Subgroups — #20288514

Order 6784.

1100 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811333	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811332	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811332	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811329	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811329	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811328	C₂	1	2	yes	maximal, complemented
7	#16811327	C₂	1	2	yes	maximal, complemented
8	#16811327	C₂	1	2	yes	maximal, complemented
9	D₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811288	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811286	C₂	1	2	yes	maximal, complemented
12	#16811253	C₂	1	1	yes	maximal, characteristic, complemented
13	#16811250	C₂	1	1	yes	maximal, characteristic, complemented
14	#16811248	C₂	1	2	yes	maximal, complemented
15	#16811248	C₂	1	2	yes	maximal, complemented
16	#16811179	C₂	1	1	yes	maximal, characteristic, complemented
17	#16811177	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#13133195	C₂	1	2	yes	maximal, complemented
19	C₂²⋊Q₈×D₅₃	C₂	1	2	yes	maximal, complemented
20	#16811095	C₂	1	1	yes	maximal, characteristic
21	#16811095	C₂	1	1	yes	maximal, characteristic
22	C₄×D₄×D₅₃	C₂	1	1	yes	maximal, characteristic
23	C₅₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂²×Dic₁₀₆	C₂²	1	2	yes	complemented
25	C₂×Q₈×D₅₃	C₂²	1	2	yes	complemented
26	C₂×D₄×D₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₅₃	C₂²	1	2	yes	—
28	#13133217	C₂²	1	1	yes	characteristic, complemented
29	#13133216	C₂²	1	1	yes	characteristic, complemented
30	#13133216	C₂²	1	2	yes	complemented
31	#13133216	C₂²	1	1	yes	characteristic, complemented
32	#13133216	C₂²	1	1	yes	characteristic, complemented
33	#13133216	C₂²	1	1	yes	characteristic, complemented
34	#13133216	C₂²	1	1	yes	characteristic, complemented
35	#13133215	C₂²	1	2	yes	complemented
36	#13133215	C₂²	1	1	yes	characteristic, complemented
37	#13133209	C₂²	1	1	yes	characteristic, complemented
38	#13133209	C₂²	1	1	yes	characteristic, complemented
39	#13133208	C₂²	1	1	yes	characteristic, complemented
40	#13133208	C₂²	1	1	yes	characteristic, complemented
41	#13133207	C₂²	1	1	yes	characteristic, complemented
42	#13133207	C₂²	1	2	yes	complemented
43	#13133207	C₂²	1	2	yes	complemented
44	#13133207	C₂²	1	1	yes	characteristic, complemented
45	#13133197	C₂²	1	2	yes	—
46	#13133197	C₂²	1	1	yes	characteristic
47	#13133197	C₂²	1	2	yes	complemented
48	#13133197	C₂²	1	2	yes	complemented
49	#13133196	C₂²	1	2	yes	complemented
50	#13133196	C₂²	1	2	yes	complemented
51	#13133196	C₂²	1	2	yes	complemented
52	#13133195	C₂²	1	1	yes	characteristic
53	#13133195	C₂²	1	2	yes	—
54	#13133195	C₂²	1	1	yes	characteristic
55	#13133195	C₂²	1	2	yes	—
56	#13133195	C₂²	1	2	yes	—
57	#13133195	C₂²	1	2	yes	—
58	#13133195	C₂²	1	2	yes	complemented
59	#13133195	C₂²	1	2	yes	complemented
60	#13133195	C₂²	1	2	yes	complemented
61	#13133195	C₂²	1	2	yes	complemented
62	#13133195	C₂²	1	2	yes	complemented
63	#13133195	C₂²	1	2	yes	complemented
64	#13133195	C₂²	1	2	yes	complemented
65	#13133195	C₂²	1	2	yes	complemented
66	#13133194	C₂²	1	2	yes	complemented
67	#13133194	C₂²	1	2	yes	complemented
68	#13133194	C₂²	1	2	yes	complemented
69	#13133194	C₂²	1	2	yes	complemented
70	#13133194	C₂²	1	2	yes	complemented
71	#13133194	C₂²	1	2	yes	complemented
72	#13133193	C₂²	1	2	yes	—
73	#13133193	C₂²	1	1	yes	characteristic
74	#13133193	C₂²	1	2	yes	complemented
75	#13133193	C₂²	1	2	yes	complemented
76	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
77	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
78	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
79	#13133182	C₂²	1	2	yes	complemented
80	#13133182	C₂²	1	2	yes	complemented
81	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
82	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
83	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
84	D₄×Dic₅₃	C₂²	1	1	yes	characteristic
85	D₄×Dic₅₃	C₂²	1	1	yes	characteristic
86	D₄×Dic₅₃	C₂²	1	1	yes	characteristic
87	#13133179	C₂²	1	1	yes	characteristic
88	#13133179	C₂²	1	1	yes	characteristic
89	#13133179	C₂²	1	1	yes	characteristic
90	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
91	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
92	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
93	#13133177	C₂²	1	2	yes	—
94	#13133177	C₂²	1	2	yes	—
95	#13133177	C₂²	1	2	yes	—
96	#13133175	C₂²	1	1	yes	characteristic
97	#13133175	C₂²	1	1	yes	characteristic
98	#13133174	C₂²	1	1	yes	characteristic
99	#13133174	C₂²	1	1	yes	characteristic
100	C₂×#11719919	C₂²	1	2	yes	complemented
101	C₂×#11719919	C₂²	1	2	yes	complemented
102	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic
103	C₄⋊C₄×D₅₃	C₂²	1	2	yes	complemented
104	C₄⋊C₄×D₅₃	C₂²	1	2	yes	complemented
105	C₄⋊C₄×D₅₃	C₂²	1	2	yes	complemented
106	C₂×#11719921	C₂²	1	2	yes	—
107	C₂×#11719921	C₂²	1	2	yes	—
108	C₂²⋊C₄×D₅₃	C₂²	1	2	yes	—
109	C₂²⋊C₄×D₅₃	C₂²	1	2	yes	—
110	C₂²⋊C₄×D₅₃	C₂²	1	2	yes	—
111	C₄²×D₅₃	C₂²	1	1	yes	characteristic
112	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
113	#125×C₅₃	C₂²	1	1	yes	characteristic, complemented
114	#125×C₅₃	C₂²	1	1	yes	characteristic, complemented
115	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
116	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
117	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
118	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
119	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
120	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
121	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
122	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
123	C₂³×D₅₃	C₂³	1	2	yes	—
124	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
125	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
126	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
128	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
129	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
130	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
131	C₂²×Dic₅₃	C₂³	1	2	yes	—
132	C₂²×Dic₅₃	C₂³	1	2	yes	—
133	C₂²×Dic₅₃	C₂³	1	2	yes	—
134	Q₈×D₅₃	—	4	4	no	—
135	D₄×D₅₃	Q₈	1	8	yes	complemented
136	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
137	C₂×C₄×D₅₃	—	4	4	no	—
138	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
139	C₂×C₄×D₅₃	C₂³	1	2	yes	—
140	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
141	C₂×C₄×D₅₃	C₂³	1	2	yes	—
142	C₂×C₄×D₅₃	C₂³	1	2	yes	—
143	C₂×C₄×D₅₃	C₂³	1	2	yes	—
144	C₂×C₄×D₅₃	C₂³	1	2	yes	—
145	C₂×C₄×D₅₃	C₂³	1	2	yes	—
146	C₂×C₄×D₅₃	C₂³	1	2	yes	—
147	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
148	C₂×Dic₁₀₆	—	4	4	no	—
149	C₂×Dic₁₀₆	C₂³	1	2	yes	—
150	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
151	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
152	C₂×Dic₁₀₆	C₂³	1	2	yes	—
153	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
154	C₂×Dic₁₀₆	C₂³	1	2	yes	—
155	C₂×Dic₁₀₆	C₂³	1	2	yes	—
156	C₂×Dic₁₀₆	C₂³	1	2	yes	—
157	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
158	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
159	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
160	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
161	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
162	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
163	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
164	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
165	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
166	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
167	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
168	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
169	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
170	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
171	C₄²×C₅₃	C₂³	1	1	yes	characteristic
172	C₄²×C₅₃	C₂³	1	1	yes	characteristic
173	C₄²×C₅₃	C₂³	1	1	yes	characteristic
174	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
175	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
176	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
177	#11719921	C₂³	1	1	yes	characteristic
178	#11719921	C₂³	1	1	yes	characteristic
179	#11719921	C₂³	1	2	yes	—
180	#11719921	C₂³	1	2	yes	—
181	#11719921	C₂³	1	2	yes	—
182	#11719921	C₂³	1	2	yes	—
183	#11719921	C₂³	1	1	yes	characteristic
184	#11719921	C₂³	1	1	yes	characteristic
185	#11719921	C₂³	1	2	yes	—
186	#11719921	C₂³	1	2	yes	—
187	#11719921	C₂³	1	2	yes	—
188	#11719921	C₂³	1	2	yes	—
189	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
190	C₄⋊Dic₅₃	C₂³	1	2	yes	—
191	C₄⋊Dic₅₃	C₂³	1	2	yes	—
192	C₄⋊Dic₅₃	C₂³	1	2	yes	—
193	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
194	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
195	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
196	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
197	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
198	#11719919	C₂³	1	2	yes	—
199	#11719919	C₂³	1	1	yes	characteristic
200	#11719919	C₂³	1	1	yes	characteristic
201	#11719919	C₂³	1	2	yes	—
202	#11719919	C₂³	1	2	yes	—
203	#11719919	C₂³	1	2	yes	—
204	#11719919	C₂³	1	2	yes	—
205	#11719919	C₂³	1	1	yes	characteristic
206	#11719919	C₂³	1	1	yes	characteristic
207	#11719919	C₂³	1	2	yes	—
208	#11719919	C₂³	1	2	yes	—
209	#11719919	C₂³	1	2	yes	—
210	#11719919	C₂³	1	2	yes	—
211	#11719919	C₂³	1	1	yes	characteristic
212	#11719919	C₂³	1	1	yes	characteristic
213	#11719919	C₂³	1	1	yes	characteristic
214	#11719919	C₂³	1	1	yes	characteristic
215	#11719919	C₂³	1	2	yes	—
216	#11719919	C₂³	1	2	yes	—
217	#11719919	C₂³	1	1	yes	characteristic
218	#11719919	C₂³	1	1	yes	characteristic
219	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
220	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
221	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
222	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
223	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
224	C₄×Dic₅₃	C₂³	1	2	yes	—
225	C₄×Dic₅₃	C₂³	1	2	yes	—
226	C₂³×C₅₃	C₂⁴	1	2	yes	—
227	C₂²×D₅₃	—	4	2	no	—
228	C₂²×D₅₃	—	4	2	no	—
229	C₂²×D₅₃	C₂×Q₈	1	8	yes	complemented
230	C₂²×D₅₃	C₂⁴	1	2	yes	—
231	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
232	Q₈×C₅₃	—	4	2	no	—
233	Q₈×C₅₃	—	4	1	no	—
234	D₄×C₅₃	C₂×Q₈	1	4	yes	complemented
235	C₂×C₄×C₅₃	—	4	2	no	—
236	C₂×C₄×C₅₃	—	4	2	no	—
237	C₂×C₄×C₅₃	—	4	2	no	—
238	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
239	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
240	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
241	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
242	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
243	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
244	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
245	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
246	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
247	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
248	C₂²⋊D₅₃	C₂×Q₈	1	8	yes	complemented
249	C₂×Dic₅₃	—	2	4	no	—
250	C₂×Dic₅₃	—	4	2	no	—
251	C₂×Dic₅₃	—	4	2	no	—
252	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
253	C₂×Dic₅₃	C₂⁴	1	2	yes	—
254	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
255	C₂×Dic₅₃	C₂⁴	1	2	yes	—
256	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
257	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
258	C₂×Dic₅₃	C₂⁴	1	2	yes	—
259	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
260	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
261	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
262	D₂₁₂	C₂×Q₈	1	4	yes	complemented
263	C₄×D₅₃	—	2	2	no	—
264	C₄×D₅₃	—	2	4	no	—
265	C₄×D₅₃	—	4	2	no	—
266	C₄×D₅₃	—	4	2	no	—
267	C₄×D₅₃	—	4	1	no	—
268	C₄×D₅₃	—	4	2	no	—
269	C₄×D₅₃	—	4	2	no	—
270	C₄×D₅₃	—	4	2	no	—
271	C₄×D₅₃	—	4	2	no	—
272	C₄×D₅₃	C₂×Q₈	1	4	yes	complemented
273	Dic₁₀₆	—	2	2	no	—
274	Dic₁₀₆	—	2	4	no	—
275	Dic₁₀₆	—	2	2	no	—
276	Dic₁₀₆	—	4	2	no	—
277	Dic₁₀₆	—	4	2	no	—
278	Dic₁₀₆	—	4	2	no	—
279	Dic₁₀₆	—	4	1	no	—
280	Dic₁₀₆	—	4	2	no	—
281	C₂²×C₅₃	—	2	2	no	—
282	C₂²×C₅₃	—	2	2	no	—
283	C₂²×C₅₃	C₂²×Q₈	1	4	yes	—
284	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
285	C₂×D₅₃	—	2	2	no	—
286	C₂×D₅₃	—	2	1	no	—
287	C₂×D₅₃	—	2	2	no	—
288	C₂×D₅₃	—	2	1	no	—
289	C₂×D₅₃	—	4	8	no	—
290	C₂×D₅₃	C₂²×Q₈	1	2	yes	—
291	C₂×D₅₃	C₂²×Q₈	1	4	yes	—
292	C₄×C₅₃	—	2	1	no	—
293	C₄×C₅₃	—	2	2	no	—
294	C₄×C₅₃	—	2	1	no	—
295	C₄×C₅₃	—	2	2	no	—
296	C₄×C₅₃	—	2	1	no	—
297	C₄×C₅₃	—	2	2	no	—
298	C₄×C₅₃	—	2	1	no	—
299	C₄×C₅₃	—	2	1	no	—
300	C₄×C₅₃	—	2	1	no	—
301	C₄×C₅₃	C₂²×Q₈	1	2	yes	—
302	Dic₅₃	—	2	1	no	—
303	Dic₅₃	—	2	2	no	—
304	Dic₅₃	—	2	1	no	—
305	Dic₅₃	—	2	2	no	—
306	Dic₅₃	—	2	1	no	—
307	Dic₅₃	—	2	1	no	—
308	Dic₅₃	—	2	2	no	—
309	Dic₅₃	—	2	1	no	—
310	Dic₅₃	C₂²×Q₈	1	2	yes	—
311	Dic₅₃	H(5,𝔽₂)	1	2	yes	—
312	#3501	—	53	1	no	maximal, Sylow
313	C₂×C₅₃	—	2	4	no	—
314	C₂×C₅₃	C₂³×Q₈	1	1	yes	characteristic
315	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
316	C₂×C₅₃	#585	1	1	yes	characteristic
317	D₅₃	—	2	2	no	—
318	D₅₃	—	2	4	no	—
319	#554	—	53	1	no	—
320	#554	—	53	1	no	—
321	#554	—	53	2	no	—
322	#554	—	53	2	no	—
323	#554	—	53	1	no	—
324	#554	—	53	1	no	—
325	#554	—	53	1	no	—
326	#554	—	53	2	no	—
327	#554	D₅₃	1	1	yes	characteristic, complemented
328	D₄×Q₈	—	53	1	no	—
329	D₄×Q₈	—	53	2	no	—
330	D₄×Q₈	—	53	1	no	—
331	#544	—	53	1	no	—
332	#544	—	53	1	no	—
333	#543	—	53	2	no	—
334	#543	—	53	2	no	—
335	#533	—	53	1	no	—
336	#533	—	53	1	no	—
337	C₂×C₂²⋊Q₈	—	53	2	no	—
338	C₂×C₂²⋊Q₈	—	53	2	no	—
339	#518	—	53	1	no	—
340	#518	—	53	1	no	—
341	C₂×C₄×D₄	—	53	1	no	—
342	C₅₃	#3501	1	1	yes	characteristic, complemented, Sylow
343	C₂²×Q₈	—	53	2	no	—
344	C₂²×Q₈	—	53	2	no	—
345	C₂²×D₄	—	53	1	no	—
346	C₂³×C₄	—	53	2	no	—
347	C₄⋊Q₈	—	53	1	no	—
348	C₄⋊Q₈	—	53	2	no	—
349	C₄⋊Q₈	—	53	1	no	—
350	C₄⋊Q₈	—	53	1	no	—
351	C₄⋊Q₈	—	53	1	no	—
352	C₄⋊Q₈	—	53	2	no	—
353	C₄⋊Q₈	—	53	1	no	—
354	C₄⋊Q₈	—	53	1	no	—
355	C₄⋊Q₈	—	53	1	no	—
356	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
357	#125	—	53	1	no	—
358	#125	—	53	1	no	—
359	#125	—	53	2	no	—
360	#125	—	53	2	no	—
361	#125	—	53	1	no	—
362	#125	—	53	1	no	—
363	#125	—	53	1	no	—
364	#125	—	53	1	no	—
365	#125	C₂×D₅₃	1	1	yes	characteristic, complemented
366	#125	C₂×D₅₃	1	1	yes	characteristic, complemented
367	C₂²⋊Q₈	—	53	2	no	—
368	C₂²⋊Q₈	—	53	1	no	—
369	C₂²⋊Q₈	—	53	2	no	—
370	C₂²⋊Q₈	—	53	2	no	—
371	C₂²⋊Q₈	—	53	2	no	—
372	C₂²⋊Q₈	—	53	2	no	—
373	C₂²⋊Q₈	—	53	2	no	—
374	C₂²⋊Q₈	—	53	1	no	—
375	C₂²⋊Q₈	—	53	2	no	—
376	C₂²⋊Q₈	—	53	1	no	—
377	C₂²⋊Q₈	—	53	2	no	—
378	C₂²⋊Q₈	—	53	2	no	—
379	C₂²⋊Q₈	—	53	2	no	—
380	C₂²⋊Q₈	—	53	2	no	—
381	C₂²⋊Q₈	—	53	2	no	—
382	C₂²⋊Q₈	—	53	2	no	—
383	C₂²⋊Q₈	—	53	2	no	—
384	C₂²⋊Q₈	—	53	2	no	—
385	C₂²⋊Q₈	—	53	1	no	—
386	C₂²⋊Q₈	—	53	2	no	—
387	C₂²⋊Q₈	—	53	2	no	—
388	C₂²⋊Q₈	—	53	2	no	—
389	C₂²⋊Q₈	—	53	2	no	—
390	C₂²⋊Q₈	—	53	2	no	—
391	C₂²⋊Q₈	—	53	2	no	—
392	C₂²⋊Q₈	—	53	2	no	—
393	C₂²⋊Q₈	—	53	2	no	—
394	C₂²⋊Q₈	—	53	2	no	—
395	C₂²⋊Q₈	—	53	2	no	—
396	C₂²⋊Q₈	—	53	2	no	—
397	C₂²⋊Q₈	—	53	2	no	—
398	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	complemented
399	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	complemented
400	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	complemented
401	C₄×Q₈	—	53	2	no	—
402	C₄×Q₈	—	53	2	no	—
403	C₄×Q₈	—	53	1	no	—
404	C₄×Q₈	—	53	1	no	—
405	C₄×Q₈	—	53	1	no	—
406	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
407	C₄×D₄	—	53	1	no	—
408	C₄×D₄	—	53	1	no	—
409	C₄×D₄	—	53	1	no	—
410	C₄×D₄	—	53	2	no	—
411	C₄×D₄	—	53	2	no	—
412	C₄×D₄	—	53	2	no	—
413	C₄×D₄	—	53	1	no	—
414	C₄×D₄	—	53	1	no	—
415	C₄×D₄	—	53	1	no	—
416	C₄×D₄	—	53	2	no	—
417	C₄×D₄	—	53	2	no	—
418	C₄×D₄	—	53	2	no	—
419	C₄×D₄	—	53	1	no	—
420	C₄×D₄	—	53	1	no	—
421	C₄×D₄	—	53	1	no	—
422	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
423	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
424	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
425	#117	—	53	1	no	—
426	#117	—	53	1	no	—
427	#117	—	53	1	no	—
428	#117	—	53	1	no	—
429	C₂×C₄⋊C₄	—	53	2	no	—
430	C₂×C₄⋊C₄	—	53	2	no	—
431	C₂×C₄⋊C₄	—	53	1	no	—
432	C₂×C₄⋊C₄	—	53	2	no	—
433	C₂×C₄⋊C₄	—	53	2	no	—
434	C₂×C₄⋊C₄	—	53	2	no	—
435	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	complemented
436	C₂×C₂²⋊C₄	—	53	2	no	—
437	C₂×C₂²⋊C₄	—	53	2	no	—
438	C₂×C₂²⋊C₄	—	53	2	no	—
439	C₂×C₂²⋊C₄	—	53	2	no	—
440	C₂×C₂²⋊C₄	—	53	2	no	—
441	C₂×C₄²	—	53	1	no	—
442	C₂⁴	—	53	2	no	—
443	C₂×Q₈	—	53	2	no	—
444	C₂×Q₈	—	53	1	no	—
445	C₂×Q₈	—	53	1	no	—
446	C₂×Q₈	—	53	2	no	—
447	C₂×Q₈	—	53	1	no	—
448	C₂×Q₈	—	53	2	no	—
449	C₂×Q₈	—	53	2	no	—
450	C₂×Q₈	—	53	2	no	—
451	C₂×Q₈	—	212	4	no	—
452	C₂×Q₈	—	212	4	no	—
453	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
454	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
455	C₂×D₄	—	53	2	no	—
456	C₂×D₄	—	53	8	no	—
457	C₂×D₄	—	53	1	no	—
458	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
459	C₂²×C₄	—	53	1	no	—
460	C₂²×C₄	—	53	2	no	—
461	C₂²×C₄	—	53	2	no	—
462	C₂²×C₄	—	53	2	no	—
463	C₂²×C₄	—	53	1	no	—
464	C₂²×C₄	—	53	2	no	—
465	C₂²×C₄	—	53	1	no	—
466	C₂²×C₄	—	53	2	no	—
467	C₂²×C₄	—	53	2	no	—
468	C₂²×C₄	—	53	2	no	—
469	C₂²×C₄	—	53	2	no	—
470	C₂²×C₄	—	53	2	no	—
471	C₂²×C₄	—	53	2	no	—
472	C₂²×C₄	—	212	4	no	—
473	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
474	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
475	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
476	C₄⋊C₄	—	53	1	no	—
477	C₄⋊C₄	—	53	2	no	—
478	C₄⋊C₄	—	53	1	no	—
479	C₄⋊C₄	—	53	2	no	—
480	C₄⋊C₄	—	53	1	no	—
481	C₄⋊C₄	—	53	1	no	—
482	C₄⋊C₄	—	53	2	no	—
483	C₄⋊C₄	—	53	1	no	—
484	C₄⋊C₄	—	53	2	no	—
485	C₄⋊C₄	—	53	2	no	—
486	C₄⋊C₄	—	53	2	no	—
487	C₄⋊C₄	—	53	2	no	—
488	C₄⋊C₄	—	53	2	no	—
489	C₄⋊C₄	—	53	2	no	—
490	C₄⋊C₄	—	53	2	no	—
491	C₄⋊C₄	—	53	2	no	—
492	C₄⋊C₄	—	53	2	no	—
493	C₄⋊C₄	—	53	1	no	—
494	C₄⋊C₄	—	53	1	no	—
495	C₄⋊C₄	—	53	1	no	—
496	C₄⋊C₄	—	53	1	no	—
497	C₄⋊C₄	—	53	1	no	—
498	C₄⋊C₄	—	53	1	no	—
499	C₄⋊C₄	—	53	1	no	—
500	C₄⋊C₄	—	53	1	no	—
501	C₄⋊C₄	—	53	2	no	—
502	C₄⋊C₄	—	53	1	no	—
503	C₄⋊C₄	—	53	1	no	—
504	C₄⋊C₄	—	53	2	no	—
505	C₄⋊C₄	—	53	1	no	—
506	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
507	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
508	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
509	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
510	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
511	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
512	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
513	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
514	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
515	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
516	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
517	C₂²⋊C₄	—	53	2	no	—
518	C₂²⋊C₄	—	53	2	no	—
519	C₂²⋊C₄	—	53	2	no	—
520	C₂²⋊C₄	—	53	2	no	—
521	C₂²⋊C₄	—	53	1	no	—
522	C₂²⋊C₄	—	53	1	no	—
523	C₂²⋊C₄	—	53	2	no	—
524	C₂²⋊C₄	—	53	1	no	—
525	C₂²⋊C₄	—	53	1	no	—
526	C₂²⋊C₄	—	53	2	no	—
527	C₂²⋊C₄	—	53	2	no	—
528	C₂²⋊C₄	—	53	2	no	—
529	C₂²⋊C₄	—	53	2	no	—
530	C₂²⋊C₄	—	53	2	no	—
531	C₂²⋊C₄	—	53	2	no	—
532	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
533	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
534	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
535	C₄²	—	53	1	no	—
536	C₄²	—	53	1	no	—
537	C₄²	—	53	1	no	—
538	C₄²	—	53	2	no	—
539	C₄²	—	53	2	no	—
540	C₄²	—	53	1	no	—
541	C₄²	—	53	1	no	—
542	C₄²	C₂²×D₅₃	1	1	yes	characteristic
543	C₄²	C₂²×D₅₃	1	1	yes	characteristic
544	C₄²	C₂²×D₅₃	1	1	yes	characteristic
545	C₂³	—	53	1	no	—
546	C₂³	—	53	2	no	—
547	C₂³	—	53	8	no	—
548	C₂³	—	212	2	no	—
549	C₂³	—	212	2	no	—
550	C₂³	C₂³×D₅₃	1	2	yes	—
551	Q₈	—	4	2	no	—
552	Q₈	—	4	1	no	—
553	Q₈	—	106	2	no	—
554	Q₈	—	106	4	no	—
555	Q₈	—	106	2	no	—
556	Q₈	—	212	2	no	—
557	Q₈	—	212	1	no	—
558	Q₈	—	212	2	no	—
559	Q₈	—	212	2	no	—
560	Q₈	—	212	2	no	—
561	D₄	—	53	8	no	—
562	D₄	—	53	4	no	—
563	D₄	Q₈×D₅₃	1	4	yes	complemented
564	C₂×C₄	—	4	2	no	—
565	C₂×C₄	—	4	2	no	—
566	C₂×C₄	—	4	2	no	—
567	C₂×C₄	—	53	1	no	—
568	C₂×C₄	—	53	1	no	—
569	C₂×C₄	—	53	4	no	—
570	C₂×C₄	—	53	1	no	—
571	C₂×C₄	—	53	1	no	—
572	C₂×C₄	—	53	2	no	—
573	C₂×C₄	—	53	1	no	—
574	C₂×C₄	—	53	1	no	—
575	C₂×C₄	—	53	2	no	—
576	C₂×C₄	—	53	1	no	—
577	C₂×C₄	—	53	2	no	—
578	C₂×C₄	—	106	2	no	—
579	C₂×C₄	—	106	4	no	—
580	C₂×C₄	—	106	4	no	—
581	C₂×C₄	—	212	2	no	—
582	C₂×C₄	—	212	2	no	—
583	C₂×C₄	—	212	1	no	—
584	C₂×C₄	—	212	2	no	—
585	C₂×C₄	—	212	2	no	—
586	C₂×C₄	—	212	2	no	—
587	C₂×C₄	—	212	2	no	—
588	C₂×C₄	—	212	2	no	—
589	C₂×C₄	—	212	2	no	—
590	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
591	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
592	C₂×C₄	C₂³×D₅₃	1	2	yes	—
593	C₂×C₄	C₂³×D₅₃	1	2	yes	—
594	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
595	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
596	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
597	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
598	C₂×C₄	C₂³×D₅₃	1	2	yes	—
599	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
600	C₂²	—	2	2	no	—
601	C₂²	—	2	2	no	—
602	C₂²	—	53	2	no	—
603	C₂²	—	53	4	no	—
604	C₂²	—	106	2	no	—
605	C₂²	—	106	1	no	—
606	C₂²	—	106	2	no	—
607	C₂²	—	106	1	no	—
608	C₂²	—	212	8	no	—
609	C₂²	C₂×Q₈×D₅₃	1	4	yes	—
610	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
611	C₄	—	2	1	no	—
612	C₄	—	2	1	no	—
613	C₄	—	2	2	no	—
614	C₄	—	2	2	no	—
615	C₄	—	2	1	no	—
616	C₄	—	2	1	no	—
617	C₄	—	2	2	no	—
618	C₄	—	2	1	no	—
619	C₄	—	2	1	no	—
620	C₄	—	53	2	no	—
621	C₄	—	53	2	no	—
622	C₄	—	106	2	no	—
623	C₄	—	106	1	no	—
624	C₄	—	106	2	no	—
625	C₄	—	106	1	no	—
626	C₄	—	106	1	no	—
627	C₄	—	106	2	no	—
628	C₄	—	106	1	no	—
629	C₄	—	106	1	no	—
630	C₄	C₂×Q₈×D₅₃	1	2	yes	—
631	C₂	—	2	4	no	—
632	C₂	—	106	2	no	—
633	C₂	—	106	4	no	—
634	C₂	C₂²×Q₈×D₅₃	1	1	yes	characteristic
635	C₂	H(5,𝔽₂)×D₅₃	1	1	yes	characteristic
636	C₂	#16811459	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	#3501
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. #16811333	C₂	semidirect
2. #16811332	C₂	semidirect
3. #16811332	C₂	semidirect
4. #16811329	C₂	semidirect
5. #16811329	C₂	semidirect
6. #16811328	C₂	semidirect
7. #16811327	C₂	semidirect
8. #16811327	C₂	semidirect
9. D₄×Dic₁₀₆	C₂	semidirect
10. #16811288	C₂	semidirect
11. #16811286	C₂	semidirect
12. #16811253	C₂	semidirect
13. #16811250	C₂	semidirect
14. #16811248	C₂	semidirect
15. #16811248	C₂	semidirect
16. #16811179	C₂	semidirect
17. #16811177	C₂	semidirect
18. C₂×#13133195	C₂	semidirect
19. C₂²⋊Q₈×D₅₃	C₂	semidirect
23. C₅₃×#554	C₂	semidirect
24. C₂²×Dic₁₀₆	C₂²	semidirect
25. C₂×Q₈×D₅₃	C₂²	semidirect
28. #13133217	C₂²	semidirect
29. #13133216	C₂²	semidirect
30. #13133216	C₂²	semidirect
31. #13133216	C₂²	semidirect
32. #13133216	C₂²	semidirect
33. #13133216	C₂²	semidirect
34. #13133216	C₂²	semidirect
35. #13133215	C₂²	semidirect
36. #13133215	C₂²	semidirect
37. #13133209	C₂²	semidirect
38. #13133209	C₂²	semidirect
39. #13133208	C₂²	semidirect
40. #13133208	C₂²	semidirect
41. #13133207	C₂²	semidirect
42. #13133207	C₂²	semidirect
43. #13133207	C₂²	semidirect
44. #13133207	C₂²	semidirect
47. #13133197	C₂²	semidirect
48. #13133197	C₂²	semidirect
49. #13133196	C₂²	semidirect
50. #13133196	C₂²	semidirect
51. #13133196	C₂²	semidirect
58. #13133195	C₂²	semidirect
59. #13133195	C₂²	semidirect
60. #13133195	C₂²	semidirect
61. #13133195	C₂²	semidirect
62. #13133195	C₂²	semidirect
63. #13133195	C₂²	semidirect
64. #13133195	C₂²	semidirect
65. #13133195	C₂²	semidirect
66. #13133194	C₂²	semidirect
67. #13133194	C₂²	semidirect
68. #13133194	C₂²	semidirect
69. #13133194	C₂²	semidirect
70. #13133194	C₂²	semidirect
71. #13133194	C₂²	semidirect
74. #13133193	C₂²	semidirect
75. #13133193	C₂²	semidirect
76. C₄×Dic₁₀₆	C₂²	semidirect
77. C₄×Dic₁₀₆	C₂²	semidirect
78. C₄×Dic₁₀₆	C₂²	semidirect
79. #13133182	C₂²	semidirect
80. #13133182	C₂²	semidirect
100. C₂×#11719919	C₂²	semidirect
101. C₂×#11719919	C₂²	semidirect
103. C₄⋊C₄×D₅₃	C₂²	semidirect
104. C₄⋊C₄×D₅₃	C₂²	semidirect
105. C₄⋊C₄×D₅₃	C₂²	semidirect
112. C₄⋊Q₈×C₅₃	C₂²	semidirect
113. #125×C₅₃	C₂²	semidirect
114. #125×C₅₃	C₂²	semidirect
115. C₂²⋊Q₈×C₅₃	C₂²	semidirect
116. C₂²⋊Q₈×C₅₃	C₂²	semidirect
117. C₂²⋊Q₈×C₅₃	C₂²	semidirect
118. C₄×Q₈×C₅₃	C₂²	semidirect
122. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
135. D₄×D₅₃	Q₈	semidirect
229. C₂²×D₅₃	C₂×Q₈	semidirect
234. D₄×C₅₃	C₂×Q₈	semidirect
248. C₂²⋊D₅₃	C₂×Q₈	semidirect
262. D₂₁₂	C₂×Q₈	semidirect
272. C₄×D₅₃	C₂×Q₈	semidirect
327. #554	D₅₃	semidirect
342. C₅₃	#3501	semidirect
356. C₄⋊Q₈	C₂×D₅₃	semidirect
365. #125	C₂×D₅₃	semidirect
366. #125	C₂×D₅₃	semidirect
398. C₂²⋊Q₈	C₂×D₅₃	semidirect
399. C₂²⋊Q₈	C₂×D₅₃	semidirect
400. C₂²⋊Q₈	C₂×D₅₃	semidirect
406. C₄×Q₈	C₂×D₅₃	semidirect
435. C₂×C₄⋊C₄	C₂×D₅₃	semidirect
563. D₄	Q₈×D₅₃	semidirect