Subgroups — #27353103

Order 13568.

762 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283348	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	#20273476	C₂	1	4	yes	maximal, complemented
4	#20273473	C₂	1	2	yes	maximal, complemented
5	#20273470	C₂	1	2	yes	maximal, complemented
6	#20273461	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic
8	C₂³×#88322	C₄	1	2	yes	—
9	#16810680	—	2	1	no	—
10	#16810680	C₂²	1	2	yes	—
11	#16810676	—	2	1	no	—
12	#16810676	C₂²	1	2	yes	—
13	#16810647	C₂²	1	2	yes	—
14	#16810587	—	2	1	no	—
15	C₄×#11719917	—	2	1	no	—
16	C₂×C₄×#88322	C₄	1	2	yes	—
17	C₂×C₄×#88322	C₄	1	1	yes	characteristic
18	C₂×C₄×#88322	C₄	1	1	yes	characteristic
19	C₂×C₄×#88322	C₄	1	1	yes	characteristic
20	C₂×C₄×#88322	C₄	1	1	yes	characteristic
21	C₂×C₄×#88322	C₂²	1	2	yes	—
22	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
23	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
24	C₂×C₄×#88322	C₂²	1	2	yes	—
25	#16810375	C₂²	1	8	yes	complemented
26	#16810373	C₂²	1	8	yes	complemented
27	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#432	—	2	1	no	—
29	C₅₃×#432	C₂²	1	2	yes	—
30	C₅₃×#431	—	2	1	no	—
31	C₅₃×#431	C₂²	1	2	yes	—
32	C₅₃×#423	C₂²	1	2	yes	—
33	C₅₃×#407	—	2	1	no	—
34	C₄×M₁₆×C₅₃	—	2	1	no	—
35	C₂×#11719917	—	2	2	no	—
36	C₂²×#88322	—	2	8	no	—
37	C₂²×#88322	C₂×C₄	1	2	yes	—
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	C₄⋊#88322	—	2	2	no	—
44	C₄⋊#88322	C₂³	1	4	yes	—
45	C₂²⋊#88322	—	2	2	no	—
46	C₂²⋊#88322	—	2	2	no	—
47	#13133128	—	2	2	no	—
48	C₄×#88322	—	2	2	no	—
49	C₄×#88322	C₂×C₄	1	1	yes	characteristic
50	C₄×#88322	C₂×C₄	1	1	yes	characteristic
51	C₄×#88322	C₂×C₄	1	2	yes	—
52	C₄×#88322	C₂×C₄	1	2	yes	—
53	C₄×#88322	C₂×C₄	1	2	yes	—
54	C₄×#88322	C₂×C₄	1	1	yes	characteristic
55	C₄×#88322	C₂×C₄	1	1	yes	characteristic
56	C₄×#88322	C₂×C₄	1	2	yes	—
57	C₄×#88322	C₂³	1	2	yes	—
58	C₄×#88322	C₂³	1	2	yes	—
59	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
61	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
62	C₂×M₁₆×C₅₃	—	2	2	no	—
63	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
70	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
71	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
72	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₅₃	—	2	2	no	—
76	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
77	C₂²⋊C₈×C₅₃	—	2	2	no	—
78	C₂²⋊C₈×C₅₃	—	2	2	no	—
79	#97×C₅₃	—	2	2	no	—
80	C₄×C₈×C₅₃	—	2	2	no	—
81	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	—	2	4	no	—
83	C₂²×C₄×C₅₃	—	2	4	no	—
84	C₂²×C₄×C₅₃	—	2	4	no	—
85	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
103	M₁₆×C₅₃	—	4	4	no	—
104	C₂×C₈×C₅₃	—	2	4	no	—
105	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
118	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
119	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
120	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
121	#11719917	—	4	4	no	—
122	C₂×#88322	—	2	4	no	—
123	C₂×#88322	—	2	4	no	—
124	C₂×#88322	—	2	4	no	—
125	C₂×#88322	—	2	4	no	—
126	C₂×#88322	—	2	4	no	—
127	C₂×#88322	—	2	4	no	—
128	C₂×#88322	M₁₆	1	8	yes	complemented
129	C₂×#88322	C₂²×C₄	1	2	yes	—
130	C₂×#88322	C₂²×C₄	1	2	yes	—
131	C₂×#88322	C₂²×C₄	1	2	yes	—
132	C₂×#88322	C₂²×C₄	1	2	yes	—
133	C₂³×C₅₃	—	2	2	no	—
134	C₂³×C₅₃	—	2	2	no	—
135	C₂³×C₅₃	—	2	2	no	—
136	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	2	no	—
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	2	2	no	—
148	C₂×C₄×C₅₃	—	2	2	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	—	2	2	no	—
152	C₂×C₄×C₅₃	—	2	2	no	—
153	C₂×C₄×C₅₃	—	2	2	no	—
154	C₂×C₄×C₅₃	#108	1	4	yes	—
155	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
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×M₁₆	1	4	yes	—
165	C₂×C₄×C₅₃	#135	1	4	yes	—
166	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
169	C₈×C₅₃	—	4	4	no	—
170	#88322	—	2	2	no	—
171	#88322	—	2	2	no	—
172	#88322	—	4	4	no	—
173	#88322	C₂×M₁₆	1	4	yes	—
174	#88322	C₂×M₁₆	1	4	yes	—
175	#17793	—	53	1	no	maximal, Sylow
176	C₂²×C₅₃	—	2	8	no	—
177	C₂²×C₅₃	—	2	1	no	—
178	C₂²×C₅₃	—	2	1	no	—
179	C₂²×C₅₃	—	2	1	no	—
180	C₂²×C₅₃	—	2	1	no	—
181	C₂²×C₅₃	—	2	1	no	—
182	C₂²×C₅₃	—	2	1	no	—
183	C₂²×C₅₃	#423	1	2	yes	—
184	C₂²×C₅₃	#427	1	2	yes	—
185	C₂²×C₅₃	C₂×#108	1	2	yes	—
186	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
187	C₄×C₅₃	—	2	1	no	—
188	C₄×C₅₃	—	2	1	no	—
189	C₄×C₅₃	—	2	1	no	—
190	C₄×C₅₃	—	2	1	no	—
191	C₄×C₅₃	—	2	1	no	—
192	C₄×C₅₃	—	2	1	no	—
193	C₄×C₅₃	#423	1	2	yes	—
194	C₄×C₅₃	#427	1	2	yes	—
195	C₄×C₅₃	C₂×#108	1	2	yes	—
196	C₄×C₅₃	C₂×#108	1	2	yes	—
197	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
198	C₄×C₅₃	C₂×#135	1	2	yes	—
199	#2920	—	53	1	no	—
200	#2920	D₅₃	1	1	yes	characteristic
201	C₂²×C₄×C₈	—	53	1	no	—
202	#1568	—	53	4	no	—
203	#1567	—	53	2	no	—
204	#1566	—	53	2	no	—
205	#1564	—	53	4	no	—
206	C₂×C₅₃	—	2	2	no	—
207	C₂×C₅₃	—	2	2	no	—
208	C₂×C₅₃	—	2	2	no	—
209	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
210	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
211	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
212	C₂³×C₈	—	53	2	no	—
213	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
214	#432	—	2	1	no	—
215	#432	—	53	2	no	—
216	#432	—	106	1	no	—
217	#432	C₂×D₅₃	1	2	yes	—
218	#431	—	2	1	no	—
219	#431	—	53	2	no	—
220	#431	—	106	1	no	—
221	#431	C₂×D₅₃	1	2	yes	—
222	#423	—	53	2	no	—
223	#423	C₂×D₅₃	1	2	yes	—
224	#407	—	2	1	no	—
225	#407	—	106	1	no	—
226	C₄×M₁₆	—	2	1	no	—
227	C₄×M₁₆	—	106	1	no	—
228	C₂×C₄×C₈	—	53	1	no	—
229	C₂×C₄×C₈	—	53	1	no	—
230	C₂×C₄×C₈	—	53	1	no	—
231	C₂×C₄×C₈	—	53	2	no	—
232	C₂×C₄×C₈	—	53	1	no	—
233	C₂×C₄×C₈	—	53	2	no	—
234	C₂×C₄×C₈	—	53	1	no	—
235	C₂×C₄×C₈	—	53	1	no	—
236	C₂×C₄×C₈	—	53	2	no	—
237	#335	—	53	8	no	—
238	#334	—	53	8	no	—
239	C₅₃	#17793	1	1	yes	characteristic, complemented, Sylow
240	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
241	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
242	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
243	C₂×M₁₆	—	2	2	no	—
244	C₂×M₁₆	—	106	2	no	—
245	C₂²×C₈	—	53	2	no	—
246	C₂²×C₈	—	53	2	no	—
247	C₂²×C₈	—	53	2	no	—
248	C₂²×C₈	—	53	2	no	—
249	C₂²×C₈	—	53	2	no	—
250	C₂²×C₈	—	53	2	no	—
251	C₂²×C₈	—	106	8	no	—
252	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
253	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
254	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
255	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
256	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
257	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
258	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
259	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
260	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
261	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
262	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
263	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
264	C₄⋊C₈	—	2	2	no	—
265	C₄⋊C₈	—	53	4	no	—
266	C₄⋊C₈	—	106	2	no	—
267	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
268	C₂²⋊C₈	—	2	2	no	—
269	C₂²⋊C₈	—	2	2	no	—
270	C₂²⋊C₈	—	106	2	no	—
271	C₂²⋊C₈	—	106	2	no	—
272	#97	—	2	2	no	—
273	#97	—	106	2	no	—
274	C₄×C₈	—	2	2	no	—
275	C₄×C₈	—	53	1	no	—
276	C₄×C₈	—	53	1	no	—
277	C₄×C₈	—	53	2	no	—
278	C₄×C₈	—	53	2	no	—
279	C₄×C₈	—	53	2	no	—
280	C₄×C₈	—	53	1	no	—
281	C₄×C₈	—	53	1	no	—
282	C₄×C₈	—	53	2	no	—
283	C₄×C₈	—	53	2	no	—
284	C₄×C₈	—	53	2	no	—
285	C₄×C₈	—	106	2	no	—
286	C₂⁴	#11719919	1	1	yes	characteristic
287	C₂²×C₄	—	2	4	no	—
288	C₂²×C₄	—	2	4	no	—
289	C₂²×C₄	—	2	4	no	—
290	C₂²×C₄	#11719919	1	1	yes	characteristic
291	C₂²×C₄	#11719919	1	1	yes	characteristic
292	C₂²×C₄	#11719919	1	1	yes	characteristic
293	C₂²×C₄	#11719919	1	1	yes	characteristic
294	C₂²×C₄	#11719919	1	1	yes	characteristic
295	C₂²×C₄	#11719919	1	1	yes	characteristic
296	C₂²×C₄	#11719919	1	1	yes	characteristic
297	C₂²×C₄	#11719919	1	1	yes	characteristic
298	C₂²×C₄	#11719919	1	1	yes	characteristic
299	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
300	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
301	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
302	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
303	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
304	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
305	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
306	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
307	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
308	M₁₆	—	4	4	no	—
309	M₁₆	—	212	4	no	—
310	C₂×C₈	—	2	4	no	—
311	C₂×C₈	—	53	8	no	—
312	C₂×C₈	—	53	2	no	—
313	C₂×C₈	—	53	2	no	—
314	C₂×C₈	—	53	2	no	—
315	C₂×C₈	—	53	2	no	—
316	C₂×C₈	—	106	4	no	—
317	C₂×C₈	—	106	4	no	—
318	C₂×C₈	—	106	4	no	—
319	C₂×C₈	—	106	4	no	—
320	C₂×C₈	—	106	4	no	—
321	C₂×C₈	—	106	4	no	—
322	C₄²	#11719919	1	1	yes	characteristic
323	C₄²	#11719919	1	1	yes	characteristic
324	C₄²	#11719919	1	1	yes	characteristic
325	C₄²	#11719919	1	1	yes	characteristic
326	C₄²	#11719919	1	1	yes	characteristic
327	C₄²	#11719919	1	1	yes	characteristic
328	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
329	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
330	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
331	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
332	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
333	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
334	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
335	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
336	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
337	C₄²	C₂³×D₅₃	1	1	yes	characteristic
338	C₂³	—	2	2	no	—
339	C₂³	—	2	2	no	—
340	C₂³	—	2	2	no	—
341	C₂³	C₂×#11719919	1	1	yes	characteristic
342	C₂³	C₂×#11719919	1	1	yes	characteristic
343	C₂³	C₂×#11719919	1	1	yes	characteristic
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	#13133152	1	4	yes	—
360	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
361	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
362	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
363	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
364	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
365	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
366	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
367	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
368	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
369	C₂×C₄	M₁₆×D₅₃	1	4	yes	—
370	C₂×C₄	#13133237	1	4	yes	—
371	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
372	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
373	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
374	C₈	—	4	4	no	—
375	C₈	—	53	4	no	—
376	C₈	—	53	4	no	—
377	C₈	—	106	2	no	—
378	C₈	—	106	2	no	—
379	C₈	—	212	4	no	—
380	C₂²	—	2	8	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	2	1	no	—
386	C₂²	—	2	1	no	—
387	C₂²	#16810646	1	2	yes	—
388	C₂²	#16810659	1	2	yes	—
389	C₂²	C₂×#13133152	1	2	yes	—
390	C₂²	C₂²×#11719919	1	1	yes	characteristic
391	C₄	—	2	1	no	—
392	C₄	—	2	1	no	—
393	C₄	—	2	1	no	—
394	C₄	—	2	1	no	—
395	C₄	—	2	1	no	—
396	C₄	—	2	1	no	—
397	C₄	#16810646	1	2	yes	—
398	C₄	#16810659	1	2	yes	—
399	C₄	C₂×#13133152	1	2	yes	—
400	C₄	C₂×#13133152	1	2	yes	—
401	C₄	C₂×M₁₆×D₅₃	1	2	yes	—
402	C₄	C₂×#13133237	1	2	yes	—
403	C₂	—	2	2	no	—
404	C₂	—	2	2	no	—
405	C₂	—	2	2	no	—
406	C₂	C₂×#16810646	1	1	yes	characteristic
407	C₂	C₂×#16810659	1	1	yes	characteristic
408	C₂	C₂²×#13133152	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#2920	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#2920	D₅₃
hypocenter	C₅₃	#17793
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
3. #20273476	C₂	semidirect
4. #20273473	C₂	semidirect
5. #20273470	C₂	semidirect
6. #20273461	C₂	semidirect
25. #16810375	C₂²	semidirect
26. #16810373	C₂²	semidirect
128. C₂×#88322	M₁₆	semidirect
239. C₅₃	#17793	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
C₂²×C₄×#88322	2
#20273476	2
#20273473	2
#20273470	2
#20273461	2
C₅₃×#2920	2
#17793	53