Subgroups — #27537612

Order 13568.

558 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11719947	C₂	1	1	yes	maximal, characteristic
2	#20276622	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276621	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276545	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276544	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276541	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276540	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276537	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276513	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276511	C₂	1	1	yes	maximal, characteristic, complemented
11	#20276511	C₂	1	1	yes	maximal, characteristic, complemented
12	#20276308	C₂	1	1	yes	maximal, characteristic, complemented
13	M₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#2168	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×#11719947	C₄	1	1	yes	characteristic
17	C₂×#13133157	C₂²	1	1	yes	characteristic
18	C₂×#13133156	C₂²	1	1	yes	characteristic
19	C₂×#13133156	C₂²	1	1	yes	characteristic
20	C₂²×#11719908	C₂²	1	1	yes	characteristic
21	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
22	#16810747	C₄	1	1	yes	characteristic
23	D₄×#88322	C₄	1	1	yes	characteristic
24	C₈×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
25	C₈×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
26	#16810686	C₄	1	1	yes	characteristic
27	#16810686	C₄	1	1	yes	characteristic
28	#16810675	C₄	1	1	yes	characteristic
29	#16810673	C₄	1	1	yes	characteristic
30	#16810673	C₄	1	1	yes	characteristic
31	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
32	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
34	C₄⋊#11719908	C₂²	1	1	yes	characteristic
35	#16810441	C₂²	1	1	yes	characteristic
36	#16810441	C₂²	1	1	yes	characteristic
37	#16810441	C₂²	1	1	yes	characteristic
38	#16810441	C₂²	1	1	yes	characteristic
39	C₂²⋊#11719908	C₂²	1	1	yes	characteristic
40	C₂²⋊#11719908	C₂²	1	1	yes	characteristic
41	#16810402	C₂²	1	1	yes	characteristic
42	#16810402	C₂²	1	1	yes	characteristic
43	#16810399	C₂²	1	1	yes	characteristic
44	#16810398	C₂²	1	1	yes	characteristic
45	#16810398	C₂²	1	1	yes	characteristic
46	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
47	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
48	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
49	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
50	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
51	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
52	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
53	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
54	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
55	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
56	C₅₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
57	C₅₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
58	C₅₃×#346	C₂²	1	1	yes	characteristic
59	C₂×#11719947	C₈	1	1	yes	characteristic
60	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
61	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
62	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
64	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
65	#13133182	C₈	1	1	yes	characteristic
66	D₄×Dic₅₃	C₈	1	1	yes	characteristic
67	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
68	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
69	#13133177	C₈	1	1	yes	characteristic
70	#13133177	C₈	1	1	yes	characteristic
71	#13133175	C₈	1	1	yes	characteristic
72	#13133173	C₈	1	1	yes	characteristic
73	#13133173	C₈	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	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
77	#13133157	—	2	2	no	—
78	#13133156	—	2	2	no	—
79	#13133156	—	2	2	no	—
80	C₂×#11719908	—	2	2	no	—
81	C₂×#11719908	C₂³	1	1	yes	characteristic
82	C₂×#11719908	C₂³	1	1	yes	characteristic
83	C₂×#11719908	C₂³	1	1	yes	characteristic
84	C₂×#11719908	C₂³	1	1	yes	characteristic
85	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
86	#13133146	C₂×C₄	1	1	yes	characteristic
87	#13133146	C₂×C₄	1	1	yes	characteristic
88	#13133146	C₂×C₄	1	1	yes	characteristic
89	#13133146	C₂×C₄	1	1	yes	characteristic
90	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
91	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
92	#13133129	C₂×C₄	1	1	yes	characteristic
93	#13133129	C₂×C₄	1	1	yes	characteristic
94	C₄×#88322	C₂×C₄	1	1	yes	characteristic
95	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
96	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
97	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
98	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
99	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
100	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
101	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
102	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
103	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
104	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
105	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
106	M₃₂×C₅₃	—	2	2	no	—
107	C₂×C₁₆×C₅₃	—	2	2	no	—
108	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
109	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
110	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
111	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
112	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
113	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
114	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
115	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
116	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
120	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
121	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
122	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
123	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
124	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
125	#11719947	—	2	4	no	—
126	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
127	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
128	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
130	C₂×C₈×C₅₃	—	2	1	no	—
131	C₂×C₈×C₅₃	—	2	1	no	—
132	C₂×C₈×C₅₃	—	2	1	no	—
133	C₂×C₈×C₅₃	—	2	1	no	—
134	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
138	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
139	C₂²⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
140	C₂²⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
141	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₂²⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
143	C₂²⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
144	#11719921	C₂×C₈	1	1	yes	characteristic
145	#11719921	C₂×C₈	1	1	yes	characteristic
146	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
147	#11719919	C₂×C₈	1	1	yes	characteristic
148	#11719919	C₂×C₈	1	1	yes	characteristic
149	#11719919	C₂×C₈	1	1	yes	characteristic
150	#11719919	C₂×C₈	1	1	yes	characteristic
151	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
152	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
153	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
154	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
155	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
156	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
157	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
158	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
159	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
160	C₂×#88322	—	2	2	no	—
161	C₂×#88322	—	2	2	no	—
162	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
163	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
164	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
165	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
166	C₈×D₅₃	—	2	1	no	—
167	C₈×D₅₃	—	2	1	no	—
168	C₈×D₅₃	—	2	2	no	—
169	C₁₆×C₅₃	—	2	1	no	—
170	C₁₆×C₅₃	—	2	1	no	—
171	C₁₆×C₅₃	—	2	1	no	—
172	C₁₆×C₅₃	#41	1	2	yes	complemented
173	#11719908	—	2	1	no	—
174	#11719908	—	2	1	no	—
175	#11719908	—	2	1	no	—
176	#11719908	#41	1	2	yes	complemented
177	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
178	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
179	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
180	D₄×C₅₃	—	2	2	no	—
181	C₂×C₄×C₅₃	—	2	1	no	—
182	C₂×C₄×C₅₃	—	2	1	no	—
183	C₂×C₄×C₅₃	—	2	1	no	—
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
186	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
187	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
188	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
190	C₂²⋊D₅₃	—	2	2	no	—
191	C₂²⋊D₅₃	—	2	2	no	—
192	C₂×Dic₅₃	—	2	2	no	—
193	C₂×Dic₅₃	—	2	2	no	—
194	C₂×Dic₅₃	—	2	2	no	—
195	C₂×Dic₅₃	M₃₂	1	4	yes	complemented
196	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
197	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
198	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
199	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
200	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
201	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
202	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
203	C₄×D₅₃	—	2	1	no	—
204	C₄×D₅₃	—	2	1	no	—
205	C₄×D₅₃	—	2	2	no	—
206	C₄×D₅₃	—	2	2	no	—
207	Dic₁₀₆	—	2	2	no	—
208	C₈×C₅₃	—	2	1	no	—
209	C₈×C₅₃	—	2	1	no	—
210	C₈×C₅₃	—	2	1	no	—
211	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
212	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
213	#88322	—	2	1	no	—
214	#88322	—	2	1	no	—
215	#88322	—	2	1	no	—
216	#88322	#131	1	2	yes	—
217	#33479	—	53	1	no	maximal, Sylow
218	C₂²×C₅₃	—	2	1	no	—
219	C₂²×C₅₃	—	2	1	no	—
220	C₂²×C₅₃	—	2	1	no	—
221	C₂²×C₅₃	—	2	1	no	—
222	C₂²×C₅₃	—	2	1	no	—
223	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
224	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
225	C₂×D₅₃	—	2	1	no	—
226	C₂×D₅₃	—	2	1	no	—
227	C₂×D₅₃	—	2	1	no	—
228	C₄×C₅₃	—	2	1	no	—
229	C₄×C₅₃	—	2	1	no	—
230	C₄×C₅₃	—	2	1	no	—
231	C₄×C₅₃	—	2	1	no	—
232	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
233	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
234	Dic₅₃	—	2	1	no	—
235	Dic₅₃	—	2	1	no	—
236	Dic₅₃	—	2	1	no	—
237	Dic₅₃	—	2	1	no	—
238	Dic₅₃	C₂×M₃₂	1	2	yes	—
239	Dic₅₃	C₂×M₃₂	1	2	yes	—
240	Dic₅₃	#504	1	2	yes	—
241	C₈×#41	—	53	1	no	—
242	#2183	—	53	1	no	—
243	#2183	—	53	1	no	—
244	#2169	—	53	1	no	—
245	#2169	—	53	1	no	—
246	#2168	—	53	1	no	—
247	#2168	—	53	1	no	—
248	#2168	—	53	1	no	—
249	#2168	D₅₃	1	1	yes	characteristic, complemented
250	#2162	—	53	1	no	—
251	#2162	—	53	1	no	—
252	#2162	—	53	1	no	—
253	C₄×M₃₂	—	53	1	no	—
254	C₄×M₃₂	—	53	1	no	—
255	C₂×C₄×C₁₆	—	53	1	no	—
256	C₂×C₅₃	—	2	2	no	—
257	C₂×C₅₃	—	4	1	no	—
258	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
259	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
260	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
261	D₅₃	—	4	1	no	—
262	C₄×#41	—	53	1	no	—
263	C₂×M₃₂	—	53	1	no	—
264	C₂×M₃₂	—	53	1	no	—
265	C₂×M₃₂	—	53	1	no	—
266	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
267	C₂²×C₁₆	—	53	1	no	—
268	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
269	C₈×Q₈	—	53	1	no	—
270	C₈×Q₈	—	53	1	no	—
271	C₈×D₄	—	53	1	no	—
272	C₈×D₄	—	53	1	no	—
273	C₈×D₄	—	53	1	no	—
274	C₈×D₄	—	53	1	no	—
275	C₈×D₄	—	53	1	no	—
276	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
277	#431	—	53	1	no	—
278	#431	—	53	1	no	—
279	#431	—	53	1	no	—
280	C₂×C₄×C₈	—	53	1	no	—
281	C₂×C₄×C₈	—	53	1	no	—
282	C₂×C₄×C₈	—	53	1	no	—
283	C₄⋊C₁₆	—	53	1	no	—
284	C₄⋊C₁₆	—	53	1	no	—
285	C₄⋊C₁₆	—	53	1	no	—
286	C₄⋊C₁₆	—	53	1	no	—
287	C₄⋊C₁₆	—	53	1	no	—
288	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
289	C₂²⋊C₁₆	—	53	1	no	—
290	C₂²⋊C₁₆	—	53	1	no	—
291	C₂²⋊C₁₆	—	53	1	no	—
292	C₂²⋊C₁₆	—	53	1	no	—
293	C₂²⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
294	C₂²⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
295	#346	—	53	1	no	—
296	#346	—	53	1	no	—
297	#346	—	53	1	no	—
298	#346	C₂×D₅₃	1	1	yes	characteristic
299	C₄×C₁₆	—	53	1	no	—
300	C₄×C₁₆	—	53	1	no	—
301	C₄×C₁₆	—	53	1	no	—
302	C₄×C₁₆	—	53	1	no	—
303	C₄×C₁₆	—	53	1	no	—
304	C₄×C₁₆	—	53	1	no	—
305	C₅₃	#33479	1	1	yes	characteristic, complemented, Sylow
306	C₂×#41	—	53	1	no	—
307	C₂²×C₈	—	53	1	no	—
308	C₂²×C₈	—	53	1	no	—
309	C₂²×C₈	—	53	1	no	—
310	C₂²×C₈	—	53	1	no	—
311	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
312	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
313	C₄×Q₈	—	53	1	no	—
314	C₄×Q₈	—	53	1	no	—
315	C₄×D₄	—	53	1	no	—
316	C₄×D₄	—	53	1	no	—
317	C₄×D₄	—	53	1	no	—
318	C₄×D₄	—	53	1	no	—
319	C₄×D₄	—	53	1	no	—
320	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
321	#117	—	53	1	no	—
322	#117	—	53	1	no	—
323	#117	—	53	1	no	—
324	C₂×C₄²	—	53	1	no	—
325	C₂×C₄²	—	53	1	no	—
326	C₂×C₄²	—	53	1	no	—
327	M₃₂	—	2	2	no	—
328	M₃₂	—	106	2	no	—
329	M₃₂	—	106	2	no	—
330	M₃₂	—	106	2	no	—
331	C₂×C₁₆	—	2	2	no	—
332	C₂×C₁₆	—	53	1	no	—
333	C₂×C₁₆	—	53	1	no	—
334	C₂×C₁₆	—	53	1	no	—
335	C₂×C₁₆	—	53	1	no	—
336	C₂×C₁₆	—	106	2	no	—
337	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
338	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
339	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
340	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
341	C₄⋊C₈	—	53	1	no	—
342	C₄⋊C₈	—	53	1	no	—
343	C₄⋊C₈	—	53	1	no	—
344	C₄⋊C₈	—	53	1	no	—
345	C₄⋊C₈	—	53	1	no	—
346	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
347	C₂²⋊C₈	—	53	1	no	—
348	C₂²⋊C₈	—	53	1	no	—
349	C₂²⋊C₈	—	53	1	no	—
350	C₂²⋊C₈	—	53	1	no	—
351	C₂²⋊C₈	C₄×D₅₃	1	1	yes	characteristic
352	C₂²⋊C₈	C₄×D₅₃	1	1	yes	characteristic
353	C₄×C₈	—	53	1	no	—
354	C₄×C₈	—	53	1	no	—
355	C₄×C₈	—	53	1	no	—
356	C₄×C₈	—	53	1	no	—
357	C₄×C₈	—	53	1	no	—
358	C₄×C₈	—	53	1	no	—
359	C₄×C₈	—	53	1	no	—
360	C₄×C₈	—	53	1	no	—
361	C₄×C₈	—	53	1	no	—
362	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
363	#41	—	106	4	no	—
364	C₂×Q₈	—	53	1	no	—
365	C₂×D₄	—	53	1	no	—
366	C₂×D₄	—	53	1	no	—
367	C₂×D₄	C₈×D₅₃	1	1	yes	characteristic
368	C₂²×C₄	—	53	1	no	—
369	C₂²×C₄	—	53	1	no	—
370	C₂²×C₄	—	53	1	no	—
371	C₂²×C₄	—	53	1	no	—
372	C₂²×C₄	—	53	1	no	—
373	C₂²×C₄	—	53	1	no	—
374	C₂²×C₄	—	53	1	no	—
375	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
376	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	53	1	no	—
382	C₂×C₈	—	53	1	no	—
383	C₂×C₈	—	53	1	no	—
384	C₂×C₈	—	53	1	no	—
385	C₂×C₈	—	106	2	no	—
386	C₂×C₈	—	106	2	no	—
387	C₂×C₈	—	106	2	no	—
388	C₂×C₈	—	106	1	no	—
389	C₂×C₈	—	106	1	no	—
390	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
391	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
392	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
393	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
394	C₄⋊C₄	—	53	1	no	—
395	C₄⋊C₄	—	53	1	no	—
396	C₄⋊C₄	—	53	1	no	—
397	C₄⋊C₄	—	53	1	no	—
398	C₄⋊C₄	—	53	1	no	—
399	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
400	C₂²⋊C₄	—	53	1	no	—
401	C₂²⋊C₄	—	53	1	no	—
402	C₂²⋊C₄	—	53	1	no	—
403	C₂²⋊C₄	—	53	1	no	—
404	C₂²⋊C₄	C₈×D₅₃	1	1	yes	characteristic
405	C₂²⋊C₄	C₈×D₅₃	1	1	yes	characteristic
406	C₄²	—	53	1	no	—
407	C₄²	—	53	1	no	—
408	C₄²	—	53	1	no	—
409	C₄²	—	53	1	no	—
410	C₄²	—	53	1	no	—
411	C₄²	—	53	1	no	—
412	C₄²	—	53	1	no	—
413	C₄²	—	53	1	no	—
414	C₄²	—	53	1	no	—
415	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
416	C₁₆	—	2	1	no	—
417	C₁₆	—	2	1	no	—
418	C₁₆	—	2	1	no	—
419	C₁₆	—	53	2	no	—
420	C₁₆	—	106	1	no	—
421	C₁₆	—	106	1	no	—
422	C₁₆	—	106	1	no	—
423	C₁₆	#11719947	1	2	yes	complemented
424	C₂³	—	53	1	no	—
425	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
426	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
427	Q₈	—	106	2	no	—
428	D₄	—	2	2	no	—
429	D₄	—	106	2	no	—
430	D₄	—	106	2	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	53	4	no	—
436	C₂×C₄	—	53	1	no	—
437	C₂×C₄	—	53	1	no	—
438	C₂×C₄	—	53	1	no	—
439	C₂×C₄	—	53	1	no	—
440	C₂×C₄	—	53	1	no	—
441	C₂×C₄	—	53	1	no	—
442	C₂×C₄	—	53	1	no	—
443	C₂×C₄	—	106	2	no	—
444	C₂×C₄	—	106	2	no	—
445	C₂×C₄	—	106	2	no	—
446	C₂×C₄	—	106	2	no	—
447	C₂×C₄	—	106	2	no	—
448	C₂×C₄	—	106	1	no	—
449	C₂×C₄	—	106	1	no	—
450	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
451	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
452	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
453	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
454	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
455	C₈	—	2	1	no	—
456	C₈	—	2	1	no	—
457	C₈	—	2	1	no	—
458	C₈	—	53	2	no	—
459	C₈	—	106	1	no	—
460	C₈	—	106	1	no	—
461	C₈	—	106	1	no	—
462	C₈	#41×D₅₃	1	1	yes	characteristic
463	C₈	C₂×#11719947	1	1	yes	characteristic
464	C₂²	—	2	1	no	—
465	C₂²	—	2	1	no	—
466	C₂²	—	2	1	no	—
467	C₂²	—	2	1	no	—
468	C₂²	—	2	1	no	—
469	C₂²	—	106	1	no	—
470	C₂²	—	106	1	no	—
471	C₂²	—	106	1	no	—
472	C₂²	M₃₂×D₅₃	1	2	yes	—
473	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
474	C₄	—	2	1	no	—
475	C₄	—	2	1	no	—
476	C₄	—	2	1	no	—
477	C₄	—	2	1	no	—
478	C₄	—	53	2	no	—
479	C₄	—	53	2	no	—
480	C₄	—	53	2	no	—
481	C₄	—	106	1	no	—
482	C₄	—	106	1	no	—
483	C₄	—	106	1	no	—
484	C₄	—	106	1	no	—
485	C₄	C₄×#11719947	1	1	yes	characteristic
486	C₄	#131×D₅₃	1	1	yes	characteristic
487	C₂	—	2	2	no	—
488	C₂	—	4	1	no	—
489	C₂	—	212	1	no	—
490	C₂	C₈×#11719947	1	1	yes	characteristic
491	C₂	C₂×M₃₂×D₅₃	1	1	yes	characteristic
492	C₂	#504×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2168	C₂
prime radical	C₂×#11719947	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2168	D₅₃
hypocenter	C₅₃	#33479
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2168

Complements

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

N	H ≅ G/N	type
2. #20276622	C₂	semidirect
3. #20276621	C₂	semidirect
4. #20276545	C₂	semidirect
5. #20276544	C₂	semidirect
6. #20276541	C₂	semidirect
7. #20276540	C₂	semidirect
8. #20276537	C₂	semidirect
9. #20276513	C₂	semidirect
10. #20276511	C₂	semidirect
11. #20276511	C₂	semidirect
12. #20276308	C₂	semidirect
13. M₃₂×Dic₅₃	C₂	semidirect
14. C₂×C₁₆×Dic₅₃	C₂	semidirect
15. C₅₃×#2168	C₂	semidirect
172. C₁₆×C₅₃	#41	semidirect
176. #11719908	#41	semidirect
195. C₂×Dic₅₃	M₃₂	semidirect
249. #2168	D₅₃	semidirect
305. C₅₃	#33479	semidirect
423. C₁₆	#11719947	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11719947	2
#20276622	2
#20276621	2
#20276545	2
#20276544	2
#20276541	2
#20276540	2
#20276537	2
#20276513	2
#20276511	2
#20276308	2
M₃₂×Dic₅₃	2
C₂×C₁₆×Dic₅₃	2
C₅₃×#2168	2
#33479	53