Subgroups — #27354080

Order 13568.

700 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284775	C₂	1	1	yes	maximal, characteristic
2	#41×#88322	C₂	1	1	yes	maximal, characteristic, complemented
3	#20283426	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273590	C₂	1	2	yes	maximal, complemented
5	#20273576	C₂	1	2	yes	maximal, complemented
6	#20273512	C₂	1	2	yes	maximal, complemented
7	#20273507	C₂	1	2	yes	maximal, complemented
8	#20273506	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273133	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×#88322	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1489	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133225	—	2	1	no	—
13	#16811173	C₂²	1	1	yes	characteristic
14	#16810947	C₄	1	2	yes	—
15	#16810918	C₄	1	1	yes	characteristic
16	#16810913	C₄	1	1	yes	characteristic
17	#16810823	C₄	1	1	yes	characteristic
18	#16810818	C₄	1	1	yes	characteristic
19	#16810768	C₄	1	2	yes	—
20	Q₈×#88322	C₂²	1	1	yes	characteristic
21	Q₈×#88322	C₂²	1	1	yes	characteristic
22	C₄×#11719924	C₂²	1	2	yes	—
23	C₄×#11719922	C₂²	1	2	yes	—
24	#16810700	—	2	1	no	—
25	#16810700	C₂²	1	2	yes	—
26	D₄×#88322	—	2	1	no	—
27	D₄×#88322	—	2	1	no	—
28	D₄×#88322	C₂²	1	1	yes	characteristic
29	D₄×#88322	C₂²	1	1	yes	characteristic
30	#16810688	—	2	1	no	—
31	#16810688	C₂²	1	2	yes	—
32	#16810676	—	2	1	no	—
33	#16810676	C₂²	1	1	yes	characteristic
34	#16810676	C₂²	1	1	yes	characteristic
35	#16810591	—	2	1	no	—
36	C₂×C₄×#88322	—	2	1	no	—
37	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
38	#16810375	C₂²	1	2	yes	—
39	#16810373	C₂²	1	2	yes	—
40	#16810352	C₂²	1	2	yes	—
41	#16810352	C₂²	1	2	yes	—
42	#16810344	C₂²	1	2	yes	—
43	C₈×#88322	C₂²	1	2	yes	—
44	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#423	C₂²	1	1	yes	characteristic
46	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
47	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
48	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
49	C₂×#11719945	—	2	1	no	—
50	C₂×#11719924	C₂×C₄	1	2	yes	—
51	C₂×#11719922	C₂×C₄	1	2	yes	—
52	#13133225	—	4	4	no	—
53	C₂×#11719912	—	2	2	no	—
54	C₂²×#88322	—	2	1	no	—
55	C₂²×#88322	—	2	1	no	—
56	C₂²×#88322	D₄	1	1	yes	characteristic
57	C₂²×#88322	D₄	1	1	yes	characteristic
58	C₂×C₈×D₅₃	—	2	2	no	—
59	#13133217	C₂×C₄	1	1	yes	characteristic
60	#13133214	C₂×C₄	1	1	yes	characteristic
61	#13133205	—	2	1	no	—
62	#13133191	—	2	1	no	—
63	#13133191	C₂×C₄	1	2	yes	—
64	C₄×D₂₁₂	—	2	1	no	—
65	C₄×D₂₁₂	C₂³	1	2	yes	—
66	#13133150	C₂×C₄	1	2	yes	—
67	#13133148	C₂×C₄	1	2	yes	—
68	C₄⋊#88322	—	2	1	no	—
69	C₄⋊#88322	—	2	1	no	—
70	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
71	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
72	C₄⋊#88322	C₂³	1	1	yes	characteristic
73	C₄⋊#88322	C₂³	1	1	yes	characteristic
74	#13133141	C₂×C₄	1	2	yes	—
75	#13133138	C₂×C₄	1	2	yes	—
76	#13133137	C₂×C₄	1	2	yes	—
77	#13133137	C₂×C₄	1	2	yes	—
78	C₂²⋊#88322	—	2	1	no	—
79	C₂²⋊#88322	—	2	1	no	—
80	C₂²⋊#88322	—	2	1	no	—
81	C₂²⋊#88322	C₂×C₄	1	2	yes	—
82	#13133129	—	2	2	no	—
83	#13133129	—	2	2	no	—
84	C₄×#88322	—	2	1	no	—
85	C₄×#88322	—	2	1	no	—
86	C₄×#88322	—	2	1	no	—
87	C₄×#88322	D₄	1	1	yes	characteristic
88	C₄×#88322	D₄	1	1	yes	characteristic
89	C₄×#88322	C₂³	1	2	yes	—
90	C₄×#88322	C₂³	1	1	yes	characteristic
91	C₄×#88322	C₂³	1	1	yes	characteristic
92	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
93	C₂×M₁₆×C₅₃	—	2	1	no	—
94	C₂×M₁₆×C₅₃	—	2	1	no	—
95	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
96	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
97	C₄×D₄×C₅₃	—	2	1	no	—
98	C₄×D₄×C₅₃	—	2	1	no	—
99	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
100	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
101	#117×C₅₃	—	2	1	no	—
102	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
103	C₂×C₄²×C₅₃	—	2	1	no	—
104	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
105	C₄⋊C₈×C₅₃	—	2	1	no	—
106	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
107	#97×C₅₃	—	2	1	no	—
108	C₄×C₈×C₅₃	C₂³	1	2	yes	—
109	#41×C₅₃	—	2	2	no	—
110	#41×C₅₃	—	2	2	no	—
111	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×D₄×C₅₃	—	2	1	no	—
113	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	—	2	1	no	—
115	C₂²×C₄×C₅₃	—	2	1	no	—
116	C₂²×C₄×C₅₃	—	2	1	no	—
117	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
120	C₂×C₂²⋊D₅₃	—	2	2	no	—
121	#11719945	—	4	2	no	—
122	#11719945	—	4	2	no	—
123	C₂×D₂₁₂	—	2	1	no	—
124	C₂×D₂₁₂	—	2	1	no	—
125	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
126	C₂×C₄×D₅₃	—	2	2	no	—
127	C₂×Dic₁₀₆	—	2	1	no	—
128	M₁₆×C₅₃	—	2	4	no	—
129	M₁₆×C₅₃	—	4	2	no	—
130	C₂×C₈×C₅₃	—	2	2	no	—
131	C₂×C₈×C₅₃	—	2	2	no	—
132	C₄⋊C₄×C₅₃	—	2	1	no	—
133	C₄⋊C₄×C₅₃	—	2	1	no	—
134	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
136	C₂²⋊C₄×C₅₃	—	2	1	no	—
137	C₂²⋊C₄×C₅₃	—	2	1	no	—
138	C₂²⋊C₄×C₅₃	—	2	1	no	—
139	C₄²×C₅₃	—	2	1	no	—
140	C₄²×C₅₃	—	2	1	no	—
141	C₄²×C₅₃	—	2	1	no	—
142	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
143	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
144	C₄²×C₅₃	#41	1	1	yes	characteristic
145	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
146	#11719924	—	2	4	no	—
147	#11719922	—	2	4	no	—
148	#11719921	—	2	2	no	—
149	#11719921	—	2	2	no	—
150	C₄⋊Dic₅₃	—	2	1	no	—
151	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
152	C₂×#88322	—	2	2	no	—
153	C₂×#88322	—	2	1	no	—
154	C₂×#88322	—	2	1	no	—
155	C₂×#88322	—	4	1	no	—
156	C₂×#88322	—	4	2	no	—
157	C₂×#88322	—	4	2	no	—
158	C₂×#88322	C₂²×C₄	1	2	yes	—
159	C₂×#88322	C₂²×C₄	1	2	yes	—
160	C₂×#88322	C₂×D₄	1	1	yes	characteristic
161	C₂×#88322	C₂×D₄	1	1	yes	characteristic
162	C₂×#88322	C₂×D₄	1	1	yes	characteristic
163	C₂×#88322	C₂×D₄	1	1	yes	characteristic
164	#11719912	—	4	4	no	—
165	C₈×D₅₃	—	4	4	no	—
166	C₂³×C₅₃	—	2	1	no	—
167	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
168	C₂²×D₅₃	—	2	2	no	—
169	Q₈×C₅₃	—	2	1	no	—
170	Q₈×C₅₃	#131	1	2	yes	complemented
171	D₄×C₅₃	—	2	4	no	—
172	D₄×C₅₃	—	2	1	no	—
173	D₄×C₅₃	#131	1	2	yes	complemented
174	C₂×C₄×C₅₃	—	2	4	no	—
175	C₂×C₄×C₅₃	—	2	1	no	—
176	C₂×C₄×C₅₃	—	2	1	no	—
177	C₂×C₄×C₅₃	—	2	1	no	—
178	C₂×C₄×C₅₃	—	2	1	no	—
179	C₂×C₄×C₅₃	—	2	1	no	—
180	C₂×C₄×C₅₃	—	2	2	no	—
181	C₂×C₄×C₅₃	—	2	1	no	—
182	C₂×C₄×C₅₃	—	2	1	no	—
183	C₂×C₄×C₅₃	—	4	1	no	—
184	C₂×C₄×C₅₃	—	4	1	no	—
185	C₂×C₄×C₅₃	—	4	2	no	—
186	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
188	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
192	C₂²⋊D₅₃	—	4	4	no	—
193	C₂×Dic₅₃	—	2	2	no	—
194	D₂₁₂	—	2	2	no	—
195	D₂₁₂	—	2	2	no	—
196	D₂₁₂	—	4	2	no	—
197	D₂₁₂	—	4	2	no	—
198	C₄×D₅₃	—	4	2	no	—
199	C₄×D₅₃	—	4	2	no	—
200	Dic₁₀₆	—	4	2	no	—
201	C₈×C₅₃	—	2	4	no	—
202	C₈×C₅₃	—	4	2	no	—
203	#88322	—	2	2	no	—
204	#88322	—	2	2	no	—
205	#88322	—	2	1	no	—
206	#88322	—	2	1	no	—
207	#88322	—	2	1	no	—
208	#88322	—	4	1	no	—
209	#88322	—	4	1	no	—
210	#88322	AGL(1,ℤ/8)	1	2	yes	—
211	#17912	—	53	1	no	maximal, Sylow
212	C₂²×C₅₃	—	2	1	no	—
213	C₂²×C₅₃	—	2	2	no	—
214	C₂²×C₅₃	—	2	1	no	—
215	C₂²×C₅₃	—	2	1	no	—
216	C₂²×C₅₃	—	4	1	no	—
217	C₂²×C₅₃	—	4	1	no	—
218	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
219	C₂×D₅₃	—	4	2	no	—
220	C₂×D₅₃	—	4	2	no	—
221	C₂×D₅₃	—	4	2	no	—
222	C₄×C₅₃	—	2	1	no	—
223	C₄×C₅₃	—	2	2	no	—
224	C₄×C₅₃	—	2	1	no	—
225	C₄×C₅₃	—	2	1	no	—
226	C₄×C₅₃	—	4	1	no	—
227	C₄×C₅₃	—	4	1	no	—
228	C₄×C₅₃	#443	1	2	yes	—
229	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
230	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
231	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
232	C₄×C₅₃	#575	1	1	yes	characteristic
233	Dic₅₃	—	4	2	no	—
234	#3106	—	53	1	no	—
235	C₈×#41	—	53	1	no	—
236	#2931	—	53	1	no	—
237	#1588	—	53	2	no	—
238	#1586	—	53	2	no	—
239	C₈×SD₁₆	—	53	2	no	—
240	C₈×D₈	—	53	2	no	—
241	#1574	—	53	1	no	—
242	#1489	D₅₃	1	1	yes	characteristic, complemented
243	#1484	—	53	1	no	—
244	C₈×M₁₆	—	53	1	no	—
245	C₂×C₅₃	—	4	1	no	—
246	C₂×C₅₃	—	4	2	no	—
247	C₂×C₅₃	#2929	1	1	yes	characteristic
248	C₂×C₅₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
249	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
250	D₅₃	—	8	2	no	—
251	C₂×#131	—	106	1	no	—
252	#532	—	53	1	no	—
253	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
254	#482	—	53	2	no	—
255	#477	—	53	1	no	—
256	#476	—	53	1	no	—
257	#460	—	53	1	no	—
258	#459	—	53	1	no	—
259	#449	—	53	2	no	—
260	C₈×Q₈	—	53	1	no	—
261	C₈×Q₈	—	53	1	no	—
262	C₄×SD₁₆	—	53	2	no	—
263	C₄×D₈	—	53	2	no	—
264	#436	—	53	2	no	—
265	#436	—	106	1	no	—
266	C₈×D₄	—	53	1	no	—
267	C₈×D₄	—	53	1	no	—
268	C₈×D₄	—	53	2	no	—
269	C₈×D₄	—	106	1	no	—
270	C₈×D₄	—	106	1	no	—
271	C₈×D₄	—	106	1	no	—
272	#431	—	53	1	no	—
273	#431	—	53	1	no	—
274	#431	—	106	1	no	—
275	#423	C₂×D₅₃	1	1	yes	characteristic
276	#408	—	106	1	no	—
277	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
278	C₂×C₄×C₈	—	53	1	no	—
279	C₂×C₄×C₈	—	106	1	no	—
280	#335	—	53	2	no	—
281	#334	—	53	2	no	—
282	Q₈⋊C₈	C₂×D₅₃	1	2	yes	—
283	D₄⋊C₈	—	53	2	no	—
284	D₄⋊C₈	—	53	2	no	—
285	D₄⋊C₈	C₂×D₅₃	1	2	yes	—
286	#322	—	53	2	no	—
287	C₈²	—	53	2	no	—
288	C₅₃	#17912	1	1	yes	characteristic, complemented, Sylow
289	C₂×#41	—	106	1	no	—
290	C₂×#41	C₄×D₅₃	1	1	yes	characteristic
291	C₂×SD₁₆	—	53	2	no	—
292	C₂×D₈	—	53	2	no	—
293	#131	—	212	4	no	—
294	C₂×M₁₆	—	2	1	no	—
295	C₂×M₁₆	—	2	1	no	—
296	C₂×M₁₆	—	106	2	no	—
297	C₂²×C₈	—	53	1	no	—
298	C₂²×C₈	—	53	1	no	—
299	C₂²×C₈	—	106	2	no	—
300	C₂²×C₈	—	106	1	no	—
301	C₂²×C₈	—	106	1	no	—
302	C₄⋊Q₈	—	53	1	no	—
303	#127	—	53	1	no	—
304	#124	—	106	1	no	—
305	#121	—	53	2	no	—
306	#121	—	106	1	no	—
307	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
308	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
309	C₄×D₄	—	2	1	no	—
310	C₄×D₄	—	2	1	no	—
311	C₄×D₄	—	53	2	no	—
312	C₄×D₄	—	106	1	no	—
313	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
314	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
315	#117	—	2	1	no	—
316	#117	C₄×D₅₃	1	1	yes	characteristic
317	C₂×C₄²	—	2	1	no	—
318	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
319	#107	—	53	2	no	—
320	#106	—	53	2	no	—
321	C₄⋊C₈	—	2	1	no	—
322	C₄⋊C₈	—	53	1	no	—
323	C₄⋊C₈	—	53	1	no	—
324	C₄⋊C₈	—	53	1	no	—
325	C₄⋊C₈	—	53	1	no	—
326	C₄⋊C₈	—	106	1	no	—
327	C₄⋊C₈	—	106	1	no	—
328	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
329	#103	—	53	2	no	—
330	#102	—	53	2	no	—
331	#102	—	53	2	no	—
332	#102	—	53	2	no	—
333	C₂²⋊C₈	—	53	2	no	—
334	C₂²⋊C₈	—	106	2	no	—
335	C₂²⋊C₈	—	106	2	no	—
336	C₂²⋊C₈	—	106	1	no	—
337	C₂²⋊C₈	—	106	1	no	—
338	C₂²⋊C₈	—	106	1	no	—
339	#97	—	2	1	no	—
340	C₄×C₈	—	53	1	no	—
341	C₄×C₈	—	53	1	no	—
342	C₄×C₈	—	53	1	no	—
343	C₄×C₈	—	53	2	no	—
344	C₄×C₈	—	53	1	no	—
345	C₄×C₈	—	106	1	no	—
346	C₄×C₈	—	106	1	no	—
347	C₄×C₈	—	106	1	no	—
348	C₄×C₈	C₂²×D₅₃	1	2	yes	—
349	#41	—	2	2	no	—
350	#41	—	2	2	no	—
351	#41	—	212	2	no	—
352	#41	—	212	2	no	—
353	C₂×Q₈	—	106	1	no	—
354	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	53	2	no	—
357	C₂×D₄	—	106	2	no	—
358	C₂×D₄	—	106	1	no	—
359	C₂×D₄	—	106	1	no	—
360	C₂×D₄	C₂×C₄×D₅₃	1	1	yes	characteristic
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	106	2	no	—
365	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
366	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
367	C₂²×C₄	#11719947	1	1	yes	characteristic
368	SD₁₆	—	106	4	no	—
369	D₈	—	106	4	no	—
370	M₁₆	—	2	4	no	—
371	M₁₆	—	4	2	no	—
372	M₁₆	—	212	4	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	53	1	no	—
376	C₂×C₈	—	53	2	no	—
377	C₂×C₈	—	53	2	no	—
378	C₂×C₈	—	53	1	no	—
379	C₂×C₈	—	53	1	no	—
380	C₂×C₈	—	53	1	no	—
381	C₂×C₈	—	106	2	no	—
382	C₂×C₈	—	106	1	no	—
383	C₂×C₈	—	106	1	no	—
384	C₂×C₈	—	212	4	no	—
385	C₂×C₈	—	212	2	no	—
386	C₂×C₈	—	212	1	no	—
387	C₂×C₈	—	212	2	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	53	2	no	—
391	C₄⋊C₄	—	106	1	no	—
392	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
393	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
394	C₂²⋊C₄	—	2	1	no	—
395	C₂²⋊C₄	—	2	1	no	—
396	C₂²⋊C₄	—	2	1	no	—
397	C₂²⋊C₄	—	106	2	no	—
398	C₂²⋊C₄	—	106	2	no	—
399	C₄²	—	2	1	no	—
400	C₄²	—	2	1	no	—
401	C₄²	—	2	1	no	—
402	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
403	C₄²	D₄×D₅₃	1	1	yes	characteristic
404	C₄²	#11719947	1	1	yes	characteristic
405	C₄²	C₂³×D₅₃	1	1	yes	characteristic
406	C₂³	—	2	1	no	—
407	C₂³	—	106	2	no	—
408	C₂³	#13133177	1	1	yes	characteristic
409	Q₈	—	2	1	no	—
410	Q₈	—	212	2	no	—
411	Q₈	#13133225	1	2	yes	complemented
412	D₄	—	2	4	no	—
413	D₄	—	2	1	no	—
414	D₄	—	106	2	no	—
415	D₄	—	106	2	no	—
416	D₄	—	212	2	no	—
417	D₄	—	212	4	no	—
418	D₄	—	212	2	no	—
419	D₄	#13133225	1	2	yes	complemented
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	106	2	no	—
433	C₂×C₄	—	212	2	no	—
434	C₂×C₄	—	212	2	no	—
435	C₂×C₄	#13133177	1	1	yes	characteristic
436	C₂×C₄	#13133177	1	1	yes	characteristic
437	C₂×C₄	#13133177	1	1	yes	characteristic
438	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
439	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
440	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
441	C₈	—	2	4	no	—
442	C₈	—	4	2	no	—
443	C₈	—	53	2	no	—
444	C₈	—	106	2	no	—
445	C₈	—	106	2	no	—
446	C₈	—	106	1	no	—
447	C₈	—	106	1	no	—
448	C₈	—	106	1	no	—
449	C₈	—	212	1	no	—
450	C₈	—	212	1	no	—
451	C₂²	—	2	1	no	—
452	C₂²	—	2	2	no	—
453	C₂²	—	2	1	no	—
454	C₂²	—	2	1	no	—
455	C₂²	—	4	1	no	—
456	C₂²	—	4	1	no	—
457	C₂²	—	212	2	no	—
458	C₂²	—	212	2	no	—
459	C₂²	—	212	2	no	—
460	C₂²	C₂×#13133177	1	1	yes	characteristic
461	C₄	—	2	1	no	—
462	C₄	—	2	2	no	—
463	C₄	—	2	1	no	—
464	C₄	—	2	1	no	—
465	C₄	—	4	1	no	—
466	C₄	—	4	1	no	—
467	C₄	—	212	2	no	—
468	C₄	#16810737	1	2	yes	—
469	C₄	C₂×#13133225	1	1	yes	characteristic
470	C₄	C₂×#13133225	1	1	yes	characteristic
471	C₄	AGL(1,ℤ/8)×D₅₃	1	1	yes	characteristic
472	C₄	#16811422	1	1	yes	characteristic
473	C₂	—	4	1	no	—
474	C₂	—	4	2	no	—
475	C₂	—	424	2	no	—
476	C₂	#20283405	1	1	yes	characteristic
477	C₂	#20283528	1	1	yes	characteristic
478	C₂	C₂×#16810737	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1489	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#20284775	C₂
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1489	D₅₃
hypocenter	C₅₃	#17912
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1489

Complements

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

N	H ≅ G/N	type
2. #41×#88322	C₂	semidirect
3. #20283426	C₂	semidirect
4. #20273590	C₂	semidirect
5. #20273576	C₂	semidirect
6. #20273512	C₂	semidirect
7. #20273507	C₂	semidirect
8. #20273506	C₂	semidirect
9. #20273133	C₂	semidirect
10. M₁₆×#88322	C₂	semidirect
11. C₅₃×#1489	C₂	semidirect
170. Q₈×C₅₃	#131	semidirect
173. D₄×C₅₃	#131	semidirect
242. #1489	D₅₃	semidirect
288. C₅₃	#17912	semidirect
411. Q₈	#13133225	semidirect
419. D₄	#13133225	semidirect

Maximal subgroups

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

subgroup	index
#20284775	2
#41×#88322	2
#20283426	2
#20273590	2
#20273576	2
#20273512	2
#20273507	2
#20273506	2
#20273133	2
M₁₆×#88322	2
C₅₃×#1489	2
#17912	53