Subgroups — #27300944

Order 13568.

578 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×C₂²⋊D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16810402	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272792	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272792	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₂²⋊D₅₃	C₄	1	1	yes	characteristic
10	C₂²×#11719908	C₂²	1	1	yes	characteristic
11	C₂²×#11719908	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×D₅₃	—	2	2	no	—
13	C₂×C₁₆×D₅₃	—	2	2	no	—
14	C₂×C₁₆×D₅₃	—	2	2	no	—
15	C₈×C₂²⋊D₅₃	—	2	2	no	—
16	C₈×C₂²⋊D₅₃	—	2	2	no	—
17	C₂×#13133146	C₄	1	1	yes	characteristic
18	C₂×C₂²⋊#88322	C₄	1	1	yes	characteristic
19	C₂×#13133129	C₄	1	1	yes	characteristic
20	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
21	C₂²⋊#11719908	—	2	2	no	—
22	#16810402	—	2	2	no	—
23	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×C₂²⋊C₁₆	—	2	2	no	—
27	C₂²×C₂²⋊D₅₃	C₈	1	1	yes	characteristic
28	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
30	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₅₃	—	2	1	no	—
32	C₂×C₈×D₅₃	—	2	1	no	—
33	C₂×C₈×D₅₃	—	2	1	no	—
34	C₂×C₈×D₅₃	—	2	1	no	—
35	C₂×C₈×D₅₃	D₄	1	2	yes	—
36	C₂×C₈×D₅₃	D₄	1	2	yes	—
37	C₄×C₂²⋊D₅₃	—	2	2	no	—
38	C₄×C₂²⋊D₅₃	—	2	2	no	—
39	C₂×#11719919	C₈	1	1	yes	characteristic
40	C₂×C₂²⋊Dic₅₃	C₈	1	1	yes	characteristic
41	C₂×#11719921	C₈	1	1	yes	characteristic
42	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×#11719908	—	2	1	no	—
44	C₂×#11719908	—	2	1	no	—
45	C₂×#11719908	—	2	1	no	—
46	C₂×#11719908	—	2	1	no	—
47	C₂×#11719908	—	2	1	no	—
48	C₂×#11719908	D₄	1	2	yes	complemented
49	C₁₆×D₅₃	—	2	8	no	—
50	#13133146	—	2	2	no	—
51	C₂²⋊#88322	—	2	2	no	—
52	#13133129	—	2	2	no	—
53	C₈×Dic₅₃	—	2	1	no	—
54	C₈×Dic₅₃	—	2	1	no	—
55	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	—	2	1	no	—
57	C₂²×C₈×C₅₃	—	2	1	no	—
58	C₂²×C₈×C₅₃	—	2	1	no	—
59	C₂²×C₈×C₅₃	—	2	1	no	—
60	C₂²×C₈×C₅₃	—	2	1	no	—
61	C₂²×C₈×C₅₃	—	2	1	no	—
62	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×C₁₆×C₅₃	—	2	1	no	—
65	C₂×C₁₆×C₅₃	—	2	1	no	—
66	C₂×C₁₆×C₅₃	—	2	1	no	—
67	C₂×C₁₆×C₅₃	—	2	1	no	—
68	C₂×C₁₆×C₅₃	—	2	1	no	—
69	C₂×C₁₆×C₅₃	D₄	1	2	yes	complemented
70	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
71	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	—	2	1	no	—
73	C₂²×C₄×C₅₃	—	2	1	no	—
74	C₂²×C₄×C₅₃	—	2	1	no	—
75	C₂²×C₄×C₅₃	—	2	1	no	—
76	C₂²×C₄×C₅₃	—	2	1	no	—
77	C₂²×C₄×C₅₃	—	2	1	no	—
78	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₂²⋊D₅₃	—	2	2	no	—
81	C₂×C₂²⋊D₅₃	—	2	2	no	—
82	C₂×C₂²⋊D₅₃	C₁₆	1	4	yes	complemented
83	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
84	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
85	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₅₃	—	2	1	no	—
87	C₂×C₄×D₅₃	—	2	1	no	—
88	C₂×C₄×D₅₃	—	2	1	no	—
89	C₂×C₄×D₅₃	—	2	1	no	—
90	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
91	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
92	C₂×C₈×C₅₃	—	2	1	no	—
93	C₂×C₈×C₅₃	—	2	1	no	—
94	C₂×C₈×C₅₃	—	2	1	no	—
95	C₂×C₈×C₅₃	—	4	1	no	—
96	C₂×C₈×C₅₃	—	4	1	no	—
97	C₂×C₈×C₅₃	—	4	1	no	—
98	C₂×C₈×C₅₃	—	4	1	no	—
99	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
104	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
105	C₂²⋊Dic₅₃	—	2	2	no	—
106	#11719921	—	2	2	no	—
107	#11719919	—	2	2	no	—
108	C₄×Dic₅₃	—	2	1	no	—
109	C₄×Dic₅₃	—	2	1	no	—
110	C₂×#88322	—	2	1	no	—
111	C₂×#88322	—	2	1	no	—
112	C₂×#88322	—	2	1	no	—
113	C₂×#88322	—	2	1	no	—
114	C₂×#88322	—	2	1	no	—
115	C₂×#88322	C₂²⋊C₄	1	2	yes	—
116	C₈×D₅₃	—	2	2	no	—
117	C₈×D₅₃	—	2	2	no	—
118	C₈×D₅₃	—	2	2	no	—
119	C₈×D₅₃	—	2	2	no	—
120	C₁₆×C₅₃	—	2	2	no	—
121	C₁₆×C₅₃	—	4	1	no	—
122	#11719908	—	2	2	no	—
123	#11719908	—	4	1	no	—
124	C₂³×C₅₃	—	2	1	no	—
125	C₂³×C₅₃	—	2	1	no	—
126	C₂³×C₅₃	—	2	1	no	—
127	C₂³×C₅₃	—	2	1	no	—
128	C₂³×C₅₃	—	2	1	no	—
129	C₂³×C₅₃	—	2	1	no	—
130	C₂³×C₅₃	C₂×C₁₆	1	2	yes	complemented
131	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
132	C₂²×D₅₃	—	2	1	no	—
133	C₂²×D₅₃	—	2	1	no	—
134	C₂²×D₅₃	—	2	1	no	—
135	C₂²×D₅₃	—	2	1	no	—
136	C₂²×D₅₃	C₂²⋊C₈	1	2	yes	—
137	C₂²×D₅₃	C₂×C₁₆	1	2	yes	—
138	C₂²×D₅₃	M₃₂	1	2	yes	—
139	C₂×C₄×C₅₃	—	2	1	no	—
140	C₂×C₄×C₅₃	—	2	1	no	—
141	C₂×C₄×C₅₃	—	2	1	no	—
142	C₂×C₄×C₅₃	—	4	1	no	—
143	C₂×C₄×C₅₃	—	4	1	no	—
144	C₂×C₄×C₅₃	—	4	1	no	—
145	C₂×C₄×C₅₃	—	4	1	no	—
146	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
150	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
151	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
152	C₂²⋊D₅₃	—	2	8	no	—
153	C₂×Dic₅₃	—	2	1	no	—
154	C₂×Dic₅₃	—	2	1	no	—
155	C₂×Dic₅₃	—	2	1	no	—
156	C₂×Dic₅₃	—	2	1	no	—
157	C₂×Dic₅₃	—	2	1	no	—
158	C₂×Dic₅₃	—	2	1	no	—
159	C₂×Dic₅₃	—	2	1	no	—
160	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
161	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	complemented
162	C₄×D₅₃	—	2	2	no	—
163	C₄×D₅₃	—	2	2	no	—
164	C₄×D₅₃	—	2	2	no	—
165	C₄×D₅₃	—	2	2	no	—
166	C₈×C₅₃	—	4	1	no	—
167	C₈×C₅₃	#120	1	1	yes	characteristic
168	C₈×C₅₃	#121	1	1	yes	characteristic
169	C₈×C₅₃	#121	1	1	yes	characteristic
170	C₈×C₅₃	#123	1	1	yes	characteristic
171	#88322	—	2	2	no	—
172	#88322	—	4	1	no	—
173	#12976	—	53	1	no	maximal, Sylow
174	C₂²×C₅₃	—	2	2	no	—
175	C₂²×C₅₃	—	2	2	no	—
176	C₂²×C₅₃	—	2	2	no	—
177	C₂²×C₅₃	—	4	1	no	—
178	C₂²×C₅₃	—	4	1	no	—
179	C₂²×C₅₃	—	4	1	no	—
180	C₂²×C₅₃	—	4	1	no	—
181	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
182	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
183	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
184	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
185	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
186	C₂²×C₅₃	#504	1	1	yes	characteristic
187	C₂²×C₅₃	#504	1	1	yes	characteristic
188	C₂×D₅₃	—	2	2	no	—
189	C₂×D₅₃	—	2	2	no	—
190	C₂×D₅₃	—	2	2	no	—
191	C₂×D₅₃	—	2	2	no	—
192	C₂×D₅₃	—	2	2	no	—
193	C₂×D₅₃	—	2	2	no	—
194	C₂×D₅₃	C₂²⋊C₁₆	1	4	yes	complemented
195	C₄×C₅₃	—	4	1	no	—
196	C₄×C₅₃	#386	1	1	yes	characteristic
197	C₄×C₅₃	#408	1	1	yes	characteristic
198	C₄×C₅₃	#435	1	1	yes	characteristic
199	C₄×C₅₃	#435	1	1	yes	characteristic
200	Dic₅₃	—	2	2	no	—
201	Dic₅₃	—	2	2	no	—
202	Dic₅₃	—	4	1	no	—
203	C₂³×C₁₆	—	53	1	no	—
204	C₂×C₈×D₄	—	53	1	no	—
205	C₂×C₂²⋊C₁₆	—	53	1	no	—
206	C₂×C₂²⋊C₁₆	—	53	1	no	—
207	C₂×C₂²⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
208	#1374	—	53	1	no	—
209	#1374	—	53	1	no	—
210	C₂×C₅₃	—	4	2	no	—
211	C₂×C₅₃	#1869	1	1	yes	characteristic
212	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
213	C₂×C₅₃	#2113	1	1	yes	characteristic
214	C₂×C₅₃	D₄×C₁₆	1	1	yes	characteristic
215	C₂×C₅₃	D₄×C₁₆	1	1	yes	characteristic
216	C₂×C₅₃	#2168	1	1	yes	characteristic
217	C₂×C₅₃	#2168	1	1	yes	characteristic
218	D₅₃	—	2	4	no	—
219	C₂³×C₈	—	53	1	no	—
220	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
221	C₂×C₄×D₄	—	53	1	no	—
222	C₂²×C₁₆	—	53	1	no	—
223	C₂²×C₁₆	—	53	1	no	—
224	C₂²×C₁₆	—	106	2	no	—
225	C₂²×C₁₆	—	106	2	no	—
226	C₂²×C₁₆	—	106	2	no	—
227	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
228	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
229	C₈×D₄	—	106	2	no	—
230	C₈×D₄	—	106	2	no	—
231	C₂×C₄⋊C₈	—	53	1	no	—
232	C₂×C₂²⋊C₈	—	53	1	no	—
233	C₂×C₂²⋊C₈	—	53	1	no	—
234	C₂×C₄×C₈	—	53	1	no	—
235	C₂²⋊C₁₆	—	2	2	no	—
236	C₂²⋊C₁₆	—	106	2	no	—
237	C₂²⋊C₁₆	—	106	2	no	—
238	C₅₃	#12976	1	1	yes	characteristic, complemented, Sylow
239	C₂²×D₄	—	53	1	no	—
240	C₂³×C₄	—	53	1	no	—
241	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
242	C₂²×C₈	—	2	1	no	—
243	C₂²×C₈	—	2	1	no	—
244	C₂²×C₈	—	2	1	no	—
245	C₂²×C₈	—	2	1	no	—
246	C₂²×C₈	—	2	1	no	—
247	C₂²×C₈	—	2	1	no	—
248	C₂²×C₈	—	53	1	no	—
249	C₂²×C₈	—	53	2	no	—
250	C₂²×C₈	—	53	2	no	—
251	C₂²×C₈	—	53	1	no	—
252	C₂²×C₈	—	106	1	no	—
253	C₂²×C₈	—	106	1	no	—
254	C₂²×C₈	—	106	1	no	—
255	C₂²×C₈	—	106	1	no	—
256	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
257	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
258	C₄×D₄	—	106	2	no	—
259	C₄×D₄	—	106	2	no	—
260	C₂×C₄⋊C₄	—	53	1	no	—
261	C₂×C₂²⋊C₄	—	53	1	no	—
262	C₂×C₂²⋊C₄	—	53	1	no	—
263	C₂×C₄²	—	53	1	no	—
264	C₂×C₁₆	—	2	1	no	—
265	C₂×C₁₆	—	2	1	no	—
266	C₂×C₁₆	—	2	1	no	—
267	C₂×C₁₆	—	2	1	no	—
268	C₂×C₁₆	—	2	1	no	—
269	C₂×C₁₆	—	53	2	no	—
270	C₂×C₁₆	—	106	8	no	—
271	C₂×C₁₆	—	106	1	no	—
272	C₂×C₁₆	—	106	1	no	—
273	C₂×C₁₆	—	106	1	no	—
274	C₂×C₁₆	—	106	1	no	—
275	C₂×C₁₆	—	106	1	no	—
276	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	complemented
277	C₄⋊C₈	—	106	2	no	—
278	C₂²⋊C₈	—	106	2	no	—
279	C₂²⋊C₈	—	106	2	no	—
280	C₄×C₈	—	106	1	no	—
281	C₄×C₈	—	106	1	no	—
282	C₂⁴	—	53	1	no	—
283	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
284	C₂×D₄	—	53	4	no	—
285	C₂×D₄	—	106	2	no	—
286	C₂×D₄	—	106	2	no	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	1	no	—
289	C₂²×C₄	—	2	1	no	—
290	C₂²×C₄	—	2	1	no	—
291	C₂²×C₄	—	2	1	no	—
292	C₂²×C₄	—	2	1	no	—
293	C₂²×C₄	—	53	1	no	—
294	C₂²×C₄	—	53	1	no	—
295	C₂²×C₄	—	53	2	no	—
296	C₂²×C₄	—	53	2	no	—
297	C₂²×C₄	—	53	1	no	—
298	C₂²×C₄	—	106	1	no	—
299	C₂²×C₄	—	106	1	no	—
300	C₂²×C₄	—	106	1	no	—
301	C₂²×C₄	—	106	1	no	—
302	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
303	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	4	1	no	—
308	C₂×C₈	—	4	1	no	—
309	C₂×C₈	—	4	1	no	—
310	C₂×C₈	—	4	1	no	—
311	C₂×C₈	—	53	2	no	—
312	C₂×C₈	—	106	1	no	—
313	C₂×C₈	—	106	1	no	—
314	C₂×C₈	—	106	2	no	—
315	C₂×C₈	—	106	2	no	—
316	C₂×C₈	—	106	2	no	—
317	C₂×C₈	—	106	2	no	—
318	C₂×C₈	—	106	1	no	—
319	C₂×C₈	—	106	1	no	—
320	C₂×C₈	—	106	1	no	—
321	C₂×C₈	#11719945	1	1	yes	characteristic
322	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
323	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
324	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
325	C₂×C₈	#11719947	1	1	yes	characteristic
326	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
327	C₄⋊C₄	—	106	2	no	—
328	C₂²⋊C₄	—	106	2	no	—
329	C₂²⋊C₄	—	106	2	no	—
330	C₄²	—	106	1	no	—
331	C₄²	—	106	1	no	—
332	C₁₆	—	2	2	no	—
333	C₁₆	—	4	1	no	—
334	C₁₆	—	106	2	no	—
335	C₁₆	—	212	1	no	—
336	C₂³	—	2	1	no	—
337	C₂³	—	2	1	no	—
338	C₂³	—	2	1	no	—
339	C₂³	—	2	1	no	—
340	C₂³	—	2	1	no	—
341	C₂³	—	2	1	no	—
342	C₂³	—	53	2	no	—
343	C₂³	—	53	2	no	—
344	C₂³	—	53	2	no	—
345	C₂³	—	106	1	no	—
346	C₂³	—	106	1	no	—
347	C₂³	—	106	1	no	—
348	C₂³	—	106	1	no	—
349	C₂³	C₁₆×D₅₃	1	2	yes	complemented
350	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
351	D₄	—	106	8	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	4	1	no	—
356	C₂×C₄	—	4	1	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	4	1	no	—
359	C₂×C₄	—	53	2	no	—
360	C₂×C₄	—	53	2	no	—
361	C₂×C₄	—	106	1	no	—
362	C₂×C₄	—	106	1	no	—
363	C₂×C₄	—	106	1	no	—
364	C₂×C₄	—	106	1	no	—
365	C₂×C₄	—	106	2	no	—
366	C₂×C₄	—	106	2	no	—
367	C₂×C₄	—	106	2	no	—
368	C₂×C₄	—	106	2	no	—
369	C₂×C₄	—	106	1	no	—
370	C₂×C₄	—	106	1	no	—
371	C₂×C₄	—	106	1	no	—
372	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
373	C₂×C₄	#13133177	1	1	yes	characteristic
374	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
375	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
376	C₂×C₄	#13133223	1	1	yes	characteristic
377	C₂×C₄	#13133225	1	1	yes	characteristic
378	C₈	—	4	1	no	—
379	C₈	—	106	2	no	—
380	C₈	—	212	1	no	—
381	C₈	#13133185	1	1	yes	characteristic
382	C₈	#13133188	1	1	yes	characteristic
383	C₈	#13133190	1	1	yes	characteristic
384	C₈	#13133199	1	1	yes	characteristic
385	C₂²	—	2	2	no	—
386	C₂²	—	2	2	no	—
387	C₂²	—	2	2	no	—
388	C₂²	—	4	1	no	—
389	C₂²	—	4	1	no	—
390	C₂²	—	4	1	no	—
391	C₂²	—	4	1	no	—
392	C₂²	—	53	4	no	—
393	C₂²	—	106	2	no	—
394	C₂²	—	106	2	no	—
395	C₂²	—	106	2	no	—
396	C₂²	—	106	2	no	—
397	C₂²	—	106	2	no	—
398	C₂²	—	106	2	no	—
399	C₂²	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
400	C₂²	#16810686	1	1	yes	characteristic
401	C₂²	C₈×C₂²⋊D₅₃	1	1	yes	characteristic
402	C₂²	C₂×C₁₆×D₅₃	1	1	yes	characteristic
403	C₂²	M₃₂×D₅₃	1	1	yes	characteristic
404	C₂²	#16811039	1	1	yes	characteristic
405	C₂²	#16811041	1	1	yes	characteristic
406	C₄	—	4	1	no	—
407	C₄	—	106	2	no	—
408	C₄	—	106	2	no	—
409	C₄	—	212	1	no	—
410	C₄	#16810499	1	1	yes	characteristic
411	C₄	#16810588	1	1	yes	characteristic
412	C₄	#16810692	1	1	yes	characteristic
413	C₄	#16810697	1	1	yes	characteristic
414	C₂	—	4	2	no	—
415	C₂	—	106	4	no	—
416	C₂	#20275004	1	1	yes	characteristic
417	C₂	C₂²⋊C₁₆×D₅₃	1	1	yes	characteristic
418	C₂	#20276327	1	1	yes	characteristic
419	C₂	#20276532	1	1	yes	characteristic
420	C₂	C₁₆×C₂²⋊D₅₃	1	1	yes	characteristic
421	C₂	#20276538	1	1	yes	characteristic
422	C₂	#20276543	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₂×C₅₃×C₂²⋊C₁₆	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂²×C₂²⋊D₅₃	C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₅₃
hypocenter	C₅₃	#12976
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₆×D₅₃	C₂	semidirect
3. C₂×C₂²⋊#11719908	C₂	semidirect
4. C₂×#16810402	C₂	semidirect
5. #20272792	C₂	semidirect
6. #20272792	C₂	semidirect
7. C₂×C₅₃×C₂²⋊C₁₆	C₂	semidirect
48. C₂×#11719908	D₄	semidirect
69. C₂×C₁₆×C₅₃	D₄	semidirect
82. C₂×C₂²⋊D₅₃	C₁₆	semidirect
130. C₂³×C₅₃	C₂×C₁₆	semidirect
161. C₂×Dic₅₃	C₂×C₁₆	semidirect
194. C₂×D₅₃	C₂²⋊C₁₆	semidirect
207. C₂×C₂²⋊C₁₆	D₅₃	semidirect
238. C₅₃	#12976	semidirect
276. C₂×C₁₆	C₂²⋊D₅₃	semidirect
349. C₂³	C₁₆×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₅₃	2
C₂×C₈×C₂²⋊D₅₃	2
C₂×C₂²⋊#11719908	2
C₂×#16810402	2
#20272792	2
C₂×C₅₃×C₂²⋊C₁₆	2
#12976	53