Subgroups — #27391978

Order 13568.

884 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810754	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810695	C₂	1	1	yes	maximal, characteristic
3	#20275656	C₂	1	1	yes	maximal, characteristic, complemented
4	#20275514	C₂	1	1	yes	maximal, characteristic
5	#20275330	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275330	C₂	1	1	yes	maximal, characteristic, complemented
7	#20275323	C₂	1	1	yes	maximal, characteristic, complemented
8	#20275323	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274832	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274499	C₂	1	1	yes	maximal, characteristic, complemented
11	#20274499	C₂	1	1	yes	maximal, characteristic, complemented
12	#20274493	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²⋊C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#13133128	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#124×C₅₃	C₂	1	1	yes	maximal, characteristic
16	C₂²×#11719917	C₂²	1	1	yes	characteristic
17	C₂³×#88322	C₂²	1	1	yes	characteristic
18	#16810754	—	2	4	no	—
19	#16810695	—	2	4	no	—
20	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
30	C₂×#13133128	C₂²	1	1	yes	characteristic
31	C₂×#13133128	C₂²	1	1	yes	characteristic
32	C₂×#13133128	C₂²	1	1	yes	characteristic
33	C₂×#13133128	C₂²	1	1	yes	characteristic
34	C₂×#13133128	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×#124×C₅₃	C₄	1	1	yes	characteristic
47	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
48	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
49	C₅₃×#391	C₄	1	1	yes	characteristic
50	C₅₃×#390	C₄	1	1	yes	characteristic
51	C₅₃×#388	C₄	1	1	yes	characteristic
52	C₅₃×#388	C₄	1	1	yes	characteristic
53	C₅₃×#388	C₄	1	1	yes	characteristic
54	C₅₃×#388	C₄	1	1	yes	characteristic
55	C₅₃×#382	C₄	1	1	yes	characteristic
56	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
57	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
58	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
59	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
60	C₄³×C₅₃	C₂²	1	1	yes	characteristic
61	C₂×#11719917	—	2	2	no	—
62	C₂×#11719917	—	2	2	no	—
63	C₂×#11719917	—	2	2	no	—
64	C₂²×#88322	—	2	2	no	—
65	C₂²×#88322	—	2	2	no	—
66	C₂²×#88322	—	2	2	no	—
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	C₂²×#88322	C₂³	1	1	yes	characteristic
72	C₂²×#88322	C₂³	1	1	yes	characteristic
73	C₂²×#88322	C₂³	1	1	yes	characteristic
74	C₂²×#88322	C₂³	1	1	yes	characteristic
75	C₄⋊#88322	—	2	2	no	—
76	C₄⋊#88322	—	2	2	no	—
77	C₄⋊#88322	—	2	2	no	—
78	C₄⋊#88322	—	2	2	no	—
79	C₂²⋊#88322	—	2	2	no	—
80	C₂²⋊#88322	—	2	2	no	—
81	C₂²⋊#88322	—	2	2	no	—
82	C₂²⋊#88322	—	2	2	no	—
83	C₂²⋊#88322	—	2	2	no	—
84	C₂²⋊#88322	—	2	2	no	—
85	#13133128	—	2	2	no	—
86	#13133128	—	2	2	no	—
87	#13133128	—	2	2	no	—
88	#13133128	—	2	2	no	—
89	#13133128	D₄	1	4	yes	—
90	C₄×#88322	—	2	2	no	—
91	C₄×#88322	—	2	2	no	—
92	C₄×#88322	—	2	2	no	—
93	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
94	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
95	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
96	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
97	#124×C₅₃	—	2	4	no	—
98	C₄×Q₈×C₅₃	—	2	2	no	—
99	C₄×Q₈×C₅₃	—	2	2	no	—
100	C₄×D₄×C₅₃	—	2	2	no	—
101	C₄×D₄×C₅₃	—	2	2	no	—
102	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
106	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
107	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
108	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
109	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
110	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
111	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
112	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
113	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
114	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
115	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
116	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
117	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
118	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
119	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
120	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
125	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
126	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
127	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
128	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
129	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
130	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
131	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
132	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
133	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
134	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₂×Q₈×C₅₃	—	2	2	no	—
136	C₂×Q₈×C₅₃	—	2	2	no	—
137	C₂×Q₈×C₅₃	—	4	1	no	—
138	C₂×D₄×C₅₃	—	2	2	no	—
139	C₂×D₄×C₅₃	—	2	2	no	—
140	C₂×D₄×C₅₃	—	4	1	no	—
141	C₂²×C₄×C₅₃	—	2	1	no	—
142	C₂²×C₄×C₅₃	—	2	1	no	—
143	C₂²×C₄×C₅₃	—	2	1	no	—
144	C₂²×C₄×C₅₃	—	2	1	no	—
145	C₂²×C₄×C₅₃	—	2	1	no	—
146	C₂²×C₄×C₅₃	—	2	1	no	—
147	C₂²×C₄×C₅₃	—	2	1	no	—
148	C₂²×C₄×C₅₃	—	2	1	no	—
149	C₂²×C₄×C₅₃	—	2	1	no	—
150	C₂²×C₄×C₅₃	—	2	1	no	—
151	C₂²×C₄×C₅₃	—	2	1	no	—
152	C₂²×C₄×C₅₃	—	2	1	no	—
153	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
154	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
164	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
165	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
166	C₄⋊C₄×C₅₃	—	2	2	no	—
167	C₄⋊C₄×C₅₃	—	2	2	no	—
168	C₄⋊C₄×C₅₃	—	2	2	no	—
169	C₄⋊C₄×C₅₃	—	2	2	no	—
170	C₂²⋊C₄×C₅₃	—	2	2	no	—
171	C₂²⋊C₄×C₅₃	—	2	2	no	—
172	C₂²⋊C₄×C₅₃	—	2	2	no	—
173	C₂²⋊C₄×C₅₃	—	2	2	no	—
174	C₂²⋊C₄×C₅₃	—	2	2	no	—
175	C₂²⋊C₄×C₅₃	—	2	2	no	—
176	C₂²⋊C₄×C₅₃	—	2	2	no	—
177	C₂²⋊C₄×C₅₃	—	4	1	no	—
178	C₂²⋊C₄×C₅₃	—	4	1	no	—
179	C₂²⋊C₄×C₅₃	M₁₆	1	4	yes	complemented
180	C₄²×C₅₃	—	2	2	no	—
181	C₄²×C₅₃	—	2	2	no	—
182	C₄²×C₅₃	—	2	1	no	—
183	C₄²×C₅₃	—	2	1	no	—
184	C₄²×C₅₃	—	2	1	no	—
185	C₄²×C₅₃	—	2	1	no	—
186	C₄²×C₅₃	—	2	1	no	—
187	C₄²×C₅₃	—	2	1	no	—
188	C₄²×C₅₃	—	2	1	no	—
189	C₄²×C₅₃	—	2	1	no	—
190	C₄²×C₅₃	—	4	1	no	—
191	C₄²×C₅₃	—	4	1	no	—
192	C₄²×C₅₃	C₂×D₄	1	2	yes	—
193	C₄²×C₅₃	#41	1	2	yes	—
194	#11719917	—	4	4	no	—
195	C₂×#88322	—	2	1	no	—
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	—	2	1	no	—
208	C₂×#88322	—	2	1	no	—
209	C₂×#88322	—	2	1	no	—
210	C₂×#88322	—	2	1	no	—
211	C₂×#88322	—	2	1	no	—
212	C₂×#88322	—	2	1	no	—
213	C₂×#88322	—	4	4	no	—
214	C₂×#88322	C₂×D₄	1	2	yes	—
215	C₂×#88322	C₂×D₄	1	2	yes	—
216	C₂×#88322	#41	1	2	yes	—
217	C₂×#88322	#41	1	2	yes	—
218	C₂×#88322	#41	1	2	yes	—
219	C₂×#88322	#41	1	2	yes	—
220	C₂³×C₅₃	—	2	1	no	—
221	C₂³×C₅₃	—	2	1	no	—
222	C₂³×C₅₃	—	2	1	no	—
223	C₂³×C₅₃	—	2	1	no	—
224	C₂³×C₅₃	—	2	1	no	—
225	C₂³×C₅₃	—	2	1	no	—
226	C₂³×C₅₃	—	2	1	no	—
227	C₂³×C₅₃	—	2	1	no	—
228	C₂³×C₅₃	—	2	1	no	—
229	C₂³×C₅₃	—	2	1	no	—
230	C₂³×C₅₃	—	2	1	no	—
231	C₂³×C₅₃	—	2	1	no	—
232	C₂³×C₅₃	—	2	1	no	—
233	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
234	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
235	Q₈×C₅₃	—	4	4	no	—
236	D₄×C₅₃	—	4	4	no	—
237	C₂×C₄×C₅₃	—	2	1	no	—
238	C₂×C₄×C₅₃	—	2	1	no	—
239	C₂×C₄×C₅₃	—	2	1	no	—
240	C₂×C₄×C₅₃	—	2	1	no	—
241	C₂×C₄×C₅₃	—	2	1	no	—
242	C₂×C₄×C₅₃	—	2	1	no	—
243	C₂×C₄×C₅₃	—	2	1	no	—
244	C₂×C₄×C₅₃	—	2	1	no	—
245	C₂×C₄×C₅₃	—	2	1	no	—
246	C₂×C₄×C₅₃	—	2	1	no	—
247	C₂×C₄×C₅₃	—	2	1	no	—
248	C₂×C₄×C₅₃	—	2	1	no	—
249	C₂×C₄×C₅₃	—	2	1	no	—
250	C₂×C₄×C₅₃	—	2	1	no	—
251	C₂×C₄×C₅₃	—	2	1	no	—
252	C₂×C₄×C₅₃	—	2	1	no	—
253	C₂×C₄×C₅₃	—	2	1	no	—
254	C₂×C₄×C₅₃	—	2	1	no	—
255	C₂×C₄×C₅₃	—	2	1	no	—
256	C₂×C₄×C₅₃	—	2	1	no	—
257	C₂×C₄×C₅₃	—	2	1	no	—
258	C₂×C₄×C₅₃	—	2	1	no	—
259	C₂×C₄×C₅₃	—	2	1	no	—
260	C₂×C₄×C₅₃	—	2	1	no	—
261	C₂×C₄×C₅₃	—	2	1	no	—
262	C₂×C₄×C₅₃	—	2	1	no	—
263	C₂×C₄×C₅₃	—	2	1	no	—
264	C₂×C₄×C₅₃	—	2	1	no	—
265	C₂×C₄×C₅₃	—	2	1	no	—
266	C₂×C₄×C₅₃	—	2	1	no	—
267	C₂×C₄×C₅₃	—	2	1	no	—
268	C₂×C₄×C₅₃	—	4	1	no	—
269	C₂×C₄×C₅₃	—	4	1	no	—
270	C₂×C₄×C₅₃	—	4	1	no	—
271	C₂×C₄×C₅₃	—	4	1	no	—
272	C₂×C₄×C₅₃	—	4	1	no	—
273	C₂×C₄×C₅₃	—	4	1	no	—
274	C₂×C₄×C₅₃	—	4	1	no	—
275	C₂×C₄×C₅₃	—	4	1	no	—
276	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
277	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
278	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
279	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
280	C₂×C₄×C₅₃	#131	1	2	yes	—
281	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
282	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
283	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
284	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
285	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
286	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
287	#88322	—	2	2	no	—
288	#88322	—	2	2	no	—
289	#88322	—	2	2	no	—
290	#88322	—	4	1	no	—
291	#88322	—	4	1	no	—
292	#88322	—	4	1	no	—
293	#88322	—	4	1	no	—
294	#88322	#124	1	4	yes	complemented
295	#21632	—	53	1	no	maximal, Sylow
296	C₂²×C₅₃	—	2	2	no	—
297	C₂²×C₅₃	—	2	2	no	—
298	C₂²×C₅₃	—	2	2	no	—
299	C₂²×C₅₃	—	4	1	no	—
300	C₂²×C₅₃	—	4	1	no	—
301	C₂²×C₅₃	—	4	1	no	—
302	C₂²×C₅₃	—	4	1	no	—
303	C₂²×C₅₃	—	4	1	no	—
304	C₂²×C₅₃	—	4	1	no	—
305	C₂²×C₅₃	—	4	1	no	—
306	C₂²×C₅₃	—	4	1	no	—
307	C₂²×C₅₃	—	4	1	no	—
308	C₂²×C₅₃	—	4	1	no	—
309	C₂²×C₅₃	—	4	1	no	—
310	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
311	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
312	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
313	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
314	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
315	C₂²×C₅₃	#568	1	1	yes	characteristic
316	C₂²×C₅₃	#568	1	1	yes	characteristic
317	C₄×C₅₃	—	2	2	no	—
318	C₄×C₅₃	—	2	2	no	—
319	C₄×C₅₃	—	2	2	no	—
320	C₄×C₅₃	—	2	2	no	—
321	C₄×C₅₃	—	2	2	no	—
322	C₄×C₅₃	—	4	1	no	—
323	C₄×C₅₃	—	4	1	no	—
324	C₄×C₅₃	—	4	1	no	—
325	C₄×C₅₃	—	4	1	no	—
326	C₄×C₅₃	—	4	1	no	—
327	C₄×C₅₃	—	4	1	no	—
328	C₄×C₅₃	—	4	1	no	—
329	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
330	C₄×C₅₃	#529	1	1	yes	characteristic
331	C₄×C₅₃	#529	1	1	yes	characteristic
332	C₄×C₅₃	#532	1	1	yes	characteristic
333	C₂×#446	—	53	1	no	—
334	C₂×#435	—	53	1	no	—
335	C₄×#124	D₅₃	1	1	yes	characteristic
336	#1985	—	53	1	no	—
337	#1955	—	53	1	no	—
338	#1924	—	53	1	no	—
339	#1924	—	53	1	no	—
340	#1923	—	53	1	no	—
341	#1923	—	53	1	no	—
342	#1831	—	53	1	no	—
343	#1753	—	53	1	no	—
344	#1753	—	53	1	no	—
345	#1752	—	53	1	no	—
346	C₈×C₂²⋊C₄	—	53	1	no	—
347	C₄×#97	—	53	1	no	—
348	C₂×C₅₃	—	4	2	no	—
349	C₂×C₅₃	—	8	1	no	—
350	C₂×C₅₃	C₄×#124	1	1	yes	characteristic
351	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
352	C₂×C₅₃	#2935	1	1	yes	characteristic
353	C₂×C₅₃	#2965	1	1	yes	characteristic
354	C₂×C₅₃	#2967	1	1	yes	characteristic
355	C₂×C₅₃	#2968	1	1	yes	characteristic
356	C₂×C₅₃	#2970	1	1	yes	characteristic
357	C₂²×M₁₆	—	53	1	no	—
358	C₂³×C₈	—	53	1	no	—
359	C₂×#124	Dic₅₃	1	1	yes	characteristic
360	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
361	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
362	#446	—	106	4	no	—
363	#435	—	106	4	no	—
364	C₂×C₄⋊C₈	—	53	1	no	—
365	C₂×C₄⋊C₈	—	53	1	no	—
366	C₂×C₄⋊C₈	—	53	1	no	—
367	C₂×C₄⋊C₈	—	53	1	no	—
368	C₂×C₂²⋊C₈	—	53	1	no	—
369	C₂×C₂²⋊C₈	—	53	1	no	—
370	C₂×C₂²⋊C₈	—	53	1	no	—
371	C₂×C₂²⋊C₈	—	53	1	no	—
372	C₂×C₂²⋊C₈	—	53	1	no	—
373	C₂×C₂²⋊C₈	—	53	1	no	—
374	C₂×#97	—	53	1	no	—
375	C₂×#97	—	53	1	no	—
376	C₂×#97	—	53	1	no	—
377	C₂×#97	—	53	1	no	—
378	C₂×#97	—	53	1	no	—
379	C₂×C₄×C₈	—	53	1	no	—
380	C₂×C₄×C₈	—	53	1	no	—
381	C₂×C₄×C₈	—	53	1	no	—
382	#391	Dic₅₃	1	1	yes	characteristic
383	#390	Dic₅₃	1	1	yes	characteristic
384	#388	Dic₅₃	1	1	yes	characteristic
385	#388	Dic₅₃	1	1	yes	characteristic
386	#388	Dic₅₃	1	1	yes	characteristic
387	#388	Dic₅₃	1	1	yes	characteristic
388	#382	Dic₅₃	1	1	yes	characteristic
389	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
390	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
391	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
392	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
393	C₄³	C₂×D₅₃	1	1	yes	characteristic
394	#336	—	53	1	no	—
395	#336	—	53	1	no	—
396	#336	—	53	1	no	—
397	#336	—	53	1	no	—
398	#336	—	53	1	no	—
399	#336	—	53	1	no	—
400	#336	—	53	1	no	—
401	#336	—	53	1	no	—
402	C₅₃	#21632	1	1	yes	characteristic, complemented, Sylow
403	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
404	C₂²×D₄	C₂×Dic₅₃	1	1	yes	characteristic
405	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
406	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
407	C₂×M₁₆	—	106	2	no	—
408	C₂×M₁₆	—	106	2	no	—
409	C₂×M₁₆	—	106	2	no	—
410	C₂²×C₈	—	53	1	no	—
411	C₂²×C₈	—	53	1	no	—
412	C₂²×C₈	—	53	1	no	—
413	C₂²×C₈	—	53	1	no	—
414	C₂²×C₈	—	53	1	no	—
415	C₂²×C₈	—	53	1	no	—
416	C₂²×C₈	—	53	1	no	—
417	C₂²×C₈	—	53	1	no	—
418	C₂²×C₈	—	106	2	no	—
419	C₂²×C₈	—	106	2	no	—
420	C₂²×C₈	—	106	2	no	—
421	#124	—	2	4	no	—
422	C₄×Q₈	—	2	2	no	—
423	C₄×Q₈	—	2	2	no	—
424	C₄×D₄	—	2	2	no	—
425	C₄×D₄	—	2	2	no	—
426	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
427	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
428	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
429	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
430	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
431	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
432	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
433	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
434	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
435	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
436	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
437	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
438	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
439	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
440	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
441	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
442	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
443	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
444	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
445	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
446	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
447	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
448	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
449	C₄⋊C₈	—	106	2	no	—
450	C₄⋊C₈	—	106	2	no	—
451	C₄⋊C₈	—	106	2	no	—
452	C₄⋊C₈	—	106	2	no	—
453	C₂²⋊C₈	—	106	2	no	—
454	C₂²⋊C₈	—	106	2	no	—
455	C₂²⋊C₈	—	106	2	no	—
456	C₂²⋊C₈	—	106	2	no	—
457	C₂²⋊C₈	—	106	2	no	—
458	C₂²⋊C₈	—	106	2	no	—
459	#97	—	53	4	no	—
460	#97	—	106	2	no	—
461	#97	—	106	2	no	—
462	#97	—	106	2	no	—
463	#97	—	106	2	no	—
464	C₄×C₈	—	106	2	no	—
465	C₄×C₈	—	106	2	no	—
466	C₄×C₈	—	106	2	no	—
467	#95	C₂×Dic₅₃	1	1	yes	characteristic
468	#95	C₂×Dic₅₃	1	1	yes	characteristic
469	#95	C₂×Dic₅₃	1	1	yes	characteristic
470	#95	C₂×Dic₅₃	1	1	yes	characteristic
471	#95	C₂×Dic₅₃	1	1	yes	characteristic
472	#95	C₂×Dic₅₃	1	1	yes	characteristic
473	#95	C₂×Dic₅₃	1	1	yes	characteristic
474	#95	C₂×Dic₅₃	1	1	yes	characteristic
475	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
476	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
477	C₂×Q₈	—	2	2	no	—
478	C₂×Q₈	—	2	2	no	—
479	C₂×Q₈	—	4	1	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	—	2	2	no	—
482	C₂×D₄	—	4	1	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	1	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
496	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
497	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
498	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
499	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
500	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
501	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
502	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
503	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
504	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
505	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
506	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
507	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
508	M₁₆	—	212	4	no	—
509	C₂×C₈	—	53	2	no	—
510	C₂×C₈	—	53	2	no	—
511	C₂×C₈	—	53	2	no	—
512	C₂×C₈	—	53	2	no	—
513	C₂×C₈	—	53	2	no	—
514	C₂×C₈	—	53	2	no	—
515	C₂×C₈	—	106	1	no	—
516	C₂×C₈	—	106	1	no	—
517	C₂×C₈	—	106	1	no	—
518	C₂×C₈	—	106	1	no	—
519	C₂×C₈	—	106	1	no	—
520	C₂×C₈	—	106	1	no	—
521	C₂×C₈	—	106	1	no	—
522	C₂×C₈	—	106	1	no	—
523	C₂×C₈	—	106	1	no	—
524	C₂×C₈	—	106	1	no	—
525	C₂×C₈	—	106	1	no	—
526	C₂×C₈	—	106	1	no	—
527	C₂×C₈	—	106	1	no	—
528	C₂×C₈	—	106	1	no	—
529	C₂×C₈	—	106	1	no	—
530	C₂×C₈	—	106	1	no	—
531	C₂×C₈	—	106	1	no	—
532	C₂×C₈	—	106	1	no	—
533	C₂×C₈	—	212	4	no	—
534	C₄⋊C₄	—	2	2	no	—
535	C₄⋊C₄	—	2	2	no	—
536	C₄⋊C₄	—	2	2	no	—
537	C₄⋊C₄	—	2	2	no	—
538	C₂²⋊C₄	—	2	2	no	—
539	C₂²⋊C₄	—	2	2	no	—
540	C₂²⋊C₄	—	2	2	no	—
541	C₂²⋊C₄	—	2	2	no	—
542	C₂²⋊C₄	—	2	2	no	—
543	C₂²⋊C₄	—	2	2	no	—
544	C₂²⋊C₄	—	2	2	no	—
545	C₂²⋊C₄	—	4	1	no	—
546	C₂²⋊C₄	—	4	1	no	—
547	C₂²⋊C₄	#11719917	1	4	yes	complemented
548	C₄²	—	2	2	no	—
549	C₄²	—	2	2	no	—
550	C₄²	—	2	1	no	—
551	C₄²	—	2	1	no	—
552	C₄²	—	2	1	no	—
553	C₄²	—	2	1	no	—
554	C₄²	—	2	1	no	—
555	C₄²	—	2	1	no	—
556	C₄²	—	2	1	no	—
557	C₄²	—	2	1	no	—
558	C₄²	—	4	1	no	—
559	C₄²	—	4	1	no	—
560	C₄²	D₄×D₅₃	1	2	yes	—
561	C₄²	#11719947	1	2	yes	—
562	C₂³	—	2	1	no	—
563	C₂³	—	2	1	no	—
564	C₂³	—	2	1	no	—
565	C₂³	—	2	1	no	—
566	C₂³	—	2	1	no	—
567	C₂³	—	2	1	no	—
568	C₂³	—	2	1	no	—
569	C₂³	—	2	1	no	—
570	C₂³	—	2	1	no	—
571	C₂³	—	2	1	no	—
572	C₂³	—	2	1	no	—
573	C₂³	—	2	1	no	—
574	C₂³	—	2	1	no	—
575	C₂³	C₂×#11719917	1	2	yes	—
576	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
577	Q₈	—	4	4	no	—
578	D₄	—	4	4	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	1	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	1	no	—
603	C₂×C₄	—	2	1	no	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	2	1	no	—
608	C₂×C₄	—	2	1	no	—
609	C₂×C₄	—	2	1	no	—
610	C₂×C₄	—	4	1	no	—
611	C₂×C₄	—	4	1	no	—
612	C₂×C₄	—	4	1	no	—
613	C₂×C₄	—	4	1	no	—
614	C₂×C₄	—	4	1	no	—
615	C₂×C₄	—	4	1	no	—
616	C₂×C₄	—	4	1	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
619	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
620	C₂×C₄	C₂×#11719917	1	2	yes	—
621	C₂×C₄	C₂×#11719917	1	2	yes	—
622	C₂×C₄	#13133224	1	2	yes	—
623	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
624	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
625	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
626	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
627	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
628	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
629	C₈	—	53	4	no	—
630	C₈	—	106	2	no	—
631	C₈	—	106	2	no	—
632	C₈	—	106	2	no	—
633	C₈	—	212	1	no	—
634	C₈	—	212	1	no	—
635	C₈	—	212	1	no	—
636	C₈	—	212	1	no	—
637	C₂²	—	2	2	no	—
638	C₂²	—	2	2	no	—
639	C₂²	—	2	2	no	—
640	C₂²	—	4	1	no	—
641	C₂²	—	4	1	no	—
642	C₂²	—	4	1	no	—
643	C₂²	—	4	1	no	—
644	C₂²	—	4	1	no	—
645	C₂²	—	4	1	no	—
646	C₂²	—	4	1	no	—
647	C₂²	—	4	1	no	—
648	C₂²	—	4	1	no	—
649	C₂²	—	4	1	no	—
650	C₂²	—	4	1	no	—
651	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
652	C₂²	#41×Dic₅₃	1	1	yes	characteristic
653	C₂²	#41×Dic₅₃	1	1	yes	characteristic
654	C₂²	C₂²×#11719917	1	1	yes	characteristic
655	C₂²	C₂×#13133224	1	1	yes	characteristic
656	C₂²	#16811390	1	1	yes	characteristic
657	C₂²	#16811390	1	1	yes	characteristic
658	C₄	—	2	2	no	—
659	C₄	—	2	2	no	—
660	C₄	—	2	2	no	—
661	C₄	—	2	2	no	—
662	C₄	—	2	2	no	—
663	C₄	—	4	1	no	—
664	C₄	—	4	1	no	—
665	C₄	—	4	1	no	—
666	C₄	—	4	1	no	—
667	C₄	—	4	1	no	—
668	C₄	—	4	1	no	—
669	C₄	—	4	1	no	—
670	C₄	#124×D₅₃	1	1	yes	characteristic
671	C₄	#16811152	1	1	yes	characteristic
672	C₄	#16811152	1	1	yes	characteristic
673	C₄	#16811172	1	1	yes	characteristic
674	C₂	—	4	2	no	—
675	C₂	—	8	1	no	—
676	C₂	#124×Dic₅₃	1	1	yes	characteristic
677	C₂	D₄×#11719917	1	1	yes	characteristic
678	C₂	#20283459	1	1	yes	characteristic
679	C₂	#20283682	1	1	yes	characteristic
680	C₂	#20283702	1	1	yes	characteristic
681	C₂	#20283711	1	1	yes	characteristic
682	C₂	#20283729	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₄×#124×C₅₃	C₂
prime radical	C₂²×D₄×C₅₃	C₂×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#124	D₅₃
hypocenter	C₅₃	#21632
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#124

Complements

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

N	H ≅ G/N	type
1. C₂×#16810754	C₂	semidirect
3. #20275656	C₂	semidirect
5. #20275330	C₂	semidirect
6. #20275330	C₂	semidirect
7. #20275323	C₂	semidirect
8. #20275323	C₂	semidirect
9. #20274832	C₂	semidirect
10. #20274499	C₂	semidirect
11. #20274499	C₂	semidirect
12. #20274493	C₂	semidirect
13. C₂²⋊C₄×#88322	C₂	semidirect
14. C₄×#13133128	C₂	semidirect
179. C₂²⋊C₄×C₅₃	M₁₆	semidirect
294. #88322	#124	semidirect
402. C₅₃	#21632	semidirect
547. C₂²⋊C₄	#11719917	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810754	2
C₂×#16810695	2
#20275656	2
#20275514	2
#20275330	2
#20275323	2
#20274832	2
#20274499	2
#20274493	2
C₂²⋊C₄×#88322	2
C₄×#13133128	2
C₄×#124×C₅₃	2
#21632	53