Subgroups — #20289319

Order 6784.

912 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃⋊#564	C₂	1	2	yes	maximal, complemented
2	#16811374	C₂	1	2	yes	maximal, complemented
3	#16811349	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811348	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811328	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811315	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811315	C₂	1	2	yes	maximal, complemented
8	#16811313	C₂	1	1	yes	maximal, characteristic, complemented
9	#16811313	C₂	1	2	yes	maximal, complemented
10	#16811308	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811307	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811254	C₂	1	1	yes	maximal, characteristic, complemented
13	#16811252	C₂	1	2	yes	maximal, complemented
14	#16811250	C₂	1	1	yes	maximal, characteristic, complemented
15	#16811233	C₂	1	1	yes	maximal, characteristic, complemented
16	#16811233	C₂	1	2	yes	maximal, complemented
17	#16811232	C₂	1	2	yes	maximal, complemented
18	#16811231	C₂	1	1	yes	maximal, characteristic, complemented
19	#16811215	C₂	1	2	yes	maximal
20	#16811175	C₂	1	1	yes	maximal, characteristic, complemented
21	#16811153	C₂	1	1	yes	maximal, characteristic
22	C₂×#13133208	C₂	1	1	yes	maximal, characteristic, complemented
23	C₅₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
24	#13133216	C₂²	1	1	yes	characteristic
25	#13133216	C₂²	1	2	yes	—
26	#13133216	C₂²	1	1	yes	characteristic
27	#13133216	C₂²	1	2	yes	—
28	#13133215	C₂²	1	2	yes	—
29	#13133215	C₂²	1	1	yes	characteristic
30	#13133215	C₂²	1	2	yes	—
31	#13133212	C₂²	1	2	yes	—
32	#13133212	C₂²	1	2	yes	—
33	#13133212	C₂²	1	2	yes	—
34	#13133212	C₂²	1	1	yes	characteristic
35	#13133211	C₂²	1	2	yes	—
36	#13133210	C₂²	1	2	yes	—
37	#13133210	C₂²	1	2	yes	—
38	#13133210	C₂²	1	2	yes	—
39	#13133210	C₂²	1	1	yes	characteristic
40	#13133210	C₂²	1	2	yes	—
41	#13133209	C₂²	1	1	yes	characteristic
42	#13133209	C₂²	1	2	yes	—
43	#13133209	C₂²	1	2	yes	—
44	#13133208	C₂²	1	1	yes	characteristic
45	#13133208	C₂²	1	1	yes	characteristic
46	#13133208	C₂²	1	1	yes	characteristic
47	#13133208	C₂²	1	2	yes	—
48	#13133208	C₂²	1	2	yes	—
49	#13133208	C₂²	1	2	yes	—
50	#13133208	C₂²	1	2	yes	—
51	#13133208	C₂²	1	2	yes	—
52	#13133208	C₂²	1	1	yes	characteristic
53	#13133208	C₂²	1	2	yes	—
54	#13133208	C₂²	1	1	yes	characteristic
55	#13133207	C₂²	1	2	yes	—
56	#13133207	C₂²	1	2	yes	—
57	#13133207	C₂²	1	2	yes	—
58	#13133207	C₂²	1	2	yes	—
59	#13133207	C₂²	1	2	yes	—
60	#13133207	C₂²	1	2	yes	—
61	#13133207	C₂²	1	2	yes	—
62	#13133207	C₂²	1	2	yes	—
63	#13133207	C₂²	1	1	yes	characteristic
64	#13133207	C₂²	1	2	yes	—
65	#13133207	C₂²	1	2	yes	—
66	#13133203	C₂²	1	1	yes	characteristic
67	#13133201	C₂²	1	1	yes	characteristic
68	#13133201	C₂²	1	2	yes	—
69	#13133200	C₂²	1	2	yes	—
70	#13133199	C₂²	1	2	yes	—
71	#13133198	C₂²	1	1	yes	characteristic
72	#13133198	C₂²	1	2	yes	—
73	#13133197	C₂²	1	1	yes	characteristic
74	#13133196	C₂²	1	1	yes	characteristic
75	#13133196	C₂²	1	1	yes	characteristic
76	#13133196	C₂²	1	2	yes	—
77	#13133196	C₂²	1	2	yes	—
78	#13133196	C₂²	1	2	yes	—
79	#13133195	C₂²	1	2	yes	—
80	#13133195	C₂²	1	2	yes	—
81	#13133194	C₂²	1	1	yes	characteristic
82	#13133194	C₂²	1	2	yes	—
83	#13133193	C₂²	1	1	yes	characteristic
84	#13133193	C₂²	1	2	yes	—
85	#13133189	C₂²	1	1	yes	characteristic
86	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic
87	Q₈×Dic₅₃	C₂²	1	2	yes	—
88	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic
89	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
90	C₄×Dic₁₀₆	C₂²	1	2	yes	—
91	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
92	#13133182	C₂²	1	1	yes	characteristic
93	#13133182	C₂²	1	1	yes	characteristic
94	#13133182	C₂²	1	2	yes	—
95	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
96	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
97	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
98	#13133176	C₂²	1	1	yes	characteristic
99	#13133176	C₂²	1	2	yes	—
100	#13133176	C₂²	1	1	yes	characteristic
101	#13133175	C₂²	1	2	yes	—
102	#13133175	C₂²	1	1	yes	characteristic
103	#13133174	C₂²	1	1	yes	characteristic
104	#13133173	C₂²	1	2	yes	—
105	#13133173	C₂²	1	2	yes	—
106	#13133173	C₂²	1	2	yes	—
107	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
108	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
109	C₂×#11719919	C₂²	1	2	yes	—
110	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic
111	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
112	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
113	#125×C₅₃	C₂²	1	2	yes	—
114	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
115	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
116	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
117	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
118	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
119	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
120	C₄×D₄×C₅₃	C₂²	1	2	yes	—
121	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
122	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
123	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
124	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
125	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
129	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
130	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
131	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
132	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
133	C₂²×Dic₅₃	C₂³	1	2	yes	—
134	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
135	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
136	C₂×C₄×D₅₃	C₂³	1	2	yes	—
137	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
138	C₂×Dic₁₀₆	C₂³	1	2	yes	—
139	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
140	C₂×Dic₁₀₆	C₂³	1	2	yes	—
141	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
142	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
143	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
144	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
145	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
146	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
147	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
148	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
149	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
150	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
151	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
152	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
153	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
154	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
155	C₄²×C₅₃	C₂³	1	2	yes	—
156	C₄²×C₅₃	C₂³	1	1	yes	characteristic
157	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
158	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
159	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
160	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
161	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
162	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
163	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
164	#11719921	C₂³	1	1	yes	characteristic
165	#11719921	C₂³	1	1	yes	characteristic
166	#11719921	C₂³	1	1	yes	characteristic
167	#11719921	C₂³	1	2	yes	—
168	#11719921	C₂³	1	2	yes	—
169	#11719921	C₂³	1	1	yes	characteristic
170	#11719921	C₂³	1	2	yes	—
171	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
172	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
173	C₄⋊Dic₅₃	C₂³	1	2	yes	—
174	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
175	C₄⋊Dic₅₃	C₂³	1	2	yes	—
176	C₄⋊Dic₅₃	C₂³	1	2	yes	—
177	C₄⋊Dic₅₃	C₂³	1	2	yes	—
178	C₄⋊Dic₅₃	C₂³	1	2	yes	—
179	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
180	C₄⋊Dic₅₃	C₂³	1	2	yes	—
181	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
182	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
183	C₄⋊Dic₅₃	C₂³	1	2	yes	—
184	C₄⋊Dic₅₃	C₂³	1	2	yes	—
185	C₄⋊Dic₅₃	C₂³	1	2	yes	—
186	#11719919	C₂³	1	2	yes	—
187	#11719919	C₂³	1	2	yes	—
188	#11719919	C₂³	1	2	yes	—
189	#11719919	C₂³	1	2	yes	—
190	#11719919	C₂³	1	2	yes	—
191	#11719919	C₂³	1	2	yes	—
192	#11719919	C₂³	1	2	yes	—
193	#11719919	C₂³	1	2	yes	—
194	#11719919	C₂³	1	2	yes	—
195	#11719919	C₂³	1	2	yes	—
196	#11719919	C₂³	1	2	yes	—
197	#11719919	C₂³	1	1	yes	characteristic
198	#11719919	C₂³	1	1	yes	characteristic
199	#11719919	C₂³	1	1	yes	characteristic
200	#11719919	C₂³	1	2	yes	—
201	#11719919	C₂³	1	1	yes	characteristic
202	#11719919	C₂³	1	2	yes	—
203	#11719919	C₂³	1	2	yes	—
204	#11719919	C₂³	1	2	yes	—
205	#11719919	C₂³	1	2	yes	—
206	#11719919	C₂³	1	1	yes	characteristic
207	#11719919	C₂³	1	1	yes	characteristic
208	#11719919	C₂³	1	2	yes	—
209	#11719919	C₂³	1	2	yes	—
210	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
211	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
212	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
213	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
214	C₄×Dic₅₃	C₂³	1	2	yes	—
215	C₄×Dic₅₃	C₂³	1	2	yes	—
216	C₄×Dic₅₃	C₂³	1	2	yes	—
217	C₄×Dic₅₃	C₂³	1	2	yes	—
218	C₄×Dic₅₃	C₂³	1	2	yes	—
219	C₄×Dic₅₃	C₂³	1	2	yes	—
220	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
221	C₄×Dic₅₃	C₂³	1	2	yes	—
222	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
223	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
224	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
225	Q₈×C₅₃	—	2	2	no	—
226	Q₈×C₅₃	—	4	2	no	—
227	D₄×C₅₃	—	2	2	no	—
228	C₂×C₄×C₅₃	—	2	2	no	—
229	C₂×C₄×C₅₃	—	2	2	no	—
230	C₂×C₄×C₅₃	—	4	2	no	—
231	C₂×C₄×C₅₃	—	4	2	no	—
232	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
233	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
234	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
235	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
236	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
237	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
238	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
239	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
240	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
241	C₂²⋊D₅₃	—	2	2	no	—
242	C₂²⋊D₅₃	—	4	1	no	—
243	C₂×Dic₅₃	—	2	2	no	—
244	C₂×Dic₅₃	—	2	4	no	—
245	C₂×Dic₅₃	—	4	1	no	—
246	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
247	C₂×Dic₅₃	C₂⁴	1	2	yes	—
248	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
249	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
250	C₂×Dic₅₃	C₂⁴	1	2	yes	—
251	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
252	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
253	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
254	C₂×Dic₅₃	C₂⁴	1	2	yes	—
255	C₂×Dic₅₃	C₂⁴	1	2	yes	—
256	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
257	C₄×D₅₃	—	2	2	no	—
258	C₄×D₅₃	—	4	2	no	—
259	Dic₁₀₆	—	2	2	no	—
260	Dic₁₀₆	—	4	2	no	—
261	Dic₁₀₆	—	4	1	no	—
262	Dic₁₀₆	—	4	2	no	—
263	C₂²×C₅₃	—	2	1	no	—
264	C₂²×C₅₃	—	2	1	no	—
265	C₂²×C₅₃	—	2	1	no	—
266	C₂²×C₅₃	—	2	1	no	—
267	C₂²×C₅₃	—	2	1	no	—
268	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
269	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
270	C₂×D₅₃	—	2	1	no	—
271	C₂×D₅₃	—	2	1	no	—
272	C₂×D₅₃	—	2	1	no	—
273	C₄×C₅₃	—	2	1	no	—
274	C₄×C₅₃	—	2	1	no	—
275	C₄×C₅₃	—	2	1	no	—
276	C₄×C₅₃	—	2	2	no	—
277	C₄×C₅₃	—	2	2	no	—
278	C₄×C₅₃	—	2	2	no	—
279	C₄×C₅₃	—	2	1	no	—
280	C₄×C₅₃	—	2	2	no	—
281	C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
282	Dic₅₃	—	2	2	no	—
283	Dic₅₃	—	2	1	no	—
284	Dic₅₃	—	2	1	no	—
285	Dic₅₃	—	2	1	no	—
286	Dic₅₃	—	2	2	no	—
287	Dic₅₃	—	2	2	no	—
288	Dic₅₃	—	2	1	no	—
289	Dic₅₃	—	2	1	no	—
290	Dic₅₃	—	2	2	no	—
291	Dic₅₃	—	2	1	no	—
292	Dic₅₃	—	2	1	no	—
293	#3563	—	53	1	no	maximal, Sylow
294	C₂×C₅₃	—	2	2	no	—
295	C₂×C₅₃	—	4	1	no	—
296	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
297	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
298	C₂×C₅₃	#585	1	1	yes	characteristic
299	D₅₃	—	4	1	no	—
300	#564	—	53	2	no	—
301	#563	—	53	2	no	—
302	#557	—	53	1	no	—
303	#557	—	53	1	no	—
304	#554	—	53	1	no	—
305	#554	D₅₃	1	1	yes	characteristic, complemented
306	#552	—	53	1	no	—
307	#552	—	53	2	no	—
308	#552	—	53	1	no	—
309	#552	—	53	2	no	—
310	#552	—	53	1	no	—
311	#552	—	53	1	no	—
312	#544	—	53	1	no	—
313	#544	—	53	2	no	—
314	#544	—	53	1	no	—
315	#541	—	53	2	no	—
316	#541	—	53	1	no	—
317	#541	—	53	2	no	—
318	#541	—	53	1	no	—
319	#539	—	53	2	no	—
320	#533	—	53	1	no	—
321	#529	—	53	1	no	—
322	C₂×#125	—	53	1	no	—
323	C₅₃	#3563	1	1	yes	characteristic, complemented, Sylow
324	C₄⋊Q₈	—	53	1	no	—
325	C₄⋊Q₈	—	53	2	no	—
326	C₄⋊Q₈	—	53	2	no	—
327	C₄⋊Q₈	—	53	1	no	—
328	C₄⋊Q₈	—	53	2	no	—
329	C₄⋊Q₈	—	53	1	no	—
330	C₄⋊Q₈	—	53	2	no	—
331	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
332	#126	—	53	1	no	—
333	#126	—	53	2	no	—
334	#126	—	53	2	no	—
335	#126	—	53	2	no	—
336	#126	—	53	1	no	—
337	#126	—	53	2	no	—
338	#126	—	53	2	no	—
339	#126	—	53	2	no	—
340	#126	—	53	2	no	—
341	#126	—	53	2	no	—
342	#125	—	53	2	no	—
343	#125	—	53	1	no	—
344	#125	—	53	1	no	—
345	#125	—	53	1	no	—
346	#125	—	53	1	no	—
347	#125	—	53	2	no	—
348	#125	—	53	1	no	—
349	#125	—	53	2	no	—
350	#125	—	53	2	no	—
351	#125	—	53	2	no	—
352	#125	—	53	2	no	—
353	#125	—	53	2	no	—
354	#125	—	53	2	no	—
355	#125	—	53	2	no	—
356	#125	—	53	2	no	—
357	#125	—	53	2	no	—
358	#125	—	53	2	no	—
359	#125	—	53	1	no	—
360	#125	—	53	2	no	—
361	#125	—	53	2	no	—
362	#125	—	53	2	no	—
363	#125	—	53	2	no	—
364	#125	—	53	2	no	—
365	#125	—	53	2	no	—
366	#125	—	53	1	no	—
367	#125	C₂×D₅₃	1	2	yes	—
368	#124	—	53	1	no	—
369	#123	—	53	1	no	—
370	#123	—	53	2	no	—
371	#123	—	53	2	no	—
372	#123	—	53	1	no	—
373	#123	—	53	2	no	—
374	#123	—	53	2	no	—
375	C₂²⋊Q₈	—	53	1	no	—
376	C₂²⋊Q₈	—	53	2	no	—
377	C₂²⋊Q₈	—	53	1	no	—
378	C₂²⋊Q₈	—	53	2	no	—
379	C₂²⋊Q₈	—	53	1	no	—
380	C₂²⋊Q₈	—	53	2	no	—
381	C₂²⋊Q₈	—	53	2	no	—
382	C₂²⋊Q₈	—	53	2	no	—
383	C₂²⋊Q₈	—	53	1	no	—
384	C₂²⋊Q₈	—	53	2	no	—
385	C₂²⋊Q₈	—	53	1	no	—
386	C₂²⋊Q₈	—	53	2	no	—
387	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
388	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
389	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	—
390	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	—
391	#121	—	53	1	no	—
392	C₄×Q₈	—	53	1	no	—
393	C₄×Q₈	—	53	2	no	—
394	C₄×Q₈	—	53	1	no	—
395	C₄×Q₈	—	53	2	no	—
396	C₄×Q₈	—	53	1	no	—
397	C₄×Q₈	—	53	1	no	—
398	C₄×Q₈	—	53	1	no	—
399	C₄×Q₈	—	53	2	no	—
400	C₄×Q₈	—	53	1	no	—
401	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
402	C₄×D₄	—	53	1	no	—
403	C₄×D₄	—	53	1	no	—
404	C₄×D₄	—	53	2	no	—
405	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
406	C₄×D₄	C₂×D₅₃	1	2	yes	—
407	#117	—	53	1	no	—
408	#117	—	53	2	no	—
409	#117	—	53	1	no	—
410	#117	—	53	2	no	—
411	#117	—	53	1	no	—
412	#117	—	53	2	no	—
413	#117	—	53	2	no	—
414	#117	—	53	2	no	—
415	#117	—	53	1	no	—
416	C₂×C₄⋊C₄	—	53	2	no	—
417	C₂×C₄⋊C₄	—	53	1	no	—
418	C₂×C₄⋊C₄	—	53	2	no	—
419	C₂×C₄⋊C₄	—	53	1	no	—
420	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
422	C₂×C₄²	—	53	1	no	—
423	C₂×Q₈	—	53	1	no	—
424	C₂×Q₈	—	53	2	no	—
425	C₂×Q₈	—	53	2	no	—
426	C₂×Q₈	—	53	1	no	—
427	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
428	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
429	C₂×D₄	—	53	1	no	—
430	C₂×D₄	—	53	1	no	—
431	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
432	C₂²×C₄	—	53	1	no	—
433	C₂²×C₄	—	53	2	no	—
434	C₂²×C₄	—	53	1	no	—
435	C₂²×C₄	—	53	1	no	—
436	C₂²×C₄	—	53	2	no	—
437	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
438	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
439	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
440	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
441	C₄⋊C₄	—	53	1	no	—
442	C₄⋊C₄	—	53	1	no	—
443	C₄⋊C₄	—	53	1	no	—
444	C₄⋊C₄	—	53	2	no	—
445	C₄⋊C₄	—	53	2	no	—
446	C₄⋊C₄	—	53	2	no	—
447	C₄⋊C₄	—	53	2	no	—
448	C₄⋊C₄	—	53	2	no	—
449	C₄⋊C₄	—	53	2	no	—
450	C₄⋊C₄	—	53	2	no	—
451	C₄⋊C₄	—	53	2	no	—
452	C₄⋊C₄	—	53	2	no	—
453	C₄⋊C₄	—	53	2	no	—
454	C₄⋊C₄	—	53	2	no	—
455	C₄⋊C₄	—	53	2	no	—
456	C₄⋊C₄	—	53	2	no	—
457	C₄⋊C₄	—	53	2	no	—
458	C₄⋊C₄	—	53	2	no	—
459	C₄⋊C₄	—	53	2	no	—
460	C₄⋊C₄	—	53	1	no	—
461	C₄⋊C₄	—	53	1	no	—
462	C₄⋊C₄	—	53	1	no	—
463	C₄⋊C₄	—	53	1	no	—
464	C₄⋊C₄	—	53	2	no	—
465	C₄⋊C₄	—	53	1	no	—
466	C₄⋊C₄	—	53	2	no	—
467	C₄⋊C₄	—	53	1	no	—
468	C₄⋊C₄	—	53	1	no	—
469	C₄⋊C₄	—	53	1	no	—
470	C₄⋊C₄	—	53	1	no	—
471	C₄⋊C₄	—	53	2	no	—
472	C₄⋊C₄	—	53	2	no	—
473	C₄⋊C₄	—	53	2	no	—
474	C₄⋊C₄	—	53	2	no	—
475	C₄⋊C₄	—	53	2	no	—
476	C₄⋊C₄	—	53	2	no	—
477	C₄⋊C₄	—	53	2	no	—
478	C₄⋊C₄	—	53	2	no	—
479	C₄⋊C₄	—	53	2	no	—
480	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
481	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
482	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
483	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
484	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
485	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
486	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
487	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
488	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
489	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
490	C₂²⋊C₄	—	53	2	no	—
491	C₂²⋊C₄	—	53	1	no	—
492	C₂²⋊C₄	—	53	1	no	—
493	C₂²⋊C₄	—	53	1	no	—
494	C₂²⋊C₄	—	53	2	no	—
495	C₂²⋊C₄	—	53	1	no	—
496	C₂²⋊C₄	—	53	2	no	—
497	C₂²⋊C₄	—	53	1	no	—
498	C₂²⋊C₄	—	53	1	no	—
499	C₂²⋊C₄	—	53	2	no	—
500	C₂²⋊C₄	—	53	1	no	—
501	C₂²⋊C₄	—	53	2	no	—
502	C₂²⋊C₄	—	53	2	no	—
503	C₂²⋊C₄	—	53	1	no	—
504	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
505	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
506	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
507	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
508	C₄²	—	53	1	no	—
509	C₄²	—	53	1	no	—
510	C₄²	—	53	1	no	—
511	C₄²	—	53	1	no	—
512	C₄²	—	53	2	no	—
513	C₄²	—	53	2	no	—
514	C₄²	—	53	2	no	—
515	C₄²	—	53	2	no	—
516	C₄²	—	53	1	no	—
517	C₄²	—	53	2	no	—
518	C₄²	—	53	2	no	—
519	C₄²	—	53	2	no	—
520	C₄²	C₂²×D₅₃	1	1	yes	characteristic
521	C₄²	C₂²×D₅₃	1	2	yes	—
522	C₂³	—	53	1	no	—
523	C₂³	C₂³×D₅₃	1	1	yes	characteristic
524	C₂³	C₂³×D₅₃	1	1	yes	characteristic
525	Q₈	—	2	2	no	—
526	Q₈	—	4	2	no	—
527	Q₈	—	106	2	no	—
528	Q₈	—	212	2	no	—
529	Q₈	—	212	2	no	—
530	Q₈	—	212	1	no	—
531	D₄	—	2	2	no	—
532	D₄	—	106	2	no	—
533	D₄	—	212	1	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	4	2	no	—
537	C₂×C₄	—	4	2	no	—
538	C₂×C₄	—	53	1	no	—
539	C₂×C₄	—	53	2	no	—
540	C₂×C₄	—	53	1	no	—
541	C₂×C₄	—	53	1	no	—
542	C₂×C₄	—	53	1	no	—
543	C₂×C₄	—	53	1	no	—
544	C₂×C₄	—	53	2	no	—
545	C₂×C₄	—	53	2	no	—
546	C₂×C₄	—	53	2	no	—
547	C₂×C₄	—	53	1	no	—
548	C₂×C₄	—	53	1	no	—
549	C₂×C₄	—	106	2	no	—
550	C₂×C₄	—	106	4	no	—
551	C₂×C₄	—	106	2	no	—
552	C₂×C₄	—	212	2	no	—
553	C₂×C₄	—	212	1	no	—
554	C₂×C₄	C₂³×D₅₃	1	2	yes	—
555	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
556	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
557	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
558	C₂×C₄	C₂³×D₅₃	1	2	yes	—
559	C₂×C₄	C₂³×D₅₃	1	2	yes	—
560	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
561	C₂×C₄	C₂³×D₅₃	1	2	yes	—
562	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
563	C₂²	—	2	1	no	—
564	C₂²	—	2	1	no	—
565	C₂²	—	2	1	no	—
566	C₂²	—	2	1	no	—
567	C₂²	—	2	1	no	—
568	C₂²	—	106	1	no	—
569	C₂²	—	106	1	no	—
570	C₂²	—	106	1	no	—
571	C₂²	#13133265	1	2	yes	—
572	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
573	C₄	—	2	1	no	—
574	C₄	—	2	1	no	—
575	C₄	—	2	2	no	—
576	C₄	—	2	1	no	—
577	C₄	—	2	2	no	—
578	C₄	—	2	2	no	—
579	C₄	—	2	2	no	—
580	C₄	—	2	1	no	—
581	C₄	—	106	1	no	—
582	C₄	—	106	2	no	—
583	C₄	—	106	1	no	—
584	C₄	—	106	1	no	—
585	C₄	—	106	2	no	—
586	C₄	—	106	2	no	—
587	C₄	—	106	1	no	—
588	C₄	—	106	1	no	—
589	C₄	—	106	2	no	—
590	C₄	—	106	1	no	—
591	C₄	—	106	1	no	—
592	C₄	#13133264	1	2	yes	—
593	C₂	—	2	2	no	—
594	C₂	—	4	1	no	—
595	C₂	—	212	1	no	—
596	C₂	C₂×#13133264	1	1	yes	characteristic
597	C₂	C₂×#13133265	1	1	yes	characteristic
598	C₂	#16811458	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#554	C₂
hypercenter	#554	D₅₃
hypocenter	C₅₃	#3563
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #554

Complements

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

N	H ≅ G/N	type
1. C₅₃⋊#564	C₂	semidirect
2. #16811374	C₂	semidirect
3. #16811349	C₂	semidirect
4. #16811348	C₂	semidirect
5. #16811328	C₂	semidirect
6. #16811315	C₂	semidirect
7. #16811315	C₂	semidirect
8. #16811313	C₂	semidirect
9. #16811313	C₂	semidirect
10. #16811308	C₂	semidirect
11. #16811307	C₂	semidirect
12. #16811254	C₂	semidirect
13. #16811252	C₂	semidirect
14. #16811250	C₂	semidirect
15. #16811233	C₂	semidirect
16. #16811233	C₂	semidirect
17. #16811232	C₂	semidirect
18. #16811231	C₂	semidirect
20. #16811175	C₂	semidirect
22. C₂×#13133208	C₂	semidirect
23. C₅₃×#554	C₂	semidirect
305. #554	D₅₃	semidirect
323. C₅₃	#3563	semidirect