Subgroups — M₁₆×#11719912 #27349800

Order 13568.

900 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	#20283343	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11719912	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20273019	C₂	1	2	yes	maximal, complemented
5	#20273017	C₂	1	2	yes	maximal, complemented
6	#20273005	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273004	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273001	C₂	1	2	yes	maximal, complemented
9	M₁₆×#88322	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11719912	C₂	1	2	yes	maximal, complemented
11	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#11719912	C₄	1	2	yes	—
13	C₂×M₁₆×D₅₃	C₄	1	2	yes	—
14	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
15	#16810679	C₄	1	2	yes	—
16	#16810677	C₄	1	2	yes	—
17	#16810676	C₂²	1	1	yes	characteristic
18	#16810674	C₂²	1	2	yes	—
19	#16810645	C₄	1	2	yes	—
20	#16810585	C₄	1	2	yes	—
21	C₄×#11719917	C₂²	1	1	yes	characteristic
22	#16810575	C₄	1	2	yes	—
23	M₁₆×Dic₅₃	C₄	1	2	yes	—
24	C₄×#11719912	C₄	1	2	yes	—
25	C₄×#11719912	C₄	1	2	yes	—
26	C₄×#11719912	C₂²	1	1	yes	characteristic
27	C₄×#11719912	C₂²	1	1	yes	characteristic
28	C₄×#11719912	C₂²	1	1	yes	characteristic
29	C₄×#11719912	C₂²	1	2	yes	—
30	C₄×#11719912	C₂²	1	2	yes	—
31	C₄×#11719912	C₂²	1	1	yes	characteristic
32	C₂×#13133127	C₄	1	2	yes	—
33	#97×D₅₃	C₄	1	2	yes	—
34	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
35	C₄×C₈×D₅₃	C₂²	1	2	yes	—
36	#16810345	C₂²	1	2	yes	complemented
37	#16810345	C₂²	1	2	yes	complemented
38	#16810344	C₂²	1	2	yes	complemented
39	#16810344	C₂²	1	2	yes	complemented
40	#16810344	C₂²	1	2	yes	complemented
41	C₈×#88322	C₂²	1	2	yes	complemented
42	C₅₃×#431	C₂²	1	1	yes	characteristic
43	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₅₃×#322	C₂²	1	2	yes	complemented
46	C₈²×C₅₃	C₂²	1	2	yes	complemented
47	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×#11719917	C₂×C₄	1	2	yes	—
51	C₂×#11719912	—	2	8	no	—
52	C₂×#11719912	C₂×C₄	1	2	yes	—
53	C₂×#11719912	C₂×C₄	1	2	yes	—
54	C₂×#11719912	C₂×C₄	1	2	yes	—
55	C₂×#11719912	C₂×C₄	1	4	yes	—
56	C₂×#11719912	C₂×C₄	1	4	yes	—
57	C₂×#11719912	C₂×C₄	1	2	yes	—
58	M₁₆×D₅₃	—	2	8	no	—
59	C₂²×#88322	C₂×C₄	1	2	yes	—
60	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
61	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
67	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
68	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
69	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄²×D₅₃	C₂³	1	1	yes	characteristic
71	C₄²×D₅₃	C₂³	1	1	yes	characteristic
72	C₄⋊#88322	C₂×C₄	1	2	yes	—
73	#13133146	C₂×C₄	1	4	yes	—
74	C₂²⋊#88322	C₂×C₄	1	2	yes	—
75	#13133129	C₂×C₄	1	4	yes	—
76	#13133128	C₂×C₄	1	2	yes	—
77	#13133127	C₂×C₄	1	4	yes	—
78	#13133127	C₂×C₄	1	2	yes	—
79	#13133127	C₂×C₄	1	2	yes	—
80	#13133127	C₂×C₄	1	2	yes	—
81	#13133127	C₂×C₄	1	2	yes	—
82	#13133127	C₂×C₄	1	4	yes	—
83	C₄×#88322	C₂×C₄	1	2	yes	—
84	C₄×#88322	C₂³	1	2	yes	—
85	C₄×#88322	C₂³	1	1	yes	characteristic
86	C₄×#88322	C₂³	1	1	yes	characteristic
87	C₄×#88322	C₂³	1	2	yes	—
88	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
89	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
90	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
91	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
92	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
93	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
94	#97×C₅₃	C₂×C₄	1	2	yes	—
95	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
96	C₄×C₈×C₅₃	C₂³	1	2	yes	—
97	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
98	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₅₃	C₂³	1	2	yes	—
100	C₂³×D₅₃	C₄²	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
105	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
106	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
107	C₂×C₄×D₅₃	—	2	4	no	—
108	C₂×C₄×D₅₃	—	2	4	no	—
109	C₂×C₄×D₅₃	—	4	2	no	—
110	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
111	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
112	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
113	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
114	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
115	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
116	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
122	M₁₆×C₅₃	M₁₆	1	8	yes	complemented
123	C₂×C₈×C₅₃	—	2	4	no	—
124	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
125	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
126	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
127	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
128	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
132	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
133	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
134	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
135	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
136	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
137	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
138	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
141	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
143	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
144	#11719917	—	2	4	no	—
145	C₂×#88322	—	2	4	no	—
146	C₂×#88322	C₂²×C₄	1	4	yes	—
147	C₂×#88322	C₂²×C₄	1	2	yes	—
148	C₂×#88322	C₂²×C₄	1	4	yes	—
149	C₂×#88322	C₂²×C₄	1	2	yes	—
150	#11719912	—	2	4	no	—
151	#11719912	—	2	8	no	—
152	#11719912	—	2	4	no	—
153	#11719912	—	2	4	no	—
154	#11719912	—	4	4	no	—
155	#11719912	M₁₆	1	8	yes	direct product factor
156	C₈×D₅₃	—	2	8	no	—
157	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
158	C₂²×D₅₃	—	2	2	no	—
159	C₂²×D₅₃	—	2	2	no	—
160	C₂²×D₅₃	—	4	1	no	—
161	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
162	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	2	2	no	—
165	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
169	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
170	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
172	C₂×Dic₅₃	—	2	2	no	—
173	C₂×Dic₅₃	—	2	2	no	—
174	C₂×Dic₅₃	—	4	1	no	—
175	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
176	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
177	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
178	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
179	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
180	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
181	C₄×D₅₃	—	2	2	no	—
182	C₄×D₅₃	—	2	2	no	—
183	C₄×D₅₃	—	2	2	no	—
184	C₄×D₅₃	—	2	2	no	—
185	C₄×D₅₃	—	2	2	no	—
186	C₄×D₅₃	—	2	2	no	—
187	C₄×D₅₃	—	2	2	no	—
188	C₄×D₅₃	—	2	2	no	—
189	C₄×D₅₃	—	2	2	no	—
190	C₄×D₅₃	—	4	1	no	—
191	C₄×D₅₃	—	4	1	no	—
192	C₄×D₅₃	C₂×M₁₆	1	4	yes	—
193	C₈×C₅₃	—	2	2	no	—
194	C₈×C₅₃	—	2	4	no	—
195	C₈×C₅₃	C₂×M₁₆	1	4	yes	complemented
196	C₈×C₅₃	C₂×M₁₆	1	8	yes	complemented
197	#88322	—	2	4	no	—
198	#88322	—	2	2	no	—
199	#88322	—	2	4	no	—
200	#88322	C₂×M₁₆	1	4	yes	complemented
201	M₁₆²	—	53	1	no	maximal, Sylow
202	C₂²×C₅₃	—	2	1	no	—
203	C₂²×C₅₃	—	2	1	no	—
204	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
205	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
206	C₂×D₅₃	—	2	1	no	—
207	C₂×D₅₃	—	2	1	no	—
208	C₂×D₅₃	—	2	1	no	—
209	C₂×D₅₃	—	2	1	no	—
210	C₂×D₅₃	—	2	1	no	—
211	C₂×D₅₃	—	4	4	no	—
212	C₂×D₅₃	C₄×M₁₆	1	2	yes	—
213	C₄×C₅₃	—	2	1	no	—
214	C₄×C₅₃	—	2	1	no	—
215	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
216	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
217	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
218	C₄×C₅₃	#568	1	2	yes	—
219	Dic₅₃	—	2	1	no	—
220	Dic₅₃	—	2	1	no	—
221	Dic₅₃	—	2	1	no	—
222	Dic₅₃	—	2	1	no	—
223	Dic₅₃	—	2	1	no	—
224	Dic₅₃	C₄×M₁₆	1	2	yes	—
225	#2920	—	53	1	no	—
226	C₂×C₄×M₁₆	—	53	1	no	—
227	C₂×C₄×M₁₆	—	53	1	no	—
228	#1455	—	53	2	no	—
229	#1455	—	53	2	no	—
230	#1451	—	53	1	no	—
231	#1451	—	53	1	no	—
232	#1451	—	53	2	no	—
233	C₈×M₁₆	—	53	1	no	—
234	C₈×M₁₆	—	53	2	no	—
235	C₈×M₁₆	D₅₃	1	1	yes	characteristic, complemented
236	C₂×C₅₃	—	2	2	no	—
237	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
238	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
239	C₂×C₅₃	#2873	1	1	yes	characteristic
240	D₅₃	—	2	2	no	—
241	D₅₃	—	4	1	no	—
242	C₂²×M₁₆	—	53	2	no	—
243	C₂²×M₁₆	—	53	2	no	—
244	C₂²×C₄²	—	53	1	no	—
245	#432	—	53	2	no	—
246	#432	—	53	2	no	—
247	#431	—	53	1	no	—
248	#431	—	53	2	no	—
249	#431	C₂×D₅₃	1	1	yes	characteristic
250	#423	—	53	2	no	—
251	#407	—	53	2	no	—
252	C₄×M₁₆	—	53	2	no	—
253	C₄×M₁₆	—	53	1	no	—
254	C₄×M₁₆	—	53	2	no	—
255	C₄×M₁₆	—	53	2	no	—
256	C₄×M₁₆	—	53	2	no	—
257	C₄×M₁₆	—	53	1	no	—
258	C₄×M₁₆	—	53	1	no	—
259	C₄×M₁₆	—	53	1	no	—
260	C₄×M₁₆	—	53	2	no	—
261	C₄×M₁₆	—	53	1	no	—
262	C₄×M₁₆	—	53	2	no	—
263	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
264	C₂×#97	—	53	2	no	—
265	C₂×#97	—	53	2	no	—
266	C₂×C₄×C₈	—	53	1	no	—
267	C₂×C₄×C₈	—	53	2	no	—
268	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
269	#322	—	53	2	no	—
270	#322	—	53	2	no	—
271	#322	—	53	2	no	—
272	#322	—	53	2	no	—
273	#322	—	53	2	no	—
274	#322	C₂×D₅₃	1	2	yes	complemented
275	C₈²	—	53	2	no	—
276	C₈²	C₂×D₅₃	1	2	yes	complemented
277	C₅₃	M₁₆²	1	1	yes	characteristic, complemented, Sylow
278	C₂³×C₄	—	53	1	no	—
279	C₂³×C₄	—	53	1	no	—
280	C₂³×C₄	—	53	1	no	—
281	C₂×M₁₆	—	53	2	no	—
282	C₂×M₁₆	—	53	2	no	—
283	C₂×M₁₆	—	53	2	no	—
284	C₂×M₁₆	—	53	2	no	—
285	C₂×M₁₆	—	53	2	no	—
286	C₂×M₁₆	—	53	4	no	—
287	C₂×M₁₆	—	53	4	no	—
288	C₂×M₁₆	—	106	8	no	—
289	C₂×M₁₆	—	106	8	no	—
290	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
291	C₂²×C₈	—	53	2	no	—
292	C₂²×C₈	—	53	4	no	—
293	C₂²×C₈	C₄×D₅₃	1	2	yes	—
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	—	53	1	no	—
301	C₂×C₄²	—	53	1	no	—
302	C₂×C₄²	—	53	1	no	—
303	C₂×C₄²	—	53	1	no	—
304	C₂×C₄²	—	53	1	no	—
305	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
306	C₄⋊C₈	—	53	4	no	—
307	C₄⋊C₈	—	53	2	no	—
308	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
309	C₂²⋊C₈	—	53	2	no	—
310	C₂²⋊C₈	—	53	4	no	—
311	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
312	#97	—	53	2	no	—
313	#97	—	53	2	no	—
314	#97	—	53	2	no	—
315	#97	—	53	2	no	—
316	#97	—	53	4	no	—
317	#97	—	53	4	no	—
318	#97	—	53	2	no	—
319	#97	C₄×D₅₃	1	2	yes	—
320	C₄×C₈	—	53	4	no	—
321	C₄×C₈	—	53	2	no	—
322	C₄×C₈	—	53	1	no	—
323	C₄×C₈	—	53	2	no	—
324	C₄×C₈	—	53	1	no	—
325	C₄×C₈	—	53	2	no	—
326	C₄×C₈	C₄×D₅₃	1	2	yes	—
327	C₄×C₈	C₂²×D₅₃	1	2	yes	—
328	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
329	C₄×C₈	C₂²×D₅₃	1	2	yes	—
330	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
331	C₂⁴	—	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	—	53	1	no	—
342	C₂²×C₄	—	53	1	no	—
343	C₂²×C₄	—	53	1	no	—
344	C₂²×C₄	—	53	1	no	—
345	C₂²×C₄	—	53	1	no	—
346	C₂²×C₄	—	53	1	no	—
347	C₂²×C₄	—	106	4	no	—
348	C₂²×C₄	—	106	4	no	—
349	C₂²×C₄	—	212	2	no	—
350	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
351	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
352	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
353	M₁₆	—	53	8	no	—
354	M₁₆	—	106	4	no	—
355	M₁₆	—	106	4	no	—
356	M₁₆	—	106	4	no	—
357	M₁₆	—	106	4	no	—
358	M₁₆	—	106	8	no	—
359	M₁₆	—	212	4	no	—
360	M₁₆	#11719912	1	8	yes	direct product factor
361	C₂×C₈	—	2	4	no	—
362	C₂×C₈	—	53	2	no	—
363	C₂×C₈	—	53	2	no	—
364	C₂×C₈	—	53	4	no	—
365	C₂×C₈	—	53	4	no	—
366	C₂×C₈	—	106	4	no	—
367	C₂×C₈	—	106	8	no	—
368	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
369	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
370	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
371	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
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	—	53	1	no	—
381	C₄²	—	53	1	no	—
382	C₄²	—	53	1	no	—
383	C₄²	—	53	1	no	—
384	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
385	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
386	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
387	C₄²	C₂³×D₅₃	1	1	yes	characteristic
388	C₂³	—	53	1	no	—
389	C₂³	—	53	1	no	—
390	C₂³	—	106	2	no	—
391	C₂³	—	106	2	no	—
392	C₂³	—	212	1	no	—
393	C₂³	C₄²×D₅₃	1	1	yes	characteristic
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	53	1	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	—	53	1	no	—
403	C₂×C₄	—	106	2	no	—
404	C₂×C₄	—	106	2	no	—
405	C₂×C₄	—	106	2	no	—
406	C₂×C₄	—	106	2	no	—
407	C₂×C₄	—	106	2	no	—
408	C₂×C₄	—	106	2	no	—
409	C₂×C₄	—	106	2	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	C₄²×D₅₃	1	1	yes	characteristic
418	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
419	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
420	C₂×C₄	C₂×#11719912	1	4	yes	—
421	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
422	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
423	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
424	C₈	—	2	2	no	—
425	C₈	—	2	4	no	—
426	C₈	—	53	4	no	—
427	C₈	—	106	2	no	—
428	C₈	—	106	4	no	—
429	C₈	—	106	4	no	—
430	C₈	M₁₆×D₅₃	1	4	yes	complemented
431	C₈	C₂×#11719912	1	8	yes	complemented
432	C₂²	—	2	1	no	—
433	C₂²	—	2	1	no	—
434	C₂²	—	53	2	no	—
435	C₂²	—	106	1	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	C₄×#11719912	1	2	yes	—
442	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
443	C₄	—	2	1	no	—
444	C₄	—	2	1	no	—
445	C₄	—	53	2	no	—
446	C₄	—	106	1	no	—
447	C₄	—	106	1	no	—
448	C₄	—	106	1	no	—
449	C₄	—	106	1	no	—
450	C₄	—	106	1	no	—
451	C₄	C₄×#11719912	1	2	yes	—
452	C₄	C₂×M₁₆×D₅₃	1	2	yes	—
453	C₄	C₂²×#11719912	1	2	yes	—
454	C₄	#16811388	1	2	yes	—
455	C₂	—	2	2	no	—
456	C₂	—	106	2	no	—
457	C₂	—	212	1	no	—
458	C₂	C₄×M₁₆×D₅₃	1	1	yes	characteristic
459	C₂	C₂×C₄×#11719912	1	1	yes	characteristic
460	C₂	#20283131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₈×M₁₆×C₅₃	C₂
prime radical	C₂³×D₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₈×M₁₆	D₅₃
hypocenter	C₅₃	M₁₆²
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
4. #20273019	C₂	semidirect
5. #20273017	C₂	semidirect
6. #20273005	C₂	semidirect
7. #20273004	C₂	semidirect
8. #20273001	C₂	semidirect
9. M₁₆×#88322	C₂	semidirect
10. C₈×#11719912	C₂	semidirect
11. C₈×M₁₆×C₅₃	C₂	semidirect
36. #16810345	C₂²	semidirect
37. #16810345	C₂²	semidirect
38. #16810344	C₂²	semidirect
39. #16810344	C₂²	semidirect
40. #16810344	C₂²	semidirect
41. C₈×#88322	C₂²	semidirect
45. C₅₃×#322	C₂²	semidirect
46. C₈²×C₅₃	C₂²	semidirect
122. M₁₆×C₅₃	M₁₆	semidirect
155. #11719912	M₁₆	direct (N × H)
195. C₈×C₅₃	C₂×M₁₆	semidirect
196. C₈×C₅₃	C₂×M₁₆	semidirect
200. #88322	C₂×M₁₆	semidirect
235. C₈×M₁₆	D₅₃	semidirect
274. #322	C₂×D₅₃	semidirect
276. C₈²	C₂×D₅₃	semidirect
277. C₅₃	M₁₆²	semidirect
360. M₁₆	#11719912	direct (N × H)
430. C₈	M₁₆×D₅₃	semidirect
431. C₈	C₂×#11719912	semidirect

Maximal subgroups

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

subgroup	index
#20283343	2
C₂×C₄×#11719912	2
C₄×M₁₆×D₅₃	2
#20273019	2
#20273017	2
#20273005	2
#20273004	2
#20273001	2
M₁₆×#88322	2
C₈×#11719912	2
C₈×M₁₆×C₅₃	2
M₁₆²	53

Subgroups — M16×#11719912 #27349800

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×#11719912 #27349800