Subgroups — #27289986

Order 13568.

982 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133149	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810385	C₂	1	2	yes	maximal
4	#20272944	C₂	1	2	yes	maximal
5	C₂²×#106×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×#88322	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
9	C₂×#13133149	—	2	2	no	—
10	C₂×#13133149	—	2	2	no	—
11	C₂×#13133149	C₄	1	4	yes	complemented
12	C₂×C₈×Dic₅₃	—	2	2	no	—
13	C₂×C₈×Dic₅₃	—	2	2	no	—
14	C₂×C₈×Dic₅₃	C₄	1	4	yes	complemented
15	C₂×#13133124	—	2	4	no	—
16	#16810385	—	2	4	no	—
17	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₂×#106×C₅₃	—	2	2	no	—
20	C₂×#106×C₅₃	—	2	2	no	—
21	C₂×#106×C₅₃	C₄	1	4	yes	complemented
22	C₂³×Dic₅₃	—	2	2	no	—
23	C₂³×Dic₅₃	D₄	1	2	yes	—
24	C₂²×#88322	—	2	1	no	—
25	C₂²×#88322	—	2	1	no	—
26	C₂²×#88322	—	2	1	no	—
27	C₂²×#88322	—	2	1	no	—
28	C₂²×#88322	C₂×C₄	1	2	yes	—
29	C₂²×#88322	D₄	1	2	yes	—
30	C₂²×#88322	Q₈	1	2	yes	—
31	C₂×C₄⋊Dic₅₃	—	2	2	no	—
32	C₂×C₄⋊Dic₅₃	—	2	2	no	—
33	C₂×C₄⋊Dic₅₃	—	2	2	no	—
34	C₂×C₄⋊Dic₅₃	—	2	2	no	—
35	C₂×C₄⋊Dic₅₃	—	2	2	no	—
36	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₅₃	—	2	1	no	—
38	C₂×C₄×Dic₅₃	—	2	1	no	—
39	C₂×C₄×Dic₅₃	—	2	1	no	—
40	C₂×C₄×Dic₅₃	—	2	1	no	—
41	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₄×Dic₅₃	Q₈	1	2	yes	—
43	#13133149	—	2	8	no	—
44	C₈×Dic₅₃	—	2	8	no	—
45	#13133124	—	4	8	no	—
46	C₂³×C₄×C₅₃	—	2	2	no	—
47	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₅₃	D₄	1	2	yes	—
52	C₂²×C₈×C₅₃	D₄	1	2	yes	—
53	C₂²×C₈×C₅₃	D₄	1	2	yes	—
54	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
55	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
58	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
60	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
61	#106×C₅₃	—	2	8	no	—
62	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	—	2	4	no	—
64	C₂²×C₄×C₅₃	—	4	2	no	—
65	C₂²×C₄×C₅₃	—	4	2	no	—
66	C₂²×C₄×C₅₃	—	4	4	no	—
67	C₂²×C₄×C₅₃	—	4	4	no	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
82	C₂²×Dic₅₃	—	2	2	no	—
83	C₂²×Dic₅₃	—	2	2	no	—
84	C₂²×Dic₅₃	—	2	4	no	—
85	C₂²×Dic₅₃	—	2	4	no	—
86	C₂²×Dic₅₃	—	4	2	no	—
87	C₂²×Dic₅₃	—	4	2	no	—
88	C₂²×Dic₅₃	—	4	4	no	—
89	C₂²×Dic₅₃	—	4	4	no	—
90	C₂²×Dic₅₃	C₄⋊C₄	1	4	yes	—
91	C₂²×Dic₅₃	SD₁₆	1	8	yes	—
92	C₂×C₈×C₅₃	C₄²	1	4	yes	—
93	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
94	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
95	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
96	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
97	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
98	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
99	C₄⋊C₄×C₅₃	—	2	4	no	—
100	C₄⋊C₄×C₅₃	—	2	4	no	—
101	C₄⋊C₄×C₅₃	—	4	2	no	—
102	C₄⋊C₄×C₅₃	—	4	2	no	—
103	C₄⋊Dic₅₃	—	2	4	no	—
104	C₄⋊Dic₅₃	—	2	4	no	—
105	C₄⋊Dic₅₃	—	4	2	no	—
106	C₄⋊Dic₅₃	—	4	2	no	—
107	C₄×Dic₅₃	—	2	2	no	—
108	C₄×Dic₅₃	—	2	2	no	—
109	C₄×Dic₅₃	—	2	2	no	—
110	C₄×Dic₅₃	—	2	2	no	—
111	C₂×#88322	—	2	2	no	—
112	C₂×#88322	—	2	2	no	—
113	C₂×#88322	—	2	2	no	—
114	C₂×#88322	—	2	2	no	—
115	C₂×#88322	—	2	2	no	—
116	C₂×#88322	—	2	2	no	—
117	C₂×#88322	C₄⋊C₄	1	4	yes	—
118	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
120	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₂³×C₅₃	#121	1	1	yes	characteristic
122	C₂³×C₅₃	#121	1	1	yes	characteristic
123	C₂³×C₅₃	#127	1	1	yes	characteristic
124	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
125	C₂³×C₅₃	C₂×SD₁₆	1	4	yes	—
126	C₂³×C₅₃	#135	1	4	yes	—
127	C₂×C₄×C₅₃	—	4	4	no	—
128	C₂×C₄×C₅₃	—	4	8	no	—
129	C₂×C₄×C₅₃	—	8	4	no	—
130	C₂×C₄×C₅₃	—	8	2	no	—
131	C₂×C₄×C₅₃	—	8	2	no	—
132	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
141	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
148	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
149	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
154	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
155	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
156	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
157	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
160	C₂×Dic₅₃	—	2	4	no	—
161	C₂×Dic₅₃	—	2	4	no	—
162	C₂×Dic₅₃	—	2	8	no	—
163	C₂×Dic₅₃	—	2	8	no	—
164	C₂×Dic₅₃	—	4	4	no	—
165	C₂×Dic₅₃	—	4	8	no	—
166	C₂×Dic₅₃	—	8	4	no	—
167	C₂×Dic₅₃	—	8	2	no	—
168	C₂×Dic₅₃	—	8	2	no	—
169	C₂×Dic₅₃	#106	1	8	yes	complemented
170	C₈×C₅₃	#95	1	8	yes	complemented
171	#88322	—	2	4	no	—
172	#10861	—	53	1	no	maximal, Sylow
173	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
174	C₂²×C₅₃	#380	1	1	yes	characteristic
175	C₂²×C₅₃	#384	1	1	yes	characteristic
176	C₂²×C₅₃	#389	1	1	yes	characteristic
177	C₂²×C₅₃	#389	1	1	yes	characteristic
178	C₂²×C₅₃	#390	1	1	yes	characteristic
179	C₂²×C₅₃	#390	1	1	yes	characteristic
180	C₂²×C₅₃	C₂×#106	1	2	yes	—
181	C₂²×C₅₃	#427	1	2	yes	—
182	C₂²×C₅₃	C₄×SD₁₆	1	4	yes	—
183	C₂²×C₅₃	C₄×SD₁₆	1	4	yes	—
184	C₂²×C₅₃	#465	1	4	yes	—
185	C₂²×C₅₃	#465	1	4	yes	—
186	C₂²×C₅₃	#492	1	2	yes	—
187	C₂²×C₅₃	#495	1	2	yes	—
188	C₂²×C₅₃	#498	1	2	yes	—
189	C₂²×C₅₃	#499	1	2	yes	—
190	C₄×C₅₃	—	8	4	no	—
191	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
192	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
193	C₄×C₅₃	#380	1	1	yes	characteristic
194	C₄×C₅₃	#382	1	1	yes	characteristic
195	C₄×C₅₃	#389	1	1	yes	characteristic
196	C₄×C₅₃	#389	1	1	yes	characteristic
197	C₄×C₅₃	#391	1	1	yes	characteristic
198	C₄×C₅₃	#391	1	1	yes	characteristic
199	Dic₅₃	—	2	8	no	—
200	Dic₅₃	—	8	4	no	—
201	C₂²×#106	—	53	1	no	—
202	C₂²×#106	D₅₃	1	1	yes	characteristic
203	C₂²×C₄×C₈	—	53	1	no	—
204	C₂×#340	—	53	2	no	—
205	C₄⋊#95	—	53	2	no	—
206	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
207	C₂×C₅₃	C₄×#106	1	2	yes	—
208	C₂×C₅₃	#1807	1	2	yes	—
209	C₂×C₅₃	#1844	1	1	yes	characteristic
210	C₂×C₅₃	#1846	1	1	yes	characteristic
211	C₂×C₅₃	#1941	1	2	yes	—
212	C₂×C₅₃	#1941	1	2	yes	—
213	C₂×C₅₃	#1968	1	2	yes	—
214	C₂×C₅₃	#1968	1	2	yes	—
215	C₂³×C₈	—	53	1	no	—
216	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
217	C₂²×C₄⋊C₄	—	53	2	no	—
218	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
219	C₂²×C₄²	—	53	1	no	—
220	C₂×#106	—	2	2	no	—
221	C₂×#106	—	2	2	no	—
222	C₂×#106	—	53	4	no	—
223	C₂×#106	—	106	2	no	—
224	C₂×#106	—	106	2	no	—
225	C₂×#106	Dic₅₃	1	4	yes	complemented
226	C₂×C₄×C₈	—	53	4	no	—
227	C₂×C₄×C₈	—	106	2	no	—
228	C₂×C₄×C₈	—	106	2	no	—
229	C₂×#95	—	106	4	no	—
230	#340	—	106	4	no	—
231	C₅₃	#10861	1	1	yes	characteristic, complemented, Sylow
232	C₂³×C₄	—	2	2	no	—
233	C₂³×C₄	—	53	2	no	—
234	C₂³×C₄	—	106	2	no	—
235	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
236	C₂²×C₈	—	53	2	no	—
237	C₂²×C₈	—	53	2	no	—
238	C₂²×C₈	—	53	2	no	—
239	C₂²×C₈	—	106	1	no	—
240	C₂²×C₈	—	106	1	no	—
241	C₂²×C₈	—	106	1	no	—
242	C₂²×C₈	—	106	1	no	—
243	C₂²×C₈	Dic₁₀₆	1	2	yes	—
244	C₂²×C₈	C₄×D₅₃	1	2	yes	—
245	C₂²×C₈	C₄×D₅₃	1	2	yes	—
246	C₂²×C₈	D₂₁₂	1	2	yes	—
247	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
248	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
249	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
250	C₂×C₄⋊C₄	—	2	2	no	—
251	C₂×C₄⋊C₄	—	2	2	no	—
252	C₂×C₄⋊C₄	—	2	2	no	—
253	C₂×C₄⋊C₄	—	2	2	no	—
254	C₂×C₄⋊C₄	—	2	2	no	—
255	C₂×C₄⋊C₄	—	53	4	no	—
256	C₂×C₄⋊C₄	—	106	2	no	—
257	C₂×C₄⋊C₄	—	106	2	no	—
258	C₂×C₄⋊C₄	—	106	2	no	—
259	C₂×C₄⋊C₄	—	106	2	no	—
260	C₂×C₄⋊C₄	—	106	2	no	—
261	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
262	C₂×C₄²	—	53	2	no	—
263	C₂×C₄²	—	53	2	no	—
264	C₂×C₄²	—	106	1	no	—
265	C₂×C₄²	—	106	1	no	—
266	C₂×C₄²	—	106	1	no	—
267	C₂×C₄²	—	106	1	no	—
268	#106	—	2	8	no	—
269	#106	—	106	8	no	—
270	C₄×C₈	—	106	8	no	—
271	#95	—	212	8	no	—
272	C₂⁴	D₄×D₅₃	1	1	yes	characteristic
273	C₂²×C₄	—	2	4	no	—
274	C₂²×C₄	—	4	2	no	—
275	C₂²×C₄	—	4	2	no	—
276	C₂²×C₄	—	4	4	no	—
277	C₂²×C₄	—	4	4	no	—
278	C₂²×C₄	—	53	4	no	—
279	C₂²×C₄	—	53	8	no	—
280	C₂²×C₄	—	106	2	no	—
281	C₂²×C₄	—	106	2	no	—
282	C₂²×C₄	—	106	4	no	—
283	C₂²×C₄	—	106	4	no	—
284	C₂²×C₄	—	212	2	no	—
285	C₂²×C₄	—	212	2	no	—
286	C₂²×C₄	—	212	4	no	—
287	C₂²×C₄	—	212	4	no	—
288	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
289	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
290	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
291	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
292	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
293	C₂²×C₄	#11719947	1	1	yes	characteristic
294	C₂²×C₄	#11719947	1	1	yes	characteristic
295	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
296	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
297	C₂²×C₄	#11719949	1	1	yes	characteristic
298	C₂²×C₄	#11719949	1	1	yes	characteristic
299	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
300	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
301	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
302	C₂×C₈	—	53	4	no	—
303	C₂×C₈	—	106	2	no	—
304	C₂×C₈	—	106	2	no	—
305	C₂×C₈	—	106	2	no	—
306	C₂×C₈	—	106	2	no	—
307	C₂×C₈	—	106	2	no	—
308	C₂×C₈	—	106	2	no	—
309	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
310	C₂×C₈	#11719919	1	4	yes	—
311	C₂×C₈	#11719919	1	4	yes	—
312	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
313	C₂×C₈	#11719921	1	4	yes	—
314	C₂×C₈	#11719921	1	4	yes	—
315	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
316	C₄⋊C₄	—	2	4	no	—
317	C₄⋊C₄	—	2	4	no	—
318	C₄⋊C₄	—	4	2	no	—
319	C₄⋊C₄	—	4	2	no	—
320	C₄⋊C₄	—	106	4	no	—
321	C₄⋊C₄	—	106	4	no	—
322	C₄⋊C₄	—	212	2	no	—
323	C₄⋊C₄	—	212	2	no	—
324	C₄²	—	106	2	no	—
325	C₄²	—	106	2	no	—
326	C₄²	—	106	2	no	—
327	C₄²	—	106	2	no	—
328	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
329	C₂³	#13133179	1	1	yes	characteristic
330	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
331	C₂³	#13133188	1	1	yes	characteristic
332	C₂³	#13133192	1	1	yes	characteristic
333	C₂³	#13133213	1	1	yes	characteristic
334	C₂³	#13133216	1	1	yes	characteristic
335	C₂³	SD₁₆×D₅₃	1	4	yes	—
336	C₂³	#13133238	1	4	yes	—
337	C₂×C₄	—	4	4	no	—
338	C₂×C₄	—	4	8	no	—
339	C₂×C₄	—	8	4	no	—
340	C₂×C₄	—	8	2	no	—
341	C₂×C₄	—	8	2	no	—
342	C₂×C₄	—	53	8	no	—
343	C₂×C₄	—	106	8	no	—
344	C₂×C₄	—	106	8	no	—
345	C₂×C₄	—	106	4	no	—
346	C₂×C₄	—	106	4	no	—
347	C₂×C₄	—	424	4	no	—
348	C₂×C₄	—	424	2	no	—
349	C₂×C₄	—	424	2	no	—
350	C₂×C₄	—	212	4	no	—
351	C₂×C₄	—	212	8	no	—
352	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
353	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
354	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
355	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
356	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
357	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
358	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
359	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
360	C₂×C₄	#13133175	1	1	yes	characteristic
361	C₂×C₄	#13133175	1	1	yes	characteristic
362	C₂×C₄	#13133179	1	1	yes	characteristic
363	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
364	C₂×C₄	#13133182	1	1	yes	characteristic
365	C₂×C₄	#13133182	1	1	yes	characteristic
366	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
367	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
368	C₂×C₄	#13133188	1	1	yes	characteristic
369	C₂×C₄	#13133192	1	1	yes	characteristic
370	C₂×C₄	#13133193	1	1	yes	characteristic
371	C₂×C₄	#13133193	1	1	yes	characteristic
372	C₂×C₄	#13133197	1	1	yes	characteristic
373	C₂×C₄	#13133197	1	1	yes	characteristic
374	C₂×C₄	#13133204	1	1	yes	characteristic
375	C₂×C₄	#13133206	1	1	yes	characteristic
376	C₂×C₄	#13133208	1	1	yes	characteristic
377	C₂×C₄	#13133208	1	1	yes	characteristic
378	C₂×C₄	#13133215	1	1	yes	characteristic
379	C₂×C₄	#13133216	1	1	yes	characteristic
380	C₈	—	106	4	no	—
381	C₈	#13133124	1	8	yes	complemented
382	C₂²	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
383	C₂²	#16810480	1	1	yes	characteristic
384	C₂²	#16810492	1	1	yes	characteristic
385	C₂²	#16810519	1	1	yes	characteristic
386	C₂²	#16810522	1	1	yes	characteristic
387	C₂²	#16810524	1	1	yes	characteristic
388	C₂²	#16810526	1	1	yes	characteristic
389	C₂²	#106×D₅₃	1	2	yes	—
390	C₂²	#16810658	1	2	yes	—
391	C₂²	#16810710	1	4	yes	—
392	C₂²	SD₁₆×Dic₅₃	1	4	yes	—
393	C₂²	#16810849	1	4	yes	—
394	C₂²	#16810853	1	4	yes	—
395	C₂²	#16810994	1	2	yes	—
396	C₂²	#16811004	1	2	yes	—
397	C₂²	#16811020	1	2	yes	—
398	C₂²	#16811023	1	2	yes	—
399	C₄	—	8	4	no	—
400	C₄	—	106	8	no	—
401	C₄	—	424	4	no	—
402	C₄	C₂×#13133124	1	1	yes	characteristic
403	C₄	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
404	C₄	#16810480	1	1	yes	characteristic
405	C₄	#16810488	1	1	yes	characteristic
406	C₄	#16810519	1	1	yes	characteristic
407	C₄	#16810522	1	1	yes	characteristic
408	C₄	#16810528	1	1	yes	characteristic
409	C₄	#16810530	1	1	yes	characteristic
410	C₂	#20272944	1	1	yes	characteristic
411	C₂	#106×Dic₅₃	1	2	yes	—
412	C₂	#20274737	1	2	yes	—
413	C₂	#20274870	1	1	yes	characteristic
414	C₂	#20274876	1	1	yes	characteristic
415	C₂	#20275441	1	2	yes	—
416	C₂	#20275444	1	2	yes	—
417	C₂	#20275567	1	2	yes	—
418	C₂	#20275571	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂²×#106×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂²×#106	D₅₃
hypocenter	C₅₃	#10861
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
11. C₂×#13133149	C₄	semidirect
14. C₂×C₈×Dic₅₃	C₄	semidirect
21. C₂×#106×C₅₃	C₄	semidirect
169. C₂×Dic₅₃	#106	semidirect
170. C₈×C₅₃	#95	semidirect
225. C₂×#106	Dic₅₃	semidirect
231. C₅₃	#10861	semidirect
381. 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×#13133149	2
C₂²×C₈×Dic₅₃	2
C₂×#16810385	2
#20272944	2
C₂²×#106×C₅₃	2
#10861	53