Subgroups — #27401441

Order 13568.

894 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283373	C₂	1	1	yes	maximal, characteristic, complemented
2	#20283164	C₂	1	1	yes	maximal, characteristic
3	#20275428	C₂	1	2	yes	maximal, complemented
4	#20275365	C₂	1	2	yes	maximal, complemented
5	#20274992	C₂	1	2	yes	maximal, complemented
6	#20274814	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274753	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274659	C₂	1	1	yes	maximal, characteristic
9	#20274562	C₂	1	2	yes	maximal, complemented
10	#20274491	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×#416	C₂	1	1	yes	maximal, characteristic
12	C₂×#13133224	C₂²	1	2	yes	—
13	#16810684	—	2	2	no	—
14	#16810680	—	2	1	no	—
15	#16810676	—	2	1	no	—
16	C₂×#13133153	C₂²	1	2	yes	complemented
17	#16810650	C₂²	1	2	yes	complemented
18	C₂×#13133144	C₂²	1	2	yes	—
19	C₂×#13133144	C₂²	1	2	yes	—
20	#16810590	—	2	2	no	—
21	#16810590	C₂²	1	2	yes	—
22	#16810587	—	2	1	no	—
23	#16810587	C₂²	1	1	yes	characteristic
24	#16810587	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊#88322	—	2	1	no	—
26	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
27	#16810579	C₂²	1	2	yes	complemented
28	C₂×#13133127	C₂²	1	2	yes	complemented
29	C₄×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic
30	#16810388	C₂²	1	2	yes	complemented
31	#16810382	C₂²	1	2	yes	complemented
32	#16810382	C₂²	1	2	yes	complemented
33	#16810377	C₂²	1	2	yes	complemented
34	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
35	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#416	C₄	1	1	yes	characteristic, complemented
38	C₅₃×#416	C₄	1	1	yes	characteristic, complemented
39	C₅₃×#416	C₄	1	1	yes	characteristic, complemented
40	C₅₃×#416	C₄	1	1	yes	characteristic, complemented
41	C₅₃×#416	C₂²	1	1	yes	characteristic
42	C₅₃×#416	C₂²	1	1	yes	characteristic
43	C₅₃×#416	C₂²	1	1	yes	characteristic
44	C₅₃×#416	C₂²	1	1	yes	characteristic
45	C₂×#103×C₅₃	C₄	1	1	yes	characteristic, complemented
46	C₂×#103×C₅₃	C₄	1	1	yes	characteristic, complemented
47	C₂×#102×C₅₃	C₄	1	1	yes	characteristic, complemented
48	C₂×#102×C₅₃	C₄	1	1	yes	characteristic, complemented
49	#13133224	—	2	4	no	—
50	#13133224	—	4	2	no	—
51	C₂×#11719917	—	2	2	no	—
52	C₂×#11719917	C₂³	1	2	yes	—
53	C₂×#11719917	C₂³	1	2	yes	—
54	C₂²×#88322	—	2	2	no	—
55	C₂²×#88322	C₂³	1	2	yes	—
56	C₂×C₂²⋊Dic₅₃	—	2	1	no	—
57	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
58	#13133153	—	4	2	no	—
59	C₄⋊#88322	—	2	2	no	—
60	C₄⋊#88322	—	2	2	no	—
61	#13133144	—	2	4	no	—
62	#13133144	—	4	2	no	—
63	C₂²⋊#88322	—	2	2	no	—
64	C₂²⋊#88322	—	2	2	no	—
65	C₂²⋊#88322	—	2	1	no	—
66	C₂²⋊#88322	—	2	1	no	—
67	C₂²⋊#88322	—	2	1	no	—
68	C₂²⋊#88322	—	2	1	no	—
69	C₂²⋊#88322	D₄	1	1	yes	characteristic
70	C₂²⋊#88322	D₄	1	1	yes	characteristic
71	C₂²⋊#88322	D₄	1	2	yes	—
72	#13133128	—	2	2	no	—
73	#13133127	—	4	2	no	—
74	C₄×#88322	—	2	2	no	—
75	#13133124	—	2	1	no	—
76	C₂×#41×C₅₃	—	2	1	no	—
77	C₂×#41×C₅₃	—	2	1	no	—
78	C₂×#41×C₅₃	—	2	4	no	—
79	C₂×#41×C₅₃	—	4	2	no	—
80	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
81	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
83	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
84	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₂²×D₄×C₅₃	—	2	1	no	—
86	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₂³×C₄×C₅₃	—	2	1	no	—
88	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₅₃	—	4	2	no	—
90	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
91	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
92	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
93	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
94	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
96	#117×C₅₃	—	2	1	no	—
97	#117×C₅₃	—	2	1	no	—
98	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
99	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
100	#117×C₅₃	C₂³	1	1	yes	characteristic
101	#117×C₅₃	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
104	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
105	C₂×C₄²×C₅₃	—	2	1	no	—
106	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
107	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
108	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
109	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
110	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
111	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
112	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
113	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
114	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
115	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
116	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
117	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
118	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
119	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
120	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
121	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
122	C₂⁴×C₅₃	—	2	1	no	—
123	C₂⁴×C₅₃	#41	1	1	yes	characteristic
124	#41×C₅₃	—	2	2	no	—
125	#41×C₅₃	—	2	2	no	—
126	#41×C₅₃	—	2	2	no	—
127	#41×C₅₃	—	2	2	no	—
128	#41×C₅₃	—	2	2	no	—
129	#41×C₅₃	—	4	4	no	—
130	#41×C₅₃	—	4	1	no	—
131	#41×C₅₃	—	4	1	no	—
132	#41×C₅₃	—	4	4	no	—
133	#41×C₅₃	#41	1	4	yes	—
134	C₂×Q₈×C₅₃	—	2	1	no	—
135	C₂×Q₈×C₅₃	—	2	2	no	—
136	C₂×Q₈×C₅₃	—	4	1	no	—
137	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
138	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₂×D₄×C₅₃	—	2	1	no	—
140	C₂×D₄×C₅₃	—	2	2	no	—
141	C₂×D₄×C₅₃	—	2	1	no	—
142	C₂×D₄×C₅₃	—	2	1	no	—
143	C₂×D₄×C₅₃	—	2	1	no	—
144	C₂×D₄×C₅₃	—	2	1	no	—
145	C₂×D₄×C₅₃	—	4	1	no	—
146	C₂×D₄×C₅₃	—	4	4	no	—
147	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄×C₅₃	—	2	1	no	—
150	C₂²×C₄×C₅₃	—	2	2	no	—
151	C₂²×C₄×C₅₃	—	2	2	no	—
152	C₂²×C₄×C₅₃	—	2	1	no	—
153	C₂²×C₄×C₅₃	—	2	1	no	—
154	C₂²×C₄×C₅₃	—	2	1	no	—
155	C₂²×C₄×C₅₃	—	2	1	no	—
156	C₂²×C₄×C₅₃	—	2	1	no	—
157	C₂²×C₄×C₅₃	—	2	1	no	—
158	C₂²×C₄×C₅₃	—	4	4	no	—
159	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
164	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
165	C₂²×Dic₅₃	—	2	2	no	—
166	C₂×C₈×C₅₃	—	2	2	no	—
167	C₂×C₈×C₅₃	—	2	2	no	—
168	C₂×C₈×C₅₃	—	4	1	no	—
169	C₂×C₈×C₅₃	—	4	1	no	—
170	C₂×C₈×C₅₃	—	4	1	no	—
171	C₂×C₈×C₅₃	—	4	1	no	—
172	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
173	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
174	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
175	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
176	C₄⋊C₄×C₅₃	—	2	1	no	—
177	C₄⋊C₄×C₅₃	—	2	1	no	—
178	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
179	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
180	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
181	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
182	C₂²⋊C₄×C₅₃	—	2	1	no	—
183	C₂²⋊C₄×C₅₃	—	2	1	no	—
184	C₂²⋊C₄×C₅₃	—	2	1	no	—
185	C₂²⋊C₄×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	C₂²⋊Dic₅₃	—	4	2	no	—
191	C₄×Dic₅₃	—	2	2	no	—
192	C₄×Dic₅₃	—	2	2	no	—
193	#11719917	—	2	2	no	—
194	#11719917	—	2	2	no	—
195	#11719917	—	2	2	no	—
196	#11719917	—	2	2	no	—
197	#11719917	—	4	4	no	—
198	C₂×#88322	—	2	2	no	—
199	C₂×#88322	—	2	2	no	—
200	C₂×#88322	—	2	2	no	—
201	C₂×#88322	—	2	2	no	—
202	C₂×#88322	—	4	4	no	—
203	C₂×#88322	C₂×D₄	1	2	yes	—
204	C₂×#88322	C₂×D₄	1	2	yes	—
205	C₂³×C₅₃	—	2	1	no	—
206	C₂³×C₅₃	—	2	2	no	—
207	C₂³×C₅₃	—	2	2	no	—
208	C₂³×C₅₃	—	2	1	no	—
209	C₂³×C₅₃	—	4	1	no	—
210	C₂³×C₅₃	—	4	2	no	—
211	C₂³×C₅₃	—	4	2	no	—
212	C₂³×C₅₃	—	8	1	no	—
213	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
214	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
215	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
216	Q₈×C₅₃	—	2	1	no	—
217	Q₈×C₅₃	—	2	1	no	—
218	Q₈×C₅₃	—	2	1	no	—
219	Q₈×C₅₃	—	4	2	no	—
220	Q₈×C₅₃	#117	1	2	yes	—
221	D₄×C₅₃	—	2	4	no	—
222	D₄×C₅₃	—	2	1	no	—
223	D₄×C₅₃	—	2	1	no	—
224	D₄×C₅₃	—	2	1	no	—
225	D₄×C₅₃	—	4	2	no	—
226	D₄×C₅₃	—	4	2	no	—
227	D₄×C₅₃	—	4	2	no	—
228	D₄×C₅₃	—	4	2	no	—
229	D₄×C₅₃	#117	1	2	yes	—
230	C₂×C₄×C₅₃	—	2	1	no	—
231	C₂×C₄×C₅₃	—	2	2	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	2	no	—
237	C₂×C₄×C₅₃	—	2	2	no	—
238	C₂×C₄×C₅₃	—	2	4	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	—	4	2	no	—
245	C₂×C₄×C₅₃	—	4	2	no	—
246	C₂×C₄×C₅₃	—	4	2	no	—
247	C₂×C₄×C₅₃	—	4	2	no	—
248	C₂×C₄×C₅₃	—	4	2	no	—
249	C₂×C₄×C₅₃	—	8	1	no	—
250	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
251	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
252	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
253	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
255	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
256	C₂×Dic₅₃	—	4	2	no	—
257	C₂×Dic₅₃	—	4	2	no	—
258	C₂×Dic₅₃	—	4	2	no	—
259	C₈×C₅₃	—	2	1	no	—
260	C₈×C₅₃	—	2	1	no	—
261	C₈×C₅₃	—	2	1	no	—
262	C₈×C₅₃	—	2	1	no	—
263	#88322	—	2	2	no	—
264	#88322	—	2	2	no	—
265	#88322	—	4	2	no	—
266	#88322	—	4	2	no	—
267	#22586	—	53	1	no	maximal, Sylow
268	C₂²×C₅₃	—	2	1	no	—
269	C₂²×C₅₃	—	2	1	no	—
270	C₂²×C₅₃	—	2	1	no	—
271	C₂²×C₅₃	—	2	1	no	—
272	C₂²×C₅₃	—	2	1	no	—
273	C₂²×C₅₃	—	2	2	no	—
274	C₂²×C₅₃	—	2	1	no	—
275	C₂²×C₅₃	—	2	1	no	—
276	C₂²×C₅₃	—	4	4	no	—
277	C₂²×C₅₃	—	4	1	no	—
278	C₂²×C₅₃	—	4	1	no	—
279	C₂²×C₅₃	—	4	1	no	—
280	C₂²×C₅₃	—	4	1	no	—
281	C₂²×C₅₃	—	4	1	no	—
282	C₂²×C₅₃	—	8	4	no	—
283	C₂²×C₅₃	#444	1	2	yes	—
284	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
285	C₂²×C₅₃	#551	1	2	yes	—
286	C₄×C₅₃	—	2	1	no	—
287	C₄×C₅₃	—	2	1	no	—
288	C₄×C₅₃	—	2	2	no	—
289	C₄×C₅₃	—	4	1	no	—
290	C₄×C₅₃	—	4	1	no	—
291	C₄×C₅₃	—	4	1	no	—
292	C₄×C₅₃	—	4	1	no	—
293	C₄×C₅₃	—	4	1	no	—
294	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
295	C₄×C₅₃	#518	1	1	yes	characteristic
296	C₄×C₅₃	#546	1	2	yes	—
297	Dic₅₃	—	8	2	no	—
298	#2924	—	53	1	no	—
299	C₂×#416	D₅₃	1	1	yes	characteristic
300	#2880	—	53	1	no	—
301	#1938	—	53	2	no	—
302	#1929	—	53	2	no	—
303	#1866	—	53	2	no	—
304	#1826	—	53	1	no	—
305	#1810	—	53	1	no	—
306	#1789	—	53	1	no	—
307	#1765	—	53	2	no	—
308	#1752	—	53	1	no	—
309	C₂×C₅₃	—	2	2	no	—
310	C₂×C₅₃	—	2	2	no	—
311	C₂×C₅₃	—	4	1	no	—
312	C₂×C₅₃	—	4	2	no	—
313	C₂×C₅₃	—	8	1	no	—
314	C₂×C₅₃	#2342	1	1	yes	characteristic
315	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
316	C₂×C₅₃	#2957	1	1	yes	characteristic
317	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
318	C₂×#131	—	53	2	no	—
319	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
320	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
321	#433	—	106	2	no	—
322	#432	—	106	1	no	—
323	#431	—	106	1	no	—
324	C₂×#108	—	53	2	no	—
325	#424	—	53	2	no	—
326	C₂×#104	—	53	2	no	—
327	C₂×#104	—	53	2	no	—
328	#416	Dic₅₃	1	1	yes	characteristic, complemented
329	#416	Dic₅₃	1	1	yes	characteristic, complemented
330	#416	Dic₅₃	1	1	yes	characteristic, complemented
331	#416	Dic₅₃	1	1	yes	characteristic, complemented
332	#416	C₂×D₅₃	1	1	yes	characteristic
333	#416	C₂×D₅₃	1	1	yes	characteristic
334	#416	C₂×D₅₃	1	1	yes	characteristic
335	#416	C₂×D₅₃	1	1	yes	characteristic
336	C₂×#103	Dic₅₃	1	1	yes	characteristic, complemented
337	C₂×#103	Dic₅₃	1	1	yes	characteristic, complemented
338	C₂×#102	Dic₅₃	1	1	yes	characteristic, complemented
339	C₂×#102	Dic₅₃	1	1	yes	characteristic, complemented
340	#408	—	53	2	no	—
341	#408	—	106	2	no	—
342	#407	—	53	1	no	—
343	#407	—	53	1	no	—
344	#407	—	106	1	no	—
345	C₂×C₂²⋊C₈	—	53	1	no	—
346	C₂×C₂²⋊C₈	—	106	1	no	—
347	#405	—	53	2	no	—
348	C₂×#97	—	53	2	no	—
349	C₄×C₂²⋊C₄	—	53	1	no	—
350	#341	—	53	2	no	—
351	#339	—	53	2	no	—
352	#339	—	53	2	no	—
353	#336	—	53	2	no	—
354	C₅₃	#22586	1	1	yes	characteristic, complemented, Sylow
355	C₂×#41	—	2	1	no	—
356	C₂×#41	—	2	1	no	—
357	C₂×#41	—	2	4	no	—
358	C₂×#41	—	4	2	no	—
359	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
360	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
361	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
362	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
363	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
364	C₂²×D₄	—	2	1	no	—
365	C₂²×D₄	C₂×Dic₅₃	1	1	yes	characteristic
366	C₂³×C₄	—	2	1	no	—
367	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
368	#131	—	106	4	no	—
369	#131	—	212	2	no	—
370	C₂×M₁₆	—	53	2	no	—
371	C₂×M₁₆	—	53	2	no	—
372	C₂×M₁₆	—	106	2	no	—
373	C₂²×C₈	—	4	2	no	—
374	C₂²×C₈	—	53	2	no	—
375	C₂²×C₈	—	106	2	no	—
376	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
377	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
378	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
379	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
380	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
381	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
382	#117	—	2	1	no	—
383	#117	—	2	1	no	—
384	#117	C₂×Dic₅₃	1	1	yes	characteristic
385	#117	C₂×Dic₅₃	1	1	yes	characteristic
386	#117	C₂²×D₅₃	1	1	yes	characteristic
387	#117	C₂²×D₅₃	1	1	yes	characteristic
388	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
389	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
390	C₂×C₂²⋊C₄	—	2	1	no	—
391	C₂×C₂²⋊C₄	—	106	1	no	—
392	C₂×C₄²	—	2	1	no	—
393	C₂×C₄²	—	53	2	no	—
394	#108	—	212	2	no	—
395	C₄⋊C₈	—	106	2	no	—
396	C₄⋊C₈	—	106	2	no	—
397	#104	—	106	4	no	—
398	#104	—	212	2	no	—
399	#103	C₂×Dic₅₃	1	1	yes	characteristic
400	#103	C₂×Dic₅₃	1	1	yes	characteristic
401	#103	C₂×Dic₅₃	1	1	yes	characteristic
402	#103	C₂×Dic₅₃	1	1	yes	characteristic
403	#103	C₂×Dic₅₃	1	1	yes	characteristic
404	#103	C₂×Dic₅₃	1	1	yes	characteristic
405	#103	C₂×Dic₅₃	1	1	yes	characteristic
406	#103	C₂×Dic₅₃	1	1	yes	characteristic
407	#102	C₂×Dic₅₃	1	1	yes	characteristic
408	#102	C₂×Dic₅₃	1	1	yes	characteristic
409	#102	C₂×Dic₅₃	1	1	yes	characteristic
410	#102	C₂×Dic₅₃	1	1	yes	characteristic
411	#102	C₂×Dic₅₃	1	1	yes	characteristic
412	#102	C₂×Dic₅₃	1	1	yes	characteristic
413	#102	C₂×Dic₅₃	1	1	yes	characteristic
414	#102	C₂×Dic₅₃	1	1	yes	characteristic
415	C₂²⋊C₈	—	53	1	no	—
416	C₂²⋊C₈	—	53	1	no	—
417	C₂²⋊C₈	—	53	2	no	—
418	C₂²⋊C₈	—	106	2	no	—
419	C₂²⋊C₈	—	106	2	no	—
420	C₂²⋊C₈	—	106	1	no	—
421	C₂²⋊C₈	—	106	1	no	—
422	C₂²⋊C₈	—	106	1	no	—
423	C₂²⋊C₈	—	106	1	no	—
424	#97	—	106	2	no	—
425	#97	—	212	2	no	—
426	C₄×C₈	—	106	2	no	—
427	#95	—	106	1	no	—
428	C₂⁴	—	2	1	no	—
429	C₂⁴	#11719947	1	1	yes	characteristic
430	#41	—	2	2	no	—
431	#41	—	2	2	no	—
432	#41	—	2	2	no	—
433	#41	—	2	2	no	—
434	#41	—	2	2	no	—
435	#41	—	4	4	no	—
436	#41	—	4	1	no	—
437	#41	—	4	1	no	—
438	#41	—	4	4	no	—
439	#41	#11719945	1	4	yes	—
440	C₂×Q₈	—	2	1	no	—
441	C₂×Q₈	—	2	2	no	—
442	C₂×Q₈	—	4	1	no	—
443	C₂×Q₈	C₂²×Dic₅₃	1	1	yes	characteristic
444	C₂×Q₈	C₂²×Dic₅₃	1	1	yes	characteristic
445	C₂×D₄	—	2	1	no	—
446	C₂×D₄	—	2	2	no	—
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	4	1	no	—
452	C₂×D₄	—	4	4	no	—
453	C₂×D₄	C₂²×Dic₅₃	1	1	yes	characteristic
454	C₂×D₄	C₂²×Dic₅₃	1	1	yes	characteristic
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	4	4	no	—
465	C₂²×C₄	—	106	2	no	—
466	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
467	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
468	C₂²×C₄	#11719947	1	1	yes	characteristic
469	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
470	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
471	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
472	M₁₆	—	106	2	no	—
473	M₁₆	—	106	2	no	—
474	M₁₆	—	106	2	no	—
475	M₁₆	—	106	2	no	—
476	M₁₆	—	212	4	no	—
477	C₂×C₈	—	2	2	no	—
478	C₂×C₈	—	2	2	no	—
479	C₂×C₈	—	4	1	no	—
480	C₂×C₈	—	4	1	no	—
481	C₂×C₈	—	4	1	no	—
482	C₂×C₈	—	4	1	no	—
483	C₂×C₈	—	53	2	no	—
484	C₂×C₈	—	53	2	no	—
485	C₂×C₈	—	106	2	no	—
486	C₂×C₈	—	106	2	no	—
487	C₂×C₈	—	106	2	no	—
488	C₂×C₈	—	106	2	no	—
489	C₂×C₈	—	212	4	no	—
490	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
491	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
492	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
493	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
497	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
498	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
499	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
500	C₂²⋊C₄	—	2	1	no	—
501	C₂²⋊C₄	—	2	1	no	—
502	C₂²⋊C₄	—	2	1	no	—
503	C₂²⋊C₄	—	2	1	no	—
504	C₂²⋊C₄	—	212	2	no	—
505	C₄²	—	2	1	no	—
506	C₄²	—	2	1	no	—
507	C₄²	—	2	1	no	—
508	C₄²	—	2	1	no	—
509	C₄²	—	106	2	no	—
510	C₄²	—	106	2	no	—
511	C₂³	—	2	2	no	—
512	C₂³	—	2	1	no	—
513	C₂³	—	2	2	no	—
514	C₂³	—	2	1	no	—
515	C₂³	—	4	1	no	—
516	C₂³	—	4	2	no	—
517	C₂³	—	4	2	no	—
518	C₂³	—	8	1	no	—
519	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
520	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
521	C₂³	C₂×#11719947	1	1	yes	characteristic
522	Q₈	—	2	1	no	—
523	Q₈	—	2	1	no	—
524	Q₈	—	2	1	no	—
525	Q₈	—	4	2	no	—
526	Q₈	#13133176	1	2	yes	—
527	D₄	—	2	4	no	—
528	D₄	—	2	1	no	—
529	D₄	—	2	1	no	—
530	D₄	—	2	1	no	—
531	D₄	—	4	2	no	—
532	D₄	—	4	2	no	—
533	D₄	—	4	2	no	—
534	D₄	—	4	2	no	—
535	D₄	#13133176	1	2	yes	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	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	—	4	2	no	—
551	C₂×C₄	—	4	2	no	—
552	C₂×C₄	—	4	2	no	—
553	C₂×C₄	—	4	2	no	—
554	C₂×C₄	—	4	2	no	—
555	C₂×C₄	—	8	1	no	—
556	C₂×C₄	—	212	2	no	—
557	C₂×C₄	—	212	2	no	—
558	C₂×C₄	—	212	2	no	—
559	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
560	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
561	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
562	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
563	C₂×C₄	C₂×#11719945	1	2	yes	—
564	C₂×C₄	#13133262	1	2	yes	—
565	C₈	—	2	1	no	—
566	C₈	—	2	1	no	—
567	C₈	—	2	1	no	—
568	C₈	—	2	1	no	—
569	C₈	—	106	2	no	—
570	C₈	—	106	2	no	—
571	C₈	—	212	2	no	—
572	C₈	—	212	2	no	—
573	C₂²	—	2	1	no	—
574	C₂²	—	2	1	no	—
575	C₂²	—	2	1	no	—
576	C₂²	—	2	1	no	—
577	C₂²	—	2	1	no	—
578	C₂²	—	2	2	no	—
579	C₂²	—	2	1	no	—
580	C₂²	—	2	1	no	—
581	C₂²	—	4	4	no	—
582	C₂²	—	4	1	no	—
583	C₂²	—	4	1	no	—
584	C₂²	—	4	1	no	—
585	C₂²	—	4	1	no	—
586	C₂²	—	4	1	no	—
587	C₂²	—	8	4	no	—
588	C₂²	#16810745	1	2	yes	—
589	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
590	C₂²	#16811299	1	2	yes	—
591	C₄	—	2	1	no	—
592	C₄	—	2	1	no	—
593	C₄	—	2	2	no	—
594	C₄	—	4	1	no	—
595	C₄	—	4	1	no	—
596	C₄	—	4	1	no	—
597	C₄	—	4	1	no	—
598	C₄	—	4	1	no	—
599	C₄	—	424	2	no	—
600	C₄	C₂×#13133176	1	1	yes	characteristic
601	C₄	#16811098	1	1	yes	characteristic
602	C₄	#16811263	1	2	yes	—
603	C₂	—	2	2	no	—
604	C₂	—	2	2	no	—
605	C₂	—	4	1	no	—
606	C₂	—	4	2	no	—
607	C₂	—	8	1	no	—
608	C₂	#20277564	1	1	yes	characteristic
609	C₂	C₂×#16810745	1	1	yes	characteristic
610	C₂	#20283632	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#416	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×#41×C₅₃	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#416	D₅₃
hypocenter	C₅₃	#22586
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#416

Complements

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

N	H ≅ G/N	type
1. #20283373	C₂	semidirect
3. #20275428	C₂	semidirect
4. #20275365	C₂	semidirect
5. #20274992	C₂	semidirect
6. #20274814	C₂	semidirect
7. #20274753	C₂	semidirect
9. #20274562	C₂	semidirect
10. #20274491	C₂	semidirect
16. C₂×#13133153	C₂²	semidirect
17. #16810650	C₂²	semidirect
27. #16810579	C₂²	semidirect
28. C₂×#13133127	C₂²	semidirect
30. #16810388	C₂²	semidirect
31. #16810382	C₂²	semidirect
32. #16810382	C₂²	semidirect
33. #16810377	C₂²	semidirect
37. C₅₃×#416	C₄	semidirect
38. C₅₃×#416	C₄	semidirect
39. C₅₃×#416	C₄	semidirect
40. C₅₃×#416	C₄	semidirect
45. C₂×#103×C₅₃	C₄	semidirect
46. C₂×#103×C₅₃	C₄	semidirect
47. C₂×#102×C₅₃	C₄	semidirect
48. C₂×#102×C₅₃	C₄	semidirect
328. #416	Dic₅₃	semidirect
329. #416	Dic₅₃	semidirect
330. #416	Dic₅₃	semidirect
331. #416	Dic₅₃	semidirect
336. C₂×#103	Dic₅₃	semidirect
337. C₂×#103	Dic₅₃	semidirect
338. C₂×#102	Dic₅₃	semidirect
339. C₂×#102	Dic₅₃	semidirect
354. C₅₃	#22586	semidirect

Maximal subgroups

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

subgroup	index
#20283373	2
#20283164	2
#20275428	2
#20275365	2
#20274992	2
#20274814	2
#20274753	2
#20274659	2
#20274562	2
#20274491	2
C₂×C₅₃×#416	2
#22586	53