Subgroups — #27406161

Order 13568.

720 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284505	C₂	1	1	yes	maximal, characteristic
2	#20283658	C₂	1	1	yes	maximal, characteristic, complemented
3	#20283280	C₂	1	1	yes	maximal, characteristic
4	#20275692	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275491	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275463	C₂	1	2	yes	maximal, complemented
7	#20275315	C₂	1	2	yes	maximal, complemented
8	#20274993	C₂	1	2	yes	maximal
9	#20274552	C₂	1	2	yes	maximal
10	C₄×#13133145	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133223	C₂²	1	2	yes	—
13	#16811180	C₂²	1	1	yes	characteristic
14	C₄×#11719945	C₂²	1	1	yes	characteristic
15	#16810931	C₄	1	2	yes	complemented
16	#16810924	C₄	1	2	yes	complemented
17	#16810919	C₄	1	1	yes	characteristic, complemented
18	#16810914	C₄	1	1	yes	characteristic, complemented
19	#16810909	C₄	1	1	yes	characteristic, complemented
20	#16810904	C₄	1	1	yes	characteristic, complemented
21	#16810752	—	2	2	no	—
22	#16810750	—	2	1	no	—
23	#16810747	—	2	1	no	—
24	C₂×#13133153	C₂²	1	2	yes	—
25	#16810661	C₂²	1	2	yes	—
26	#16810649	C₂²	1	2	yes	—
27	#16810649	C₂²	1	2	yes	—
28	#16810648	—	2	2	no	—
29	#16810647	C₂²	1	1	yes	characteristic
30	#16810645	—	2	1	no	—
31	C₄⋊C₈×D₅₃	—	2	1	no	—
32	C₂×#13133145	C₂²	1	1	yes	characteristic
33	C₂×#13133145	C₂²	1	1	yes	characteristic
34	#16810620	C₂²	1	2	yes	—
35	#16810578	C₂²	1	2	yes	—
36	C₄×#11719917	C₂²	1	1	yes	characteristic
37	#16810388	C₂²	1	2	yes	—
38	#16810383	C₂²	1	1	yes	characteristic
39	#16810383	C₂²	1	2	yes	—
40	#16810383	C₂²	1	2	yes	—
41	#16810383	C₂²	1	1	yes	characteristic
42	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
43	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
45	C₄³×C₅₃	C₂²	1	1	yes	characteristic
46	C₅₃×#336	C₂²	1	2	yes	—
47	C₂×#11719945	C₂³	1	1	yes	characteristic
48	#13133223	—	2	4	no	—
49	#13133223	—	4	2	no	—
50	C₂×#11719917	C₂³	1	1	yes	characteristic
51	C₂×#11719917	C₂³	1	2	yes	—
52	C₂×#11719917	C₂³	1	1	yes	characteristic
53	C₂×#11719912	—	2	2	no	—
54	C₂×C₈×D₅₃	—	2	2	no	—
55	#13133217	C₂×C₄	1	1	yes	characteristic
56	#13133217	C₂×C₄	1	1	yes	characteristic
57	#13133209	C₂×C₄	1	2	yes	—
58	#13133196	—	2	1	no	—
59	#13133196	—	2	1	no	—
60	C₄×Dic₁₀₆	—	2	1	no	—
61	C₄×Dic₁₀₆	—	2	1	no	—
62	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
64	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
65	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
66	C₄×C₂²⋊D₅₃	—	2	1	no	—
67	C₄×C₂²⋊D₅₃	—	2	1	no	—
68	#13133176	C₂³	1	2	yes	—
69	#13133176	C₂³	1	1	yes	characteristic
70	#13133174	—	2	1	no	—
71	C₄²×D₅₃	—	2	1	no	—
72	#13133153	—	4	2	no	—
73	#13133151	C₂×C₄	1	2	yes	—
74	#13133151	C₂×C₄	1	2	yes	—
75	#13133149	C₂×C₄	1	2	yes	—
76	#13133149	C₂×C₄	1	2	yes	—
77	C₄⋊#88322	—	2	1	no	—
78	C₄⋊#88322	—	2	1	no	—
79	#13133146	—	2	2	no	—
80	#13133146	—	2	2	no	—
81	#13133146	—	2	2	no	—
82	#13133146	—	2	2	no	—
83	#13133145	—	2	2	no	—
84	#13133145	Q₈	1	4	yes	complemented
85	#13133142	C₂×C₄	1	2	yes	—
86	#13133142	C₂×C₄	1	2	yes	—
87	#13133139	C₂×C₄	1	2	yes	—
88	#13133139	C₂×C₄	1	2	yes	—
89	#13133128	—	2	1	no	—
90	#13133127	—	2	2	no	—
91	C₄×#88322	—	2	1	no	—
92	C₈×Dic₅₃	—	2	2	no	—
93	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
94	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
95	C₂×C₄²×C₅₃	—	2	1	no	—
96	C₂×C₄²×C₅₃	—	2	1	no	—
97	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
100	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
101	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
102	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
103	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
104	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
105	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
106	C₄×C₈×C₅₃	—	4	2	no	—
107	C₂²×C₄×C₅₃	—	2	1	no	—
108	C₂²×C₄×C₅₃	—	2	1	no	—
109	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
111	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
112	C₂×C₂²⋊D₅₃	—	2	1	no	—
113	#11719945	—	2	1	no	—
114	#11719945	—	2	1	no	—
115	#11719945	C₂×Q₈	1	2	yes	—
116	#11719945	#41	1	2	yes	—
117	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×D₅₃	—	2	1	no	—
119	C₂×C₄×D₅₃	—	2	1	no	—
120	C₂×C₄×D₅₃	—	2	1	no	—
121	C₂×Dic₁₀₆	—	2	1	no	—
122	C₂×Dic₁₀₆	—	2	1	no	—
123	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₅₃	—	2	2	no	—
125	C₂×C₈×C₅₃	—	2	2	no	—
126	C₂×C₈×C₅₃	—	2	2	no	—
127	C₂×C₈×C₅₃	—	2	2	no	—
128	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
129	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
130	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
131	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
132	C₄²×C₅₃	—	2	4	no	—
133	C₄²×C₅₃	—	2	1	no	—
134	C₄²×C₅₃	—	2	1	no	—
135	C₄²×C₅₃	—	2	1	no	—
136	C₄²×C₅₃	—	4	2	no	—
137	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
138	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
141	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
142	C₂²⋊Dic₅₃	—	2	1	no	—
143	C₂²⋊Dic₅₃	—	2	2	no	—
144	#11719921	—	2	1	no	—
145	#11719921	—	2	1	no	—
146	C₄⋊Dic₅₃	—	2	1	no	—
147	C₄⋊Dic₅₃	—	2	1	no	—
148	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
149	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
150	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
152	#11719919	—	2	1	no	—
153	#11719919	—	2	1	no	—
154	#11719919	—	2	2	no	—
155	#11719919	—	2	2	no	—
156	C₄×Dic₅₃	—	2	1	no	—
157	C₄×Dic₅₃	—	2	1	no	—
158	C₄×Dic₅₃	—	2	2	no	—
159	C₄×Dic₅₃	—	2	1	no	—
160	#11719917	—	2	1	no	—
161	#11719917	—	2	1	no	—
162	#11719917	—	2	2	no	—
163	#11719917	—	2	1	no	—
164	#11719917	—	2	2	no	—
165	#11719917	C₂×Q₈	1	2	yes	—
166	C₂×#88322	—	2	1	no	—
167	C₂×#88322	—	2	1	no	—
168	C₂×#88322	—	2	2	no	—
169	#11719912	—	4	4	no	—
170	C₈×D₅₃	—	4	4	no	—
171	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
172	C₂²×D₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	2	2	no	—
174	C₂×C₄×C₅₃	—	2	2	no	—
175	C₂×C₄×C₅₃	—	2	2	no	—
176	C₂×C₄×C₅₃	—	2	2	no	—
177	C₂×C₄×C₅₃	—	2	2	no	—
178	C₂×C₄×C₅₃	—	4	1	no	—
179	C₂×C₄×C₅₃	—	4	1	no	—
180	C₂×C₄×C₅₃	—	4	1	no	—
181	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
182	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
186	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
188	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
189	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
190	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
191	C₂²⋊D₅₃	—	2	2	no	—
192	C₂²⋊D₅₃	—	2	2	no	—
193	C₂×Dic₅₃	—	2	1	no	—
194	C₂×Dic₅₃	—	2	2	no	—
195	C₂×Dic₅₃	—	2	2	no	—
196	C₂×Dic₅₃	—	2	1	no	—
197	C₂×Dic₅₃	—	2	1	no	—
198	D₂₁₂	—	2	1	no	—
199	D₂₁₂	C₄×Q₈	1	2	yes	—
200	C₄×D₅₃	—	2	2	no	—
201	C₄×D₅₃	—	2	2	no	—
202	C₄×D₅₃	—	4	2	no	—
203	C₄×D₅₃	—	8	1	no	—
204	Dic₁₀₆	—	2	1	no	—
205	Dic₁₀₆	—	4	2	no	—
206	Dic₁₀₆	—	8	1	no	—
207	Dic₁₀₆	C₄×Q₈	1	2	yes	—
208	C₈×C₅₃	—	2	2	no	—
209	C₈×C₅₃	—	2	2	no	—
210	C₈×C₅₃	—	2	2	no	—
211	C₈×C₅₃	—	2	2	no	—
212	#88322	—	2	2	no	—
213	#88322	—	4	2	no	—
214	#88322	—	4	1	no	—
215	#22982	—	53	1	no	maximal, Sylow
216	C₂²×C₅₃	—	2	1	no	—
217	C₂²×C₅₃	—	2	1	no	—
218	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
219	C₂²×C₅₃	#552	1	1	yes	characteristic
220	C₂²×C₅₃	#554	1	1	yes	characteristic
221	C₂×D₅₃	—	2	2	no	—
222	C₂×D₅₃	—	4	1	no	—
223	C₂×D₅₃	—	4	1	no	—
224	C₄×C₅₃	—	2	4	no	—
225	C₄×C₅₃	—	2	1	no	—
226	C₄×C₅₃	—	2	1	no	—
227	C₄×C₅₃	—	2	1	no	—
228	C₄×C₅₃	—	4	2	no	—
229	C₄×C₅₃	#443	1	2	yes	—
230	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
231	C₄×C₅₃	#520	1	1	yes	characteristic
232	C₄×C₅₃	#546	1	1	yes	characteristic
233	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
234	Dic₅₃	—	2	2	no	—
235	Dic₅₃	—	4	2	no	—
236	Dic₅₃	—	4	1	no	—
237	Dic₅₃	—	4	2	no	—
238	Dic₅₃	—	4	1	no	—
239	#3076	—	53	1	no	—
240	#2962	—	53	1	no	—
241	#2906	—	53	1	no	—
242	#1994	—	53	1	no	—
243	#1950	—	53	1	no	—
244	#1945	—	53	2	no	—
245	#1922	—	53	2	no	—
246	#1866	—	53	2	no	—
247	C₄×C₄⋊C₈	D₅₃	1	1	yes	characteristic, complemented
248	#1764	—	53	2	no	—
249	C₄×#104	—	53	1	no	—
250	C₂×C₅₃	—	2	1	no	—
251	C₂×C₅₃	—	2	1	no	—
252	C₂×C₅₃	#2360	1	1	yes	characteristic
253	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
254	C₂×C₅₃	#2956	1	1	yes	characteristic
255	D₅₃	—	4	2	no	—
256	C₂×#131	—	53	2	no	—
257	#533	—	53	1	no	—
258	C₄×#41	—	53	1	no	—
259	#479	—	53	2	no	—
260	#478	—	53	2	no	—
261	#477	—	53	1	no	—
262	#476	—	53	1	no	—
263	#475	—	53	1	no	—
264	#474	—	53	1	no	—
265	#446	—	106	2	no	—
266	#446	—	106	1	no	—
267	C₈×Q₈	—	106	1	no	—
268	C₂×#108	—	53	2	no	—
269	#427	—	53	2	no	—
270	#424	—	53	2	no	—
271	#424	—	53	2	no	—
272	#424	—	106	2	no	—
273	#423	—	53	1	no	—
274	#423	—	106	1	no	—
275	C₂×C₄⋊C₈	—	106	1	no	—
276	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
277	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
278	C₂×#104	—	53	1	no	—
279	C₂×#104	—	53	1	no	—
280	#416	—	53	2	no	—
281	#405	—	53	2	no	—
282	C₄×M₁₆	—	53	1	no	—
283	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
284	C₄³	C₂×D₅₃	1	1	yes	characteristic
285	#341	—	53	2	no	—
286	#339	—	53	2	no	—
287	#339	—	53	1	no	—
288	#339	—	53	1	no	—
289	#339	—	53	2	no	—
290	#336	C₂×D₅₃	1	2	yes	—
291	C₅₃	#22982	1	1	yes	characteristic, complemented, Sylow
292	C₂×#41	—	53	1	no	—
293	#131	—	106	4	no	—
294	#131	—	212	2	no	—
295	C₂×M₁₆	—	53	1	no	—
296	C₂×M₁₆	—	53	2	no	—
297	C₂×M₁₆	—	53	1	no	—
298	C₂×M₁₆	—	106	2	no	—
299	C₂²×C₈	—	106	2	no	—
300	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
301	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
302	C₄⋊Q₈	—	53	1	no	—
303	C₄⋊Q₈	—	53	1	no	—
304	#125	—	53	2	no	—
305	C₂²⋊Q₈	—	106	1	no	—
306	C₂²⋊Q₈	—	106	1	no	—
307	C₄×Q₈	—	53	1	no	—
308	C₄×Q₈	—	53	1	no	—
309	C₄×Q₈	—	106	1	no	—
310	C₄×Q₈	—	106	1	no	—
311	C₄×D₄	—	53	1	no	—
312	C₄×D₄	—	53	1	no	—
313	C₄×D₄	—	106	1	no	—
314	C₄×D₄	—	106	1	no	—
315	#117	—	53	1	no	—
316	#117	—	53	2	no	—
317	#117	—	106	1	no	—
318	C₂×C₄²	—	2	1	no	—
319	C₂×C₄²	—	2	1	no	—
320	C₂×C₄²	—	106	1	no	—
321	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
322	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
323	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
324	#108	—	212	2	no	—
325	#107	—	53	2	no	—
326	#107	—	53	2	no	—
327	#106	—	53	2	no	—
328	#106	—	53	2	no	—
329	C₄⋊C₈	—	106	2	no	—
330	C₄⋊C₈	—	106	2	no	—
331	C₄⋊C₈	—	106	2	no	—
332	C₄⋊C₈	—	106	2	no	—
333	C₄⋊C₈	—	106	1	no	—
334	C₄⋊C₈	—	106	1	no	—
335	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
336	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
337	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
338	C₄⋊C₈	D₂₁₂	1	1	yes	characteristic
339	C₄⋊C₈	D₂₁₂	1	1	yes	characteristic
340	C₄⋊C₈	D₂₁₂	1	2	yes	—
341	#104	—	53	4	no	—
342	#104	—	106	2	no	—
343	#103	—	53	2	no	—
344	#103	—	53	2	no	—
345	#102	—	53	2	no	—
346	#102	—	53	2	no	—
347	#97	—	106	2	no	—
348	#97	—	106	1	no	—
349	C₄×C₈	—	4	2	no	—
350	C₄×C₈	—	106	2	no	—
351	C₄×C₈	—	106	1	no	—
352	#41	—	53	2	no	—
353	#41	—	53	2	no	—
354	#41	—	106	1	no	—
355	#41	—	106	1	no	—
356	C₂×Q₈	—	53	1	no	—
357	C₂×Q₈	—	106	1	no	—
358	C₂×Q₈	—	106	1	no	—
359	C₂×D₄	—	53	1	no	—
360	C₂×D₄	—	106	1	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	106	1	no	—
364	C₂²×C₄	—	106	1	no	—
365	C₂²×C₄	—	106	1	no	—
366	C₂²×C₄	#11719945	1	1	yes	characteristic
367	C₂²×C₄	#11719945	1	1	yes	characteristic
368	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
369	M₁₆	—	53	2	no	—
370	M₁₆	—	106	1	no	—
371	M₁₆	—	106	1	no	—
372	M₁₆	—	106	2	no	—
373	M₁₆	—	106	1	no	—
374	M₁₆	—	106	2	no	—
375	M₁₆	—	212	4	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	106	1	no	—
381	C₂×C₈	—	106	1	no	—
382	C₂×C₈	—	106	2	no	—
383	C₂×C₈	—	212	4	no	—
384	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
385	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
386	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
387	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
388	C₄⋊C₄	—	53	2	no	—
389	C₄⋊C₄	—	53	2	no	—
390	C₄⋊C₄	—	53	1	no	—
391	C₄⋊C₄	—	53	1	no	—
392	C₄⋊C₄	—	106	1	no	—
393	C₄⋊C₄	—	106	1	no	—
394	C₄⋊C₄	—	106	2	no	—
395	C₄⋊C₄	—	106	2	no	—
396	C₄⋊C₄	—	106	1	no	—
397	C₄⋊C₄	—	106	1	no	—
398	C₂²⋊C₄	—	106	1	no	—
399	C₂²⋊C₄	—	106	2	no	—
400	C₂²⋊C₄	—	106	1	no	—
401	C₂²⋊C₄	—	106	1	no	—
402	C₄²	—	2	4	no	—
403	C₄²	—	2	1	no	—
404	C₄²	—	2	1	no	—
405	C₄²	—	2	1	no	—
406	C₄²	—	4	2	no	—
407	C₄²	—	106	1	no	—
408	C₄²	—	106	1	no	—
409	C₄²	—	106	2	no	—
410	C₄²	—	106	1	no	—
411	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
412	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
413	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
414	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
415	C₄²	Q₈×D₅₃	1	2	yes	—
416	C₂³	—	106	1	no	—
417	C₂³	C₂×#11719945	1	1	yes	characteristic
418	Q₈	—	53	2	no	—
419	Q₈	—	106	1	no	—
420	Q₈	—	424	1	no	—
421	Q₈	—	212	2	no	—
422	D₄	—	53	2	no	—
423	D₄	—	106	2	no	—
424	D₄	—	106	2	no	—
425	D₄	—	106	1	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	106	1	no	—
435	C₂×C₄	—	106	2	no	—
436	C₂×C₄	—	106	2	no	—
437	C₂×C₄	—	106	1	no	—
438	C₂×C₄	—	106	1	no	—
439	C₂×C₄	—	106	2	no	—
440	C₂×C₄	—	106	2	no	—
441	C₂×C₄	—	424	1	no	—
442	C₂×C₄	—	212	2	no	—
443	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
444	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
445	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
446	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
447	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
448	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
449	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
450	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
451	C₂×C₄	#13133262	1	1	yes	characteristic
452	C₂×C₄	#13133264	1	1	yes	characteristic
453	C₈	—	2	2	no	—
454	C₈	—	2	2	no	—
455	C₈	—	2	2	no	—
456	C₈	—	2	2	no	—
457	C₈	—	106	2	no	—
458	C₈	—	212	2	no	—
459	C₈	—	212	1	no	—
460	C₂²	—	2	1	no	—
461	C₂²	—	2	1	no	—
462	C₂²	—	106	2	no	—
463	C₂²	—	212	1	no	—
464	C₂²	—	212	1	no	—
465	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
466	C₂²	#16811312	1	1	yes	characteristic
467	C₂²	#16811328	1	1	yes	characteristic
468	C₄	—	2	4	no	—
469	C₄	—	2	1	no	—
470	C₄	—	2	1	no	—
471	C₄	—	2	1	no	—
472	C₄	—	4	2	no	—
473	C₄	—	106	2	no	—
474	C₄	—	212	1	no	—
475	C₄	—	212	2	no	—
476	C₄	—	212	1	no	—
477	C₄	—	212	2	no	—
478	C₄	#16810741	1	2	yes	—
479	C₄	C₄×Q₈×D₅₃	1	1	yes	characteristic
480	C₄	#16811102	1	1	yes	characteristic
481	C₄	#16811268	1	1	yes	characteristic
482	C₄	Q₈×D₂₁₂	1	1	yes	characteristic
483	C₂	—	2	1	no	—
484	C₂	—	2	1	no	—
485	C₂	—	212	2	no	—
486	C₂	#20277722	1	1	yes	characteristic
487	C₂	C₂×#16810741	1	1	yes	characteristic
488	C₂	#20283626	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×C₄⋊C₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#11719945	C₂³
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#22982
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #20283658	C₂	semidirect
4. #20275692	C₂	semidirect
5. #20275491	C₂	semidirect
6. #20275463	C₂	semidirect
7. #20275315	C₂	semidirect
11. C₄×C₄⋊C₈×C₅₃	C₂	semidirect
15. #16810931	C₄	semidirect
16. #16810924	C₄	semidirect
17. #16810919	C₄	semidirect
18. #16810914	C₄	semidirect
19. #16810909	C₄	semidirect
20. #16810904	C₄	semidirect
84. #13133145	Q₈	semidirect
247. C₄×C₄⋊C₈	D₅₃	semidirect
291. C₅₃	#22982	semidirect

Maximal subgroups

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

subgroup	index
#20284505	2
#20283658	2
#20283280	2
#20275692	2
#20275491	2
#20275463	2
#20275315	2
#20274993	2
#20274552	2
C₄×#13133145	2
C₄×C₄⋊C₈×C₅₃	2
#22982	53