Subgroups — #20285184

Order 6784.

564 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	C₂×#13133246	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13133244	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13133238	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×SD₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811198	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811186	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811015	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811014	C₂	1	1	yes	maximal, characteristic, complemented
9	#16811010	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811009	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811007	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811004	C₂	1	1	yes	maximal, characteristic, complemented
13	#16810994	C₂	1	1	yes	maximal, characteristic, complemented
14	#16810577	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11719949	C₂²	1	1	yes	characteristic
17	C₂×#11719947	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₅₃	C₂²	1	1	yes	characteristic
19	C₂×D₄×D₅₃	C₂²	1	1	yes	characteristic
20	#13133246	—	2	4	no	—
21	#13133244	—	2	4	no	—
22	#13133238	—	2	4	no	—
23	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic, complemented
24	C₂×#11719925	C₂²	1	1	yes	characteristic, complemented
25	C₂×#11719925	C₂²	1	1	yes	characteristic, complemented
26	C₂×#11719913	C₂²	1	1	yes	characteristic
27	C₂×#11719913	C₂²	1	1	yes	characteristic, complemented
28	C₂×#11719923	C₂²	1	1	yes	characteristic
29	C₂×#11719923	C₂²	1	1	yes	characteristic, complemented
30	C₂×#11719924	C₂²	1	1	yes	characteristic
31	C₂×#11719924	C₂²	1	1	yes	characteristic
32	SD₁₆×D₅₃	—	2	4	no	—
33	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
34	C₂×#11719922	C₂²	1	1	yes	characteristic
35	C₂×#11719922	C₂²	1	1	yes	characteristic, complemented
36	C₂×#11719912	C₂²	1	1	yes	characteristic
37	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
38	#13133215	C₂²	1	1	yes	characteristic, complemented
39	#13133213	C₂²	1	1	yes	characteristic
40	#13133206	C₂²	1	1	yes	characteristic, complemented
41	#13133205	C₂²	1	1	yes	characteristic
42	#13133205	C₂²	1	1	yes	characteristic, complemented
43	#13133204	C₂²	1	1	yes	characteristic
44	#13133202	—	2	2	no	—
45	#13133195	—	2	2	no	—
46	#13133189	—	2	2	no	—
47	#13133185	—	2	2	no	—
48	#13133174	C₂²	1	1	yes	characteristic
49	#13133128	C₂²	1	1	yes	characteristic, complemented
50	#13133127	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
52	C₂×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
55	C₂×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
57	#127×C₅₃	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₅₃	—	2	1	no	—
60	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₅₃	—	2	2	no	—
63	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₅₃	—	2	1	no	—
66	C₂×C₂²⋊D₅₃	—	2	1	no	—
67	C₂×C₂²⋊D₅₃	—	2	2	no	—
68	C₂²×Dic₅₃	—	2	1	no	—
69	#11719949	—	2	2	no	—
70	#11719949	—	2	2	no	—
71	Q₈×D₅₃	—	2	2	no	—
72	Q₈×D₅₃	—	2	2	no	—
73	#11719947	—	2	2	no	—
74	#11719947	—	2	2	no	—
75	D₄×D₅₃	—	2	2	no	—
76	D₄×D₅₃	—	2	2	no	—
77	C₂×D₂₁₂	—	2	1	no	—
78	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
79	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
80	C₂×C₄×D₅₃	—	2	1	no	—
81	C₂×C₄×D₅₃	—	2	1	no	—
82	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
83	C₂×Dic₁₀₆	—	2	1	no	—
84	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
85	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
86	SD₁₆×C₅₃	—	2	2	no	—
87	SD₁₆×C₅₃	—	2	2	no	—
88	SD₁₆×C₅₃	—	2	2	no	—
89	SD₁₆×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₄⋊C₄×C₅₃	—	2	2	no	—
93	C₄²×C₅₃	C₂³	1	1	yes	characteristic
94	C₂²⋊Dic₅₃	—	2	2	no	—
95	#11719925	—	2	2	no	—
96	#11719925	—	2	2	no	—
97	#11719924	—	2	2	no	—
98	#11719924	—	2	2	no	—
99	#11719923	—	2	2	no	—
100	#11719923	—	2	2	no	—
101	#11719922	—	2	2	no	—
102	#11719922	—	2	2	no	—
103	#11719921	—	2	2	no	—
104	#11719921	—	2	2	no	—
105	#11719921	—	2	2	no	—
106	#11719921	D₄	1	2	yes	—
107	#11719919	—	2	2	no	—
108	#11719919	D₄	1	2	yes	—
109	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
110	C₂×#88322	C₂³	1	1	yes	characteristic
111	C₂×#88322	C₂³	1	1	yes	characteristic
112	Dic₂₁₂	—	2	2	no	—
113	D₄₂₄	—	2	2	no	—
114	#11719913	—	2	2	no	—
115	#11719913	—	2	2	no	—
116	#11719912	D₄	1	4	yes	complemented
117	C₈×D₅₃	D₄	1	4	yes	complemented
118	C₂³×C₅₃	—	2	1	no	—
119	C₂³×C₅₃	—	2	1	no	—
120	C₂²×D₅₃	—	2	1	no	—
121	C₂²×D₅₃	—	2	1	no	—
122	C₂²×D₅₃	—	4	2	no	—
123	C₂²×D₅₃	—	8	2	no	—
124	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
125	Q₈×C₅₃	—	2	1	no	—
126	Q₈×C₅₃	—	2	1	no	—
127	Q₈×C₅₃	—	2	1	no	—
128	Q₈×C₅₃	—	2	1	no	—
129	D₄×C₅₃	—	2	1	no	—
130	D₄×C₅₃	—	2	1	no	—
131	D₄×C₅₃	—	2	1	no	—
132	D₄×C₅₃	—	2	1	no	—
133	D₄×C₅₃	—	8	2	no	—
134	C₂×C₄×C₅₃	—	2	1	no	—
135	C₂×C₄×C₅₃	—	2	1	no	—
136	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
137	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
138	C₂²⋊D₅₃	—	4	2	no	—
139	C₂²⋊D₅₃	—	4	2	no	—
140	C₂²⋊D₅₃	—	8	2	no	—
141	C₂×Dic₅₃	—	2	1	no	—
142	C₂×Dic₅₃	—	2	1	no	—
143	C₂×Dic₅₃	—	4	2	no	—
144	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
145	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
146	D₂₁₂	—	2	1	no	—
147	D₂₁₂	—	2	1	no	—
148	D₂₁₂	—	2	1	no	—
149	D₂₁₂	—	2	1	no	—
150	D₂₁₂	—	4	2	no	—
151	C₄×D₅₃	—	4	2	no	—
152	C₄×D₅₃	—	4	2	no	—
153	C₄×D₅₃	C₂×D₄	1	2	yes	—
154	C₄×D₅₃	C₂×D₄	1	2	yes	—
155	Dic₁₀₆	—	2	1	no	—
156	Dic₁₀₆	—	2	1	no	—
157	Dic₁₀₆	—	2	1	no	—
158	Dic₁₀₆	—	2	1	no	—
159	Dic₁₀₆	—	4	2	no	—
160	C₈×C₅₃	C₂×D₄	1	2	yes	—
161	C₈×C₅₃	C₂×D₄	1	2	yes	—
162	#88322	C₂×D₄	1	2	yes	—
163	#88322	C₂×D₄	1	2	yes	—
164	C₂²×C₅₃	—	4	1	no	—
165	C₂²×C₅₃	—	4	1	no	—
166	C₂²×C₅₃	—	4	2	no	—
167	C₂²×C₅₃	—	4	2	no	—
168	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
169	C₂×D₅₃	—	2	2	no	—
170	C₂×D₅₃	—	4	1	no	—
171	C₂×D₅₃	—	4	1	no	—
172	C₂×D₅₃	—	4	2	no	—
173	C₂×D₅₃	—	4	2	no	—
174	C₂×D₅₃	—	8	4	no	—
175	C₂×D₅₃	#127	1	2	yes	—
176	C₄×C₅₃	—	2	2	no	—
177	C₄×C₅₃	—	4	1	no	—
178	C₄×C₅₃	—	4	1	no	—
179	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
180	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
181	Dic₅₃	—	2	2	no	—
182	Dic₅₃	—	4	1	no	—
183	Dic₅₃	—	4	1	no	—
184	Dic₅₃	#127	1	2	yes	—
185	#3154	—	53	1	no	maximal, Sylow
186	C₂×C₅₃	—	8	1	no	—
187	C₂×C₅₃	—	8	1	no	—
188	C₂×C₅₃	C₂×#127	1	1	yes	characteristic
189	C₂×C₅₃	#577	1	2	yes	—
190	D₅₃	—	2	2	no	—
191	D₅₃	—	8	1	no	—
192	D₅₃	—	8	1	no	—
193	C₂×#137	—	53	1	no	—
194	C₂×AGL(1,ℤ/8)	—	53	1	no	—
195	C₂×#135	—	53	1	no	—
196	C₂²×SD₁₆	—	53	1	no	—
197	#536	—	53	1	no	—
198	#535	—	53	1	no	—
199	#497	—	53	1	no	—
200	#497	—	53	1	no	—
201	#496	—	53	1	no	—
202	#496	—	53	1	no	—
203	#495	—	53	1	no	—
204	#495	—	53	1	no	—
205	#492	—	53	1	no	—
206	#492	D₅₃	1	1	yes	characteristic, complemented
207	#405	—	53	1	no	—
208	C₅₃	#3154	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	—	53	1	no	—
210	C₂×#41	—	53	1	no	—
211	C₂²×Q₈	—	53	1	no	—
212	C₂²×D₄	—	53	1	no	—
213	#137	—	106	4	no	—
214	AGL(1,ℤ/8)	—	106	4	no	—
215	#135	—	106	4	no	—
216	C₂×Q₁₆	—	53	1	no	—
217	C₂×Q₁₆	—	53	1	no	—
218	C₂×Q₁₆	—	53	1	no	—
219	C₂×SD₁₆	—	53	1	no	—
220	C₂×SD₁₆	—	53	1	no	—
221	C₂×SD₁₆	—	53	1	no	—
222	C₂×SD₁₆	—	53	1	no	—
223	C₂×SD₁₆	—	53	1	no	—
224	C₂×SD₁₆	—	53	1	no	—
225	C₂×SD₁₆	—	106	4	no	—
226	C₂×SD₁₆	C₂×D₅₃	1	1	yes	characteristic
227	C₂×SD₁₆	C₂×D₅₃	1	1	yes	characteristic
228	C₂×SD₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
229	C₂×SD₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
230	C₂×D₈	—	53	1	no	—
231	C₂×D₈	—	53	1	no	—
232	C₂×D₈	—	53	1	no	—
233	C₂×M₁₆	—	53	1	no	—
234	C₂²×C₈	—	53	1	no	—
235	C₄⋊Q₈	—	53	1	no	—
236	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
237	#127	—	53	1	no	—
238	#127	C₂×D₅₃	1	1	yes	characteristic
239	#124	—	53	1	no	—
240	#124	—	53	1	no	—
241	#124	—	53	1	no	—
242	#124	—	53	1	no	—
243	#123	—	106	2	no	—
244	C₂²⋊Q₈	—	106	2	no	—
245	#121	—	106	2	no	—
246	#120	—	106	2	no	—
247	#117	—	53	1	no	—
248	#97	—	53	1	no	—
249	#97	—	53	1	no	—
250	C₄×C₈	—	53	1	no	—
251	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
252	C₂⁴	—	106	1	no	—
253	#41	—	106	2	no	—
254	#41	—	106	2	no	—
255	#41	—	106	2	no	—
256	#41	—	106	2	no	—
257	C₂×Q₈	—	53	1	no	—
258	C₂×Q₈	—	53	1	no	—
259	C₂×Q₈	—	106	1	no	—
260	C₂×Q₈	—	106	2	no	—
261	C₂×Q₈	—	106	2	no	—
262	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
263	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	53	1	no	—
266	C₂×D₄	—	53	1	no	—
267	C₂×D₄	—	106	1	no	—
268	C₂×D₄	—	106	2	no	—
269	C₂×D₄	—	106	2	no	—
270	C₂×D₄	—	106	2	no	—
271	C₂×D₄	—	106	1	no	—
272	C₂×D₄	—	106	1	no	—
273	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
274	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
275	C₂²×C₄	—	53	1	no	—
276	C₂²×C₄	—	106	1	no	—
277	C₂²×C₄	—	106	1	no	—
278	C₂²×C₄	—	106	1	no	—
279	Q₁₆	—	106	2	no	—
280	Q₁₆	—	106	2	no	—
281	Q₁₆	—	106	2	no	—
282	SD₁₆	—	2	2	no	—
283	SD₁₆	—	2	2	no	—
284	SD₁₆	—	2	2	no	—
285	SD₁₆	—	2	2	no	—
286	SD₁₆	—	106	2	no	—
287	SD₁₆	—	106	2	no	—
288	SD₁₆	—	106	2	no	—
289	SD₁₆	—	106	2	no	—
290	SD₁₆	—	106	2	no	—
291	SD₁₆	—	106	2	no	—
292	D₈	—	106	2	no	—
293	D₈	—	106	2	no	—
294	D₈	—	106	2	no	—
295	M₁₆	—	53	4	no	—
296	C₂×C₈	—	53	1	no	—
297	C₂×C₈	—	53	1	no	—
298	C₂×C₈	—	53	4	no	—
299	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
300	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	53	2	no	—
303	C₄⋊C₄	—	106	2	no	—
304	C₂²⋊C₄	—	53	2	no	—
305	C₂²⋊C₄	—	106	2	no	—
306	C₂²⋊C₄	—	106	2	no	—
307	C₂²⋊C₄	—	106	2	no	—
308	C₂²⋊C₄	—	106	2	no	—
309	C₄²	—	53	1	no	—
310	C₄²	C₂²×D₅₃	1	1	yes	characteristic
311	C₂³	—	2	1	no	—
312	C₂³	—	2	1	no	—
313	C₂³	—	53	1	no	—
314	C₂³	—	106	1	no	—
315	C₂³	—	106	1	no	—
316	C₂³	—	424	2	no	—
317	C₂³	—	212	2	no	—
318	Q₈	—	2	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	2	1	no	—
321	Q₈	—	2	1	no	—
322	Q₈	—	106	1	no	—
323	Q₈	—	106	1	no	—
324	Q₈	—	106	1	no	—
325	Q₈	—	106	1	no	—
326	Q₈	—	212	2	no	—
327	D₄	—	2	1	no	—
328	D₄	—	2	1	no	—
329	D₄	—	2	1	no	—
330	D₄	—	2	1	no	—
331	D₄	—	8	2	no	—
332	D₄	—	106	1	no	—
333	D₄	—	106	1	no	—
334	D₄	—	106	1	no	—
335	D₄	—	106	1	no	—
336	D₄	—	424	2	no	—
337	D₄	—	212	2	no	—
338	D₄	—	212	2	no	—
339	D₄	—	212	2	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	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	—	106	1	no	—
347	C₂×C₄	—	106	1	no	—
348	C₂×C₄	—	212	2	no	—
349	C₂×C₄	—	212	2	no	—
350	C₂×C₄	—	212	2	no	—
351	C₂×C₄	D₄×D₅₃	1	2	yes	—
352	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
353	C₈	—	53	2	no	—
354	C₈	—	53	2	no	—
355	C₈	D₄×D₅₃	1	2	yes	—
356	C₈	D₄×D₅₃	1	2	yes	—
357	C₂²	—	4	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	2	no	—
361	C₂²	—	53	2	no	—
362	C₂²	—	106	2	no	—
363	C₂²	—	424	4	no	—
364	C₂²	—	212	1	no	—
365	C₂²	—	212	1	no	—
366	C₂²	—	212	2	no	—
367	C₂²	—	212	2	no	—
368	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
369	C₄	—	2	2	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	53	2	no	—
373	C₄	—	106	2	no	—
374	C₄	—	212	1	no	—
375	C₄	—	212	1	no	—
376	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
377	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
378	C₂	—	8	1	no	—
379	C₂	—	8	1	no	—
380	C₂	—	106	2	no	—
381	C₂	—	424	1	no	—
382	C₂	—	424	1	no	—
383	C₂	#127×D₅₃	1	1	yes	characteristic
384	C₂	#16811429	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#492	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#492	D₅₃
hypocenter	C₅₃	#3154
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

Complements

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

N	H ≅ G/N	type
1. C₂×#13133246	C₂	semidirect
2. C₂×#13133244	C₂	semidirect
3. C₂×#13133238	C₂	semidirect
4. C₂×SD₁₆×D₅₃	C₂	semidirect
5. #16811198	C₂	semidirect
6. #16811186	C₂	semidirect
7. #16811015	C₂	semidirect
8. #16811014	C₂	semidirect
9. #16811010	C₂	semidirect
10. #16811009	C₂	semidirect
11. #16811007	C₂	semidirect
12. #16811004	C₂	semidirect
13. #16810994	C₂	semidirect
14. #16810577	C₂	semidirect
15. C₅₃×#492	C₂	semidirect
23. C₂×Dic₂₁₂	C₂²	semidirect
24. C₂×#11719925	C₂²	semidirect
25. C₂×#11719925	C₂²	semidirect
27. C₂×#11719913	C₂²	semidirect
29. C₂×#11719923	C₂²	semidirect
35. C₂×#11719922	C₂²	semidirect
38. #13133215	C₂²	semidirect
40. #13133206	C₂²	semidirect
42. #13133205	C₂²	semidirect
49. #13133128	C₂²	semidirect
50. #13133127	C₂²	semidirect
51. C₈×Dic₅₃	C₂²	semidirect
54. C₂×SD₁₆×C₅₃	C₂²	semidirect
55. C₂×SD₁₆×C₅₃	C₂²	semidirect
56. C₄⋊Q₈×C₅₃	C₂²	semidirect
58. C₄×C₈×C₅₃	C₂²	semidirect
116. #11719912	D₄	semidirect
117. C₈×D₅₃	D₄	semidirect
206. #492	D₅₃	semidirect
208. C₅₃	#3154	semidirect
228. C₂×SD₁₆	C₂×D₅₃	semidirect
229. C₂×SD₁₆	C₂×D₅₃	semidirect
236. C₄⋊Q₈	C₂×D₅₃	semidirect
251. C₄×C₈	C₂×D₅₃	semidirect