Subgroups — #20283725

Order 6784.

406 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	C₄×#11719947	C₂	1	1	yes	maximal, characteristic
2	#16810752	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810751	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810697	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810695	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810692	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₂²⋊D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810686	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810679	C₂	1	1	yes	maximal, characteristic, complemented
10	#16810677	C₂	1	1	yes	maximal, characteristic, complemented
11	#16810677	C₂	1	1	yes	maximal, characteristic, complemented
12	M₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#13133127	C₂	1	1	yes	maximal, characteristic, complemented
14	#97×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#435	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11719947	C₄	1	1	yes	characteristic
17	C₂×#11719917	C₂²	1	1	yes	characteristic
18	C₂²×#88322	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
21	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
22	#13133182	C₄	1	1	yes	characteristic
23	D₄×Dic₅₃	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
26	#13133177	C₄	1	1	yes	characteristic
27	#13133177	C₄	1	1	yes	characteristic
28	#13133175	C₄	1	1	yes	characteristic
29	#13133173	C₄	1	1	yes	characteristic
30	#13133173	C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
33	C₄²×D₅₃	C₂²	1	1	yes	characteristic
34	C₄⋊#88322	C₂²	1	1	yes	characteristic
35	#13133146	C₂²	1	1	yes	characteristic
36	#13133146	C₂²	1	1	yes	characteristic
37	#13133146	C₂²	1	1	yes	characteristic
38	#13133146	C₂²	1	1	yes	characteristic
39	C₂²⋊#88322	C₂²	1	1	yes	characteristic
40	C₂²⋊#88322	C₂²	1	1	yes	characteristic
41	#13133129	C₂²	1	1	yes	characteristic
42	#13133129	C₂²	1	1	yes	characteristic
43	#13133128	C₂²	1	1	yes	characteristic
44	#13133127	C₂²	1	1	yes	characteristic
45	#13133127	C₂²	1	1	yes	characteristic
46	#13133127	C₂²	1	1	yes	characteristic
47	#13133127	C₂²	1	1	yes	characteristic
48	#13133127	C₂²	1	1	yes	characteristic
49	#13133127	C₂²	1	1	yes	characteristic
50	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
58	#97×C₅₃	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
68	#11719947	—	2	4	no	—
69	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
72	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
73	M₁₆×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
81	C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂³	1	1	yes	characteristic
83	C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
85	#11719921	C₂×C₄	1	1	yes	characteristic
86	#11719921	C₂×C₄	1	1	yes	characteristic
87	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
88	#11719919	C₂×C₄	1	1	yes	characteristic
89	#11719919	C₂×C₄	1	1	yes	characteristic
90	#11719919	C₂×C₄	1	1	yes	characteristic
91	#11719919	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₅₃	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	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
101	#11719917	—	2	2	no	—
102	C₂×#88322	—	2	2	no	—
103	C₂×#88322	C₂³	1	1	yes	characteristic
104	C₂×#88322	C₂³	1	1	yes	characteristic
105	C₂×#88322	C₂³	1	1	yes	characteristic
106	C₂×#88322	C₂³	1	1	yes	characteristic
107	C₈×D₅₃	—	2	2	no	—
108	C₈×D₅₃	—	2	2	no	—
109	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
112	D₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
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	C₂²⋊D₅₃	—	2	2	no	—
123	C₂²⋊D₅₃	M₁₆	1	4	yes	complemented
124	C₂×Dic₅₃	—	2	2	no	—
125	C₂×Dic₅₃	—	2	2	no	—
126	C₂×Dic₅₃	—	4	1	no	—
127	C₂×Dic₅₃	—	4	1	no	—
128	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
132	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
133	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
134	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₄×D₅₃	—	2	2	no	—
136	C₄×D₅₃	—	2	1	no	—
137	C₄×D₅₃	—	2	1	no	—
138	C₄×D₅₃	—	4	1	no	—
139	Dic₁₀₆	—	2	2	no	—
140	C₈×C₅₃	—	2	1	no	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	—	2	1	no	—
143	C₈×C₅₃	#41	1	2	yes	complemented
144	#88322	—	2	1	no	—
145	#88322	—	2	1	no	—
146	#88322	—	2	1	no	—
147	#88322	#41	1	2	yes	complemented
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	—	2	1	no	—
152	C₂²×C₅₃	—	2	1	no	—
153	C₂²×C₅₃	C₂×M₁₆	1	2	yes	—
154	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
155	C₂×D₅₃	—	2	1	no	—
156	C₂×D₅₃	—	2	1	no	—
157	C₂×D₅₃	C₂×M₁₆	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	C₂×#41	1	1	yes	characteristic
163	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
164	Dic₅₃	—	2	1	no	—
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	C₂×M₁₆	1	2	yes	—
171	#2970	—	53	1	no	maximal, Sylow
172	C₂×C₅₃	—	2	2	no	—
173	C₂×C₅₃	—	4	1	no	—
174	C₂×C₅₃	C₄×#41	1	1	yes	characteristic
175	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
176	C₂×C₅₃	#568	1	1	yes	characteristic
177	D₅₃	—	2	2	no	—
178	C₄×#41	—	53	1	no	—
179	#446	—	53	1	no	—
180	#446	—	53	1	no	—
181	#435	—	53	1	no	—
182	#435	—	53	1	no	—
183	#435	—	53	1	no	—
184	#435	D₅₃	1	1	yes	characteristic, complemented
185	C₈×D₄	—	53	1	no	—
186	C₈×D₄	—	53	1	no	—
187	#432	—	53	1	no	—
188	#432	—	53	1	no	—
189	#432	—	53	1	no	—
190	C₄×M₁₆	—	53	1	no	—
191	C₂×#97	—	53	1	no	—
192	C₂×#97	—	53	1	no	—
193	C₅₃	#2970	1	1	yes	characteristic, complemented, Sylow
194	C₂×#41	—	53	1	no	—
195	C₂×M₁₆	—	53	1	no	—
196	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
197	C₂²×C₈	—	53	1	no	—
198	C₂²×C₈	—	53	1	no	—
199	C₂²×C₈	—	53	1	no	—
200	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
201	C₄×Q₈	—	53	1	no	—
202	C₄×Q₈	—	53	1	no	—
203	C₄×D₄	—	53	1	no	—
204	C₄×D₄	—	53	1	no	—
205	C₄×D₄	—	53	1	no	—
206	C₄×D₄	—	53	1	no	—
207	C₄×D₄	—	53	1	no	—
208	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
209	#117	—	53	1	no	—
210	#117	—	53	1	no	—
211	#117	—	53	1	no	—
212	C₂×C₄²	—	53	1	no	—
213	C₂×C₄²	—	53	1	no	—
214	C₂×C₄²	—	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	—	53	1	no	—
220	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
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	C₂×D₅₃	1	1	yes	characteristic
226	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
227	#97	—	53	1	no	—
228	#97	—	53	1	no	—
229	#97	—	53	1	no	—
230	#97	—	53	1	no	—
231	#97	—	53	1	no	—
232	#97	—	53	1	no	—
233	#97	—	53	1	no	—
234	#97	C₂×D₅₃	1	1	yes	characteristic
235	C₄×C₈	—	53	1	no	—
236	C₄×C₈	—	53	1	no	—
237	#41	—	106	4	no	—
238	C₂×Q₈	—	53	1	no	—
239	C₂×D₄	—	53	1	no	—
240	C₂×D₄	—	53	1	no	—
241	C₂×D₄	C₄×D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
248	C₂²×C₄	—	53	1	no	—
249	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
251	M₁₆	—	2	2	no	—
252	M₁₆	—	106	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	53	1	no	—
255	C₂×C₈	—	53	1	no	—
256	C₂×C₈	—	53	1	no	—
257	C₂×C₈	—	53	1	no	—
258	C₂×C₈	—	106	2	no	—
259	C₂×C₈	—	106	2	no	—
260	C₂×C₈	—	106	2	no	—
261	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
265	C₄⋊C₄	—	53	1	no	—
266	C₄⋊C₄	—	53	1	no	—
267	C₄⋊C₄	—	53	1	no	—
268	C₄⋊C₄	—	53	1	no	—
269	C₄⋊C₄	—	53	1	no	—
270	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
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	C₄×D₅₃	1	1	yes	characteristic
276	C₂²⋊C₄	C₄×D₅₃	1	1	yes	characteristic
277	C₄²	—	53	1	no	—
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	—	53	1	no	—
285	C₄²	—	53	1	no	—
286	C₄²	C₂²×D₅₃	1	1	yes	characteristic
287	C₂³	—	53	1	no	—
288	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
289	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
290	Q₈	—	106	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	53	4	no	—
293	D₄	—	106	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	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	—	53	1	no	—
303	C₂×C₄	—	53	1	no	—
304	C₂×C₄	—	53	1	no	—
305	C₂×C₄	—	106	2	no	—
306	C₂×C₄	—	106	2	no	—
307	C₂×C₄	—	106	2	no	—
308	C₂×C₄	—	106	1	no	—
309	C₂×C₄	—	106	1	no	—
310	C₂×C₄	—	212	1	no	—
311	C₂×C₄	—	212	1	no	—
312	C₂×C₄	—	212	1	no	—
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	C₂×C₄×D₅₃	1	1	yes	characteristic
317	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	53	2	no	—
322	C₈	—	106	1	no	—
323	C₈	—	106	1	no	—
324	C₈	—	106	1	no	—
325	C₈	#11719947	1	2	yes	complemented
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	53	2	no	—
332	C₂²	—	106	1	no	—
333	C₂²	—	106	1	no	—
334	C₂²	M₁₆×D₅₃	1	2	yes	—
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	—	53	2	no	—
341	C₄	—	106	1	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	C₂×#11719947	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	4	1	no	—
351	C₂	—	106	2	no	—
352	C₂	C₄×#11719947	1	1	yes	characteristic
353	C₂	C₂×M₁₆×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×#435	C₂
hypercenter	#435	D₅₃
hypocenter	C₅₃	#2970
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #435

Complements

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

N	H ≅ G/N	type
2. #16810752	C₂	semidirect
3. #16810751	C₂	semidirect
4. #16810697	C₂	semidirect
5. #16810695	C₂	semidirect
6. #16810692	C₂	semidirect
7. C₈×C₂²⋊D₅₃	C₂	semidirect
8. #16810686	C₂	semidirect
9. #16810679	C₂	semidirect
10. #16810677	C₂	semidirect
11. #16810677	C₂	semidirect
12. M₁₆×Dic₅₃	C₂	semidirect
13. C₂×#13133127	C₂	semidirect
14. #97×D₅₃	C₂	semidirect
15. C₅₃×#435	C₂	semidirect
123. C₂²⋊D₅₃	M₁₆	semidirect
143. C₈×C₅₃	#41	semidirect
147. #88322	#41	semidirect
184. #435	D₅₃	semidirect
193. C₅₃	#2970	semidirect
325. C₈	#11719947	semidirect