Subgroups — #28310260

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₃⋊M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#20289770	C₂	1	2	yes	maximal, complemented
4	#20289769	C₂	1	2	yes	maximal, complemented
5	#405×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133292	—	2	1	no	—
7	D₅₃⋊M₃₂	—	2	2	no	—
8	D₅₃⋊M₃₂	C₄	1	4	yes	complemented
9	C₂×#13133291	C₂²	1	2	yes	—
10	C₂×D₅₃⋊C₁₆	—	2	1	no	—
11	C₂×D₅₃⋊C₁₆	—	2	1	no	—
12	C₂×D₅₃⋊C₁₆	—	2	2	no	—
13	C₂×D₅₃⋊C₁₆	—	2	1	no	—
14	C₂×D₅₃⋊C₁₆	C₄	1	4	yes	complemented
15	C₂²×#11719910	C₂²	1	2	yes	—
16	C₂×M₁₆×D₅₃	C₄	1	1	yes	characteristic
17	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
18	#117×D₅₃	C₄	1	1	yes	characteristic
19	#16810581	C₂²	1	1	yes	characteristic
20	#16810580	—	2	1	no	—
21	#16810579	C₂²	1	1	yes	characteristic
22	#16810578	C₄	1	1	yes	characteristic
23	#16810577	—	2	1	no	—
24	#97×D₅₃	—	2	1	no	—
25	C₄×C₈×D₅₃	—	2	1	no	—
26	C₅₃×#405	C₄	1	1	yes	characteristic
27	#13133292	—	2	4	no	—
28	#13133291	—	2	2	no	—
29	#13133291	C₂×C₄	1	4	yes	complemented
30	D₅₃⋊C₁₆	—	2	4	no	—
31	D₅₃⋊C₁₆	—	2	4	no	—
32	D₅₃⋊C₁₆	—	2	4	no	—
33	C₂×#11719910	—	2	2	no	—
34	C₂×#11719910	—	2	2	no	—
35	C₂×#11719910	—	2	2	no	—
36	C₂×#11719910	C₂×C₄	1	4	yes	complemented
37	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
39	C₂×#11719912	—	2	1	no	—
40	M₁₆×D₅₃	—	2	2	no	—
41	M₁₆×D₅₃	C₈	1	4	yes	—
42	C₂²×#88322	C₂³	1	1	yes	characteristic
43	C₂×C₈×D₅₃	—	2	1	no	—
44	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
47	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
48	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
49	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
50	C₂×C₈×D₅₃	D₄	1	2	yes	—
51	C₂×C₈×D₅₃	Q₈	1	2	yes	—
52	#13133176	C₂×C₄	1	1	yes	characteristic
53	#13133175	C₈	1	2	yes	—
54	#13133174	—	2	1	no	—
55	#13133173	—	2	1	no	—
56	C₄⋊C₄×D₅₃	C₈	1	2	yes	—
57	C₂²⋊C₄×D₅₃	—	2	1	no	—
58	C₄²×D₅₃	—	2	1	no	—
59	#13133128	—	2	1	no	—
60	#13133127	—	2	1	no	—
61	#13133127	—	2	1	no	—
62	C₄×#88322	—	2	1	no	—
63	C₈×Dic₅₃	—	2	1	no	—
64	C₈×Dic₅₃	—	2	1	no	—
65	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
67	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
68	#97×C₅₃	—	2	1	no	—
69	C₄×C₈×C₅₃	—	2	1	no	—
70	C₂³×D₅₃	M₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×Dic₅₃	M₁₆	1	1	yes	characteristic
73	C₂×C₄×D₅₃	—	2	2	no	—
74	C₂×C₄×D₅₃	C₄²	1	2	yes	—
75	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
78	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
79	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
82	M₁₆×C₅₃	—	2	1	no	—
83	M₁₆×C₅₃	C₂×C₈	1	2	yes	—
84	C₂×C₈×C₅₃	—	2	1	no	—
85	C₂×C₈×C₅₃	C₄²	1	2	yes	—
86	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
89	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
90	C₂²⋊C₄×C₅₃	—	2	1	no	—
91	C₄²×C₅₃	—	2	1	no	—
92	C₂²⋊Dic₅₃	—	2	1	no	—
93	#11719921	—	2	2	no	—
94	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
95	#11719919	—	2	2	no	—
96	C₄×Dic₅₃	—	2	1	no	—
97	C₄×Dic₅₃	—	2	2	no	—
98	#11719917	—	2	1	no	—
99	#11719917	C₂×C₈	1	2	yes	—
100	C₂×#88322	—	2	1	no	—
101	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
102	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
103	C₂×#88322	C₂×D₄	1	1	yes	characteristic
104	C₂×#88322	C₂×D₄	1	1	yes	characteristic
105	C₂×#88322	C₂×D₄	1	1	yes	characteristic
106	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
107	#11719912	—	2	4	no	—
108	C₈×D₅₃	—	2	4	no	—
109	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
110	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
111	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
112	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
113	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
114	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
115	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
116	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
117	#11719910	—	2	4	no	—
118	#11719910	—	2	4	no	—
119	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
120	C₂²×D₅₃	—	2	4	no	—
121	C₂²×D₅₃	#97	1	2	yes	—
122	C₂²×D₅₃	C₂²⋊C₈	1	1	yes	characteristic
123	C₂²×D₅₃	C₂²⋊C₈	1	1	yes	characteristic
124	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
132	C₂×Dic₅₃	—	2	2	no	—
133	C₂×Dic₅₃	#97	1	2	yes	—
134	C₂×Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
135	C₂×Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
136	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
137	C₄×D₅₃	—	4	4	no	—
138	C₄×D₅₃	#95	1	2	yes	—
139	C₄×D₅₃	#95	1	2	yes	—
140	C₄×D₅₃	C₄×C₈	1	2	yes	—
141	C₄×D₅₃	C₄×C₈	1	2	yes	—
142	C₄×D₅₃	C₂²⋊C₈	1	2	yes	—
143	C₄×D₅₃	C₂²⋊C₈	1	2	yes	—
144	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
145	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
146	C₈×C₅₃	—	2	2	no	—
147	C₈×C₅₃	#95	1	2	yes	—
148	C₈×C₅₃	#95	1	2	yes	—
149	#88322	—	2	2	no	—
150	#88322	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	#88322	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
153	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×#1379	—	53	1	no	maximal, Sylow
155	C₂²×C₅₃	—	2	2	no	—
156	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
157	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
158	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
159	C₂×D₅₃	—	2	2	no	—
160	C₂×D₅₃	—	2	4	no	—
161	C₂×D₅₃	—	2	4	no	—
162	C₂×D₅₃	#336	1	2	yes	—
163	C₂×D₅₃	#336	1	2	yes	—
164	C₂×D₅₃	#350	1	4	yes	—
165	C₄×C₅₃	—	4	2	no	—
166	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
167	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
168	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
169	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
170	Dic₅₃	—	4	2	no	—
171	Dic₅₃	#336	1	2	yes	—
172	Dic₅₃	#336	1	2	yes	—
173	C₂²×M₃₂	—	53	1	no	—
174	C₂³×C₁₆	—	53	1	no	—
175	C₂×#405	—	53	1	no	—
176	#1379	—	53	2	no	—
177	#1379	—	53	2	no	—
178	C₂×C₅₃	—	2	1	no	—
179	C₂×C₅₃	—	2	1	no	—
180	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
181	C₂×C₅₃	C₂×#350	1	2	yes	—
182	D₅₃	—	2	2	no	—
183	D₅₃	#1379	1	4	yes	complemented
184	C₂²×M₁₆	—	53	1	no	—
185	C₂³×C₈	—	53	1	no	—
186	C₂×#117	—	53	1	no	—
187	C₂×M₃₂	—	53	2	no	—
188	C₂×M₃₂	—	53	4	no	—
189	C₂×M₃₂	—	106	2	no	—
190	C₂×M₃₂	—	106	1	no	—
191	C₂²×C₁₆	—	53	2	no	—
192	C₂²×C₁₆	—	53	4	no	—
193	C₂²×C₁₆	—	106	2	no	—
194	C₂²×C₁₆	—	106	1	no	—
195	C₂²×C₁₆	—	106	1	no	—
196	C₂²×C₁₆	—	106	1	no	—
197	#405	—	53	1	no	—
198	#405	—	53	1	no	—
199	#405	—	53	1	no	—
200	#405	—	106	1	no	—
201	#405	—	106	1	no	—
202	#405	#6157	1	1	yes	characteristic
203	C₂×#97	—	106	1	no	—
204	C₂×C₄×C₈	—	106	1	no	—
205	C₅₃	C₂×#1379	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₄	—	53	1	no	—
207	C₂×M₁₆	—	53	1	no	—
208	C₂×M₁₆	—	53	4	no	—
209	C₂×M₁₆	—	106	1	no	—
210	C₂×M₁₆	—	106	2	no	—
211	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
212	C₂²×C₈	—	53	1	no	—
213	C₂²×C₈	—	53	2	no	—
214	C₂²×C₈	—	53	2	no	—
215	C₂²×C₈	—	53	2	no	—
216	C₂²×C₈	—	53	2	no	—
217	C₂²×C₈	—	53	1	no	—
218	C₂²×C₈	—	53	1	no	—
219	C₂²×C₈	—	53	1	no	—
220	C₂²×C₈	—	53	1	no	—
221	C₂²×C₈	—	106	1	no	—
222	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
223	#117	—	53	1	no	—
224	#117	—	53	2	no	—
225	#117	—	106	1	no	—
226	#117	—	106	1	no	—
227	#117	C₂×#6157	1	1	yes	characteristic
228	C₂×C₄⋊C₄	—	53	2	no	—
229	C₂×C₂²⋊C₄	—	106	1	no	—
230	C₂×C₄²	—	106	1	no	—
231	M₃₂	—	53	4	no	—
232	M₃₂	—	106	2	no	—
233	M₃₂	—	106	4	no	—
234	C₂×C₁₆	—	53	4	no	—
235	C₂×C₁₆	—	106	2	no	—
236	C₂×C₁₆	—	106	2	no	—
237	C₂×C₁₆	—	106	2	no	—
238	C₂×C₁₆	—	106	4	no	—
239	C₂×C₁₆	—	106	4	no	—
240	C₂×C₁₆	—	106	4	no	—
241	#97	—	2	1	no	—
242	#97	—	106	1	no	—
243	#97	—	106	1	no	—
244	#97	—	106	1	no	—
245	C₄×C₈	—	2	1	no	—
246	C₄×C₈	—	106	1	no	—
247	C₄×C₈	—	106	1	no	—
248	C₄×C₈	—	106	1	no	—
249	C₂⁴	—	53	1	no	—
250	C₂²×C₄	—	53	1	no	—
251	C₂²×C₄	—	53	2	no	—
252	C₂²×C₄	—	53	1	no	—
253	C₂²×C₄	—	53	1	no	—
254	C₂²×C₄	—	53	2	no	—
255	C₂²×C₄	—	53	1	no	—
256	C₂²×C₄	—	53	1	no	—
257	C₂²×C₄	—	53	2	no	—
258	C₂²×C₄	—	53	2	no	—
259	C₂²×C₄	—	106	2	no	—
260	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
261	M₁₆	—	2	1	no	—
262	M₁₆	—	53	2	no	—
263	M₁₆	—	106	4	no	—
264	M₁₆	—	106	1	no	—
265	M₁₆	D₅₃⋊C₈	1	2	yes	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	53	1	no	—
268	C₂×C₈	—	53	1	no	—
269	C₂×C₈	—	53	1	no	—
270	C₂×C₈	—	53	1	no	—
271	C₂×C₈	—	53	1	no	—
272	C₂×C₈	—	53	1	no	—
273	C₂×C₈	—	53	2	no	—
274	C₂×C₈	—	53	2	no	—
275	C₂×C₈	—	53	2	no	—
276	C₂×C₈	—	53	2	no	—
277	C₂×C₈	—	53	2	no	—
278	C₂×C₈	—	53	2	no	—
279	C₂×C₈	—	53	2	no	—
280	C₂×C₈	—	53	2	no	—
281	C₂×C₈	—	106	4	no	—
282	C₂×C₈	—	106	1	no	—
283	C₂×C₈	C₄×#6157	1	2	yes	—
284	C₂×C₈	C₄⋊#6157	1	2	yes	—
285	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
286	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
287	C₄⋊C₄	—	53	2	no	—
288	C₄⋊C₄	—	106	2	no	—
289	C₄⋊C₄	D₅₃⋊C₈	1	2	yes	—
290	C₂²⋊C₄	—	2	1	no	—
291	C₂²⋊C₄	—	106	2	no	—
292	C₂²⋊C₄	—	106	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	106	2	no	—
295	C₄²	—	106	1	no	—
296	C₁₆	—	106	4	no	—
297	C₁₆	—	106	4	no	—
298	C₂³	—	53	1	no	—
299	C₂³	—	53	2	no	—
300	C₂³	—	53	1	no	—
301	C₂³	—	53	2	no	—
302	C₂³	—	106	4	no	—
303	C₂³	D₅₃⋊M₁₆	1	1	yes	characteristic
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	53	2	no	—
306	C₂×C₄	—	53	1	no	—
307	C₂×C₄	—	53	1	no	—
308	C₂×C₄	—	53	2	no	—
309	C₂×C₄	—	53	2	no	—
310	C₂×C₄	—	53	2	no	—
311	C₂×C₄	—	53	2	no	—
312	C₂×C₄	—	53	2	no	—
313	C₂×C₄	—	53	2	no	—
314	C₂×C₄	—	53	2	no	—
315	C₂×C₄	—	53	2	no	—
316	C₂×C₄	—	53	2	no	—
317	C₂×C₄	—	106	2	no	—
318	C₂×C₄	—	212	4	no	—
319	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
320	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
321	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
322	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
323	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
324	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	53	1	no	—
327	C₈	—	53	1	no	—
328	C₈	—	53	1	no	—
329	C₈	—	53	1	no	—
330	C₈	—	106	2	no	—
331	C₈	(C₂×#6157)⋊C₄	1	2	yes	—
332	C₈	(C₂×#6157)⋊C₄	1	2	yes	—
333	C₂²	—	2	2	no	—
334	C₂²	—	53	2	no	—
335	C₂²	—	53	4	no	—
336	C₂²	—	53	2	no	—
337	C₂²	—	106	4	no	—
338	C₂²	—	106	4	no	—
339	C₂²	—	106	2	no	—
340	C₂²	#16811582	1	1	yes	characteristic
341	C₂²	D₅₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
342	C₂²	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
343	C₄	—	4	2	no	—
344	C₄	—	53	2	no	—
345	C₄	—	53	2	no	—
346	C₄	—	212	2	no	—
347	C₄	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic
348	C₄	C₄×D₅₃⋊C₈	1	1	yes	characteristic
349	C₄	D₅₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
350	C₄	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	53	4	no	—
354	C₂	—	106	2	no	—
355	C₂	#20290376	1	1	yes	characteristic
356	C₂	D₅₃⋊#350	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₅₃×#405	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₂²×#88322	C₂³
hypercenter	#405	#6157
hypocenter	C₅₃	C₂×#1379
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
3. #20289770	C₂	semidirect
4. #20289769	C₂	semidirect
8. D₅₃⋊M₃₂	C₄	semidirect
14. C₂×D₅₃⋊C₁₆	C₄	semidirect
29. #13133291	C₂×C₄	semidirect
36. C₂×#11719910	C₂×C₄	semidirect
183. D₅₃	#1379	semidirect
205. C₅₃	C₂×#1379	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅₃⋊M₃₂	2
C₂²×D₅₃⋊C₁₆	2
#20289770	2
#20289769	2
#405×D₅₃	2
C₂×#1379	53