Subgroups — #27349809

Order 13568.

806 conjugacy classes of subgroups.

The subgroup lattice has 386 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	C₂×C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	#20273018	C₂	1	4	yes	maximal, complemented
4	#20273010	C₂	1	4	yes	maximal, complemented
5	#20273008	C₂	1	2	yes	maximal, complemented
6	#20272999	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11719917	C₄	1	2	yes	—
9	#16810680	C₄	1	2	yes	—
10	#16810680	C₄	1	2	yes	—
11	#16810676	C₂²	1	2	yes	—
12	#16810676	C₂²	1	2	yes	—
13	#16810647	C₄	1	2	yes	—
14	#16810587	C₄	1	2	yes	—
15	C₄×#11719917	C₄	1	2	yes	—
16	C₄×#11719917	C₄	1	2	yes	—
17	C₄×#11719917	C₂²	1	2	yes	—
18	C₄×#11719917	C₂²	1	2	yes	—
19	C₄×#11719917	C₂²	1	2	yes	—
20	C₂×#13133128	C₄	1	2	yes	—
21	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
22	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
23	#16810345	C₂²	1	4	yes	complemented
24	#16810345	C₂²	1	4	yes	complemented
25	#16810344	C₂²	1	4	yes	complemented
26	C₈×#88322	C₂²	1	4	yes	complemented
27	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#432	C₄	1	2	yes	—
29	C₅₃×#432	C₄	1	2	yes	—
30	C₅₃×#431	C₂²	1	2	yes	—
31	C₅₃×#431	C₂²	1	2	yes	—
32	C₅₃×#423	C₄	1	2	yes	—
33	C₅₃×#407	C₄	1	2	yes	—
34	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
35	C₂×#11719917	—	2	8	no	—
36	C₂×#11719917	C₂×C₄	1	4	yes	—
37	C₂×#11719917	C₂×C₄	1	4	yes	—
38	C₂×#11719917	C₂×C₄	1	4	yes	—
39	C₂²×#88322	C₂×C₄	1	2	yes	—
40	C₂²×#88322	C₂×C₄	1	2	yes	—
41	C₄⋊#88322	C₂×C₄	1	4	yes	—
42	C₄⋊#88322	C₂×C₄	1	4	yes	—
43	C₂²⋊#88322	C₂×C₄	1	4	yes	—
44	C₂²⋊#88322	C₂×C₄	1	4	yes	—
45	#13133128	C₂×C₄	1	4	yes	—
46	#13133128	C₂×C₄	1	4	yes	—
47	#13133128	C₂×C₄	1	4	yes	—
48	C₄×#88322	C₂×C₄	1	2	yes	—
49	C₄×#88322	C₂×C₄	1	2	yes	—
50	C₄×#88322	C₂³	1	2	yes	—
51	C₄×#88322	C₂³	1	2	yes	—
52	C₄×#88322	C₂³	1	4	yes	—
53	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
57	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
70	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
71	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
72	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
73	#97×C₅₃	C₂×C₄	1	4	yes	—
74	C₄×C₈×C₅₃	C₂³	1	4	yes	—
75	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	—	2	4	no	—
77	C₂²×C₄×C₅₃	—	4	2	no	—
78	C₂²×C₄×C₅₃	—	4	2	no	—
79	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
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	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	M₁₆×C₅₃	—	4	4	no	—
98	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
99	C₄²×C₅₃	C₄²	1	1	yes	characteristic
100	C₄²×C₅₃	C₄²	1	1	yes	characteristic
101	C₄²×C₅₃	C₄²	1	1	yes	characteristic
102	C₄²×C₅₃	C₄²	1	1	yes	characteristic
103	C₄²×C₅₃	C₄²	1	1	yes	characteristic
104	C₄²×C₅₃	C₄²	1	1	yes	characteristic
105	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₂²×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	C₂⁴	1	1	yes	characteristic
115	#11719917	—	2	8	no	—
116	#11719917	—	2	8	no	—
117	#11719917	—	4	4	no	—
118	C₂×#88322	—	2	4	no	—
119	C₂×#88322	M₁₆	1	8	yes	complemented
120	C₂×#88322	C₂²×C₄	1	4	yes	—
121	C₂×#88322	C₂²×C₄	1	8	yes	—
122	C₂×#88322	C₂²×C₄	1	4	yes	—
123	C₂³×C₅₃	—	2	2	no	—
124	C₂³×C₅₃	—	4	1	no	—
125	C₂³×C₅₃	—	4	1	no	—
126	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
127	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
128	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	4	1	no	—
141	C₂×C₄×C₅₃	—	4	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×C₄²	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
156	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
159	C₈×C₅₃	—	2	8	no	—
160	#88322	—	2	4	no	—
161	#88322	—	2	8	no	—
162	#88322	C₂×M₁₆	1	8	yes	—
163	#17334	—	53	1	no	maximal, Sylow
164	C₂²×C₅₃	—	2	1	no	—
165	C₂²×C₅₃	—	2	1	no	—
166	C₂²×C₅₃	—	2	1	no	—
167	C₂²×C₅₃	—	2	1	no	—
168	C₂²×C₅₃	—	2	1	no	—
169	C₂²×C₅₃	—	2	1	no	—
170	C₂²×C₅₃	—	2	1	no	—
171	C₂²×C₅₃	—	2	1	no	—
172	C₂²×C₅₃	—	4	4	no	—
173	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
174	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
175	C₄×C₅₃	—	2	1	no	—
176	C₄×C₅₃	—	2	1	no	—
177	C₄×C₅₃	—	2	1	no	—
178	C₄×C₅₃	—	2	1	no	—
179	C₄×C₅₃	—	2	1	no	—
180	C₄×C₅₃	—	2	1	no	—
181	C₄×C₅₃	—	2	1	no	—
182	C₄×C₅₃	—	2	1	no	—
183	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
184	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
185	C₄×C₅₃	#568	1	2	yes	—
186	C₄×C₅₃	#568	1	2	yes	—
187	#2920	—	53	1	no	—
188	#2920	D₅₃	1	1	yes	characteristic
189	C₂×C₄×M₁₆	—	53	1	no	—
190	#1455	—	53	4	no	—
191	#1453	—	53	4	no	—
192	#1452	—	53	2	no	—
193	#1450	—	53	2	no	—
194	C₂×C₅₃	—	2	2	no	—
195	C₂×C₅₃	—	4	1	no	—
196	C₂×C₅₃	—	4	1	no	—
197	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
198	C₂×C₅₃	#2873	1	1	yes	characteristic
199	C₂×C₅₃	#2873	1	1	yes	characteristic
200	C₂²×M₁₆	—	53	2	no	—
201	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
202	#432	—	53	2	no	—
203	#432	—	53	2	no	—
204	#432	Dic₅₃	1	2	yes	—
205	#432	Dic₅₃	1	2	yes	—
206	#431	—	53	2	no	—
207	#431	—	53	2	no	—
208	#431	C₂×D₅₃	1	2	yes	—
209	#431	C₂×D₅₃	1	2	yes	—
210	#423	—	53	2	no	—
211	#423	Dic₅₃	1	2	yes	—
212	#407	—	53	2	no	—
213	#407	Dic₅₃	1	2	yes	—
214	C₄×M₁₆	—	53	2	no	—
215	C₄×M₁₆	—	53	2	no	—
216	C₄×M₁₆	—	53	2	no	—
217	C₄×M₁₆	—	53	2	no	—
218	C₄×M₁₆	—	53	2	no	—
219	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
220	C₂×#97	—	53	2	no	—
221	C₂×C₄×C₈	—	53	1	no	—
222	C₂×C₄×C₈	—	53	1	no	—
223	#322	—	53	4	no	—
224	#322	—	53	4	no	—
225	#322	—	53	4	no	—
226	C₈²	—	53	4	no	—
227	C₅₃	#17334	1	1	yes	characteristic, complemented, Sylow
228	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
229	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
230	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
231	C₂×M₁₆	—	53	4	no	—
232	C₂×M₁₆	—	53	4	no	—
233	C₂×M₁₆	—	53	4	no	—
234	C₂×M₁₆	—	106	8	no	—
235	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
236	C₂²×C₈	—	53	2	no	—
237	C₂²×C₈	—	53	2	no	—
238	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
239	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
240	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
241	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
242	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
243	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
244	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
245	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
246	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
247	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
248	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
249	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
250	C₄⋊C₈	—	53	4	no	—
251	C₄⋊C₈	—	53	4	no	—
252	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
253	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
254	C₂²⋊C₈	—	53	4	no	—
255	C₂²⋊C₈	—	53	4	no	—
256	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
257	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
258	#97	—	53	4	no	—
259	#97	—	53	4	no	—
260	#97	—	53	4	no	—
261	#97	C₂×Dic₅₃	1	4	yes	—
262	C₄×C₈	—	53	2	no	—
263	C₄×C₈	—	53	2	no	—
264	C₄×C₈	—	53	2	no	—
265	C₄×C₈	—	53	4	no	—
266	C₄×C₈	—	53	2	no	—
267	C₄×C₈	C₂²×D₅₃	1	4	yes	—
268	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
269	C₂²×C₄	—	2	4	no	—
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	—	4	2	no	—
272	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
273	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
274	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
275	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
276	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
277	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
278	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
279	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
280	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
281	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
282	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
283	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
284	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
285	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
286	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
287	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
288	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
289	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
290	M₁₆	—	4	4	no	—
291	M₁₆	—	106	8	no	—
292	M₁₆	—	106	8	no	—
293	M₁₆	—	212	4	no	—
294	C₂×C₈	—	53	8	no	—
295	C₂×C₈	—	53	4	no	—
296	C₂×C₈	—	53	4	no	—
297	C₂×C₈	—	53	8	no	—
298	C₂×C₈	—	106	4	no	—
299	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
300	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
301	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
302	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
303	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
304	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
305	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
306	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
307	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
308	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
309	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
310	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
311	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
312	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
313	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
314	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
315	C₄²	C₂³×D₅₃	1	1	yes	characteristic
316	C₂³	—	2	2	no	—
317	C₂³	—	4	1	no	—
318	C₂³	—	4	1	no	—
319	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
320	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
321	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
340	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
341	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
342	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
343	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
344	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
345	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
346	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
347	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
348	C₂×C₄	M₁₆×D₅₃	1	4	yes	—
349	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
350	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
351	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
352	C₈	—	2	8	no	—
353	C₈	—	53	8	no	—
354	C₈	—	106	4	no	—
355	C₈	—	106	8	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	4	4	no	—
365	C₂²	M₁₆×Dic₅₃	1	2	yes	—
366	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	M₁₆×Dic₅₃	1	2	yes	—
376	C₄	C₂×M₁₆×D₅₃	1	2	yes	—
377	C₄	#16811389	1	2	yes	—
378	C₄	#16811390	1	2	yes	—
379	C₂	—	2	2	no	—
380	C₂	—	4	1	no	—
381	C₂	—	4	1	no	—
382	C₂	C₂×M₁₆×Dic₅₃	1	1	yes	characteristic
383	C₂	#20283133	1	1	yes	characteristic
384	C₂	#20283134	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₅₃×#2920	C₂
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#2920	D₅₃
hypocenter	C₅₃	#17334
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
3. #20273018	C₂	semidirect
4. #20273010	C₂	semidirect
5. #20273008	C₂	semidirect
6. #20272999	C₂	semidirect
23. #16810345	C₂²	semidirect
24. #16810345	C₂²	semidirect
25. #16810344	C₂²	semidirect
26. C₈×#88322	C₂²	semidirect
119. C₂×#88322	M₁₆	semidirect
227. C₅₃	#17334	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
C₂×C₄×#11719917	2
#20273018	2
#20273010	2
#20273008	2
#20272999	2
C₅₃×#2920	2
#17334	53