Subgroups — #27382362

Order 13568.

926 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810684	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810579	C₂	1	1	yes	maximal, characteristic
3	#20274590	C₂	1	1	yes	maximal, characteristic, complemented
4	#20274590	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274584	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄⋊C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274499	C₂	1	1	yes	maximal, characteristic
8	#20274499	C₂	1	1	yes	maximal, characteristic
9	#20274493	C₂	1	1	yes	maximal, characteristic
10	C₂²⋊C₄×#88322	C₂	1	1	yes	maximal, characteristic
11	#20274467	C₂	1	1	yes	maximal, characteristic, complemented
12	#20274461	C₂	1	1	yes	maximal, characteristic, complemented
13	#20274402	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#13133128	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#2298	C₂	1	1	yes	maximal, characteristic
16	C₂²×#11719917	C₂²	1	1	yes	characteristic
17	C₂³×#88322	C₂²	1	1	yes	characteristic
18	#16810684	C₄	1	8	yes	complemented
19	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
23	#16810579	C₄	1	8	yes	complemented
24	C₂×#13133128	C₂²	1	1	yes	characteristic
25	C₂×#13133128	C₂²	1	1	yes	characteristic
26	C₂×#13133128	C₂²	1	1	yes	characteristic
27	C₂×#13133128	C₂²	1	1	yes	characteristic
28	C₂×#13133128	C₂²	1	1	yes	characteristic
29	C₂×#13133128	C₂²	1	1	yes	characteristic
30	C₂×#13133128	C₂²	1	1	yes	characteristic
31	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
32	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
33	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
34	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
35	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
36	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
37	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
38	#16810378	C₂²	1	1	yes	characteristic
39	#16810378	C₂²	1	1	yes	characteristic
40	#16810378	C₂²	1	1	yes	characteristic
41	#16810378	C₂²	1	1	yes	characteristic
42	#16810378	C₂²	1	1	yes	characteristic
43	#16810378	C₂²	1	1	yes	characteristic
44	#16810378	C₂²	1	1	yes	characteristic
45	#16810378	C₂²	1	1	yes	characteristic
46	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
47	C₅₃×#388	C₄	1	2	yes	—
48	C₅₃×#388	C₄	1	2	yes	—
49	C₅₃×#387	C₄	1	2	yes	—
50	C₅₃×#387	C₄	1	2	yes	—
51	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
52	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
53	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
54	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
55	C₅₃×#376	C₂²	1	1	yes	characteristic
56	C₄³×C₅₃	C₂²	1	1	yes	characteristic
57	C₂×#11719917	—	2	2	no	—
58	C₂×#11719917	—	2	2	no	—
59	C₂×#11719917	C₂×C₄	1	4	yes	—
60	C₂²×#88322	—	2	2	no	—
61	C₂²×#88322	—	2	2	no	—
62	C₂²×#88322	C₂×C₄	1	4	yes	—
63	C₂²×#88322	C₂³	1	1	yes	characteristic
64	C₂²×#88322	C₂³	1	1	yes	characteristic
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	C₂²×#88322	C₂³	1	1	yes	characteristic
70	C₂²×#88322	C₂³	1	1	yes	characteristic
71	C₄⋊#88322	C₂×C₄	1	4	yes	—
72	C₄⋊#88322	C₂×C₄	1	4	yes	—
73	C₂²⋊#88322	C₂×C₄	1	4	yes	—
74	C₂²⋊#88322	C₂×C₄	1	4	yes	—
75	#13133128	—	2	2	no	—
76	#13133128	—	2	2	no	—
77	#13133128	—	2	2	no	—
78	#13133128	—	2	2	no	—
79	#13133128	C₂×C₄	1	4	yes	—
80	#13133128	C₂×C₄	1	4	yes	—
81	#13133128	C₂×C₄	1	4	yes	—
82	C₄×#88322	—	2	2	no	—
83	C₄×#88322	—	2	2	no	—
84	C₄×#88322	—	2	2	no	—
85	C₄×#88322	—	2	2	no	—
86	C₄×#88322	C₂×C₄	1	4	yes	—
87	C₄×#88322	C₂×C₄	1	4	yes	—
88	C₄×#88322	C₂×C₄	1	4	yes	—
89	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
90	#117×C₅₃	C₂×C₄	1	4	yes	—
91	#117×C₅₃	C₂×C₄	1	4	yes	—
92	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
93	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
94	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
96	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
97	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
98	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
99	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
100	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
101	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
102	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
108	#95×C₅₃	C₂×C₄	1	2	yes	—
109	#95×C₅₃	C₂×C₄	1	2	yes	—
110	#95×C₅₃	C₂×C₄	1	2	yes	—
111	#95×C₅₃	C₂×C₄	1	2	yes	—
112	#95×C₅₃	C₂×C₄	1	2	yes	—
113	#95×C₅₃	C₂×C₄	1	2	yes	—
114	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₅₃	—	2	1	no	—
116	C₂²×C₄×C₅₃	—	2	1	no	—
117	C₂²×C₄×C₅₃	—	2	1	no	—
118	C₂²×C₄×C₅₃	—	2	1	no	—
119	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
120	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
121	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
122	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
128	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
129	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
130	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
131	C₄⋊C₄×C₅₃	—	2	4	no	—
132	C₄⋊C₄×C₅₃	—	4	2	no	—
133	C₄⋊C₄×C₅₃	C₄²	1	4	yes	—
134	C₄⋊C₄×C₅₃	C₄²	1	4	yes	—
135	C₂²⋊C₄×C₅₃	—	2	4	no	—
136	C₂²⋊C₄×C₅₃	—	4	2	no	—
137	C₂²⋊C₄×C₅₃	C₄²	1	4	yes	—
138	C₂²⋊C₄×C₅₃	C₄²	1	4	yes	—
139	C₄²×C₅₃	—	2	4	no	—
140	C₄²×C₅₃	—	2	1	no	—
141	C₄²×C₅₃	—	2	1	no	—
142	C₄²×C₅₃	—	2	1	no	—
143	C₄²×C₅₃	—	2	1	no	—
144	C₄²×C₅₃	—	2	1	no	—
145	C₄²×C₅₃	—	2	1	no	—
146	C₄²×C₅₃	—	2	1	no	—
147	C₄²×C₅₃	—	2	1	no	—
148	C₄²×C₅₃	—	4	2	no	—
149	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
150	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
151	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
152	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
153	#11719917	—	2	8	no	—
154	C₂×#88322	—	2	1	no	—
155	C₂×#88322	—	2	1	no	—
156	C₂×#88322	—	2	1	no	—
157	C₂×#88322	—	2	1	no	—
158	C₂×#88322	—	2	1	no	—
159	C₂×#88322	—	2	1	no	—
160	C₂×#88322	—	2	1	no	—
161	C₂×#88322	—	2	1	no	—
162	C₂×#88322	—	2	1	no	—
163	C₂×#88322	—	2	1	no	—
164	C₂×#88322	—	2	1	no	—
165	C₂×#88322	—	2	1	no	—
166	C₂×#88322	—	2	8	no	—
167	C₂×#88322	C₂²×C₄	1	2	yes	—
168	C₂×#88322	C₂²×C₄	1	2	yes	—
169	C₂×#88322	C₂²×C₄	1	2	yes	—
170	C₂×#88322	C₂²×C₄	1	2	yes	—
171	C₂×#88322	C₂²×C₄	1	2	yes	—
172	C₂×#88322	C₂²×C₄	1	2	yes	—
173	C₂×#88322	C₂²×C₄	1	2	yes	—
174	C₂×#88322	C₂²×C₄	1	2	yes	—
175	C₂×#88322	#41	1	2	yes	—
176	C₂×#88322	#41	1	2	yes	—
177	C₂×#88322	#41	1	2	yes	—
178	C₂×#88322	#41	1	2	yes	—
179	C₂³×C₅₃	—	2	1	no	—
180	C₂³×C₅₃	—	2	1	no	—
181	C₂³×C₅₃	—	2	2	no	—
182	C₂³×C₅₃	—	2	2	no	—
183	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
184	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	—	2	1	no	—
186	C₂×C₄×C₅₃	—	2	1	no	—
187	C₂×C₄×C₅₃	—	2	2	no	—
188	C₂×C₄×C₅₃	—	2	2	no	—
189	C₂×C₄×C₅₃	—	2	2	no	—
190	C₂×C₄×C₅₃	—	2	2	no	—
191	C₂×C₄×C₅₃	—	2	2	no	—
192	C₂×C₄×C₅₃	—	2	2	no	—
193	C₂×C₄×C₅₃	—	2	2	no	—
194	C₂×C₄×C₅₃	—	2	2	no	—
195	C₂×C₄×C₅₃	—	2	2	no	—
196	C₂×C₄×C₅₃	—	2	2	no	—
197	C₂×C₄×C₅₃	—	2	2	no	—
198	C₂×C₄×C₅₃	—	2	2	no	—
199	C₂×C₄×C₅₃	—	2	2	no	—
200	C₂×C₄×C₅₃	—	2	2	no	—
201	C₂×C₄×C₅₃	—	2	2	no	—
202	C₂×C₄×C₅₃	—	2	2	no	—
203	C₂×C₄×C₅₃	—	2	2	no	—
204	C₂×C₄×C₅₃	—	2	2	no	—
205	C₂×C₄×C₅₃	—	2	2	no	—
206	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
207	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
208	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
209	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
210	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
211	C₂×C₄×C₅₃	#131	1	4	yes	—
212	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
214	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
217	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
218	#88322	—	2	2	no	—
219	#88322	—	2	2	no	—
220	#88322	—	2	2	no	—
221	#88322	—	2	2	no	—
222	#88322	—	2	2	no	—
223	#88322	—	2	2	no	—
224	#88322	#117	1	4	yes	—
225	#88322	#117	1	4	yes	—
226	#20478	—	53	1	no	maximal, Sylow
227	C₂²×C₅₃	—	2	2	no	—
228	C₂²×C₅₃	—	2	2	no	—
229	C₂²×C₅₃	—	2	2	no	—
230	C₂²×C₅₃	—	4	2	no	—
231	C₂²×C₅₃	—	4	2	no	—
232	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
233	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
234	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
235	C₂²×C₅₃	C₂×#131	1	2	yes	—
236	C₂²×C₅₃	#568	1	2	yes	—
237	C₄×C₅₃	—	2	2	no	—
238	C₄×C₅₃	—	2	2	no	—
239	C₄×C₅₃	—	2	2	no	—
240	C₄×C₅₃	—	2	2	no	—
241	C₄×C₅₃	—	2	4	no	—
242	C₄×C₅₃	—	2	2	no	—
243	C₄×C₅₃	—	4	2	no	—
244	C₄×C₅₃	—	4	2	no	—
245	C₄×C₅₃	—	4	2	no	—
246	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
247	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
248	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
249	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
250	C₂×#433	—	53	1	no	—
251	C₂×#405	—	53	1	no	—
252	#2298	D₅₃	1	1	yes	characteristic
253	#1771	—	53	1	no	—
254	#1771	—	53	1	no	—
255	#1770	—	53	1	no	—
256	C₈×C₄⋊C₄	—	53	1	no	—
257	#1753	—	53	1	no	—
258	#1753	—	53	1	no	—
259	#1752	—	53	1	no	—
260	C₈×C₂²⋊C₄	—	53	1	no	—
261	#1746	—	53	1	no	—
262	#1744	—	53	1	no	—
263	#1726	—	53	1	no	—
264	C₄×#97	—	53	1	no	—
265	C₂×C₅₃	—	4	2	no	—
266	C₂×C₅₃	C₄×#117	1	1	yes	characteristic
267	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
268	C₂×C₅₃	#2875	1	1	yes	characteristic
269	C₂×C₅₃	#2968	1	2	yes	—
270	C₂×C₅₃	#2969	1	2	yes	—
271	C₂²×M₁₆	—	53	1	no	—
272	C₂³×C₈	—	53	1	no	—
273	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
274	#433	—	53	8	no	—
275	C₂×C₄⋊C₈	—	53	1	no	—
276	C₂×C₄⋊C₈	—	53	1	no	—
277	C₂×C₂²⋊C₈	—	53	1	no	—
278	C₂×C₂²⋊C₈	—	53	1	no	—
279	#405	—	53	8	no	—
280	C₂×#97	—	53	1	no	—
281	C₂×#97	—	53	1	no	—
282	C₂×#97	—	53	1	no	—
283	C₂×#97	—	53	1	no	—
284	C₂×#97	—	53	1	no	—
285	C₂×#97	—	53	1	no	—
286	C₂×#97	—	53	1	no	—
287	C₂×C₄×C₈	—	53	1	no	—
288	C₂×C₄×C₈	—	53	1	no	—
289	C₂×C₄×C₈	—	53	1	no	—
290	C₂×C₄×C₈	—	53	1	no	—
291	C₂×C₄×C₈	—	53	1	no	—
292	C₂×C₄×C₈	—	53	1	no	—
293	C₂×C₄×C₈	—	53	1	no	—
294	#388	Dic₅₃	1	2	yes	—
295	#388	Dic₅₃	1	2	yes	—
296	#387	Dic₅₃	1	2	yes	—
297	#387	Dic₅₃	1	2	yes	—
298	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
299	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
300	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
301	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
302	#376	C₂×D₅₃	1	1	yes	characteristic
303	C₄³	C₂×D₅₃	1	1	yes	characteristic
304	#336	—	53	1	no	—
305	#336	—	53	1	no	—
306	#336	—	53	1	no	—
307	#336	—	53	1	no	—
308	#336	—	53	1	no	—
309	#336	—	53	1	no	—
310	#336	—	53	1	no	—
311	#336	—	53	1	no	—
312	C₅₃	#20478	1	1	yes	characteristic, complemented, Sylow
313	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
314	C₂×M₁₆	—	53	4	no	—
315	C₂×M₁₆	—	106	2	no	—
316	C₂×M₁₆	—	106	2	no	—
317	C₂²×C₈	—	53	1	no	—
318	C₂²×C₈	—	53	1	no	—
319	C₂²×C₈	—	53	4	no	—
320	C₂²×C₈	—	53	1	no	—
321	C₂²×C₈	—	53	1	no	—
322	C₂²×C₈	—	53	1	no	—
323	C₂²×C₈	—	53	1	no	—
324	C₂²×C₈	—	53	1	no	—
325	C₂²×C₈	—	53	1	no	—
326	C₂²×C₈	—	106	2	no	—
327	C₂²×C₈	—	106	2	no	—
328	#117	C₄×D₅₃	1	4	yes	—
329	#117	C₄×D₅₃	1	4	yes	—
330	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
331	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
332	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
333	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
334	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
335	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
336	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
337	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
338	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
339	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
340	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
341	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
342	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
343	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
344	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
345	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
346	C₄⋊C₈	—	53	4	no	—
347	C₄⋊C₈	—	53	4	no	—
348	C₂²⋊C₈	—	53	4	no	—
349	C₂²⋊C₈	—	53	4	no	—
350	#97	—	53	4	no	—
351	#97	—	53	4	no	—
352	#97	—	53	4	no	—
353	#97	—	106	2	no	—
354	#97	—	106	2	no	—
355	#97	—	106	2	no	—
356	#97	—	106	2	no	—
357	C₄×C₈	—	53	4	no	—
358	C₄×C₈	—	53	4	no	—
359	C₄×C₈	—	53	4	no	—
360	C₄×C₈	—	106	2	no	—
361	C₄×C₈	—	106	2	no	—
362	C₄×C₈	—	106	2	no	—
363	C₄×C₈	—	106	2	no	—
364	#95	C₂×Dic₅₃	1	2	yes	—
365	#95	C₂×Dic₅₃	1	2	yes	—
366	#95	C₂×Dic₅₃	1	2	yes	—
367	#95	C₂×Dic₅₃	1	2	yes	—
368	#95	C₂×Dic₅₃	1	2	yes	—
369	#95	C₂×Dic₅₃	1	2	yes	—
370	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
376	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
377	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
378	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
379	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
380	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
381	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
382	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
383	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
384	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
385	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
386	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
387	M₁₆	—	106	8	no	—
388	C₂×C₈	—	53	2	no	—
389	C₂×C₈	—	53	2	no	—
390	C₂×C₈	—	53	2	no	—
391	C₂×C₈	—	53	2	no	—
392	C₂×C₈	—	53	2	no	—
393	C₂×C₈	—	53	2	no	—
394	C₂×C₈	—	53	2	no	—
395	C₂×C₈	—	53	2	no	—
396	C₂×C₈	—	53	2	no	—
397	C₂×C₈	—	53	2	no	—
398	C₂×C₈	—	53	2	no	—
399	C₂×C₈	—	53	2	no	—
400	C₂×C₈	—	106	8	no	—
401	C₂×C₈	—	106	1	no	—
402	C₂×C₈	—	106	1	no	—
403	C₂×C₈	—	106	1	no	—
404	C₂×C₈	—	106	1	no	—
405	C₂×C₈	—	106	1	no	—
406	C₂×C₈	—	106	1	no	—
407	C₂×C₈	—	106	1	no	—
408	C₂×C₈	—	106	1	no	—
409	C₂×C₈	—	106	1	no	—
410	C₂×C₈	—	106	1	no	—
411	C₂×C₈	—	106	1	no	—
412	C₂×C₈	—	106	1	no	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	4	2	no	—
415	C₄⋊C₄	C₄×Dic₅₃	1	4	yes	—
416	C₄⋊C₄	C₄×Dic₅₃	1	4	yes	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₂²⋊C₄	—	4	2	no	—
419	C₂²⋊C₄	C₄×Dic₅₃	1	4	yes	—
420	C₂²⋊C₄	C₄×Dic₅₃	1	4	yes	—
421	C₄²	—	2	4	no	—
422	C₄²	—	2	1	no	—
423	C₄²	—	2	1	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	2	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	2	1	no	—
428	C₄²	—	2	1	no	—
429	C₄²	—	2	1	no	—
430	C₄²	—	4	2	no	—
431	C₄²	C₂×C₄×D₅₃	1	2	yes	—
432	C₄²	C₂×C₄×D₅₃	1	2	yes	—
433	C₄²	C₂×C₄×D₅₃	1	2	yes	—
434	C₄²	C₂×C₄×D₅₃	1	2	yes	—
435	C₂³	—	2	1	no	—
436	C₂³	—	2	1	no	—
437	C₂³	—	2	2	no	—
438	C₂³	—	2	2	no	—
439	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
440	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
463	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
464	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
465	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
466	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
467	C₂×C₄	#13133224	1	4	yes	—
468	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
469	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
470	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
471	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
472	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
473	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
474	C₈	—	53	4	no	—
475	C₈	—	53	4	no	—
476	C₈	—	106	2	no	—
477	C₈	—	106	2	no	—
478	C₈	—	106	2	no	—
479	C₈	—	106	2	no	—
480	C₈	—	106	2	no	—
481	C₈	—	106	2	no	—
482	C₂²	—	2	2	no	—
483	C₂²	—	2	2	no	—
484	C₂²	—	2	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
488	C₂²	#41×Dic₅₃	1	1	yes	characteristic
489	C₂²	#41×Dic₅₃	1	1	yes	characteristic
490	C₂²	C₂×#13133224	1	2	yes	—
491	C₂²	#16811390	1	2	yes	—
492	C₄	—	2	2	no	—
493	C₄	—	2	2	no	—
494	C₄	—	2	2	no	—
495	C₄	—	2	2	no	—
496	C₄	—	2	4	no	—
497	C₄	—	2	2	no	—
498	C₄	—	4	2	no	—
499	C₄	—	4	2	no	—
500	C₄	—	4	2	no	—
501	C₄	#117×D₅₃	1	1	yes	characteristic
502	C₄	#117×D₅₃	1	1	yes	characteristic
503	C₄	#16811091	1	1	yes	characteristic
504	C₄	#16811091	1	1	yes	characteristic
505	C₂	—	4	2	no	—
506	C₂	#117×Dic₅₃	1	1	yes	characteristic
507	C₂	C₄×#13133224	1	1	yes	characteristic
508	C₂	#20283143	1	1	yes	characteristic
509	C₂	#20283711	1	2	yes	—
510	C₂	#20283720	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂²×C₄²
Fitting	C₅₃×#2298	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#2298	D₅₃
hypocenter	C₅₃	#20478
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2298

Complements

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

N	H ≅ G/N	type
3. #20274590	C₂	semidirect
4. #20274590	C₂	semidirect
5. #20274584	C₂	semidirect
6. C₄⋊C₄×#88322	C₂	semidirect
11. #20274467	C₂	semidirect
12. #20274461	C₂	semidirect
13. #20274402	C₂	semidirect
14. C₄×#13133128	C₂	semidirect
18. #16810684	C₄	semidirect
23. #16810579	C₄	semidirect
312. C₅₃	#20478	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810684	2
C₂×#16810579	2
#20274590	2
#20274584	2
C₄⋊C₄×#88322	2
#20274499	2
#20274493	2
C₂²⋊C₄×#88322	2
#20274467	2
#20274461	2
#20274402	2
C₄×#13133128	2
C₅₃×#2298	2
#20478	53