Subgroups — #27289251

Order 13568.

1078 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133127	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810378	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810377	C₂	1	1	yes	maximal, characteristic
5	C₂×#16810377	C₂	1	1	yes	maximal, characteristic
6	C₄×#13133124	C₂	1	1	yes	maximal, characteristic
7	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂³×#88322	C₂²	1	1	yes	characteristic
9	C₂³×#88322	C₂²	1	1	yes	characteristic
10	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×#13133127	—	2	2	no	—
13	C₂×#13133127	—	2	2	no	—
14	C₂×#13133127	C₄	1	4	yes	complemented
15	C₂×C₈×Dic₅₃	—	2	2	no	—
16	C₂×C₈×Dic₅₃	—	2	2	no	—
17	C₂×C₈×Dic₅₃	C₄	1	4	yes	complemented
18	C₂×#13133124	C₄	1	4	yes	—
19	#16810378	C₄	1	4	yes	complemented
20	#16810377	C₄	1	4	yes	complemented
21	#16810377	C₄	1	4	yes	complemented
22	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#97×C₅₃	—	2	2	no	—
26	C₂×#97×C₅₃	—	2	2	no	—
27	C₂×#97×C₅₃	C₄	1	4	yes	complemented
28	C₂³×Dic₅₃	C₂×C₄	1	2	yes	—
29	C₂³×Dic₅₃	C₂×C₄	1	2	yes	—
30	C₂²×#88322	—	2	1	no	—
31	C₂²×#88322	—	2	1	no	—
32	C₂²×#88322	—	2	1	no	—
33	C₂²×#88322	—	2	1	no	—
34	C₂²×#88322	—	2	1	no	—
35	C₂²×#88322	—	2	1	no	—
36	C₂²×#88322	—	2	1	no	—
37	C₂²×#88322	—	2	1	no	—
38	C₂²×#88322	C₂×C₄	1	2	yes	—
39	C₂²×#88322	C₂×C₄	1	2	yes	—
40	C₂²×#88322	C₂×C₄	1	2	yes	—
41	C₂²×#88322	C₂×C₄	1	2	yes	—
42	C₂²×#88322	C₂×C₄	1	2	yes	—
43	C₂²×#88322	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₅₃	—	2	1	no	—
45	C₂×C₄×Dic₅₃	—	2	1	no	—
46	C₂×C₄×Dic₅₃	—	2	1	no	—
47	C₂×C₄×Dic₅₃	—	2	1	no	—
48	C₂×C₄×Dic₅₃	—	2	1	no	—
49	C₂×C₄×Dic₅₃	—	2	1	no	—
50	C₂×C₄×Dic₅₃	—	2	1	no	—
51	C₂×C₄×Dic₅₃	—	2	1	no	—
52	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
54	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
55	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
56	#13133127	—	2	8	no	—
57	C₈×Dic₅₃	—	2	8	no	—
58	#13133124	—	2	8	no	—
59	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
60	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₅₃	—	2	1	no	—
62	C₂²×C₈×C₅₃	—	2	1	no	—
63	C₂²×C₈×C₅₃	—	2	1	no	—
64	C₂²×C₈×C₅₃	—	2	1	no	—
65	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
66	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
67	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
68	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
69	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
70	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
71	C₂²×C₈×C₅₃	D₄	1	2	yes	—
72	C₂²×C₈×C₅₃	D₄	1	2	yes	—
73	C₂²×C₈×C₅₃	D₄	1	2	yes	—
74	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
75	C₂×C₄²×C₅₃	—	2	1	no	—
76	C₂×C₄²×C₅₃	—	2	1	no	—
77	C₂×C₄²×C₅₃	—	2	1	no	—
78	C₂×C₄²×C₅₃	—	2	1	no	—
79	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
80	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
81	#97×C₅₃	—	2	8	no	—
82	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	—	2	2	no	—
84	C₂²×C₄×C₅₃	—	2	2	no	—
85	C₂²×C₄×C₅₃	—	2	4	no	—
86	C₂²×C₄×C₅₃	—	2	4	no	—
87	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
88	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
102	C₂²×Dic₅₃	—	2	2	no	—
103	C₂²×Dic₅₃	—	2	2	no	—
104	C₂²×Dic₅₃	—	2	4	no	—
105	C₂²×Dic₅₃	—	2	2	no	—
106	C₂²×Dic₅₃	—	2	2	no	—
107	C₂²×Dic₅₃	—	2	4	no	—
108	C₂²×Dic₅₃	—	2	4	no	—
109	C₂²×Dic₅₃	C₄²	1	4	yes	—
110	C₂²×Dic₅₃	C₄²	1	4	yes	—
111	C₂²×Dic₅₃	M₁₆	1	8	yes	—
112	C₂×C₈×C₅₃	—	2	2	no	—
113	C₂×C₈×C₅₃	—	2	2	no	—
114	C₂×C₈×C₅₃	—	2	2	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	—	2	2	no	—
117	C₂×C₈×C₅₃	—	2	2	no	—
118	C₂×C₈×C₅₃	C₄²	1	4	yes	—
119	C₂×C₈×C₅₃	C₄²	1	4	yes	—
120	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
121	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
122	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
123	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
124	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
125	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
126	C₄²×C₅₃	—	2	2	no	—
127	C₄²×C₅₃	—	2	2	no	—
128	C₄²×C₅₃	—	2	2	no	—
129	C₄²×C₅₃	—	2	2	no	—
130	C₄×Dic₅₃	—	2	2	no	—
131	C₄×Dic₅₃	—	2	2	no	—
132	C₄×Dic₅₃	—	2	2	no	—
133	C₄×Dic₅₃	—	2	2	no	—
134	C₄×Dic₅₃	—	2	2	no	—
135	C₄×Dic₅₃	—	2	2	no	—
136	C₄×Dic₅₃	—	2	2	no	—
137	C₄×Dic₅₃	—	2	2	no	—
138	C₂×#88322	—	2	2	no	—
139	C₂×#88322	—	2	2	no	—
140	C₂×#88322	—	2	2	no	—
141	C₂×#88322	—	2	2	no	—
142	C₂×#88322	—	2	2	no	—
143	C₂×#88322	—	2	2	no	—
144	C₂×#88322	—	2	2	no	—
145	C₂×#88322	—	2	2	no	—
146	C₂×#88322	—	2	2	no	—
147	C₂×#88322	—	2	2	no	—
148	C₂×#88322	—	2	2	no	—
149	C₂×#88322	—	2	2	no	—
150	C₂×#88322	C₄²	1	4	yes	—
151	C₂×#88322	C₄²	1	4	yes	—
152	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
153	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
154	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
155	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
156	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
157	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
158	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
159	C₂³×C₅₃	C₂×M₁₆	1	4	yes	—
160	C₂³×C₅₃	#131	1	4	yes	—
161	C₂×C₄×C₅₃	—	2	4	no	—
162	C₂×C₄×C₅₃	—	2	4	no	—
163	C₂×C₄×C₅₃	—	2	4	no	—
164	C₂×C₄×C₅₃	—	4	4	no	—
165	C₂×C₄×C₅₃	—	4	4	no	—
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×C₄²	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
170	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
177	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
178	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
179	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
180	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
181	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
182	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
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	C₄×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
194	C₂×Dic₅₃	—	2	4	no	—
195	C₂×Dic₅₃	—	2	4	no	—
196	C₂×Dic₅₃	—	2	8	no	—
197	C₂×Dic₅₃	—	2	8	no	—
198	C₂×Dic₅₃	—	2	4	no	—
199	C₂×Dic₅₃	—	2	4	no	—
200	C₂×Dic₅₃	—	2	4	no	—
201	C₂×Dic₅₃	—	4	4	no	—
202	C₂×Dic₅₃	—	4	4	no	—
203	C₂×Dic₅₃	#97	1	8	yes	complemented
204	C₈×C₅₃	—	2	4	no	—
205	C₈×C₅₃	#95	1	8	yes	complemented
206	#88322	—	2	4	no	—
207	#88322	—	2	4	no	—
208	#10696	—	53	1	no	maximal, Sylow
209	C₂²×C₅₃	C₄³	1	1	yes	characteristic
210	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
211	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
212	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
213	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
214	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
215	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
216	C₂²×C₅₃	C₂×#97	1	2	yes	—
217	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
218	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
219	C₂²×C₅₃	#405	1	2	yes	—
220	C₂²×C₅₃	#405	1	2	yes	—
221	C₂²×C₅₃	#405	1	2	yes	—
222	C₂²×C₅₃	#435	1	4	yes	—
223	C₂²×C₅₃	#435	1	4	yes	—
224	C₂²×C₅₃	#436	1	4	yes	—
225	C₂²×C₅₃	#446	1	4	yes	—
226	C₄×C₅₃	—	4	4	no	—
227	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
228	C₄×C₅₃	#376	1	1	yes	characteristic
229	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
230	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
231	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
232	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
233	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
234	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
235	Dic₅₃	—	2	8	no	—
236	Dic₅₃	—	4	4	no	—
237	C₂²×#97	—	53	1	no	—
238	C₂²×#97	D₅₃	1	1	yes	characteristic
239	C₂²×C₄×C₈	—	53	1	no	—
240	C₂×#336	—	53	1	no	—
241	C₂×#336	—	53	1	no	—
242	C₂×#336	—	53	1	no	—
243	C₄×#95	—	53	1	no	—
244	C₂×C₅₃	C₄×#95	1	1	yes	characteristic
245	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
246	C₂×C₅₃	#1726	1	1	yes	characteristic
247	C₂×C₅₃	#1752	1	2	yes	—
248	C₂×C₅₃	#1753	1	2	yes	—
249	C₂×C₅₃	#1753	1	2	yes	—
250	C₂×C₅₃	#1770	1	2	yes	—
251	C₂×C₅₃	#1771	1	2	yes	—
252	C₂×C₅₃	#1771	1	2	yes	—
253	C₂³×C₈	—	53	1	no	—
254	C₂³×C₈	—	53	1	no	—
255	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
256	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
257	C₂²×C₄²	—	53	1	no	—
258	C₂²×C₄²	—	53	1	no	—
259	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
260	C₂×#97	—	2	2	no	—
261	C₂×#97	—	2	2	no	—
262	C₂×#97	—	53	4	no	—
263	C₂×#97	—	106	2	no	—
264	C₂×#97	—	106	2	no	—
265	C₂×#97	Dic₅₃	1	4	yes	complemented
266	C₂×C₄×C₈	—	53	4	no	—
267	C₂×C₄×C₈	—	106	2	no	—
268	C₂×C₄×C₈	—	106	2	no	—
269	C₂×#95	—	53	4	no	—
270	#336	—	53	4	no	—
271	#336	—	53	4	no	—
272	#336	—	53	4	no	—
273	C₅₃	#10696	1	1	yes	characteristic, complemented, Sylow
274	C₂³×C₄	—	53	2	no	—
275	C₂³×C₄	—	53	2	no	—
276	C₂³×C₄	C₄×D₅₃	1	2	yes	—
277	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
278	C₂²×C₈	—	2	1	no	—
279	C₂²×C₈	—	2	1	no	—
280	C₂²×C₈	—	2	1	no	—
281	C₂²×C₈	—	2	1	no	—
282	C₂²×C₈	—	53	2	no	—
283	C₂²×C₈	—	53	2	no	—
284	C₂²×C₈	—	53	2	no	—
285	C₂²×C₈	—	53	2	no	—
286	C₂²×C₈	—	53	2	no	—
287	C₂²×C₈	—	53	2	no	—
288	C₂²×C₈	—	106	1	no	—
289	C₂²×C₈	—	106	1	no	—
290	C₂²×C₈	—	106	1	no	—
291	C₂²×C₈	—	106	1	no	—
292	C₂²×C₈	—	106	1	no	—
293	C₂²×C₈	—	106	1	no	—
294	C₂²×C₈	—	106	1	no	—
295	C₂²×C₈	—	106	1	no	—
296	C₂²×C₈	Dic₁₀₆	1	2	yes	—
297	C₂²×C₈	C₄×D₅₃	1	2	yes	—
298	C₂²×C₈	C₄×D₅₃	1	2	yes	—
299	C₂²×C₈	C₄×D₅₃	1	2	yes	—
300	C₂²×C₈	C₄×D₅₃	1	2	yes	—
301	C₂²×C₈	D₂₁₂	1	2	yes	—
302	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
303	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
304	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
305	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
306	C₂×C₄²	—	2	1	no	—
307	C₂×C₄²	—	2	1	no	—
308	C₂×C₄²	—	2	1	no	—
309	C₂×C₄²	—	2	1	no	—
310	C₂×C₄²	—	53	2	no	—
311	C₂×C₄²	—	53	2	no	—
312	C₂×C₄²	—	53	2	no	—
313	C₂×C₄²	—	53	2	no	—
314	C₂×C₄²	—	106	1	no	—
315	C₂×C₄²	—	106	1	no	—
316	C₂×C₄²	—	106	1	no	—
317	C₂×C₄²	—	106	1	no	—
318	C₂×C₄²	—	106	1	no	—
319	C₂×C₄²	—	106	1	no	—
320	C₂×C₄²	—	106	1	no	—
321	C₂×C₄²	—	106	1	no	—
322	C₂×C₄²	C₄×D₅₃	1	2	yes	—
323	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
324	#97	—	2	8	no	—
325	#97	—	106	8	no	—
326	C₄×C₈	—	106	8	no	—
327	#95	—	106	8	no	—
328	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	—	2	4	no	—
333	C₂²×C₄	—	53	4	no	—
334	C₂²×C₄	—	53	8	no	—
335	C₂²×C₄	—	53	4	no	—
336	C₂²×C₄	—	106	2	no	—
337	C₂²×C₄	—	106	2	no	—
338	C₂²×C₄	—	106	4	no	—
339	C₂²×C₄	—	106	2	no	—
340	C₂²×C₄	—	106	2	no	—
341	C₂²×C₄	—	106	4	no	—
342	C₂²×C₄	—	106	4	no	—
343	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
344	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
345	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
346	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
347	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
348	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
349	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
350	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
351	C₂²×C₄	#11719945	1	1	yes	characteristic
352	C₂²×C₄	#11719945	1	1	yes	characteristic
353	C₂²×C₄	#11719945	1	1	yes	characteristic
354	C₂²×C₄	#11719945	1	1	yes	characteristic
355	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
356	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
357	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	53	4	no	—
365	C₂×C₈	—	53	4	no	—
366	C₂×C₈	—	106	2	no	—
367	C₂×C₈	—	106	2	no	—
368	C₂×C₈	—	106	2	no	—
369	C₂×C₈	—	106	2	no	—
370	C₂×C₈	—	106	2	no	—
371	C₂×C₈	—	106	2	no	—
372	C₂×C₈	—	106	2	no	—
373	C₂×C₈	—	106	2	no	—
374	C₂×C₈	—	106	2	no	—
375	C₂×C₈	—	106	2	no	—
376	C₂×C₈	—	106	2	no	—
377	C₂×C₈	—	106	2	no	—
378	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
379	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
380	C₂×C₈	#11719919	1	4	yes	—
381	C₂×C₈	#11719919	1	4	yes	—
382	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
383	C₂×C₈	#11719921	1	4	yes	—
384	C₂×C₈	#11719921	1	4	yes	—
385	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	2	no	—
388	C₄²	—	2	2	no	—
389	C₄²	—	2	2	no	—
390	C₄²	—	106	2	no	—
391	C₄²	—	106	2	no	—
392	C₄²	—	106	2	no	—
393	C₄²	—	106	2	no	—
394	C₄²	—	106	2	no	—
395	C₄²	—	106	2	no	—
396	C₄²	—	106	2	no	—
397	C₄²	—	106	2	no	—
398	C₂³	C₄²×D₅₃	1	1	yes	characteristic
399	C₂³	C₄²×D₅₃	1	1	yes	characteristic
400	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
401	C₂³	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
402	C₂³	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
403	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
404	C₂³	C₄×Dic₁₀₆	1	1	yes	characteristic
405	C₂³	M₁₆×D₅₃	1	4	yes	—
406	C₂³	#13133225	1	4	yes	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	4	4	no	—
411	C₂×C₄	—	4	4	no	—
412	C₂×C₄	—	53	8	no	—
413	C₂×C₄	—	106	4	no	—
414	C₂×C₄	—	106	8	no	—
415	C₂×C₄	—	106	8	no	—
416	C₂×C₄	—	106	4	no	—
417	C₂×C₄	—	106	4	no	—
418	C₂×C₄	—	106	4	no	—
419	C₂×C₄	—	106	4	no	—
420	C₂×C₄	—	212	4	no	—
421	C₂×C₄	—	212	4	no	—
422	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
423	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
424	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
425	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
426	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
427	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
428	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
429	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
430	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
431	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
432	C₂×C₄	#13133174	1	1	yes	characteristic
433	C₂×C₄	#13133174	1	1	yes	characteristic
434	C₂×C₄	#13133174	1	1	yes	characteristic
435	C₂×C₄	#13133174	1	1	yes	characteristic
436	C₂×C₄	#13133176	1	1	yes	characteristic
437	C₂×C₄	#13133176	1	1	yes	characteristic
438	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
439	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
440	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
441	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
442	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
443	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
444	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
445	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
446	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
447	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
448	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
449	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
450	C₈	—	2	4	no	—
451	C₈	—	106	4	no	—
452	C₈	—	106	4	no	—
453	C₈	#13133124	1	8	yes	complemented
454	C₂²	C₄²×Dic₅₃	1	1	yes	characteristic
455	C₂²	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
456	C₂²	C₄×#11719921	1	1	yes	characteristic
457	C₂²	C₄×#11719921	1	1	yes	characteristic
458	C₂²	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
459	C₂²	C₄×#11719919	1	1	yes	characteristic
460	C₂²	C₄×#11719919	1	1	yes	characteristic
461	C₂²	#97×D₅₃	1	2	yes	—
462	C₂²	M₁₆×Dic₅₃	1	2	yes	—
463	C₂²	#16810575	1	2	yes	—
464	C₂²	#16810580	1	2	yes	—
465	C₂²	#16810580	1	2	yes	—
466	C₂²	#16810581	1	2	yes	—
467	C₂²	#16810692	1	4	yes	—
468	C₂²	#16810696	1	4	yes	—
469	C₂²	#16810699	1	4	yes	—
470	C₂²	#16810751	1	4	yes	—
471	C₄	—	4	4	no	—
472	C₄	—	106	8	no	—
473	C₄	—	212	4	no	—
474	C₄	C₂×#13133124	1	1	yes	characteristic
475	C₄	#16810469	1	1	yes	characteristic
476	C₄	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
477	C₄	C₄×#11719921	1	1	yes	characteristic
478	C₄	C₄×#11719921	1	1	yes	characteristic
479	C₄	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
480	C₄	C₄×#11719919	1	1	yes	characteristic
481	C₄	C₄×#11719919	1	1	yes	characteristic
482	C₂	C₄×#13133124	1	1	yes	characteristic
483	C₂	#97×Dic₅₃	1	1	yes	characteristic
484	C₂	#20274403	1	1	yes	characteristic
485	C₂	#20274492	1	2	yes	—
486	C₂	#20274496	1	2	yes	—
487	C₂	#20274500	1	2	yes	—
488	C₂	#20274583	1	2	yes	—
489	C₂	#20274587	1	2	yes	—
490	C₂	#20274591	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₄³
Fitting	C₂²×#97×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂²×#97	D₅₃
hypocenter	C₅₃	#10696
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
14. C₂×#13133127	C₄	semidirect
17. C₂×C₈×Dic₅₃	C₄	semidirect
19. #16810378	C₄	semidirect
20. #16810377	C₄	semidirect
21. #16810377	C₄	semidirect
27. C₂×#97×C₅₃	C₄	semidirect
203. C₂×Dic₅₃	#97	semidirect
205. C₈×C₅₃	#95	semidirect
265. C₂×#97	Dic₅₃	semidirect
273. C₅₃	#10696	semidirect
453. C₈	#13133124	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133127	2
C₂²×C₈×Dic₅₃	2
C₂×#16810378	2
C₂×#16810377	2
C₄×#13133124	2
C₂²×#97×C₅₃	2
#10696	53