Subgroups — #20283807

Order 6784.

322 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811176	C₂	1	1	yes	maximal, characteristic
2	#16810753	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810752	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810752	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810751	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810751	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810750	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810747	C₂	1	1	yes	maximal, characteristic, complemented
10	#16810684	C₂	1	1	yes	maximal, characteristic
11	#16810681	C₂	1	1	yes	maximal, characteristic
12	#16810681	C₂	1	1	yes	maximal, characteristic
13	#16810677	C₂	1	1	yes	maximal, characteristic
14	#16810673	C₂	1	1	yes	maximal, characteristic
15	C₅₃×#433	C₂	1	1	yes	maximal, characteristic
16	#13133216	C₄	1	1	yes	characteristic
17	#13133216	C₄	1	1	yes	characteristic
18	#13133208	C₄	1	1	yes	characteristic
19	#13133208	C₄	1	1	yes	characteristic
20	#13133194	C₄	1	1	yes	characteristic
21	#13133194	C₄	1	1	yes	characteristic
22	#13133194	C₄	1	1	yes	characteristic
23	#13133194	C₄	1	1	yes	characteristic
24	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
26	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
27	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
28	#13133176	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
30	C₄⋊#88322	C₂²	1	1	yes	characteristic
31	C₄⋊#88322	C₂²	1	1	yes	characteristic
32	#13133146	C₂²	1	1	yes	characteristic
33	#13133146	C₂²	1	1	yes	characteristic
34	#13133146	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	#13133146	C₂²	1	1	yes	characteristic
40	C₂²⋊#88322	C₂²	1	1	yes	characteristic
41	C₂²⋊#88322	C₂²	1	1	yes	characteristic
42	#13133128	C₂²	1	1	yes	characteristic
43	#13133127	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	C₄×#88322	C₂²	1	1	yes	characteristic
48	C₈×Dic₅₃	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	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
63	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
64	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
65	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂³	1	1	yes	characteristic
75	C₄²×C₅₃	C₂³	1	1	yes	characteristic
76	C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
77	C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
78	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
79	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
80	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
81	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
83	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
84	#11719919	C₂×C₄	1	1	yes	characteristic
85	#11719919	C₂×C₄	1	1	yes	characteristic
86	#11719919	C₂×C₄	1	1	yes	characteristic
87	#11719919	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	C₂³	1	1	yes	characteristic
95	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
96	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
97	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
98	C₂×#88322	C₂³	1	1	yes	characteristic
99	C₂×#88322	C₂³	1	1	yes	characteristic
100	C₂×#88322	C₂³	1	1	yes	characteristic
101	C₂×#88322	C₂³	1	1	yes	characteristic
102	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
111	C₂×Dic₅₃	—	2	2	no	—
112	C₂×Dic₅₃	—	4	1	no	—
113	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
121	Dic₁₀₆	—	2	2	no	—
122	Dic₁₀₆	—	2	2	no	—
123	Dic₁₀₆	—	4	1	no	—
124	Dic₁₀₆	—	4	1	no	—
125	C₈×C₅₃	—	2	1	no	—
126	C₈×C₅₃	—	2	1	no	—
127	C₈×C₅₃	—	2	1	no	—
128	C₈×C₅₃	—	2	1	no	—
129	#88322	—	2	1	no	—
130	#88322	—	2	1	no	—
131	#88322	—	2	1	no	—
132	#88322	—	2	1	no	—
133	C₂²×C₅₃	—	2	1	no	—
134	C₂²×C₅₃	—	2	1	no	—
135	C₂²×C₅₃	—	2	1	no	—
136	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
137	C₄×C₅₃	—	2	1	no	—
138	C₄×C₅₃	—	2	1	no	—
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	—	2	1	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
143	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
144	Dic₅₃	—	2	1	no	—
145	Dic₅₃	—	2	1	no	—
146	Dic₅₃	—	2	1	no	—
147	Dic₅₃	—	2	1	no	—
148	Dic₅₃	—	2	1	no	—
149	Dic₅₃	—	2	1	no	—
150	Dic₅₃	—	2	1	no	—
151	Dic₅₃	#131	1	2	yes	—
152	#2985	—	53	1	no	maximal, Sylow
153	C₂×C₅₃	—	4	1	no	—
154	C₂×C₅₃	#519	1	1	yes	characteristic
155	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
156	C₂×C₅₃	#568	1	1	yes	characteristic
157	#533	—	53	1	no	—
158	#446	—	53	1	no	—
159	#446	—	53	1	no	—
160	#446	—	53	1	no	—
161	#446	—	53	1	no	—
162	#446	—	53	1	no	—
163	#446	—	53	1	no	—
164	C₈×Q₈	—	53	1	no	—
165	C₈×Q₈	—	53	1	no	—
166	#433	—	53	1	no	—
167	#433	—	53	1	no	—
168	#433	—	53	1	no	—
169	#433	D₅₃	1	1	yes	characteristic
170	#432	—	53	1	no	—
171	#431	—	53	1	no	—
172	C₅₃	#2985	1	1	yes	characteristic, complemented, Sylow
173	C₄⋊Q₈	—	53	1	no	—
174	C₄⋊Q₈	—	53	1	no	—
175	#125	—	53	1	no	—
176	#125	—	53	1	no	—
177	C₂²⋊Q₈	—	53	1	no	—
178	C₂²⋊Q₈	—	53	1	no	—
179	C₂²⋊Q₈	—	53	1	no	—
180	C₂²⋊Q₈	—	53	1	no	—
181	C₄×Q₈	—	53	1	no	—
182	C₄×Q₈	—	53	1	no	—
183	C₄×Q₈	—	53	1	no	—
184	C₄×Q₈	—	53	1	no	—
185	#117	—	53	1	no	—
186	#117	C₂×D₅₃	1	1	yes	characteristic
187	C₂×C₄²	—	53	1	no	—
188	C₄⋊C₈	—	53	1	no	—
189	C₄⋊C₈	—	53	1	no	—
190	C₄⋊C₈	—	53	1	no	—
191	C₄⋊C₈	—	53	1	no	—
192	C₄⋊C₈	—	53	1	no	—
193	C₄⋊C₈	—	53	1	no	—
194	C₄⋊C₈	—	53	1	no	—
195	C₄⋊C₈	—	53	1	no	—
196	C₄⋊C₈	—	53	1	no	—
197	C₄⋊C₈	—	53	1	no	—
198	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
199	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
200	C₂²⋊C₈	—	53	1	no	—
201	C₂²⋊C₈	—	53	1	no	—
202	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
203	C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
204	#97	—	53	1	no	—
205	#97	—	53	1	no	—
206	#97	—	53	1	no	—
207	#97	—	53	1	no	—
208	#97	—	53	1	no	—
209	#97	C₂×D₅₃	1	1	yes	characteristic
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	C₂×D₅₃	1	1	yes	characteristic
216	C₂×Q₈	—	53	1	no	—
217	C₂×Q₈	—	53	1	no	—
218	C₂×Q₈	—	53	1	no	—
219	C₂×Q₈	—	53	1	no	—
220	C₂²×C₄	—	53	1	no	—
221	C₂²×C₄	—	53	1	no	—
222	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
223	C₂×C₈	—	53	1	no	—
224	C₂×C₈	—	53	1	no	—
225	C₂×C₈	—	53	1	no	—
226	C₂×C₈	—	53	1	no	—
227	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
228	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
229	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
230	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
231	C₄⋊C₄	—	53	1	no	—
232	C₄⋊C₄	—	53	1	no	—
233	C₄⋊C₄	—	53	1	no	—
234	C₄⋊C₄	—	53	1	no	—
235	C₄⋊C₄	—	53	1	no	—
236	C₄⋊C₄	—	53	1	no	—
237	C₄⋊C₄	—	53	1	no	—
238	C₄⋊C₄	—	53	1	no	—
239	C₄⋊C₄	—	53	1	no	—
240	C₄⋊C₄	—	53	1	no	—
241	C₄⋊C₄	—	53	1	no	—
242	C₄⋊C₄	—	53	1	no	—
243	C₄⋊C₄	—	53	1	no	—
244	C₄⋊C₄	—	53	1	no	—
245	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
246	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
247	C₂²⋊C₄	—	53	1	no	—
248	C₂²⋊C₄	—	53	1	no	—
249	C₂²⋊C₄	C₄×D₅₃	1	1	yes	characteristic
250	C₂²⋊C₄	C₄×D₅₃	1	1	yes	characteristic
251	C₄²	—	53	1	no	—
252	C₄²	—	53	1	no	—
253	C₄²	—	53	1	no	—
254	C₄²	—	53	1	no	—
255	C₄²	—	53	1	no	—
256	C₄²	—	53	1	no	—
257	C₄²	C₂²×D₅₃	1	1	yes	characteristic
258	C₄²	C₂²×D₅₃	1	1	yes	characteristic
259	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
260	Q₈	—	106	2	no	—
261	Q₈	—	106	2	no	—
262	Q₈	—	212	1	no	—
263	Q₈	—	212	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	53	1	no	—
267	C₂×C₄	—	53	1	no	—
268	C₂×C₄	—	53	1	no	—
269	C₂×C₄	—	53	1	no	—
270	C₂×C₄	—	53	1	no	—
271	C₂×C₄	—	53	1	no	—
272	C₂×C₄	—	53	1	no	—
273	C₂×C₄	—	53	1	no	—
274	C₂×C₄	—	106	2	no	—
275	C₂×C₄	—	212	1	no	—
276	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
281	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	106	1	no	—
287	C₈	—	106	1	no	—
288	C₈	—	106	1	no	—
289	C₈	—	106	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	53	2	no	—
300	C₄	—	106	1	no	—
301	C₄	—	106	1	no	—
302	C₄	—	106	1	no	—
303	C₄	—	106	1	no	—
304	C₄	—	106	1	no	—
305	C₄	—	106	1	no	—
306	C₄	—	106	1	no	—
307	C₄	#13133264	1	1	yes	characteristic
308	C₄	#13133264	1	1	yes	characteristic
309	C₂	—	4	1	no	—
310	C₂	#16811100	1	1	yes	characteristic
311	C₂	#131×D₅₃	1	1	yes	characteristic
312	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	#2985
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. #16810753	C₂	semidirect
3. #16810752	C₂	semidirect
4. #16810752	C₂	semidirect
5. #16810751	C₂	semidirect
6. #16810751	C₂	semidirect
7. #16810750	C₂	semidirect
8. C₈×Dic₁₀₆	C₂	semidirect
9. #16810747	C₂	semidirect
172. C₅₃	#2985	semidirect