Subgroups — #28316129

Order 13568.

756 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291392	C₂	1	1	yes	maximal, characteristic
2	D₅₃⋊#423	C₂	1	1	yes	maximal, characteristic
3	#20290279	C₂	1	4	yes	maximal, complemented
4	#20290274	C₂	1	1	yes	maximal, characteristic, complemented
5	#20290273	C₂	1	1	yes	maximal, characteristic, complemented
6	#20290271	C₂	1	1	yes	maximal, characteristic, complemented
7	#20290270	C₂	1	1	yes	maximal, characteristic, complemented
8	#20290267	C₂	1	4	yes	maximal, complemented
9	C₄×M₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
10	C₂×D₅₃⋊M₁₆	—	2	1	no	—
11	#16811645	—	2	1	no	—
12	#16811644	C₂²	1	2	yes	—
13	#16811642	C₂²	1	2	yes	—
14	#16811639	—	2	1	no	—
15	#16811628	—	2	1	no	—
16	#16811627	C₂²	1	1	yes	characteristic
17	#16811627	C₂²	1	1	yes	characteristic
18	#16811627	C₂²	1	1	yes	characteristic
19	#16811627	C₂²	1	1	yes	characteristic
20	#16811626	—	2	1	no	—
21	#16811626	C₂²	1	2	yes	—
22	D₅₃⋊(C₄⋊C₈)	—	2	1	no	—
23	D₅₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
24	#16811594	—	2	1	no	—
25	C₄×#11719936	—	2	1	no	—
26	#16811487	C₂²	1	4	yes	complemented
27	#16811486	C₂²	1	4	yes	complemented
28	#16811485	C₂²	1	4	yes	complemented
29	#16811484	C₂²	1	4	yes	complemented
30	C₂×M₁₆×D₅₃	C₄	1	2	yes	—
31	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
32	C₄×#11719917	C₄	1	1	yes	characteristic
33	#16810575	C₂²	1	2	yes	—
34	M₁₆×Dic₅₃	C₂²	1	2	yes	—
35	C₄×#11719912	C₄	1	1	yes	characteristic
36	C₄×#11719912	C₄	1	1	yes	characteristic
37	#97×D₅₃	C₄	1	2	yes	—
38	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
39	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
40	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
41	C₂×#11719936	—	2	1	no	—
42	C₂×#11719936	—	2	1	no	—
43	C₂×#11719936	—	2	1	no	—
44	C₂×#11719936	—	2	1	no	—
45	D₅₃⋊M₁₆	—	4	4	no	—
46	C₂×#11719940	—	2	2	no	—
47	C₂×D₅₃⋊C₈	—	2	2	no	—
48	#13133284	—	2	2	no	—
49	#13133284	—	2	2	no	—
50	C₄⋊#88324	—	2	1	no	—
51	C₄⋊#88324	—	2	1	no	—
52	C₄⋊#88324	—	2	1	no	—
53	C₄⋊#88324	—	2	1	no	—
54	C₄⋊#88324	C₂³	1	2	yes	—
55	C₄⋊#88324	C₂³	1	2	yes	—
56	C₄⋊#88324	C₂³	1	2	yes	—
57	C₄⋊#88324	C₂³	1	2	yes	—
58	#13133274	—	2	2	no	—
59	C₂²⋊#88324	—	2	2	no	—
60	#13133272	—	2	2	no	—
61	C₄×#88324	—	2	2	no	—
62	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
64	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
65	C₂×#11719917	C₂×C₄	1	2	yes	—
66	C₂×#11719912	C₂×C₄	1	2	yes	—
67	C₂×#11719912	C₂×C₄	1	2	yes	—
68	M₁₆×D₅₃	—	2	8	no	—
69	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
70	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
71	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
72	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
73	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
74	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
75	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
76	C₄²×D₅₃	D₄	1	1	yes	characteristic
77	C₄²×D₅₃	D₄	1	1	yes	characteristic
78	C₄²×D₅₃	Q₈	1	1	yes	characteristic
79	C₄²×D₅₃	Q₈	1	1	yes	characteristic
80	C₄²×D₅₃	C₂³	1	1	yes	characteristic
81	C₄²×D₅₃	C₂³	1	1	yes	characteristic
82	#13133128	C₂×C₄	1	2	yes	—
83	#13133127	C₂×C₄	1	2	yes	—
84	#13133127	C₂×C₄	1	2	yes	—
85	C₄×#88322	C₂×C₄	1	1	yes	characteristic
86	C₄×#88322	C₂×C₄	1	1	yes	characteristic
87	C₈×Dic₅₃	C₂³	1	2	yes	—
88	C₈×Dic₅₃	C₂³	1	2	yes	—
89	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
90	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
91	#97×C₅₃	C₂×C₄	1	2	yes	—
92	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
93	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
94	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₄×D₅₃	—	2	4	no	—
102	C₂×C₄×D₅₃	—	2	2	no	—
103	C₂×C₄×D₅₃	—	2	2	no	—
104	C₂×C₄×D₅₃	—	4	2	no	—
105	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
116	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
117	#11719940	—	4	4	no	—
118	C₂×#88324	—	2	2	no	—
119	C₂×#88324	—	2	2	no	—
120	C₂×#88324	—	2	2	no	—
121	C₂×#88324	—	2	2	no	—
122	#11719936	—	4	2	no	—
123	#11719936	—	4	2	no	—
124	#11719936	—	4	2	no	—
125	#11719936	—	4	2	no	—
126	D₅₃⋊C₈	—	4	4	no	—
127	M₁₆×C₅₃	—	2	4	no	—
128	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
129	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
130	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
131	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
132	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
134	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
135	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
136	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
137	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
138	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
140	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
141	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
142	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
143	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
144	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
145	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
146	#11719917	—	2	4	no	—
147	C₂×#88322	C₂²×C₄	1	2	yes	—
148	C₂×#88322	C₂²×C₄	1	2	yes	—
149	#11719912	—	2	4	no	—
150	#11719912	M₁₆	1	8	yes	complemented
151	C₈×D₅₃	—	2	4	no	—
152	C₈×D₅₃	—	2	4	no	—
153	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂²×D₅₃	—	2	2	no	—
155	C₂²×D₅₃	—	2	2	no	—
156	C₂²×D₅₃	—	4	1	no	—
157	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	2	no	—
162	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
168	C₂×Dic₅₃	—	2	1	no	—
169	C₂×Dic₅₃	—	2	1	no	—
170	C₂×Dic₅₃	—	2	2	no	—
171	C₂×Dic₅₃	—	4	1	no	—
172	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
177	C₂×Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
178	C₄×D₅₃	—	2	2	no	—
179	C₄×D₅₃	—	2	2	no	—
180	C₄×D₅₃	—	2	2	no	—
181	C₄×D₅₃	—	2	2	no	—
182	C₄×D₅₃	—	2	2	no	—
183	C₄×D₅₃	—	2	2	no	—
184	C₄×D₅₃	—	2	1	no	—
185	C₄×D₅₃	—	2	1	no	—
186	C₄×D₅₃	—	2	2	no	—
187	C₄×D₅₃	—	4	1	no	—
188	C₄×D₅₃	—	4	1	no	—
189	C₄×D₅₃	C₂×M₁₆	1	4	yes	—
190	C₄×D₅₃	AGL(1,ℤ/8)	1	2	yes	—
191	C₄×D₅₃	#137	1	2	yes	—
192	#88324	—	4	2	no	—
193	#88324	—	4	2	no	—
194	#88324	—	4	2	no	—
195	#88324	—	4	2	no	—
196	C₈×C₅₃	—	2	2	no	—
197	C₈×C₅₃	C₂×M₁₆	1	4	yes	complemented
198	#88322	—	2	2	no	—
199	#88322	C₂×M₁₆	1	4	yes	complemented
200	#17796	—	53	1	no	maximal, Sylow
201	C₂²×C₅₃	—	2	1	no	—
202	C₂²×C₅₃	—	2	1	no	—
203	C₂²×C₅₃	—	2	1	no	—
204	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
205	C₂×D₅₃	—	2	1	no	—
206	C₂×D₅₃	—	2	1	no	—
207	C₂×D₅₃	—	2	1	no	—
208	C₂×D₅₃	—	2	1	no	—
209	C₂×D₅₃	—	4	4	no	—
210	C₂×D₅₃	#423	1	2	yes	—
211	C₂×D₅₃	#428	1	2	yes	—
212	C₄×C₅₃	—	2	1	no	—
213	C₄×C₅₃	—	2	1	no	—
214	C₄×C₅₃	—	2	1	no	—
215	C₄×C₅₃	#428	1	2	yes	—
216	C₄×C₅₃	#430	1	2	yes	—
217	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
218	Dic₅₃	—	2	1	no	—
219	Dic₅₃	—	2	1	no	—
220	Dic₅₃	—	2	1	no	—
221	Dic₅₃	—	2	1	no	—
222	Dic₅₃	#423	1	2	yes	—
223	Dic₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
224	Dic₅₃	C₂×#137	1	1	yes	characteristic
225	#2920	—	53	1	no	—
226	C₂×#423	—	53	1	no	—
227	C₂×C₄×M₁₆	—	53	1	no	—
228	#1569	—	53	4	no	—
229	#1567	—	53	1	no	—
230	#1567	—	53	1	no	—
231	#1566	—	53	1	no	—
232	#1566	—	53	1	no	—
233	#1565	—	53	4	no	—
234	C₂×C₅₃	—	4	1	no	—
235	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
236	C₂×C₅₃	C₂×#428	1	1	yes	characteristic
237	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
238	D₅₃	—	2	2	no	—
239	D₅₃	—	2	2	no	—
240	C₂²×M₁₆	—	53	2	no	—
241	C₂²×M₁₆	—	106	1	no	—
242	C₂²×C₄²	—	53	1	no	—
243	#432	—	53	2	no	—
244	#432	—	106	1	no	—
245	#431	—	53	2	no	—
246	#431	—	106	1	no	—
247	#423	—	53	1	no	—
248	#423	—	53	1	no	—
249	#423	—	53	1	no	—
250	#423	—	53	1	no	—
251	#423	—	53	2	no	—
252	#423	—	106	1	no	—
253	#423	—	106	1	no	—
254	C₂×C₄⋊C₈	—	53	2	no	—
255	C₂×C₄⋊C₈	—	106	1	no	—
256	#407	—	106	1	no	—
257	C₄×M₁₆	—	53	2	no	—
258	C₄×M₁₆	—	53	1	no	—
259	C₄×M₁₆	—	53	2	no	—
260	C₄×M₁₆	—	53	1	no	—
261	C₄×M₁₆	—	53	1	no	—
262	C₄×M₁₆	—	106	1	no	—
263	C₄×M₁₆	#6157	1	1	yes	characteristic
264	C₂×#97	—	53	2	no	—
265	C₂×C₄×C₈	—	53	1	no	—
266	C₂×C₄×C₈	—	53	1	no	—
267	#335	—	53	4	no	—
268	#335	—	53	4	no	—
269	#334	—	53	4	no	—
270	#334	—	53	4	no	—
271	C₅₃	#17796	1	1	yes	characteristic, complemented, Sylow
272	C₂³×C₄	—	53	1	no	—
273	C₂³×C₄	—	53	1	no	—
274	C₂³×C₄	—	53	1	no	—
275	C₂×M₁₆	—	53	2	no	—
276	C₂×M₁₆	—	53	2	no	—
277	C₂×M₁₆	—	53	2	no	—
278	C₂×M₁₆	—	106	8	no	—
279	C₂×M₁₆	—	106	2	no	—
280	C₂×M₁₆	—	106	1	no	—
281	C₂×M₁₆	—	106	1	no	—
282	C₂×M₁₆	—	106	1	no	—
283	C₂×M₁₆	—	106	1	no	—
284	C₂×M₁₆	—	212	4	no	—
285	C₂×M₁₆	C₂×#6157	1	2	yes	—
286	C₂²×C₈	—	53	2	no	—
287	C₂²×C₈	—	53	2	no	—
288	C₂²×C₈	—	106	2	no	—
289	C₂×C₄²	—	53	1	no	—
290	C₂×C₄²	—	53	1	no	—
291	C₂×C₄²	—	53	1	no	—
292	C₂×C₄²	—	53	1	no	—
293	C₂×C₄²	—	53	1	no	—
294	C₂×C₄²	—	53	1	no	—
295	C₂×C₄²	—	53	1	no	—
296	C₂×C₄²	—	53	1	no	—
297	C₂×C₄²	—	53	1	no	—
298	C₂×C₄²	—	53	1	no	—
299	C₂×C₄²	—	53	1	no	—
300	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
301	C₄⋊C₈	—	53	2	no	—
302	C₄⋊C₈	—	53	2	no	—
303	C₄⋊C₈	—	53	2	no	—
304	C₄⋊C₈	—	53	2	no	—
305	C₄⋊C₈	—	106	2	no	—
306	C₄⋊C₈	—	106	1	no	—
307	C₄⋊C₈	—	106	1	no	—
308	C₄⋊C₈	—	106	2	no	—
309	C₄⋊C₈	—	106	1	no	—
310	C₄⋊C₈	—	106	1	no	—
311	C₂²⋊C₈	—	106	2	no	—
312	C₂²⋊C₈	—	106	2	no	—
313	#97	—	53	2	no	—
314	#97	—	53	2	no	—
315	#97	—	53	2	no	—
316	#97	—	106	2	no	—
317	#97	C₂×#6157	1	2	yes	—
318	C₄×C₈	—	53	2	no	—
319	C₄×C₈	—	53	2	no	—
320	C₄×C₈	—	53	1	no	—
321	C₄×C₈	—	53	1	no	—
322	C₄×C₈	—	106	2	no	—
323	C₄×C₈	C₂×#6157	1	1	yes	characteristic
324	C₄×C₈	C₂×#6157	1	1	yes	characteristic
325	C₂⁴	—	53	1	no	—
326	C₂²×C₄	—	53	1	no	—
327	C₂²×C₄	—	53	1	no	—
328	C₂²×C₄	—	53	1	no	—
329	C₂²×C₄	—	53	1	no	—
330	C₂²×C₄	—	53	1	no	—
331	C₂²×C₄	—	53	1	no	—
332	C₂²×C₄	—	53	1	no	—
333	C₂²×C₄	—	53	1	no	—
334	C₂²×C₄	—	53	1	no	—
335	C₂²×C₄	—	53	1	no	—
336	C₂²×C₄	—	53	1	no	—
337	C₂²×C₄	—	53	1	no	—
338	C₂²×C₄	—	53	1	no	—
339	C₂²×C₄	—	53	1	no	—
340	C₂²×C₄	—	53	1	no	—
341	C₂²×C₄	—	106	4	no	—
342	C₂²×C₄	—	106	2	no	—
343	C₂²×C₄	—	106	2	no	—
344	C₂²×C₄	—	212	2	no	—
345	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
346	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
347	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
348	M₁₆	—	2	4	no	—
349	M₁₆	—	53	8	no	—
350	M₁₆	—	106	4	no	—
351	M₁₆	—	106	4	no	—
352	M₁₆	—	212	2	no	—
353	M₁₆	—	212	2	no	—
354	M₁₆	—	212	2	no	—
355	M₁₆	—	212	2	no	—
356	M₁₆	—	212	4	no	—
357	C₂×C₈	—	53	2	no	—
358	C₂×C₈	—	53	2	no	—
359	C₂×C₈	—	106	4	no	—
360	C₂×C₈	—	106	4	no	—
361	C₂×C₈	—	106	2	no	—
362	C₂×C₈	—	106	2	no	—
363	C₂×C₈	—	106	2	no	—
364	C₂×C₈	—	106	2	no	—
365	C₂×C₈	—	212	4	no	—
366	C₂×C₈	C₂²×#6157	1	2	yes	—
367	C₂×C₈	C₂²×#6157	1	2	yes	—
368	C₄²	—	53	1	no	—
369	C₄²	—	53	1	no	—
370	C₄²	—	53	1	no	—
371	C₄²	—	53	1	no	—
372	C₄²	—	53	1	no	—
373	C₄²	—	53	1	no	—
374	C₄²	—	53	1	no	—
375	C₄²	—	53	1	no	—
376	C₄²	—	53	1	no	—
377	C₄²	—	53	1	no	—
378	C₄²	—	53	1	no	—
379	C₄²	—	53	1	no	—
380	C₄²	C₄⋊#6157	1	1	yes	characteristic
381	C₄²	C₄⋊#6157	1	1	yes	characteristic
382	C₄²	C₂²×#6157	1	1	yes	characteristic
383	C₄²	C₂²×#6157	1	1	yes	characteristic
384	C₂³	—	53	1	no	—
385	C₂³	—	53	1	no	—
386	C₂³	—	106	2	no	—
387	C₂³	—	106	2	no	—
388	C₂³	—	212	1	no	—
389	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	53	1	no	—
394	C₂×C₄	—	53	1	no	—
395	C₂×C₄	—	53	2	no	—
396	C₂×C₄	—	53	2	no	—
397	C₂×C₄	—	53	4	no	—
398	C₂×C₄	—	53	1	no	—
399	C₂×C₄	—	53	1	no	—
400	C₂×C₄	—	53	1	no	—
401	C₂×C₄	—	53	1	no	—
402	C₂×C₄	—	106	1	no	—
403	C₂×C₄	—	106	2	no	—
404	C₂×C₄	—	106	1	no	—
405	C₂×C₄	—	106	2	no	—
406	C₂×C₄	—	106	2	no	—
407	C₂×C₄	—	106	1	no	—
408	C₂×C₄	—	106	2	no	—
409	C₂×C₄	—	106	1	no	—
410	C₂×C₄	—	106	2	no	—
411	C₂×C₄	—	106	2	no	—
412	C₂×C₄	—	106	2	no	—
413	C₂×C₄	—	106	2	no	—
414	C₂×C₄	—	212	1	no	—
415	C₂×C₄	—	212	1	no	—
416	C₂×C₄	—	212	1	no	—
417	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
418	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
419	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
420	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
421	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
422	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
423	C₈	—	2	2	no	—
424	C₈	—	53	4	no	—
425	C₈	—	106	2	no	—
426	C₈	—	212	2	no	—
427	C₈	—	212	2	no	—
428	C₈	—	212	2	no	—
429	C₈	—	212	2	no	—
430	C₈	D₅₃⋊M₁₆	1	4	yes	complemented
431	C₂²	—	2	1	no	—
432	C₂²	—	2	1	no	—
433	C₂²	—	2	1	no	—
434	C₂²	—	53	2	no	—
435	C₂²	—	53	2	no	—
436	C₂²	—	106	1	no	—
437	C₂²	—	106	1	no	—
438	C₂²	—	106	1	no	—
439	C₂²	—	106	1	no	—
440	C₂²	—	212	4	no	—
441	C₂²	C₂²×C₄⋊#6157	1	1	yes	characteristic
442	C₄	—	2	1	no	—
443	C₄	—	2	1	no	—
444	C₄	—	2	1	no	—
445	C₄	—	53	2	no	—
446	C₄	—	53	1	no	—
447	C₄	—	53	1	no	—
448	C₄	—	106	1	no	—
449	C₄	—	106	1	no	—
450	C₄	—	106	1	no	—
451	C₄	—	106	1	no	—
452	C₄	#16811634	1	2	yes	—
453	C₄	#16811638	1	2	yes	—
454	C₄	C₂×D₅₃⋊M₁₆	1	2	yes	—
455	C₂	—	4	1	no	—
456	C₂	—	106	2	no	—
457	C₂	—	106	2	no	—
458	C₂	D₅₃⋊#423	1	1	yes	characteristic
459	C₂	C₂×#16811634	1	1	yes	characteristic
460	C₂	C₂×#16811638	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₄×M₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₄⋊C₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₄×M₁₆	#6157
hypocenter	C₅₃	#17796
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20290279	C₂	semidirect
4. #20290274	C₂	semidirect
5. #20290273	C₂	semidirect
6. #20290271	C₂	semidirect
7. #20290270	C₂	semidirect
8. #20290267	C₂	semidirect
26. #16811487	C₂²	semidirect
27. #16811486	C₂²	semidirect
28. #16811485	C₂²	semidirect
29. #16811484	C₂²	semidirect
150. #11719912	M₁₆	semidirect
197. C₈×C₅₃	C₂×M₁₆	semidirect
199. #88322	C₂×M₁₆	semidirect
271. C₅₃	#17796	semidirect
430. C₈	D₅₃⋊M₁₆	semidirect

Maximal subgroups

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

subgroup	index
#20291392	2
D₅₃⋊#423	2
#20290279	2
#20290274	2
#20290273	2
#20290271	2
#20290270	2
#20290267	2
C₄×M₁₆×D₅₃	2
#17796	53