Subgroups — #27388925

Order 13568.

830 conjugacy classes of subgroups.

The subgroup lattice has 620 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	#20275439	C₂	1	1	yes	maximal, characteristic, complemented
4	#20275439	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275434	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275434	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274900	C₂	1	1	yes	maximal, characteristic
8	#20274900	C₂	1	1	yes	maximal, characteristic
9	#20274895	C₂	1	1	yes	maximal, characteristic
10	#20274895	C₂	1	1	yes	maximal, characteristic
11	#20274572	C₂	1	1	yes	maximal, characteristic, complemented
12	#20274572	C₂	1	1	yes	maximal, characteristic, complemented
13	#20274568	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₄⋊#88322	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	—	2	4	no	—
19	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
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×C₄⋊#88322	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
31	#16810579	—	2	4	no	—
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×#117×C₅₃	C₂²	1	1	yes	characteristic
47	C₅₃×#388	C₄	1	1	yes	characteristic
48	C₅₃×#388	C₄	1	1	yes	characteristic
49	C₅₃×#388	C₄	1	1	yes	characteristic
50	C₅₃×#388	C₄	1	1	yes	characteristic
51	C₅₃×#387	C₄	1	1	yes	characteristic
52	C₅₃×#387	C₄	1	1	yes	characteristic
53	C₅₃×#387	C₄	1	1	yes	characteristic
54	C₅₃×#387	C₄	1	1	yes	characteristic
55	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
56	C₄×C₄⋊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	C₅₃×#376	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	C₄⋊#88322	—	2	2	no	—
80	C₄⋊#88322	—	2	2	no	—
81	C₄⋊#88322	—	2	2	no	—
82	C₄⋊#88322	—	2	2	no	—
83	C₄⋊#88322	—	2	2	no	—
84	C₄⋊#88322	—	2	2	no	—
85	C₂²⋊#88322	—	2	2	no	—
86	C₂²⋊#88322	—	2	2	no	—
87	#13133128	—	2	2	no	—
88	#13133128	—	2	2	no	—
89	#13133128	—	2	2	no	—
90	C₄×#88322	—	2	2	no	—
91	C₄×#88322	—	2	2	no	—
92	C₄×#88322	—	2	2	no	—
93	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
94	#117×C₅₃	D₄	1	4	yes	—
95	#117×C₅₃	Q₈	1	4	yes	—
96	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
102	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
104	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
105	C₂×C₂²⋊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×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
109	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
110	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
111	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
112	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
118	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
119	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
120	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
121	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
122	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
123	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
124	#95×C₅₃	C₂×C₄	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	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	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
149	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
150	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
151	C₄⋊C₄×C₅₃	—	2	2	no	—
152	C₄⋊C₄×C₅₃	—	2	2	no	—
153	C₄⋊C₄×C₅₃	—	4	1	no	—
154	C₄⋊C₄×C₅₃	—	4	1	no	—
155	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
156	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
157	C₂²⋊C₄×C₅₃	—	2	2	no	—
158	C₂²⋊C₄×C₅₃	—	2	2	no	—
159	C₂²⋊C₄×C₅₃	—	4	1	no	—
160	C₂²⋊C₄×C₅₃	—	4	1	no	—
161	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
162	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
163	C₄²×C₅₃	—	2	2	no	—
164	C₄²×C₅₃	—	2	2	no	—
165	C₄²×C₅₃	—	2	1	no	—
166	C₄²×C₅₃	—	2	1	no	—
167	C₄²×C₅₃	—	2	1	no	—
168	C₄²×C₅₃	—	2	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	C₄²×C₅₃	—	4	1	no	—
174	C₄²×C₅₃	—	4	1	no	—
175	C₄²×C₅₃	C₂×D₄	1	2	yes	—
176	C₄²×C₅₃	C₂×D₄	1	2	yes	—
177	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
178	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
179	#11719917	—	2	8	no	—
180	C₂×#88322	—	2	8	no	—
181	C₂×#88322	—	2	1	no	—
182	C₂×#88322	—	2	1	no	—
183	C₂×#88322	—	2	1	no	—
184	C₂×#88322	—	2	1	no	—
185	C₂×#88322	—	2	1	no	—
186	C₂×#88322	—	2	1	no	—
187	C₂×#88322	—	2	1	no	—
188	C₂×#88322	—	2	1	no	—
189	C₂×#88322	—	2	1	no	—
190	C₂×#88322	—	2	1	no	—
191	C₂×#88322	—	2	1	no	—
192	C₂×#88322	—	2	1	no	—
193	C₂×#88322	—	2	1	no	—
194	C₂×#88322	—	2	1	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	#41	1	2	yes	—
202	C₂×#88322	#41	1	2	yes	—
203	C₂×#88322	#41	1	2	yes	—
204	C₂×#88322	#41	1	2	yes	—
205	C₂³×C₅₃	—	2	1	no	—
206	C₂³×C₅₃	—	2	1	no	—
207	C₂³×C₅₃	—	2	1	no	—
208	C₂³×C₅₃	—	2	1	no	—
209	C₂³×C₅₃	—	2	1	no	—
210	C₂³×C₅₃	—	2	1	no	—
211	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
212	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₅₃	—	2	1	no	—
214	C₂×C₄×C₅₃	—	2	1	no	—
215	C₂×C₄×C₅₃	—	2	1	no	—
216	C₂×C₄×C₅₃	—	2	1	no	—
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	1	no	—
222	C₂×C₄×C₅₃	—	2	1	no	—
223	C₂×C₄×C₅₃	—	2	1	no	—
224	C₂×C₄×C₅₃	—	2	1	no	—
225	C₂×C₄×C₅₃	—	2	1	no	—
226	C₂×C₄×C₅₃	—	2	1	no	—
227	C₂×C₄×C₅₃	—	2	2	no	—
228	C₂×C₄×C₅₃	—	2	2	no	—
229	C₂×C₄×C₅₃	—	2	2	no	—
230	C₂×C₄×C₅₃	—	2	2	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	—	2	1	no	—
235	C₂×C₄×C₅₃	—	2	1	no	—
236	C₂×C₄×C₅₃	—	2	1	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	C₂×C₄⋊C₄	1	2	yes	—
250	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
251	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
252	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
253	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
254	C₂×C₄×C₅₃	#131	1	2	yes	—
255	C₂×C₄×C₅₃	#131	1	2	yes	—
256	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
257	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
258	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
259	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
260	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
261	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
262	#88322	—	2	2	no	—
263	#88322	—	2	2	no	—
264	#88322	—	2	2	no	—
265	#88322	—	2	2	no	—
266	#88322	—	4	1	no	—
267	#88322	—	4	1	no	—
268	#88322	—	4	1	no	—
269	#88322	—	4	1	no	—
270	#21357	—	53	1	no	maximal, Sylow
271	C₂²×C₅₃	—	2	2	no	—
272	C₂²×C₅₃	—	2	2	no	—
273	C₂²×C₅₃	—	2	2	no	—
274	C₂²×C₅₃	—	4	1	no	—
275	C₂²×C₅₃	—	4	1	no	—
276	C₂²×C₅₃	—	4	1	no	—
277	C₂²×C₅₃	—	4	1	no	—
278	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
279	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
280	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
281	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
282	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
283	C₂²×C₅₃	#568	1	1	yes	characteristic
284	C₂²×C₅₃	#568	1	1	yes	characteristic
285	C₄×C₅₃	—	2	2	no	—
286	C₄×C₅₃	—	2	2	no	—
287	C₄×C₅₃	—	2	2	no	—
288	C₄×C₅₃	—	2	2	no	—
289	C₄×C₅₃	—	2	2	no	—
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	—	4	1	no	—
296	C₄×C₅₃	—	4	1	no	—
297	C₄×C₅₃	—	4	1	no	—
298	C₄×C₅₃	#525	1	1	yes	characteristic
299	C₄×C₅₃	#525	1	1	yes	characteristic
300	C₄×C₅₃	#533	1	1	yes	characteristic
301	C₄×C₅₃	#533	1	1	yes	characteristic
302	C₂×#433	—	53	1	no	—
303	C₂×#405	—	53	1	no	—
304	#2298	D₅₃	1	1	yes	characteristic
305	#1940	—	53	1	no	—
306	#1940	—	53	1	no	—
307	#1939	—	53	1	no	—
308	#1939	—	53	1	no	—
309	#1851	—	53	1	no	—
310	#1851	—	53	1	no	—
311	#1850	—	53	1	no	—
312	#1850	—	53	1	no	—
313	#1768	—	53	1	no	—
314	#1768	—	53	1	no	—
315	#1767	—	53	1	no	—
316	C₄×C₄⋊C₈	—	53	1	no	—
317	C₂×C₅₃	—	4	2	no	—
318	C₂×C₅₃	#2296	1	1	yes	characteristic
319	C₂×C₅₃	#2906	1	1	yes	characteristic
320	C₂×C₅₃	#2906	1	1	yes	characteristic
321	C₂×C₅₃	#2966	1	1	yes	characteristic
322	C₂×C₅₃	#2967	1	1	yes	characteristic
323	C₂×C₅₃	#2968	1	1	yes	characteristic
324	C₂×C₅₃	#2969	1	1	yes	characteristic
325	C₂²×M₁₆	—	53	1	no	—
326	C₂³×C₈	—	53	1	no	—
327	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
328	#433	—	106	4	no	—
329	C₂×C₄⋊C₈	—	53	1	no	—
330	C₂×C₄⋊C₈	—	53	1	no	—
331	C₂×C₄⋊C₈	—	53	1	no	—
332	C₂×C₄⋊C₈	—	53	1	no	—
333	C₂×C₄⋊C₈	—	53	1	no	—
334	C₂×C₄⋊C₈	—	53	1	no	—
335	C₂×C₄⋊C₈	—	53	1	no	—
336	C₂×C₄⋊C₈	—	53	1	no	—
337	C₂×C₄⋊C₈	—	53	1	no	—
338	C₂×C₄⋊C₈	—	53	1	no	—
339	C₂×C₂²⋊C₈	—	53	1	no	—
340	C₂×C₂²⋊C₈	—	53	1	no	—
341	#405	—	106	4	no	—
342	C₂×#97	—	53	1	no	—
343	C₂×#97	—	53	1	no	—
344	C₂×#97	—	53	1	no	—
345	C₂×C₄×C₈	—	53	1	no	—
346	C₂×C₄×C₈	—	53	1	no	—
347	C₂×C₄×C₈	—	53	1	no	—
348	#388	Dic₅₃	1	1	yes	characteristic
349	#388	Dic₅₃	1	1	yes	characteristic
350	#388	Dic₅₃	1	1	yes	characteristic
351	#388	Dic₅₃	1	1	yes	characteristic
352	#387	Dic₅₃	1	1	yes	characteristic
353	#387	Dic₅₃	1	1	yes	characteristic
354	#387	Dic₅₃	1	1	yes	characteristic
355	#387	Dic₅₃	1	1	yes	characteristic
356	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
357	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
358	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
359	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
360	#376	C₂×D₅₃	1	1	yes	characteristic
361	C₄³	C₂×D₅₃	1	1	yes	characteristic
362	#336	—	53	1	no	—
363	#336	—	53	1	no	—
364	#336	—	53	1	no	—
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	C₅₃	#21357	1	1	yes	characteristic, complemented, Sylow
371	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
372	C₂×M₁₆	—	106	2	no	—
373	C₂×M₁₆	—	106	2	no	—
374	C₂×M₁₆	—	106	2	no	—
375	C₂²×C₈	—	53	1	no	—
376	C₂²×C₈	—	53	1	no	—
377	C₂²×C₈	—	53	1	no	—
378	C₂²×C₈	—	53	1	no	—
379	C₂²×C₈	—	53	1	no	—
380	C₂²×C₈	—	53	1	no	—
381	C₂²×C₈	—	53	1	no	—
382	C₂²×C₈	—	53	1	no	—
383	C₂²×C₈	—	106	2	no	—
384	C₂²×C₈	—	106	2	no	—
385	C₂²×C₈	—	106	2	no	—
386	#117	Dic₁₀₆	1	4	yes	—
387	#117	D₂₁₂	1	4	yes	—
388	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
389	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
390	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
391	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
394	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
395	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
396	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
397	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
398	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
399	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
400	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
401	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
402	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
403	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
404	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
405	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
406	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
407	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
408	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
409	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
410	C₄⋊C₈	—	106	2	no	—
411	C₄⋊C₈	—	106	2	no	—
412	C₄⋊C₈	—	106	2	no	—
413	C₄⋊C₈	—	106	2	no	—
414	C₄⋊C₈	—	106	2	no	—
415	C₄⋊C₈	—	106	2	no	—
416	C₄⋊C₈	—	106	2	no	—
417	C₄⋊C₈	—	106	2	no	—
418	C₄⋊C₈	—	106	2	no	—
419	C₄⋊C₈	—	106	2	no	—
420	C₂²⋊C₈	—	106	2	no	—
421	C₂²⋊C₈	—	106	2	no	—
422	#97	—	106	2	no	—
423	#97	—	106	2	no	—
424	#97	—	106	2	no	—
425	C₄×C₈	—	106	2	no	—
426	C₄×C₈	—	106	2	no	—
427	C₄×C₈	—	106	2	no	—
428	#95	C₂×Dic₅₃	1	1	yes	characteristic
429	#95	C₂×Dic₅₃	1	1	yes	characteristic
430	#95	C₂×Dic₅₃	1	1	yes	characteristic
431	#95	C₂×Dic₅₃	1	1	yes	characteristic
432	#95	C₂×Dic₅₃	1	1	yes	characteristic
433	#95	C₂×Dic₅₃	1	1	yes	characteristic
434	#95	C₂×Dic₅₃	1	1	yes	characteristic
435	#95	C₂×Dic₅₃	1	1	yes	characteristic
436	#95	C₂×Dic₅₃	1	1	yes	characteristic
437	#95	C₂×Dic₅₃	1	1	yes	characteristic
438	#95	C₂×Dic₅₃	1	1	yes	characteristic
439	#95	C₂×Dic₅₃	1	1	yes	characteristic
440	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
446	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
447	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
448	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
449	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
450	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
451	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
452	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
453	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
454	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
455	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
456	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
457	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
458	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
459	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
460	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
461	M₁₆	—	106	8	no	—
462	C₂×C₈	—	53	2	no	—
463	C₂×C₈	—	53	2	no	—
464	C₂×C₈	—	53	2	no	—
465	C₂×C₈	—	53	2	no	—
466	C₂×C₈	—	106	8	no	—
467	C₂×C₈	—	106	1	no	—
468	C₂×C₈	—	106	1	no	—
469	C₂×C₈	—	106	1	no	—
470	C₂×C₈	—	106	1	no	—
471	C₂×C₈	—	106	1	no	—
472	C₂×C₈	—	106	1	no	—
473	C₂×C₈	—	106	1	no	—
474	C₂×C₈	—	106	1	no	—
475	C₂×C₈	—	106	1	no	—
476	C₂×C₈	—	106	1	no	—
477	C₂×C₈	—	106	1	no	—
478	C₂×C₈	—	106	1	no	—
479	C₂×C₈	—	106	1	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	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	4	1	no	—
490	C₄⋊C₄	—	4	1	no	—
491	C₄⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
492	C₄⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
493	C₂²⋊C₄	—	2	2	no	—
494	C₂²⋊C₄	—	2	2	no	—
495	C₂²⋊C₄	—	4	1	no	—
496	C₂²⋊C₄	—	4	1	no	—
497	C₂²⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
498	C₂²⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
499	C₄²	—	2	2	no	—
500	C₄²	—	2	2	no	—
501	C₄²	—	2	1	no	—
502	C₄²	—	2	1	no	—
503	C₄²	—	2	1	no	—
504	C₄²	—	2	1	no	—
505	C₄²	—	2	1	no	—
506	C₄²	—	2	1	no	—
507	C₄²	—	2	1	no	—
508	C₄²	—	2	1	no	—
509	C₄²	—	4	1	no	—
510	C₄²	—	4	1	no	—
511	C₄²	C₂×Dic₁₀₆	1	2	yes	—
512	C₄²	C₂×Dic₁₀₆	1	2	yes	—
513	C₄²	C₂×D₂₁₂	1	2	yes	—
514	C₄²	C₂×D₂₁₂	1	2	yes	—
515	C₂³	—	2	1	no	—
516	C₂³	—	2	1	no	—
517	C₂³	—	2	1	no	—
518	C₂³	—	2	1	no	—
519	C₂³	—	2	1	no	—
520	C₂³	—	2	1	no	—
521	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
522	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	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	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
560	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
561	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
562	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
563	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
564	C₂×C₄	#13133224	1	2	yes	—
565	C₂×C₄	#13133224	1	2	yes	—
566	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
567	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
568	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
569	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
570	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
571	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
572	C₈	—	106	2	no	—
573	C₈	—	106	2	no	—
574	C₈	—	106	2	no	—
575	C₈	—	106	2	no	—
576	C₈	—	212	1	no	—
577	C₈	—	212	1	no	—
578	C₈	—	212	1	no	—
579	C₈	—	212	1	no	—
580	C₂²	—	2	2	no	—
581	C₂²	—	2	2	no	—
582	C₂²	—	2	2	no	—
583	C₂²	—	4	1	no	—
584	C₂²	—	4	1	no	—
585	C₂²	—	4	1	no	—
586	C₂²	—	4	1	no	—
587	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
588	C₂²	#41×Dic₅₃	1	1	yes	characteristic
589	C₂²	#41×Dic₅₃	1	1	yes	characteristic
590	C₂²	C₂×#13133224	1	1	yes	characteristic
591	C₂²	C₂×#13133224	1	1	yes	characteristic
592	C₂²	#16811390	1	1	yes	characteristic
593	C₂²	#16811390	1	1	yes	characteristic
594	C₄	—	2	2	no	—
595	C₄	—	2	2	no	—
596	C₄	—	2	2	no	—
597	C₄	—	2	2	no	—
598	C₄	—	2	2	no	—
599	C₄	—	2	2	no	—
600	C₄	—	2	2	no	—
601	C₄	—	4	1	no	—
602	C₄	—	4	1	no	—
603	C₄	—	4	1	no	—
604	C₄	—	4	1	no	—
605	C₄	—	4	1	no	—
606	C₄	—	4	1	no	—
607	C₄	#16811135	1	1	yes	characteristic
608	C₄	#16811135	1	1	yes	characteristic
609	C₄	#16811176	1	1	yes	characteristic
610	C₄	#16811176	1	1	yes	characteristic
611	C₂	—	4	2	no	—
612	C₂	#20277199	1	1	yes	characteristic
613	C₂	#20283283	1	1	yes	characteristic
614	C₂	#20283283	1	1	yes	characteristic
615	C₂	#20283691	1	1	yes	characteristic
616	C₂	#20283702	1	1	yes	characteristic
617	C₂	#20283711	1	1	yes	characteristic
618	C₂	#20283720	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#2298	C₂
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#2298	D₅₃
hypocenter	C₅₃	#21357
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2298

Complements

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

N	H ≅ G/N	type
3. #20275439	C₂	semidirect
4. #20275439	C₂	semidirect
5. #20275434	C₂	semidirect
6. #20275434	C₂	semidirect
11. #20274572	C₂	semidirect
12. #20274572	C₂	semidirect
13. #20274568	C₂	semidirect
14. C₄×C₄⋊#88322	C₂	semidirect
370. C₅₃	#21357	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
#20275439	2
#20275434	2
#20274900	2
#20274895	2
#20274572	2
#20274568	2
C₄×C₄⋊#88322	2
C₅₃×#2298	2
#21357	53