Subgroups — #27537083

Order 13568.

706 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811041	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811039	C₂	1	1	yes	maximal, characteristic
3	#20283137	C₂	1	1	yes	maximal, characteristic
4	#20276474	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276472	C₂	1	2	yes	maximal, complemented
6	#20276472	C₂	1	2	yes	maximal, complemented
7	#20276470	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276469	C₂	1	2	yes	maximal, complemented
9	C₂×#16810441	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₁₆×D₅₃	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133225	C₄	1	1	yes	characteristic
13	C₂×#13133223	C₂²	1	1	yes	characteristic
14	#16811091	C₄	1	1	yes	characteristic
15	#16811041	—	2	4	no	—
16	#16811039	—	2	4	no	—
17	C₂×#13133157	C₂²	1	1	yes	characteristic
18	C₂×#13133156	C₂²	1	2	yes	—
19	C₂×#13133156	C₂²	1	2	yes	—
20	C₂²×#11719908	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
22	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
23	#16810580	C₄	1	2	yes	—
24	#16810579	C₄	1	1	yes	characteristic
25	#16810577	C₂²	1	2	yes	—
26	#16810575	C₄	1	2	yes	—
27	M₁₆×Dic₅₃	C₄	1	1	yes	characteristic
28	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
29	C₄×C₈×D₅₃	C₂²	1	2	yes	—
30	C₄⋊#11719908	C₂²	1	2	yes	—
31	C₄⋊#11719908	C₂²	1	2	yes	—
32	#16810441	C₂²	1	1	yes	characteristic
33	#16810441	C₂²	1	2	yes	—
34	#16810441	C₂²	1	1	yes	characteristic
35	#16810441	C₂²	1	1	yes	characteristic
36	#16810441	C₂²	1	1	yes	characteristic
37	#16810441	C₂²	1	2	yes	—
38	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
39	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#405	C₂²	1	1	yes	characteristic
41	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
42	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
43	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
44	#13133225	—	2	4	no	—
45	#13133223	D₄	1	4	yes	—
46	#13133223	Q₈	1	4	yes	—
47	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
48	C₂×#11719912	C₂×C₄	1	2	yes	—
49	C₂×#11719912	C₂×C₄	1	2	yes	—
50	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₅₃	C₂³	1	2	yes	—
53	#13133182	C₈	1	1	yes	characteristic
54	#13133182	C₈	1	1	yes	characteristic
55	#13133179	C₈	1	1	yes	characteristic
56	#13133179	C₈	1	1	yes	characteristic
57	#13133177	C₈	1	2	yes	—
58	#13133177	C₈	1	2	yes	—
59	#13133174	C₂×C₄	1	2	yes	—
60	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄²×D₅₃	C₂×C₄	1	2	yes	—
62	#13133157	—	2	2	no	—
63	#13133156	—	2	4	no	—
64	#13133156	—	2	4	no	—
65	C₂×#11719908	—	2	2	no	—
66	C₂×#11719908	C₂³	1	1	yes	characteristic
67	C₂×#11719908	C₂³	1	2	yes	—
68	C₂×#11719908	C₂³	1	1	yes	characteristic
69	C₁₆×D₅₃	—	2	4	no	—
70	C₁₆×D₅₃	—	2	4	no	—
71	#13133128	C₂×C₄	1	2	yes	—
72	#13133127	C₂×C₄	1	2	yes	—
73	C₄×#88322	C₂×C₄	1	2	yes	—
74	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
75	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
76	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
77	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
78	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
79	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
81	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
82	M₃₂×C₅₃	—	2	2	no	—
83	C₂×C₁₆×C₅₃	—	2	2	no	—
84	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
85	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
86	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
87	#97×C₅₃	C₂×C₄	1	2	yes	—
88	C₄×C₈×C₅₃	C₂³	1	2	yes	—
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
91	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
92	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
93	#11719945	C₄⋊C₄	1	4	yes	—
94	#11719945	M₁₆	1	4	yes	—
95	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
96	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
97	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
98	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
99	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
100	M₁₆×C₅₃	—	2	2	no	—
101	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
105	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
107	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
108	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
109	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
110	#11719921	C₂×C₈	1	2	yes	—
111	#11719921	C₂×C₈	1	2	yes	—
112	#11719919	C₂×C₈	1	2	yes	—
113	#11719919	C₂×C₈	1	2	yes	—
114	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
115	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
116	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
117	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
118	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
120	#11719917	C₄⋊C₄	1	4	yes	—
121	C₂×#88322	—	2	2	no	—
122	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
123	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
124	C₂×#88322	C₂²×C₄	1	2	yes	—
125	#11719912	—	2	4	no	—
126	#11719912	C₄⋊C₄	1	8	yes	—
127	C₈×D₅₃	—	2	4	no	—
128	C₈×D₅₃	C₂×D₄	1	4	yes	—
129	C₈×D₅₃	C₂×Q₈	1	4	yes	—
130	C₁₆×C₅₃	—	2	1	no	—
131	C₁₆×C₅₃	—	2	1	no	—
132	C₁₆×C₅₃	—	2	2	no	—
133	#11719908	—	2	1	no	—
134	#11719908	—	2	2	no	—
135	#11719908	—	2	1	no	—
136	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
137	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
138	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
140	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
141	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
143	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
144	C₂²⋊D₅₃	C₄⋊C₈	1	8	yes	—
145	C₂×Dic₅₃	—	2	2	no	—
146	C₂×Dic₅₃	—	4	1	no	—
147	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
148	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
149	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
150	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
151	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
152	D₂₁₂	C₄⋊C₈	1	4	yes	—
153	C₄×D₅₃	—	2	4	no	—
154	C₄×D₅₃	—	4	2	no	—
155	C₄×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
156	C₄×D₅₃	C₂×M₁₆	1	4	yes	—
157	Dic₁₀₆	C₄⋊C₈	1	4	yes	—
158	C₈×C₅₃	—	2	2	no	—
159	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
160	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
161	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
162	#88322	—	2	1	no	—
163	#88322	—	2	1	no	—
164	#88322	C₂×C₄⋊C₄	1	4	yes	—
165	#33398	—	53	1	no	maximal, Sylow
166	C₂²×C₅₃	—	2	1	no	—
167	C₂²×C₅₃	—	2	1	no	—
168	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
169	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
170	C₂×D₅₃	—	2	2	no	—
171	C₂×D₅₃	—	2	2	no	—
172	C₂×D₅₃	C₂×C₄⋊C₈	1	4	yes	—
173	C₄×C₅₃	—	2	2	no	—
174	C₄×C₅₃	—	2	2	no	—
175	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
176	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
177	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
178	Dic₅₃	—	2	1	no	—
179	Dic₅₃	—	2	1	no	—
180	Dic₅₃	—	2	1	no	—
181	Dic₅₃	C₂×C₄⋊C₈	1	4	yes	—
182	Dic₅₃	#504	1	2	yes	complemented
183	C₂×#504	—	53	1	no	—
184	C₂×#504	—	53	1	no	—
185	C₄×#131	—	53	1	no	—
186	#2151	—	53	1	no	—
187	#2151	—	53	2	no	—
188	#2151	—	53	2	no	—
189	#2151	D₅₃	1	1	yes	characteristic, complemented
190	#2150	—	53	1	no	—
191	#2150	—	53	2	no	—
192	C₂×C₄⋊C₁₆	—	53	1	no	—
193	C₂×C₄⋊C₁₆	—	53	2	no	—
194	C₂×C₅₃	—	2	2	no	—
195	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
196	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
197	C₂×C₅₃	#3407	1	1	yes	characteristic
198	D₅₃	—	2	4	no	—
199	C₂×#131	—	53	1	no	—
200	C₂×#131	—	53	1	no	—
201	C₄×#41	—	53	1	no	—
202	#504	—	106	4	no	—
203	#504	—	106	4	no	—
204	C₂×M₃₂	—	53	1	no	—
205	C₂×M₃₂	—	53	2	no	—
206	C₂×M₃₂	—	53	2	no	—
207	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
208	C₂²×C₁₆	—	53	1	no	—
209	C₂²×C₁₆	—	53	2	no	—
210	C₂²×C₁₆	—	53	2	no	—
211	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
212	#405	—	53	1	no	—
213	#405	—	53	2	no	—
214	#405	—	53	2	no	—
215	#405	C₂×D₅₃	1	1	yes	characteristic
216	C₄×M₁₆	—	53	1	no	—
217	C₄×M₁₆	—	53	2	no	—
218	C₂×C₄×C₈	—	53	1	no	—
219	C₂×C₄×C₈	—	53	2	no	—
220	C₄⋊C₁₆	—	53	1	no	—
221	C₄⋊C₁₆	—	53	2	no	—
222	C₄⋊C₁₆	—	53	1	no	—
223	C₄⋊C₁₆	—	53	1	no	—
224	C₄⋊C₁₆	—	53	1	no	—
225	C₄⋊C₁₆	—	53	2	no	—
226	C₄⋊C₁₆	—	53	2	no	—
227	C₄⋊C₁₆	—	53	2	no	—
228	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
229	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
230	C₅₃	#33398	1	1	yes	characteristic, complemented, Sylow
231	C₂×#41	—	53	1	no	—
232	#131	—	53	4	no	—
233	#131	—	53	4	no	—
234	#131	—	106	4	no	—
235	C₂×M₁₆	—	53	1	no	—
236	C₂×M₁₆	—	53	2	no	—
237	C₂×M₁₆	—	53	2	no	—
238	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
239	C₂²×C₈	—	53	2	no	—
240	C₂²×C₈	—	53	1	no	—
241	C₂²×C₈	—	53	2	no	—
242	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
243	C₄×Q₈	—	53	1	no	—
244	C₄×Q₈	—	53	1	no	—
245	C₄×D₄	—	53	1	no	—
246	C₄×D₄	—	53	1	no	—
247	C₄×D₄	—	53	2	no	—
248	C₄×D₄	—	53	2	no	—
249	#117	—	53	2	no	—
250	#117	C₄×D₅₃	1	1	yes	characteristic
251	C₂×C₄²	—	53	1	no	—
252	C₂×C₄²	—	53	2	no	—
253	M₃₂	—	2	2	no	—
254	M₃₂	—	106	4	no	—
255	M₃₂	—	106	4	no	—
256	M₃₂	—	106	2	no	—
257	C₂×C₁₆	—	2	2	no	—
258	C₂×C₁₆	—	53	1	no	—
259	C₂×C₁₆	—	53	2	no	—
260	C₂×C₁₆	—	53	1	no	—
261	C₂×C₁₆	—	106	4	no	—
262	C₂×C₁₆	—	106	4	no	—
263	C₂×C₁₆	—	106	2	no	—
264	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
265	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
266	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
267	#97	—	53	2	no	—
268	#97	—	53	2	no	—
269	#97	C₄×D₅₃	1	2	yes	—
270	C₄×C₈	—	53	2	no	—
271	C₄×C₈	—	53	1	no	—
272	C₄×C₈	—	53	1	no	—
273	C₄×C₈	—	53	1	no	—
274	C₄×C₈	—	53	1	no	—
275	C₄×C₈	—	53	2	no	—
276	C₄×C₈	C₂²×D₅₃	1	2	yes	—
277	#41	—	53	4	no	—
278	#41	—	53	4	no	—
279	C₂×Q₈	—	53	1	no	—
280	C₂×D₄	—	53	2	no	—
281	C₂×D₄	—	53	1	no	—
282	C₂²×C₄	—	53	2	no	—
283	C₂²×C₄	—	53	1	no	—
284	C₂²×C₄	—	53	1	no	—
285	C₂²×C₄	—	53	2	no	—
286	C₂²×C₄	—	53	2	no	—
287	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
288	M₁₆	—	2	2	no	—
289	M₁₆	—	53	4	no	—
290	M₁₆	—	53	8	no	—
291	M₁₆	—	106	4	no	—
292	C₂×C₈	—	53	2	no	—
293	C₂×C₈	—	53	4	no	—
294	C₂×C₈	—	53	4	no	—
295	C₂×C₈	—	53	1	no	—
296	C₂×C₈	—	53	1	no	—
297	C₂×C₈	—	106	4	no	—
298	C₂×C₈	—	106	2	no	—
299	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
300	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
301	C₂×C₈	D₄×D₅₃	1	2	yes	—
302	C₂×C₈	Q₈×D₅₃	1	2	yes	—
303	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
304	C₄⋊C₄	—	53	2	no	—
305	C₄⋊C₄	—	53	2	no	—
306	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
307	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
308	C₂²⋊C₄	—	53	2	no	—
309	C₂²⋊C₄	—	53	2	no	—
310	C₂²⋊C₄	C₈×D₅₃	1	2	yes	—
311	C₄²	—	53	2	no	—
312	C₄²	—	53	1	no	—
313	C₄²	—	53	2	no	—
314	C₄²	—	53	1	no	—
315	C₄²	—	53	1	no	—
316	C₄²	—	53	1	no	—
317	C₄²	C₂×C₄×D₅₃	1	2	yes	—
318	C₁₆	—	2	1	no	—
319	C₁₆	—	2	1	no	—
320	C₁₆	—	2	2	no	—
321	C₁₆	—	106	1	no	—
322	C₁₆	—	106	2	no	—
323	C₁₆	—	106	1	no	—
324	C₂³	—	53	2	no	—
325	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
326	Q₈	—	53	4	no	—
327	D₄	—	53	8	no	—
328	D₄	—	53	4	no	—
329	C₂×C₄	—	53	1	no	—
330	C₂×C₄	—	53	2	no	—
331	C₂×C₄	—	53	4	no	—
332	C₂×C₄	—	53	4	no	—
333	C₂×C₄	—	53	1	no	—
334	C₂×C₄	—	53	1	no	—
335	C₂×C₄	—	53	1	no	—
336	C₂×C₄	—	106	4	no	—
337	C₂×C₄	—	106	2	no	—
338	C₂×C₄	—	212	2	no	—
339	C₂×C₄	—	212	1	no	—
340	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
341	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
342	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
343	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
344	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
345	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
346	C₈	—	2	2	no	—
347	C₈	—	53	4	no	—
348	C₈	—	106	1	no	—
349	C₈	—	106	1	no	—
350	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
351	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
352	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	53	4	no	—
356	C₂²	—	106	2	no	—
357	C₂²	—	106	2	no	—
358	C₂²	C₄⋊C₈×D₅₃	1	2	yes	—
359	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
360	C₄	—	2	2	no	—
361	C₄	—	2	2	no	—
362	C₄	—	53	2	no	—
363	C₄	—	53	4	no	—
364	C₄	—	106	1	no	—
365	C₄	—	106	1	no	—
366	C₄	—	106	1	no	—
367	C₄	C₄⋊C₈×D₅₃	1	2	yes	—
368	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
369	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
370	C₂	—	2	2	no	—
371	C₂	—	106	4	no	—
372	C₂	C₂×C₄⋊C₈×D₅₃	1	1	yes	characteristic
373	C₂	#504×D₅₃	1	1	yes	characteristic
374	C₂	#20287816	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2151	C₂
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2151	D₅₃
hypocenter	C₅₃	#33398
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2151

Complements

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

N	H ≅ G/N	type
4. #20276474	C₂	semidirect
5. #20276472	C₂	semidirect
6. #20276472	C₂	semidirect
7. #20276470	C₂	semidirect
8. #20276469	C₂	semidirect
9. C₂×#16810441	C₂	semidirect
10. C₄⋊C₁₆×D₅₃	C₂	semidirect
11. C₅₃×#2151	C₂	semidirect
182. Dic₅₃	#504	semidirect
189. #2151	D₅₃	semidirect
230. C₅₃	#33398	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811041	2
C₂×#16811039	2
#20283137	2
#20276474	2
#20276472	2
#20276470	2
#20276469	2
C₂×#16810441	2
C₄⋊C₁₆×D₅₃	2
C₅₃×#2151	2
#33398	53