Subgroups — #27405883

Order 13568.

788 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810751	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810651	C₂	1	1	yes	maximal, characteristic
3	#20277989	C₂	1	1	yes	maximal, characteristic
4	#20275653	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275437	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275433	C₂	1	1	yes	maximal, characteristic, complemented
7	#20275432	C₂	1	1	yes	maximal, characteristic, complemented
8	#20275279	C₂	1	1	yes	maximal, characteristic, complemented
9	#20275278	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274898	C₂	1	1	yes	maximal, characteristic
11	#20274895	C₂	1	1	yes	maximal, characteristic
12	#20274589	C₂	1	1	yes	maximal, characteristic, complemented
13	#20274494	C₂	1	1	yes	maximal, characteristic
14	#20274483	C₂	1	1	yes	maximal, characteristic
15	C₅₃×#1753	C₂	1	1	yes	maximal, characteristic
16	C₂³×#88322	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
18	C₂×#13133176	C₂²	1	1	yes	characteristic
19	#16810751	—	2	4	no	—
20	#16810651	—	2	4	no	—
21	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
23	C₂×#13133146	C₂²	1	1	yes	characteristic
24	C₂×#13133146	C₂²	1	1	yes	characteristic
25	C₂×#13133146	C₂²	1	1	yes	characteristic
26	C₂×#13133146	C₂²	1	1	yes	characteristic
27	C₂×#13133146	C₂²	1	1	yes	characteristic
28	C₂×#13133146	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
30	C₂×#13133127	C₂²	1	1	yes	characteristic
31	C₂×#13133127	C₂²	1	1	yes	characteristic
32	C₂×#13133127	C₂²	1	1	yes	characteristic
33	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
34	#16810565	C₄	1	1	yes	characteristic
35	#16810559	C₄	1	1	yes	characteristic
36	#16810557	C₄	1	1	yes	characteristic
37	#16810555	C₄	1	1	yes	characteristic
38	#16810546	C₄	1	1	yes	characteristic
39	#16810546	C₄	1	1	yes	characteristic
40	#16810544	C₄	1	1	yes	characteristic
41	#16810535	C₄	1	1	yes	characteristic
42	C₄×#11719919	C₂²	1	1	yes	characteristic
43	C₄×#11719919	C₂²	1	1	yes	characteristic
44	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
45	C₄×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic
46	#16810378	C₂²	1	1	yes	characteristic
47	#16810378	C₂²	1	1	yes	characteristic
48	#16810377	C₂²	1	1	yes	characteristic
49	#16810377	C₂²	1	1	yes	characteristic
50	#16810377	C₂²	1	1	yes	characteristic
51	#16810377	C₂²	1	1	yes	characteristic
52	#16810377	C₂²	1	1	yes	characteristic
53	#16810377	C₂²	1	1	yes	characteristic
54	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
55	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
56	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
57	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
58	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
59	C₅₃×#336	C₂²	1	1	yes	characteristic
60	C₅₃×#336	C₂²	1	1	yes	characteristic
61	C₂²×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
62	C₂²×#88322	—	2	2	no	—
63	C₂²×#88322	—	2	2	no	—
64	C₂²×#88322	—	4	1	no	—
65	C₂²×#88322	C₂³	1	1	yes	characteristic
66	C₂²×#88322	C₂³	1	1	yes	characteristic
67	C₂²×#88322	C₂³	1	1	yes	characteristic
68	C₂²×#88322	C₂³	1	1	yes	characteristic
69	C₄×Dic₁₀₆	—	2	2	no	—
70	C₄×Dic₁₀₆	—	2	2	no	—
71	#13133176	—	2	2	no	—
72	#13133176	—	2	2	no	—
73	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
79	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
80	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
81	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
82	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
83	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
84	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
85	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
86	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
87	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
89	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
90	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
91	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
92	C₄⋊#88322	—	2	2	no	—
93	C₄⋊#88322	—	4	1	no	—
94	#13133146	—	2	2	no	—
95	#13133146	—	2	2	no	—
96	#13133146	—	2	2	no	—
97	#13133146	—	2	2	no	—
98	#13133146	—	4	1	no	—
99	#13133146	D₄	1	4	yes	—
100	C₂²⋊#88322	—	2	2	no	—
101	#13133127	—	2	2	no	—
102	#13133127	—	2	2	no	—
103	#13133127	—	4	1	no	—
104	C₈×Dic₅₃	—	2	2	no	—
105	#13133124	C₂×C₄	1	1	yes	characteristic
106	#13133124	C₂×C₄	1	1	yes	characteristic
107	#13133124	C₂×C₄	1	1	yes	characteristic
108	#13133124	C₂×C₄	1	1	yes	characteristic
109	#13133124	C₂×C₄	1	1	yes	characteristic
110	#13133124	C₂×C₄	1	1	yes	characteristic
111	#13133124	C₂×C₄	1	1	yes	characteristic
112	#13133124	C₂×C₄	1	1	yes	characteristic
113	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
114	C₂×M₁₆×C₅₃	—	2	2	no	—
115	C₂×M₁₆×C₅₃	—	2	2	no	—
116	C₂×M₁₆×C₅₃	—	4	1	no	—
117	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
118	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
119	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
120	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
121	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
122	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
123	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
125	C₂²⋊C₈×C₅₃	Q₈	1	4	yes	complemented
126	#97×C₅₃	—	2	2	no	—
127	C₄×C₈×C₅₃	—	4	1	no	—
128	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
129	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
130	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₅₃	—	2	1	no	—
132	C₂²×C₄×C₅₃	—	2	1	no	—
133	C₂²×C₄×C₅₃	—	2	1	no	—
134	C₂²×C₄×C₅₃	—	2	1	no	—
135	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
141	C₂²×C₄×C₅₃	#41	1	2	yes	—
142	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
143	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₂×Dic₁₀₆	—	2	2	no	—
152	C₂×Dic₁₀₆	—	4	1	no	—
153	C₂×Dic₁₀₆	—	4	1	no	—
154	M₁₆×C₅₃	—	4	4	no	—
155	C₂×C₈×C₅₃	—	2	1	no	—
156	C₂×C₈×C₅₃	—	2	1	no	—
157	C₂×C₈×C₅₃	—	2	1	no	—
158	C₂×C₈×C₅₃	—	2	1	no	—
159	C₂×C₈×C₅₃	—	2	1	no	—
160	C₂×C₈×C₅₃	—	2	1	no	—
161	C₂×C₈×C₅₃	—	2	1	no	—
162	C₂×C₈×C₅₃	—	2	1	no	—
163	C₂×C₈×C₅₃	—	2	1	no	—
164	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
165	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
166	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
167	C₂²⋊C₄×C₅₃	—	2	2	no	—
168	C₂²⋊C₄×C₅₃	—	4	1	no	—
169	C₄²×C₅₃	—	2	1	no	—
170	C₄²×C₅₃	—	2	1	no	—
171	C₄²×C₅₃	—	2	1	no	—
172	C₄²×C₅₃	—	2	1	no	—
173	C₂²⋊Dic₅₃	—	2	2	no	—
174	C₂²⋊Dic₅₃	—	2	2	no	—
175	C₂²⋊Dic₅₃	—	4	1	no	—
176	C₂²⋊Dic₅₃	—	4	1	no	—
177	C₄⋊Dic₅₃	—	2	2	no	—
178	C₄⋊Dic₅₃	—	2	2	no	—
179	C₄⋊Dic₅₃	—	2	2	no	—
180	C₄⋊Dic₅₃	—	2	2	no	—
181	C₄⋊Dic₅₃	—	4	1	no	—
182	#11719919	—	2	2	no	—
183	#11719919	—	2	2	no	—
184	#11719919	—	2	2	no	—
185	#11719919	—	4	1	no	—
186	#11719919	—	4	1	no	—
187	#11719919	—	4	1	no	—
188	C₄×Dic₅₃	—	2	1	no	—
189	C₄×Dic₅₃	—	2	1	no	—
190	C₄×Dic₅₃	—	2	1	no	—
191	C₄×Dic₅₃	—	2	1	no	—
192	C₄×Dic₅₃	—	2	1	no	—
193	C₄×Dic₅₃	—	2	1	no	—
194	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
195	C₄×Dic₅₃	#41	1	2	yes	—
196	C₂×#88322	—	2	1	no	—
197	C₂×#88322	—	2	1	no	—
198	C₂×#88322	—	2	1	no	—
199	C₂×#88322	—	2	1	no	—
200	C₂×#88322	—	2	1	no	—
201	C₂×#88322	—	2	1	no	—
202	C₂×#88322	—	2	1	no	—
203	C₂×#88322	—	2	1	no	—
204	C₂×#88322	—	2	1	no	—
205	C₂×#88322	—	2	1	no	—
206	C₂×#88322	—	2	1	no	—
207	C₂×#88322	—	4	4	no	—
208	C₂×#88322	C₂×D₄	1	2	yes	—
209	C₂³×C₅₃	—	2	1	no	—
210	C₂³×C₅₃	—	2	1	no	—
211	C₂³×C₅₃	—	2	1	no	—
212	C₂³×C₅₃	—	2	1	no	—
213	C₂³×C₅₃	—	2	1	no	—
214	C₂³×C₅₃	—	2	1	no	—
215	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
216	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
217	C₂×C₄×C₅₃	—	2	1	no	—
218	C₂×C₄×C₅₃	—	2	1	no	—
219	C₂×C₄×C₅₃	—	2	1	no	—
220	C₂×C₄×C₅₃	—	2	1	no	—
221	C₂×C₄×C₅₃	—	2	2	no	—
222	C₂×C₄×C₅₃	—	2	2	no	—
223	C₂×C₄×C₅₃	—	2	2	no	—
224	C₂×C₄×C₅₃	—	2	2	no	—
225	C₂×C₄×C₅₃	—	2	1	no	—
226	C₂×C₄×C₅₃	—	2	1	no	—
227	C₂×C₄×C₅₃	—	2	1	no	—
228	C₂×C₄×C₅₃	—	2	1	no	—
229	C₂×C₄×C₅₃	—	2	1	no	—
230	C₂×C₄×C₅₃	—	2	1	no	—
231	C₂×C₄×C₅₃	—	2	1	no	—
232	C₂×C₄×C₅₃	—	2	1	no	—
233	C₂×C₄×C₅₃	—	2	1	no	—
234	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
235	C₂×C₄×C₅₃	#131	1	2	yes	—
236	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
238	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
239	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
240	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
241	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
242	C₂×Dic₅₃	—	2	1	no	—
243	C₂×Dic₅₃	—	2	1	no	—
244	C₂×Dic₅₃	—	2	1	no	—
245	C₂×Dic₅₃	—	2	1	no	—
246	C₂×Dic₅₃	—	2	1	no	—
247	C₂×Dic₅₃	—	2	1	no	—
248	C₂×Dic₅₃	—	2	1	no	—
249	C₂×Dic₅₃	—	2	1	no	—
250	C₂×Dic₅₃	—	2	1	no	—
251	C₂×Dic₅₃	—	2	1	no	—
252	C₂×Dic₅₃	—	2	1	no	—
253	C₂×Dic₅₃	—	2	1	no	—
254	C₂×Dic₅₃	—	2	1	no	—
255	C₂×Dic₅₃	—	2	1	no	—
256	C₂×Dic₅₃	—	2	1	no	—
257	C₂×Dic₅₃	—	2	1	no	—
258	C₂×Dic₅₃	—	2	1	no	—
259	C₂×Dic₅₃	—	2	1	no	—
260	C₂×Dic₅₃	—	2	1	no	—
261	C₂×Dic₅₃	—	2	1	no	—
262	C₂×Dic₅₃	—	2	1	no	—
263	C₂×Dic₅₃	—	2	1	no	—
264	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
265	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
266	Dic₁₀₆	—	4	4	no	—
267	C₈×C₅₃	—	2	2	no	—
268	C₈×C₅₃	—	2	2	no	—
269	C₈×C₅₃	—	2	2	no	—
270	C₈×C₅₃	—	4	1	no	—
271	#88322	—	2	2	no	—
272	#88322	—	4	1	no	—
273	#88322	—	4	1	no	—
274	#88322	—	4	1	no	—
275	#22961	—	53	1	no	maximal, Sylow
276	C₂²×C₅₃	—	2	2	no	—
277	C₂²×C₅₃	—	2	2	no	—
278	C₂²×C₅₃	—	2	2	no	—
279	C₂²×C₅₃	—	4	1	no	—
280	C₂²×C₅₃	—	4	1	no	—
281	C₂²×C₅₃	—	4	1	no	—
282	C₂²×C₅₃	—	4	1	no	—
283	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
284	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
285	C₂²×C₅₃	#520	1	1	yes	characteristic
286	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
287	C₂²×C₅₃	#568	1	1	yes	characteristic
288	C₂²×C₅₃	#568	1	1	yes	characteristic
289	C₂²×C₅₃	#568	1	1	yes	characteristic
290	C₄×C₅₃	—	2	2	no	—
291	C₄×C₅₃	—	2	2	no	—
292	C₄×C₅₃	—	4	1	no	—
293	C₄×C₅₃	—	4	1	no	—
294	C₄×C₅₃	—	4	1	no	—
295	C₄×C₅₃	#546	1	1	yes	characteristic
296	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
297	C₄×C₅₃	#552	1	1	yes	characteristic
298	C₄×C₅₃	#554	1	1	yes	characteristic
299	Dic₅₃	—	2	2	no	—
300	Dic₅₃	—	2	2	no	—
301	Dic₅₃	—	4	1	no	—
302	Dic₅₃	—	4	1	no	—
303	Dic₅₃	—	4	1	no	—
304	Dic₅₃	—	4	1	no	—
305	Dic₅₃	—	4	1	no	—
306	Dic₅₃	—	4	1	no	—
307	C₂×#446	—	53	1	no	—
308	C₂×#424	—	53	1	no	—
309	#2396	—	53	1	no	—
310	#1985	—	53	1	no	—
311	#1940	—	53	1	no	—
312	#1939	—	53	1	no	—
313	#1939	—	53	1	no	—
314	#1917	—	53	1	no	—
315	#1917	—	53	1	no	—
316	#1851	—	53	1	no	—
317	#1850	—	53	1	no	—
318	#1771	—	53	1	no	—
319	#1753	D₅₃	1	1	yes	characteristic
320	#1752	—	53	1	no	—
321	#1750	—	53	1	no	—
322	C₂×C₅₃	—	4	2	no	—
323	C₂×C₅₃	#2360	1	1	yes	characteristic
324	C₂×C₅₃	#2930	1	1	yes	characteristic
325	C₂×C₅₃	#2935	1	1	yes	characteristic
326	C₂×C₅₃	#2961	1	1	yes	characteristic
327	C₂×C₅₃	#2962	1	1	yes	characteristic
328	C₂×C₅₃	#2992	1	1	yes	characteristic
329	C₂×C₅₃	#2995	1	1	yes	characteristic
330	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
331	C₂³×C₈	—	53	1	no	—
332	C₂×C₄×Q₈	—	53	1	no	—
333	C₂×#117	—	53	1	no	—
334	#446	—	106	4	no	—
335	#424	—	106	4	no	—
336	C₂×C₄⋊C₈	—	53	1	no	—
337	C₂×C₄⋊C₈	—	53	1	no	—
338	C₂×C₄⋊C₈	—	53	1	no	—
339	C₂×C₄⋊C₈	—	53	1	no	—
340	C₂×C₄⋊C₈	—	53	1	no	—
341	C₂×C₄⋊C₈	—	53	1	no	—
342	C₂×C₄⋊C₈	—	53	1	no	—
343	C₂×C₄⋊C₈	—	53	1	no	—
344	C₂×C₂²⋊C₈	—	53	1	no	—
345	C₂×C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
346	C₂×#97	—	53	1	no	—
347	C₂×#97	—	53	1	no	—
348	C₂×#97	—	53	1	no	—
349	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
350	C₂×C₄×C₈	—	53	1	no	—
351	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
352	#400	—	53	1	no	—
353	#399	—	53	1	no	—
354	#398	—	53	1	no	—
355	#398	—	53	1	no	—
356	#396	—	53	1	no	—
357	#396	—	53	1	no	—
358	#395	—	53	1	no	—
359	#393	—	53	1	no	—
360	C₄×C₄⋊C₄	—	53	1	no	—
361	C₄×C₄⋊C₄	—	53	1	no	—
362	C₄×C₄⋊C₄	—	53	1	no	—
363	C₄×C₂²⋊C₄	—	53	1	no	—
364	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
365	#336	—	53	1	no	—
366	#336	—	53	1	no	—
367	#336	—	53	1	no	—
368	#336	—	53	1	no	—
369	#336	—	53	1	no	—
370	#336	—	53	1	no	—
371	#336	—	53	1	no	—
372	#336	—	53	1	no	—
373	#336	C₂×D₅₃	1	1	yes	characteristic
374	#336	C₂×D₅₃	1	1	yes	characteristic
375	C₅₃	#22961	1	1	yes	characteristic, complemented, Sylow
376	C₂²×Q₈	—	53	1	no	—
377	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
378	C₂×M₁₆	—	2	2	no	—
379	C₂×M₁₆	—	2	2	no	—
380	C₂×M₁₆	—	4	1	no	—
381	C₂²×C₈	—	53	1	no	—
382	C₂²×C₈	—	53	1	no	—
383	C₂²×C₈	—	53	1	no	—
384	C₂²×C₈	—	53	1	no	—
385	C₂²×C₈	—	106	2	no	—
386	C₂²×C₈	—	106	2	no	—
387	C₂²×C₈	—	212	1	no	—
388	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
389	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
390	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
391	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
392	C₄×Q₈	—	106	2	no	—
393	C₄×Q₈	—	106	2	no	—
394	#117	—	106	2	no	—
395	#117	—	106	2	no	—
396	C₂×C₄⋊C₄	—	53	1	no	—
397	C₂×C₄⋊C₄	—	53	1	no	—
398	C₂×C₄⋊C₄	—	53	1	no	—
399	C₂×C₄⋊C₄	—	53	1	no	—
400	C₂×C₄⋊C₄	—	53	1	no	—
401	C₂×C₄⋊C₄	—	53	1	no	—
402	C₂×C₄⋊C₄	—	53	1	no	—
403	C₂×C₄⋊C₄	—	53	1	no	—
404	C₂×C₄⋊C₄	—	53	1	no	—
405	C₂×C₄⋊C₄	—	53	1	no	—
406	C₂×C₄⋊C₄	—	53	1	no	—
407	C₂×C₂²⋊C₄	—	53	1	no	—
408	C₂×C₂²⋊C₄	—	53	1	no	—
409	C₂×C₂²⋊C₄	—	53	1	no	—
410	C₂×C₂²⋊C₄	—	53	1	no	—
411	C₂×C₂²⋊C₄	C₄×D₅₃	1	1	yes	characteristic
412	C₂×C₂²⋊C₄	C₄×D₅₃	1	1	yes	characteristic
413	C₂×C₄²	—	53	1	no	—
414	C₂×C₄²	—	53	1	no	—
415	C₂×C₄²	—	53	1	no	—
416	C₂×C₄²	—	53	1	no	—
417	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
418	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
419	C₄⋊C₈	—	53	4	no	—
420	C₄⋊C₈	—	106	2	no	—
421	C₄⋊C₈	—	106	2	no	—
422	C₄⋊C₈	—	106	2	no	—
423	C₄⋊C₈	—	106	2	no	—
424	C₄⋊C₈	—	106	2	no	—
425	C₄⋊C₈	—	212	1	no	—
426	C₄⋊C₈	—	212	1	no	—
427	C₂²⋊C₈	—	106	2	no	—
428	C₂²⋊C₈	Dic₁₀₆	1	4	yes	complemented
429	#97	—	2	2	no	—
430	#97	—	106	2	no	—
431	#97	—	106	2	no	—
432	#97	—	212	1	no	—
433	C₄×C₈	—	4	1	no	—
434	C₄×C₈	—	106	2	no	—
435	#95	—	53	1	no	—
436	#95	—	53	1	no	—
437	#95	—	53	1	no	—
438	#95	—	53	1	no	—
439	#95	—	53	1	no	—
440	#95	—	53	1	no	—
441	#95	—	53	1	no	—
442	#95	—	53	1	no	—
443	#95	C₄×D₅₃	1	1	yes	characteristic
444	#95	C₄×D₅₃	1	1	yes	characteristic
445	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
446	C₂×Q₈	—	106	2	no	—
447	C₂×Q₈	—	212	1	no	—
448	C₂×Q₈	—	212	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	53	1	no	—
454	C₂²×C₄	—	53	1	no	—
455	C₂²×C₄	—	53	1	no	—
456	C₂²×C₄	—	53	1	no	—
457	C₂²×C₄	—	53	1	no	—
458	C₂²×C₄	—	53	1	no	—
459	C₂²×C₄	—	53	1	no	—
460	C₂²×C₄	—	53	1	no	—
461	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
462	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
463	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
464	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
465	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
466	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
467	C₂²×C₄	#11719945	1	2	yes	—
468	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
469	M₁₆	—	4	4	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	53	2	no	—
480	C₂×C₈	—	106	1	no	—
481	C₂×C₈	—	106	1	no	—
482	C₂×C₈	—	106	1	no	—
483	C₂×C₈	—	106	1	no	—
484	C₂×C₈	—	106	1	no	—
485	C₂×C₈	—	106	1	no	—
486	C₂×C₈	—	106	1	no	—
487	C₂×C₈	—	106	1	no	—
488	C₂×C₈	—	106	1	no	—
489	C₂×C₈	—	106	1	no	—
490	C₂×C₈	—	106	1	no	—
491	C₂×C₈	—	212	4	no	—
492	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
493	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
494	C₂×C₈	D₄×D₅₃	1	2	yes	—
495	C₄⋊C₄	—	106	2	no	—
496	C₄⋊C₄	—	106	2	no	—
497	C₄⋊C₄	—	106	2	no	—
498	C₄⋊C₄	—	106	2	no	—
499	C₄⋊C₄	—	106	2	no	—
500	C₄⋊C₄	—	106	2	no	—
501	C₄⋊C₄	—	106	2	no	—
502	C₄⋊C₄	—	212	1	no	—
503	C₄⋊C₄	—	212	1	no	—
504	C₄⋊C₄	—	212	1	no	—
505	C₄⋊C₄	—	212	1	no	—
506	C₂²⋊C₄	—	2	2	no	—
507	C₂²⋊C₄	—	4	1	no	—
508	C₂²⋊C₄	—	106	2	no	—
509	C₂²⋊C₄	—	106	2	no	—
510	C₂²⋊C₄	—	212	1	no	—
511	C₂²⋊C₄	—	212	1	no	—
512	C₄²	—	2	1	no	—
513	C₄²	—	2	1	no	—
514	C₄²	—	2	1	no	—
515	C₄²	—	2	1	no	—
516	C₄²	—	53	2	no	—
517	C₄²	—	53	2	no	—
518	C₄²	—	106	1	no	—
519	C₄²	—	106	1	no	—
520	C₄²	—	106	1	no	—
521	C₄²	—	106	1	no	—
522	C₄²	—	106	1	no	—
523	C₄²	—	106	1	no	—
524	C₂³	—	2	1	no	—
525	C₂³	—	2	1	no	—
526	C₂³	—	2	1	no	—
527	C₂³	—	2	1	no	—
528	C₂³	—	2	1	no	—
529	C₂³	—	2	1	no	—
530	C₂³	C₄×Dic₁₀₆	1	2	yes	—
531	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
532	Q₈	—	212	4	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	53	2	no	—
551	C₂×C₄	—	53	2	no	—
552	C₂×C₄	—	106	1	no	—
553	C₂×C₄	—	106	1	no	—
554	C₂×C₄	—	106	1	no	—
555	C₂×C₄	—	106	1	no	—
556	C₂×C₄	—	106	1	no	—
557	C₂×C₄	—	106	1	no	—
558	C₂×C₄	—	106	1	no	—
559	C₂×C₄	—	106	1	no	—
560	C₂×C₄	—	106	1	no	—
561	C₂×C₄	—	106	1	no	—
562	C₂×C₄	—	106	1	no	—
563	C₂×C₄	—	106	1	no	—
564	C₂×C₄	—	106	1	no	—
565	C₂×C₄	—	106	1	no	—
566	C₂×C₄	—	106	1	no	—
567	C₂×C₄	—	106	1	no	—
568	C₂×C₄	—	106	1	no	—
569	C₂×C₄	—	106	1	no	—
570	C₂×C₄	—	106	1	no	—
571	C₂×C₄	—	106	1	no	—
572	C₂×C₄	—	106	1	no	—
573	C₂×C₄	—	106	1	no	—
574	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
575	C₂×C₄	#13133225	1	2	yes	—
576	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
577	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
578	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
579	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
580	C₂×C₄	#13133262	1	1	yes	characteristic
581	C₂×C₄	#13133264	1	1	yes	characteristic
582	C₈	—	2	2	no	—
583	C₈	—	2	2	no	—
584	C₈	—	2	2	no	—
585	C₈	—	4	1	no	—
586	C₈	—	106	2	no	—
587	C₈	—	212	1	no	—
588	C₈	—	212	1	no	—
589	C₈	—	212	1	no	—
590	C₂²	—	2	2	no	—
591	C₂²	—	2	2	no	—
592	C₂²	—	2	2	no	—
593	C₂²	—	4	1	no	—
594	C₂²	—	4	1	no	—
595	C₂²	—	4	1	no	—
596	C₂²	—	4	1	no	—
597	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
598	C₂²	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
599	C₂²	#16811102	1	1	yes	characteristic
600	C₂²	C₂×#13133225	1	1	yes	characteristic
601	C₂²	#16811388	1	1	yes	characteristic
602	C₂²	#16811388	1	1	yes	characteristic
603	C₂²	#16811389	1	1	yes	characteristic
604	C₄	—	2	2	no	—
605	C₄	—	2	2	no	—
606	C₄	—	4	1	no	—
607	C₄	—	4	1	no	—
608	C₄	—	4	1	no	—
609	C₄	—	106	2	no	—
610	C₄	—	106	2	no	—
611	C₄	—	212	1	no	—
612	C₄	—	212	1	no	—
613	C₄	—	212	1	no	—
614	C₄	—	212	1	no	—
615	C₄	—	212	1	no	—
616	C₄	—	212	1	no	—
617	C₄	#16811263	1	1	yes	characteristic
618	C₄	D₄×Dic₁₀₆	1	1	yes	characteristic
619	C₄	#16811306	1	1	yes	characteristic
620	C₄	#16811332	1	1	yes	characteristic
621	C₂	—	4	2	no	—
622	C₂	#20277721	1	1	yes	characteristic
623	C₂	#20283415	1	1	yes	characteristic
624	C₂	#20283460	1	1	yes	characteristic
625	C₂	#20283653	1	1	yes	characteristic
626	C₂	#20283662	1	1	yes	characteristic
627	C₂	#20283855	1	1	yes	characteristic
628	C₂	#20283881	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#1753	C₂
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1753	D₅₃
hypocenter	C₅₃	#22961
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1753

Complements

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

N	H ≅ G/N	type
1. C₂×#16810751	C₂	semidirect
4. #20275653	C₂	semidirect
5. #20275437	C₂	semidirect
6. #20275433	C₂	semidirect
7. #20275432	C₂	semidirect
8. #20275279	C₂	semidirect
9. #20275278	C₂	semidirect
12. #20274589	C₂	semidirect
125. C₂²⋊C₈×C₅₃	Q₈	semidirect
375. C₅₃	#22961	semidirect
428. C₂²⋊C₈	Dic₁₀₆	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16810751	2
C₂×#16810651	2
#20277989	2
#20275653	2
#20275437	2
#20275433	2
#20275432	2
#20275279	2
#20275278	2
#20274898	2
#20274895	2
#20274589	2
#20274494	2
#20274483	2
C₅₃×#1753	2
#22961	53