Subgroups — #20283858

Order 6784.

354 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811151	C₂	1	1	yes	maximal, characteristic
2	#16810751	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810702	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810698	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810698	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810696	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810691	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810686	C₂	1	1	yes	maximal, characteristic, complemented
10	#16810683	C₂	1	1	yes	maximal, characteristic, complemented
11	#16810675	C₂	1	1	yes	maximal, characteristic, complemented
12	#16810650	C₂	1	1	yes	maximal, characteristic, complemented
13	#16810648	C₂	1	1	yes	maximal, characteristic, complemented
14	#16810645	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11719917	C₂²	1	1	yes	characteristic
17	C₂×#11719912	C₂²	1	1	yes	characteristic
18	C₂×#11719912	C₂²	1	1	yes	characteristic
19	C₂×#11719912	C₂²	1	1	yes	characteristic
20	C₂²×#88322	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
22	#13133212	C₄	1	1	yes	characteristic
23	#13133210	C₄	1	1	yes	characteristic
24	#13133207	C₄	1	1	yes	characteristic
25	#13133203	C₄	1	1	yes	characteristic
26	#13133202	C₄	1	1	yes	characteristic
27	#13133199	C₄	1	1	yes	characteristic
28	#13133195	C₄	1	1	yes	characteristic
29	#13133190	C₄	1	1	yes	characteristic
30	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
31	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
34	#13133174	C₂²	1	1	yes	characteristic
35	C₄²×D₅₃	C₂²	1	1	yes	characteristic
36	C₄⋊#88322	C₂²	1	1	yes	characteristic
37	C₄⋊#88322	C₂²	1	1	yes	characteristic
38	C₄⋊#88322	C₂²	1	1	yes	characteristic
39	C₄⋊#88322	C₂²	1	1	yes	characteristic
40	#13133146	C₂²	1	1	yes	characteristic
41	#13133146	C₂²	1	1	yes	characteristic
42	#13133146	C₂²	1	1	yes	characteristic
43	#13133146	C₂²	1	1	yes	characteristic
44	#13133129	C₂²	1	1	yes	characteristic
45	#13133129	C₂²	1	1	yes	characteristic
46	#13133129	C₂²	1	1	yes	characteristic
47	#13133129	C₂²	1	1	yes	characteristic
48	#13133127	C₂²	1	1	yes	characteristic
49	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
50	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
52	#117×C₅₃	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
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
58	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
67	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂³	1	1	yes	characteristic
77	C₄²×C₅₃	C₂³	1	1	yes	characteristic
78	C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
79	C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
80	#11719921	C₂×C₄	1	1	yes	characteristic
81	#11719921	C₂×C₄	1	1	yes	characteristic
82	#11719921	C₂×C₄	1	1	yes	characteristic
83	#11719921	C₂×C₄	1	1	yes	characteristic
84	#11719921	C₂×C₄	1	1	yes	characteristic
85	#11719921	C₂×C₄	1	1	yes	characteristic
86	C₄⋊Dic₅₃	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	#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	C₂³	1	1	yes	characteristic
97	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
98	#11719917	—	4	1	no	—
99	C₂×#88322	—	4	1	no	—
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	—	4	1	no	—
105	#11719912	—	4	1	no	—
106	#11719912	—	4	1	no	—
107	C₈×D₅₃	—	4	1	no	—
108	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₅₃	—	2	2	no	—
120	C₂²⋊D₅₃	—	4	1	no	—
121	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
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	D₂₁₂	—	4	1	no	—
128	C₄×D₅₃	—	2	1	no	—
129	C₄×D₅₃	—	2	1	no	—
130	C₄×D₅₃	—	2	1	no	—
131	C₄×D₅₃	—	2	1	no	—
132	C₄×D₅₃	—	2	2	no	—
133	C₄×D₅₃	—	4	1	no	—
134	Dic₁₀₆	—	2	2	no	—
135	C₈×C₅₃	—	2	1	no	—
136	C₈×C₅₃	—	2	1	no	—
137	C₈×C₅₃	—	2	1	no	—
138	C₈×C₅₃	—	2	1	no	—
139	#88322	—	2	1	no	—
140	#88322	—	2	1	no	—
141	#88322	—	2	1	no	—
142	#88322	—	2	1	no	—
143	C₂²×C₅₃	—	2	1	no	—
144	C₂²×C₅₃	—	2	1	no	—
145	C₂²×C₅₃	—	2	1	no	—
146	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
147	C₂×D₅₃	—	2	1	no	—
148	C₂×D₅₃	—	2	1	no	—
149	C₂×D₅₃	—	2	1	no	—
150	C₂×D₅₃	—	2	1	no	—
151	C₂×D₅₃	—	2	1	no	—
152	C₂×D₅₃	—	2	1	no	—
153	C₄×C₅₃	—	2	1	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
159	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
160	Dic₅₃	—	2	1	no	—
161	Dic₅₃	—	2	1	no	—
162	Dic₅₃	—	2	1	no	—
163	Dic₅₃	—	2	1	no	—
164	Dic₅₃	—	2	1	no	—
165	Dic₅₃	#131	1	2	yes	—
166	#2993	—	53	1	no	maximal, Sylow
167	C₂×C₅₃	—	4	1	no	—
168	C₂×C₅₃	#520	1	1	yes	characteristic
169	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
170	C₂×C₅₃	#568	1	1	yes	characteristic
171	D₅₃	—	4	1	no	—
172	D₅₃	—	4	1	no	—
173	#529	—	53	1	no	—
174	#446	—	53	1	no	—
175	C₈×Q₈	—	53	1	no	—
176	#436	—	53	1	no	—
177	#436	—	53	1	no	—
178	#436	—	53	1	no	—
179	#435	—	53	1	no	—
180	#435	—	53	1	no	—
181	C₈×D₄	—	53	1	no	—
182	#433	—	53	1	no	—
183	#433	D₅₃	1	1	yes	characteristic, complemented
184	#431	—	53	1	no	—
185	#424	—	53	1	no	—
186	#424	—	53	1	no	—
187	#423	—	53	1	no	—
188	C₅₃	#2993	1	1	yes	characteristic, complemented, Sylow
189	C₂×M₁₆	—	53	1	no	—
190	C₂×M₁₆	—	53	1	no	—
191	C₂×M₁₆	—	53	1	no	—
192	C₂×M₁₆	—	53	1	no	—
193	C₂²×C₈	—	53	1	no	—
194	C₂²×C₈	—	53	1	no	—
195	#126	—	53	1	no	—
196	#126	—	53	1	no	—
197	#125	—	53	1	no	—
198	#124	—	53	1	no	—
199	#123	—	53	1	no	—
200	#123	—	53	1	no	—
201	C₂²⋊Q₈	—	53	1	no	—
202	#121	—	53	1	no	—
203	C₄×Q₈	—	53	1	no	—
204	C₄×D₄	—	53	1	no	—
205	C₄×D₄	—	53	1	no	—
206	C₄×D₄	—	53	1	no	—
207	#117	—	53	1	no	—
208	#117	C₂×D₅₃	1	1	yes	characteristic
209	C₂×C₄²	—	53	1	no	—
210	C₄⋊C₈	—	53	1	no	—
211	C₄⋊C₈	—	53	1	no	—
212	C₄⋊C₈	—	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	C₂×D₅₃	1	1	yes	characteristic
219	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
220	C₂²⋊C₈	—	53	1	no	—
221	C₂²⋊C₈	—	53	1	no	—
222	C₂²⋊C₈	—	53	1	no	—
223	C₂²⋊C₈	—	53	1	no	—
224	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
225	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
226	#97	—	53	1	no	—
227	#97	C₂×D₅₃	1	1	yes	characteristic
228	C₄×C₈	—	53	1	no	—
229	C₄×C₈	—	53	1	no	—
230	C₄×C₈	—	53	1	no	—
231	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
232	C₂×Q₈	—	53	1	no	—
233	C₂×D₄	—	53	1	no	—
234	C₂×D₄	—	53	1	no	—
235	C₂×D₄	—	53	1	no	—
236	C₂²×C₄	—	53	1	no	—
237	C₂²×C₄	—	53	1	no	—
238	C₂²×C₄	—	53	1	no	—
239	C₂²×C₄	—	53	1	no	—
240	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
241	M₁₆	—	212	1	no	—
242	M₁₆	—	212	1	no	—
243	M₁₆	—	212	1	no	—
244	M₁₆	—	212	1	no	—
245	C₂×C₈	—	53	1	no	—
246	C₂×C₈	—	53	1	no	—
247	C₂×C₈	—	53	1	no	—
248	C₂×C₈	—	53	1	no	—
249	C₂×C₈	—	212	1	no	—
250	C₂×C₈	—	212	1	no	—
251	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
254	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
255	C₄⋊C₄	—	53	1	no	—
256	C₄⋊C₄	—	53	1	no	—
257	C₄⋊C₄	—	53	1	no	—
258	C₄⋊C₄	—	53	1	no	—
259	C₄⋊C₄	—	53	1	no	—
260	C₄⋊C₄	—	53	1	no	—
261	C₄⋊C₄	—	53	1	no	—
262	C₄⋊C₄	—	53	1	no	—
263	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
264	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
265	C₂²⋊C₄	—	53	1	no	—
266	C₂²⋊C₄	—	53	1	no	—
267	C₂²⋊C₄	—	53	1	no	—
268	C₂²⋊C₄	—	53	1	no	—
269	C₂²⋊C₄	—	53	1	no	—
270	C₂²⋊C₄	—	53	1	no	—
271	C₂²⋊C₄	—	53	1	no	—
272	C₂²⋊C₄	—	53	1	no	—
273	C₂²⋊C₄	C₄×D₅₃	1	1	yes	characteristic
274	C₂²⋊C₄	C₄×D₅₃	1	1	yes	characteristic
275	C₄²	—	53	1	no	—
276	C₄²	—	53	1	no	—
277	C₄²	—	53	1	no	—
278	C₄²	—	53	1	no	—
279	C₄²	C₂²×D₅₃	1	1	yes	characteristic
280	C₄²	C₂²×D₅₃	1	1	yes	characteristic
281	C₂³	—	53	1	no	—
282	C₂³	—	53	1	no	—
283	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
284	Q₈	—	106	2	no	—
285	D₄	—	106	2	no	—
286	D₄	—	212	1	no	—
287	D₄	—	212	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	53	1	no	—
291	C₂×C₄	—	53	1	no	—
292	C₂×C₄	—	53	1	no	—
293	C₂×C₄	—	53	1	no	—
294	C₂×C₄	—	53	1	no	—
295	C₂×C₄	—	53	1	no	—
296	C₂×C₄	—	106	2	no	—
297	C₂×C₄	—	106	1	no	—
298	C₂×C₄	—	106	1	no	—
299	C₂×C₄	—	106	1	no	—
300	C₂×C₄	—	106	1	no	—
301	C₂×C₄	—	212	1	no	—
302	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
304	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
305	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	106	1	no	—
313	C₈	—	106	1	no	—
314	C₈	—	106	1	no	—
315	C₈	—	106	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	106	1	no	—
320	C₂²	—	106	1	no	—
321	C₂²	—	106	1	no	—
322	C₂²	—	106	1	no	—
323	C₂²	—	106	1	no	—
324	C₂²	—	106	1	no	—
325	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	53	2	no	—
332	C₄	—	106	1	no	—
333	C₄	—	106	1	no	—
334	C₄	—	106	1	no	—
335	C₄	—	106	1	no	—
336	C₄	—	106	1	no	—
337	C₄	#13133263	1	1	yes	characteristic
338	C₄	#13133264	1	1	yes	characteristic
339	C₂	—	4	1	no	—
340	C₂	—	212	1	no	—
341	C₂	—	212	1	no	—
342	C₂	#16811103	1	1	yes	characteristic
343	C₂	#131×D₅₃	1	1	yes	characteristic
344	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	#2993
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. #16810751	C₂	semidirect
3. C₈×Dic₁₀₆	C₂	semidirect
4. #16810702	C₂	semidirect
5. #16810698	C₂	semidirect
6. #16810698	C₂	semidirect
7. #16810696	C₂	semidirect
8. #16810691	C₂	semidirect
9. #16810686	C₂	semidirect
10. #16810683	C₂	semidirect
11. #16810675	C₂	semidirect
12. #16810650	C₂	semidirect
13. #16810648	C₂	semidirect
14. #16810645	C₂	semidirect
15. C₅₃×#433	C₂	semidirect
183. #433	D₅₃	semidirect
188. C₅₃	#2993	semidirect