Subgroups — #20289014

Order 6784.

1032 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811377	C₂	1	2	yes	maximal, complemented
2	#16811371	C₂	1	2	yes	maximal, complemented
3	#16811346	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811333	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811329	C₂	1	2	yes	maximal, complemented
6	#16811326	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811326	C₂	1	2	yes	maximal, complemented
8	#16811311	C₂	1	1	yes	maximal, characteristic, complemented
9	#16811288	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811285	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811285	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811276	C₂	1	1	yes	maximal, characteristic, complemented
13	#16811244	C₂	1	1	yes	maximal, characteristic, complemented
14	#16811239	C₂	1	2	yes	maximal, complemented
15	#16811214	C₂	1	2	yes	maximal, complemented
16	#16811212	C₂	1	2	yes	maximal, complemented
17	#16811212	C₂	1	2	yes	maximal, complemented
18	#16811203	C₂	1	1	yes	maximal, characteristic, complemented
19	#16811196	C₂	1	1	yes	maximal, characteristic, complemented
20	#16811136	C₂	1	1	yes	maximal, characteristic, complemented
21	#16811135	C₂	1	1	yes	maximal, characteristic, complemented
22	C₂²⋊Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
23	C₅₃×#541	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#11719949	C₂²	1	1	yes	characteristic
25	C₂×#11719949	C₂²	1	1	yes	characteristic, complemented
26	C₂×#11719945	C₂²	1	1	yes	characteristic
27	C₂×Q₈×D₅₃	C₂²	1	1	yes	characteristic
28	C₂×Q₈×D₅₃	C₂²	1	1	yes	characteristic, complemented
29	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
30	#13133217	C₂²	1	1	yes	characteristic, complemented
31	#13133216	C₂²	1	2	yes	complemented
32	#13133216	C₂²	1	2	yes	complemented
33	#13133212	C₂²	1	2	yes	—
34	#13133212	C₂²	1	2	yes	—
35	#13133212	C₂²	1	2	yes	complemented
36	#13133212	C₂²	1	2	yes	complemented
37	#13133212	C₂²	1	2	yes	complemented
38	#13133209	C₂²	1	2	yes	complemented
39	#13133208	C₂²	1	2	yes	complemented
40	#13133205	C₂²	1	2	yes	—
41	#13133205	C₂²	1	2	yes	—
42	#13133205	C₂²	1	1	yes	characteristic, complemented
43	#13133205	C₂²	1	2	yes	complemented
44	#13133203	C₂²	1	2	yes	—
45	#13133203	C₂²	1	2	yes	—
46	#13133202	C₂²	1	2	yes	—
47	#13133202	C₂²	1	2	yes	—
48	#13133202	C₂²	1	2	yes	—
49	#13133202	C₂²	1	2	yes	complemented
50	#13133202	C₂²	1	2	yes	complemented
51	#13133202	C₂²	1	2	yes	complemented
52	#13133201	C₂²	1	1	yes	characteristic
53	#13133201	C₂²	1	1	yes	characteristic
54	#13133199	C₂²	1	2	yes	—
55	#13133199	C₂²	1	2	yes	—
56	#13133197	C₂²	1	1	yes	characteristic
57	#13133197	C₂²	1	1	yes	characteristic
58	#13133197	C₂²	1	1	yes	characteristic
59	#13133197	C₂²	1	1	yes	characteristic, complemented
60	#13133197	C₂²	1	2	yes	complemented
61	#13133197	C₂²	1	2	yes	complemented
62	#13133196	C₂²	1	1	yes	characteristic
63	#13133195	C₂²	1	2	yes	—
64	#13133195	C₂²	1	2	yes	—
65	#13133195	C₂²	1	2	yes	—
66	#13133195	C₂²	1	2	yes	complemented
67	#13133195	C₂²	1	2	yes	complemented
68	#13133195	C₂²	1	2	yes	complemented
69	#13133195	C₂²	1	2	yes	complemented
70	#13133194	C₂²	1	2	yes	—
71	#13133193	C₂²	1	1	yes	characteristic
72	#13133193	C₂²	1	1	yes	characteristic
73	#13133193	C₂²	1	1	yes	characteristic, complemented
74	#13133193	C₂²	1	1	yes	characteristic, complemented
75	#13133192	C₂²	1	2	yes	—
76	#13133192	C₂²	1	1	yes	characteristic
77	#13133192	C₂²	1	2	yes	—
78	#13133192	C₂²	1	2	yes	—
79	#13133192	C₂²	1	1	yes	characteristic
80	#13133192	C₂²	1	2	yes	—
81	#13133192	C₂²	1	1	yes	characteristic
82	#13133192	C₂²	1	1	yes	characteristic, complemented
83	#13133192	C₂²	1	2	yes	complemented
84	#13133191	C₂²	1	1	yes	characteristic
85	#13133190	C₂²	1	2	yes	—
86	#13133190	C₂²	1	2	yes	—
87	#13133187	C₂²	1	1	yes	characteristic
88	#13133187	C₂²	1	1	yes	characteristic
89	#13133187	C₂²	1	2	yes	—
90	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
91	C₄×Dic₁₀₆	C₂²	1	2	yes	complemented
92	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
93	C₄×D₂₁₂	C₂²	1	2	yes	—
94	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
95	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
96	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
97	C₄×D₂₁₂	C₂²	1	2	yes	—
98	C₄×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
99	C₄×D₂₁₂	C₂²	1	2	yes	complemented
100	#13133179	C₂²	1	2	yes	—
101	#13133179	C₂²	1	2	yes	—
102	#13133179	C₂²	1	2	yes	—
103	#13133177	C₂²	1	2	yes	—
104	#13133175	C₂²	1	1	yes	characteristic
105	#13133175	C₂²	1	2	yes	complemented
106	#13133175	C₂²	1	1	yes	characteristic, complemented
107	C₄⋊C₄×D₅₃	C₂²	1	2	yes	—
108	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic
109	C₄⋊C₄×D₅₃	C₂²	1	2	yes	complemented
110	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic, complemented
111	C₂²⋊C₄×D₅₃	C₂²	1	2	yes	—
112	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
113	#126×C₅₃	C₂²	1	2	yes	—
114	#126×C₅₃	C₂²	1	2	yes	—
115	#125×C₅₃	C₂²	1	1	yes	characteristic, complemented
116	#125×C₅₃	C₂²	1	2	yes	complemented
117	#125×C₅₃	C₂²	1	2	yes	complemented
118	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
119	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
120	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
121	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
122	#117×C₅₃	C₂²	1	1	yes	characteristic
123	C₂³×D₅₃	C₂³	1	1	yes	characteristic
124	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
125	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
127	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
128	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
129	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
130	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
131	#11719949	—	4	2	no	—
132	#11719949	—	4	2	no	—
133	Q₈×D₅₃	—	4	2	no	—
134	Q₈×D₅₃	—	4	2	no	—
135	#11719945	—	4	2	no	—
136	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
137	C₂×D₂₁₂	C₂³	1	2	yes	—
138	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
139	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
140	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
141	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
142	C₂×D₂₁₂	C₂³	1	2	yes	—
143	C₂×D₂₁₂	C₂³	1	2	yes	—
144	C₂×D₂₁₂	C₂³	1	2	yes	—
145	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
146	C₂×C₄×D₅₃	—	4	2	no	—
147	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
148	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
149	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
150	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
151	C₂×C₄×D₅₃	C₂³	1	2	yes	—
152	C₂×C₄×D₅₃	C₂³	1	2	yes	—
153	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
154	C₂×C₄×D₅₃	C₂³	1	2	yes	—
155	C₂×C₄×D₅₃	C₂³	1	2	yes	—
156	C₂×C₄×D₅₃	C₂³	1	2	yes	—
157	C₂×C₄×D₅₃	C₂³	1	2	yes	—
158	C₂×C₄×D₅₃	C₂³	1	2	yes	—
159	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
160	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
161	C₂×Dic₁₀₆	C₂³	1	2	yes	—
162	C₂×Dic₁₀₆	C₂³	1	2	yes	—
163	C₂×Dic₁₀₆	C₂³	1	2	yes	—
164	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
165	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
166	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
167	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
168	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
169	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
170	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
171	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
172	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
173	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
174	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
175	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
176	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
177	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
178	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
179	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
180	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
181	C₄²×C₅₃	C₂³	1	1	yes	characteristic
182	C₄²×C₅₃	C₂³	1	1	yes	characteristic
183	C₄²×C₅₃	C₂³	1	2	yes	—
184	C₄²×C₅₃	C₂³	1	2	yes	—
185	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
186	#11719921	C₂³	1	2	yes	—
187	#11719921	C₂³	1	2	yes	—
188	#11719921	C₂³	1	2	yes	—
189	#11719921	C₂³	1	1	yes	characteristic
190	#11719921	C₂³	1	2	yes	—
191	#11719921	C₂³	1	2	yes	—
192	#11719921	C₂³	1	2	yes	—
193	#11719921	C₂³	1	2	yes	—
194	#11719921	C₂³	1	1	yes	characteristic
195	#11719921	C₂³	1	2	yes	—
196	#11719921	C₂³	1	2	yes	—
197	#11719921	C₂³	1	2	yes	—
198	#11719921	C₂³	1	1	yes	characteristic
199	#11719921	C₂³	1	1	yes	characteristic
200	#11719921	C₂³	1	1	yes	characteristic
201	#11719921	C₂³	1	1	yes	characteristic
202	#11719921	C₂³	1	2	yes	—
203	#11719921	C₂³	1	1	yes	characteristic
204	#11719921	C₂³	1	1	yes	characteristic
205	#11719921	C₂³	1	2	yes	—
206	#11719921	C₂³	1	2	yes	—
207	#11719921	C₂³	1	2	yes	—
208	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
209	C₄⋊Dic₅₃	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	1	yes	characteristic
215	C₄⋊Dic₅₃	C₂³	1	2	yes	—
216	C₄⋊Dic₅₃	C₂³	1	2	yes	—
217	#11719919	C₂³	1	2	yes	—
218	#11719919	C₂³	1	2	yes	—
219	#11719919	C₂³	1	2	yes	—
220	#11719919	C₂³	1	2	yes	—
221	#11719919	C₂³	1	2	yes	—
222	#11719919	C₂³	1	2	yes	—
223	#11719919	C₂³	1	2	yes	—
224	#11719919	C₂³	1	2	yes	—
225	C₄×Dic₅₃	C₂³	1	2	yes	—
226	C₄×Dic₅₃	C₂³	1	2	yes	—
227	C₄×Dic₅₃	C₂³	1	2	yes	—
228	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
229	C₂²×D₅₃	—	2	2	no	—
230	C₂²×D₅₃	—	4	1	no	—
231	C₂²×D₅₃	—	4	1	no	—
232	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
233	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
234	C₂²×D₅₃	C₂⁴	1	2	yes	—
235	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
236	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
237	Q₈×C₅₃	—	2	2	no	—
238	Q₈×C₅₃	—	2	2	no	—
239	C₂×C₄×C₅₃	—	2	2	no	—
240	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
241	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
242	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
243	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
244	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
245	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
246	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
247	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
248	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
249	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
250	C₂²⋊D₅₃	—	4	1	no	—
251	C₂²⋊D₅₃	—	4	1	no	—
252	C₂²⋊D₅₃	—	4	2	no	—
253	C₂×Dic₅₃	—	4	1	no	—
254	C₂×Dic₅₃	C₂⁴	1	2	yes	—
255	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
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	2	yes	—
260	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
261	D₂₁₂	—	2	2	no	—
262	D₂₁₂	—	2	4	no	—
263	D₂₁₂	—	2	2	no	—
264	D₂₁₂	—	2	2	no	—
265	D₂₁₂	—	4	1	no	—
266	D₂₁₂	—	4	1	no	—
267	D₂₁₂	—	4	2	no	—
268	D₂₁₂	—	4	1	no	—
269	D₂₁₂	—	4	2	no	—
270	D₂₁₂	—	4	2	no	—
271	C₄×D₅₃	—	2	2	no	—
272	C₄×D₅₃	—	2	2	no	—
273	C₄×D₅₃	—	2	2	no	—
274	C₄×D₅₃	—	2	2	no	—
275	C₄×D₅₃	—	4	1	no	—
276	C₄×D₅₃	—	4	1	no	—
277	C₄×D₅₃	—	4	2	no	—
278	C₄×D₅₃	—	4	2	no	—
279	C₄×D₅₃	—	4	2	no	—
280	C₄×D₅₃	—	4	2	no	—
281	C₄×D₅₃	—	4	2	no	—
282	C₄×D₅₃	—	4	2	no	—
283	C₄×D₅₃	—	4	2	no	—
284	Dic₁₀₆	—	2	2	no	—
285	Dic₁₀₆	—	4	2	no	—
286	Dic₁₀₆	—	4	1	no	—
287	Dic₁₀₆	—	4	2	no	—
288	Dic₁₀₆	—	4	2	no	—
289	C₂²×C₅₃	—	2	1	no	—
290	C₂²×C₅₃	—	2	1	no	—
291	C₂²×C₅₃	—	2	1	no	—
292	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
293	C₂×D₅₃	—	2	1	no	—
294	C₂×D₅₃	—	2	1	no	—
295	C₂×D₅₃	—	2	1	no	—
296	C₂×D₅₃	—	2	1	no	—
297	C₂×D₅₃	—	2	1	no	—
298	C₂×D₅₃	—	2	2	no	—
299	C₂×D₅₃	—	2	1	no	—
300	C₂×D₅₃	—	2	1	no	—
301	C₂×D₅₃	—	2	2	no	—
302	C₂×D₅₃	—	2	1	no	—
303	C₂×D₅₃	—	2	1	no	—
304	C₂×D₅₃	—	2	2	no	—
305	C₂×D₅₃	—	2	1	no	—
306	C₂×D₅₃	—	2	1	no	—
307	C₂×D₅₃	—	4	4	no	—
308	C₂×D₅₃	2₋¹⁺⁴	1	2	yes	—
309	C₄×C₅₃	—	2	1	no	—
310	C₄×C₅₃	—	2	2	no	—
311	C₄×C₅₃	—	2	2	no	—
312	C₄×C₅₃	—	2	2	no	—
313	C₄×C₅₃	—	2	1	no	—
314	C₄×C₅₃	—	2	1	no	—
315	C₄×C₅₃	—	2	1	no	—
316	C₄×C₅₃	—	2	2	no	—
317	C₄×C₅₃	—	2	1	no	—
318	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
319	Dic₅₃	—	2	2	no	—
320	Dic₅₃	—	2	1	no	—
321	Dic₅₃	—	2	1	no	—
322	Dic₅₃	—	2	1	no	—
323	Dic₅₃	—	2	1	no	—
324	Dic₅₃	—	2	2	no	—
325	Dic₅₃	—	2	2	no	—
326	#3541	—	53	1	no	maximal, Sylow
327	C₂×C₅₃	—	4	1	no	—
328	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
329	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
330	C₂×C₅₃	#585	1	1	yes	characteristic
331	D₅₃	—	2	2	no	—
332	D₅₃	—	4	1	no	—
333	D₅₃	—	4	1	no	—
334	D₅₃	—	4	2	no	—
335	D₅₃	—	4	1	no	—
336	#563	—	53	2	no	—
337	#562	—	53	2	no	—
338	#556	—	53	1	no	—
339	#554	—	53	2	no	—
340	#554	—	53	1	no	—
341	#553	—	53	1	no	—
342	#553	—	53	2	no	—
343	#552	—	53	1	no	—
344	D₄×Q₈	—	53	1	no	—
345	#548	—	53	1	no	—
346	#548	—	53	1	no	—
347	#547	—	53	1	no	—
348	#542	—	53	2	no	—
349	#542	—	53	1	no	—
350	#541	D₅₃	1	1	yes	characteristic, complemented
351	#539	—	53	2	no	—
352	#538	—	53	2	no	—
353	#538	—	53	2	no	—
354	#537	—	53	1	no	—
355	#536	—	53	1	no	—
356	#525	—	53	1	no	—
357	#525	—	53	1	no	—
358	C₂×C₂²⋊Q₈	—	53	1	no	—
359	C₅₃	#3541	1	1	yes	characteristic, complemented, Sylow
360	C₂×#41	—	53	1	no	—
361	C₂×#41	—	53	1	no	—
362	C₂×#41	—	53	1	no	—
363	C₂²×Q₈	—	53	1	no	—
364	C₂²×Q₈	—	53	1	no	—
365	C₂³×C₄	—	53	1	no	—
366	C₄⋊Q₈	—	53	2	no	—
367	C₄⋊Q₈	—	53	2	no	—
368	C₄⋊Q₈	—	53	1	no	—
369	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
370	#126	—	53	2	no	—
371	#126	—	53	2	no	—
372	#126	—	53	2	no	—
373	#126	—	53	2	no	—
374	#126	—	53	2	no	—
375	#126	C₂×D₅₃	1	2	yes	—
376	#126	C₂×D₅₃	1	2	yes	—
377	#125	—	53	2	no	—
378	#125	—	53	2	no	—
379	#125	C₂×D₅₃	1	1	yes	characteristic, complemented
380	#125	C₂×D₅₃	1	2	yes	complemented
381	#125	C₂×D₅₃	1	2	yes	complemented
382	#124	—	53	2	no	—
383	#124	—	53	2	no	—
384	#124	—	53	2	no	—
385	#124	—	53	2	no	—
386	#124	—	53	1	no	—
387	#124	—	53	2	no	—
388	#123	—	53	2	no	—
389	#123	—	53	2	no	—
390	#123	—	53	2	no	—
391	#123	—	53	2	no	—
392	#123	—	53	2	no	—
393	#123	—	53	2	no	—
394	#123	—	53	1	no	—
395	#123	—	53	1	no	—
396	#123	—	53	2	no	—
397	#123	—	53	2	no	—
398	C₂²⋊Q₈	—	53	1	no	—
399	C₂²⋊Q₈	—	53	1	no	—
400	C₂²⋊Q₈	—	53	1	no	—
401	C₂²⋊Q₈	—	53	1	no	—
402	C₂²⋊Q₈	—	53	1	no	—
403	C₂²⋊Q₈	—	53	1	no	—
404	C₂²⋊Q₈	—	53	2	no	—
405	C₂²⋊Q₈	—	53	1	no	—
406	C₂²⋊Q₈	—	53	1	no	—
407	C₂²⋊Q₈	—	53	1	no	—
408	C₂²⋊Q₈	—	53	2	no	—
409	C₂²⋊Q₈	—	53	2	no	—
410	C₂²⋊Q₈	—	53	2	no	—
411	C₂²⋊Q₈	—	53	2	no	—
412	C₂²⋊Q₈	—	53	2	no	—
413	C₂²⋊Q₈	—	53	2	no	—
414	C₂²⋊Q₈	—	53	2	no	—
415	C₂²⋊Q₈	—	53	2	no	—
416	C₂²⋊Q₈	—	53	2	no	—
417	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
418	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
419	#121	—	53	1	no	—
420	#121	—	53	1	no	—
421	#121	—	53	1	no	—
422	#121	—	53	1	no	—
423	#121	—	53	1	no	—
424	#121	—	53	2	no	—
425	#121	—	53	2	no	—
426	#121	—	53	2	no	—
427	#121	—	53	2	no	—
428	#121	—	53	2	no	—
429	#121	—	53	2	no	—
430	#121	—	53	2	no	—
431	#120	—	53	2	no	—
432	#120	—	53	1	no	—
433	#120	—	53	1	no	—
434	C₄×Q₈	—	53	1	no	—
435	C₄×Q₈	—	53	2	no	—
436	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
437	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
438	C₄×D₄	—	53	2	no	—
439	C₄×D₄	—	53	2	no	—
440	C₄×D₄	—	53	2	no	—
441	C₄×D₄	—	53	2	no	—
442	C₄×D₄	—	53	1	no	—
443	C₄×D₄	—	53	1	no	—
444	C₄×D₄	—	53	1	no	—
445	C₄×D₄	—	53	1	no	—
446	C₄×D₄	—	53	2	no	—
447	C₄×D₄	—	53	2	no	—
448	C₄×D₄	—	53	2	no	—
449	C₄×D₄	—	53	1	no	—
450	#117	—	53	1	no	—
451	#117	—	53	1	no	—
452	#117	—	53	2	no	—
453	#117	C₂×D₅₃	1	1	yes	characteristic
454	C₂×C₄⋊C₄	—	53	2	no	—
455	C₂×C₄⋊C₄	—	53	1	no	—
456	C₂×C₄⋊C₄	—	53	1	no	—
457	C₂×C₄⋊C₄	—	53	2	no	—
458	C₂×C₂²⋊C₄	—	53	2	no	—
459	C₂⁴	—	53	1	no	—
460	#41	—	212	2	no	—
461	#41	—	212	2	no	—
462	#41	—	212	2	no	—
463	C₂×Q₈	—	53	1	no	—
464	C₂×Q₈	—	53	2	no	—
465	C₂×Q₈	—	53	2	no	—
466	C₂×Q₈	—	53	1	no	—
467	C₂×Q₈	—	53	2	no	—
468	C₂×Q₈	—	212	2	no	—
469	C₂×Q₈	—	212	2	no	—
470	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
471	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
472	C₂×D₄	—	53	2	no	—
473	C₂×D₄	—	53	1	no	—
474	C₂×D₄	—	53	1	no	—
475	C₂×D₄	—	53	2	no	—
476	C₂×D₄	—	53	1	no	—
477	C₂×D₄	—	53	1	no	—
478	C₂×D₄	—	53	1	no	—
479	C₂×D₄	—	53	1	no	—
480	C₂×D₄	—	53	1	no	—
481	C₂×D₄	—	53	2	no	—
482	C₂×D₄	—	53	2	no	—
483	C₂×D₄	—	53	1	no	—
484	C₂×D₄	—	53	2	no	—
485	C₂²×C₄	—	53	1	no	—
486	C₂²×C₄	—	53	1	no	—
487	C₂²×C₄	—	53	1	no	—
488	C₂²×C₄	—	53	1	no	—
489	C₂²×C₄	—	53	1	no	—
490	C₂²×C₄	—	53	2	no	—
491	C₂²×C₄	—	53	2	no	—
492	C₂²×C₄	—	53	1	no	—
493	C₂²×C₄	—	53	2	no	—
494	C₂²×C₄	—	53	2	no	—
495	C₂²×C₄	—	53	2	no	—
496	C₂²×C₄	—	53	2	no	—
497	C₂²×C₄	—	53	2	no	—
498	C₂²×C₄	—	53	1	no	—
499	C₂²×C₄	—	212	2	no	—
500	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
501	C₄⋊C₄	—	53	1	no	—
502	C₄⋊C₄	—	53	2	no	—
503	C₄⋊C₄	—	53	2	no	—
504	C₄⋊C₄	—	53	1	no	—
505	C₄⋊C₄	—	53	1	no	—
506	C₄⋊C₄	—	53	2	no	—
507	C₄⋊C₄	—	53	2	no	—
508	C₄⋊C₄	—	53	2	no	—
509	C₄⋊C₄	—	53	2	no	—
510	C₄⋊C₄	—	53	1	no	—
511	C₄⋊C₄	—	53	1	no	—
512	C₄⋊C₄	—	53	2	no	—
513	C₄⋊C₄	—	53	2	no	—
514	C₄⋊C₄	—	53	1	no	—
515	C₄⋊C₄	—	53	2	no	—
516	C₄⋊C₄	—	53	2	no	—
517	C₄⋊C₄	—	53	2	no	—
518	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
519	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
520	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
521	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
522	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
523	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
524	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
525	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
526	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
527	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
528	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
529	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
530	C₂²⋊C₄	—	53	2	no	—
531	C₂²⋊C₄	—	53	2	no	—
532	C₂²⋊C₄	—	53	2	no	—
533	C₂²⋊C₄	—	53	1	no	—
534	C₂²⋊C₄	—	53	2	no	—
535	C₂²⋊C₄	—	53	2	no	—
536	C₂²⋊C₄	—	53	2	no	—
537	C₂²⋊C₄	—	53	2	no	—
538	C₂²⋊C₄	—	53	1	no	—
539	C₂²⋊C₄	—	53	1	no	—
540	C₂²⋊C₄	—	53	1	no	—
541	C₂²⋊C₄	—	53	1	no	—
542	C₂²⋊C₄	—	53	1	no	—
543	C₂²⋊C₄	—	53	2	no	—
544	C₂²⋊C₄	—	53	2	no	—
545	C₂²⋊C₄	—	53	2	no	—
546	C₂²⋊C₄	—	53	1	no	—
547	C₂²⋊C₄	—	53	2	no	—
548	C₂²⋊C₄	—	53	1	no	—
549	C₂²⋊C₄	—	53	2	no	—
550	C₂²⋊C₄	—	53	2	no	—
551	C₂²⋊C₄	—	53	2	no	—
552	C₂²⋊C₄	—	53	2	no	—
553	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
554	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
555	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
556	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
557	C₄²	—	53	2	no	—
558	C₄²	—	53	2	no	—
559	C₄²	—	53	2	no	—
560	C₄²	C₂²×D₅₃	1	1	yes	characteristic
561	C₄²	C₂²×D₅₃	1	1	yes	characteristic
562	C₄²	C₂²×D₅₃	1	2	yes	—
563	C₄²	C₂²×D₅₃	1	2	yes	—
564	C₂³	—	53	1	no	—
565	C₂³	—	53	1	no	—
566	C₂³	—	53	1	no	—
567	C₂³	—	53	1	no	—
568	C₂³	—	53	2	no	—
569	C₂³	—	106	2	no	—
570	C₂³	—	212	1	no	—
571	C₂³	—	212	1	no	—
572	C₂³	C₂³×D₅₃	1	1	yes	characteristic
573	Q₈	—	2	2	no	—
574	Q₈	—	2	2	no	—
575	Q₈	—	106	2	no	—
576	Q₈	—	212	2	no	—
577	Q₈	—	212	2	no	—
578	Q₈	—	212	2	no	—
579	Q₈	—	212	1	no	—
580	D₄	—	106	4	no	—
581	D₄	—	106	2	no	—
582	D₄	—	106	2	no	—
583	D₄	—	106	2	no	—
584	D₄	—	212	1	no	—
585	D₄	—	212	2	no	—
586	D₄	—	212	1	no	—
587	D₄	—	212	1	no	—
588	D₄	—	212	2	no	—
589	D₄	—	212	2	no	—
590	D₄	—	212	2	no	—
591	D₄	—	212	1	no	—
592	D₄	—	212	1	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	53	2	no	—
595	C₂×C₄	—	53	1	no	—
596	C₂×C₄	—	53	1	no	—
597	C₂×C₄	—	53	2	no	—
598	C₂×C₄	—	53	2	no	—
599	C₂×C₄	—	53	1	no	—
600	C₂×C₄	—	53	1	no	—
601	C₂×C₄	—	106	2	no	—
602	C₂×C₄	—	106	2	no	—
603	C₂×C₄	—	106	2	no	—
604	C₂×C₄	—	106	2	no	—
605	C₂×C₄	—	212	2	no	—
606	C₂×C₄	—	212	2	no	—
607	C₂×C₄	—	212	2	no	—
608	C₂×C₄	—	212	1	no	—
609	C₂×C₄	—	212	1	no	—
610	C₂×C₄	—	212	2	no	—
611	C₂×C₄	—	212	2	no	—
612	C₂×C₄	—	212	2	no	—
613	C₂×C₄	—	212	2	no	—
614	C₂×C₄	—	212	1	no	—
615	C₂×C₄	C₂³×D₅₃	1	2	yes	—
616	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
617	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
618	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
619	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
620	C₂×C₄	C₂³×D₅₃	1	2	yes	—
621	C₂×C₄	C₂³×D₅₃	1	2	yes	—
622	C₂×C₄	C₂³×D₅₃	1	2	yes	—
623	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
624	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
625	C₂²	—	2	1	no	—
626	C₂²	—	2	1	no	—
627	C₂²	—	2	1	no	—
628	C₂²	—	53	2	no	—
629	C₂²	—	106	1	no	—
630	C₂²	—	106	1	no	—
631	C₂²	—	106	1	no	—
632	C₂²	—	106	1	no	—
633	C₂²	—	106	1	no	—
634	C₂²	—	106	2	no	—
635	C₂²	—	106	1	no	—
636	C₂²	—	106	1	no	—
637	C₂²	—	106	2	no	—
638	C₂²	—	106	1	no	—
639	C₂²	—	106	1	no	—
640	C₂²	—	106	2	no	—
641	C₂²	—	106	1	no	—
642	C₂²	—	106	1	no	—
643	C₂²	—	212	4	no	—
644	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
645	C₄	—	2	1	no	—
646	C₄	—	2	2	no	—
647	C₄	—	2	2	no	—
648	C₄	—	2	2	no	—
649	C₄	—	2	1	no	—
650	C₄	—	2	1	no	—
651	C₄	—	2	2	no	—
652	C₄	—	2	1	no	—
653	C₄	—	2	1	no	—
654	C₄	—	106	1	no	—
655	C₄	—	106	1	no	—
656	C₄	—	106	1	no	—
657	C₄	—	106	1	no	—
658	C₄	—	106	2	no	—
659	C₄	—	106	2	no	—
660	C₄	—	106	2	no	—
661	C₄	#13133263	1	2	yes	—
662	C₂	—	4	1	no	—
663	C₂	—	106	2	no	—
664	C₂	—	212	1	no	—
665	C₂	—	212	1	no	—
666	C₂	—	212	1	no	—
667	C₂	—	212	2	no	—
668	C₂	C₂×#13133263	1	1	yes	characteristic
669	C₂	2₋¹⁺⁴×D₅₃	1	1	yes	characteristic
670	C₂	#16811460	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#541	C₂
hypercenter	#541	D₅₃
hypocenter	C₅₃	#3541
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #541

Complements

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

N	H ≅ G/N	type
1. #16811377	C₂	semidirect
2. #16811371	C₂	semidirect
3. #16811346	C₂	semidirect
4. #16811333	C₂	semidirect
5. #16811329	C₂	semidirect
6. #16811326	C₂	semidirect
7. #16811326	C₂	semidirect
8. #16811311	C₂	semidirect
9. #16811288	C₂	semidirect
10. #16811285	C₂	semidirect
11. #16811285	C₂	semidirect
12. #16811276	C₂	semidirect
13. #16811244	C₂	semidirect
14. #16811239	C₂	semidirect
15. #16811214	C₂	semidirect
16. #16811212	C₂	semidirect
17. #16811212	C₂	semidirect
18. #16811203	C₂	semidirect
19. #16811196	C₂	semidirect
20. #16811136	C₂	semidirect
21. #16811135	C₂	semidirect
22. C₂²⋊Q₈×D₅₃	C₂	semidirect
23. C₅₃×#541	C₂	semidirect
25. C₂×#11719949	C₂²	semidirect
28. C₂×Q₈×D₅₃	C₂²	semidirect
30. #13133217	C₂²	semidirect
31. #13133216	C₂²	semidirect
32. #13133216	C₂²	semidirect
35. #13133212	C₂²	semidirect
36. #13133212	C₂²	semidirect
37. #13133212	C₂²	semidirect
38. #13133209	C₂²	semidirect
39. #13133208	C₂²	semidirect
42. #13133205	C₂²	semidirect
43. #13133205	C₂²	semidirect
49. #13133202	C₂²	semidirect
50. #13133202	C₂²	semidirect
51. #13133202	C₂²	semidirect
59. #13133197	C₂²	semidirect
60. #13133197	C₂²	semidirect
61. #13133197	C₂²	semidirect
66. #13133195	C₂²	semidirect
67. #13133195	C₂²	semidirect
68. #13133195	C₂²	semidirect
69. #13133195	C₂²	semidirect
73. #13133193	C₂²	semidirect
74. #13133193	C₂²	semidirect
82. #13133192	C₂²	semidirect
83. #13133192	C₂²	semidirect
90. C₄×Dic₁₀₆	C₂²	semidirect
91. C₄×Dic₁₀₆	C₂²	semidirect
98. C₄×D₂₁₂	C₂²	semidirect
99. C₄×D₂₁₂	C₂²	semidirect
105. #13133175	C₂²	semidirect
106. #13133175	C₂²	semidirect
109. C₄⋊C₄×D₅₃	C₂²	semidirect
110. C₄⋊C₄×D₅₃	C₂²	semidirect
112. C₄⋊Q₈×C₅₃	C₂²	semidirect
115. #125×C₅₃	C₂²	semidirect
116. #125×C₅₃	C₂²	semidirect
117. #125×C₅₃	C₂²	semidirect
120. C₄×Q₈×C₅₃	C₂²	semidirect
121. C₄×Q₈×C₅₃	C₂²	semidirect
350. #541	D₅₃	semidirect
359. C₅₃	#3541	semidirect
369. C₄⋊Q₈	C₂×D₅₃	semidirect
379. #125	C₂×D₅₃	semidirect
380. #125	C₂×D₅₃	semidirect
381. #125	C₂×D₅₃	semidirect
436. C₄×Q₈	C₂×D₅₃	semidirect
437. C₄×Q₈	C₂×D₅₃	semidirect