Subgroups — #27355015

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284746	C₂	1	1	yes	maximal, characteristic
2	#20283461	C₂	1	1	yes	maximal, characteristic
3	#20273605	C₂	1	2	yes	maximal, complemented
4	#20273558	C₂	1	2	yes	maximal, complemented
5	#20273549	C₂	1	2	yes	maximal, complemented
6	C₈×#11719924	C₂	1	2	yes	maximal, complemented
7	#20273487	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273165	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×#88322	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133225	—	2	1	no	—
13	#16811173	C₂²	1	1	yes	characteristic
14	#16810941	C₄	1	2	yes	—
15	#16810908	C₄	1	1	yes	characteristic
16	#16810908	C₄	1	1	yes	characteristic
17	#16810823	C₄	1	1	yes	characteristic
18	#16810823	C₄	1	1	yes	characteristic
19	#16810762	C₄	1	2	yes	—
20	C₄×#11719924	C₂²	1	2	yes	—
21	C₄×#11719924	C₂²	1	2	yes	—
22	#16810702	C₂²	1	2	yes	—
23	#16810696	—	2	2	no	—
24	C₈×D₂₁₂	C₂²	1	2	yes	—
25	#16810676	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
27	#16810591	—	2	1	no	—
28	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
29	#16810373	C₂²	1	2	yes	—
30	#16810373	C₂²	1	2	yes	—
31	Q₈⋊#88322	C₂²	1	1	yes	characteristic
32	Q₈⋊#88322	C₂²	1	1	yes	characteristic
33	Q₈⋊#88322	C₂²	1	1	yes	characteristic
34	Q₈⋊#88322	C₂²	1	1	yes	characteristic
35	#16810350	C₂²	1	2	yes	—
36	#16810350	C₂²	1	2	yes	—
37	#16810344	C₂²	1	2	yes	—
38	C₈×#88322	C₂²	1	2	yes	—
39	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#446	—	2	2	no	—
41	C₅₃×#446	C₂²	1	2	yes	—
42	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
43	C₅₃×#423	—	2	1	no	—
44	C₅₃×#423	C₂²	1	1	yes	characteristic
45	C₄×M₁₆×C₅₃	—	2	1	no	—
46	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
47	C₂×#11719945	—	2	1	no	—
48	C₂×#11719924	C₂×C₄	1	2	yes	—
49	C₂×#11719924	C₂×C₄	1	2	yes	—
50	#13133225	—	4	4	no	—
51	C₂×#11719912	—	2	2	no	—
52	C₂²×#88322	—	2	1	no	—
53	C₂²×#88322	—	2	1	no	—
54	C₂×C₈×D₅₃	—	2	2	no	—
55	#13133217	C₂×C₄	1	1	yes	characteristic
56	#13133214	C₂×C₄	1	1	yes	characteristic
57	#13133205	—	2	1	no	—
58	#13133191	—	2	1	no	—
59	#13133191	C₂×C₄	1	2	yes	—
60	C₄×D₂₁₂	—	2	1	no	—
61	C₄×D₂₁₂	C₂³	1	2	yes	—
62	#13133148	C₂×C₄	1	2	yes	—
63	#13133148	C₂×C₄	1	2	yes	—
64	C₄⋊#88322	—	2	1	no	—
65	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
66	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
67	C₄⋊#88322	C₂³	1	1	yes	characteristic
68	C₄⋊#88322	C₂³	1	1	yes	characteristic
69	#13133141	C₂×C₄	1	2	yes	—
70	#13133141	C₂×C₄	1	2	yes	—
71	#13133137	C₂×C₄	1	2	yes	—
72	#13133137	C₂×C₄	1	2	yes	—
73	C₂²⋊#88322	C₂×C₄	1	2	yes	—
74	#13133129	—	2	2	no	—
75	#13133129	—	2	2	no	—
76	C₄×#88322	—	2	1	no	—
77	C₄×#88322	C₂³	1	1	yes	characteristic
78	C₄×#88322	C₂³	1	1	yes	characteristic
79	C₄×#88322	C₂³	1	2	yes	—
80	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
81	C₂×M₁₆×C₅₃	—	2	1	no	—
82	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
83	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
84	C₄×Q₈×C₅₃	—	2	1	no	—
85	C₄×Q₈×C₅₃	—	2	1	no	—
86	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
91	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₅₃	—	2	1	no	—
93	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₅₃	—	2	2	no	—
95	C₄⋊C₈×C₅₃	—	2	2	no	—
96	C₄⋊C₈×C₅₃	—	2	1	no	—
97	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
98	#97×C₅₃	—	2	2	no	—
99	#97×C₅₃	D₄	1	2	yes	—
100	C₄×C₈×C₅₃	—	2	2	no	—
101	C₄×C₈×C₅₃	C₂³	1	2	yes	—
102	C₂×Q₈×C₅₃	—	2	1	no	—
103	C₂×Q₈×C₅₃	—	2	2	no	—
104	C₂×Q₈×C₅₃	—	4	1	no	—
105	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	—	2	1	no	—
108	C₂²×C₄×C₅₃	—	2	1	no	—
109	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
112	C₂×C₂²⋊D₅₃	—	2	2	no	—
113	#11719945	—	4	2	no	—
114	#11719945	—	4	2	no	—
115	C₂×D₂₁₂	—	2	1	no	—
116	C₂×D₂₁₂	—	2	1	no	—
117	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
118	C₂×C₄×D₅₃	—	2	2	no	—
119	C₂×Dic₁₀₆	—	2	1	no	—
120	M₁₆×C₅₃	—	2	2	no	—
121	M₁₆×C₅₃	—	4	2	no	—
122	M₁₆×C₅₃	SD₁₆	1	4	yes	complemented
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
126	C₄⋊C₄×C₅₃	—	2	1	no	—
127	C₄⋊C₄×C₅₃	—	2	1	no	—
128	C₄⋊C₄×C₅₃	—	2	1	no	—
129	C₄⋊C₄×C₅₃	—	2	1	no	—
130	C₄⋊C₄×C₅₃	—	2	1	no	—
131	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
132	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₅₃	—	2	1	no	—
134	C₄²×C₅₃	—	2	1	no	—
135	C₄²×C₅₃	—	2	1	no	—
136	C₄²×C₅₃	—	2	1	no	—
137	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
138	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
139	C₄²×C₅₃	#41	1	1	yes	characteristic
140	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
141	#11719924	—	2	4	no	—
142	#11719924	—	4	2	no	—
143	#11719921	—	2	2	no	—
144	#11719921	—	2	2	no	—
145	C₄⋊Dic₅₃	—	2	1	no	—
146	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
147	C₂×#88322	—	2	1	no	—
148	C₂×#88322	—	2	1	no	—
149	C₂×#88322	—	2	1	no	—
150	C₂×#88322	—	2	1	no	—
151	C₂×#88322	—	4	2	no	—
152	C₂×#88322	—	4	2	no	—
153	C₂×#88322	C₂²×C₄	1	2	yes	—
154	C₂×#88322	C₂²×C₄	1	2	yes	—
155	#11719912	—	4	4	no	—
156	C₈×D₅₃	—	4	4	no	—
157	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
158	C₂²×D₅₃	—	2	2	no	—
159	Q₈×C₅₃	—	2	1	no	—
160	Q₈×C₅₃	—	2	1	no	—
161	Q₈×C₅₃	—	2	1	no	—
162	Q₈×C₅₃	—	4	2	no	—
163	Q₈×C₅₃	#131	1	2	yes	complemented
164	C₂×C₄×C₅₃	—	2	1	no	—
165	C₂×C₄×C₅₃	—	2	1	no	—
166	C₂×C₄×C₅₃	—	2	1	no	—
167	C₂×C₄×C₅₃	—	2	1	no	—
168	C₂×C₄×C₅₃	—	2	1	no	—
169	C₂×C₄×C₅₃	—	2	1	no	—
170	C₂×C₄×C₅₃	—	4	1	no	—
171	C₂×C₄×C₅₃	—	4	2	no	—
172	C₂×C₄×C₅₃	—	8	1	no	—
173	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
177	C₂×C₄×C₅₃	#135	1	2	yes	—
178	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
179	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
181	C₂²⋊D₅₃	—	4	4	no	—
182	C₂×Dic₅₃	—	2	2	no	—
183	D₂₁₂	—	2	2	no	—
184	D₂₁₂	—	2	2	no	—
185	D₂₁₂	—	4	2	no	—
186	D₂₁₂	—	8	1	no	—
187	C₄×D₅₃	—	4	2	no	—
188	C₄×D₅₃	—	4	2	no	—
189	Dic₁₀₆	—	4	2	no	—
190	C₈×C₅₃	—	2	2	no	—
191	C₈×C₅₃	—	4	2	no	—
192	C₈×C₅₃	C₂×SD₁₆	1	4	yes	—
193	#88322	—	2	2	no	—
194	#88322	—	2	2	no	—
195	#88322	—	2	2	no	—
196	#88322	—	4	1	no	—
197	#88322	—	4	1	no	—
198	#88322	—	4	1	no	—
199	#17997	—	53	1	no	maximal, Sylow
200	C₂²×C₅₃	—	2	1	no	—
201	C₂²×C₅₃	—	2	1	no	—
202	C₂²×C₅₃	C₄×SD₁₆	1	2	yes	—
203	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
204	C₂×D₅₃	—	4	2	no	—
205	C₂×D₅₃	—	4	2	no	—
206	C₂×D₅₃	—	4	2	no	—
207	C₄×C₅₃	—	2	1	no	—
208	C₄×C₅₃	—	2	2	no	—
209	C₄×C₅₃	—	2	1	no	—
210	C₄×C₅₃	—	2	1	no	—
211	C₄×C₅₃	—	4	1	no	—
212	C₄×C₅₃	—	4	1	no	—
213	C₄×C₅₃	—	4	1	no	—
214	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
215	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
216	C₄×C₅₃	#568	1	1	yes	characteristic
217	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
218	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
219	Dic₅₃	—	4	2	no	—
220	#3103	—	53	1	no	—
221	Q₈×M₁₆	D₅₃	1	1	yes	characteristic, complemented
222	#2935	—	53	1	no	—
223	#1590	—	53	2	no	—
224	#1583	—	53	2	no	—
225	#1582	—	53	2	no	—
226	C₈×SD₁₆	—	53	2	no	—
227	#1570	—	53	1	no	—
228	#1492	—	53	1	no	—
229	C₂×Q₈⋊C₈	—	53	1	no	—
230	C₈×M₁₆	—	53	1	no	—
231	C₂×C₅₃	—	2	2	no	—
232	C₂×C₅₃	#2933	1	1	yes	characteristic
233	C₂×C₅₃	C₂×C₄×SD₁₆	1	1	yes	characteristic
234	C₂×C₅₃	#2956	1	1	yes	characteristic
235	D₅₃	—	8	2	no	—
236	C₂×#131	—	106	1	no	—
237	#532	—	53	1	no	—
238	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
239	#481	—	53	2	no	—
240	#475	—	53	1	no	—
241	#475	—	53	1	no	—
242	#460	—	53	1	no	—
243	#460	—	53	1	no	—
244	#448	—	53	2	no	—
245	#446	—	2	2	no	—
246	#446	C₂×D₅₃	1	2	yes	—
247	C₈×Q₈	C₂×D₅₃	1	2	yes	—
248	C₄×SD₁₆	—	53	2	no	—
249	C₄×SD₁₆	—	53	2	no	—
250	#436	—	53	2	no	—
251	#435	—	106	2	no	—
252	C₈×D₄	—	53	2	no	—
253	#431	—	53	1	no	—
254	#423	—	2	1	no	—
255	#423	C₂×D₅₃	1	1	yes	characteristic
256	C₂×C₄⋊C₈	—	53	1	no	—
257	#408	—	106	1	no	—
258	C₄×M₁₆	—	2	1	no	—
259	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
260	C₂×C₄×C₈	—	53	1	no	—
261	#334	—	53	2	no	—
262	#334	—	53	2	no	—
263	Q₈⋊C₈	—	53	1	no	—
264	Q₈⋊C₈	—	53	1	no	—
265	Q₈⋊C₈	—	53	1	no	—
266	Q₈⋊C₈	—	53	1	no	—
267	D₄⋊C₈	—	53	2	no	—
268	D₄⋊C₈	—	53	2	no	—
269	#322	—	53	2	no	—
270	C₈²	—	53	2	no	—
271	C₅₃	#17997	1	1	yes	characteristic, complemented, Sylow
272	C₂×#41	—	106	1	no	—
273	C₂²×Q₈	C₄×D₅₃	1	1	yes	characteristic
274	C₂×SD₁₆	—	53	2	no	—
275	C₂×SD₁₆	—	53	2	no	—
276	#131	—	212	4	no	—
277	C₂×M₁₆	—	2	1	no	—
278	C₂×M₁₆	—	106	2	no	—
279	C₂×M₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
280	C₂×M₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
281	C₂²×C₈	—	106	1	no	—
282	C₂²×C₈	—	106	1	no	—
283	C₂²×C₈	—	106	2	no	—
284	C₄⋊Q₈	—	53	1	no	—
285	#127	—	53	1	no	—
286	#124	—	106	1	no	—
287	#121	—	53	2	no	—
288	#121	—	106	1	no	—
289	C₄×Q₈	—	2	1	no	—
290	C₄×Q₈	—	2	1	no	—
291	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
292	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
293	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
294	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
295	C₄×D₄	—	53	2	no	—
296	C₄×D₄	—	106	1	no	—
297	C₂×C₄⋊C₄	—	2	1	no	—
298	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
299	C₂×C₄²	—	2	1	no	—
300	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
301	#106	—	53	2	no	—
302	#106	—	53	2	no	—
303	C₄⋊C₈	—	2	2	no	—
304	C₄⋊C₈	—	2	2	no	—
305	C₄⋊C₈	—	2	1	no	—
306	C₄⋊C₈	—	53	1	no	—
307	C₄⋊C₈	—	53	1	no	—
308	C₄⋊C₈	—	53	1	no	—
309	C₄⋊C₈	—	53	1	no	—
310	C₄⋊C₈	—	106	1	no	—
311	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
312	#103	—	53	2	no	—
313	#103	—	53	2	no	—
314	#102	—	53	2	no	—
315	#102	—	53	2	no	—
316	C₂²⋊C₈	—	53	2	no	—
317	C₂²⋊C₈	—	106	2	no	—
318	C₂²⋊C₈	—	106	2	no	—
319	#97	—	2	2	no	—
320	#97	C₂²⋊D₅₃	1	2	yes	—
321	C₄×C₈	—	2	2	no	—
322	C₄×C₈	—	53	2	no	—
323	C₄×C₈	—	53	1	no	—
324	C₄×C₈	—	53	1	no	—
325	C₄×C₈	—	106	1	no	—
326	C₄×C₈	C₂²×D₅₃	1	2	yes	—
327	#41	—	212	2	no	—
328	#41	—	212	2	no	—
329	C₂×Q₈	—	2	1	no	—
330	C₂×Q₈	—	2	2	no	—
331	C₂×Q₈	—	4	1	no	—
332	C₂×Q₈	—	106	1	no	—
333	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
334	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
335	C₂×D₄	—	53	2	no	—
336	C₂×D₄	—	106	2	no	—
337	C₂×D₄	—	106	1	no	—
338	C₂×D₄	—	106	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	106	2	no	—
342	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
343	C₂²×C₄	#11719945	1	1	yes	characteristic
344	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
345	SD₁₆	—	106	4	no	—
346	SD₁₆	—	212	2	no	—
347	M₁₆	—	2	2	no	—
348	M₁₆	—	4	2	no	—
349	M₁₆	—	212	4	no	—
350	M₁₆	#11719924	1	4	yes	complemented
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	53	2	no	—
353	C₂×C₈	—	53	2	no	—
354	C₂×C₈	—	106	1	no	—
355	C₂×C₈	—	106	1	no	—
356	C₂×C₈	—	106	1	no	—
357	C₂×C₈	—	106	1	no	—
358	C₂×C₈	—	212	2	no	—
359	C₂×C₈	—	212	2	no	—
360	C₂×C₈	—	212	4	no	—
361	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
362	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	53	2	no	—
369	C₄⋊C₄	—	106	1	no	—
370	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
371	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
372	C₂²⋊C₄	—	106	2	no	—
373	C₂²⋊C₄	—	106	2	no	—
374	C₄²	—	2	1	no	—
375	C₄²	—	2	1	no	—
376	C₄²	—	2	1	no	—
377	C₄²	—	2	1	no	—
378	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
379	C₄²	#11719945	1	1	yes	characteristic
380	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
381	C₄²	C₂³×D₅₃	1	1	yes	characteristic
382	C₂³	—	106	2	no	—
383	C₂³	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
384	Q₈	—	2	1	no	—
385	Q₈	—	2	1	no	—
386	Q₈	—	2	1	no	—
387	Q₈	—	4	2	no	—
388	Q₈	—	212	2	no	—
389	Q₈	#13133225	1	2	yes	complemented
390	D₄	—	106	2	no	—
391	D₄	—	106	2	no	—
392	D₄	—	424	1	no	—
393	D₄	—	212	4	no	—
394	D₄	—	212	2	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	8	1	no	—
404	C₂×C₄	—	106	2	no	—
405	C₂×C₄	—	212	2	no	—
406	C₂×C₄	—	212	2	no	—
407	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
408	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
409	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
410	C₂×C₄	C₂×#11719924	1	2	yes	—
411	C₂×C₄	#13133237	1	2	yes	—
412	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
413	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
414	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
415	C₈	—	2	2	no	—
416	C₈	—	4	2	no	—
417	C₈	—	106	2	no	—
418	C₈	—	106	2	no	—
419	C₈	—	106	2	no	—
420	C₈	—	212	1	no	—
421	C₈	—	212	1	no	—
422	C₈	—	212	1	no	—
423	C₈	C₂×#11719924	1	4	yes	—
424	C₂²	—	2	1	no	—
425	C₂²	—	2	1	no	—
426	C₂²	—	212	2	no	—
427	C₂²	—	212	2	no	—
428	C₂²	—	212	2	no	—
429	C₂²	C₄×#11719924	1	2	yes	—
430	C₂²	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
431	C₄	—	2	1	no	—
432	C₄	—	2	2	no	—
433	C₄	—	2	1	no	—
434	C₄	—	2	1	no	—
435	C₄	—	4	1	no	—
436	C₄	—	4	1	no	—
437	C₄	—	4	1	no	—
438	C₄	—	212	2	no	—
439	C₄	C₄×#11719924	1	2	yes	—
440	C₄	C₂×#13133225	1	1	yes	characteristic
441	C₄	#16811389	1	1	yes	characteristic
442	C₄	C₂²×#11719924	1	1	yes	characteristic
443	C₄	C₂×#13133237	1	1	yes	characteristic
444	C₂	—	2	2	no	—
445	C₂	—	424	2	no	—
446	C₂	#20283440	1	1	yes	characteristic
447	C₂	C₂×C₄×#11719924	1	1	yes	characteristic
448	C₂	#20283622	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	Q₈×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#16811173	C₂²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	Q₈×M₁₆	D₅₃
hypocenter	C₅₃	#17997
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ Q₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20273605	C₂	semidirect
4. #20273558	C₂	semidirect
5. #20273549	C₂	semidirect
6. C₈×#11719924	C₂	semidirect
7. #20273487	C₂	semidirect
9. C₂×Q₈⋊#88322	C₂	semidirect
10. M₁₆×#88322	C₂	semidirect
11. Q₈×M₁₆×C₅₃	C₂	semidirect
122. M₁₆×C₅₃	SD₁₆	semidirect
163. Q₈×C₅₃	#131	semidirect
221. Q₈×M₁₆	D₅₃	semidirect
271. C₅₃	#17997	semidirect
350. M₁₆	#11719924	semidirect
389. Q₈	#13133225	semidirect

Maximal subgroups

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

subgroup	index
#20284746	2
#20283461	2
#20273605	2
#20273558	2
#20273549	2
C₈×#11719924	2
#20273487	2
#20273165	2
C₂×Q₈⋊#88322	2
M₁₆×#88322	2
Q₈×M₁₆×C₅₃	2
#17997	53