Subgroups — #27537187

Order 13568.

582 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133223	C₂	1	1	yes	maximal, characteristic
2	#20276521	C₂	1	2	yes	maximal, complemented
3	#20276519	C₂	1	2	yes	maximal, complemented
4	#20276518	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276516	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276516	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276472	C₂	1	2	yes	maximal, complemented
8	#20276327	C₂	1	2	yes	maximal
9	#20276308	C₂	1	1	yes	maximal, characteristic, complemented
10	#346×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133223	C₄	1	1	yes	characteristic
13	C₂×#13133223	C₂²	1	1	yes	characteristic
14	C₄×#11719945	C₄	1	1	yes	characteristic
15	C₂×#13133156	C₂²	1	2	yes	—
16	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
17	#16810578	C₄	1	1	yes	characteristic
18	#16810578	C₂²	1	1	yes	characteristic
19	#16810577	C₄	1	1	yes	characteristic
20	#16810577	C₄	1	1	yes	characteristic
21	#16810577	C₂²	1	1	yes	characteristic
22	#16810577	C₂²	1	1	yes	characteristic
23	C₄×#11719917	C₄	1	1	yes	characteristic
24	C₄×#11719912	C₄	1	1	yes	characteristic
25	C₄×#11719912	C₄	1	1	yes	characteristic
26	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
27	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
28	C₄⋊#11719908	C₂²	1	2	yes	—
29	#16810441	C₂²	1	2	yes	—
30	#16810441	C₂²	1	2	yes	—
31	C₂²⋊#11719908	C₂²	1	2	yes	—
32	#16810402	C₂²	1	2	yes	—
33	#16810402	C₂²	1	2	yes	—
34	#16810399	C₂²	1	1	yes	characteristic
35	#16810399	C₂²	1	1	yes	characteristic
36	#16810398	C₂²	1	2	yes	—
37	C₁₆×Dic₅₃	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
40	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
41	C₅₃×#346	C₂²	1	1	yes	characteristic
42	C₅₃×#346	C₂²	1	1	yes	characteristic
43	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
44	#13133223	—	2	4	no	—
45	#13133223	—	2	4	no	—
46	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
47	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
48	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
49	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
50	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
51	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
56	C₄×Dic₁₀₆	C₈	1	2	yes	—
57	C₄×D₂₁₂	C₈	1	2	yes	—
58	C₄×C₂²⋊D₅₃	C₈	1	2	yes	—
59	C₄×C₂²⋊D₅₃	C₈	1	2	yes	—
60	#13133176	C₂×C₄	1	1	yes	characteristic
61	#13133174	C₂×C₄	1	1	yes	characteristic
62	#13133174	C₂×C₄	1	1	yes	characteristic
63	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
65	#13133156	—	2	4	no	—
66	C₂×#11719908	C₂³	1	2	yes	—
67	C₂×#11719908	C₂³	1	2	yes	—
68	C₁₆×D₅₃	—	2	4	no	—
69	#13133128	C₂×C₄	1	1	yes	characteristic
70	#13133128	C₂×C₄	1	1	yes	characteristic
71	#13133127	C₂×C₄	1	1	yes	characteristic
72	#13133127	C₂×C₄	1	1	yes	characteristic
73	#13133127	C₂×C₄	1	1	yes	characteristic
74	#13133127	C₂×C₄	1	1	yes	characteristic
75	C₄×#88322	C₂×C₄	1	1	yes	characteristic
76	C₄×#88322	C₂×C₄	1	1	yes	characteristic
77	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
78	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
79	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
80	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
84	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
85	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
86	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
91	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
93	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
94	#11719945	—	2	2	no	—
95	#11719945	—	2	2	no	—
96	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
97	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
98	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
99	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
102	C₂×C₈×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	—	2	2	no	—
104	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
108	C₄²×C₅₃	C₂×C₈	1	2	yes	—
109	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
112	#11719921	C₂×C₈	1	2	yes	—
113	#11719921	C₂×C₈	1	2	yes	—
114	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
115	#11719919	C₂×C₈	1	2	yes	—
116	#11719919	C₂×C₈	1	2	yes	—
117	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
118	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
119	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
121	#11719917	—	2	2	no	—
122	#11719917	—	2	2	no	—
123	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
124	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
125	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
126	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
127	#11719912	—	2	2	no	—
128	#11719912	—	2	2	no	—
129	#11719912	—	2	2	no	—
130	#11719912	—	2	2	no	—
131	C₈×D₅₃	—	2	2	no	—
132	C₈×D₅₃	—	2	2	no	—
133	C₈×D₅₃	M₁₆	1	4	yes	—
134	C₈×D₅₃	#41	1	4	yes	—
135	C₁₆×C₅₃	—	2	2	no	—
136	C₁₆×C₅₃	—	2	2	no	—
137	#11719908	—	2	2	no	—
138	#11719908	—	2	2	no	—
139	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
140	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
141	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	1	no	—
144	C₂×C₄×C₅₃	—	4	2	no	—
145	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
147	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
148	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
149	C₂²⋊D₅₃	—	2	2	no	—
150	C₂²⋊D₅₃	—	2	2	no	—
151	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
152	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
153	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
154	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
155	D₂₁₂	—	2	2	no	—
156	C₄×D₅₃	—	2	1	no	—
157	C₄×D₅₃	—	2	1	no	—
158	C₄×D₅₃	—	4	2	no	—
159	C₄×D₅₃	—	4	2	no	—
160	C₄×D₅₃	#117	1	2	yes	—
161	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
162	Dic₁₀₆	—	2	2	no	—
163	C₈×C₅₃	—	2	1	no	—
164	C₈×C₅₃	—	2	1	no	—
165	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
166	C₈×C₅₃	C₂×#41	1	2	yes	—
167	#88322	—	2	1	no	—
168	#88322	—	2	1	no	—
169	#88322	#117	1	2	yes	—
170	#88322	C₂×M₁₆	1	2	yes	—
171	#33421	—	53	1	no	maximal, Sylow
172	C₂²×C₅₃	—	2	2	no	—
173	C₂²×C₅₃	—	2	1	no	—
174	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
175	C₂×D₅₃	—	2	2	no	—
176	C₂×D₅₃	—	2	2	no	—
177	C₂×D₅₃	—	2	1	no	—
178	C₂×D₅₃	#431	1	2	yes	—
179	C₄×C₅₃	—	2	2	no	—
180	C₄×C₅₃	—	2	2	no	—
181	C₄×C₅₃	—	2	1	no	—
182	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
183	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
184	Dic₅₃	—	2	2	no	—
185	Dic₅₃	—	2	2	no	—
186	Dic₅₃	—	2	1	no	—
187	Dic₅₃	#431	1	2	yes	—
188	C₄×#131	—	53	1	no	—
189	#2164	—	53	2	no	—
190	#2164	—	53	2	no	—
191	#2163	—	53	1	no	—
192	#2163	—	53	1	no	—
193	#2163	—	53	1	no	—
194	#2163	D₅₃	1	1	yes	characteristic, complemented
195	#2151	—	53	2	no	—
196	#2113	—	53	2	no	—
197	C₄×M₃₂	—	53	1	no	—
198	C₂×#346	—	53	1	no	—
199	C₂×C₅₃	—	4	1	no	—
200	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
201	C₂×C₅₃	#3407	1	2	yes	—
202	D₅₃	—	2	2	no	—
203	D₅₃	—	4	1	no	—
204	C₂×#131	—	53	1	no	—
205	C₂×#131	—	53	1	no	—
206	C₄×#41	—	53	1	no	—
207	C₂×M₃₂	—	53	2	no	—
208	C₂²×C₁₆	—	53	2	no	—
209	#405	—	53	1	no	—
210	#405	—	53	1	no	—
211	#405	—	53	1	no	—
212	#405	—	53	1	no	—
213	#405	—	53	1	no	—
214	#405	—	53	1	no	—
215	C₄×M₁₆	—	53	1	no	—
216	C₄×M₁₆	—	53	1	no	—
217	C₄×M₁₆	—	53	1	no	—
218	C₂×C₄×C₈	—	53	1	no	—
219	C₂×C₄×C₈	—	53	1	no	—
220	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
221	C₄⋊C₁₆	—	53	2	no	—
222	C₄⋊C₁₆	—	53	2	no	—
223	C₄⋊C₁₆	—	53	2	no	—
224	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
225	C₂²⋊C₁₆	—	53	2	no	—
226	C₂²⋊C₁₆	—	53	2	no	—
227	C₂²⋊C₁₆	—	53	2	no	—
228	C₂²⋊C₁₆	C₂×D₅₃	1	2	yes	—
229	#346	—	53	2	no	—
230	#346	—	53	1	no	—
231	#346	—	53	1	no	—
232	#346	C₂×D₅₃	1	1	yes	characteristic
233	#346	C₂×D₅₃	1	1	yes	characteristic
234	C₄×C₁₆	—	53	2	no	—
235	C₅₃	#33421	1	1	yes	characteristic, complemented, Sylow
236	C₂×#41	—	53	1	no	—
237	#131	—	106	4	no	—
238	#131	—	106	4	no	—
239	C₂×M₁₆	—	53	1	no	—
240	C₂×M₁₆	—	53	1	no	—
241	C₂×M₁₆	—	53	1	no	—
242	C₂×M₁₆	—	53	1	no	—
243	C₂×M₁₆	—	53	1	no	—
244	C₂×M₁₆	—	53	1	no	—
245	C₂²×C₈	—	53	1	no	—
246	C₂²×C₈	—	53	1	no	—
247	C₂²×C₈	—	53	1	no	—
248	C₂²×C₈	—	53	1	no	—
249	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
250	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
251	C₄×Q₈	—	53	2	no	—
252	C₄×D₄	—	53	2	no	—
253	C₄×D₄	—	53	2	no	—
254	C₄×D₄	—	53	2	no	—
255	#117	—	53	1	no	—
256	#117	—	53	1	no	—
257	#117	—	53	1	no	—
258	C₂×C₄²	—	53	1	no	—
259	C₂×C₄²	—	53	1	no	—
260	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
261	M₃₂	—	106	4	no	—
262	C₂×C₁₆	—	53	2	no	—
263	C₂×C₁₆	—	53	2	no	—
264	C₂×C₁₆	—	106	4	no	—
265	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
266	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
267	#97	—	53	1	no	—
268	#97	—	53	1	no	—
269	#97	—	53	1	no	—
270	#97	—	53	1	no	—
271	#97	—	53	1	no	—
272	#97	—	53	1	no	—
273	C₄×C₈	—	53	1	no	—
274	C₄×C₈	—	53	1	no	—
275	C₄×C₈	—	53	1	no	—
276	C₄×C₈	—	53	1	no	—
277	C₄×C₈	—	53	1	no	—
278	C₄×C₈	—	53	1	no	—
279	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
280	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
281	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
282	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
283	#41	—	106	2	no	—
284	#41	—	106	2	no	—
285	C₂×Q₈	—	53	1	no	—
286	C₂×D₄	—	53	1	no	—
287	C₂×D₄	—	53	1	no	—
288	C₂×D₄	—	53	1	no	—
289	C₂²×C₄	—	53	1	no	—
290	C₂²×C₄	—	53	1	no	—
291	C₂²×C₄	—	53	2	no	—
292	C₂²×C₄	—	53	2	no	—
293	C₂²×C₄	C₈×D₅₃	1	2	yes	—
294	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
295	M₁₆	—	106	2	no	—
296	M₁₆	—	106	2	no	—
297	M₁₆	—	106	2	no	—
298	M₁₆	—	106	2	no	—
299	M₁₆	—	106	2	no	—
300	M₁₆	—	106	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	53	1	no	—
304	C₂×C₈	—	53	1	no	—
305	C₂×C₈	—	53	4	no	—
306	C₂×C₈	—	53	4	no	—
307	C₂×C₈	—	53	1	no	—
308	C₂×C₈	—	53	1	no	—
309	C₂×C₈	—	106	2	no	—
310	C₂×C₈	—	106	2	no	—
311	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
312	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
313	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
314	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
315	C₄⋊C₄	—	53	2	no	—
316	C₄⋊C₄	—	53	2	no	—
317	C₄⋊C₄	—	53	2	no	—
318	C₂²⋊C₄	—	53	2	no	—
319	C₂²⋊C₄	—	53	2	no	—
320	C₂²⋊C₄	—	53	2	no	—
321	C₄²	—	53	2	no	—
322	C₄²	—	53	2	no	—
323	C₄²	—	53	1	no	—
324	C₄²	—	53	1	no	—
325	C₄²	C₈×D₅₃	1	2	yes	—
326	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
327	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
328	C₁₆	—	2	2	no	—
329	C₁₆	—	2	2	no	—
330	C₁₆	—	106	2	no	—
331	C₁₆	—	106	2	no	—
332	C₂³	—	53	1	no	—
333	C₂³	—	53	1	no	—
334	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
335	Q₈	—	106	2	no	—
336	D₄	—	106	2	no	—
337	D₄	—	106	2	no	—
338	D₄	—	106	2	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	53	1	no	—
343	C₂×C₄	—	53	2	no	—
344	C₂×C₄	—	53	2	no	—
345	C₂×C₄	—	53	2	no	—
346	C₂×C₄	—	53	2	no	—
347	C₂×C₄	—	53	1	no	—
348	C₂×C₄	—	106	1	no	—
349	C₂×C₄	—	106	1	no	—
350	C₂×C₄	—	212	2	no	—
351	C₂×C₄	—	212	2	no	—
352	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
353	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
354	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	53	2	no	—
359	C₈	—	53	2	no	—
360	C₈	—	106	1	no	—
361	C₈	—	106	1	no	—
362	C₈	M₁₆×D₅₃	1	2	yes	—
363	C₈	#41×D₅₃	1	2	yes	—
364	C₂²	—	2	2	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	53	2	no	—
367	C₂²	—	106	2	no	—
368	C₂²	—	106	2	no	—
369	C₂²	—	106	1	no	—
370	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
371	C₄	—	2	2	no	—
372	C₄	—	2	2	no	—
373	C₄	—	2	1	no	—
374	C₄	—	53	2	no	—
375	C₄	—	106	2	no	—
376	C₄	—	106	2	no	—
377	C₄	—	106	1	no	—
378	C₄	#117×D₅₃	1	1	yes	characteristic
379	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
380	C₂	—	4	1	no	—
381	C₂	—	106	2	no	—
382	C₂	—	212	1	no	—
383	C₂	#431×D₅₃	1	1	yes	characteristic
384	C₂	#20287817	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2163	C₂
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2163	D₅₃
hypocenter	C₅₃	#33421
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
2. #20276521	C₂	semidirect
3. #20276519	C₂	semidirect
4. #20276518	C₂	semidirect
5. #20276516	C₂	semidirect
6. #20276516	C₂	semidirect
7. #20276472	C₂	semidirect
9. #20276308	C₂	semidirect
10. #346×D₅₃	C₂	semidirect
11. C₅₃×#2163	C₂	semidirect
194. #2163	D₅₃	semidirect
235. C₅₃	#33421	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133223	2
#20276521	2
#20276519	2
#20276518	2
#20276516	2
#20276472	2
#20276327	2
#20276308	2
#346×D₅₃	2
C₅₃×#2163	2
#33421	53