Subgroups — #20288786

Order 6784.

1042 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811347	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811346	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811326	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811321	C₂	1	2	yes	maximal, complemented
5	#16811315	C₂	1	2	yes	maximal, complemented
6	#16811311	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811300	C₂	1	2	yes	maximal, complemented
8	#16811300	C₂	1	2	yes	maximal, complemented
9	#16811277	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811268	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811268	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811247	C₂	1	1	yes	maximal, characteristic, complemented
13	#16811236	C₂	1	1	yes	maximal, characteristic, complemented
14	#16811222	C₂	1	2	yes	maximal, complemented
15	#16811213	C₂	1	2	yes	maximal, complemented
16	#16811160	C₂	1	1	yes	maximal, characteristic, complemented
17	#16811159	C₂	1	2	yes	maximal, complemented
18	#16811158	C₂	1	1	yes	maximal, characteristic, complemented
19	C₂×#13133212	C₂	1	2	yes	maximal, complemented
20	#16811102	C₂	1	1	yes	maximal, characteristic, complemented
21	#16811095	C₂	1	1	yes	maximal, characteristic, complemented
22	C₄×#11719947	C₂	1	1	yes	maximal, characteristic, complemented
23	C₅₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#11719947	C₂²	1	1	yes	characteristic
25	C₂×#11719947	C₂²	1	1	yes	characteristic
26	C₂×D₄×D₅₃	C₂²	1	1	yes	characteristic
27	#13133217	C₂²	1	1	yes	characteristic, complemented
28	#13133212	C₂²	1	2	yes	—
29	#13133212	C₂²	1	2	yes	—
30	#13133212	C₂²	1	2	yes	—
31	#13133212	C₂²	1	1	yes	characteristic
32	#13133212	C₂²	1	2	yes	—
33	#13133212	C₂²	1	2	yes	complemented
34	#13133212	C₂²	1	2	yes	complemented
35	#13133212	C₂²	1	2	yes	complemented
36	#13133212	C₂²	1	2	yes	complemented
37	#13133212	C₂²	1	2	yes	complemented
38	#13133212	C₂²	1	2	yes	complemented
39	#13133212	C₂²	1	1	yes	characteristic, complemented
40	#13133208	C₂²	1	2	yes	complemented
41	#13133208	C₂²	1	1	yes	characteristic, complemented
42	#13133208	C₂²	1	2	yes	complemented
43	#13133208	C₂²	1	2	yes	complemented
44	#13133206	C₂²	1	1	yes	characteristic
45	#13133206	C₂²	1	2	yes	—
46	#13133206	C₂²	1	2	yes	complemented
47	#13133205	C₂²	1	1	yes	characteristic
48	#13133205	C₂²	1	1	yes	characteristic
49	#13133202	C₂²	1	1	yes	characteristic
50	#13133202	C₂²	1	2	yes	—
51	#13133202	C₂²	1	2	yes	complemented
52	#13133202	C₂²	1	2	yes	complemented
53	#13133202	C₂²	1	1	yes	characteristic, complemented
54	#13133202	C₂²	1	2	yes	complemented
55	#13133201	C₂²	1	2	yes	—
56	#13133201	C₂²	1	2	yes	—
57	#13133201	C₂²	1	2	yes	complemented
58	#13133201	C₂²	1	2	yes	complemented
59	#13133197	C₂²	1	1	yes	characteristic
60	#13133197	C₂²	1	1	yes	characteristic
61	#13133197	C₂²	1	1	yes	characteristic, complemented
62	#13133197	C₂²	1	1	yes	characteristic, complemented
63	#13133193	C₂²	1	2	yes	complemented
64	#13133193	C₂²	1	2	yes	complemented
65	#13133193	C₂²	1	1	yes	characteristic, complemented
66	#13133192	C₂²	1	1	yes	characteristic
67	#13133192	C₂²	1	1	yes	characteristic
68	#13133192	C₂²	1	1	yes	characteristic
69	#13133192	C₂²	1	2	yes	complemented
70	#13133191	C₂²	1	2	yes	—
71	#13133191	C₂²	1	2	yes	—
72	#13133191	C₂²	1	2	yes	complemented
73	#13133187	C₂²	1	2	yes	—
74	#13133187	C₂²	1	2	yes	—
75	Q₈×Dic₅₃	C₂²	1	2	yes	complemented
76	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
77	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
78	#13133182	C₂²	1	1	yes	characteristic, complemented
79	#13133182	C₂²	1	1	yes	characteristic, complemented
80	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
81	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
82	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
83	C₄×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
84	D₄×Dic₅₃	C₂²	1	1	yes	characteristic
85	#13133179	C₂²	1	1	yes	characteristic
86	#13133179	C₂²	1	2	yes	complemented
87	#13133179	C₂²	1	2	yes	complemented
88	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
89	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
90	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
91	#13133177	C₂²	1	2	yes	—
92	#13133177	C₂²	1	2	yes	complemented
93	#13133177	C₂²	1	2	yes	complemented
94	#13133175	C₂²	1	1	yes	characteristic
95	#13133175	C₂²	1	2	yes	complemented
96	#13133175	C₂²	1	2	yes	complemented
97	#13133175	C₂²	1	2	yes	complemented
98	#13133175	C₂²	1	1	yes	characteristic, complemented
99	#13133174	C₂²	1	1	yes	characteristic
100	#13133174	C₂²	1	1	yes	characteristic, complemented
101	#13133173	C₂²	1	2	yes	complemented
102	#13133173	C₂²	1	2	yes	complemented
103	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	complemented
104	C₂×#11719919	C₂²	1	2	yes	complemented
105	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic, complemented
106	C₂×#11719921	C₂²	1	2	yes	—
107	C₂×#11719921	C₂²	1	2	yes	—
108	C₂×#11719921	C₂²	1	2	yes	—
109	C₂²⋊C₄×D₅₃	C₂²	1	2	yes	—
110	C₂×C₄×Dic₅₃	C₂²	1	2	yes	complemented
111	C₄²×D₅₃	C₂²	1	1	yes	characteristic, complemented
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	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
132	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
133	C₂²×Dic₅₃	C₂³	1	2	yes	—
134	C₂²×Dic₅₃	C₂³	1	2	yes	—
135	C₂²×Dic₅₃	C₂³	1	2	yes	—
136	#11719947	—	2	4	no	—
137	#11719947	—	2	4	no	—
138	D₄×D₅₃	—	2	4	no	—
139	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
140	C₂×D₂₁₂	C₂³	1	2	yes	—
141	C₂×D₂₁₂	C₂³	1	2	yes	—
142	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
143	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
144	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
145	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
146	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
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	2	yes	—
151	C₂×C₄×D₅₃	C₂³	1	2	yes	—
152	C₂×C₄×D₅₃	C₂³	1	2	yes	—
153	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
154	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
155	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
156	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
157	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
158	C₄⋊C₄×C₅₃	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	1	yes	characteristic
164	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
165	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
166	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
167	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
168	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
169	C₄²×C₅₃	C₂³	1	1	yes	characteristic
170	C₄²×C₅₃	C₂³	1	1	yes	characteristic
171	C₄²×C₅₃	C₂³	1	1	yes	characteristic
172	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
173	#11719921	C₂³	1	1	yes	characteristic
174	#11719921	C₂³	1	2	yes	—
175	#11719921	C₂³	1	1	yes	characteristic
176	#11719921	C₂³	1	1	yes	characteristic
177	#11719921	C₂³	1	1	yes	characteristic
178	#11719921	C₂³	1	2	yes	—
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	2	yes	—
184	#11719921	C₂³	1	2	yes	—
185	#11719921	C₂³	1	2	yes	—
186	#11719921	C₂³	1	1	yes	characteristic
187	#11719921	C₂³	1	1	yes	characteristic
188	#11719921	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
194	#11719921	C₂³	1	2	yes	—
195	#11719921	C₂³	1	1	yes	characteristic
196	#11719921	C₂³	1	1	yes	characteristic
197	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
198	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
199	C₄⋊Dic₅₃	C₂³	1	2	yes	—
200	C₄⋊Dic₅₃	C₂³	1	2	yes	—
201	C₄⋊Dic₅₃	C₂³	1	2	yes	—
202	C₄⋊Dic₅₃	C₂³	1	2	yes	—
203	C₄⋊Dic₅₃	C₂³	1	2	yes	—
204	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
205	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
206	C₄⋊Dic₅₃	C₂³	1	2	yes	—
207	#11719919	C₂³	1	2	yes	—
208	#11719919	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	2	yes	—
213	#11719919	C₂³	1	2	yes	—
214	#11719919	C₂³	1	1	yes	characteristic
215	#11719919	C₂³	1	1	yes	characteristic
216	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
217	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
218	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
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	2	yes	—
223	C₄×Dic₅₃	C₂³	1	2	yes	—
224	C₄×Dic₅₃	C₂³	1	2	yes	—
225	C₂³×C₅₃	C₂⁴	1	2	yes	—
226	C₂²×D₅₃	—	2	4	no	—
227	C₂²×D₅₃	—	4	2	no	—
228	C₂²×D₅₃	—	4	2	no	—
229	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
230	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
231	C₂²×D₅₃	C₂⁴	1	2	yes	—
232	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
233	Q₈×C₅₃	—	4	1	no	—
234	Q₈×C₅₃	—	4	2	no	—
235	D₄×C₅₃	#41	1	4	yes	—
236	C₂×C₄×C₅₃	—	4	2	no	—
237	C₂×C₄×C₅₃	—	4	2	no	—
238	C₂×C₄×C₅₃	—	4	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	1	yes	characteristic
242	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
243	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
244	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
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	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
249	C₂²⋊D₅₃	—	2	4	no	—
250	C₂²⋊D₅₃	—	2	4	no	—
251	C₂²⋊D₅₃	—	2	4	no	—
252	C₂×Dic₅₃	—	2	4	no	—
253	C₂×Dic₅₃	—	2	4	no	—
254	C₂×Dic₅₃	—	4	2	no	—
255	C₂×Dic₅₃	C₂⁴	1	2	yes	—
256	C₂×Dic₅₃	C₂⁴	1	2	yes	—
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	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
263	D₂₁₂	—	2	2	no	—
264	D₂₁₂	—	4	2	no	—
265	D₂₁₂	—	4	1	no	—
266	D₂₁₂	—	4	1	no	—
267	D₂₁₂	—	4	2	no	—
268	D₂₁₂	—	4	1	no	—
269	C₄×D₅₃	—	2	2	no	—
270	C₄×D₅₃	—	2	2	no	—
271	C₄×D₅₃	—	2	2	no	—
272	C₄×D₅₃	—	4	1	no	—
273	C₄×D₅₃	—	4	2	no	—
274	C₄×D₅₃	—	4	2	no	—
275	C₄×D₅₃	—	4	2	no	—
276	C₄×D₅₃	—	4	1	no	—
277	Dic₁₀₆	—	2	2	no	—
278	Dic₁₀₆	—	2	2	no	—
279	C₂²×C₅₃	—	2	2	no	—
280	C₂²×C₅₃	—	2	2	no	—
281	C₂²×C₅₃	C₂×#41	1	4	yes	—
282	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
283	C₂×D₅₃	—	2	1	no	—
284	C₂×D₅₃	—	2	1	no	—
285	C₂×D₅₃	—	2	1	no	—
286	C₂×D₅₃	—	2	2	no	—
287	C₂×D₅₃	—	2	1	no	—
288	C₂×D₅₃	—	2	1	no	—
289	C₂×D₅₃	—	2	1	no	—
290	C₂×D₅₃	—	2	2	no	—
291	C₂×D₅₃	—	2	2	no	—
292	C₂×D₅₃	—	2	1	no	—
293	C₂×D₅₃	—	2	1	no	—
294	C₂×D₅₃	—	2	1	no	—
295	C₂×D₅₃	—	8	4	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	2	no	—
301	C₄×C₅₃	—	2	1	no	—
302	C₄×C₅₃	—	2	2	no	—
303	C₄×C₅₃	—	2	1	no	—
304	C₄×C₅₃	—	2	1	no	—
305	C₄×C₅₃	C₂×#41	1	2	yes	—
306	Dic₅₃	—	2	2	no	—
307	Dic₅₃	—	2	2	no	—
308	Dic₅₃	—	2	1	no	—
309	Dic₅₃	—	2	1	no	—
310	Dic₅₃	—	2	1	no	—
311	Dic₅₃	—	2	1	no	—
312	Dic₅₃	—	2	2	no	—
313	Dic₅₃	—	2	1	no	—
314	#3521	—	53	1	no	maximal, Sylow
315	C₂×C₅₃	—	2	4	no	—
316	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
317	C₂×C₅₃	#585	1	1	yes	characteristic
318	C₂×C₅₃	#585	1	1	yes	characteristic
319	D₅₃	—	4	1	no	—
320	D₅₃	—	4	1	no	—
321	D₅₃	—	4	2	no	—
322	D₅₃	—	4	1	no	—
323	#556	—	53	1	no	—
324	#556	—	53	1	no	—
325	#554	D₅₃	1	1	yes	characteristic, complemented
326	#553	—	53	1	no	—
327	#553	—	53	2	no	—
328	#552	—	53	1	no	—
329	#552	—	53	2	no	—
330	#551	—	53	2	no	—
331	#551	—	53	2	no	—
332	#547	—	53	1	no	—
333	#546	—	53	1	no	—
334	#546	—	53	1	no	—
335	#542	—	53	1	no	—
336	#541	—	53	1	no	—
337	#539	—	53	2	no	—
338	#538	—	53	2	no	—
339	#529	—	53	1	no	—
340	#529	—	53	2	no	—
341	#529	—	53	1	no	—
342	C₂×#126	—	53	2	no	—
343	#520	—	53	1	no	—
344	#518	—	53	1	no	—
345	C₄×#41	—	53	1	no	—
346	C₅₃	#3521	1	1	yes	characteristic, complemented, Sylow
347	C₂×#41	—	53	1	no	—
348	C₂×#41	—	53	1	no	—
349	C₂²×D₄	—	53	1	no	—
350	C₄⋊Q₈	—	53	1	no	—
351	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
352	#126	—	53	1	no	—
353	#126	—	53	2	no	—
354	#126	—	53	2	no	—
355	#126	—	53	2	no	—
356	#126	—	53	2	no	—
357	#126	—	53	2	no	—
358	#126	—	53	2	no	—
359	#126	—	53	2	no	—
360	#126	—	53	1	no	—
361	#126	—	53	2	no	—
362	#126	—	53	2	no	—
363	#126	—	53	2	no	—
364	#125	—	53	2	no	—
365	#125	—	53	1	no	—
366	#125	—	53	2	no	—
367	#125	—	53	2	no	—
368	#125	C₂×D₅₃	1	1	yes	characteristic, complemented
369	#125	C₂×D₅₃	1	1	yes	characteristic, complemented
370	#124	—	53	2	no	—
371	#124	—	53	1	no	—
372	#124	—	53	2	no	—
373	#124	—	53	1	no	—
374	#124	—	53	1	no	—
375	#123	—	53	2	no	—
376	#123	—	53	1	no	—
377	#123	—	53	1	no	—
378	#123	—	53	2	no	—
379	#123	—	53	2	no	—
380	#123	—	53	2	no	—
381	#123	—	53	2	no	—
382	#123	—	53	2	no	—
383	#123	—	53	2	no	—
384	#123	—	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	1	no	—
389	C₂²⋊Q₈	—	53	1	no	—
390	C₂²⋊Q₈	—	53	1	no	—
391	C₂²⋊Q₈	—	53	1	no	—
392	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	complemented
393	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	complemented
394	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	complemented
395	#121	—	53	2	no	—
396	#121	—	53	2	no	—
397	#121	—	53	2	no	—
398	#121	—	53	1	no	—
399	#121	—	53	1	no	—
400	#121	—	53	2	no	—
401	#121	—	53	1	no	—
402	#120	—	53	2	no	—
403	#120	—	53	2	no	—
404	C₄×Q₈	—	53	2	no	—
405	C₄×Q₈	—	53	1	no	—
406	C₄×Q₈	—	53	1	no	—
407	C₄×Q₈	—	53	1	no	—
408	C₄×Q₈	—	53	1	no	—
409	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
410	C₄×D₄	—	53	1	no	—
411	C₄×D₄	—	53	2	no	—
412	C₄×D₄	—	53	2	no	—
413	C₄×D₄	—	53	2	no	—
414	C₄×D₄	—	53	1	no	—
415	C₄×D₄	—	53	2	no	—
416	C₄×D₄	—	53	2	no	—
417	C₄×D₄	—	53	2	no	—
418	C₄×D₄	—	53	2	no	—
419	C₄×D₄	—	53	2	no	—
420	C₄×D₄	—	53	1	no	—
421	C₄×D₄	—	53	1	no	—
422	C₄×D₄	—	53	1	no	—
423	C₄×D₄	—	53	1	no	—
424	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
425	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
426	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
427	#117	—	53	1	no	—
428	#117	—	53	1	no	—
429	#117	—	53	2	no	—
430	#117	—	53	2	no	—
431	#117	—	53	2	no	—
432	#117	—	53	2	no	—
433	#117	—	53	2	no	—
434	#117	—	53	1	no	—
435	#117	—	53	1	no	—
436	C₂×C₄⋊C₄	—	53	2	no	—
437	C₂×C₄⋊C₄	—	53	2	no	—
438	C₂×C₄⋊C₄	—	53	1	no	—
439	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	complemented
440	C₂×C₂²⋊C₄	—	53	2	no	—
441	C₂×C₂²⋊C₄	—	53	2	no	—
442	C₂×C₂²⋊C₄	—	53	2	no	—
443	C₂×C₂²⋊C₄	—	53	2	no	—
444	C₂×C₄²	—	53	2	no	—
445	C₂×C₄²	—	53	1	no	—
446	C₂⁴	—	53	2	no	—
447	#41	—	106	4	no	—
448	#41	—	106	4	no	—
449	C₂×Q₈	—	53	1	no	—
450	C₂×Q₈	—	53	1	no	—
451	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
452	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
453	C₂×D₄	—	53	2	no	—
454	C₂×D₄	—	53	2	no	—
455	C₂×D₄	—	53	2	no	—
456	C₂×D₄	—	53	1	no	—
457	C₂×D₄	—	53	1	no	—
458	C₂×D₄	—	53	1	no	—
459	C₂×D₄	—	53	2	no	—
460	C₂×D₄	—	53	2	no	—
461	C₂×D₄	—	53	1	no	—
462	C₂×D₄	—	106	4	no	—
463	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
464	C₂²×C₄	—	53	2	no	—
465	C₂²×C₄	—	53	2	no	—
466	C₂²×C₄	—	53	2	no	—
467	C₂²×C₄	—	53	1	no	—
468	C₂²×C₄	—	53	1	no	—
469	C₂²×C₄	—	53	1	no	—
470	C₂²×C₄	—	53	1	no	—
471	C₂²×C₄	—	53	1	no	—
472	C₂²×C₄	—	53	2	no	—
473	C₂²×C₄	—	53	2	no	—
474	C₂²×C₄	—	53	2	no	—
475	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
476	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
477	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
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	2	no	—
484	C₄⋊C₄	—	53	2	no	—
485	C₄⋊C₄	—	53	2	no	—
486	C₄⋊C₄	—	53	2	no	—
487	C₄⋊C₄	—	53	1	no	—
488	C₄⋊C₄	—	53	1	no	—
489	C₄⋊C₄	—	53	1	no	—
490	C₄⋊C₄	—	53	2	no	—
491	C₄⋊C₄	—	53	2	no	—
492	C₄⋊C₄	—	53	2	no	—
493	C₄⋊C₄	—	53	2	no	—
494	C₄⋊C₄	—	53	1	no	—
495	C₄⋊C₄	—	53	1	no	—
496	C₄⋊C₄	—	53	2	no	—
497	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
498	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
499	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
500	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
501	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
502	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
503	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
504	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
505	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
506	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
507	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
508	C₂²⋊C₄	—	53	2	no	—
509	C₂²⋊C₄	—	53	1	no	—
510	C₂²⋊C₄	—	53	2	no	—
511	C₂²⋊C₄	—	53	1	no	—
512	C₂²⋊C₄	—	53	1	no	—
513	C₂²⋊C₄	—	53	1	no	—
514	C₂²⋊C₄	—	53	2	no	—
515	C₂²⋊C₄	—	53	2	no	—
516	C₂²⋊C₄	—	53	2	no	—
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	2	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	1	no	—
530	C₂²⋊C₄	—	53	1	no	—
531	C₂²⋊C₄	—	53	2	no	—
532	C₂²⋊C₄	—	53	1	no	—
533	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
534	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
535	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
536	C₄²	—	53	1	no	—
537	C₄²	—	53	1	no	—
538	C₄²	—	53	1	no	—
539	C₄²	—	53	2	no	—
540	C₄²	—	53	2	no	—
541	C₄²	—	53	2	no	—
542	C₄²	—	53	2	no	—
543	C₄²	—	53	2	no	—
544	C₄²	—	53	2	no	—
545	C₄²	C₂²×D₅₃	1	1	yes	characteristic
546	C₄²	C₂²×D₅₃	1	1	yes	characteristic
547	C₄²	C₂²×D₅₃	1	1	yes	characteristic
548	C₂³	—	53	1	no	—
549	C₂³	—	53	2	no	—
550	C₂³	—	53	1	no	—
551	C₂³	—	53	1	no	—
552	C₂³	—	106	4	no	—
553	C₂³	—	212	2	no	—
554	C₂³	—	212	2	no	—
555	C₂³	C₂³×D₅₃	1	2	yes	—
556	Q₈	—	4	2	no	—
557	Q₈	—	4	1	no	—
558	Q₈	—	106	2	no	—
559	Q₈	—	106	2	no	—
560	D₄	—	106	4	no	—
561	D₄	—	106	4	no	—
562	D₄	—	106	4	no	—
563	D₄	—	106	2	no	—
564	D₄	—	212	1	no	—
565	D₄	—	212	2	no	—
566	D₄	—	212	1	no	—
567	D₄	—	212	2	no	—
568	D₄	—	212	1	no	—
569	D₄	#11719949	1	4	yes	—
570	C₂×C₄	—	4	2	no	—
571	C₂×C₄	—	4	2	no	—
572	C₂×C₄	—	4	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	—	53	2	no	—
579	C₂×C₄	—	53	1	no	—
580	C₂×C₄	—	53	1	no	—
581	C₂×C₄	—	106	4	no	—
582	C₂×C₄	—	106	4	no	—
583	C₂×C₄	—	106	2	no	—
584	C₂×C₄	—	106	2	no	—
585	C₂×C₄	—	106	2	no	—
586	C₂×C₄	—	212	1	no	—
587	C₂×C₄	—	212	2	no	—
588	C₂×C₄	—	212	2	no	—
589	C₂×C₄	—	212	2	no	—
590	C₂×C₄	—	212	1	no	—
591	C₂×C₄	—	212	2	no	—
592	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
593	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
594	C₂×C₄	C₂³×D₅₃	1	2	yes	—
595	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
596	C₂×C₄	C₂³×D₅₃	1	2	yes	—
597	C₂×C₄	C₂³×D₅₃	1	2	yes	—
598	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
599	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
600	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
601	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
602	C₂²	—	2	2	no	—
603	C₂²	—	2	2	no	—
604	C₂²	—	106	1	no	—
605	C₂²	—	106	1	no	—
606	C₂²	—	106	1	no	—
607	C₂²	—	106	2	no	—
608	C₂²	—	106	1	no	—
609	C₂²	—	106	2	no	—
610	C₂²	—	106	1	no	—
611	C₂²	—	106	1	no	—
612	C₂²	—	106	1	no	—
613	C₂²	—	106	1	no	—
614	C₂²	—	106	2	no	—
615	C₂²	—	106	1	no	—
616	C₂²	—	424	4	no	—
617	C₂²	C₂×#11719949	1	4	yes	—
618	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
619	C₄	—	2	1	no	—
620	C₄	—	2	1	no	—
621	C₄	—	2	2	no	—
622	C₄	—	2	2	no	—
623	C₄	—	2	1	no	—
624	C₄	—	2	1	no	—
625	C₄	—	2	2	no	—
626	C₄	—	2	1	no	—
627	C₄	—	2	1	no	—
628	C₄	—	106	1	no	—
629	C₄	—	106	1	no	—
630	C₄	—	106	1	no	—
631	C₄	—	106	2	no	—
632	C₄	—	106	2	no	—
633	C₄	—	106	1	no	—
634	C₄	—	106	2	no	—
635	C₄	—	106	1	no	—
636	C₄	C₂×#11719949	1	2	yes	—
637	C₂	—	2	4	no	—
638	C₂	—	212	1	no	—
639	C₂	—	212	2	no	—
640	C₂	—	212	1	no	—
641	C₂	—	212	1	no	—
642	C₂	C₂²×#11719949	1	1	yes	characteristic
643	C₂	#16811458	1	1	yes	characteristic
644	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	#3521
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. #16811347	C₂	semidirect
2. #16811346	C₂	semidirect
3. #16811326	C₂	semidirect
4. #16811321	C₂	semidirect
5. #16811315	C₂	semidirect
6. #16811311	C₂	semidirect
7. #16811300	C₂	semidirect
8. #16811300	C₂	semidirect
9. #16811277	C₂	semidirect
10. #16811268	C₂	semidirect
11. #16811268	C₂	semidirect
12. #16811247	C₂	semidirect
13. #16811236	C₂	semidirect
14. #16811222	C₂	semidirect
15. #16811213	C₂	semidirect
16. #16811160	C₂	semidirect
17. #16811159	C₂	semidirect
18. #16811158	C₂	semidirect
19. C₂×#13133212	C₂	semidirect
20. #16811102	C₂	semidirect
21. #16811095	C₂	semidirect
22. C₄×#11719947	C₂	semidirect
23. C₅₃×#554	C₂	semidirect
27. #13133217	C₂²	semidirect
33. #13133212	C₂²	semidirect
34. #13133212	C₂²	semidirect
35. #13133212	C₂²	semidirect
36. #13133212	C₂²	semidirect
37. #13133212	C₂²	semidirect
38. #13133212	C₂²	semidirect
39. #13133212	C₂²	semidirect
40. #13133208	C₂²	semidirect
41. #13133208	C₂²	semidirect
42. #13133208	C₂²	semidirect
43. #13133208	C₂²	semidirect
46. #13133206	C₂²	semidirect
51. #13133202	C₂²	semidirect
52. #13133202	C₂²	semidirect
53. #13133202	C₂²	semidirect
54. #13133202	C₂²	semidirect
57. #13133201	C₂²	semidirect
58. #13133201	C₂²	semidirect
61. #13133197	C₂²	semidirect
62. #13133197	C₂²	semidirect
63. #13133193	C₂²	semidirect
64. #13133193	C₂²	semidirect
65. #13133193	C₂²	semidirect
69. #13133192	C₂²	semidirect
72. #13133191	C₂²	semidirect
75. Q₈×Dic₅₃	C₂²	semidirect
76. C₄×Dic₁₀₆	C₂²	semidirect
77. C₄×Dic₁₀₆	C₂²	semidirect
78. #13133182	C₂²	semidirect
79. #13133182	C₂²	semidirect
83. C₄×D₂₁₂	C₂²	semidirect
86. #13133179	C₂²	semidirect
87. #13133179	C₂²	semidirect
89. C₄×C₂²⋊D₅₃	C₂²	semidirect
90. C₄×C₂²⋊D₅₃	C₂²	semidirect
92. #13133177	C₂²	semidirect
93. #13133177	C₂²	semidirect
95. #13133175	C₂²	semidirect
96. #13133175	C₂²	semidirect
97. #13133175	C₂²	semidirect
98. #13133175	C₂²	semidirect
100. #13133174	C₂²	semidirect
101. #13133173	C₂²	semidirect
102. #13133173	C₂²	semidirect
103. C₂×C₄⋊Dic₅₃	C₂²	semidirect
104. C₂×#11719919	C₂²	semidirect
105. C₄⋊C₄×D₅₃	C₂²	semidirect
110. C₂×C₄×Dic₅₃	C₂²	semidirect
111. C₄²×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
325. #554	D₅₃	semidirect
346. C₅₃	#3521	semidirect
351. C₄⋊Q₈	C₂×D₅₃	semidirect
368. #125	C₂×D₅₃	semidirect
369. #125	C₂×D₅₃	semidirect
392. C₂²⋊Q₈	C₂×D₅₃	semidirect
393. C₂²⋊Q₈	C₂×D₅₃	semidirect
394. C₂²⋊Q₈	C₂×D₅₃	semidirect
409. C₄×Q₈	C₂×D₅₃	semidirect
439. C₂×C₄⋊C₄	C₂×D₅₃	semidirect