
193	C₄×M₃₂	—	53	1	no	—
194	C₂×C₅₃	—	2	1	no	—
195	C₂×C₅₃	—	4	2	no	—
196	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
197	D₅₃	—	2	1	no	—
198	D₅₃	—	2	1	no	—
199	D₅₃	—	4	2	no	—
200	C₂×#135	—	53	1	no	—
201	C₂×#131	—	53	2	no	—
202	#504	—	53	4	no	—
203	#504	—	53	4	no	—
204	#504	—	106	4	no	—
205	C₂×M₃₂	—	53	1	no	—
206	C₂×M₃₂	—	53	1	no	—
207	C₂×M₃₂	—	53	2	no	—
208	C₂×M₃₂	—	106	2	no	—
209	C₂×M₃₂	—	106	2	no	—
210	#443	—	53	1	no	—
211	#443	—	53	2	no	—
212	#443	—	53	1	no	—
213	#443	—	53	2	no	—
214	#443	—	53	1	no	—
215	#443	#6157	1	1	yes	characteristic
216	C₂×#108	—	53	1	no	—
217	C₂×#104	—	53	2	no	—
218	C₂×C₄×C₈	—	53	1	no	—
219	#364	—	53	2	no	—
220	#364	—	53	2	no	—
221	#350	—	53	4	no	—
222	#350	—	53	4	no	—
223	#346	—	53	2	no	—
224	C₄×C₁₆	—	53	2	no	—
225	C₅₃	#33443	1	1	yes	characteristic, complemented, Sylow
226	C₂×#41	—	53	2	no	—
227	#135	—	53	2	no	—
228	#135	—	53	1	no	—
229	#135	—	53	2	no	—
230	#135	—	53	1	no	—
231	#135	—	53	1	no	—
232	#135	C₂×#6157	1	1	yes	characteristic
233	C₂×Q₁₆	—	53	1	no	—
234	C₂×SD₁₆	—	53	2	no	—
235	C₂×D₈	—	53	1	no	—
236	#131	—	53	2	no	—
237	#131	—	53	2	no	—
238	#131	—	53	2	no	—
239	#131	—	106	2	no	—
240	#131	—	106	2	no	—
241	#131	C₂×#6157	1	2	yes	—
242	C₂×M₁₆	—	53	2	no	—
243	C₂×M₁₆	—	106	2	no	—
244	C₂²×C₈	—	53	1	no	—
245	C₂²×C₈	—	53	1	no	—
246	C₂²×C₈	—	106	2	no	—
247	C₂×C₄²	—	53	1	no	—
248	M₃₂	—	53	1	no	—
249	M₃₂	—	53	1	no	—
250	M₃₂	—	53	2	no	—
251	M₃₂	—	53	4	no	—
252	M₃₂	—	106	4	no	—
253	M₃₂	—	106	4	no	—
254	M₃₂	—	106	1	no	—
255	M₃₂	—	106	1	no	—
256	M₃₂	—	106	2	no	—
257	C₂×C₁₆	—	53	2	no	—
258	C₂×C₁₆	—	53	2	no	—
259	C₂×C₁₆	—	106	4	no	—
260	C₂×C₁₆	—	106	4	no	—
261	C₂×C₁₆	—	106	2	no	—
262	#108	—	53	2	no	—
263	#108	—	53	1	no	—
264	#108	C₂×#6157	1	1	yes	characteristic
265	#104	—	53	2	no	—
266	#104	—	53	2	no	—
267	#104	—	53	2	no	—
268	#104	C₂×#6157	1	2	yes	—
269	C₄×C₈	—	53	1	no	—
270	C₄×C₈	—	53	1	no	—
271	C₄×C₈	—	53	1	no	—
272	C₄×C₈	C₂×#6157	1	1	yes	characteristic
273	#41	—	53	2	no	—
274	#41	—	53	2	no	—
275	#41	—	53	2	no	—
276	#41	—	106	2	no	—
277	#41	—	106	2	no	—
278	#41	C₂²×#6157	1	2	yes	—
279	C₂×Q₈	—	53	2	no	—
280	C₂×D₄	—	53	2	no	—
281	C₂×D₄	—	106	2	no	—
282	C₂²×C₄	—	53	1	no	—
283	C₂²×C₄	—	106	2	no	—
284	C₂²×C₄	—	106	1	no	—
285	Q₁₆	—	53	2	no	—
286	Q₁₆	—	53	1	no	—
287	Q₁₆	D₅₃⋊C₈	1	1	yes	characteristic
288	SD₁₆	—	53	2	no	—
289	SD₁₆	—	53	2	no	—
290	SD₁₆	—	53	2	no	—
291	SD₁₆	D₅₃⋊C₈	1	2	yes	—
292	D₈	—	53	2	no	—
293	D₈	—	53	1	no	—
294	D₈	D₅₃⋊C₈	1	1	yes	characteristic
295	M₁₆	—	2	2	no	—
296	M₁₆	—	53	2	no	—
297	M₁₆	—	106	2	no	—
298	M₁₆	—	106	2	no	—
299	M₁₆	—	106	2	no	—
300	M₁₆	—	106	2	no	—
301	M₁₆	C₂²×#6157	1	2	yes	—
302	C₂×C₈	—	2	2	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	—	53	1	no	—
311	C₂×C₈	—	53	1	no	—
312	C₂×C₈	—	53	1	no	—
313	C₂×C₈	—	106	2	no	—
314	C₂×C₈	—	106	2	no	—
315	C₂×C₈	—	106	2	no	—
316	C₂×C₈	—	106	2	no	—
317	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
318	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
319	C₄²	—	53	1	no	—
320	C₄²	—	106	1	no	—
321	C₄²	C₂²×#6157	1	1	yes	characteristic
322	C₁₆	—	53	2	no	—
323	C₁₆	—	53	2	no	—
324	C₁₆	—	106	2	no	—
325	C₁₆	—	106	4	no	—
326	C₂³	—	53	1	no	—
327	C₂³	—	106	2	no	—
328	Q₈	—	53	2	no	—
329	Q₈	—	106	2	no	—
330	Q₈	C₂×D₅₃⋊C₈	1	2	yes	—
331	D₄	—	2	2	no	—
332	D₄	—	53	2	no	—
333	D₄	—	106	2	no	—
334	D₄	—	106	2	no	—
335	D₄	—	106	2	no	—
336	D₄	—	106	2	no	—
337	D₄	C₂×D₅₃⋊C₈	1	2	yes	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	53	1	no	—
341	C₂×C₄	—	53	1	no	—
342	C₂×C₄	—	53	1	no	—
343	C₂×C₄	—	53	1	no	—
344	C₂×C₄	—	53	1	no	—
345	C₂×C₄	—	106	2	no	—
346	C₂×C₄	—	106	2	no	—
347	C₂×C₄	—	106	2	no	—
348	C₂×C₄	—	106	1	no	—
349	C₂×C₄	—	106	2	no	—
350	C₂×C₄	—	212	2	no	—
351	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
352	C₈	—	2	2	no	—
353	C₈	—	53	1	no	—
354	C₈	—	53	1	no	—
355	C₈	—	53	1	no	—
356	C₈	—	53	1	no	—
357	C₈	—	106	2	no	—
358	C₈	D₄×#6157	1	1	yes	characteristic
359	C₈	D₄×#6157	1	1	yes	characteristic
360	C₈	C₂×D₅₃⋊C₈	1	1	yes	characteristic
361	C₈	C₂×D₅₃⋊C₈	1	1	yes	characteristic
362	C₂²	—	2	2	no	—
363	C₂²	—	53	1	no	—
364	C₂²	—	53	1	no	—
365	C₂²	—	106	2	no	—
366	C₂²	—	424	2	no	—
367	C₂²	—	212	1	no	—
368	C₂²	D₅₃⋊#131	1	1	yes	characteristic
369	C₄	—	2	2	no	—
370	C₄	—	4	1	no	—
371	C₄	—	53	1	no	—
372	C₄	—	53	1	no	—
373	C₄	—	106	2	no	—
374	C₄	—	106	2	no	—
375	C₄	C₂×D₄×#6157	1	1	yes	characteristic
376	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
377	C₂	—	2	1	no	—
378	C₂	—	4	2	no	—
379	C₂	—	106	1	no	—
380	C₂	—	106	1	no	—
381	C₂	—	212	2	no	—
382	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. #20291106	C₂	semidirect
3. #20291104	C₂	semidirect
4. #20291102	C₂	semidirect
5. D₅₃⋊#364	C₂	semidirect
6. #20290958	C₂	semidirect
7. #20290922	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. #16811511	C₂²	semidirect
22. #16811509	C₂²	semidirect
225. 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
#20291106	2
#20291104	2
#20291102	2
D₅₃⋊#364	2
#20290958	2
#20290922	2
#443×D₅₃	2
#33443	53