Subgroups — #20289301

Order 6784.

1050 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811367	C₂	1	2	yes	maximal, complemented
2	#16811363	C₂	1	2	yes	maximal, complemented
3	#16811346	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811319	C₂	1	2	yes	maximal, complemented
5	#16811296	C₂	1	2	yes	maximal, complemented
6	#16811293	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811285	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811281	C₂	1	1	yes	maximal, characteristic, complemented
9	#16811277	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811268	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811266	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811231	C₂	1	1	yes	maximal, characteristic, complemented
13	#16811225	C₂	1	2	yes	maximal, complemented
14	#16811224	C₂	1	1	yes	maximal, characteristic, complemented
15	#16811217	C₂	1	2	yes	maximal, complemented
16	#16811207	C₂	1	2	yes	maximal, complemented
17	#16811206	C₂	1	2	yes	maximal, complemented
18	#16811193	C₂	1	1	yes	maximal, characteristic, complemented
19	#16811186	C₂	1	1	yes	maximal, characteristic, complemented
20	#16811172	C₂	1	1	yes	maximal, characteristic, complemented
21	#16811160	C₂	1	1	yes	maximal, characteristic, complemented
22	#16811155	C₂	1	1	yes	maximal, characteristic, complemented
23	C₅₃×#559	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#11719949	C₂²	1	1	yes	characteristic
25	C₂×#11719947	C₂²	1	1	yes	characteristic
26	C₂×#11719947	C₂²	1	1	yes	characteristic
27	C₂×#11719947	C₂²	1	1	yes	characteristic, complemented
28	C₂×Q₈×D₅₃	C₂²	1	1	yes	characteristic, complemented
29	C₂×D₄×D₅₃	C₂²	1	1	yes	characteristic
30	#13133217	C₂²	1	1	yes	characteristic, complemented
31	#13133215	C₂²	1	1	yes	characteristic, complemented
32	#13133213	C₂²	1	1	yes	characteristic
33	#13133213	C₂²	1	2	yes	complemented
34	#13133212	C₂²	1	1	yes	characteristic
35	#13133212	C₂²	1	2	yes	—
36	#13133212	C₂²	1	1	yes	characteristic, complemented
37	#13133212	C₂²	1	2	yes	complemented
38	#13133212	C₂²	1	2	yes	complemented
39	#13133211	C₂²	1	2	yes	—
40	#13133210	C₂²	1	2	yes	complemented
41	#13133209	C₂²	1	2	yes	complemented
42	#13133208	C₂²	1	1	yes	characteristic, complemented
43	#13133208	C₂²	1	1	yes	characteristic, complemented
44	#13133207	C₂²	1	2	yes	complemented
45	#13133206	C₂²	1	1	yes	characteristic
46	#13133205	C₂²	1	1	yes	characteristic
47	#13133205	C₂²	1	1	yes	characteristic
48	#13133205	C₂²	1	2	yes	complemented
49	#13133204	C₂²	1	1	yes	characteristic
50	#13133203	C₂²	1	2	yes	—
51	#13133202	C₂²	1	1	yes	characteristic
52	#13133202	C₂²	1	1	yes	characteristic, complemented
53	#13133201	C₂²	1	2	yes	—
54	#13133201	C₂²	1	2	yes	—
55	#13133201	C₂²	1	2	yes	complemented
56	#13133200	C₂²	1	2	yes	complemented
57	#13133199	C₂²	1	2	yes	—
58	#13133199	C₂²	1	2	yes	complemented
59	#13133199	C₂²	1	2	yes	complemented
60	#13133199	C₂²	1	2	yes	complemented
61	#13133199	C₂²	1	2	yes	complemented
62	#13133197	C₂²	1	1	yes	characteristic
63	#13133197	C₂²	1	1	yes	characteristic, complemented
64	#13133197	C₂²	1	2	yes	complemented
65	#13133194	C₂²	1	2	yes	complemented
66	#13133194	C₂²	1	2	yes	complemented
67	#13133194	C₂²	1	2	yes	complemented
68	#13133193	C₂²	1	1	yes	characteristic
69	#13133193	C₂²	1	2	yes	complemented
70	#13133193	C₂²	1	1	yes	characteristic, complemented
71	#13133192	C₂²	1	1	yes	characteristic
72	#13133192	C₂²	1	1	yes	characteristic
73	#13133192	C₂²	1	2	yes	complemented
74	#13133191	C₂²	1	2	yes	—
75	#13133191	C₂²	1	2	yes	—
76	#13133191	C₂²	1	2	yes	complemented
77	#13133190	C₂²	1	2	yes	—
78	#13133190	C₂²	1	2	yes	—
79	#13133190	C₂²	1	2	yes	complemented
80	#13133190	C₂²	1	2	yes	complemented
81	#13133190	C₂²	1	2	yes	complemented
82	#13133189	C₂²	1	2	yes	—
83	#13133189	C₂²	1	2	yes	—
84	#13133189	C₂²	1	2	yes	complemented
85	#13133189	C₂²	1	2	yes	complemented
86	#13133188	C₂²	1	1	yes	characteristic
87	#13133188	C₂²	1	1	yes	characteristic, complemented
88	#13133187	C₂²	1	2	yes	—
89	#13133187	C₂²	1	2	yes	—
90	#13133185	C₂²	1	2	yes	—
91	#13133185	C₂²	1	2	yes	—
92	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
93	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
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	1	yes	characteristic, complemented
98	C₄×D₂₁₂	C₂²	1	2	yes	complemented
99	D₄×Dic₅₃	C₂²	1	2	yes	complemented
100	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
101	#13133177	C₂²	1	2	yes	complemented
102	#13133177	C₂²	1	2	yes	complemented
103	#13133177	C₂²	1	2	yes	complemented
104	#13133176	C₂²	1	2	yes	complemented
105	#13133175	C₂²	1	2	yes	complemented
106	#13133174	C₂²	1	1	yes	characteristic
107	#13133174	C₂²	1	1	yes	characteristic, complemented
108	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	complemented
109	C₂×#11719921	C₂²	1	2	yes	—
110	C₂²⋊C₄×D₅₃	C₂²	1	2	yes	—
111	C₄²×D₅₃	C₂²	1	1	yes	characteristic, complemented
112	#127×C₅₃	C₂²	1	1	yes	characteristic
113	#124×C₅₃	C₂²	1	1	yes	characteristic
114	#124×C₅₃	C₂²	1	1	yes	characteristic, complemented
115	#124×C₅₃	C₂²	1	2	yes	complemented
116	#123×C₅₃	C₂²	1	2	yes	complemented
117	#123×C₅₃	C₂²	1	2	yes	complemented
118	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
119	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
120	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
121	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
122	C₄×D₄×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	1	yes	characteristic
126	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
127	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
128	C₂×D₄×C₅₃	C₂³	1	2	yes	—
129	C₂×D₄×C₅₃	C₂³	1	2	yes	—
130	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
131	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
132	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
133	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
134	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
135	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
136	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
137	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
138	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
139	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
140	C₂²×Dic₅₃	C₂³	1	2	yes	—
141	C₂²×Dic₅₃	C₂³	1	2	yes	—
142	C₂²×Dic₅₃	C₂³	1	2	yes	—
143	#11719949	—	4	2	no	—
144	Q₈×D₅₃	—	4	2	no	—
145	#11719947	—	4	2	no	—
146	#11719947	—	4	2	no	—
147	#11719947	—	4	2	no	—
148	D₄×D₅₃	—	4	2	no	—
149	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
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×D₂₁₂	C₂³	1	1	yes	characteristic
154	C₂×D₂₁₂	C₂³	1	2	yes	—
155	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
156	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
157	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
158	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
159	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
160	C₂×C₄×D₅₃	C₂³	1	2	yes	—
161	C₂×C₄×D₅₃	C₂³	1	2	yes	—
162	C₂×C₄×D₅₃	C₂³	1	2	yes	—
163	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
164	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
165	C₂×Dic₁₀₆	C₂³	1	2	yes	—
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	1	yes	characteristic
170	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
171	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
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	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
177	C₄²×C₅₃	C₂³	1	1	yes	characteristic
178	C₄²×C₅₃	C₂³	1	1	yes	characteristic
179	C₄²×C₅₃	C₂³	1	2	yes	—
180	C₄²×C₅₃	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	#11719921	C₂³	1	1	yes	characteristic
186	#11719921	C₂³	1	2	yes	—
187	#11719921	C₂³	1	1	yes	characteristic
188	#11719921	C₂³	1	2	yes	—
189	#11719921	C₂³	1	2	yes	—
190	#11719921	C₂³	1	2	yes	—
191	#11719921	C₂³	1	1	yes	characteristic
192	#11719921	C₂³	1	1	yes	characteristic
193	#11719921	C₂³	1	1	yes	characteristic
194	#11719921	C₂³	1	1	yes	characteristic
195	#11719921	C₂³	1	2	yes	—
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	1	yes	characteristic
201	#11719921	C₂³	1	1	yes	characteristic
202	#11719921	C₂³	1	2	yes	—
203	#11719921	C₂³	1	1	yes	characteristic
204	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
205	C₄⋊Dic₅₃	C₂³	1	2	yes	—
206	C₄⋊Dic₅₃	C₂³	1	2	yes	—
207	C₄⋊Dic₅₃	C₂³	1	2	yes	—
208	C₄⋊Dic₅₃	C₂³	1	2	yes	—
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	#11719919	C₂³	1	1	yes	characteristic
214	#11719919	C₂³	1	1	yes	characteristic
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	2	yes	—
220	#11719919	C₂³	1	2	yes	—
221	#11719919	C₂³	1	2	yes	—
222	#11719919	C₂³	1	2	yes	—
223	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
224	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
225	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
226	C₄×Dic₅₃	C₂³	1	2	yes	—
227	C₄×Dic₅₃	C₂³	1	2	yes	—
228	C₂³×C₅₃	C₂⁴	1	2	yes	—
229	C₂³×C₅₃	C₂⁴	1	2	yes	—
230	C₂²×D₅₃	—	4	2	no	—
231	C₂²×D₅₃	—	4	2	no	—
232	C₂²×D₅₃	—	4	2	no	—
233	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
234	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
235	C₂²×D₅₃	C₂⁴	1	2	yes	—
236	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
237	Q₈×C₅₃	—	2	2	no	—
238	Q₈×C₅₃	—	4	1	no	—
239	D₄×C₅₃	—	2	2	no	—
240	D₄×C₅₃	—	2	2	no	—
241	D₄×C₅₃	—	4	2	no	—
242	D₄×C₅₃	—	4	2	no	—
243	C₂×C₄×C₅₃	—	4	2	no	—
244	C₂×C₄×C₅₃	—	4	2	no	—
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	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
250	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
251	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
252	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
253	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
254	C₂²⋊D₅₃	—	4	2	no	—
255	C₂²⋊D₅₃	—	4	2	no	—
256	C₂²⋊D₅₃	—	4	2	no	—
257	C₂²⋊D₅₃	—	4	2	no	—
258	C₂²⋊D₅₃	—	4	2	no	—
259	C₂²⋊D₅₃	—	4	2	no	—
260	C₂²⋊D₅₃	—	4	2	no	—
261	C₂²⋊D₅₃	—	4	2	no	—
262	C₂×Dic₅₃	—	4	2	no	—
263	C₂×Dic₅₃	—	4	2	no	—
264	C₂×Dic₅₃	—	4	2	no	—
265	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
266	C₂×Dic₅₃	C₂⁴	1	2	yes	—
267	C₂×Dic₅₃	C₂⁴	1	2	yes	—
268	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
269	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
270	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
271	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
272	C₂×Dic₅₃	C₂⁴	1	2	yes	—
273	D₂₁₂	—	2	2	no	—
274	D₂₁₂	—	4	1	no	—
275	D₂₁₂	—	4	1	no	—
276	D₂₁₂	—	4	2	no	—
277	D₂₁₂	—	4	1	no	—
278	D₂₁₂	—	4	2	no	—
279	C₄×D₅₃	—	2	2	no	—
280	C₄×D₅₃	—	2	2	no	—
281	C₄×D₅₃	—	2	2	no	—
282	C₄×D₅₃	—	4	2	no	—
283	C₄×D₅₃	—	4	1	no	—
284	C₄×D₅₃	—	4	1	no	—
285	C₄×D₅₃	—	4	2	no	—
286	C₄×D₅₃	—	4	2	no	—
287	Dic₁₀₆	—	2	2	no	—
288	Dic₁₀₆	—	2	2	no	—
289	Dic₁₀₆	—	4	2	no	—
290	C₂²×C₅₃	—	2	2	no	—
291	C₂²×C₅₃	—	2	2	no	—
292	C₂²×C₅₃	—	2	2	no	—
293	C₂²×C₅₃	—	2	2	no	—
294	C₂²×C₅₃	—	2	2	no	—
295	C₂²×C₅₃	—	2	2	no	—
296	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
297	C₂×D₅₃	—	2	1	no	—
298	C₂×D₅₃	—	2	1	no	—
299	C₂×D₅₃	—	2	1	no	—
300	C₂×D₅₃	—	2	2	no	—
301	C₂×D₅₃	—	2	1	no	—
302	C₂×D₅₃	—	2	1	no	—
303	C₂×D₅₃	—	2	2	no	—
304	C₂×D₅₃	—	2	1	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	—	8	4	no	—
310	C₄×C₅₃	—	2	2	no	—
311	C₄×C₅₃	—	2	1	no	—
312	C₄×C₅₃	—	2	1	no	—
313	C₄×C₅₃	—	2	1	no	—
314	C₄×C₅₃	—	2	1	no	—
315	C₄×C₅₃	—	2	2	no	—
316	C₄×C₅₃	—	2	1	no	—
317	C₄×C₅₃	—	2	1	no	—
318	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
319	Dic₅₃	—	2	2	no	—
320	Dic₅₃	—	2	2	no	—
321	Dic₅₃	—	2	1	no	—
322	Dic₅₃	—	2	1	no	—
323	Dic₅₃	—	2	1	no	—
324	Dic₅₃	—	2	1	no	—
325	Dic₅₃	—	2	2	no	—
326	Dic₅₃	—	2	1	no	—
327	#3562	—	53	1	no	maximal, Sylow
328	C₂×C₅₃	—	4	2	no	—
329	C₂×C₅₃	—	4	2	no	—
330	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
331	C₂×C₅₃	#585	1	1	yes	characteristic
332	C₂×C₅₃	#585	1	1	yes	characteristic
333	D₅₃	—	4	1	no	—
334	D₅₃	—	4	1	no	—
335	D₅₃	—	4	2	no	—
336	D₅₃	—	4	1	no	—
337	#562	—	53	2	no	—
338	#560	—	53	2	no	—
339	#559	D₅₃	1	1	yes	characteristic, complemented
340	#556	—	53	1	no	—
341	#553	—	53	2	no	—
342	#551	—	53	2	no	—
343	#550	—	53	1	no	—
344	#548	—	53	1	no	—
345	#548	—	53	1	no	—
346	#547	—	53	1	no	—
347	#546	—	53	1	no	—
348	#546	—	53	1	no	—
349	#541	—	53	1	no	—
350	#540	—	53	2	no	—
351	#540	—	53	1	no	—
352	#539	—	53	2	no	—
353	#538	—	53	2	no	—
354	#538	—	53	2	no	—
355	#536	—	53	1	no	—
356	#535	—	53	1	no	—
357	#532	—	53	1	no	—
358	#529	—	53	1	no	—
359	#529	—	53	1	no	—
360	C₅₃	#3562	1	1	yes	characteristic, complemented, Sylow
361	C₂×#41	—	53	1	no	—
362	C₂×#41	—	53	1	no	—
363	C₂×#41	—	53	1	no	—
364	C₂×#41	—	53	1	no	—
365	C₂²×Q₈	—	53	1	no	—
366	C₂²×D₄	—	53	1	no	—
367	C₄⋊Q₈	—	53	1	no	—
368	C₄⋊Q₈	—	53	1	no	—
369	#127	—	53	1	no	—
370	#127	—	53	2	no	—
371	#127	C₂×D₅₃	1	1	yes	characteristic
372	#126	—	53	1	no	—
373	#126	—	53	1	no	—
374	#126	—	53	2	no	—
375	#126	—	53	2	no	—
376	#126	—	53	2	no	—
377	#126	—	53	2	no	—
378	#126	—	53	2	no	—
379	#125	—	53	1	no	—
380	#125	—	53	1	no	—
381	#125	—	53	2	no	—
382	#125	—	53	2	no	—
383	#124	—	53	1	no	—
384	#124	—	53	1	no	—
385	#124	—	53	2	no	—
386	#124	—	53	1	no	—
387	#124	—	53	1	no	—
388	#124	—	53	2	no	—
389	#124	C₂×D₅₃	1	1	yes	characteristic
390	#124	C₂×D₅₃	1	1	yes	characteristic, complemented
391	#124	C₂×D₅₃	1	2	yes	complemented
392	#123	—	53	2	no	—
393	#123	—	53	1	no	—
394	#123	—	53	1	no	—
395	#123	—	53	2	no	—
396	#123	—	53	2	no	—
397	#123	—	53	2	no	—
398	#123	—	53	2	no	—
399	#123	—	53	2	no	—
400	#123	—	53	2	no	—
401	#123	—	53	2	no	—
402	#123	—	53	2	no	—
403	#123	C₂×D₅₃	1	2	yes	complemented
404	#123	C₂×D₅₃	1	2	yes	complemented
405	C₂²⋊Q₈	—	53	1	no	—
406	C₂²⋊Q₈	—	53	2	no	—
407	C₂²⋊Q₈	—	53	1	no	—
408	C₂²⋊Q₈	—	53	1	no	—
409	C₂²⋊Q₈	—	53	2	no	—
410	C₂²⋊Q₈	—	53	1	no	—
411	C₂²⋊Q₈	—	53	2	no	—
412	C₂²⋊Q₈	—	53	2	no	—
413	C₂²⋊Q₈	—	53	2	no	—
414	#121	—	53	1	no	—
415	#121	—	53	2	no	—
416	#121	—	53	2	no	—
417	#121	—	53	2	no	—
418	#121	—	53	2	no	—
419	#121	—	53	1	no	—
420	#121	—	53	2	no	—
421	#121	—	53	2	no	—
422	#121	—	53	2	no	—
423	#121	—	53	1	no	—
424	#121	—	53	1	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	2	no	—
433	#120	—	53	2	no	—
434	#120	—	53	2	no	—
435	C₄×Q₈	—	53	1	no	—
436	C₄×Q₈	—	53	1	no	—
437	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
438	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
439	C₄×D₄	—	53	2	no	—
440	C₄×D₄	—	53	2	no	—
441	C₄×D₄	—	53	2	no	—
442	C₄×D₄	—	53	2	no	—
443	C₄×D₄	—	53	2	no	—
444	C₄×D₄	—	53	1	no	—
445	C₄×D₄	—	53	1	no	—
446	C₄×D₄	—	53	1	no	—
447	C₄×D₄	—	53	2	no	—
448	C₄×D₄	—	53	1	no	—
449	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
450	C₄×D₄	C₂×D₅₃	1	2	yes	complemented
451	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
452	#117	—	53	2	no	—
453	#117	—	53	2	no	—
454	#117	—	53	1	no	—
455	#117	—	53	1	no	—
456	C₂×C₄⋊C₄	—	53	2	no	—
457	C₂×C₂²⋊C₄	—	53	2	no	—
458	C₂×C₂²⋊C₄	—	53	2	no	—
459	C₂×C₄²	—	53	1	no	—
460	C₂⁴	—	53	2	no	—
461	#41	—	212	2	no	—
462	#41	—	212	2	no	—
463	#41	—	212	2	no	—
464	#41	—	212	2	no	—
465	C₂×Q₈	—	53	1	no	—
466	C₂×Q₈	—	53	1	no	—
467	C₂×Q₈	—	53	2	no	—
468	C₂×Q₈	—	212	2	no	—
469	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
470	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
471	C₂×D₄	—	53	2	no	—
472	C₂×D₄	—	53	2	no	—
473	C₂×D₄	—	53	2	no	—
474	C₂×D₄	—	53	2	no	—
475	C₂×D₄	—	53	2	no	—
476	C₂×D₄	—	53	2	no	—
477	C₂×D₄	—	53	2	no	—
478	C₂×D₄	—	53	2	no	—
479	C₂×D₄	—	53	1	no	—
480	C₂×D₄	—	53	1	no	—
481	C₂×D₄	—	53	1	no	—
482	C₂×D₄	—	53	2	no	—
483	C₂×D₄	—	53	1	no	—
484	C₂×D₄	—	53	2	no	—
485	C₂×D₄	—	212	2	no	—
486	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
487	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
488	C₂×D₄	C₂²×D₅₃	1	2	yes	—
489	C₂×D₄	C₂²×D₅₃	1	2	yes	—
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	1	no	—
498	C₂²×C₄	—	53	2	no	—
499	C₂²×C₄	—	53	2	no	—
500	C₂²×C₄	—	53	2	no	—
501	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
502	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
503	C₄⋊C₄	—	53	1	no	—
504	C₄⋊C₄	—	53	2	no	—
505	C₄⋊C₄	—	53	2	no	—
506	C₄⋊C₄	—	53	2	no	—
507	C₄⋊C₄	—	53	1	no	—
508	C₄⋊C₄	—	53	1	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	1	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	—	53	2	no	—
519	C₄⋊C₄	—	53	1	no	—
520	C₄⋊C₄	—	53	2	no	—
521	C₄⋊C₄	—	53	2	no	—
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	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
526	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
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	1	no	—
532	C₂²⋊C₄	—	53	1	no	—
533	C₂²⋊C₄	—	53	2	no	—
534	C₂²⋊C₄	—	53	2	no	—
535	C₂²⋊C₄	—	53	2	no	—
536	C₂²⋊C₄	—	53	2	no	—
537	C₂²⋊C₄	—	53	1	no	—
538	C₂²⋊C₄	—	53	1	no	—
539	C₂²⋊C₄	—	53	2	no	—
540	C₂²⋊C₄	—	53	1	no	—
541	C₂²⋊C₄	—	53	1	no	—
542	C₂²⋊C₄	—	53	1	no	—
543	C₂²⋊C₄	—	53	1	no	—
544	C₂²⋊C₄	—	53	2	no	—
545	C₂²⋊C₄	—	53	2	no	—
546	C₂²⋊C₄	—	53	2	no	—
547	C₂²⋊C₄	—	53	1	no	—
548	C₂²⋊C₄	—	53	2	no	—
549	C₂²⋊C₄	—	53	1	no	—
550	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
551	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
552	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
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	1	no	—
558	C₄²	—	53	1	no	—
559	C₄²	—	53	2	no	—
560	C₄²	—	53	2	no	—
561	C₄²	C₂²×D₅₃	1	1	yes	characteristic
562	C₄²	C₂²×D₅₃	1	1	yes	characteristic
563	C₄²	C₂²×D₅₃	1	1	yes	characteristic
564	C₄²	C₂²×D₅₃	1	2	yes	—
565	C₂³	—	53	1	no	—
566	C₂³	—	53	2	no	—
567	C₂³	—	53	1	no	—
568	C₂³	—	53	1	no	—
569	C₂³	—	212	2	no	—
570	C₂³	—	212	2	no	—
571	C₂³	—	212	2	no	—
572	C₂³	C₂³×D₅₃	1	2	yes	—
573	C₂³	C₂³×D₅₃	1	2	yes	—
574	Q₈	—	2	2	no	—
575	Q₈	—	4	1	no	—
576	Q₈	—	106	2	no	—
577	Q₈	—	106	2	no	—
578	Q₈	—	212	2	no	—
579	D₄	—	2	2	no	—
580	D₄	—	2	2	no	—
581	D₄	—	4	2	no	—
582	D₄	—	4	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	2	no	—
592	D₄	—	212	2	no	—
593	D₄	—	212	2	no	—
594	D₄	—	212	2	no	—
595	D₄	—	212	2	no	—
596	D₄	—	212	2	no	—
597	C₂×C₄	—	4	2	no	—
598	C₂×C₄	—	4	2	no	—
599	C₂×C₄	—	53	1	no	—
600	C₂×C₄	—	53	1	no	—
601	C₂×C₄	—	53	2	no	—
602	C₂×C₄	—	53	1	no	—
603	C₂×C₄	—	53	2	no	—
604	C₂×C₄	—	53	2	no	—
605	C₂×C₄	—	53	1	no	—
606	C₂×C₄	—	53	1	no	—
607	C₂×C₄	—	106	2	no	—
608	C₂×C₄	—	106	2	no	—
609	C₂×C₄	—	106	2	no	—
610	C₂×C₄	—	212	1	no	—
611	C₂×C₄	—	212	2	no	—
612	C₂×C₄	—	212	1	no	—
613	C₂×C₄	—	212	2	no	—
614	C₂×C₄	—	212	2	no	—
615	C₂×C₄	—	212	2	no	—
616	C₂×C₄	—	212	2	no	—
617	C₂×C₄	—	212	2	no	—
618	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
619	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
620	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
621	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
622	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
623	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
624	C₂×C₄	C₂³×D₅₃	1	2	yes	—
625	C₂×C₄	C₂³×D₅₃	1	2	yes	—
626	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
627	C₂²	—	2	2	no	—
628	C₂²	—	2	2	no	—
629	C₂²	—	2	2	no	—
630	C₂²	—	2	2	no	—
631	C₂²	—	2	2	no	—
632	C₂²	—	2	2	no	—
633	C₂²	—	106	1	no	—
634	C₂²	—	106	1	no	—
635	C₂²	—	106	1	no	—
636	C₂²	—	106	2	no	—
637	C₂²	—	106	1	no	—
638	C₂²	—	106	1	no	—
639	C₂²	—	106	2	no	—
640	C₂²	—	106	1	no	—
641	C₂²	—	106	1	no	—
642	C₂²	—	106	2	no	—
643	C₂²	—	106	1	no	—
644	C₂²	—	106	1	no	—
645	C₂²	—	424	4	no	—
646	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
647	C₄	—	2	2	no	—
648	C₄	—	2	1	no	—
649	C₄	—	2	2	no	—
650	C₄	—	2	1	no	—
651	C₄	—	2	1	no	—
652	C₄	—	2	1	no	—
653	C₄	—	2	1	no	—
654	C₄	—	2	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	1	no	—
661	C₄	—	106	1	no	—
662	C₄	—	106	2	no	—
663	C₄	#13133263	1	2	yes	—
664	C₂	—	4	2	no	—
665	C₂	—	4	2	no	—
666	C₂	—	212	1	no	—
667	C₂	—	212	2	no	—
668	C₂	—	212	1	no	—
669	C₂	—	212	1	no	—
670	C₂	C₂×#13133263	1	1	yes	characteristic
671	C₂	#16811458	1	1	yes	characteristic
672	C₂	#16811459	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#559	C₂
hypercenter	#559	D₅₃
hypocenter	C₅₃	#3562
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #559

Complements

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

N	H ≅ G/N	type
1. #16811367	C₂	semidirect
2. #16811363	C₂	semidirect
3. #16811346	C₂	semidirect
4. #16811319	C₂	semidirect
5. #16811296	C₂	semidirect
6. #16811293	C₂	semidirect
7. #16811285	C₂	semidirect
8. #16811281	C₂	semidirect
9. #16811277	C₂	semidirect
10. #16811268	C₂	semidirect
11. #16811266	C₂	semidirect
12. #16811231	C₂	semidirect
13. #16811225	C₂	semidirect
14. #16811224	C₂	semidirect
15. #16811217	C₂	semidirect
16. #16811207	C₂	semidirect
17. #16811206	C₂	semidirect
18. #16811193	C₂	semidirect
19. #16811186	C₂	semidirect
20. #16811172	C₂	semidirect
21. #16811160	C₂	semidirect
22. #16811155	C₂	semidirect
23. C₅₃×#559	C₂	semidirect
27. C₂×#11719947	C₂²	semidirect
28. C₂×Q₈×D₅₃	C₂²	semidirect
30. #13133217	C₂²	semidirect
31. #13133215	C₂²	semidirect
33. #13133213	C₂²	semidirect
36. #13133212	C₂²	semidirect
37. #13133212	C₂²	semidirect
38. #13133212	C₂²	semidirect
40. #13133210	C₂²	semidirect
41. #13133209	C₂²	semidirect
42. #13133208	C₂²	semidirect
43. #13133208	C₂²	semidirect
44. #13133207	C₂²	semidirect
48. #13133205	C₂²	semidirect
52. #13133202	C₂²	semidirect
55. #13133201	C₂²	semidirect
56. #13133200	C₂²	semidirect
58. #13133199	C₂²	semidirect
59. #13133199	C₂²	semidirect
60. #13133199	C₂²	semidirect
61. #13133199	C₂²	semidirect
63. #13133197	C₂²	semidirect
64. #13133197	C₂²	semidirect
65. #13133194	C₂²	semidirect
66. #13133194	C₂²	semidirect
67. #13133194	C₂²	semidirect
69. #13133193	C₂²	semidirect
70. #13133193	C₂²	semidirect
73. #13133192	C₂²	semidirect
76. #13133191	C₂²	semidirect
79. #13133190	C₂²	semidirect
80. #13133190	C₂²	semidirect
81. #13133190	C₂²	semidirect
84. #13133189	C₂²	semidirect
85. #13133189	C₂²	semidirect
87. #13133188	C₂²	semidirect
92. C₄×Dic₁₀₆	C₂²	semidirect
93. C₄×Dic₁₀₆	C₂²	semidirect
97. C₄×D₂₁₂	C₂²	semidirect
98. C₄×D₂₁₂	C₂²	semidirect
99. D₄×Dic₅₃	C₂²	semidirect
100. C₄×C₂²⋊D₅₃	C₂²	semidirect
101. #13133177	C₂²	semidirect
102. #13133177	C₂²	semidirect
103. #13133177	C₂²	semidirect
104. #13133176	C₂²	semidirect
105. #13133175	C₂²	semidirect
107. #13133174	C₂²	semidirect
108. C₂×C₄⋊Dic₅₃	C₂²	semidirect
111. C₄²×D₅₃	C₂²	semidirect
114. #124×C₅₃	C₂²	semidirect
115. #124×C₅₃	C₂²	semidirect
116. #123×C₅₃	C₂²	semidirect
117. #123×C₅₃	C₂²	semidirect
118. C₄×Q₈×C₅₃	C₂²	semidirect
119. C₄×Q₈×C₅₃	C₂²	semidirect
121. C₄×D₄×C₅₃	C₂²	semidirect
122. C₄×D₄×C₅₃	C₂²	semidirect
339. #559	D₅₃	semidirect
360. C₅₃	#3562	semidirect
390. #124	C₂×D₅₃	semidirect
391. #124	C₂×D₅₃	semidirect
403. #123	C₂×D₅₃	semidirect
404. #123	C₂×D₅₃	semidirect
437. C₄×Q₈	C₂×D₅₃	semidirect
438. C₄×Q₈	C₂×D₅₃	semidirect
450. C₄×D₄	C₂×D₅₃	semidirect
451. C₄×D₄	C₂×D₅₃	semidirect