Subgroups — #20283371

Order 6784.

402 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#117×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#16810684	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810681	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810681	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810679	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810678	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810675	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810674	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810648	C₂	1	1	yes	maximal, characteristic, complemented
10	#16810648	C₂	1	1	yes	maximal, characteristic, complemented
11	#16810585	C₂	1	1	yes	maximal, characteristic
12	C₂²⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
13	#16810580	C₂	1	1	yes	maximal, characteristic, complemented
14	#16810577	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
17	C₂×#11719912	C₂²	1	1	yes	characteristic
18	C₂×#11719912	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
21	#13133176	C₂²	1	1	yes	characteristic
22	#13133175	C₄	1	1	yes	characteristic
23	#13133175	C₄	1	1	yes	characteristic
24	#13133174	C₂²	1	1	yes	characteristic
25	#13133174	C₂²	1	1	yes	characteristic
26	#13133173	C₄	1	1	yes	characteristic
27	#13133173	C₄	1	1	yes	characteristic
28	C₄⋊C₄×D₅₃	C₄	1	1	yes	characteristic
29	C₄⋊C₄×D₅₃	C₄	1	1	yes	characteristic
30	C₂²⋊C₄×D₅₃	C₄	1	1	yes	characteristic
31	C₂²⋊C₄×D₅₃	C₄	1	1	yes	characteristic
32	C₄²×D₅₃	C₂²	1	1	yes	characteristic
33	C₄²×D₅₃	C₂²	1	1	yes	characteristic
34	C₄⋊#88322	C₂²	1	1	yes	characteristic, complemented
35	C₄⋊#88322	C₂²	1	1	yes	characteristic, complemented
36	#13133146	C₂²	1	1	yes	characteristic, complemented
37	#13133146	C₂²	1	1	yes	characteristic, complemented
38	#13133146	C₂²	1	1	yes	characteristic, complemented
39	#13133146	C₂²	1	1	yes	characteristic, complemented
40	C₂²⋊#88322	C₂²	1	1	yes	characteristic
41	C₂²⋊#88322	C₂²	1	1	yes	characteristic
42	#13133129	C₂²	1	1	yes	characteristic
43	#13133129	C₂²	1	1	yes	characteristic
44	#13133129	C₂²	1	1	yes	characteristic
45	#13133129	C₂²	1	1	yes	characteristic
46	#13133128	C₂²	1	1	yes	characteristic, complemented
47	#13133127	C₂²	1	1	yes	characteristic, complemented
48	#13133127	C₂²	1	1	yes	characteristic, complemented
49	C₄×#88322	C₂²	1	1	yes	characteristic, complemented
50	C₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
52	#117×C₅₃	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
54	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
55	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
57	#97×C₅₃	C₂²	1	1	yes	characteristic, complemented
58	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₅₃	—	2	2	no	—
63	C₂×C₄×D₅₃	—	2	2	no	—
64	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₄²×C₅₃	C₂³	1	1	yes	characteristic
81	C₄²×C₅₃	C₂³	1	1	yes	characteristic
82	C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
84	#11719921	C₂×C₄	1	1	yes	characteristic
85	#11719921	C₂×C₄	1	1	yes	characteristic
86	#11719921	C₂×C₄	1	1	yes	characteristic
87	#11719921	C₂×C₄	1	1	yes	characteristic
88	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
89	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
90	#11719919	C₂×C₄	1	1	yes	characteristic
91	#11719919	C₂×C₄	1	1	yes	characteristic
92	#11719919	C₂×C₄	1	1	yes	characteristic
93	#11719919	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
96	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
97	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
98	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
99	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
100	C₂×#88322	C₂³	1	1	yes	characteristic
101	C₂×#88322	C₂³	1	1	yes	characteristic
102	C₂×#88322	C₂³	1	1	yes	characteristic
103	C₂×#88322	C₂³	1	1	yes	characteristic
104	#11719912	—	2	2	no	—
105	#11719912	—	2	2	no	—
106	C₈×D₅₃	—	2	2	no	—
107	C₈×D₅₃	—	2	2	no	—
108	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂²×D₅₃	—	2	2	no	—
110	C₂²×D₅₃	—	2	2	no	—
111	C₂²×D₅₃	—	4	1	no	—
112	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₅₃	—	2	2	no	—
123	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
125	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₄×D₅₃	—	2	2	no	—
130	C₄×D₅₃	—	2	2	no	—
131	C₄×D₅₃	—	2	2	no	—
132	C₄×D₅₃	—	2	1	no	—
133	C₄×D₅₃	—	2	1	no	—
134	C₄×D₅₃	—	2	2	no	—
135	C₄×D₅₃	—	4	1	no	—
136	C₄×D₅₃	—	4	1	no	—
137	C₄×D₅₃	#41	1	2	yes	—
138	C₄×D₅₃	#41	1	2	yes	—
139	C₈×C₅₃	—	2	1	no	—
140	C₈×C₅₃	—	2	1	no	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	—	2	1	no	—
143	#88322	—	2	1	no	—
144	#88322	—	2	1	no	—
145	#88322	—	2	1	no	—
146	#88322	—	2	1	no	—
147	C₂²×C₅₃	—	2	1	no	—
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
151	C₂×D₅₃	—	2	1	no	—
152	C₂×D₅₃	—	2	1	no	—
153	C₂×D₅₃	—	2	1	no	—
154	C₂×D₅₃	—	2	1	no	—
155	C₂×D₅₃	—	4	4	no	—
156	C₂×D₅₃	#117	1	2	yes	—
157	C₂×D₅₃	#131	1	2	yes	—
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	—	2	1	no	—
162	C₄×C₅₃	—	2	1	no	—
163	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
164	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
165	Dic₅₃	—	2	1	no	—
166	Dic₅₃	—	2	1	no	—
167	Dic₅₃	—	2	1	no	—
168	Dic₅₃	—	2	1	no	—
169	Dic₅₃	—	2	1	no	—
170	Dic₅₃	#117	1	2	yes	—
171	#2924	—	53	1	no	maximal, Sylow
172	C₂×C₅₃	—	4	1	no	—
173	C₂×C₅₃	C₂×#117	1	1	yes	characteristic
174	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
175	C₂×C₅₃	#568	1	1	yes	characteristic
176	D₅₃	—	2	2	no	—
177	D₅₃	—	2	2	no	—
178	C₂×#117	—	53	1	no	—
179	#433	—	53	1	no	—
180	#433	—	53	1	no	—
181	#433	—	53	1	no	—
182	#433	D₅₃	1	1	yes	characteristic, complemented
183	#432	—	53	1	no	—
184	#432	—	53	1	no	—
185	#431	—	53	1	no	—
186	#431	—	53	1	no	—
187	#424	—	53	1	no	—
188	#424	—	53	1	no	—
189	#407	—	53	1	no	—
190	C₂×C₂²⋊C₈	—	53	1	no	—
191	#405	—	53	1	no	—
192	#405	—	53	1	no	—
193	C₅₃	#2924	1	1	yes	characteristic, complemented, Sylow
194	C₂³×C₄	—	53	1	no	—
195	C₂×M₁₆	—	53	1	no	—
196	C₂×M₁₆	—	53	1	no	—
197	C₂²×C₈	—	53	1	no	—
198	C₂²×C₈	—	53	1	no	—
199	#117	—	53	1	no	—
200	#117	—	53	1	no	—
201	#117	—	53	1	no	—
202	#117	—	53	1	no	—
203	#117	—	53	1	no	—
204	#117	—	53	1	no	—
205	#117	—	53	1	no	—
206	#117	C₂×D₅₃	1	1	yes	characteristic
207	C₂×C₄⋊C₄	—	53	1	no	—
208	C₂×C₄⋊C₄	—	53	1	no	—
209	C₂×C₂²⋊C₄	—	53	1	no	—
210	C₂×C₂²⋊C₄	—	53	1	no	—
211	C₂×C₄²	—	53	1	no	—
212	C₂×C₄²	—	53	1	no	—
213	C₄⋊C₈	—	53	1	no	—
214	C₄⋊C₈	—	53	1	no	—
215	C₄⋊C₈	—	53	1	no	—
216	C₄⋊C₈	—	53	1	no	—
217	C₄⋊C₈	—	53	1	no	—
218	C₄⋊C₈	—	53	1	no	—
219	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
220	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
221	C₂²⋊C₈	—	53	1	no	—
222	C₂²⋊C₈	—	53	1	no	—
223	C₂²⋊C₈	—	53	1	no	—
224	C₂²⋊C₈	—	53	1	no	—
225	C₂²⋊C₈	—	53	1	no	—
226	C₂²⋊C₈	—	53	1	no	—
227	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
228	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
229	#97	—	53	1	no	—
230	#97	—	53	1	no	—
231	#97	—	53	1	no	—
232	#97	C₂×D₅₃	1	1	yes	characteristic, complemented
233	C₄×C₈	—	53	1	no	—
234	C₄×C₈	—	53	1	no	—
235	C₄×C₈	—	53	1	no	—
236	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
237	C₂⁴	—	53	1	no	—
238	C₂²×C₄	—	53	1	no	—
239	C₂²×C₄	—	53	1	no	—
240	C₂²×C₄	—	53	1	no	—
241	C₂²×C₄	—	53	1	no	—
242	C₂²×C₄	—	53	1	no	—
243	C₂²×C₄	—	53	1	no	—
244	C₂²×C₄	—	53	1	no	—
245	C₂²×C₄	—	53	1	no	—
246	C₂²×C₄	—	53	1	no	—
247	C₂²×C₄	—	106	2	no	—
248	C₂²×C₄	—	106	2	no	—
249	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
250	M₁₆	—	106	2	no	—
251	M₁₆	—	106	2	no	—
252	C₂×C₈	—	53	1	no	—
253	C₂×C₈	—	53	1	no	—
254	C₂×C₈	—	53	1	no	—
255	C₂×C₈	—	53	1	no	—
256	C₂×C₈	—	106	2	no	—
257	C₂×C₈	—	106	2	no	—
258	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
259	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
260	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
262	C₄⋊C₄	—	53	1	no	—
263	C₄⋊C₄	—	53	1	no	—
264	C₄⋊C₄	—	53	1	no	—
265	C₄⋊C₄	—	53	1	no	—
266	C₄⋊C₄	—	53	1	no	—
267	C₄⋊C₄	—	53	1	no	—
268	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
269	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
270	C₂²⋊C₄	—	53	1	no	—
271	C₂²⋊C₄	—	53	1	no	—
272	C₂²⋊C₄	—	53	1	no	—
273	C₂²⋊C₄	—	53	1	no	—
274	C₂²⋊C₄	—	53	1	no	—
275	C₂²⋊C₄	—	53	1	no	—
276	C₂²⋊C₄	C₄×D₅₃	1	1	yes	characteristic
277	C₂²⋊C₄	C₄×D₅₃	1	1	yes	characteristic
278	C₄²	—	53	1	no	—
279	C₄²	—	53	1	no	—
280	C₄²	—	53	1	no	—
281	C₄²	—	53	1	no	—
282	C₄²	—	53	1	no	—
283	C₄²	—	53	1	no	—
284	C₄²	C₂²×D₅₃	1	1	yes	characteristic
285	C₄²	C₂²×D₅₃	1	1	yes	characteristic
286	C₂³	—	53	1	no	—
287	C₂³	—	53	1	no	—
288	C₂³	—	106	2	no	—
289	C₂³	—	106	2	no	—
290	C₂³	—	212	1	no	—
291	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	53	1	no	—
295	C₂×C₄	—	53	2	no	—
296	C₂×C₄	—	53	2	no	—
297	C₂×C₄	—	53	1	no	—
298	C₂×C₄	—	53	1	no	—
299	C₂×C₄	—	53	1	no	—
300	C₂×C₄	—	53	1	no	—
301	C₂×C₄	—	53	1	no	—
302	C₂×C₄	—	106	2	no	—
303	C₂×C₄	—	106	2	no	—
304	C₂×C₄	—	106	2	no	—
305	C₂×C₄	—	106	2	no	—
306	C₂×C₄	—	106	2	no	—
307	C₂×C₄	—	106	1	no	—
308	C₂×C₄	—	106	1	no	—
309	C₂×C₄	—	212	1	no	—
310	C₂×C₄	—	212	1	no	—
311	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	106	1	no	—
322	C₈	—	106	1	no	—
323	C₈	—	106	1	no	—
324	C₈	—	106	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	53	2	no	—
329	C₂²	—	53	2	no	—
330	C₂²	—	106	1	no	—
331	C₂²	—	106	1	no	—
332	C₂²	—	106	1	no	—
333	C₂²	—	106	1	no	—
334	C₂²	—	212	4	no	—
335	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	53	2	no	—
342	C₄	—	106	1	no	—
343	C₄	—	106	1	no	—
344	C₄	—	106	1	no	—
345	C₄	—	106	1	no	—
346	C₄	—	106	1	no	—
347	C₄	#41×D₅₃	1	1	yes	characteristic
348	C₄	#41×D₅₃	1	1	yes	characteristic
349	C₂	—	4	1	no	—
350	C₂	—	106	2	no	—
351	C₂	—	106	2	no	—
352	C₂	#117×D₅₃	1	1	yes	characteristic
353	C₂	#131×D₅₃	1	1	yes	characteristic
354	C₂	#16811388	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#433	C₂
hypercenter	#433	D₅₃
hypocenter	C₅₃	#2924
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. #16810684	C₂	semidirect
3. #16810681	C₂	semidirect
4. #16810681	C₂	semidirect
5. #16810679	C₂	semidirect
6. #16810678	C₂	semidirect
7. #16810675	C₂	semidirect
8. #16810674	C₂	semidirect
9. #16810648	C₂	semidirect
10. #16810648	C₂	semidirect
13. #16810580	C₂	semidirect
14. #16810577	C₂	semidirect
15. C₅₃×#433	C₂	semidirect
34. C₄⋊#88322	C₂²	semidirect
35. C₄⋊#88322	C₂²	semidirect
36. #13133146	C₂²	semidirect
37. #13133146	C₂²	semidirect
38. #13133146	C₂²	semidirect
39. #13133146	C₂²	semidirect
46. #13133128	C₂²	semidirect
47. #13133127	C₂²	semidirect
48. #13133127	C₂²	semidirect
49. C₄×#88322	C₂²	semidirect
50. C₈×Dic₅₃	C₂²	semidirect
51. C₈×Dic₅₃	C₂²	semidirect
53. C₄⋊C₈×C₅₃	C₂²	semidirect
54. C₄⋊C₈×C₅₃	C₂²	semidirect
57. #97×C₅₃	C₂²	semidirect
58. C₄×C₈×C₅₃	C₂²	semidirect
182. #433	D₅₃	semidirect
193. C₅₃	#2924	semidirect
219. C₄⋊C₈	C₂×D₅₃	semidirect
220. C₄⋊C₈	C₂×D₅₃	semidirect
232. #97	C₂×D₅₃	semidirect
236. C₄×C₈	C₂×D₅₃	semidirect