Subgroups — #20288558

Order 6784.

1140 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811357	C₂	1	2	yes	maximal, complemented
2	#16811356	C₂	1	2	yes	maximal, complemented
3	#16811344	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811341	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811337	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811336	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811322	C₂	1	2	yes	maximal, complemented
8	#16811318	C₂	1	2	yes	maximal, complemented
9	#16811292	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811291	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811275	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811272	C₂	1	1	yes	maximal, characteristic, complemented
13	#16811208	C₂	1	2	yes	maximal, complemented
14	#16811206	C₂	1	2	yes	maximal, complemented
15	#16811188	C₂	1	2	yes	maximal, complemented
16	#16811175	C₂	1	1	yes	maximal, characteristic, complemented
17	#16811168	C₂	1	1	yes	maximal, characteristic, complemented
18	#16811136	C₂	1	1	yes	maximal, characteristic, complemented
19	#16811134	C₂	1	1	yes	maximal, characteristic, complemented
20	#16811133	C₂	1	1	yes	maximal, characteristic, complemented
21	#16811102	C₂	1	2	yes	maximal, complemented
22	C₂×C₄×C₂²⋊D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
23	C₅₃×#525	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#11719949	C₂²	1	1	yes	characteristic, complemented
25	C₂×#11719945	C₂²	1	1	yes	characteristic, complemented
26	C₂×#11719947	C₂²	1	2	yes	—
27	C₂×#11719947	C₂²	1	1	yes	characteristic, complemented
28	C₂²×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
30	#13133216	C₂²	1	1	yes	characteristic, complemented
31	#13133216	C₂²	1	1	yes	characteristic, complemented
32	#13133215	C₂²	1	1	yes	characteristic, complemented
33	#13133213	C₂²	1	2	yes	—
34	#13133213	C₂²	1	2	yes	—
35	#13133213	C₂²	1	1	yes	characteristic, complemented
36	#13133212	C₂²	1	2	yes	complemented
37	#13133211	C₂²	1	2	yes	complemented
38	#13133210	C₂²	1	2	yes	complemented
39	#13133210	C₂²	1	2	yes	complemented
40	#13133207	C₂²	1	2	yes	complemented
41	#13133205	C₂²	1	2	yes	complemented
42	#13133204	C₂²	1	2	yes	complemented
43	#13133203	C₂²	1	2	yes	—
44	#13133203	C₂²	1	2	yes	—
45	#13133203	C₂²	1	2	yes	complemented
46	#13133203	C₂²	1	2	yes	complemented
47	#13133203	C₂²	1	2	yes	complemented
48	#13133202	C₂²	1	2	yes	complemented
49	#13133202	C₂²	1	2	yes	complemented
50	#13133200	C₂²	1	2	yes	complemented
51	#13133200	C₂²	1	2	yes	complemented
52	#13133199	C₂²	1	2	yes	complemented
53	#13133199	C₂²	1	2	yes	complemented
54	#13133198	C₂²	1	2	yes	complemented
55	#13133197	C₂²	1	1	yes	characteristic, complemented
56	#13133196	C₂²	1	1	yes	characteristic
57	#13133196	C₂²	1	1	yes	characteristic, complemented
58	#13133194	C₂²	1	2	yes	—
59	#13133193	C₂²	1	1	yes	characteristic, complemented
60	#13133192	C₂²	1	1	yes	characteristic, complemented
61	#13133191	C₂²	1	1	yes	characteristic
62	#13133191	C₂²	1	2	yes	complemented
63	#13133191	C₂²	1	1	yes	characteristic, complemented
64	#13133190	C₂²	1	2	yes	—
65	#13133190	C₂²	1	2	yes	—
66	#13133190	C₂²	1	2	yes	complemented
67	#13133189	C₂²	1	2	yes	—
68	#13133189	C₂²	1	2	yes	—
69	#13133189	C₂²	1	2	yes	—
70	#13133189	C₂²	1	2	yes	complemented
71	#13133188	C₂²	1	2	yes	complemented
72	#13133188	C₂²	1	1	yes	characteristic, complemented
73	#13133187	C₂²	1	2	yes	—
74	#13133186	C₂²	1	2	yes	—
75	#13133185	C₂²	1	2	yes	—
76	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
77	C₄×Dic₁₀₆	C₂²	1	2	yes	complemented
78	#13133182	C₂²	1	1	yes	characteristic, complemented
79	#13133182	C₂²	1	2	yes	complemented
80	#13133182	C₂²	1	2	yes	complemented
81	D₄×Dic₅₃	C₂²	1	2	yes	—
82	D₄×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
83	#13133179	C₂²	1	1	yes	characteristic, complemented
84	#13133179	C₂²	1	2	yes	complemented
85	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
86	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
87	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
88	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
89	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
90	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
91	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
92	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
93	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
94	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
95	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
96	#13133177	C₂²	1	2	yes	—
97	#13133177	C₂²	1	2	yes	complemented
98	#13133177	C₂²	1	2	yes	complemented
99	#13133177	C₂²	1	2	yes	complemented
100	#13133176	C₂²	1	1	yes	characteristic, complemented
101	#13133175	C₂²	1	1	yes	characteristic, complemented
102	#13133175	C₂²	1	1	yes	characteristic, complemented
103	#13133174	C₂²	1	2	yes	complemented
104	#13133173	C₂²	1	2	yes	—
105	#13133173	C₂²	1	2	yes	complemented
106	C₂×#11719919	C₂²	1	1	yes	characteristic
107	C₂×#11719919	C₂²	1	2	yes	complemented
108	C₄⋊C₄×D₅₃	C₂²	1	2	yes	complemented
109	C₂×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic
110	C₂×#11719921	C₂²	1	1	yes	characteristic
111	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
112	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic, complemented
113	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
114	#123×C₅₃	C₂²	1	2	yes	complemented
115	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
116	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
117	#121×C₅₃	C₂²	1	1	yes	characteristic
118	#121×C₅₃	C₂²	1	1	yes	characteristic, complemented
119	#120×C₅₃	C₂²	1	2	yes	—
120	C₄×D₄×C₅₃	C₂²	1	2	yes	—
121	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
122	#117×C₅₃	C₂²	1	1	yes	characteristic, complemented
123	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
124	C₂³×D₅₃	C₂³	1	1	yes	characteristic
125	#41×C₅₃	—	4	2	no	—
126	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
127	C₂×D₄×C₅₃	C₂³	1	2	yes	—
128	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
129	C₂×D₄×C₅₃	C₂³	1	2	yes	—
130	C₂×D₄×C₅₃	C₂³	1	2	yes	—
131	C₂²×C₄×C₅₃	—	4	2	no	—
132	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
133	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
134	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
135	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
136	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
137	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
138	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
139	C₂×C₂²⋊D₅₃	—	2	4	no	—
140	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
141	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
142	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
143	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
144	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
145	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
146	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
147	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
148	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
149	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
150	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
151	C₂²×Dic₅₃	C₂³	1	2	yes	—
152	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
153	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
154	C₂²×Dic₅₃	C₂³	1	2	yes	—
155	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
156	#11719949	—	4	2	no	—
157	#11719947	—	2	8	no	—
158	#11719947	—	4	2	no	—
159	#11719945	—	4	2	no	—
160	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
161	C₂×D₂₁₂	C₂³	1	2	yes	—
162	C₂×D₂₁₂	C₂³	1	2	yes	—
163	C₂×C₄×D₅₃	—	4	2	no	—
164	C₂×C₄×D₅₃	C₂³	1	2	yes	—
165	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
166	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
167	C₂×C₄×D₅₃	C₂³	1	2	yes	—
168	C₂×C₄×D₅₃	C₂³	1	2	yes	—
169	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
170	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
171	C₂×Dic₁₀₆	C₂³	1	2	yes	—
172	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
173	C₂×Dic₁₀₆	C₂³	1	2	yes	—
174	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
175	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
176	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
177	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
178	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
179	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
180	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
181	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
182	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
183	C₄²×C₅₃	C₂³	1	2	yes	—
184	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
185	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
186	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
187	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
188	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
189	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
190	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
191	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
192	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
193	#11719921	C₂³	1	1	yes	characteristic
194	#11719921	C₂³	1	1	yes	characteristic
195	#11719921	C₂³	1	1	yes	characteristic
196	#11719921	C₂³	1	1	yes	characteristic
197	#11719921	C₂³	1	2	yes	—
198	#11719921	C₂³	1	2	yes	—
199	#11719921	C₂³	1	2	yes	—
200	#11719921	C₂³	1	2	yes	—
201	#11719921	C₂³	1	2	yes	—
202	#11719921	C₂³	1	2	yes	—
203	#11719921	C₂³	1	2	yes	—
204	#11719921	C₂³	1	2	yes	—
205	C₄⋊Dic₅₃	C₂³	1	2	yes	—
206	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
207	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
208	C₄⋊Dic₅₃	C₂³	1	2	yes	—
209	#11719919	C₂³	1	1	yes	characteristic
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	2	yes	—
215	#11719919	C₂³	1	2	yes	—
216	#11719919	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	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	1	yes	characteristic
225	C₄×Dic₅₃	C₂³	1	2	yes	—
226	C₄×Dic₅₃	C₂³	1	2	yes	—
227	C₄×Dic₅₃	C₂³	1	2	yes	—
228	C₄×Dic₅₃	C₂³	1	2	yes	—
229	C₄×Dic₅₃	C₂³	1	2	yes	—
230	C₂³×C₅₃	—	2	2	no	—
231	C₂³×C₅₃	—	4	1	no	—
232	C₂³×C₅₃	—	4	1	no	—
233	C₂³×C₅₃	C₂⁴	1	2	yes	—
234	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
235	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
236	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
237	C₂²×D₅₃	—	2	2	no	—
238	C₂²×D₅₃	—	4	1	no	—
239	C₂²×D₅₃	—	4	1	no	—
240	C₂²×D₅₃	C₂⁴	1	2	yes	—
241	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
242	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
243	Q₈×C₅₃	—	2	2	no	—
244	D₄×C₅₃	—	2	4	no	—
245	D₄×C₅₃	—	2	2	no	—
246	D₄×C₅₃	—	4	2	no	—
247	D₄×C₅₃	—	4	2	no	—
248	C₂×C₄×C₅₃	—	2	2	no	—
249	C₂×C₄×C₅₃	—	2	2	no	—
250	C₂×C₄×C₅₃	—	2	2	no	—
251	C₂×C₄×C₅₃	—	4	1	no	—
252	C₂×C₄×C₅₃	—	4	1	no	—
253	C₂×C₄×C₅₃	—	4	1	no	—
254	C₂×C₄×C₅₃	—	4	2	no	—
255	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
256	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
257	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
258	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
259	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
260	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
261	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
262	C₂²⋊D₅₃	—	2	4	no	—
263	C₂²⋊D₅₃	—	2	4	no	—
264	C₂²⋊D₅₃	—	2	2	no	—
265	C₂²⋊D₅₃	—	2	2	no	—
266	C₂²⋊D₅₃	—	2	2	no	—
267	C₂²⋊D₅₃	—	2	2	no	—
268	C₂²⋊D₅₃	—	4	2	no	—
269	C₂²⋊D₅₃	—	4	2	no	—
270	C₂²⋊D₅₃	—	4	2	no	—
271	C₂²⋊D₅₃	—	4	2	no	—
272	C₂²⋊D₅₃	—	4	2	no	—
273	C₂×Dic₅₃	—	2	4	no	—
274	C₂×Dic₅₃	—	4	1	no	—
275	C₂×Dic₅₃	—	4	2	no	—
276	C₂×Dic₅₃	—	4	1	no	—
277	C₂×Dic₅₃	#41	1	4	yes	complemented
278	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
279	C₂×Dic₅₃	C₂⁴	1	2	yes	—
280	C₂×Dic₅₃	C₂⁴	1	2	yes	—
281	C₂×Dic₅₃	C₂⁴	1	2	yes	—
282	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
283	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
284	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
285	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
286	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
287	D₂₁₂	—	2	2	no	—
288	D₂₁₂	—	4	2	no	—
289	D₂₁₂	—	4	2	no	—
290	C₄×D₅₃	—	2	2	no	—
291	C₄×D₅₃	—	2	2	no	—
292	C₄×D₅₃	—	2	4	no	—
293	C₄×D₅₃	—	4	1	no	—
294	C₄×D₅₃	—	4	1	no	—
295	C₄×D₅₃	—	4	2	no	—
296	C₄×D₅₃	—	4	2	no	—
297	Dic₁₀₆	—	2	2	no	—
298	Dic₁₀₆	—	2	4	no	—
299	Dic₁₀₆	—	4	2	no	—
300	C₂²×C₅₃	—	2	1	no	—
301	C₂²×C₅₃	—	2	1	no	—
302	C₂²×C₅₃	—	2	2	no	—
303	C₂²×C₅₃	—	2	1	no	—
304	C₂²×C₅₃	—	2	1	no	—
305	C₂²×C₅₃	—	2	2	no	—
306	C₂²×C₅₃	—	2	1	no	—
307	C₂²×C₅₃	—	2	1	no	—
308	C₂²×C₅₃	—	2	2	no	—
309	C₂²×C₅₃	—	2	1	no	—
310	C₂²×C₅₃	—	2	1	no	—
311	C₂²×C₅₃	—	4	4	no	—
312	C₂²×C₅₃	C₂×#41	1	2	yes	—
313	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
314	C₂×D₅₃	—	2	2	no	—
315	C₂×D₅₃	—	2	1	no	—
316	C₂×D₅₃	—	2	1	no	—
317	C₂×D₅₃	—	2	1	no	—
318	C₂×D₅₃	—	2	1	no	—
319	C₂×D₅₃	—	2	2	no	—
320	C₂×D₅₃	—	2	1	no	—
321	C₂×D₅₃	—	2	2	no	—
322	C₂×D₅₃	—	2	1	no	—
323	C₂×D₅₃	—	4	4	no	—
324	C₄×C₅₃	—	2	2	no	—
325	C₄×C₅₃	—	2	1	no	—
326	C₄×C₅₃	—	2	1	no	—
327	C₄×C₅₃	—	2	2	no	—
328	C₄×C₅₃	—	2	1	no	—
329	C₄×C₅₃	—	2	2	no	—
330	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
331	Dic₅₃	—	2	2	no	—
332	Dic₅₃	—	2	2	no	—
333	Dic₅₃	—	2	1	no	—
334	Dic₅₃	—	2	1	no	—
335	Dic₅₃	—	2	1	no	—
336	Dic₅₃	—	2	2	no	—
337	Dic₅₃	—	2	1	no	—
338	Dic₅₃	C₂×#41	1	2	yes	—
339	Dic₅₃	C₂×#41	1	2	yes	—
340	#3506	—	53	1	no	maximal, Sylow
341	C₂×C₅₃	—	2	2	no	—
342	C₂×C₅₃	—	4	2	no	—
343	C₂×C₅₃	—	4	1	no	—
344	C₂×C₅₃	—	4	1	no	—
345	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
346	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
347	C₂×C₅₃	#585	1	1	yes	characteristic
348	D₅₃	—	4	2	no	—
349	D₅₃	—	4	1	no	—
350	D₅₃	—	4	1	no	—
351	#559	—	53	2	no	—
352	#559	—	53	2	no	—
353	#556	—	53	1	no	—
354	#556	—	53	1	no	—
355	#555	—	53	1	no	—
356	#555	—	53	1	no	—
357	#553	—	53	2	no	—
358	#553	—	53	2	no	—
359	#550	—	53	1	no	—
360	#550	—	53	1	no	—
361	#547	—	53	1	no	—
362	#547	—	53	1	no	—
363	#538	—	53	2	no	—
364	#538	—	53	2	no	—
365	#535	—	53	2	no	—
366	#533	—	53	1	no	—
367	#532	—	53	1	no	—
368	#525	—	53	1	no	—
369	#525	—	53	1	no	—
370	#525	—	53	1	no	—
371	#525	D₅₃	1	1	yes	characteristic, complemented
372	#520	—	53	2	no	—
373	C₂×C₄×D₄	—	53	1	no	—
374	C₅₃	#3506	1	1	yes	characteristic, complemented, Sylow
375	C₂×#41	—	53	1	no	—
376	C₂×#41	—	53	2	no	—
377	C₂×#41	—	53	1	no	—
378	C₂×#41	—	53	1	no	—
379	C₂×#41	C₂×D₅₃	1	1	yes	characteristic, complemented
380	C₂²×D₄	—	53	1	no	—
381	C₂³×C₄	—	53	1	no	—
382	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
383	C₄⋊Q₈	—	53	1	no	—
384	C₄⋊Q₈	—	53	1	no	—
385	C₄⋊Q₈	—	53	1	no	—
386	#127	—	53	1	no	—
387	#127	—	53	2	no	—
388	#127	—	53	2	no	—
389	#126	—	53	2	no	—
390	#126	—	53	2	no	—
391	#126	—	53	2	no	—
392	#126	—	53	2	no	—
393	#125	—	53	2	no	—
394	#124	—	53	2	no	—
395	#124	—	53	2	no	—
396	#124	—	53	2	no	—
397	#124	—	53	2	no	—
398	#124	—	53	2	no	—
399	#124	—	53	2	no	—
400	#124	—	53	2	no	—
401	#123	—	53	2	no	—
402	#123	—	53	2	no	—
403	#123	—	53	2	no	—
404	#123	—	53	2	no	—
405	#123	—	53	2	no	—
406	#123	—	53	2	no	—
407	#123	—	53	2	no	—
408	#123	C₂×D₅₃	1	2	yes	complemented
409	C₂²⋊Q₈	—	53	1	no	—
410	C₂²⋊Q₈	—	53	1	no	—
411	C₂²⋊Q₈	—	53	1	no	—
412	C₂²⋊Q₈	—	53	1	no	—
413	C₂²⋊Q₈	—	53	2	no	—
414	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
415	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
416	#121	—	53	1	no	—
417	#121	—	53	2	no	—
418	#121	—	53	2	no	—
419	#121	—	53	2	no	—
420	#121	—	53	2	no	—
421	#121	—	53	2	no	—
422	#121	—	53	1	no	—
423	#121	—	53	1	no	—
424	#121	—	53	2	no	—
425	#121	—	53	1	no	—
426	#121	—	53	2	no	—
427	#121	—	53	2	no	—
428	#121	—	53	2	no	—
429	#121	C₂×D₅₃	1	1	yes	characteristic
430	#121	C₂×D₅₃	1	1	yes	characteristic, complemented
431	#120	—	53	2	no	—
432	#120	—	53	2	no	—
433	#120	—	53	2	no	—
434	#120	C₂×D₅₃	1	2	yes	—
435	C₄×Q₈	—	53	1	no	—
436	C₄×Q₈	—	53	2	no	—
437	C₄×Q₈	—	53	1	no	—
438	C₄×Q₈	—	53	2	no	—
439	C₄×Q₈	—	53	2	no	—
440	C₄×D₄	—	53	2	no	—
441	C₄×D₄	—	53	1	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	1	no	—
447	C₄×D₄	—	53	1	no	—
448	C₄×D₄	—	53	1	no	—
449	C₄×D₄	—	53	1	no	—
450	C₄×D₄	—	53	2	no	—
451	C₄×D₄	—	53	2	no	—
452	C₄×D₄	—	53	2	no	—
453	C₄×D₄	—	53	2	no	—
454	C₄×D₄	—	53	1	no	—
455	C₄×D₄	—	53	2	no	—
456	C₄×D₄	—	53	2	no	—
457	C₄×D₄	—	53	2	no	—
458	C₄×D₄	—	53	2	no	—
459	C₄×D₄	C₂×D₅₃	1	2	yes	—
460	C₄×D₄	C₂×D₅₃	1	2	yes	complemented
461	#117	—	53	1	no	—
462	#117	—	53	1	no	—
463	#117	—	53	1	no	—
464	#117	—	53	2	no	—
465	#117	—	53	2	no	—
466	#117	—	53	2	no	—
467	#117	C₂×D₅₃	1	1	yes	characteristic, complemented
468	C₂×C₄⋊C₄	—	53	1	no	—
469	C₂×C₄⋊C₄	—	53	2	no	—
470	C₂×C₄⋊C₄	—	53	2	no	—
471	C₂×C₂²⋊C₄	—	53	1	no	—
472	C₂×C₂²⋊C₄	—	53	1	no	—
473	C₂×C₄²	—	53	1	no	—
474	C₂⁴	—	53	1	no	—
475	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
476	#41	—	4	2	no	—
477	#41	—	106	8	no	—
478	#41	—	212	2	no	—
479	#41	—	212	2	no	—
480	#41	—	212	2	no	—
481	C₂×Q₈	—	53	2	no	—
482	C₂×Q₈	—	53	1	no	—
483	C₂×Q₈	—	53	2	no	—
484	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
485	C₂×D₄	—	53	2	no	—
486	C₂×D₄	—	53	1	no	—
487	C₂×D₄	—	53	1	no	—
488	C₂×D₄	—	53	1	no	—
489	C₂×D₄	—	53	1	no	—
490	C₂×D₄	—	53	2	no	—
491	C₂×D₄	—	53	2	no	—
492	C₂×D₄	—	53	2	no	—
493	C₂×D₄	—	53	2	no	—
494	C₂×D₄	—	53	2	no	—
495	C₂×D₄	—	53	2	no	—
496	C₂×D₄	—	53	1	no	—
497	C₂×D₄	—	53	2	no	—
498	C₂×D₄	—	53	2	no	—
499	C₂×D₄	—	106	4	no	—
500	C₂×D₄	C₂²×D₅₃	1	2	yes	—
501	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
502	C₂×D₄	C₂²×D₅₃	1	2	yes	—
503	C₂×D₄	C₂²×D₅₃	1	2	yes	—
504	C₂²×C₄	—	4	2	no	—
505	C₂²×C₄	—	53	1	no	—
506	C₂²×C₄	—	53	2	no	—
507	C₂²×C₄	—	53	1	no	—
508	C₂²×C₄	—	53	1	no	—
509	C₂²×C₄	—	53	2	no	—
510	C₂²×C₄	—	53	2	no	—
511	C₂²×C₄	—	53	1	no	—
512	C₂²×C₄	—	53	1	no	—
513	C₂²×C₄	—	53	2	no	—
514	C₂²×C₄	—	53	2	no	—
515	C₂²×C₄	—	53	1	no	—
516	C₂²×C₄	—	53	1	no	—
517	C₂²×C₄	—	212	2	no	—
518	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
519	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
520	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
521	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
522	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
523	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
524	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
525	C₄⋊C₄	—	53	1	no	—
526	C₄⋊C₄	—	53	1	no	—
527	C₄⋊C₄	—	53	2	no	—
528	C₄⋊C₄	—	53	1	no	—
529	C₄⋊C₄	—	53	1	no	—
530	C₄⋊C₄	—	53	2	no	—
531	C₄⋊C₄	—	53	2	no	—
532	C₄⋊C₄	—	53	2	no	—
533	C₄⋊C₄	—	53	2	no	—
534	C₄⋊C₄	—	53	1	no	—
535	C₄⋊C₄	—	53	1	no	—
536	C₄⋊C₄	—	53	2	no	—
537	C₄⋊C₄	—	53	2	no	—
538	C₄⋊C₄	—	53	2	no	—
539	C₄⋊C₄	—	53	2	no	—
540	C₄⋊C₄	—	53	2	no	—
541	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
542	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
543	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
544	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
545	C₂²⋊C₄	—	53	1	no	—
546	C₂²⋊C₄	—	53	1	no	—
547	C₂²⋊C₄	—	53	1	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	—	53	2	no	—
554	C₂²⋊C₄	—	53	1	no	—
555	C₂²⋊C₄	—	53	1	no	—
556	C₂²⋊C₄	—	53	1	no	—
557	C₂²⋊C₄	—	53	1	no	—
558	C₂²⋊C₄	—	53	2	no	—
559	C₂²⋊C₄	—	53	2	no	—
560	C₂²⋊C₄	—	53	2	no	—
561	C₂²⋊C₄	—	53	2	no	—
562	C₂²⋊C₄	—	53	2	no	—
563	C₂²⋊C₄	—	53	2	no	—
564	C₂²⋊C₄	—	53	2	no	—
565	C₂²⋊C₄	—	53	2	no	—
566	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
567	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
568	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
569	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
570	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
571	C₄²	—	53	1	no	—
572	C₄²	—	53	1	no	—
573	C₄²	—	53	1	no	—
574	C₄²	—	53	1	no	—
575	C₄²	—	53	2	no	—
576	C₄²	—	53	2	no	—
577	C₄²	—	53	2	no	—
578	C₄²	—	53	2	no	—
579	C₄²	—	53	2	no	—
580	C₄²	C₂²×D₅₃	1	2	yes	—
581	C₂³	—	2	2	no	—
582	C₂³	—	4	1	no	—
583	C₂³	—	4	1	no	—
584	C₂³	—	53	1	no	—
585	C₂³	—	53	1	no	—
586	C₂³	—	53	2	no	—
587	C₂³	—	106	2	no	—
588	C₂³	—	212	1	no	—
589	C₂³	—	212	1	no	—
590	C₂³	C₂³×D₅₃	1	1	yes	characteristic
591	C₂³	C₂³×D₅₃	1	1	yes	characteristic
592	C₂³	C₂³×D₅₃	1	2	yes	—
593	C₂³	C₂³×D₅₃	1	1	yes	characteristic
594	Q₈	—	2	2	no	—
595	Q₈	—	106	4	no	—
596	Q₈	—	106	2	no	—
597	Q₈	—	212	2	no	—
598	D₄	—	2	4	no	—
599	D₄	—	2	2	no	—
600	D₄	—	4	2	no	—
601	D₄	—	4	2	no	—
602	D₄	—	106	2	no	—
603	D₄	—	106	2	no	—
604	D₄	—	106	2	no	—
605	D₄	—	106	2	no	—
606	D₄	—	106	4	no	—
607	D₄	—	106	4	no	—
608	D₄	—	106	2	no	—
609	D₄	—	212	2	no	—
610	D₄	—	212	2	no	—
611	D₄	—	212	2	no	—
612	D₄	—	212	2	no	—
613	D₄	—	212	2	no	—
614	D₄	—	212	2	no	—
615	D₄	—	212	2	no	—
616	C₂×C₄	—	2	2	no	—
617	C₂×C₄	—	2	2	no	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	4	1	no	—
620	C₂×C₄	—	4	1	no	—
621	C₂×C₄	—	4	1	no	—
622	C₂×C₄	—	4	2	no	—
623	C₂×C₄	—	53	4	no	—
624	C₂×C₄	—	53	1	no	—
625	C₂×C₄	—	53	1	no	—
626	C₂×C₄	—	53	2	no	—
627	C₂×C₄	—	53	1	no	—
628	C₂×C₄	—	53	2	no	—
629	C₂×C₄	—	53	1	no	—
630	C₂×C₄	—	53	1	no	—
631	C₂×C₄	—	53	1	no	—
632	C₂×C₄	—	53	2	no	—
633	C₂×C₄	—	106	4	no	—
634	C₂×C₄	—	106	4	no	—
635	C₂×C₄	—	106	2	no	—
636	C₂×C₄	—	106	2	no	—
637	C₂×C₄	—	212	1	no	—
638	C₂×C₄	—	212	1	no	—
639	C₂×C₄	—	212	2	no	—
640	C₂×C₄	—	212	2	no	—
641	C₂×C₄	—	212	1	no	—
642	C₂×C₄	—	212	1	no	—
643	C₂×C₄	—	212	2	no	—
644	C₂×C₄	C₂³×D₅₃	1	2	yes	—
645	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
646	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
647	C₂×C₄	C₂³×D₅₃	1	2	yes	—
648	C₂×C₄	C₂³×D₅₃	1	2	yes	—
649	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
650	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
651	C₂²	—	2	1	no	—
652	C₂²	—	2	1	no	—
653	C₂²	—	2	2	no	—
654	C₂²	—	2	1	no	—
655	C₂²	—	2	1	no	—
656	C₂²	—	2	1	no	—
657	C₂²	—	2	1	no	—
658	C₂²	—	2	2	no	—
659	C₂²	—	2	1	no	—
660	C₂²	—	2	2	no	—
661	C₂²	—	2	1	no	—
662	C₂²	—	4	4	no	—
663	C₂²	—	106	2	no	—
664	C₂²	—	106	1	no	—
665	C₂²	—	106	1	no	—
666	C₂²	—	106	1	no	—
667	C₂²	—	106	1	no	—
668	C₂²	—	106	2	no	—
669	C₂²	—	106	2	no	—
670	C₂²	—	106	1	no	—
671	C₂²	—	106	1	no	—
672	C₂²	—	212	4	no	—
673	C₂²	#41×D₅₃	1	2	yes	—
674	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
675	C₄	—	2	2	no	—
676	C₄	—	2	1	no	—
677	C₄	—	2	1	no	—
678	C₄	—	2	1	no	—
679	C₄	—	2	2	no	—
680	C₄	—	2	2	no	—
681	C₄	—	53	2	no	—
682	C₄	—	53	2	no	—
683	C₄	—	106	2	no	—
684	C₄	—	106	1	no	—
685	C₄	—	106	1	no	—
686	C₄	—	106	2	no	—
687	C₄	—	106	1	no	—
688	C₄	—	106	1	no	—
689	C₄	—	106	2	no	—
690	C₄	#13133262	1	2	yes	—
691	C₂	—	2	2	no	—
692	C₂	—	4	1	no	—
693	C₂	—	4	1	no	—
694	C₂	—	4	2	no	—
695	C₂	—	212	1	no	—
696	C₂	—	212	1	no	—
697	C₂	—	212	2	no	—
698	C₂	C₂×#41×D₅₃	1	1	yes	characteristic
699	C₂	C₂×#13133262	1	1	yes	characteristic
700	C₂	#16811459	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#525	C₂
hypercenter	#525	D₅₃
hypocenter	C₅₃	#3506
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #525

Complements

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

N	H ≅ G/N	type
1. #16811357	C₂	semidirect
2. #16811356	C₂	semidirect
3. #16811344	C₂	semidirect
4. #16811341	C₂	semidirect
5. #16811337	C₂	semidirect
6. #16811336	C₂	semidirect
7. #16811322	C₂	semidirect
8. #16811318	C₂	semidirect
9. #16811292	C₂	semidirect
10. #16811291	C₂	semidirect
11. #16811275	C₂	semidirect
12. #16811272	C₂	semidirect
13. #16811208	C₂	semidirect
14. #16811206	C₂	semidirect
15. #16811188	C₂	semidirect
16. #16811175	C₂	semidirect
17. #16811168	C₂	semidirect
18. #16811136	C₂	semidirect
19. #16811134	C₂	semidirect
20. #16811133	C₂	semidirect
21. #16811102	C₂	semidirect
22. C₂×C₄×C₂²⋊D₅₃	C₂	semidirect
23. C₅₃×#525	C₂	semidirect
24. C₂×#11719949	C₂²	semidirect
25. C₂×#11719945	C₂²	semidirect
27. C₂×#11719947	C₂²	semidirect
30. #13133216	C₂²	semidirect
31. #13133216	C₂²	semidirect
32. #13133215	C₂²	semidirect
35. #13133213	C₂²	semidirect
36. #13133212	C₂²	semidirect
37. #13133211	C₂²	semidirect
38. #13133210	C₂²	semidirect
39. #13133210	C₂²	semidirect
40. #13133207	C₂²	semidirect
41. #13133205	C₂²	semidirect
42. #13133204	C₂²	semidirect
45. #13133203	C₂²	semidirect
46. #13133203	C₂²	semidirect
47. #13133203	C₂²	semidirect
48. #13133202	C₂²	semidirect
49. #13133202	C₂²	semidirect
50. #13133200	C₂²	semidirect
51. #13133200	C₂²	semidirect
52. #13133199	C₂²	semidirect
53. #13133199	C₂²	semidirect
54. #13133198	C₂²	semidirect
55. #13133197	C₂²	semidirect
57. #13133196	C₂²	semidirect
59. #13133193	C₂²	semidirect
60. #13133192	C₂²	semidirect
62. #13133191	C₂²	semidirect
63. #13133191	C₂²	semidirect
66. #13133190	C₂²	semidirect
70. #13133189	C₂²	semidirect
71. #13133188	C₂²	semidirect
72. #13133188	C₂²	semidirect
76. Q₈×Dic₅₃	C₂²	semidirect
77. C₄×Dic₁₀₆	C₂²	semidirect
78. #13133182	C₂²	semidirect
79. #13133182	C₂²	semidirect
80. #13133182	C₂²	semidirect
82. D₄×Dic₅₃	C₂²	semidirect
83. #13133179	C₂²	semidirect
84. #13133179	C₂²	semidirect
94. C₄×C₂²⋊D₅₃	C₂²	semidirect
95. C₄×C₂²⋊D₅₃	C₂²	semidirect
97. #13133177	C₂²	semidirect
98. #13133177	C₂²	semidirect
99. #13133177	C₂²	semidirect
100. #13133176	C₂²	semidirect
101. #13133175	C₂²	semidirect
102. #13133175	C₂²	semidirect
103. #13133174	C₂²	semidirect
105. #13133173	C₂²	semidirect
107. C₂×#11719919	C₂²	semidirect
108. C₄⋊C₄×D₅₃	C₂²	semidirect
112. C₂×#41×C₅₃	C₂²	semidirect
114. #123×C₅₃	C₂²	semidirect
116. C₂²⋊Q₈×C₅₃	C₂²	semidirect
118. #121×C₅₃	C₂²	semidirect
121. C₄×D₄×C₅₃	C₂²	semidirect
122. #117×C₅₃	C₂²	semidirect
277. C₂×Dic₅₃	#41	semidirect
371. #525	D₅₃	semidirect
374. C₅₃	#3506	semidirect
379. C₂×#41	C₂×D₅₃	semidirect
408. #123	C₂×D₅₃	semidirect
415. C₂²⋊Q₈	C₂×D₅₃	semidirect
430. #121	C₂×D₅₃	semidirect
460. C₄×D₄	C₂×D₅₃	semidirect
467. #117	C₂×D₅₃	semidirect