Subgroups — #20288429

Order 6784.

1040 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811345	C₂	1	2	yes	maximal, complemented
2	#16811344	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811339	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811336	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811335	C₂	1	2	yes	maximal, complemented
6	#16811324	C₂	1	2	yes	maximal, complemented
7	#16811319	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811317	C₂	1	1	yes	maximal, characteristic, complemented
9	#16811315	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811314	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811309	C₂	1	2	yes	maximal, complemented
12	#16811250	C₂	1	1	yes	maximal, characteristic, complemented
13	#16811212	C₂	1	2	yes	maximal, complemented
14	#16811207	C₂	1	2	yes	maximal, complemented
15	#16811203	C₂	1	1	yes	maximal, characteristic, complemented
16	#16811160	C₂	1	2	yes	maximal, complemented
17	#16811156	C₂	1	2	yes	maximal, complemented
18	#16811136	C₂	1	1	yes	maximal, characteristic, complemented
19	#16811132	C₂	1	1	yes	maximal, characteristic, complemented
20	#16811105	C₂	1	1	yes	maximal, characteristic, complemented
21	#16811103	C₂	1	1	yes	maximal, characteristic, complemented
22	#117×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
23	C₅₃×#556	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#11719949	C₂²	1	1	yes	characteristic
25	C₂×#11719945	C₂²	1	1	yes	characteristic
26	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
27	#13133216	C₂²	1	1	yes	characteristic, complemented
28	#13133215	C₂²	1	2	yes	complemented
29	#13133212	C₂²	1	2	yes	—
30	#13133212	C₂²	1	2	yes	complemented
31	#13133212	C₂²	1	2	yes	complemented
32	#13133211	C₂²	1	2	yes	—
33	#13133211	C₂²	1	2	yes	complemented
34	#13133210	C₂²	1	2	yes	—
35	#13133209	C₂²	1	1	yes	characteristic, complemented
36	#13133208	C₂²	1	2	yes	complemented
37	#13133208	C₂²	1	1	yes	characteristic, complemented
38	#13133207	C₂²	1	2	yes	complemented
39	#13133207	C₂²	1	2	yes	complemented
40	#13133206	C₂²	1	2	yes	—
41	#13133206	C₂²	1	2	yes	complemented
42	#13133205	C₂²	1	2	yes	—
43	#13133203	C₂²	1	2	yes	—
44	#13133203	C₂²	1	2	yes	complemented
45	#13133202	C₂²	1	2	yes	—
46	#13133202	C₂²	1	1	yes	characteristic
47	#13133202	C₂²	1	2	yes	—
48	#13133201	C₂²	1	2	yes	—
49	#13133199	C₂²	1	2	yes	—
50	#13133199	C₂²	1	2	yes	—
51	#13133199	C₂²	1	1	yes	characteristic
52	#13133195	C₂²	1	1	yes	characteristic
53	#13133195	C₂²	1	2	yes	—
54	#13133195	C₂²	1	2	yes	complemented
55	#13133194	C₂²	1	1	yes	characteristic
56	#13133193	C₂²	1	1	yes	characteristic
57	#13133193	C₂²	1	2	yes	—
58	#13133193	C₂²	1	1	yes	characteristic, complemented
59	#13133192	C₂²	1	2	yes	—
60	#13133192	C₂²	1	1	yes	characteristic
61	#13133192	C₂²	1	2	yes	—
62	#13133192	C₂²	1	1	yes	characteristic, complemented
63	#13133191	C₂²	1	1	yes	characteristic
64	#13133191	C₂²	1	1	yes	characteristic, complemented
65	#13133190	C₂²	1	2	yes	—
66	#13133190	C₂²	1	1	yes	characteristic
67	#13133190	C₂²	1	2	yes	—
68	#13133190	C₂²	1	2	yes	—
69	#13133190	C₂²	1	2	yes	complemented
70	#13133189	C₂²	1	2	yes	—
71	#13133189	C₂²	1	1	yes	characteristic
72	#13133187	C₂²	1	1	yes	characteristic
73	#13133187	C₂²	1	2	yes	—
74	#13133185	C₂²	1	1	yes	characteristic
75	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
76	C₄×Dic₁₀₆	C₂²	1	2	yes	complemented
77	#13133182	C₂²	1	2	yes	complemented
78	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
79	C₄×D₂₁₂	C₂²	1	2	yes	—
80	C₄×D₂₁₂	C₂²	1	2	yes	—
81	C₄×D₂₁₂	C₂²	1	2	yes	complemented
82	C₄×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
83	#13133179	C₂²	1	2	yes	—
84	#13133179	C₂²	1	2	yes	—
85	#13133179	C₂²	1	1	yes	characteristic, complemented
86	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
87	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
88	#13133177	C₂²	1	2	yes	—
89	#13133177	C₂²	1	2	yes	—
90	#13133177	C₂²	1	2	yes	complemented
91	#13133177	C₂²	1	2	yes	complemented
92	#13133176	C₂²	1	1	yes	characteristic
93	#13133175	C₂²	1	1	yes	characteristic
94	#13133175	C₂²	1	1	yes	characteristic
95	#13133175	C₂²	1	1	yes	characteristic
96	#13133175	C₂²	1	2	yes	complemented
97	#13133175	C₂²	1	1	yes	characteristic, complemented
98	#13133174	C₂²	1	2	yes	—
99	#13133174	C₂²	1	1	yes	characteristic
100	#13133174	C₂²	1	2	yes	complemented
101	#13133174	C₂²	1	2	yes	complemented
102	#13133174	C₂²	1	2	yes	complemented
103	#13133173	C₂²	1	2	yes	—
104	#13133173	C₂²	1	2	yes	complemented
105	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
106	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic
107	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic
108	C₄⋊C₄×D₅₃	C₂²	1	2	yes	complemented
109	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic, complemented
110	C₂²⋊C₄×D₅₃	C₂²	1	2	yes	—
111	C₄²×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	complemented
115	#124×C₅₃	C₂²	1	2	yes	—
116	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
117	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
118	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
119	C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
120	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
121	#117×C₅₃	C₂²	1	1	yes	characteristic
122	#117×C₅₃	C₂²	1	1	yes	characteristic, complemented
123	C₂³×D₅₃	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
129	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
130	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
131	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
132	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
133	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
134	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
135	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
136	C₂²×Dic₅₃	C₂³	1	2	yes	—
137	#11719949	—	2	4	no	—
138	#11719945	—	2	4	no	—
139	C₂×D₂₁₂	C₂³	1	2	yes	—
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	1	yes	characteristic
144	C₂×D₂₁₂	C₂³	1	2	yes	—
145	C₂×C₄×D₅₃	—	2	4	no	—
146	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
147	C₂×C₄×D₅₃	C₂³	1	2	yes	—
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	2	yes	—
154	C₂×C₄×D₅₃	C₂³	1	2	yes	—
155	C₂×C₄×D₅₃	C₂³	1	2	yes	—
156	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
157	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
158	C₂×Dic₁₀₆	C₂³	1	2	yes	—
159	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
160	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
161	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
162	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
163	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
164	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
165	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
166	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
167	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
168	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
169	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
170	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
171	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
172	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
173	C₄²×C₅₃	C₂³	1	2	yes	—
174	C₄²×C₅₃	C₂³	1	2	yes	—
175	C₄²×C₅₃	C₂³	1	1	yes	characteristic
176	C₄²×C₅₃	C₂³	1	2	yes	—
177	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
178	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
179	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
180	#11719921	C₂³	1	2	yes	—
181	#11719921	C₂³	1	2	yes	—
182	#11719921	C₂³	1	1	yes	characteristic
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	#11719921	C₂³	1	2	yes	—
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	1	yes	characteristic
196	#11719921	C₂³	1	2	yes	—
197	#11719921	C₂³	1	1	yes	characteristic
198	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
199	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
200	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
201	C₄⋊Dic₅₃	C₂³	1	2	yes	—
202	C₄⋊Dic₅₃	C₂³	1	2	yes	—
203	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
204	C₄⋊Dic₅₃	C₂³	1	2	yes	—
205	#11719919	C₂³	1	2	yes	—
206	#11719919	C₂³	1	2	yes	—
207	#11719919	C₂³	1	1	yes	characteristic
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	2	yes	—
214	#11719919	C₂³	1	2	yes	—
215	#11719919	C₂³	1	2	yes	—
216	#11719919	C₂³	1	1	yes	characteristic
217	#11719919	C₂³	1	2	yes	—
218	C₄×Dic₅₃	C₂³	1	2	yes	—
219	C₄×Dic₅₃	C₂³	1	2	yes	—
220	C₄×Dic₅₃	C₂³	1	2	yes	—
221	C₄×Dic₅₃	C₂³	1	2	yes	—
222	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
223	C₄×Dic₅₃	C₂³	1	2	yes	—
224	C₄×Dic₅₃	C₂³	1	2	yes	—
225	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
226	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
227	C₂²×D₅₃	—	2	2	no	—
228	C₂²×D₅₃	—	4	1	no	—
229	C₂²×D₅₃	—	4	1	no	—
230	C₂²×D₅₃	C₂⁴	1	2	yes	—
231	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
232	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
233	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
234	Q₈×C₅₃	—	2	2	no	—
235	Q₈×C₅₃	—	4	2	no	—
236	D₄×C₅₃	—	4	1	no	—
237	C₂×C₄×C₅₃	—	2	2	no	—
238	C₂×C₄×C₅₃	—	2	2	no	—
239	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
240	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
241	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
242	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
243	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
244	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
245	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
246	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
247	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
248	C₂²⋊D₅₃	—	2	2	no	—
249	C₂²⋊D₅₃	—	2	2	no	—
250	C₂²⋊D₅₃	—	4	1	no	—
251	C₂²⋊D₅₃	—	4	2	no	—
252	C₂²⋊D₅₃	—	4	1	no	—
253	C₂²⋊D₅₃	—	4	2	no	—
254	C₂×Dic₅₃	—	2	2	no	—
255	C₂×Dic₅₃	—	4	2	no	—
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	2	yes	—
261	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
262	C₂×Dic₅₃	C₂⁴	1	2	yes	—
263	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
264	D₂₁₂	—	2	2	no	—
265	D₂₁₂	—	2	4	no	—
266	D₂₁₂	—	2	2	no	—
267	D₂₁₂	—	4	2	no	—
268	D₂₁₂	—	4	1	no	—
269	D₂₁₂	—	4	2	no	—
270	C₄×D₅₃	—	2	2	no	—
271	C₄×D₅₃	—	2	2	no	—
272	C₄×D₅₃	—	2	2	no	—
273	C₄×D₅₃	—	2	2	no	—
274	C₄×D₅₃	—	2	4	no	—
275	C₄×D₅₃	—	4	2	no	—
276	C₄×D₅₃	—	4	2	no	—
277	C₄×D₅₃	—	4	2	no	—
278	C₄×D₅₃	—	4	2	no	—
279	C₄×D₅₃	—	4	2	no	—
280	C₄×D₅₃	#41	1	4	yes	—
281	Dic₁₀₆	—	2	2	no	—
282	Dic₁₀₆	—	4	2	no	—
283	C₂²×C₅₃	—	2	1	no	—
284	C₂²×C₅₃	—	2	1	no	—
285	C₂²×C₅₃	—	2	1	no	—
286	C₂²×C₅₃	—	2	1	no	—
287	C₂²×C₅₃	—	2	1	no	—
288	C₂²×C₅₃	—	2	1	no	—
289	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
290	C₂×D₅₃	—	2	1	no	—
291	C₂×D₅₃	—	2	1	no	—
292	C₂×D₅₃	—	2	1	no	—
293	C₂×D₅₃	—	2	2	no	—
294	C₂×D₅₃	—	2	1	no	—
295	C₂×D₅₃	—	2	1	no	—
296	C₂×D₅₃	—	2	2	no	—
297	C₂×D₅₃	—	2	2	no	—
298	C₂×D₅₃	—	2	1	no	—
299	C₂×D₅₃	—	2	1	no	—
300	C₂×D₅₃	—	2	1	no	—
301	C₂×D₅₃	—	4	4	no	—
302	C₂×D₅₃	C₂×#41	1	2	yes	—
303	C₄×C₅₃	—	2	1	no	—
304	C₄×C₅₃	—	2	1	no	—
305	C₄×C₅₃	—	2	2	no	—
306	C₄×C₅₃	—	2	2	no	—
307	C₄×C₅₃	—	2	2	no	—
308	C₄×C₅₃	—	2	1	no	—
309	C₄×C₅₃	—	2	2	no	—
310	C₄×C₅₃	—	2	1	no	—
311	C₄×C₅₃	C₂×#41	1	2	yes	—
312	Dic₅₃	—	2	1	no	—
313	Dic₅₃	—	2	1	no	—
314	Dic₅₃	—	2	1	no	—
315	Dic₅₃	—	2	2	no	—
316	Dic₅₃	—	2	2	no	—
317	Dic₅₃	—	2	1	no	—
318	Dic₅₃	—	2	2	no	—
319	Dic₅₃	C₂×#41	1	2	yes	—
320	#3493	—	53	1	no	maximal, Sylow
321	C₂×C₅₃	—	4	1	no	—
322	C₂×C₅₃	—	4	1	no	—
323	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
324	C₂×C₅₃	#585	1	1	yes	characteristic
325	C₂×C₅₃	#585	1	1	yes	characteristic
326	D₅₃	—	2	2	no	—
327	D₅₃	—	4	2	no	—
328	D₅₃	—	4	1	no	—
329	D₅₃	—	4	1	no	—
330	#556	—	53	1	no	—
331	#556	—	53	2	no	—
332	#556	D₅₃	1	1	yes	characteristic, complemented
333	#555	—	53	1	no	—
334	#555	—	53	2	no	—
335	#555	—	53	1	no	—
336	#553	—	53	2	no	—
337	#553	—	53	1	no	—
338	#553	—	53	1	no	—
339	#552	—	53	2	no	—
340	#552	—	53	1	no	—
341	#552	—	53	1	no	—
342	#544	—	53	1	no	—
343	#538	—	53	2	no	—
344	#538	—	53	2	no	—
345	#537	—	53	1	no	—
346	#529	—	53	2	no	—
347	#529	—	53	2	no	—
348	#525	—	53	1	no	—
349	#525	—	53	1	no	—
350	#520	—	53	1	no	—
351	#520	—	53	1	no	—
352	C₂×#117	—	53	1	no	—
353	C₅₃	#3493	1	1	yes	characteristic, complemented, Sylow
354	C₂×#41	—	53	1	no	—
355	C₂×#41	—	53	1	no	—
356	C₂³×C₄	—	53	1	no	—
357	C₄⋊Q₈	—	53	2	no	—
358	C₄⋊Q₈	—	53	1	no	—
359	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
360	#126	—	53	2	no	—
361	#126	—	53	2	no	—
362	#126	—	53	2	no	—
363	#126	—	53	2	no	—
364	#126	—	53	2	no	—
365	#126	—	53	2	no	—
366	#126	C₂×D₅₃	1	2	yes	—
367	#126	C₂×D₅₃	1	2	yes	complemented
368	#125	—	53	2	no	—
369	#125	—	53	2	no	—
370	#125	—	53	2	no	—
371	#125	—	53	1	no	—
372	#125	—	53	1	no	—
373	#124	—	53	2	no	—
374	#124	—	53	2	no	—
375	#124	—	53	2	no	—
376	#124	—	53	2	no	—
377	#124	—	53	2	no	—
378	#124	C₂×D₅₃	1	2	yes	—
379	#123	—	53	2	no	—
380	#123	—	53	1	no	—
381	#123	—	53	2	no	—
382	#123	—	53	1	no	—
383	#123	—	53	2	no	—
384	#123	—	53	2	no	—
385	#123	—	53	2	no	—
386	C₂²⋊Q₈	—	53	1	no	—
387	C₂²⋊Q₈	—	53	2	no	—
388	C₂²⋊Q₈	—	53	1	no	—
389	C₂²⋊Q₈	—	53	1	no	—
390	C₂²⋊Q₈	—	53	2	no	—
391	C₂²⋊Q₈	—	53	2	no	—
392	C₂²⋊Q₈	—	53	1	no	—
393	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
394	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
395	#121	—	53	1	no	—
396	#121	—	53	2	no	—
397	#121	—	53	1	no	—
398	#121	—	53	1	no	—
399	#121	—	53	1	no	—
400	#121	—	53	2	no	—
401	#121	—	53	1	no	—
402	#121	—	53	2	no	—
403	#121	—	53	1	no	—
404	#121	—	53	2	no	—
405	#121	—	53	2	no	—
406	#121	—	53	2	no	—
407	#121	—	53	2	no	—
408	#120	—	53	1	no	—
409	#120	—	53	2	no	—
410	#120	—	53	1	no	—
411	C₄×Q₈	—	53	1	no	—
412	C₄×Q₈	—	53	2	no	—
413	C₄×Q₈	—	53	2	no	—
414	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
415	C₄×Q₈	C₂×D₅₃	1	2	yes	complemented
416	C₄×D₄	—	53	1	no	—
417	C₄×D₄	—	53	1	no	—
418	C₄×D₄	—	53	2	no	—
419	C₄×D₄	—	53	1	no	—
420	C₄×D₄	—	53	2	no	—
421	C₄×D₄	—	53	2	no	—
422	C₄×D₄	—	53	2	no	—
423	C₄×D₄	—	53	2	no	—
424	C₄×D₄	—	53	2	no	—
425	C₄×D₄	—	53	2	no	—
426	C₄×D₄	—	53	2	no	—
427	C₄×D₄	—	53	1	no	—
428	C₄×D₄	—	53	1	no	—
429	C₄×D₄	—	53	2	no	—
430	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
431	#117	—	53	1	no	—
432	#117	—	53	1	no	—
433	#117	—	53	1	no	—
434	#117	—	53	1	no	—
435	#117	—	53	1	no	—
436	#117	—	53	2	no	—
437	#117	—	53	2	no	—
438	#117	—	53	2	no	—
439	#117	—	53	2	no	—
440	#117	—	53	1	no	—
441	#117	—	53	2	no	—
442	#117	—	53	2	no	—
443	#117	—	53	2	no	—
444	#117	C₂×D₅₃	1	1	yes	characteristic
445	#117	C₂×D₅₃	1	1	yes	characteristic, complemented
446	C₂×C₄⋊C₄	—	53	1	no	—
447	C₂×C₄⋊C₄	—	53	1	no	—
448	C₂×C₄⋊C₄	—	53	1	no	—
449	C₂×C₄⋊C₄	—	53	2	no	—
450	C₂×C₄⋊C₄	—	53	1	no	—
451	C₂×C₂²⋊C₄	—	53	2	no	—
452	C₂×C₄²	—	53	2	no	—
453	C₂⁴	—	53	1	no	—
454	#41	—	106	4	no	—
455	#41	—	106	4	no	—
456	C₂×Q₈	—	53	1	no	—
457	C₂×Q₈	—	53	2	no	—
458	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
459	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
460	C₂×D₄	—	53	1	no	—
461	C₂×D₄	—	53	1	no	—
462	C₂×D₄	—	53	1	no	—
463	C₂×D₄	—	53	2	no	—
464	C₂×D₄	—	53	1	no	—
465	C₂×D₄	—	53	2	no	—
466	C₂×D₄	—	53	2	no	—
467	C₂×D₄	—	53	1	no	—
468	C₂×D₄	—	53	1	no	—
469	C₂×D₄	—	53	1	no	—
470	C₂×D₄	—	53	2	no	—
471	C₂×D₄	—	53	2	no	—
472	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
473	C₂²×C₄	—	53	1	no	—
474	C₂²×C₄	—	53	1	no	—
475	C₂²×C₄	—	53	2	no	—
476	C₂²×C₄	—	53	1	no	—
477	C₂²×C₄	—	53	2	no	—
478	C₂²×C₄	—	53	1	no	—
479	C₂²×C₄	—	53	1	no	—
480	C₂²×C₄	—	53	1	no	—
481	C₂²×C₄	—	53	2	no	—
482	C₂²×C₄	—	53	2	no	—
483	C₂²×C₄	—	53	2	no	—
484	C₂²×C₄	—	53	2	no	—
485	C₂²×C₄	—	53	2	no	—
486	C₂²×C₄	—	106	4	no	—
487	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
488	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
489	C₄⋊C₄	—	53	2	no	—
490	C₄⋊C₄	—	53	1	no	—
491	C₄⋊C₄	—	53	1	no	—
492	C₄⋊C₄	—	53	1	no	—
493	C₄⋊C₄	—	53	2	no	—
494	C₄⋊C₄	—	53	1	no	—
495	C₄⋊C₄	—	53	2	no	—
496	C₄⋊C₄	—	53	2	no	—
497	C₄⋊C₄	—	53	2	no	—
498	C₄⋊C₄	—	53	1	no	—
499	C₄⋊C₄	—	53	2	no	—
500	C₄⋊C₄	—	53	1	no	—
501	C₄⋊C₄	—	53	1	no	—
502	C₄⋊C₄	—	53	2	no	—
503	C₄⋊C₄	—	53	2	no	—
504	C₄⋊C₄	—	53	2	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	C₂²×D₅₃	1	1	yes	characteristic
510	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
511	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
512	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
513	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
514	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
515	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
516	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
517	C₂²⋊C₄	—	53	2	no	—
518	C₂²⋊C₄	—	53	1	no	—
519	C₂²⋊C₄	—	53	1	no	—
520	C₂²⋊C₄	—	53	2	no	—
521	C₂²⋊C₄	—	53	2	no	—
522	C₂²⋊C₄	—	53	2	no	—
523	C₂²⋊C₄	—	53	1	no	—
524	C₂²⋊C₄	—	53	1	no	—
525	C₂²⋊C₄	—	53	2	no	—
526	C₂²⋊C₄	—	53	1	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	—	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	1	no	—
537	C₂²⋊C₄	—	53	1	no	—
538	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
539	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
540	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
541	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
542	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
543	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
544	C₄²	—	53	2	no	—
545	C₄²	—	53	2	no	—
546	C₄²	—	53	2	no	—
547	C₄²	—	53	2	no	—
548	C₄²	—	53	2	no	—
549	C₄²	—	53	2	no	—
550	C₄²	—	53	1	no	—
551	C₄²	C₂²×D₅₃	1	2	yes	—
552	C₄²	C₂²×D₅₃	1	2	yes	—
553	C₄²	C₂²×D₅₃	1	2	yes	—
554	C₄²	C₂²×D₅₃	1	1	yes	characteristic
555	C₂³	—	53	1	no	—
556	C₂³	—	53	1	no	—
557	C₂³	—	53	1	no	—
558	C₂³	—	53	2	no	—
559	C₂³	—	106	2	no	—
560	C₂³	—	212	1	no	—
561	C₂³	—	212	1	no	—
562	C₂³	C₂³×D₅₃	1	1	yes	characteristic
563	C₂³	C₂³×D₅₃	1	1	yes	characteristic
564	Q₈	—	2	2	no	—
565	Q₈	—	4	2	no	—
566	Q₈	—	106	2	no	—
567	Q₈	—	212	2	no	—
568	D₄	—	4	1	no	—
569	D₄	—	106	2	no	—
570	D₄	—	106	4	no	—
571	D₄	—	106	2	no	—
572	D₄	—	106	2	no	—
573	D₄	—	106	2	no	—
574	D₄	—	212	2	no	—
575	D₄	—	212	2	no	—
576	D₄	—	212	1	no	—
577	D₄	—	212	2	no	—
578	D₄	—	212	1	no	—
579	D₄	—	212	1	no	—
580	D₄	—	212	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	53	1	no	—
584	C₂×C₄	—	53	4	no	—
585	C₂×C₄	—	53	1	no	—
586	C₂×C₄	—	53	2	no	—
587	C₂×C₄	—	53	1	no	—
588	C₂×C₄	—	53	1	no	—
589	C₂×C₄	—	53	2	no	—
590	C₂×C₄	—	53	1	no	—
591	C₂×C₄	—	53	2	no	—
592	C₂×C₄	—	106	2	no	—
593	C₂×C₄	—	106	4	no	—
594	C₂×C₄	—	106	2	no	—
595	C₂×C₄	—	106	2	no	—
596	C₂×C₄	—	106	2	no	—
597	C₂×C₄	—	106	2	no	—
598	C₂×C₄	—	212	2	no	—
599	C₂×C₄	—	212	2	no	—
600	C₂×C₄	—	212	2	no	—
601	C₂×C₄	—	212	2	no	—
602	C₂×C₄	—	212	2	no	—
603	C₂×C₄	—	212	2	no	—
604	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
605	C₂×C₄	C₂³×D₅₃	1	2	yes	—
606	C₂×C₄	C₂³×D₅₃	1	2	yes	—
607	C₂×C₄	C₂³×D₅₃	1	2	yes	—
608	C₂×C₄	C₂³×D₅₃	1	2	yes	—
609	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
610	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
611	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
612	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
613	C₂²	—	2	1	no	—
614	C₂²	—	2	1	no	—
615	C₂²	—	2	1	no	—
616	C₂²	—	2	1	no	—
617	C₂²	—	2	1	no	—
618	C₂²	—	2	1	no	—
619	C₂²	—	53	2	no	—
620	C₂²	—	106	1	no	—
621	C₂²	—	106	1	no	—
622	C₂²	—	106	1	no	—
623	C₂²	—	106	2	no	—
624	C₂²	—	106	1	no	—
625	C₂²	—	106	1	no	—
626	C₂²	—	106	2	no	—
627	C₂²	—	106	1	no	—
628	C₂²	—	106	1	no	—
629	C₂²	—	106	2	no	—
630	C₂²	—	106	1	no	—
631	C₂²	—	212	4	no	—
632	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
633	C₄	—	2	1	no	—
634	C₄	—	2	1	no	—
635	C₄	—	2	2	no	—
636	C₄	—	2	2	no	—
637	C₄	—	2	1	no	—
638	C₄	—	2	2	no	—
639	C₄	—	2	2	no	—
640	C₄	—	2	1	no	—
641	C₄	—	53	2	no	—
642	C₄	—	106	1	no	—
643	C₄	—	106	1	no	—
644	C₄	—	106	2	no	—
645	C₄	—	106	1	no	—
646	C₄	—	106	2	no	—
647	C₄	—	106	1	no	—
648	C₄	—	106	2	no	—
649	C₄	#41×D₅₃	1	2	yes	—
650	C₂	—	4	1	no	—
651	C₂	—	4	1	no	—
652	C₂	—	106	2	no	—
653	C₂	—	212	1	no	—
654	C₂	—	212	2	no	—
655	C₂	—	212	1	no	—
656	C₂	C₂×#41×D₅₃	1	1	yes	characteristic
657	C₂	#16811459	1	1	yes	characteristic
658	C₂	#16811460	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#556	C₂
hypercenter	#556	D₅₃
hypocenter	C₅₃	#3493
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #556

Complements

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

N	H ≅ G/N	type
1. #16811345	C₂	semidirect
2. #16811344	C₂	semidirect
3. #16811339	C₂	semidirect
4. #16811336	C₂	semidirect
5. #16811335	C₂	semidirect
6. #16811324	C₂	semidirect
7. #16811319	C₂	semidirect
8. #16811317	C₂	semidirect
9. #16811315	C₂	semidirect
10. #16811314	C₂	semidirect
11. #16811309	C₂	semidirect
12. #16811250	C₂	semidirect
13. #16811212	C₂	semidirect
14. #16811207	C₂	semidirect
15. #16811203	C₂	semidirect
16. #16811160	C₂	semidirect
17. #16811156	C₂	semidirect
18. #16811136	C₂	semidirect
19. #16811132	C₂	semidirect
20. #16811105	C₂	semidirect
21. #16811103	C₂	semidirect
22. #117×D₅₃	C₂	semidirect
23. C₅₃×#556	C₂	semidirect
27. #13133216	C₂²	semidirect
28. #13133215	C₂²	semidirect
30. #13133212	C₂²	semidirect
31. #13133212	C₂²	semidirect
33. #13133211	C₂²	semidirect
35. #13133209	C₂²	semidirect
36. #13133208	C₂²	semidirect
37. #13133208	C₂²	semidirect
38. #13133207	C₂²	semidirect
39. #13133207	C₂²	semidirect
41. #13133206	C₂²	semidirect
44. #13133203	C₂²	semidirect
54. #13133195	C₂²	semidirect
58. #13133193	C₂²	semidirect
62. #13133192	C₂²	semidirect
64. #13133191	C₂²	semidirect
69. #13133190	C₂²	semidirect
75. Q₈×Dic₅₃	C₂²	semidirect
76. C₄×Dic₁₀₆	C₂²	semidirect
77. #13133182	C₂²	semidirect
81. C₄×D₂₁₂	C₂²	semidirect
82. C₄×D₂₁₂	C₂²	semidirect
85. #13133179	C₂²	semidirect
90. #13133177	C₂²	semidirect
91. #13133177	C₂²	semidirect
96. #13133175	C₂²	semidirect
97. #13133175	C₂²	semidirect
100. #13133174	C₂²	semidirect
101. #13133174	C₂²	semidirect
102. #13133174	C₂²	semidirect
104. #13133173	C₂²	semidirect
105. C₂×C₄⋊Dic₅₃	C₂²	semidirect
108. C₄⋊C₄×D₅₃	C₂²	semidirect
109. C₄⋊C₄×D₅₃	C₂²	semidirect
112. C₄⋊Q₈×C₅₃	C₂²	semidirect
114. #126×C₅₃	C₂²	semidirect
117. C₂²⋊Q₈×C₅₃	C₂²	semidirect
118. C₄×Q₈×C₅₃	C₂²	semidirect
119. C₄×Q₈×C₅₃	C₂²	semidirect
122. #117×C₅₃	C₂²	semidirect
332. #556	D₅₃	semidirect
353. C₅₃	#3493	semidirect
359. C₄⋊Q₈	C₂×D₅₃	semidirect
367. #126	C₂×D₅₃	semidirect
394. C₂²⋊Q₈	C₂×D₅₃	semidirect
414. C₄×Q₈	C₂×D₅₃	semidirect
415. C₄×Q₈	C₂×D₅₃	semidirect
445. #117	C₂×D₅₃	semidirect