Subgroups — #27389037

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283350	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810681	C₂	1	1	yes	maximal, characteristic
3	#20283133	C₂	1	1	yes	maximal, characteristic
4	#20275489	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275489	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275487	C₂	1	1	yes	maximal, characteristic, complemented
7	#20275487	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274922	C₂	1	1	yes	maximal, characteristic
9	#20274920	C₂	1	1	yes	maximal, characteristic
10	#20274919	C₂	1	1	yes	maximal, characteristic
11	#20274623	C₂	1	1	yes	maximal, characteristic, complemented
12	#20274620	C₂	1	1	yes	maximal, characteristic, complemented
13	#108×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
14	#20274616	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#1855	C₂	1	1	yes	maximal, characteristic
16	C₂²×#11719917	C₂²	1	1	yes	characteristic
17	C₂×#13133176	C₂²	1	1	yes	characteristic
18	#16810681	—	2	1	no	—
19	#16810681	—	2	1	no	—
20	#16810681	—	2	1	no	—
21	#16810681	—	2	1	no	—
22	#16810681	C₄	1	2	yes	—
23	#16810681	C₄	1	2	yes	—
24	#16810681	C₄	1	2	yes	—
25	#16810681	C₄	1	2	yes	—
26	C₂×#13133153	C₂²	1	1	yes	characteristic
27	C₂×#13133153	C₂²	1	1	yes	characteristic
28	C₂×#13133152	C₂²	1	1	yes	characteristic
29	C₂×#13133152	C₂²	1	1	yes	characteristic
30	C₂×#13133152	C₂²	1	1	yes	characteristic
31	C₂×#13133152	C₂²	1	1	yes	characteristic
32	#16810646	C₂²	1	1	yes	characteristic
33	#16810646	C₂²	1	1	yes	characteristic
34	C₂×#13133146	C₂²	1	1	yes	characteristic
35	C₂×#13133146	C₂²	1	1	yes	characteristic
36	#16810587	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
38	#16810581	—	2	1	no	—
39	#16810581	—	2	1	no	—
40	#16810578	—	2	1	no	—
41	#16810578	—	2	1	no	—
42	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
43	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
44	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
45	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
46	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
47	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
48	C₂×#13133127	C₂²	1	1	yes	characteristic
49	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
50	#16810388	C₂²	1	1	yes	characteristic
51	#16810388	C₂²	1	1	yes	characteristic
52	#16810387	C₂²	1	1	yes	characteristic
53	#16810387	C₂²	1	1	yes	characteristic
54	#16810387	C₂²	1	1	yes	characteristic
55	#16810387	C₂²	1	1	yes	characteristic
56	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
57	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
58	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
59	C₅₃×#407	C₂²	1	1	yes	characteristic
60	C₂×C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
61	C₅₃×#341	C₂²	1	1	yes	characteristic
62	C₅₃×#341	C₂²	1	1	yes	characteristic
63	C₂×#11719917	—	2	2	no	—
64	C₂×#11719917	—	2	2	no	—
65	C₂×#11719917	—	2	1	no	—
66	C₂×#11719917	C₂³	1	1	yes	characteristic
67	C₂×#11719917	C₂³	1	1	yes	characteristic
68	C₂×#11719917	C₂³	1	1	yes	characteristic
69	C₂×#11719917	C₂³	1	1	yes	characteristic
70	C₂²×#88322	—	2	1	no	—
71	C₂²×#88322	C₂³	1	1	yes	characteristic
72	C₂²×#88322	C₂³	1	1	yes	characteristic
73	#13133176	C₂×C₄	1	1	yes	characteristic
74	#13133176	C₂×C₄	1	1	yes	characteristic
75	#13133176	C₂×C₄	1	1	yes	characteristic
76	#13133176	C₂×C₄	1	1	yes	characteristic
77	#13133176	D₄	1	1	yes	characteristic
78	#13133176	D₄	1	1	yes	characteristic
79	#13133176	D₄	1	1	yes	characteristic
80	#13133176	D₄	1	1	yes	characteristic
81	C₂×C₄⋊Dic₅₃	Q₈	1	1	yes	characteristic
82	C₂×C₄⋊Dic₅₃	Q₈	1	1	yes	characteristic
83	C₂×C₂²⋊Dic₅₃	Q₈	1	1	yes	characteristic
84	C₂×C₂²⋊Dic₅₃	Q₈	1	1	yes	characteristic
85	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
86	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
87	#13133153	—	2	2	no	—
88	#13133153	—	2	2	no	—
89	#13133152	—	2	2	no	—
90	#13133152	—	2	2	no	—
91	#13133152	—	2	2	no	—
92	#13133152	—	2	2	no	—
93	#13133146	—	2	1	no	—
94	#13133146	—	2	1	no	—
95	#13133146	—	2	1	no	—
96	#13133146	—	2	1	no	—
97	#13133146	C₂×C₄	1	2	yes	—
98	#13133146	C₂×C₄	1	2	yes	—
99	#13133146	C₂×C₄	1	2	yes	—
100	#13133146	C₂×C₄	1	2	yes	—
101	C₂²⋊#88322	—	2	1	no	—
102	C₂²⋊#88322	—	2	1	no	—
103	C₂²⋊#88322	C₂×C₄	1	2	yes	—
104	C₂²⋊#88322	C₂×C₄	1	2	yes	—
105	#13133127	—	2	1	no	—
106	#13133127	—	2	1	no	—
107	#13133127	—	2	1	no	—
108	#13133127	—	2	1	no	—
109	#13133127	—	2	1	no	—
110	#13133127	—	2	1	no	—
111	#13133127	C₂×C₄	1	2	yes	—
112	#13133127	C₂×C₄	1	2	yes	—
113	C₈×Dic₅₃	—	2	1	no	—
114	C₈×Dic₅₃	—	2	1	no	—
115	C₈×Dic₅₃	—	2	1	no	—
116	C₈×Dic₅₃	—	2	1	no	—
117	C₈×Dic₅₃	—	2	1	no	—
118	C₈×Dic₅₃	—	2	1	no	—
119	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
120	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
121	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
122	C₂×M₁₆×C₅₃	—	2	2	no	—
123	C₂×M₁₆×C₅₃	—	2	2	no	—
124	C₂×M₁₆×C₅₃	—	2	1	no	—
125	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
126	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
127	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
128	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
129	C₂²×C₈×C₅₃	—	2	1	no	—
130	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
131	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
132	#108×C₅₃	—	2	2	no	—
133	#108×C₅₃	—	2	2	no	—
134	C₂²⋊C₈×C₅₃	—	2	1	no	—
135	C₂²⋊C₈×C₅₃	—	2	1	no	—
136	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
137	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
138	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
139	C₂²×C₄×C₅₃	—	2	1	no	—
140	C₂²×C₄×C₅₃	—	2	1	no	—
141	C₂²×C₄×C₅₃	—	2	1	no	—
142	C₂²×C₄×C₅₃	—	2	1	no	—
143	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
148	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
149	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
150	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
151	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
152	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
153	M₁₆×C₅₃	—	2	4	no	—
154	M₁₆×C₅₃	—	2	1	no	—
155	M₁₆×C₅₃	—	2	1	no	—
156	M₁₆×C₅₃	—	2	1	no	—
157	M₁₆×C₅₃	—	2	1	no	—
158	M₁₆×C₅₃	—	2	1	no	—
159	M₁₆×C₅₃	—	2	1	no	—
160	M₁₆×C₅₃	#41	1	2	yes	—
161	M₁₆×C₅₃	#41	1	2	yes	—
162	C₂×C₈×C₅₃	—	2	4	no	—
163	C₂×C₈×C₅₃	—	2	1	no	—
164	C₂×C₈×C₅₃	—	2	1	no	—
165	C₂×C₈×C₅₃	—	2	1	no	—
166	C₂×C₈×C₅₃	—	2	1	no	—
167	C₂×C₈×C₅₃	—	2	1	no	—
168	C₂×C₈×C₅₃	—	2	1	no	—
169	C₂×C₈×C₅₃	—	2	1	no	—
170	C₂×C₈×C₅₃	—	2	1	no	—
171	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
172	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
173	C₂²⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
174	C₂²⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
175	C₂²⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
176	C₂²⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
177	C₂²⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
178	C₂²⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
179	C₂²⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
180	C₂²⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
181	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
182	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
183	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
184	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
185	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
186	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
187	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
188	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
189	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
190	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
191	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
192	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
193	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
194	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
195	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
196	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
197	#11719917	—	2	4	no	—
198	#11719917	—	2	1	no	—
199	#11719917	—	2	1	no	—
200	#11719917	—	2	1	no	—
201	#11719917	—	2	1	no	—
202	#11719917	—	2	1	no	—
203	#11719917	—	2	1	no	—
204	#11719917	#41	1	2	yes	—
205	#11719917	#41	1	2	yes	—
206	C₂×#88322	—	2	4	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	—	2	1	no	—
214	C₂×#88322	—	2	1	no	—
215	C₂×#88322	C₂²×C₄	1	2	yes	—
216	C₂×#88322	C₂²×C₄	1	2	yes	—
217	C₂³×C₅₃	—	2	1	no	—
218	C₂³×C₅₃	—	2	1	no	—
219	C₂³×C₅₃	—	4	1	no	—
220	C₂³×C₅₃	—	4	1	no	—
221	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
222	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
223	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
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	1	no	—
228	C₂×C₄×C₅₃	—	2	1	no	—
229	C₂×C₄×C₅₃	—	2	1	no	—
230	C₂×C₄×C₅₃	—	2	1	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	C₂×C₄⋊C₄	1	1	yes	characteristic
235	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
236	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
237	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
239	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
240	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
241	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
242	C₂×Dic₅₃	—	4	1	no	—
243	C₂×Dic₅₃	—	4	1	no	—
244	C₂×Dic₅₃	—	4	1	no	—
245	C₂×Dic₅₃	—	4	1	no	—
246	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
247	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
248	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
249	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
250	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
251	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
252	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
253	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
254	C₈×C₅₃	—	2	2	no	—
255	C₈×C₅₃	—	2	2	no	—
256	C₈×C₅₃	—	2	2	no	—
257	C₈×C₅₃	—	2	2	no	—
258	C₈×C₅₃	—	4	1	no	—
259	C₈×C₅₃	—	4	1	no	—
260	#88322	—	2	2	no	—
261	#88322	—	2	2	no	—
262	#88322	—	2	2	no	—
263	#88322	—	2	2	no	—
264	#88322	—	4	1	no	—
265	#88322	—	4	1	no	—
266	#21372	—	53	1	no	maximal, Sylow
267	C₂²×C₅₃	—	2	1	no	—
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	1	no	—
274	C₂²×C₅₃	—	2	1	no	—
275	C₂²×C₅₃	—	8	1	no	—
276	C₂²×C₅₃	—	8	1	no	—
277	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
278	C₂²×C₅₃	#533	1	1	yes	characteristic
279	C₂²×C₅₃	#533	1	1	yes	characteristic
280	C₄×C₅₃	—	2	1	no	—
281	C₄×C₅₃	—	2	1	no	—
282	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
283	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
284	C₄×C₅₃	#525	1	1	yes	characteristic
285	C₄×C₅₃	#525	1	1	yes	characteristic
286	Dic₅₃	—	2	1	no	—
287	Dic₅₃	—	2	1	no	—
288	Dic₅₃	—	2	1	no	—
289	Dic₅₃	—	2	1	no	—
290	Dic₅₃	—	2	1	no	—
291	Dic₅₃	—	2	1	no	—
292	Dic₅₃	—	2	1	no	—
293	Dic₅₃	—	2	1	no	—
294	#2921	—	53	1	no	—
295	C₂×#433	—	53	1	no	—
296	#2873	—	53	1	no	—
297	#1950	—	53	1	no	—
298	#1950	—	53	1	no	—
299	#1950	—	53	1	no	—
300	#1950	—	53	1	no	—
301	#1855	—	53	1	no	—
302	#1855	—	53	1	no	—
303	#1855	—	53	1	no	—
304	#1855	D₅₃	1	1	yes	characteristic
305	#1778	—	53	1	no	—
306	#1778	—	53	1	no	—
307	C₄×#108	—	53	1	no	—
308	C₄×#108	—	53	1	no	—
309	C₂×C₅₃	—	2	1	no	—
310	C₂×C₅₃	—	2	1	no	—
311	C₂×C₅₃	—	4	1	no	—
312	C₂×C₅₃	—	4	1	no	—
313	C₂×C₅₃	#2296	1	1	yes	characteristic
314	C₂×C₅₃	#2916	1	1	yes	characteristic
315	C₂×C₅₃	#2916	1	1	yes	characteristic
316	C₂²×M₁₆	—	53	1	no	—
317	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
318	C₂×#117	—	53	1	no	—
319	#433	—	53	2	no	—
320	#433	—	53	2	no	—
321	#433	—	53	2	no	—
322	#433	—	53	2	no	—
323	#433	—	106	1	no	—
324	#433	—	106	1	no	—
325	#433	—	106	1	no	—
326	#433	—	106	1	no	—
327	C₂×#108	—	53	1	no	—
328	C₂×#108	—	53	1	no	—
329	C₂×#108	—	53	1	no	—
330	C₂×#108	—	53	1	no	—
331	C₂×#108	—	53	1	no	—
332	C₂×#108	—	53	1	no	—
333	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
334	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
335	#423	—	53	1	no	—
336	#423	—	53	1	no	—
337	C₂×C₄⋊C₈	—	53	1	no	—
338	C₂×C₄⋊C₈	—	53	1	no	—
339	#407	—	53	1	no	—
340	#407	C₂×D₅₃	1	1	yes	characteristic
341	C₂×C₂²⋊C₈	—	53	1	no	—
342	C₂×C₂²⋊C₈	C₂×D₅₃	1	1	yes	characteristic
343	#405	—	106	1	no	—
344	#405	—	106	1	no	—
345	#405	—	106	1	no	—
346	#405	—	106	1	no	—
347	C₄×M₁₆	—	53	1	no	—
348	C₄×M₁₆	—	53	1	no	—
349	C₄×M₁₆	—	53	1	no	—
350	C₄×M₁₆	—	53	1	no	—
351	C₄×M₁₆	—	53	1	no	—
352	C₄×M₁₆	—	53	1	no	—
353	C₂×#97	—	53	1	no	—
354	C₂×C₄×C₈	—	53	1	no	—
355	#341	—	53	1	no	—
356	#341	—	53	1	no	—
357	#341	—	53	1	no	—
358	#341	—	53	1	no	—
359	#341	—	53	1	no	—
360	#341	—	53	1	no	—
361	#341	C₂×D₅₃	1	1	yes	characteristic
362	#341	C₂×D₅₃	1	1	yes	characteristic
363	C₅₃	#21372	1	1	yes	characteristic, complemented, Sylow
364	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
365	C₂×M₁₆	—	2	1	no	—
366	C₂×M₁₆	—	2	2	no	—
367	C₂×M₁₆	—	2	2	no	—
368	C₂×M₁₆	—	53	1	no	—
369	C₂×M₁₆	—	53	1	no	—
370	C₂×M₁₆	—	53	1	no	—
371	C₂×M₁₆	—	53	1	no	—
372	C₂×M₁₆	—	106	2	no	—
373	C₂×M₁₆	—	106	2	no	—
374	C₂×M₁₆	—	106	1	no	—
375	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
376	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
377	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
378	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
379	C₂²×C₈	—	2	1	no	—
380	C₂²×C₈	—	53	1	no	—
381	C₂²×C₈	—	53	1	no	—
382	C₂²×C₈	—	106	1	no	—
383	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
384	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
385	#117	—	53	1	no	—
386	#117	—	53	1	no	—
387	#117	—	53	1	no	—
388	#117	—	53	1	no	—
389	#117	—	53	1	no	—
390	#117	—	53	1	no	—
391	#117	—	53	1	no	—
392	#117	—	53	1	no	—
393	C₂×C₄⋊C₄	—	53	1	no	—
394	C₂×C₄⋊C₄	—	53	1	no	—
395	C₂×C₂²⋊C₄	—	53	1	no	—
396	C₂×C₂²⋊C₄	—	53	1	no	—
397	C₂×C₄²	—	53	1	no	—
398	C₂×C₄²	—	53	1	no	—
399	#108	—	2	2	no	—
400	#108	—	2	2	no	—
401	#108	—	106	2	no	—
402	#108	—	106	2	no	—
403	#108	—	106	2	no	—
404	#108	—	106	2	no	—
405	#108	—	106	2	no	—
406	#108	—	106	2	no	—
407	C₄⋊C₈	—	53	2	no	—
408	C₄⋊C₈	—	53	2	no	—
409	C₄⋊C₈	—	53	2	no	—
410	C₄⋊C₈	—	53	2	no	—
411	C₄⋊C₈	—	106	1	no	—
412	C₄⋊C₈	—	106	1	no	—
413	C₄⋊C₈	—	106	1	no	—
414	C₄⋊C₈	—	106	1	no	—
415	C₂²⋊C₈	—	2	1	no	—
416	C₂²⋊C₈	—	2	1	no	—
417	C₂²⋊C₈	—	53	2	no	—
418	C₂²⋊C₈	—	53	2	no	—
419	C₂²⋊C₈	—	106	1	no	—
420	C₂²⋊C₈	—	106	1	no	—
421	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
422	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
423	#97	—	53	2	no	—
424	#97	—	53	2	no	—
425	#97	—	106	1	no	—
426	#97	—	106	1	no	—
427	#97	—	106	1	no	—
428	#97	—	106	1	no	—
429	#97	—	106	1	no	—
430	#97	—	106	1	no	—
431	C₄×C₈	—	53	2	no	—
432	C₄×C₈	—	53	2	no	—
433	C₄×C₈	—	106	1	no	—
434	C₄×C₈	—	106	1	no	—
435	C₄×C₈	—	106	1	no	—
436	C₄×C₈	—	106	1	no	—
437	C₄×C₈	—	106	1	no	—
438	C₄×C₈	—	106	1	no	—
439	C₂⁴	Q₈×D₅₃	1	1	yes	characteristic
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	53	1	no	—
445	C₂²×C₄	—	53	1	no	—
446	C₂²×C₄	—	53	1	no	—
447	C₂²×C₄	—	53	1	no	—
448	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
449	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
450	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
451	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
452	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
453	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
454	M₁₆	—	2	4	no	—
455	M₁₆	—	2	1	no	—
456	M₁₆	—	2	1	no	—
457	M₁₆	—	2	1	no	—
458	M₁₆	—	2	1	no	—
459	M₁₆	—	2	1	no	—
460	M₁₆	—	2	1	no	—
461	M₁₆	—	53	2	no	—
462	M₁₆	—	53	2	no	—
463	M₁₆	—	106	4	no	—
464	M₁₆	—	106	1	no	—
465	M₁₆	—	106	1	no	—
466	M₁₆	—	106	1	no	—
467	M₁₆	—	106	1	no	—
468	M₁₆	—	106	1	no	—
469	M₁₆	—	106	1	no	—
470	M₁₆	#11719947	1	2	yes	—
471	M₁₆	#11719947	1	2	yes	—
472	C₂×C₈	—	2	4	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	53	2	no	—
482	C₂×C₈	—	53	2	no	—
483	C₂×C₈	—	106	4	no	—
484	C₂×C₈	—	106	1	no	—
485	C₂×C₈	—	106	1	no	—
486	C₂×C₈	—	106	1	no	—
487	C₂×C₈	—	106	1	no	—
488	C₂×C₈	—	106	1	no	—
489	C₂×C₈	—	106	1	no	—
490	C₂×C₈	—	106	1	no	—
491	C₂×C₈	—	106	1	no	—
492	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
493	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
494	C₄⋊C₄	—	53	1	no	—
495	C₄⋊C₄	—	53	1	no	—
496	C₄⋊C₄	—	53	1	no	—
497	C₄⋊C₄	—	53	1	no	—
498	C₄⋊C₄	—	53	1	no	—
499	C₄⋊C₄	—	53	1	no	—
500	C₄⋊C₄	—	53	1	no	—
501	C₄⋊C₄	—	53	1	no	—
502	C₂²⋊C₄	—	53	1	no	—
503	C₂²⋊C₄	—	53	1	no	—
504	C₂²⋊C₄	—	53	1	no	—
505	C₂²⋊C₄	—	53	1	no	—
506	C₂²⋊C₄	—	53	1	no	—
507	C₂²⋊C₄	—	53	1	no	—
508	C₂²⋊C₄	—	53	1	no	—
509	C₂²⋊C₄	—	53	1	no	—
510	C₄²	—	53	1	no	—
511	C₄²	—	53	1	no	—
512	C₄²	—	53	1	no	—
513	C₄²	—	53	1	no	—
514	C₄²	—	53	1	no	—
515	C₄²	—	53	1	no	—
516	C₄²	—	53	1	no	—
517	C₄²	—	53	1	no	—
518	C₂³	—	2	1	no	—
519	C₂³	—	2	1	no	—
520	C₂³	—	4	1	no	—
521	C₂³	—	4	1	no	—
522	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
523	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
524	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
536	C₂×C₄	—	53	1	no	—
537	C₂×C₄	—	53	1	no	—
538	C₂×C₄	—	53	1	no	—
539	C₂×C₄	—	53	1	no	—
540	C₂×C₄	—	53	1	no	—
541	C₂×C₄	—	53	1	no	—
542	C₂×C₄	—	53	1	no	—
543	C₂×C₄	—	212	1	no	—
544	C₂×C₄	—	212	1	no	—
545	C₂×C₄	—	212	1	no	—
546	C₂×C₄	—	212	1	no	—
547	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
548	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
549	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
550	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
551	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
552	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
553	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
554	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
555	C₈	—	2	2	no	—
556	C₈	—	2	2	no	—
557	C₈	—	2	2	no	—
558	C₈	—	2	2	no	—
559	C₈	—	4	1	no	—
560	C₈	—	4	1	no	—
561	C₈	—	106	2	no	—
562	C₈	—	106	2	no	—
563	C₈	—	106	2	no	—
564	C₈	—	106	2	no	—
565	C₈	—	212	1	no	—
566	C₈	—	212	1	no	—
567	C₂²	—	2	1	no	—
568	C₂²	—	2	1	no	—
569	C₂²	—	2	1	no	—
570	C₂²	—	2	1	no	—
571	C₂²	—	2	1	no	—
572	C₂²	—	2	1	no	—
573	C₂²	—	2	1	no	—
574	C₂²	—	2	1	no	—
575	C₂²	—	8	1	no	—
576	C₂²	—	8	1	no	—
577	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
578	C₂²	#16811175	1	1	yes	characteristic
579	C₂²	#16811175	1	1	yes	characteristic
580	C₄	—	2	1	no	—
581	C₄	—	2	1	no	—
582	C₄	—	106	1	no	—
583	C₄	—	106	1	no	—
584	C₄	—	106	1	no	—
585	C₄	—	106	1	no	—
586	C₄	—	106	1	no	—
587	C₄	—	106	1	no	—
588	C₄	—	106	1	no	—
589	C₄	—	106	1	no	—
590	C₄	C₄×#11719947	1	1	yes	characteristic
591	C₄	C₄×#11719947	1	1	yes	characteristic
592	C₄	#16811129	1	1	yes	characteristic
593	C₄	#16811131	1	1	yes	characteristic
594	C₂	—	2	1	no	—
595	C₂	—	2	1	no	—
596	C₂	—	4	1	no	—
597	C₂	—	4	1	no	—
598	C₂	#20277194	1	1	yes	characteristic
599	C₂	#20283323	1	1	yes	characteristic
600	C₂	#20283325	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	C₅₃×#1855	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1855	D₅₃
hypocenter	C₅₃	#21372
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1855

Complements

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

N	H ≅ G/N	type
4. #20275489	C₂	semidirect
5. #20275489	C₂	semidirect
6. #20275487	C₂	semidirect
7. #20275487	C₂	semidirect
11. #20274623	C₂	semidirect
12. #20274620	C₂	semidirect
13. #108×Dic₅₃	C₂	semidirect
14. #20274616	C₂	semidirect
363. C₅₃	#21372	semidirect

Maximal subgroups

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

subgroup	index
#20283350	2
C₂×#16810681	2
#20283133	2
#20275489	2
#20275487	2
#20274922	2
#20274920	2
#20274919	2
#20274623	2
#20274620	2
#108×Dic₅₃	2
#20274616	2
C₅₃×#1855	2
#21372	53