Subgroups — #27353191

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 392 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	#20273500	C₂	1	4	yes	maximal, complemented
4	#20273490	C₂	1	2	yes	maximal, complemented
5	#20273486	C₂	1	2	yes	maximal, complemented
6	#20273467	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic
8	C₂³×#88322	—	2	1	no	—
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	1	yes	characteristic
14	#16810647	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#88322	—	2	1	no	—
16	C₂×C₄⋊#88322	—	2	1	no	—
17	C₂×C₄⋊#88322	—	2	1	no	—
18	C₂×C₄⋊#88322	C₂²	1	2	yes	—
19	C₂×C₄⋊#88322	C₂²	1	2	yes	—
20	C₂×C₄⋊#88322	C₂²	1	2	yes	—
21	#16810587	—	2	1	no	—
22	C₄×#11719917	—	2	1	no	—
23	#16810376	C₂²	1	8	yes	complemented
24	#16810374	C₂²	1	8	yes	complemented
25	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#432	C₄	1	2	yes	—
27	C₅₃×#432	C₄	1	2	yes	—
28	C₅₃×#431	C₂²	1	2	yes	—
29	C₅₃×#431	C₂²	1	2	yes	—
30	C₅₃×#423	C₄	1	2	yes	—
31	C₅₃×#407	C₄	1	2	yes	—
32	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
33	C₂×#11719917	—	2	1	no	—
34	C₂×#11719917	—	2	1	no	—
35	C₂²×#88322	—	2	2	no	—
36	C₂²×#88322	—	2	2	no	—
37	C₂²×#88322	—	2	2	no	—
38	C₂²×#88322	—	4	4	no	—
39	C₄⋊#88322	—	2	2	no	—
40	C₄⋊#88322	—	2	2	no	—
41	C₄⋊#88322	—	2	2	no	—
42	C₄⋊#88322	—	2	1	no	—
43	C₄⋊#88322	—	2	1	no	—
44	C₄⋊#88322	C₂³	1	2	yes	—
45	C₄⋊#88322	C₂³	1	4	yes	—
46	C₄⋊#88322	C₂³	1	2	yes	—
47	C₂²⋊#88322	—	2	2	no	—
48	C₂²⋊#88322	—	2	2	no	—
49	#13133128	—	2	2	no	—
50	C₄×#88322	—	2	2	no	—
51	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
54	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
55	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
63	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
68	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
69	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
70	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
71	#97×C₅₃	C₂×C₄	1	4	yes	—
72	C₄×C₈×C₅₃	C₂³	1	4	yes	—
73	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	—	2	4	no	—
75	C₂²×C₄×C₅₃	—	2	2	no	—
76	C₂²×C₄×C₅₃	—	2	2	no	—
77	C₂²×C₄×C₅₃	—	4	2	no	—
78	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
96	M₁₆×C₅₃	—	2	8	no	—
97	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
98	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
111	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
112	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
113	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
114	#11719917	—	4	2	no	—
115	#11719917	—	4	2	no	—
116	C₂×#88322	—	2	2	no	—
117	C₂×#88322	—	2	2	no	—
118	C₂×#88322	—	2	4	no	—
119	C₂×#88322	—	4	4	no	—
120	C₂×#88322	—	4	4	no	—
121	C₂×#88322	—	4	4	no	—
122	C₂³×C₅₃	—	2	2	no	—
123	C₂³×C₅₃	—	2	2	no	—
124	C₂³×C₅₃	—	2	2	no	—
125	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	—	2	2	no	—
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	1	no	—
135	C₂×C₄×C₅₃	—	2	1	no	—
136	C₂×C₄×C₅₃	—	2	1	no	—
137	C₂×C₄×C₅₃	—	2	1	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
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	—	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×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	#131	1	4	yes	—
156	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
157	C₂×C₄×C₅₃	#137	1	2	yes	—
158	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
161	C₈×C₅₃	—	2	8	no	—
162	#88322	—	4	4	no	—
163	#88322	—	4	2	no	—
164	#88322	—	4	2	no	—
165	#17807	—	53	1	no	maximal, Sylow
166	C₂²×C₅₃	—	2	8	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	—	2	1	no	—
173	C₂²×C₅₃	#424	1	2	yes	—
174	C₂²×C₅₃	#428	1	2	yes	—
175	C₂²×C₅₃	#430	1	2	yes	—
176	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
184	C₄×C₅₃	—	2	1	no	—
185	C₄×C₅₃	#424	1	2	yes	—
186	C₄×C₅₃	#428	1	2	yes	—
187	C₄×C₅₃	C₂×#131	1	2	yes	—
188	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
189	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
190	#2920	—	53	1	no	—
191	#2920	D₅₃	1	1	yes	characteristic
192	C₂²×C₄⋊C₈	—	53	1	no	—
193	#1573	—	53	4	no	—
194	#1571	—	53	2	no	—
195	#1570	—	53	2	no	—
196	#1565	—	53	4	no	—
197	C₂×C₅₃	—	2	2	no	—
198	C₂×C₅₃	—	2	2	no	—
199	C₂×C₅₃	—	2	2	no	—
200	C₂×C₅₃	C₂×#424	1	1	yes	characteristic
201	C₂×C₅₃	C₂×#428	1	1	yes	characteristic
202	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
203	C₂³×C₈	—	106	1	no	—
204	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
205	#432	—	53	2	no	—
206	#432	—	106	1	no	—
207	#432	Dic₅₃	1	2	yes	—
208	#432	Dic₅₃	1	2	yes	—
209	#431	—	53	2	no	—
210	#431	—	106	1	no	—
211	#431	C₂×D₅₃	1	2	yes	—
212	#431	C₂×D₅₃	1	2	yes	—
213	#423	—	53	1	no	—
214	#423	—	53	1	no	—
215	#423	Dic₅₃	1	2	yes	—
216	C₂×C₄⋊C₈	—	53	2	no	—
217	C₂×C₄⋊C₈	—	53	2	no	—
218	C₂×C₄⋊C₈	—	53	2	no	—
219	C₂×C₄⋊C₈	—	106	1	no	—
220	C₂×C₄⋊C₈	—	106	1	no	—
221	C₂×C₄⋊C₈	—	106	1	no	—
222	#407	—	106	1	no	—
223	#407	Dic₅₃	1	2	yes	—
224	C₄×M₁₆	—	106	1	no	—
225	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
226	#335	—	53	8	no	—
227	#334	—	53	8	no	—
228	C₅₃	#17807	1	1	yes	characteristic, complemented, Sylow
229	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
230	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
231	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
232	C₂×M₁₆	—	106	1	no	—
233	C₂×M₁₆	—	106	1	no	—
234	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
235	C₂²×C₈	—	106	2	no	—
236	C₂²×C₈	—	106	2	no	—
237	C₂²×C₈	—	106	2	no	—
238	C₂²×C₈	—	212	4	no	—
239	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
240	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
241	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
242	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
243	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
244	C₂×C₄²	D₂₁₂	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	C₂×Dic₅₃	1	1	yes	characteristic
248	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
249	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
250	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
251	C₄⋊C₈	—	53	2	no	—
252	C₄⋊C₈	—	53	4	no	—
253	C₄⋊C₈	—	53	2	no	—
254	C₄⋊C₈	—	106	2	no	—
255	C₄⋊C₈	—	106	2	no	—
256	C₄⋊C₈	—	106	2	no	—
257	C₄⋊C₈	—	106	1	no	—
258	C₄⋊C₈	—	106	1	no	—
259	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
260	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
261	C₂²⋊C₈	—	106	2	no	—
262	C₂²⋊C₈	—	106	2	no	—
263	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
264	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
265	#97	—	106	2	no	—
266	#97	C₂×Dic₅₃	1	4	yes	—
267	C₄×C₈	—	106	2	no	—
268	C₄×C₈	C₂²×D₅₃	1	4	yes	—
269	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
270	C₂²×C₄	—	2	4	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	4	2	no	—
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	C₄⋊Dic₅₃	1	1	yes	characteristic
282	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
283	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
284	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
285	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
287	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
288	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
289	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
290	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
291	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
292	M₁₆	—	2	8	no	—
293	M₁₆	—	212	2	no	—
294	M₁₆	—	212	2	no	—
295	C₂×C₈	—	106	2	no	—
296	C₂×C₈	—	106	2	no	—
297	C₂×C₈	—	106	4	no	—
298	C₂×C₈	—	212	4	no	—
299	C₂×C₈	—	212	4	no	—
300	C₂×C₈	—	212	4	no	—
301	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
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	C₄⋊Dic₅₃	1	1	yes	characteristic
307	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
308	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
309	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
310	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
311	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
312	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
313	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
314	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
315	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
316	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
317	C₄²	C₂³×D₅₃	1	1	yes	characteristic
318	C₂³	—	2	2	no	—
319	C₂³	—	2	2	no	—
320	C₂³	—	2	2	no	—
321	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
322	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
323	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
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	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
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	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
349	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
350	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
351	C₂×C₄	#13133224	1	4	yes	—
352	C₂×C₄	#13133245	1	2	yes	—
353	C₂×C₄	#13133250	1	2	yes	—
354	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
356	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
357	C₈	—	2	8	no	—
358	C₈	—	212	4	no	—
359	C₈	—	212	2	no	—
360	C₈	—	212	2	no	—
361	C₂²	—	2	8	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	#16810650	1	2	yes	—
369	C₂²	#16810665	1	2	yes	—
370	C₂²	#16810672	1	2	yes	—
371	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	#16810650	1	2	yes	—
381	C₄	#16810665	1	2	yes	—
382	C₄	C₂×#13133224	1	2	yes	—
383	C₄	C₂×#13133245	1	1	yes	characteristic
384	C₄	C₂×#13133250	1	1	yes	characteristic
385	C₂	—	2	2	no	—
386	C₂	—	2	2	no	—
387	C₂	—	2	2	no	—
388	C₂	C₂×#16810650	1	1	yes	characteristic
389	C₂	C₂×#16810665	1	1	yes	characteristic
390	C₂	C₂×#16810672	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
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	#17807
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
3. #20273500	C₂	semidirect
4. #20273490	C₂	semidirect
5. #20273486	C₂	semidirect
6. #20273467	C₂	semidirect
23. #16810376	C₂²	semidirect
24. #16810374	C₂²	semidirect
228. C₅₃	#17807	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
#20273500	2
#20273490	2
#20273486	2
#20273467	2
C₅₃×#2920	2
#17807	53