Subgroups — #20289291

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	#16811369	C₂	1	2	yes	maximal, complemented
2	#16811363	C₂	1	2	yes	maximal, complemented
3	#16811355	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811347	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811318	C₂	1	2	yes	maximal, complemented
6	#16811300	C₂	1	2	yes	maximal, complemented
7	#16811292	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811284	C₂	1	1	yes	maximal, characteristic, complemented
9	#16811284	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811276	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811266	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811261	C₂	1	1	yes	maximal, characteristic, complemented
13	#16811234	C₂	1	1	yes	maximal, characteristic, complemented
14	#16811228	C₂	1	1	yes	maximal, characteristic, complemented
15	#16811225	C₂	1	2	yes	maximal, complemented
16	#16811219	C₂	1	2	yes	maximal, complemented
17	#16811213	C₂	1	2	yes	maximal, complemented
18	#16811209	C₂	1	2	yes	maximal, complemented
19	#16811191	C₂	1	1	yes	maximal, characteristic, complemented
20	#16811174	C₂	1	1	yes	maximal, characteristic, complemented
21	#16811152	C₂	1	1	yes	maximal, characteristic, complemented
22	#16811151	C₂	1	1	yes	maximal, characteristic, complemented
23	C₅₃×#536	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#11719949	C₂²	1	1	yes	characteristic
25	C₂×#11719949	C₂²	1	1	yes	characteristic
26	C₂×#11719949	C₂²	1	1	yes	characteristic, complemented
27	C₂×D₄×D₅₃	C₂²	1	1	yes	characteristic
28	#13133214	C₂²	1	1	yes	characteristic
29	#13133213	C₂²	1	1	yes	characteristic
30	#13133213	C₂²	1	2	yes	complemented
31	#13133212	C₂²	1	1	yes	characteristic
32	#13133212	C₂²	1	2	yes	—
33	#13133212	C₂²	1	2	yes	—
34	#13133212	C₂²	1	2	yes	complemented
35	#13133212	C₂²	1	2	yes	complemented
36	#13133210	C₂²	1	1	yes	characteristic, complemented
37	#13133210	C₂²	1	2	yes	complemented
38	#13133209	C₂²	1	1	yes	characteristic, complemented
39	#13133207	C₂²	1	2	yes	complemented
40	#13133207	C₂²	1	1	yes	characteristic, complemented
41	#13133207	C₂²	1	2	yes	complemented
42	#13133206	C₂²	1	2	yes	—
43	#13133206	C₂²	1	1	yes	characteristic
44	#13133206	C₂²	1	1	yes	characteristic
45	#13133206	C₂²	1	1	yes	characteristic, complemented
46	#13133206	C₂²	1	2	yes	complemented
47	#13133203	C₂²	1	1	yes	characteristic
48	#13133203	C₂²	1	2	yes	complemented
49	#13133202	C₂²	1	2	yes	—
50	#13133202	C₂²	1	1	yes	characteristic
51	#13133202	C₂²	1	2	yes	complemented
52	#13133202	C₂²	1	2	yes	complemented
53	#13133202	C₂²	1	2	yes	complemented
54	#13133201	C₂²	1	2	yes	complemented
55	#13133200	C₂²	1	2	yes	—
56	#13133199	C₂²	1	2	yes	—
57	#13133199	C₂²	1	2	yes	complemented
58	#13133199	C₂²	1	2	yes	complemented
59	#13133199	C₂²	1	1	yes	characteristic, complemented
60	#13133197	C₂²	1	2	yes	complemented
61	#13133195	C₂²	1	1	yes	characteristic, complemented
62	#13133193	C₂²	1	1	yes	characteristic
63	#13133193	C₂²	1	1	yes	characteristic
64	#13133193	C₂²	1	1	yes	characteristic, complemented
65	#13133193	C₂²	1	2	yes	complemented
66	#13133193	C₂²	1	2	yes	complemented
67	#13133192	C₂²	1	2	yes	—
68	#13133192	C₂²	1	2	yes	complemented
69	#13133192	C₂²	1	2	yes	complemented
70	#13133191	C₂²	1	2	yes	—
71	#13133191	C₂²	1	1	yes	characteristic
72	#13133191	C₂²	1	1	yes	characteristic, complemented
73	#13133191	C₂²	1	2	yes	complemented
74	#13133191	C₂²	1	2	yes	complemented
75	#13133190	C₂²	1	2	yes	—
76	#13133190	C₂²	1	2	yes	—
77	#13133190	C₂²	1	1	yes	characteristic
78	#13133190	C₂²	1	2	yes	—
79	#13133190	C₂²	1	2	yes	complemented
80	#13133190	C₂²	1	2	yes	complemented
81	#13133189	C₂²	1	2	yes	—
82	#13133188	C₂²	1	1	yes	characteristic
83	#13133188	C₂²	1	2	yes	complemented
84	#13133187	C₂²	1	2	yes	—
85	#13133187	C₂²	1	2	yes	—
86	#13133187	C₂²	1	2	yes	—
87	#13133185	C₂²	1	2	yes	—
88	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
89	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
90	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
91	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
92	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
93	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
94	C₄×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
95	D₄×Dic₅₃	C₂²	1	1	yes	characteristic
96	#13133179	C₂²	1	2	yes	complemented
97	#13133179	C₂²	1	2	yes	complemented
98	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
99	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic, complemented
100	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
101	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
102	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
103	#13133177	C₂²	1	2	yes	complemented
104	#13133177	C₂²	1	2	yes	complemented
105	#13133175	C₂²	1	2	yes	complemented
106	#13133174	C₂²	1	1	yes	characteristic
107	#13133173	C₂²	1	2	yes	complemented
108	C₄⋊C₄×D₅₃	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	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
113	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
114	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
115	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
116	#124×C₅₃	C₂²	1	1	yes	characteristic
117	#124×C₅₃	C₂²	1	1	yes	characteristic
118	#123×C₅₃	C₂²	1	2	yes	complemented
119	#123×C₅₃	C₂²	1	2	yes	complemented
120	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
121	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
122	#117×C₅₃	C₂²	1	1	yes	characteristic, complemented
123	C₂³×D₅₃	C₂³	1	2	yes	—
124	#41×C₅₃	—	4	2	no	—
125	C₂×Q₈×C₅₃	—	4	2	no	—
126	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
127	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
128	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
129	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
130	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
131	C₂×D₄×C₅₃	C₂³	1	2	yes	—
132	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
133	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
134	C₂²×C₄×C₅₃	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	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	2	yes	—
141	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
142	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
143	C₂²×Dic₅₃	C₂³	1	2	yes	—
144	#11719949	—	4	2	no	—
145	#11719949	—	4	2	no	—
146	#11719949	—	4	2	no	—
147	D₄×D₅₃	—	4	2	no	—
148	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
149	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
150	C₂×D₂₁₂	C₂³	1	2	yes	—
151	C₂×D₂₁₂	C₂³	1	2	yes	—
152	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
153	C₂×D₂₁₂	C₂³	1	2	yes	—
154	C₂×D₂₁₂	C₂³	1	2	yes	—
155	C₂×D₂₁₂	C₂³	1	2	yes	—
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×C₄×D₅₃	C₂³	1	2	yes	—
164	C₂×C₄×D₅₃	C₂³	1	2	yes	—
165	C₂×Dic₁₀₆	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	2	yes	—
169	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
170	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
171	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
172	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
173	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
178	C₄²×C₅₃	C₂³	1	1	yes	characteristic
179	C₄²×C₅₃	C₂³	1	1	yes	characteristic
180	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
181	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
182	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
183	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
184	#11719921	C₂³	1	1	yes	characteristic
185	#11719921	C₂³	1	2	yes	—
186	#11719921	C₂³	1	1	yes	characteristic
187	#11719921	C₂³	1	2	yes	—
188	#11719921	C₂³	1	2	yes	—
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	1	yes	characteristic
198	#11719921	C₂³	1	2	yes	—
199	#11719921	C₂³	1	2	yes	—
200	#11719921	C₂³	1	2	yes	—
201	#11719921	C₂³	1	1	yes	characteristic
202	#11719921	C₂³	1	2	yes	—
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	1	yes	characteristic
207	C₄⋊Dic₅₃	C₂³	1	2	yes	—
208	C₄⋊Dic₅₃	C₂³	1	2	yes	—
209	C₄⋊Dic₅₃	C₂³	1	2	yes	—
210	#11719919	C₂³	1	2	yes	—
211	#11719919	C₂³	1	2	yes	—
212	#11719919	C₂³	1	2	yes	—
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	2	yes	—
217	#11719919	C₂³	1	1	yes	characteristic
218	#11719919	C₂³	1	1	yes	characteristic
219	#11719919	C₂³	1	2	yes	—
220	#11719919	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	1	yes	characteristic
224	C₄×Dic₅₃	C₂³	1	2	yes	—
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	1	yes	characteristic
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	2	yes	—
235	C₂²×D₅₃	C₂⁴	1	2	yes	—
236	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
237	Q₈×C₅₃	—	2	2	no	—
238	Q₈×C₅₃	—	2	2	no	—
239	Q₈×C₅₃	—	2	2	no	—
240	Q₈×C₅₃	—	4	2	no	—
241	D₄×C₅₃	—	2	2	no	—
242	D₄×C₅₃	—	4	2	no	—
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	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	2	yes	—
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	C₂²⋊D₅₃	—	4	2	no	—
256	C₂²⋊D₅₃	—	4	1	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	C₂²⋊D₅₃	—	4	1	no	—
263	C₂×Dic₅₃	—	4	2	no	—
264	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
265	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
266	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
267	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
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	1	yes	characteristic
272	D₂₁₂	—	2	2	no	—
273	D₂₁₂	—	2	2	no	—
274	D₂₁₂	—	4	1	no	—
275	D₂₁₂	—	4	2	no	—
276	D₂₁₂	—	4	2	no	—
277	D₂₁₂	—	4	2	no	—
278	D₂₁₂	—	4	2	no	—
279	D₂₁₂	—	4	2	no	—
280	C₄×D₅₃	—	2	2	no	—
281	C₄×D₅₃	—	2	2	no	—
282	C₄×D₅₃	—	4	1	no	—
283	C₄×D₅₃	—	4	1	no	—
284	C₄×D₅₃	—	4	2	no	—
285	C₄×D₅₃	—	4	2	no	—
286	C₄×D₅₃	—	4	2	no	—
287	C₄×D₅₃	—	4	2	no	—
288	C₄×D₅₃	—	4	2	no	—
289	Dic₁₀₆	—	4	1	no	—
290	C₂²×C₅₃	—	2	2	no	—
291	C₂²×C₅₃	—	2	1	no	—
292	C₂²×C₅₃	—	2	2	no	—
293	C₂²×C₅₃	—	2	1	no	—
294	C₂²×C₅₃	—	2	1	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	2	no	—
300	C₂×D₅₃	—	2	2	no	—
301	C₂×D₅₃	—	2	1	no	—
302	C₂×D₅₃	—	2	2	no	—
303	C₂×D₅₃	—	2	2	no	—
304	C₂×D₅₃	—	2	1	no	—
305	C₂×D₅₃	—	2	2	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	2	no	—
312	C₄×C₅₃	—	2	2	no	—
313	C₄×C₅₃	—	2	1	no	—
314	C₄×C₅₃	—	2	1	no	—
315	C₄×C₅₃	—	2	1	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	1	no	—
320	Dic₅₃	—	2	1	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	#3562	—	53	1	no	maximal, Sylow
328	C₂×C₅₃	—	4	2	no	—
329	C₂×C₅₃	—	4	1	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	2	no	—
335	D₅₃	—	4	2	no	—
336	D₅₃	—	4	1	no	—
337	#562	—	53	2	no	—
338	#560	—	53	2	no	—
339	#559	—	53	1	no	—
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	D₅₃	1	1	yes	characteristic, complemented
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	C₂×D₅₃	1	1	yes	characteristic
365	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
366	C₂²×D₄	—	53	1	no	—
367	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
368	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
369	#127	—	53	1	no	—
370	#127	—	53	2	no	—
371	#127	—	53	1	no	—
372	#126	—	53	1	no	—
373	#126	—	53	2	no	—
374	#126	—	53	2	no	—
375	#126	—	53	2	no	—
376	#126	—	53	2	no	—
377	#126	—	53	2	no	—
378	#126	—	53	1	no	—
379	#125	—	53	1	no	—
380	#125	—	53	2	no	—
381	#125	—	53	2	no	—
382	#125	—	53	1	no	—
383	#124	—	53	1	no	—
384	#124	—	53	1	no	—
385	#124	—	53	1	no	—
386	#124	—	53	2	no	—
387	#124	—	53	2	no	—
388	#124	—	53	1	no	—
389	#124	—	53	2	no	—
390	#124	C₂×D₅₃	1	1	yes	characteristic
391	#124	C₂×D₅₃	1	1	yes	characteristic
392	#123	—	53	2	no	—
393	#123	—	53	1	no	—
394	#123	—	53	2	no	—
395	#123	—	53	2	no	—
396	#123	—	53	2	no	—
397	#123	—	53	2	no	—
398	#123	—	53	1	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	2	no	—
408	C₂²⋊Q₈	—	53	1	no	—
409	C₂²⋊Q₈	—	53	1	no	—
410	C₂²⋊Q₈	—	53	1	no	—
411	C₂²⋊Q₈	—	53	2	no	—
412	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	complemented
413	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	complemented
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	1	no	—
421	#121	—	53	2	no	—
422	#121	—	53	2	no	—
423	#121	—	53	2	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	—	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	—	53	1	no	—
438	C₄×Q₈	—	53	1	no	—
439	C₄×D₄	—	53	1	no	—
440	C₄×D₄	—	53	1	no	—
441	C₄×D₄	—	53	1	no	—
442	C₄×D₄	—	53	2	no	—
443	C₄×D₄	—	53	2	no	—
444	C₄×D₄	—	53	2	no	—
445	C₄×D₄	—	53	2	no	—
446	C₄×D₄	—	53	2	no	—
447	C₄×D₄	—	53	2	no	—
448	C₄×D₄	—	53	2	no	—
449	C₄×D₄	—	53	1	no	—
450	C₄×D₄	—	53	1	no	—
451	C₄×D₄	—	53	1	no	—
452	#117	—	53	2	no	—
453	#117	—	53	2	no	—
454	#117	—	53	1	no	—
455	#117	C₂×D₅₃	1	1	yes	characteristic, complemented
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	—	4	2	no	—
462	#41	—	212	2	no	—
463	#41	—	212	2	no	—
464	#41	—	212	2	no	—
465	C₂×Q₈	—	4	2	no	—
466	C₂×Q₈	—	53	1	no	—
467	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
468	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
469	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
470	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
471	C₂×D₄	—	53	1	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	1	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	2	no	—
484	C₂×D₄	—	53	2	no	—
485	C₂×D₄	—	53	2	no	—
486	C₂×D₄	—	53	2	no	—
487	C₂×D₄	—	212	2	no	—
488	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
489	C₂×D₄	C₂²×D₅₃	1	2	yes	—
490	C₂²×C₄	—	53	2	no	—
491	C₂²×C₄	—	53	1	no	—
492	C₂²×C₄	—	53	1	no	—
493	C₂²×C₄	—	53	1	no	—
494	C₂²×C₄	—	53	1	no	—
495	C₂²×C₄	—	53	2	no	—
496	C₂²×C₄	—	53	2	no	—
497	C₂²×C₄	—	53	2	no	—
498	C₂²×C₄	—	53	2	no	—
499	C₂²×C₄	—	53	2	no	—
500	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
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	1	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	1	no	—
513	C₄⋊C₄	—	53	1	no	—
514	C₄⋊C₄	—	53	2	no	—
515	C₄⋊C₄	—	53	2	no	—
516	C₄⋊C₄	—	53	1	no	—
517	C₄⋊C₄	—	53	2	no	—
518	C₄⋊C₄	—	53	2	no	—
519	C₄⋊C₄	—	53	2	no	—
520	C₄⋊C₄	—	53	1	no	—
521	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
522	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
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	2	yes	—
527	C₂²⋊C₄	—	53	2	no	—
528	C₂²⋊C₄	—	53	1	no	—
529	C₂²⋊C₄	—	53	1	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	2	no	—
541	C₂²⋊C₄	—	53	2	no	—
542	C₂²⋊C₄	—	53	2	no	—
543	C₂²⋊C₄	—	53	2	no	—
544	C₂²⋊C₄	—	53	1	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	2	no	—
550	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
551	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
552	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
553	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
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	1	no	—
561	C₄²	—	53	2	no	—
562	C₄²	—	53	2	no	—
563	C₄²	C₂²×D₅₃	1	1	yes	characteristic
564	C₄²	C₂²×D₅₃	1	1	yes	characteristic
565	C₂³	—	53	1	no	—
566	C₂³	—	53	2	no	—
567	C₂³	—	53	1	no	—
568	C₂³	—	53	2	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	1	yes	characteristic
574	Q₈	—	2	2	no	—
575	Q₈	—	2	2	no	—
576	Q₈	—	2	2	no	—
577	Q₈	—	4	2	no	—
578	Q₈	—	212	1	no	—
579	D₄	—	2	2	no	—
580	D₄	—	4	2	no	—
581	D₄	—	106	2	no	—
582	D₄	—	106	2	no	—
583	D₄	—	212	1	no	—
584	D₄	—	212	2	no	—
585	D₄	—	212	2	no	—
586	D₄	—	212	2	no	—
587	D₄	—	212	2	no	—
588	D₄	—	212	2	no	—
589	D₄	—	212	2	no	—
590	D₄	—	212	2	no	—
591	D₄	—	212	2	no	—
592	D₄	—	212	1	no	—
593	D₄	—	212	2	no	—
594	D₄	—	212	1	no	—
595	D₄	—	212	2	no	—
596	D₄	—	212	2	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	4	2	no	—
599	C₂×C₄	—	4	2	no	—
600	C₂×C₄	—	53	1	no	—
601	C₂×C₄	—	53	1	no	—
602	C₂×C₄	—	53	1	no	—
603	C₂×C₄	—	53	1	no	—
604	C₂×C₄	—	53	2	no	—
605	C₂×C₄	—	53	1	no	—
606	C₂×C₄	—	53	2	no	—
607	C₂×C₄	—	53	1	no	—
608	C₂×C₄	—	106	2	no	—
609	C₂×C₄	—	106	2	no	—
610	C₂×C₄	—	212	2	no	—
611	C₂×C₄	—	212	2	no	—
612	C₂×C₄	—	212	1	no	—
613	C₂×C₄	—	212	1	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	2	yes	—
621	C₂×C₄	C₂³×D₅₃	1	2	yes	—
622	C₂×C₄	C₂³×D₅₃	1	2	yes	—
623	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
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	C₂²	—	2	1	no	—
628	C₂²	—	2	2	no	—
629	C₂²	—	2	2	no	—
630	C₂²	—	2	1	no	—
631	C₂²	—	2	1	no	—
632	C₂²	—	2	2	no	—
633	C₂²	—	106	1	no	—
634	C₂²	—	106	1	no	—
635	C₂²	—	106	2	no	—
636	C₂²	—	106	2	no	—
637	C₂²	—	106	1	no	—
638	C₂²	—	106	2	no	—
639	C₂²	—	106	1	no	—
640	C₂²	—	106	2	no	—
641	C₂²	—	106	2	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	1	no	—
648	C₄	—	2	2	no	—
649	C₄	—	2	2	no	—
650	C₄	—	2	2	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	1	no	—
663	C₄	#13133262	1	2	yes	—
664	C₂	—	4	2	no	—
665	C₂	—	4	1	no	—
666	C₂	—	212	1	no	—
667	C₂	—	212	2	no	—
668	C₂	—	212	1	no	—
669	C₂	—	212	2	no	—
670	C₂	C₂×#13133262	1	1	yes	characteristic
671	C₂	#16811460	1	1	yes	characteristic
672	C₂	#16811460	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#536	C₂
hypercenter	#536	D₅₃
hypocenter	C₅₃	#3562
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #536

Complements

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

N	H ≅ G/N	type
1. #16811369	C₂	semidirect
2. #16811363	C₂	semidirect
3. #16811355	C₂	semidirect
4. #16811347	C₂	semidirect
5. #16811318	C₂	semidirect
6. #16811300	C₂	semidirect
7. #16811292	C₂	semidirect
8. #16811284	C₂	semidirect
9. #16811284	C₂	semidirect
10. #16811276	C₂	semidirect
11. #16811266	C₂	semidirect
12. #16811261	C₂	semidirect
13. #16811234	C₂	semidirect
14. #16811228	C₂	semidirect
15. #16811225	C₂	semidirect
16. #16811219	C₂	semidirect
17. #16811213	C₂	semidirect
18. #16811209	C₂	semidirect
19. #16811191	C₂	semidirect
20. #16811174	C₂	semidirect
21. #16811152	C₂	semidirect
22. #16811151	C₂	semidirect
23. C₅₃×#536	C₂	semidirect
26. C₂×#11719949	C₂²	semidirect
30. #13133213	C₂²	semidirect
34. #13133212	C₂²	semidirect
35. #13133212	C₂²	semidirect
36. #13133210	C₂²	semidirect
37. #13133210	C₂²	semidirect
38. #13133209	C₂²	semidirect
39. #13133207	C₂²	semidirect
40. #13133207	C₂²	semidirect
41. #13133207	C₂²	semidirect
45. #13133206	C₂²	semidirect
46. #13133206	C₂²	semidirect
48. #13133203	C₂²	semidirect
51. #13133202	C₂²	semidirect
52. #13133202	C₂²	semidirect
53. #13133202	C₂²	semidirect
54. #13133201	C₂²	semidirect
57. #13133199	C₂²	semidirect
58. #13133199	C₂²	semidirect
59. #13133199	C₂²	semidirect
60. #13133197	C₂²	semidirect
61. #13133195	C₂²	semidirect
64. #13133193	C₂²	semidirect
65. #13133193	C₂²	semidirect
66. #13133193	C₂²	semidirect
68. #13133192	C₂²	semidirect
69. #13133192	C₂²	semidirect
72. #13133191	C₂²	semidirect
73. #13133191	C₂²	semidirect
74. #13133191	C₂²	semidirect
79. #13133190	C₂²	semidirect
80. #13133190	C₂²	semidirect
83. #13133188	C₂²	semidirect
88. Q₈×Dic₅₃	C₂²	semidirect
89. Q₈×Dic₅₃	C₂²	semidirect
90. Q₈×Dic₅₃	C₂²	semidirect
91. C₄×Dic₁₀₆	C₂²	semidirect
94. C₄×D₂₁₂	C₂²	semidirect
96. #13133179	C₂²	semidirect
97. #13133179	C₂²	semidirect
99. C₄×C₂²⋊D₅₃	C₂²	semidirect
100. C₄×C₂²⋊D₅₃	C₂²	semidirect
101. C₄×C₂²⋊D₅₃	C₂²	semidirect
102. C₄×C₂²⋊D₅₃	C₂²	semidirect
103. #13133177	C₂²	semidirect
104. #13133177	C₂²	semidirect
105. #13133175	C₂²	semidirect
107. #13133173	C₂²	semidirect
108. C₄⋊C₄×D₅₃	C₂²	semidirect
111. C₄²×D₅₃	C₂²	semidirect
113. C₂²×Q₈×C₅₃	C₂²	semidirect
114. C₄⋊Q₈×C₅₃	C₂²	semidirect
115. C₄⋊Q₈×C₅₃	C₂²	semidirect
118. #123×C₅₃	C₂²	semidirect
119. #123×C₅₃	C₂²	semidirect
120. C₂²⋊Q₈×C₅₃	C₂²	semidirect
121. C₂²⋊Q₈×C₅₃	C₂²	semidirect
122. #117×C₅₃	C₂²	semidirect
355. #536	D₅₃	semidirect
360. C₅₃	#3562	semidirect
365. C₂²×Q₈	C₂×D₅₃	semidirect
367. C₄⋊Q₈	C₂×D₅₃	semidirect
368. C₄⋊Q₈	C₂×D₅₃	semidirect
403. #123	C₂×D₅₃	semidirect
404. #123	C₂×D₅₃	semidirect
412. C₂²⋊Q₈	C₂×D₅₃	semidirect
413. C₂²⋊Q₈	C₂×D₅₃	semidirect
455. #117	C₂×D₅₃	semidirect