Subgroups — #27536736

Order 13568.

834 conjugacy classes of subgroups.

The subgroup lattice has 334 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	#20276313	C₂	1	2	yes	maximal, complemented
5	#20276312	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276310	C₂	1	2	yes	maximal, complemented
7	#20276308	C₂	1	2	yes	maximal, complemented
8	M₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₁₆×D₅₃	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2108	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	C₄	1	8	yes	complemented
16	#16811039	C₄	1	8	yes	complemented
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	#16810399	C₂²	1	2	yes	—
31	#16810398	C₂²	1	2	yes	—
32	C₄×#11719908	C₂²	1	2	yes	—
33	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₅₃	C₂²	1	2	yes	—
35	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
36	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
37	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
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×#346	C₂²	1	2	yes	—
42	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
43	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
44	#13133225	C₈	1	8	yes	—
45	#13133223	C₂×C₄	1	4	yes	—
46	#13133223	C₂×C₄	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	2	yes	—
54	#13133179	C₈	1	2	yes	—
55	#13133177	C₈	1	4	yes	—
56	#13133174	C₂×C₄	1	2	yes	—
57	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄²×D₅₃	C₂×C₄	1	2	yes	—
59	#13133157	C₂×C₄	1	4	yes	complemented
60	#13133156	C₂×C₄	1	8	yes	complemented
61	#13133156	C₂×C₄	1	8	yes	complemented
62	C₂×#11719908	C₂×C₄	1	4	yes	complemented
63	C₂×#11719908	C₂³	1	1	yes	characteristic
64	C₂×#11719908	C₂³	1	2	yes	—
65	C₂×#11719908	C₂³	1	1	yes	characteristic
66	C₁₆×D₅₃	C₂×C₄	1	8	yes	complemented
67	C₁₆×D₅₃	C₂×C₄	1	8	yes	complemented
68	#13133128	C₂×C₄	1	2	yes	—
69	#13133127	C₂×C₄	1	2	yes	—
70	C₄×#88322	C₂×C₄	1	2	yes	—
71	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
72	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
73	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
74	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
75	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
76	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
79	M₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
80	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
81	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
83	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
84	#97×C₅₃	C₂×C₄	1	2	yes	—
85	C₄×C₈×C₅₃	C₂³	1	2	yes	—
86	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
88	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
89	#11719945	C₄²	1	4	yes	—
90	#11719945	C₂×C₈	1	4	yes	—
91	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
92	C₂×C₄×D₅₃	C₂×C₈	1	4	yes	—
93	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
94	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
95	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
96	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
102	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
103	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
104	#11719921	C₂×C₈	1	4	yes	—
105	#11719919	C₂×C₈	1	4	yes	—
106	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
107	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
108	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
110	#11719917	C₄²	1	4	yes	—
111	C₂×#88322	C₂×C₈	1	4	yes	—
112	C₂×#88322	C₂²×C₄	1	2	yes	—
113	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
114	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
115	#11719912	C₄²	1	8	yes	—
116	#11719912	C₂×C₈	1	8	yes	—
117	C₈×D₅₃	C₂×C₈	1	8	yes	—
118	C₈×D₅₃	C₂²×C₄	1	4	yes	—
119	C₈×D₅₃	C₂²×C₄	1	4	yes	—
120	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
121	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
122	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
123	#11719908	C₂²×C₄	1	2	yes	—
124	#11719908	C₂²×C₄	1	4	yes	—
125	#11719908	C₂²×C₄	1	2	yes	—
126	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
127	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
130	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
133	C₂²⋊D₅₃	C₄×C₈	1	8	yes	—
134	C₂×Dic₅₃	—	2	4	no	—
135	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
136	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
137	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
138	D₂₁₂	C₄×C₈	1	4	yes	—
139	C₄×D₅₃	—	2	8	no	—
140	C₄×D₅₃	C₂×C₄²	1	4	yes	—
141	C₄×D₅₃	C₂²×C₈	1	4	yes	—
142	Dic₁₀₆	C₄×C₈	1	4	yes	—
143	C₈×C₅₃	C₂×C₄²	1	2	yes	—
144	C₈×C₅₃	C₂²×C₈	1	4	yes	—
145	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
146	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
147	#88322	C₂×C₄²	1	4	yes	—
148	#88322	C₂²×C₈	1	2	yes	—
149	#88322	C₂²×C₈	1	2	yes	—
150	C₄×#504	—	53	1	no	maximal, Sylow
151	C₂²×C₅₃	—	2	2	no	—
152	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
153	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
154	C₂×D₅₃	—	2	4	no	—
155	C₂×D₅₃	C₂×C₄×C₈	1	4	yes	—
156	C₄×C₅₃	—	2	4	no	—
157	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
158	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
159	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
160	Dic₅₃	—	2	2	no	—
161	Dic₅₃	C₂×C₄×C₈	1	4	yes	—
162	Dic₅₃	#504	1	4	yes	complemented
163	C₂×#504	—	53	1	no	—
164	C₂×#504	—	53	1	no	—
165	C₄×#131	—	53	1	no	—
166	#2108	—	53	1	no	—
167	#2108	—	53	2	no	—
168	#2108	—	53	2	no	—
169	#2108	D₅₃	1	1	yes	characteristic, complemented
170	C₄×M₃₂	—	53	1	no	—
171	C₄×M₃₂	—	53	2	no	—
172	C₂×C₄×C₁₆	—	53	1	no	—
173	C₂×C₄×C₁₆	—	53	2	no	—
174	C₂×C₅₃	—	2	2	no	—
175	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
176	C₂×C₅₃	C₂×#504	1	2	yes	—
177	D₅₃	—	2	4	no	—
178	C₂×#131	—	53	1	no	—
179	C₂×#131	—	53	1	no	—
180	C₄×#41	—	53	1	no	—
181	#504	—	53	8	no	—
182	#504	—	53	8	no	—
183	C₂×M₃₂	—	53	1	no	—
184	C₂×M₃₂	—	53	2	no	—
185	C₂×M₃₂	—	53	2	no	—
186	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
187	C₂²×C₁₆	—	53	1	no	—
188	C₂²×C₁₆	—	53	2	no	—
189	C₂²×C₁₆	—	53	2	no	—
190	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
191	#405	—	53	1	no	—
192	#405	—	53	2	no	—
193	#405	—	53	2	no	—
194	#405	C₂×D₅₃	1	1	yes	characteristic
195	C₄×M₁₆	—	53	1	no	—
196	C₄×M₁₆	—	53	2	no	—
197	C₂×C₄×C₈	—	53	1	no	—
198	C₂×C₄×C₈	—	53	2	no	—
199	#346	—	53	2	no	—
200	#346	—	53	2	no	—
201	#346	C₂×D₅₃	1	2	yes	—
202	C₄×C₁₆	—	53	1	no	—
203	C₄×C₁₆	—	53	2	no	—
204	C₄×C₁₆	—	53	1	no	—
205	C₄×C₁₆	—	53	1	no	—
206	C₄×C₁₆	—	53	1	no	—
207	C₄×C₁₆	—	53	2	no	—
208	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
209	C₅₃	C₄×#504	1	1	yes	characteristic, complemented, Sylow
210	C₂×#41	—	53	1	no	—
211	#131	—	53	8	no	—
212	#131	—	53	4	no	—
213	#131	—	53	4	no	—
214	C₂×M₁₆	—	53	1	no	—
215	C₂×M₁₆	—	53	2	no	—
216	C₂×M₁₆	—	53	2	no	—
217	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
218	C₂²×C₈	—	53	1	no	—
219	C₂²×C₈	—	53	2	no	—
220	C₂²×C₈	—	53	2	no	—
221	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
222	C₄×Q₈	—	53	2	no	—
223	C₄×D₄	—	53	2	no	—
224	C₄×D₄	—	53	4	no	—
225	#117	—	53	2	no	—
226	#117	C₄×D₅₃	1	1	yes	characteristic
227	C₂×C₄²	—	53	1	no	—
228	C₂×C₄²	—	53	2	no	—
229	M₃₂	—	53	4	no	—
230	M₃₂	—	53	8	no	—
231	M₃₂	—	53	8	no	—
232	M₃₂	C₄×D₅₃	1	4	yes	complemented
233	C₂×C₁₆	—	53	4	no	—
234	C₂×C₁₆	—	53	1	no	—
235	C₂×C₁₆	—	53	1	no	—
236	C₂×C₁₆	—	53	2	no	—
237	C₂×C₁₆	—	53	8	no	—
238	C₂×C₁₆	—	53	8	no	—
239	C₂×C₁₆	C₄×D₅₃	1	4	yes	complemented
240	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
241	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
242	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
243	#97	—	53	2	no	—
244	#97	—	53	2	no	—
245	#97	C₄×D₅₃	1	2	yes	—
246	C₄×C₈	—	53	1	no	—
247	C₄×C₈	—	53	2	no	—
248	C₄×C₈	—	53	1	no	—
249	C₄×C₈	—	53	1	no	—
250	C₄×C₈	—	53	1	no	—
251	C₄×C₈	—	53	2	no	—
252	C₄×C₈	C₂²×D₅₃	1	2	yes	—
253	#41	—	53	4	no	—
254	#41	—	53	4	no	—
255	C₂×Q₈	—	53	1	no	—
256	C₂×D₄	—	53	2	no	—
257	C₂×D₄	—	53	1	no	—
258	C₂²×C₄	—	53	2	no	—
259	C₂²×C₄	—	53	2	no	—
260	C₂²×C₄	—	53	4	no	—
261	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
262	M₁₆	—	53	4	no	—
263	M₁₆	—	53	8	no	—
264	M₁₆	—	53	8	no	—
265	M₁₆	C₈×D₅₃	1	4	yes	—
266	C₂×C₈	—	53	4	no	—
267	C₂×C₈	—	53	1	no	—
268	C₂×C₈	—	53	2	no	—
269	C₂×C₈	—	53	1	no	—
270	C₂×C₈	—	53	8	no	—
271	C₂×C₈	—	53	4	no	—
272	C₂×C₈	—	53	4	no	—
273	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
274	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
275	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
276	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
277	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
278	C₄⋊C₄	—	53	4	no	—
279	C₄⋊C₄	C₈×D₅₃	1	2	yes	—
280	C₂²⋊C₄	—	53	4	no	—
281	C₂²⋊C₄	C₈×D₅₃	1	2	yes	—
282	C₄²	—	53	4	no	—
283	C₄²	—	53	2	no	—
284	C₄²	—	53	1	no	—
285	C₄²	—	53	1	no	—
286	C₄²	C₂×C₄×D₅₃	1	2	yes	—
287	C₁₆	—	53	2	no	—
288	C₁₆	—	53	2	no	—
289	C₁₆	—	53	4	no	—
290	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
291	C₁₆	C₂×C₄×D₅₃	1	4	yes	—
292	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
293	C₂³	—	53	2	no	—
294	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
295	Q₈	—	53	4	no	—
296	D₄	—	53	8	no	—
297	D₄	—	53	4	no	—
298	C₂×C₄	—	53	2	no	—
299	C₂×C₄	—	53	2	no	—
300	C₂×C₄	—	53	4	no	—
301	C₂×C₄	—	53	4	no	—
302	C₂×C₄	—	53	2	no	—
303	C₂×C₄	—	106	8	no	—
304	C₂×C₄	—	106	4	no	—
305	C₂×C₄	C₄²×D₅₃	1	2	yes	—
306	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
307	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
309	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
310	C₈	—	53	4	no	—
311	C₈	—	53	2	no	—
312	C₈	—	53	2	no	—
313	C₈	C₄²×D₅₃	1	2	yes	—
314	C₈	C₂×C₈×D₅₃	1	4	yes	—
315	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
316	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
317	C₂²	—	2	2	no	—
318	C₂²	—	53	4	no	—
319	C₂²	—	106	4	no	—
320	C₂²	C₄×C₈×D₅₃	1	2	yes	—
321	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
322	C₄	—	2	4	no	—
323	C₄	—	53	4	no	—
324	C₄	—	53	4	no	—
325	C₄	—	106	2	no	—
326	C₄	C₄×C₈×D₅₃	1	2	yes	—
327	C₄	C₂×C₄²×D₅₃	1	1	yes	characteristic
328	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
329	C₂	—	2	2	no	—
330	C₂	—	106	4	no	—
331	C₂	C₂×C₄×C₈×D₅₃	1	1	yes	characteristic
332	C₂	#504×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2108	C₂
socle	C₂²×C₅₃	C₂³×C₈
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2108	D₅₃
hypocenter	C₅₃	C₄×#504
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
4. #20276313	C₂	semidirect
5. #20276312	C₂	semidirect
6. #20276310	C₂	semidirect
7. #20276308	C₂	semidirect
8. M₃₂×Dic₅₃	C₂	semidirect
9. C₂×C₁₆×Dic₅₃	C₂	semidirect
10. C₄×C₁₆×D₅₃	C₂	semidirect
11. C₅₃×#2108	C₂	semidirect
15. #16811041	C₄	semidirect
16. #16811039	C₄	semidirect
59. #13133157	C₂×C₄	semidirect
60. #13133156	C₂×C₄	semidirect
61. #13133156	C₂×C₄	semidirect
62. C₂×#11719908	C₂×C₄	semidirect
66. C₁₆×D₅₃	C₂×C₄	semidirect
67. C₁₆×D₅₃	C₂×C₄	semidirect
79. M₃₂×C₅₃	C₂×C₄	semidirect
80. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
162. Dic₅₃	#504	semidirect
169. #2108	D₅₃	semidirect
209. C₅₃	C₄×#504	semidirect
232. M₃₂	C₄×D₅₃	semidirect
239. C₂×C₁₆	C₄×D₅₃	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
#20276313	2
#20276312	2
#20276310	2
#20276308	2
M₃₂×Dic₅₃	2
C₂×C₁₆×Dic₅₃	2
C₄×C₁₆×D₅₃	2
C₅₃×#2108	2
C₄×#504	53