Subgroups — #20288666

Order 6784.

1142 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811356	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811341	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811338	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811336	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811302	C₂	1	2	yes	maximal, complemented
6	#16811299	C₂	1	2	yes	maximal, complemented
7	D₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811282	C₂	1	2	yes	maximal, complemented
9	#16811268	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811267	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811263	C₂	1	2	yes	maximal, complemented
12	D₄×C₂²⋊D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	#16811244	C₂	1	1	yes	maximal, characteristic, complemented
14	#16811238	C₂	1	1	yes	maximal, characteristic, complemented
15	#16811211	C₂	1	2	yes	maximal, complemented
16	#16811206	C₂	1	2	yes	maximal, complemented
17	#16811170	C₂	1	1	yes	maximal, characteristic, complemented
18	#16811153	C₂	1	2	yes	maximal, complemented
19	#16811151	C₂	1	1	yes	maximal, characteristic, complemented
20	C₂×#13133203	C₂	1	2	yes	maximal, complemented
21	#16811095	C₂	1	1	yes	maximal, characteristic, complemented
22	C₄×#11719947	C₂	1	1	yes	maximal, characteristic, complemented
23	C₅₃×#518	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#11719945	C₂²	1	2	yes	complemented
25	C₂×#11719947	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₀₆	C₂²	1	2	yes	complemented
27	C₂²×C₂²⋊D₅₃	C₂²	1	2	yes	—
28	C₂×D₄×D₅₃	C₂²	1	1	yes	characteristic
29	#13133217	C₂²	1	1	yes	characteristic, complemented
30	#13133216	C₂²	1	1	yes	characteristic, complemented
31	#13133216	C₂²	1	1	yes	characteristic, complemented
32	#13133213	C₂²	1	1	yes	characteristic
33	#13133212	C₂²	1	1	yes	characteristic, complemented
34	#13133212	C₂²	1	2	yes	complemented
35	#13133211	C₂²	1	2	yes	complemented
36	#13133210	C₂²	1	1	yes	characteristic, complemented
37	#13133210	C₂²	1	2	yes	complemented
38	#13133210	C₂²	1	2	yes	complemented
39	#13133207	C₂²	1	1	yes	characteristic, complemented
40	#13133205	C₂²	1	1	yes	characteristic, complemented
41	#13133205	C₂²	1	2	yes	complemented
42	#13133204	C₂²	1	1	yes	characteristic, complemented
43	#13133203	C₂²	1	2	yes	—
44	#13133203	C₂²	1	1	yes	characteristic
45	#13133203	C₂²	1	2	yes	complemented
46	#13133203	C₂²	1	2	yes	complemented
47	#13133203	C₂²	1	2	yes	complemented
48	#13133203	C₂²	1	2	yes	complemented
49	#13133203	C₂²	1	2	yes	complemented
50	#13133203	C₂²	1	2	yes	complemented
51	#13133203	C₂²	1	2	yes	complemented
52	#13133202	C₂²	1	1	yes	characteristic, complemented
53	#13133201	C₂²	1	2	yes	complemented
54	#13133200	C₂²	1	2	yes	complemented
55	#13133200	C₂²	1	2	yes	complemented
56	#13133199	C₂²	1	2	yes	complemented
57	#13133199	C₂²	1	2	yes	complemented
58	#13133199	C₂²	1	1	yes	characteristic, complemented
59	#13133198	C₂²	1	2	yes	complemented
60	#13133197	C₂²	1	1	yes	characteristic, complemented
61	#13133196	C₂²	1	1	yes	characteristic, complemented
62	#13133196	C₂²	1	2	yes	complemented
63	#13133196	C₂²	1	2	yes	complemented
64	#13133195	C₂²	1	1	yes	characteristic, complemented
65	#13133195	C₂²	1	2	yes	complemented
66	#13133194	C₂²	1	2	yes	complemented
67	#13133194	C₂²	1	2	yes	complemented
68	#13133192	C₂²	1	1	yes	characteristic, complemented
69	#13133191	C₂²	1	1	yes	characteristic
70	#13133191	C₂²	1	2	yes	complemented
71	#13133191	C₂²	1	2	yes	complemented
72	#13133190	C₂²	1	1	yes	characteristic
73	#13133190	C₂²	1	2	yes	complemented
74	#13133190	C₂²	1	2	yes	complemented
75	#13133189	C₂²	1	2	yes	—
76	#13133189	C₂²	1	2	yes	complemented
77	#13133188	C₂²	1	1	yes	characteristic
78	#13133188	C₂²	1	1	yes	characteristic, complemented
79	#13133187	C₂²	1	2	yes	complemented
80	#13133186	C₂²	1	2	yes	—
81	#13133186	C₂²	1	2	yes	complemented
82	#13133185	C₂²	1	2	yes	—
83	C₄×Dic₁₀₆	C₂²	1	2	yes	complemented
84	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
85	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
86	#13133182	C₂²	1	1	yes	characteristic, complemented
87	C₄×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
88	C₄×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
89	D₄×Dic₅₃	C₂²	1	1	yes	characteristic
90	D₄×Dic₅₃	C₂²	1	1	yes	characteristic
91	#13133179	C₂²	1	1	yes	characteristic, complemented
92	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
93	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic, complemented
94	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
95	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
96	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
97	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
98	#13133177	C₂²	1	2	yes	complemented
99	#13133177	C₂²	1	2	yes	complemented
100	#13133176	C₂²	1	2	yes	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	#13133174	C₂²	1	1	yes	characteristic, complemented
105	#13133173	C₂²	1	2	yes	complemented
106	C₂×C₂²⋊Dic₅₃	C₂²	1	2	yes	complemented
107	C₂×#11719921	C₂²	1	2	yes	complemented
108	C₂×#11719921	C₂²	1	2	yes	complemented
109	C₂²⋊C₄×D₅₃	C₂²	1	2	yes	complemented
110	C₂×C₄×Dic₅₃	C₂²	1	2	yes	complemented
111	C₄²×D₅₃	C₂²	1	1	yes	characteristic, complemented
112	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
113	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
114	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
115	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
116	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
117	C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
118	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
119	#117×C₅₃	C₂²	1	1	yes	characteristic, complemented
120	#117×C₅₃	C₂²	1	1	yes	characteristic, complemented
121	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
122	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
123	C₂⁴×C₅₃	C₂³	1	2	yes	—
124	C₂³×D₅₃	C₂³	1	2	yes	—
125	C₂×D₄×C₅₃	—	2	4	no	—
126	C₂×D₄×C₅₃	C₂³	1	2	yes	—
127	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
128	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
129	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
130	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
131	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
132	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
133	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
134	C₂×C₂²⋊D₅₃	—	4	4	no	—
135	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
136	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
137	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
138	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
139	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
140	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
141	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
142	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
143	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
144	C₂²×Dic₅₃	C₂³	1	2	yes	—
145	C₂²×Dic₅₃	C₂³	1	2	yes	—
146	C₂²×Dic₅₃	C₂³	1	2	yes	—
147	#11719947	—	2	4	no	—
148	D₄×D₅₃	—	2	4	no	—
149	#11719945	—	4	4	no	—
150	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
151	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
152	C₂×D₂₁₂	C₂³	1	2	yes	—
153	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
154	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
155	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
156	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
157	C₂×C₄×D₅₃	C₂³	1	2	yes	—
158	C₂×C₄×D₅₃	C₂³	1	2	yes	—
159	C₂×Dic₁₀₆	—	4	4	no	—
160	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
161	C₂×Dic₁₀₆	C₂³	1	2	yes	—
162	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
163	C₂×Dic₁₀₆	C₂³	1	2	yes	—
164	C₂×Dic₁₀₆	C₂³	1	2	yes	—
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	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
170	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
171	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
172	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
173	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
174	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
175	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
176	C₄²×C₅₃	C₂³	1	1	yes	characteristic
177	C₄²×C₅₃	C₂³	1	1	yes	characteristic
178	C₄²×C₅₃	C₂³	1	2	yes	—
179	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
180	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
181	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
182	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
183	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
184	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
185	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
186	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
187	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
188	#11719921	C₂³	1	1	yes	characteristic
189	#11719921	C₂³	1	2	yes	—
190	#11719921	C₂³	1	1	yes	characteristic
191	#11719921	C₂³	1	1	yes	characteristic
192	#11719921	C₂³	1	2	yes	—
193	#11719921	C₂³	1	2	yes	—
194	#11719921	C₂³	1	2	yes	—
195	#11719921	C₂³	1	2	yes	—
196	#11719921	C₂³	1	2	yes	—
197	#11719921	C₂³	1	2	yes	—
198	#11719921	C₂³	1	2	yes	—
199	#11719921	C₂³	1	1	yes	characteristic
200	#11719921	C₂³	1	2	yes	—
201	#11719921	C₂³	1	1	yes	characteristic
202	#11719921	C₂³	1	1	yes	characteristic
203	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
204	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
205	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
206	C₄⋊Dic₅₃	C₂³	1	2	yes	—
207	C₄⋊Dic₅₃	C₂³	1	2	yes	—
208	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
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	2	yes	—
215	#11719919	C₂³	1	1	yes	characteristic
216	#11719919	C₂³	1	1	yes	characteristic
217	#11719919	C₂³	1	2	yes	—
218	#11719919	C₂³	1	2	yes	—
219	#11719919	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	2	yes	—
224	C₄×Dic₅₃	C₂³	1	2	yes	—
225	C₄×Dic₅₃	C₂³	1	2	yes	—
226	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
227	C₄×Dic₅₃	C₂³	1	2	yes	—
228	C₂³×C₅₃	—	2	4	no	—
229	C₂³×C₅₃	—	4	2	no	—
230	C₂³×C₅₃	—	4	2	no	—
231	C₂³×C₅₃	C₂⁴	1	2	yes	—
232	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
233	C₂³×C₅₃	C₂⁴	1	2	yes	—
234	C₂²×D₅₃	—	2	4	no	—
235	C₂²×D₅₃	—	4	2	no	—
236	C₂²×D₅₃	—	4	2	no	—
237	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
238	C₂²×D₅₃	C₂⁴	1	2	yes	—
239	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
240	D₄×C₅₃	—	2	4	no	—
241	D₄×C₅₃	—	2	2	no	—
242	D₄×C₅₃	#41	1	4	yes	complemented
243	C₂×C₄×C₅₃	—	2	2	no	—
244	C₂×C₄×C₅₃	—	4	2	no	—
245	C₂×C₄×C₅₃	—	4	2	no	—
246	C₂×C₄×C₅₃	—	4	2	no	—
247	C₂×C₄×C₅₃	—	4	2	no	—
248	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
249	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
250	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
251	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
252	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
253	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
254	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
255	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
256	C₂²⋊D₅₃	—	2	4	no	—
257	C₂²⋊D₅₃	—	2	4	no	—
258	C₂²⋊D₅₃	—	2	4	no	—
259	C₂²⋊D₅₃	—	2	2	no	—
260	C₂²⋊D₅₃	—	4	2	no	—
261	C₂²⋊D₅₃	—	4	2	no	—
262	C₂²⋊D₅₃	—	4	2	no	—
263	C₂²⋊D₅₃	—	4	1	no	—
264	C₂²⋊D₅₃	—	4	2	no	—
265	C₂×Dic₅₃	—	2	4	no	—
266	C₂×Dic₅₃	—	2	4	no	—
267	C₂×Dic₅₃	—	4	2	no	—
268	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
269	C₂×Dic₅₃	C₂⁴	1	2	yes	—
270	C₂×Dic₅₃	C₂⁴	1	2	yes	—
271	C₂×Dic₅₃	C₂⁴	1	2	yes	—
272	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
273	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
274	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
275	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
276	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
277	D₂₁₂	—	2	2	no	—
278	D₂₁₂	—	4	1	no	—
279	D₂₁₂	—	4	2	no	—
280	C₄×D₅₃	—	2	2	no	—
281	C₄×D₅₃	—	2	2	no	—
282	C₄×D₅₃	—	2	2	no	—
283	C₄×D₅₃	—	4	2	no	—
284	C₄×D₅₃	—	4	1	no	—
285	C₄×D₅₃	—	4	2	no	—
286	Dic₁₀₆	—	2	2	no	—
287	Dic₁₀₆	—	2	2	no	—
288	Dic₁₀₆	—	2	4	no	—
289	Dic₁₀₆	—	4	2	no	—
290	Dic₁₀₆	—	4	2	no	—
291	C₂²×C₅₃	—	2	2	no	—
292	C₂²×C₅₃	—	2	2	no	—
293	C₂²×C₅₃	—	2	1	no	—
294	C₂²×C₅₃	—	2	2	no	—
295	C₂²×C₅₃	—	2	2	no	—
296	C₂²×C₅₃	—	2	1	no	—
297	C₂²×C₅₃	—	2	2	no	—
298	C₂²×C₅₃	—	2	1	no	—
299	C₂²×C₅₃	—	8	4	no	—
300	C₂²×C₅₃	C₂×#41	1	4	yes	—
301	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
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	2	no	—
307	C₂×D₅₃	—	2	1	no	—
308	C₂×D₅₃	—	2	1	no	—
309	C₂×D₅₃	—	2	2	no	—
310	C₂×D₅₃	—	2	1	no	—
311	C₂×D₅₃	—	8	4	no	—
312	C₄×C₅₃	—	2	1	no	—
313	C₄×C₅₃	—	2	1	no	—
314	C₄×C₅₃	—	2	2	no	—
315	C₄×C₅₃	—	2	1	no	—
316	C₄×C₅₃	—	2	2	no	—
317	C₄×C₅₃	—	2	1	no	—
318	C₄×C₅₃	—	2	1	no	—
319	C₄×C₅₃	C₂×#41	1	2	yes	—
320	Dic₅₃	—	2	2	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	Dic₅₃	—	2	1	no	—
327	Dic₅₃	—	2	1	no	—
328	Dic₅₃	H(5,𝔽₂)	1	2	yes	—
329	#3514	—	53	1	no	maximal, Sylow
330	C₂×C₅₃	—	2	4	no	—
331	C₂×C₅₃	—	4	2	no	—
332	C₂×C₅₃	—	4	1	no	—
333	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
334	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
335	C₂×C₅₃	#585	1	1	yes	characteristic
336	D₅₃	—	4	1	no	—
337	D₅₃	—	4	2	no	—
338	D₅₃	—	4	1	no	—
339	#559	—	53	1	no	—
340	#556	—	53	1	no	—
341	#555	—	53	1	no	—
342	#555	—	53	1	no	—
343	#551	—	53	2	no	—
344	#551	—	53	2	no	—
345	D₄×Q₈	—	53	1	no	—
346	#548	—	53	2	no	—
347	#546	—	53	1	no	—
348	#546	—	53	1	no	—
349	#546	—	53	2	no	—
350	D₄²	—	53	1	no	—
351	#542	—	53	1	no	—
352	#542	—	53	1	no	—
353	#538	—	53	2	no	—
354	#538	—	53	2	no	—
355	#532	—	53	1	no	—
356	#529	—	53	2	no	—
357	#529	—	53	1	no	—
358	C₂×#124	—	53	2	no	—
359	#518	—	53	1	no	—
360	#518	D₅₃	1	1	yes	characteristic, complemented
361	C₄×#41	—	53	1	no	—
362	C₅₃	#3514	1	1	yes	characteristic, complemented, Sylow
363	C₂×#41	—	53	1	no	—
364	C₂×#41	—	53	2	no	—
365	C₂²×Q₈	—	53	2	no	—
366	C₂²×D₄	—	53	2	no	—
367	C₂²×D₄	—	53	1	no	—
368	C₂²×D₄	C₂×D₅₃	1	1	yes	characteristic
369	C₄⋊Q₈	—	53	1	no	—
370	C₄⋊Q₈	—	53	1	no	—
371	C₄⋊Q₈	—	53	1	no	—
372	#127	—	53	1	no	—
373	#126	—	53	2	no	—
374	#126	—	53	1	no	—
375	#126	—	53	2	no	—
376	#126	—	53	1	no	—
377	#126	—	53	2	no	—
378	#126	—	53	2	no	—
379	#125	—	53	1	no	—
380	#124	—	53	1	no	—
381	#124	—	53	1	no	—
382	#124	—	53	2	no	—
383	#124	—	53	2	no	—
384	#124	—	53	2	no	—
385	#124	—	53	2	no	—
386	#124	—	53	2	no	—
387	#124	—	53	2	no	—
388	#124	—	53	2	no	—
389	#124	—	53	2	no	—
390	#124	—	53	2	no	—
391	#124	—	53	1	no	—
392	#123	—	53	2	no	—
393	#123	—	53	2	no	—
394	#123	—	53	2	no	—
395	#123	—	53	1	no	—
396	#123	—	53	1	no	—
397	#123	—	53	2	no	—
398	#123	—	53	2	no	—
399	#123	—	53	2	no	—
400	C₂²⋊Q₈	—	53	2	no	—
401	C₂²⋊Q₈	—	53	1	no	—
402	C₂²⋊Q₈	—	53	2	no	—
403	C₂²⋊Q₈	—	53	1	no	—
404	C₂²⋊Q₈	—	53	1	no	—
405	C₂²⋊Q₈	—	53	2	no	—
406	C₂²⋊Q₈	—	53	2	no	—
407	C₂²⋊Q₈	—	53	2	no	—
408	#121	—	53	2	no	—
409	#121	—	53	1	no	—
410	#121	—	53	2	no	—
411	#121	—	53	1	no	—
412	#121	—	53	2	no	—
413	#121	—	53	2	no	—
414	#121	—	53	1	no	—
415	#121	—	53	1	no	—
416	#121	—	53	1	no	—
417	#121	—	53	2	no	—
418	#121	—	53	2	no	—
419	#120	—	53	2	no	—
420	#120	—	53	2	no	—
421	#120	—	53	2	no	—
422	#120	—	53	2	no	—
423	C₄×Q₈	—	53	1	no	—
424	C₄×Q₈	—	53	1	no	—
425	C₄×Q₈	—	53	2	no	—
426	C₄×Q₈	—	53	1	no	—
427	C₄×D₄	—	53	1	no	—
428	C₄×D₄	—	53	1	no	—
429	C₄×D₄	—	53	2	no	—
430	C₄×D₄	—	53	2	no	—
431	C₄×D₄	—	53	1	no	—
432	C₄×D₄	—	53	1	no	—
433	C₄×D₄	—	53	2	no	—
434	C₄×D₄	—	53	2	no	—
435	C₄×D₄	—	53	1	no	—
436	C₄×D₄	—	53	2	no	—
437	C₄×D₄	—	53	2	no	—
438	C₄×D₄	—	53	1	no	—
439	C₄×D₄	—	53	1	no	—
440	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
441	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
442	C₄×D₄	C₂×D₅₃	1	2	yes	complemented
443	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
444	C₄×D₄	C₂×D₅₃	1	2	yes	complemented
445	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
446	#117	—	53	2	no	—
447	#117	—	53	1	no	—
448	#117	—	53	1	no	—
449	#117	—	53	2	no	—
450	#117	—	53	1	no	—
451	#117	—	53	2	no	—
452	#117	C₂×D₅₃	1	1	yes	characteristic, complemented
453	#117	C₂×D₅₃	1	1	yes	characteristic, complemented
454	C₂×C₂²⋊C₄	—	53	2	no	—
455	C₂×C₂²⋊C₄	—	53	2	no	—
456	C₂×C₂²⋊C₄	—	53	2	no	—
457	C₂×C₂²⋊C₄	—	53	2	no	—
458	C₂×C₂²⋊C₄	C₂×D₅₃	1	2	yes	complemented
459	C₂×C₂²⋊C₄	C₂×D₅₃	1	2	yes	complemented
460	C₂×C₄²	—	53	2	no	—
461	C₂×C₄²	—	53	1	no	—
462	C₂⁴	—	53	2	no	—
463	C₂⁴	C₂²×D₅₃	1	2	yes	—
464	#41	—	106	4	no	—
465	#41	—	212	4	no	—
466	C₂×Q₈	—	53	1	no	—
467	C₂×Q₈	—	53	1	no	—
468	C₂×Q₈	—	53	2	no	—
469	C₂×Q₈	—	53	2	no	—
470	C₂×Q₈	—	53	2	no	—
471	C₂×Q₈	—	212	4	no	—
472	C₂×D₄	—	2	4	no	—
473	C₂×D₄	—	53	2	no	—
474	C₂×D₄	—	53	2	no	—
475	C₂×D₄	—	53	1	no	—
476	C₂×D₄	—	53	2	no	—
477	C₂×D₄	—	53	1	no	—
478	C₂×D₄	—	53	2	no	—
479	C₂×D₄	—	53	2	no	—
480	C₂×D₄	—	53	2	no	—
481	C₂×D₄	—	53	2	no	—
482	C₂×D₄	—	53	1	no	—
483	C₂×D₄	—	53	1	no	—
484	C₂×D₄	—	53	2	no	—
485	C₂×D₄	—	106	4	no	—
486	C₂×D₄	—	212	4	no	—
487	C₂×D₄	C₂²×D₅₃	1	2	yes	—
488	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
489	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
490	C₂²×C₄	—	53	2	no	—
491	C₂²×C₄	—	53	2	no	—
492	C₂²×C₄	—	53	2	no	—
493	C₂²×C₄	—	53	1	no	—
494	C₂²×C₄	—	53	1	no	—
495	C₂²×C₄	—	53	1	no	—
496	C₂²×C₄	—	53	1	no	—
497	C₂²×C₄	—	53	2	no	—
498	C₂²×C₄	—	53	2	no	—
499	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
500	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
501	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
502	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
503	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
504	C₄⋊C₄	—	53	1	no	—
505	C₄⋊C₄	—	53	1	no	—
506	C₄⋊C₄	—	53	1	no	—
507	C₄⋊C₄	—	53	1	no	—
508	C₄⋊C₄	—	53	1	no	—
509	C₄⋊C₄	—	53	1	no	—
510	C₄⋊C₄	—	53	2	no	—
511	C₄⋊C₄	—	53	2	no	—
512	C₄⋊C₄	—	53	2	no	—
513	C₄⋊C₄	—	53	2	no	—
514	C₄⋊C₄	—	53	1	no	—
515	C₄⋊C₄	—	53	1	no	—
516	C₄⋊C₄	—	53	2	no	—
517	C₄⋊C₄	—	53	2	no	—
518	C₄⋊C₄	—	53	2	no	—
519	C₄⋊C₄	—	53	1	no	—
520	C₄⋊C₄	—	53	1	no	—
521	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
522	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
523	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
524	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
533	C₂²⋊C₄	—	53	2	no	—
534	C₂²⋊C₄	—	53	1	no	—
535	C₂²⋊C₄	—	53	2	no	—
536	C₂²⋊C₄	—	53	1	no	—
537	C₂²⋊C₄	—	53	1	no	—
538	C₂²⋊C₄	—	53	2	no	—
539	C₂²⋊C₄	—	53	2	no	—
540	C₂²⋊C₄	—	53	2	no	—
541	C₂²⋊C₄	—	53	2	no	—
542	C₂²⋊C₄	—	53	2	no	—
543	C₂²⋊C₄	—	53	2	no	—
544	C₂²⋊C₄	—	53	2	no	—
545	C₂²⋊C₄	—	53	1	no	—
546	C₂²⋊C₄	—	53	2	no	—
547	C₂²⋊C₄	—	53	1	no	—
548	C₂²⋊C₄	—	53	1	no	—
549	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
550	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
551	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
552	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
553	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
554	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
555	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
556	C₄²	—	53	1	no	—
557	C₄²	—	53	2	no	—
558	C₄²	—	53	1	no	—
559	C₄²	—	53	2	no	—
560	C₄²	—	53	1	no	—
561	C₄²	—	53	2	no	—
562	C₄²	—	53	2	no	—
563	C₄²	—	53	1	no	—
564	C₄²	C₂²×D₅₃	1	1	yes	characteristic
565	C₄²	C₂²×D₅₃	1	2	yes	—
566	C₄²	C₂²×D₅₃	1	1	yes	characteristic
567	C₂³	—	2	4	no	—
568	C₂³	—	4	2	no	—
569	C₂³	—	4	2	no	—
570	C₂³	—	53	1	no	—
571	C₂³	—	53	2	no	—
572	C₂³	—	53	1	no	—
573	C₂³	—	106	4	no	—
574	C₂³	—	212	2	no	—
575	C₂³	—	212	2	no	—
576	C₂³	C₂³×D₅₃	1	1	yes	characteristic
577	C₂³	C₂³×D₅₃	1	2	yes	—
578	C₂³	C₂³×D₅₃	1	2	yes	—
579	Q₈	—	106	2	no	—
580	Q₈	—	106	4	no	—
581	Q₈	—	106	2	no	—
582	Q₈	—	212	2	no	—
583	Q₈	—	212	2	no	—
584	D₄	—	2	4	no	—
585	D₄	—	2	2	no	—
586	D₄	—	106	2	no	—
587	D₄	—	106	4	no	—
588	D₄	—	106	4	no	—
589	D₄	—	106	4	no	—
590	D₄	—	106	2	no	—
591	D₄	—	212	1	no	—
592	D₄	—	212	2	no	—
593	D₄	—	212	1	no	—
594	D₄	—	212	2	no	—
595	D₄	—	212	2	no	—
596	D₄	—	212	2	no	—
597	D₄	—	212	2	no	—
598	D₄	#11719945	1	4	yes	complemented
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	4	2	no	—
601	C₂×C₄	—	4	2	no	—
602	C₂×C₄	—	4	2	no	—
603	C₂×C₄	—	4	2	no	—
604	C₂×C₄	—	53	1	no	—
605	C₂×C₄	—	53	1	no	—
606	C₂×C₄	—	53	2	no	—
607	C₂×C₄	—	53	1	no	—
608	C₂×C₄	—	53	1	no	—
609	C₂×C₄	—	53	2	no	—
610	C₂×C₄	—	53	1	no	—
611	C₂×C₄	—	53	1	no	—
612	C₂×C₄	—	53	2	no	—
613	C₂×C₄	—	106	2	no	—
614	C₂×C₄	—	106	4	no	—
615	C₂×C₄	—	106	4	no	—
616	C₂×C₄	—	106	2	no	—
617	C₂×C₄	—	106	2	no	—
618	C₂×C₄	—	212	2	no	—
619	C₂×C₄	—	212	1	no	—
620	C₂×C₄	—	212	2	no	—
621	C₂×C₄	—	212	2	no	—
622	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
623	C₂×C₄	C₂³×D₅₃	1	2	yes	—
624	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
625	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
626	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
627	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
628	C₂×C₄	C₂³×D₅₃	1	2	yes	—
629	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
630	C₂²	—	2	2	no	—
631	C₂²	—	2	2	no	—
632	C₂²	—	2	1	no	—
633	C₂²	—	2	2	no	—
634	C₂²	—	2	2	no	—
635	C₂²	—	2	1	no	—
636	C₂²	—	2	2	no	—
637	C₂²	—	2	1	no	—
638	C₂²	—	8	4	no	—
639	C₂²	—	106	1	no	—
640	C₂²	—	106	1	no	—
641	C₂²	—	106	2	no	—
642	C₂²	—	106	1	no	—
643	C₂²	—	106	2	no	—
644	C₂²	—	106	1	no	—
645	C₂²	—	106	1	no	—
646	C₂²	—	106	2	no	—
647	C₂²	—	106	1	no	—
648	C₂²	—	424	4	no	—
649	C₂²	C₂×#11719945	1	4	yes	—
650	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
651	C₄	—	2	1	no	—
652	C₄	—	2	1	no	—
653	C₄	—	2	2	no	—
654	C₄	—	2	1	no	—
655	C₄	—	2	1	no	—
656	C₄	—	2	2	no	—
657	C₄	—	2	1	no	—
658	C₄	—	53	2	no	—
659	C₄	—	106	1	no	—
660	C₄	—	106	1	no	—
661	C₄	—	106	2	no	—
662	C₄	—	106	2	no	—
663	C₄	—	106	2	no	—
664	C₄	—	106	1	no	—
665	C₄	—	106	1	no	—
666	C₄	—	106	1	no	—
667	C₄	C₂×#11719945	1	2	yes	—
668	C₂	—	2	4	no	—
669	C₂	—	4	1	no	—
670	C₂	—	4	2	no	—
671	C₂	—	212	1	no	—
672	C₂	—	212	2	no	—
673	C₂	—	212	1	no	—
674	C₂	C₂²×#11719945	1	1	yes	characteristic
675	C₂	H(5,𝔽₂)×D₅₃	1	1	yes	characteristic
676	C₂	#16811458	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#518	C₂
hypercenter	#518	D₅₃
hypocenter	C₅₃	#3514
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #518

Complements

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

N	H ≅ G/N	type
1. #16811356	C₂	semidirect
2. #16811341	C₂	semidirect
3. #16811338	C₂	semidirect
4. #16811336	C₂	semidirect
5. #16811302	C₂	semidirect
6. #16811299	C₂	semidirect
7. D₄×Dic₁₀₆	C₂	semidirect
8. #16811282	C₂	semidirect
9. #16811268	C₂	semidirect
10. #16811267	C₂	semidirect
11. #16811263	C₂	semidirect
12. D₄×C₂²⋊D₅₃	C₂	semidirect
13. #16811244	C₂	semidirect
14. #16811238	C₂	semidirect
15. #16811211	C₂	semidirect
16. #16811206	C₂	semidirect
17. #16811170	C₂	semidirect
18. #16811153	C₂	semidirect
19. #16811151	C₂	semidirect
20. C₂×#13133203	C₂	semidirect
21. #16811095	C₂	semidirect
22. C₄×#11719947	C₂	semidirect
23. C₅₃×#518	C₂	semidirect
24. C₂×#11719945	C₂²	semidirect
26. C₂²×Dic₁₀₆	C₂²	semidirect
29. #13133217	C₂²	semidirect
30. #13133216	C₂²	semidirect
31. #13133216	C₂²	semidirect
33. #13133212	C₂²	semidirect
34. #13133212	C₂²	semidirect
35. #13133211	C₂²	semidirect
36. #13133210	C₂²	semidirect
37. #13133210	C₂²	semidirect
38. #13133210	C₂²	semidirect
39. #13133207	C₂²	semidirect
40. #13133205	C₂²	semidirect
41. #13133205	C₂²	semidirect
42. #13133204	C₂²	semidirect
45. #13133203	C₂²	semidirect
46. #13133203	C₂²	semidirect
47. #13133203	C₂²	semidirect
48. #13133203	C₂²	semidirect
49. #13133203	C₂²	semidirect
50. #13133203	C₂²	semidirect
51. #13133203	C₂²	semidirect
52. #13133202	C₂²	semidirect
53. #13133201	C₂²	semidirect
54. #13133200	C₂²	semidirect
55. #13133200	C₂²	semidirect
56. #13133199	C₂²	semidirect
57. #13133199	C₂²	semidirect
58. #13133199	C₂²	semidirect
59. #13133198	C₂²	semidirect
60. #13133197	C₂²	semidirect
61. #13133196	C₂²	semidirect
62. #13133196	C₂²	semidirect
63. #13133196	C₂²	semidirect
64. #13133195	C₂²	semidirect
65. #13133195	C₂²	semidirect
66. #13133194	C₂²	semidirect
67. #13133194	C₂²	semidirect
68. #13133192	C₂²	semidirect
70. #13133191	C₂²	semidirect
71. #13133191	C₂²	semidirect
73. #13133190	C₂²	semidirect
74. #13133190	C₂²	semidirect
76. #13133189	C₂²	semidirect
78. #13133188	C₂²	semidirect
79. #13133187	C₂²	semidirect
81. #13133186	C₂²	semidirect
83. C₄×Dic₁₀₆	C₂²	semidirect
84. C₄×Dic₁₀₆	C₂²	semidirect
85. C₄×Dic₁₀₆	C₂²	semidirect
86. #13133182	C₂²	semidirect
87. C₄×D₂₁₂	C₂²	semidirect
88. C₄×D₂₁₂	C₂²	semidirect
91. #13133179	C₂²	semidirect
93. C₄×C₂²⋊D₅₃	C₂²	semidirect
94. C₄×C₂²⋊D₅₃	C₂²	semidirect
95. C₄×C₂²⋊D₅₃	C₂²	semidirect
96. C₄×C₂²⋊D₅₃	C₂²	semidirect
97. C₄×C₂²⋊D₅₃	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
104. #13133174	C₂²	semidirect
105. #13133173	C₂²	semidirect
106. C₂×C₂²⋊Dic₅₃	C₂²	semidirect
107. C₂×#11719921	C₂²	semidirect
108. C₂×#11719921	C₂²	semidirect
109. C₂²⋊C₄×D₅₃	C₂²	semidirect
110. C₂×C₄×Dic₅₃	C₂²	semidirect
111. C₄²×D₅₃	C₂²	semidirect
115. C₄×D₄×C₅₃	C₂²	semidirect
116. C₄×D₄×C₅₃	C₂²	semidirect
117. C₄×D₄×C₅₃	C₂²	semidirect
118. C₄×D₄×C₅₃	C₂²	semidirect
119. #117×C₅₃	C₂²	semidirect
120. #117×C₅₃	C₂²	semidirect
121. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
122. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
242. D₄×C₅₃	#41	semidirect
360. #518	D₅₃	semidirect
362. C₅₃	#3514	semidirect
442. C₄×D₄	C₂×D₅₃	semidirect
443. C₄×D₄	C₂×D₅₃	semidirect
444. C₄×D₄	C₂×D₅₃	semidirect
445. C₄×D₄	C₂×D₅₃	semidirect
452. #117	C₂×D₅₃	semidirect
453. #117	C₂×D₅₃	semidirect
458. C₂×C₂²⋊C₄	C₂×D₅₃	semidirect
459. C₂×C₂²⋊C₄	C₂×D₅₃	semidirect
598. D₄	#11719945	semidirect