Subgroups — #20285207

Order 6784.

532 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133249	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13133246	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13133238	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811198	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811193	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811017	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811016	C₂	1	1	yes	maximal, characteristic, complemented
9	#16811014	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811005	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811004	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811002	C₂	1	1	yes	maximal, characteristic, complemented
13	#16811001	C₂	1	1	yes	maximal, characteristic, complemented
14	#16810580	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#497	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×Q₈×D₅₃	C₂²	1	1	yes	characteristic
20	#13133249	—	2	4	no	—
21	#13133246	—	2	4	no	—
22	#13133238	—	2	4	no	—
23	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
24	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
25	C₂×#11719925	C₂²	1	1	yes	characteristic
26	C₂×#11719925	C₂²	1	1	yes	characteristic
27	C₂×#11719925	C₂²	1	1	yes	characteristic
28	C₂×#11719925	C₂²	1	1	yes	characteristic
29	C₂×#11719925	C₂²	1	1	yes	characteristic
30	Q₁₆×D₅₃	—	2	4	no	—
31	C₂×#11719913	C₂²	1	1	yes	characteristic
32	C₂×#11719913	C₂²	1	1	yes	characteristic
33	C₂×#11719923	C₂²	1	1	yes	characteristic
34	C₂×#11719924	C₂²	1	1	yes	characteristic
35	C₂×#11719922	C₂²	1	1	yes	characteristic
36	C₂×#11719912	C₂²	1	1	yes	characteristic
37	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
38	#13133217	C₂²	1	1	yes	characteristic
39	#13133215	C₂²	1	1	yes	characteristic
40	#13133215	C₂²	1	1	yes	characteristic
41	#13133206	C₂²	1	1	yes	characteristic
42	#13133205	C₂²	1	1	yes	characteristic
43	#13133204	C₂²	1	1	yes	characteristic
44	#13133202	—	2	2	no	—
45	#13133199	—	2	2	no	—
46	#13133195	—	2	2	no	—
47	#13133194	—	2	2	no	—
48	#13133174	C₂²	1	1	yes	characteristic
49	#13133127	C₂²	1	1	yes	characteristic
50	C₄×#88322	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
55	C₂×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
56	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
57	#124×C₅₃	C₂²	1	1	yes	characteristic
58	#97×C₅₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₅₃	—	2	1	no	—
64	C₂²×Dic₅₃	—	2	1	no	—
65	#11719949	—	2	2	no	—
66	#11719949	—	2	2	no	—
67	Q₈×D₅₃	—	2	2	no	—
68	Q₈×D₅₃	—	2	2	no	—
69	Q₈×D₅₃	—	2	2	no	—
70	Q₈×D₅₃	—	2	2	no	—
71	#11719947	—	2	2	no	—
72	#11719947	—	2	2	no	—
73	C₂×D₂₁₂	—	2	1	no	—
74	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₅₃	—	2	1	no	—
76	C₂×C₄×D₅₃	—	2	1	no	—
77	C₂×C₄×D₅₃	—	2	1	no	—
78	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₀₆	—	2	1	no	—
80	C₂×Dic₁₀₆	—	2	1	no	—
81	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
82	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
83	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
84	Q₁₆×C₅₃	—	2	2	no	—
85	Q₁₆×C₅₃	—	2	2	no	—
86	SD₁₆×C₅₃	—	2	2	no	—
87	SD₁₆×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
90	C₄⋊C₄×C₅₃	—	2	2	no	—
91	C₂²⋊C₄×C₅₃	—	2	2	no	—
92	C₄²×C₅₃	C₂³	1	1	yes	characteristic
93	C₂²⋊Dic₅₃	—	2	2	no	—
94	#11719925	—	2	2	no	—
95	#11719925	—	2	2	no	—
96	#11719925	—	2	2	no	—
97	#11719925	—	2	2	no	—
98	#11719925	—	2	2	no	—
99	#11719924	—	2	2	no	—
100	#11719923	—	2	2	no	—
101	#11719922	—	2	2	no	—
102	#11719921	—	2	2	no	—
103	#11719921	—	2	2	no	—
104	#11719921	D₄	1	2	yes	—
105	C₄⋊Dic₅₃	—	2	2	no	—
106	#11719919	—	2	2	no	—
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	Dic₂₁₂	—	2	2	no	—
114	#11719913	—	2	2	no	—
115	#11719913	—	2	2	no	—
116	#11719912	D₄	1	4	yes	—
117	C₈×D₅₃	D₄	1	4	yes	—
118	C₂³×C₅₃	—	2	1	no	—
119	C₂²×D₅₃	—	2	1	no	—
120	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
121	Q₈×C₅₃	—	2	1	no	—
122	Q₈×C₅₃	—	2	1	no	—
123	Q₈×C₅₃	—	2	1	no	—
124	Q₈×C₅₃	—	2	1	no	—
125	Q₈×C₅₃	—	2	1	no	—
126	Q₈×C₅₃	—	2	1	no	—
127	D₄×C₅₃	—	2	1	no	—
128	D₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	—	2	1	no	—
132	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
133	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
134	C₂²⋊D₅₃	—	4	2	no	—
135	C₂×Dic₅₃	—	2	1	no	—
136	C₂×Dic₅₃	—	2	1	no	—
137	C₂×Dic₅₃	—	2	1	no	—
138	C₂×Dic₅₃	—	4	2	no	—
139	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
140	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
141	D₂₁₂	—	2	1	no	—
142	D₂₁₂	—	2	1	no	—
143	D₂₁₂	—	4	2	no	—
144	C₄×D₅₃	—	4	2	no	—
145	C₄×D₅₃	—	4	2	no	—
146	C₄×D₅₃	—	4	2	no	—
147	C₄×D₅₃	C₂×D₄	1	2	yes	—
148	C₄×D₅₃	C₂×D₄	1	2	yes	—
149	Dic₁₀₆	—	2	1	no	—
150	Dic₁₀₆	—	2	1	no	—
151	Dic₁₀₆	—	2	1	no	—
152	Dic₁₀₆	—	2	1	no	—
153	Dic₁₀₆	—	2	1	no	—
154	Dic₁₀₆	—	2	1	no	—
155	Dic₁₀₆	—	4	2	no	—
156	Dic₁₀₆	—	4	2	no	—
157	C₈×C₅₃	C₂×D₄	1	2	yes	—
158	C₈×C₅₃	C₂×D₄	1	2	yes	—
159	#88322	C₂×D₄	1	2	yes	—
160	#88322	C₂×D₄	1	2	yes	—
161	C₂²×C₅₃	—	4	1	no	—
162	C₂²×C₅₃	—	4	2	no	—
163	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
164	C₂×D₅₃	—	2	2	no	—
165	C₂×D₅₃	—	4	1	no	—
166	C₂×D₅₃	—	4	2	no	—
167	C₂×D₅₃	#127	1	2	yes	—
168	C₄×C₅₃	—	2	2	no	—
169	C₄×C₅₃	—	4	1	no	—
170	C₄×C₅₃	—	4	1	no	—
171	C₄×C₅₃	—	4	1	no	—
172	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
173	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
174	Dic₅₃	—	2	2	no	—
175	Dic₅₃	—	4	1	no	—
176	Dic₅₃	—	4	1	no	—
177	Dic₅₃	—	4	1	no	—
178	Dic₅₃	#127	1	2	yes	—
179	#3156	—	53	1	no	maximal, Sylow
180	C₂×C₅₃	—	8	1	no	—
181	C₂×C₅₃	C₂×#127	1	1	yes	characteristic
182	C₂×C₅₃	#578	1	2	yes	—
183	D₅₃	—	2	2	no	—
184	D₅₃	—	8	1	no	—
185	C₂×#137	—	53	1	no	—
186	C₂×#137	—	53	1	no	—
187	C₂×#135	—	53	1	no	—
188	C₂²×Q₁₆	—	53	1	no	—
189	#536	—	53	1	no	—
190	#536	—	53	1	no	—
191	#497	—	53	1	no	—
192	#497	—	53	1	no	—
193	#497	—	53	1	no	—
194	#497	D₅₃	1	1	yes	characteristic, complemented
195	#495	—	53	1	no	—
196	#495	—	53	1	no	—
197	#494	—	53	1	no	—
198	#494	—	53	1	no	—
199	#405	—	53	1	no	—
200	C₅₃	#3156	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	53	1	no	—
202	C₂×#41	—	53	1	no	—
203	C₂²×Q₈	—	53	1	no	—
204	C₂²×Q₈	—	53	1	no	—
205	#137	—	106	4	no	—
206	#137	—	106	4	no	—
207	#135	—	106	4	no	—
208	C₂×Q₁₆	—	53	1	no	—
209	C₂×Q₁₆	—	53	1	no	—
210	C₂×Q₁₆	—	53	1	no	—
211	C₂×Q₁₆	—	53	1	no	—
212	C₂×Q₁₆	—	53	1	no	—
213	C₂×Q₁₆	—	53	1	no	—
214	C₂×Q₁₆	—	53	1	no	—
215	C₂×Q₁₆	—	106	4	no	—
216	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
217	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
218	C₂×SD₁₆	—	53	1	no	—
219	C₂×SD₁₆	—	53	1	no	—
220	C₂×SD₁₆	—	53	1	no	—
221	C₂×SD₁₆	—	53	1	no	—
222	C₂×SD₁₆	C₂×D₅₃	1	1	yes	characteristic
223	C₂×SD₁₆	C₂×D₅₃	1	1	yes	characteristic
224	C₂×D₈	—	53	1	no	—
225	C₂×M₁₆	—	53	1	no	—
226	C₂²×C₈	—	53	1	no	—
227	C₄⋊Q₈	—	53	1	no	—
228	C₄⋊Q₈	—	53	1	no	—
229	C₄⋊Q₈	—	53	1	no	—
230	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
231	#124	—	53	1	no	—
232	#124	—	53	1	no	—
233	#124	—	53	1	no	—
234	#124	C₂×D₅₃	1	1	yes	characteristic
235	#123	—	106	2	no	—
236	#123	—	106	2	no	—
237	C₂²⋊Q₈	—	106	2	no	—
238	C₂²⋊Q₈	—	106	2	no	—
239	#117	—	53	1	no	—
240	#97	—	53	1	no	—
241	#97	C₂×D₅₃	1	1	yes	characteristic
242	C₄×C₈	—	53	1	no	—
243	C₄×C₈	—	53	1	no	—
244	#41	—	106	2	no	—
245	#41	—	106	2	no	—
246	#41	—	106	2	no	—
247	#41	—	106	2	no	—
248	C₂×Q₈	—	53	1	no	—
249	C₂×Q₈	—	53	1	no	—
250	C₂×Q₈	—	53	1	no	—
251	C₂×Q₈	—	106	1	no	—
252	C₂×Q₈	—	106	1	no	—
253	C₂×Q₈	—	106	2	no	—
254	C₂×Q₈	—	106	2	no	—
255	C₂×Q₈	—	106	2	no	—
256	C₂×Q₈	—	106	2	no	—
257	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
258	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
259	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
260	C₂×D₄	—	53	1	no	—
261	C₂×D₄	—	106	1	no	—
262	C₂×D₄	—	106	1	no	—
263	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
264	C₂²×C₄	—	53	1	no	—
265	C₂²×C₄	—	106	1	no	—
266	C₂²×C₄	—	106	1	no	—
267	C₂²×C₄	—	106	1	no	—
268	C₂²×C₄	—	106	1	no	—
269	Q₁₆	—	2	2	no	—
270	Q₁₆	—	2	2	no	—
271	Q₁₆	—	106	2	no	—
272	Q₁₆	—	106	2	no	—
273	Q₁₆	—	106	2	no	—
274	Q₁₆	—	106	2	no	—
275	Q₁₆	—	106	2	no	—
276	Q₁₆	—	106	2	no	—
277	Q₁₆	—	106	2	no	—
278	SD₁₆	—	2	2	no	—
279	SD₁₆	—	2	2	no	—
280	SD₁₆	—	106	2	no	—
281	SD₁₆	—	106	2	no	—
282	SD₁₆	—	106	2	no	—
283	SD₁₆	—	106	2	no	—
284	D₈	—	106	2	no	—
285	M₁₆	—	53	4	no	—
286	C₂×C₈	—	53	1	no	—
287	C₂×C₈	—	53	1	no	—
288	C₂×C₈	—	53	4	no	—
289	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
290	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	53	2	no	—
293	C₄⋊C₄	—	106	2	no	—
294	C₄⋊C₄	—	106	2	no	—
295	C₄⋊C₄	—	106	2	no	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	53	2	no	—
298	C₂²⋊C₄	—	106	2	no	—
299	C₂²⋊C₄	—	106	2	no	—
300	C₂²⋊C₄	—	106	2	no	—
301	C₄²	—	53	1	no	—
302	C₄²	C₂²×D₅₃	1	1	yes	characteristic
303	C₂³	—	2	1	no	—
304	C₂³	—	53	1	no	—
305	C₂³	—	106	1	no	—
306	Q₈	—	2	1	no	—
307	Q₈	—	2	1	no	—
308	Q₈	—	2	1	no	—
309	Q₈	—	2	1	no	—
310	Q₈	—	2	1	no	—
311	Q₈	—	2	1	no	—
312	Q₈	—	106	1	no	—
313	Q₈	—	106	1	no	—
314	Q₈	—	106	1	no	—
315	Q₈	—	106	1	no	—
316	Q₈	—	106	1	no	—
317	Q₈	—	106	1	no	—
318	Q₈	—	212	2	no	—
319	Q₈	—	212	2	no	—
320	D₄	—	2	1	no	—
321	D₄	—	2	1	no	—
322	D₄	—	106	1	no	—
323	D₄	—	106	1	no	—
324	D₄	—	212	2	no	—
325	D₄	—	212	2	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	53	1	no	—
330	C₂×C₄	—	53	2	no	—
331	C₂×C₄	—	53	2	no	—
332	C₂×C₄	—	53	2	no	—
333	C₂×C₄	—	106	1	no	—
334	C₂×C₄	—	106	1	no	—
335	C₂×C₄	—	106	1	no	—
336	C₂×C₄	—	212	2	no	—
337	C₂×C₄	—	212	2	no	—
338	C₂×C₄	—	212	2	no	—
339	C₂×C₄	—	212	2	no	—
340	C₂×C₄	D₄×D₅₃	1	2	yes	—
341	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
342	C₈	—	53	2	no	—
343	C₈	—	53	2	no	—
344	C₈	D₄×D₅₃	1	2	yes	—
345	C₈	D₄×D₅₃	1	2	yes	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	2	no	—
348	C₂²	—	53	2	no	—
349	C₂²	—	106	2	no	—
350	C₂²	—	212	1	no	—
351	C₂²	—	212	2	no	—
352	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	53	2	no	—
358	C₄	—	106	2	no	—
359	C₄	—	212	1	no	—
360	C₄	—	212	1	no	—
361	C₄	—	212	1	no	—
362	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
363	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
364	C₂	—	8	1	no	—
365	C₂	—	106	2	no	—
366	C₂	—	424	1	no	—
367	C₂	#127×D₅₃	1	1	yes	characteristic
368	C₂	#16811439	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#497	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#497	D₅₃
hypocenter	C₅₃	#3156
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #497

Complements

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

N	H ≅ G/N	type
1. C₂×#13133249	C₂	semidirect
2. C₂×#13133246	C₂	semidirect
4. C₂×Q₁₆×D₅₃	C₂	semidirect
5. #16811198	C₂	semidirect
6. #16811193	C₂	semidirect
7. #16811017	C₂	semidirect
8. #16811016	C₂	semidirect
9. #16811014	C₂	semidirect
10. #16811005	C₂	semidirect
11. #16811004	C₂	semidirect
12. #16811002	C₂	semidirect
13. #16811001	C₂	semidirect
14. #16810580	C₂	semidirect
15. C₅₃×#497	C₂	semidirect
194. #497	D₅₃	semidirect
200. C₅₃	#3156	semidirect