Subgroups — C₅₃×#18093 #27225605

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3168	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2994	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1594	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1593	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1583	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1582	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1580	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1579	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1483	C₂	1	2	yes	maximal, complemented
11	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#529	C₂²	1	2	yes	—
13	C₅₃×#499	C₄	1	1	yes	characteristic
14	C₅₃×#495	C₄	1	1	yes	characteristic
15	C₅₃×#488	C₄	1	2	yes	—
16	C₅₃×#467	C₄	1	1	yes	characteristic
17	C₅₃×#466	C₄	1	1	yes	characteristic
18	C₅₃×#465	C₄	1	2	yes	—
19	C₅₃×#446	C₂²	1	2	yes	—
20	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
21	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
23	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#436	C₂²	1	2	yes	—
25	C₅₃×#435	—	2	2	no	—
26	C₈×D₄×C₅₃	—	2	2	no	—
27	C₈×D₄×C₅₃	C₂²	1	2	yes	—
28	C₅₃×#433	—	2	2	no	—
29	C₅₃×#431	C₂²	1	1	yes	characteristic
30	C₅₃×#424	—	2	2	no	—
31	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
32	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#334	C₂²	1	1	yes	characteristic
35	C₅₃×#334	C₂²	1	2	yes	—
36	C₅₃×#334	C₂²	1	1	yes	characteristic
37	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
38	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
39	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
40	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
41	C₅₃×#322	C₂²	1	1	yes	characteristic
42	C₅₃×#322	C₂²	1	1	yes	characteristic
43	C₅₃×#322	C₂²	1	1	yes	characteristic
44	C₅₃×#322	C₂²	1	1	yes	characteristic
45	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
47	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×M₁₆×C₅₃	—	2	2	no	—
49	C₂×M₁₆×C₅₃	—	2	1	no	—
50	C₂²×C₈×C₅₃	—	2	2	no	—
51	C₂²×C₈×C₅₃	—	2	2	no	—
52	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	#126×C₅₃	—	2	2	no	—
55	#125×C₅₃	C₂×C₄	1	2	yes	—
56	#124×C₅₃	C₂×C₄	1	2	yes	—
57	#123×C₅₃	—	2	2	no	—
58	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
59	#121×C₅₃	C₂×C₄	1	2	yes	—
60	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
61	C₄×D₄×C₅₃	—	2	2	no	—
62	C₄×D₄×C₅₃	C₂³	1	2	yes	—
63	#117×C₅₃	—	2	2	no	—
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
66	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
67	#106×C₅₃	C₂×C₄	1	2	yes	—
68	C₄⋊C₈×C₅₃	—	2	2	no	—
69	C₄⋊C₈×C₅₃	—	2	2	no	—
70	C₄⋊C₈×C₅₃	—	2	2	no	—
71	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
72	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
73	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
74	#103×C₅₃	C₂×C₄	1	2	yes	—
75	#103×C₅₃	C₂×C₄	1	2	yes	—
76	#102×C₅₃	C₂×C₄	1	2	yes	—
77	#102×C₅₃	C₂×C₄	1	2	yes	—
78	C₂²⋊C₈×C₅₃	—	2	2	no	—
79	C₂²⋊C₈×C₅₃	—	2	2	no	—
80	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
81	#97×C₅₃	—	2	2	no	—
82	#97×C₅₃	—	2	1	no	—
83	C₄×C₈×C₅₃	—	2	2	no	—
84	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂×D₄×C₅₃	—	2	2	no	—
94	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₅₃	—	2	2	no	—
96	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
99	Q₁₆×C₅₃	—	2	2	no	—
100	SD₁₆×C₅₃	—	4	2	no	—
101	D₈×C₅₃	—	2	2	no	—
102	M₁₆×C₅₃	—	4	2	no	—
103	M₁₆×C₅₃	—	8	2	no	—
104	C₂×C₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	—	2	1	no	—
108	C₂×C₈×C₅₃	—	2	1	no	—
109	C₂×C₈×C₅₃	—	4	1	no	—
110	C₂×C₈×C₅₃	—	8	2	no	—
111	C₂×C₈×C₅₃	—	8	2	no	—
112	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
117	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
118	C₄⋊C₄×C₅₃	—	2	2	no	—
119	C₄⋊C₄×C₅₃	—	2	2	no	—
120	C₄⋊C₄×C₅₃	—	2	2	no	—
121	C₄⋊C₄×C₅₃	—	2	2	no	—
122	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
123	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
124	C₂²⋊C₄×C₅₃	—	2	2	no	—
125	C₂²⋊C₄×C₅₃	—	2	2	no	—
126	C₂²⋊C₄×C₅₃	—	2	2	no	—
127	C₂²⋊C₄×C₅₃	—	2	2	no	—
128	C₂²⋊C₄×C₅₃	—	2	2	no	—
129	C₄²×C₅₃	—	2	2	no	—
130	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₅₃	#41	1	1	yes	characteristic
133	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
134	C₂³×C₅₃	—	2	2	no	—
135	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
136	Q₈×C₅₃	—	2	2	no	—
137	Q₈×C₅₃	—	2	2	no	—
138	D₄×C₅₃	—	2	2	no	—
139	D₄×C₅₃	—	2	2	no	—
140	D₄×C₅₃	—	4	4	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	1	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	—	4	2	no	—
149	C₂×C₄×C₅₃	—	4	2	no	—
150	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
156	C₈×C₅₃	—	2	1	no	—
157	C₈×C₅₃	—	2	1	no	—
158	C₈×C₅₃	—	2	1	no	—
159	C₈×C₅₃	—	2	2	no	—
160	C₈×C₅₃	—	2	2	no	—
161	C₈×C₅₃	—	4	1	no	—
162	C₈×C₅₃	—	4	1	no	—
163	C₈×C₅₃	—	4	2	no	—
164	C₈×C₅₃	—	4	2	no	—
165	C₈×C₅₃	#131	1	2	yes	—
166	#18093	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂²×C₅₃	—	2	1	no	—
168	C₂²×C₅₃	—	2	1	no	—
169	C₂²×C₅₃	—	4	2	no	—
170	C₂²×C₅₃	—	4	2	no	—
171	C₂²×C₅₃	—	4	2	no	—
172	C₂²×C₅₃	#444	1	2	yes	—
173	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
174	C₄×C₅₃	—	2	1	no	—
175	C₄×C₅₃	—	2	1	no	—
176	C₄×C₅₃	—	2	1	no	—
177	C₄×C₅₃	—	2	1	no	—
178	C₄×C₅₃	—	4	2	no	—
179	C₄×C₅₃	—	4	2	no	—
180	C₄×C₅₃	—	4	2	no	—
181	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
182	C₄×C₅₃	#568	1	1	yes	characteristic
183	C₄×C₅₃	#577	1	2	yes	—
184	#3168	C₂×C₅₃	1	1	yes	characteristic
185	#2994	C₂×C₅₃	1	2	yes	complemented
186	#1594	C₂×C₅₃	1	1	yes	characteristic, complemented
187	#1593	C₂×C₅₃	1	1	yes	characteristic, complemented
188	#1583	C₂×C₅₃	1	2	yes	complemented
189	#1582	C₂×C₅₃	1	2	yes	complemented
190	#1580	C₂×C₅₃	1	1	yes	characteristic, complemented
191	#1579	C₂×C₅₃	1	1	yes	characteristic, complemented
192	C₂×#334	C₂×C₅₃	1	1	yes	characteristic, complemented
193	#1483	C₂×C₅₃	1	2	yes	complemented
194	#1451	C₂×C₅₃	1	1	yes	characteristic, complemented
195	C₂×C₅₃	—	2	2	no	—
196	C₂×C₅₃	—	8	2	no	—
197	C₂×C₅₃	#2935	1	1	yes	characteristic
198	C₂×C₅₃	#2952	1	1	yes	characteristic
199	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
200	#529	C₂²×C₅₃	1	2	yes	—
201	#499	C₄×C₅₃	1	1	yes	characteristic
202	#495	C₄×C₅₃	1	1	yes	characteristic
203	#488	C₄×C₅₃	1	2	yes	—
204	#467	C₄×C₅₃	1	1	yes	characteristic
205	#466	C₄×C₅₃	1	1	yes	characteristic
206	#465	C₄×C₅₃	1	2	yes	—
207	#446	C₂²×C₅₃	1	2	yes	—
208	C₈×Q₈	C₂²×C₅₃	1	2	yes	—
209	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
210	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	—
211	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
212	#436	C₂²×C₅₃	1	2	yes	—
213	#435	—	2	2	no	—
214	C₈×D₄	—	2	2	no	—
215	C₈×D₄	C₂²×C₅₃	1	2	yes	—
216	#433	—	2	2	no	—
217	#431	C₂²×C₅₃	1	1	yes	characteristic
218	#424	—	2	2	no	—
219	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
220	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
221	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
222	#334	C₂²×C₅₃	1	1	yes	characteristic
223	#334	C₂²×C₅₃	1	2	yes	—
224	#334	C₂²×C₅₃	1	1	yes	characteristic
225	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
226	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
227	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
228	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
229	#322	C₂²×C₅₃	1	1	yes	characteristic
230	#322	C₂²×C₅₃	1	1	yes	characteristic
231	#322	C₂²×C₅₃	1	1	yes	characteristic
232	#322	C₂²×C₅₃	1	1	yes	characteristic
233	C₅₃	#18093	1	1	yes	characteristic, direct product factor, Sylow
234	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
235	C₂×SD₁₆	C₂×C₄×C₅₃	1	2	yes	—
236	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic
237	C₂×M₁₆	—	2	1	no	—
238	C₂×M₁₆	—	2	2	no	—
239	C₂²×C₈	—	2	2	no	—
240	C₂²×C₈	—	2	2	no	—
241	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
242	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
243	#126	—	2	2	no	—
244	#125	C₂×C₄×C₅₃	1	2	yes	—
245	#124	C₂×C₄×C₅₃	1	2	yes	—
246	#123	—	2	2	no	—
247	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
248	#121	C₂×C₄×C₅₃	1	2	yes	—
249	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
250	C₄×D₄	—	2	2	no	—
251	C₄×D₄	C₂³×C₅₃	1	2	yes	—
252	#117	—	2	2	no	—
253	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
254	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
255	#107	C₂×C₄×C₅₃	1	1	yes	characteristic
256	#106	C₂×C₄×C₅₃	1	2	yes	—
257	C₄⋊C₈	—	2	2	no	—
258	C₄⋊C₈	—	2	2	no	—
259	C₄⋊C₈	—	2	2	no	—
260	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
261	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
262	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
263	#103	C₂×C₄×C₅₃	1	2	yes	—
264	#103	C₂×C₄×C₅₃	1	2	yes	—
265	#102	C₂×C₄×C₅₃	1	2	yes	—
266	#102	C₂×C₄×C₅₃	1	2	yes	—
267	C₂²⋊C₈	—	2	2	no	—
268	C₂²⋊C₈	—	2	2	no	—
269	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
270	#97	—	2	2	no	—
271	#97	—	2	1	no	—
272	C₄×C₈	—	2	2	no	—
273	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
274	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
275	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
276	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
277	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
278	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
279	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
280	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
281	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
287	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
288	Q₁₆	—	2	2	no	—
289	SD₁₆	—	4	2	no	—
290	D₈	—	2	2	no	—
291	M₁₆	—	4	2	no	—
292	M₁₆	—	8	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	4	1	no	—
299	C₂×C₈	—	8	2	no	—
300	C₂×C₈	—	8	2	no	—
301	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
302	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
303	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
304	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
305	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
306	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
312	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
313	C₂²⋊C₄	—	2	2	no	—
314	C₂²⋊C₄	—	2	2	no	—
315	C₂²⋊C₄	—	2	2	no	—
316	C₂²⋊C₄	—	2	2	no	—
317	C₂²⋊C₄	—	2	2	no	—
318	C₄²	—	2	2	no	—
319	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
320	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
321	C₄²	#41×C₅₃	1	1	yes	characteristic
322	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
323	C₂³	—	2	2	no	—
324	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
325	Q₈	—	2	2	no	—
326	Q₈	—	2	2	no	—
327	D₄	—	2	2	no	—
328	D₄	—	2	2	no	—
329	D₄	—	4	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
340	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
341	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
342	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
343	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
344	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	2	1	no	—
348	C₈	—	2	2	no	—
349	C₈	—	2	2	no	—
350	C₈	—	4	1	no	—
351	C₈	—	4	1	no	—
352	C₈	—	4	2	no	—
353	C₈	—	4	2	no	—
354	C₈	#131×C₅₃	1	2	yes	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	4	2	no	—
359	C₂²	—	4	2	no	—
360	C₂²	C₅₃×#444	1	2	yes	—
361	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	4	2	no	—
367	C₄	—	4	2	no	—
368	C₄	—	4	2	no	—
369	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
370	C₄	C₅₃×#568	1	1	yes	characteristic
371	C₄	C₅₃×#577	1	2	yes	—
372	C₂	—	2	2	no	—
373	C₂	—	8	2	no	—
374	C₂	C₅₃×#2935	1	1	yes	characteristic
375	C₂	C₅₃×#2952	1	1	yes	characteristic
376	C₂	C₂×C₅₃×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₅₃×#466	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₅₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2994	C₂	semidirect
3. C₅₃×#1594	C₂	semidirect
4. C₅₃×#1593	C₂	semidirect
5. C₅₃×#1583	C₂	semidirect
6. C₅₃×#1582	C₂	semidirect
7. C₅₃×#1580	C₂	semidirect
8. C₅₃×#1579	C₂	semidirect
9. C₂×C₅₃×#334	C₂	semidirect
10. C₅₃×#1483	C₂	semidirect
11. C₅₃×#1451	C₂	semidirect
166. #18093	C₅₃	direct (N × H)
185. #2994	C₂×C₅₃	semidirect
186. #1594	C₂×C₅₃	semidirect
187. #1593	C₂×C₅₃	semidirect
188. #1583	C₂×C₅₃	semidirect
189. #1582	C₂×C₅₃	semidirect
190. #1580	C₂×C₅₃	semidirect
191. #1579	C₂×C₅₃	semidirect
192. C₂×#334	C₂×C₅₃	semidirect
193. #1483	C₂×C₅₃	semidirect
194. #1451	C₂×C₅₃	semidirect
233. C₅₃	#18093	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3168	2
C₅₃×#2994	2
C₅₃×#1594	2
C₅₃×#1593	2
C₅₃×#1583	2
C₅₃×#1582	2
C₅₃×#1580	2
C₅₃×#1579	2
C₂×C₅₃×#334	2
C₅₃×#1483	2
C₅₃×#1451	2
#18093	53

Subgroups — C53×#18093 #27225605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18093 #27225605