Subgroups — C₅₃×#60620 #27268132

Order 13568.

2014 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3458	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#536	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#3140	C₂	1	4	yes	maximal, complemented
4	C₅₃×#3139	C₂	1	4	yes	maximal, complemented
5	C₅₃×#3137	C₂	1	16	yes	maximal, complemented
6	C₅₃×#2896	C₂	1	2	yes	maximal
7	C₂×C₅₃×#419	C₂	1	2	yes	maximal, complemented
8	C₅₃×#2873	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
10	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#544	—	2	2	no	—
13	C₅₃×#544	C₂²	1	8	yes	complemented
14	C₅₃×#543	—	2	2	no	—
15	C₅₃×#536	—	2	4	no	—
16	C₅₃×#536	C₂²	1	8	yes	—
17	C₅₃×#533	—	2	4	no	—
18	C₅₃×#533	C₂²	1	4	yes	—
19	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
20	C₂×#124×C₅₃	C₂²	1	2	yes	—
21	C₂×#123×C₅₃	—	2	2	no	—
22	C₂×C₂²⋊Q₈×C₅₃	—	2	2	no	—
23	C₂×#117×C₅₃	C₂²	1	2	yes	—
24	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#491	C₂²	1	16	yes	complemented
26	C₅₃×#489	C₂²	1	16	yes	complemented
27	C₅₃×#488	C₂²	1	16	yes	complemented
28	C₅₃×#487	C₂²	1	16	yes	complemented
29	C₅₃×#419	C₂²	1	4	yes	—
30	C₅₃×#419	C₂²	1	8	yes	—
31	C₅₃×#419	C₂²	1	8	yes	complemented
32	C₅₃×#418	C₂²	1	4	yes	—
33	C₅₃×#417	C₂²	1	8	yes	—
34	C₅₃×#416	C₂²	1	8	yes	—
35	C₂×#103×C₅₃	C₂²	1	8	yes	complemented
36	C₅₃×#405	C₂²	1	8	yes	complemented
37	C₄×M₁₆×C₅₃	C₂²	1	4	yes	—
38	C₂×#41×C₅₃	—	2	2	no	—
39	C₂×#41×C₅₃	—	2	8	no	—
40	C₂×#41×C₅₃	C₂³	1	4	yes	—
41	C₂²×Q₈×C₅₃	—	2	1	no	—
42	C₂²×Q₈×C₅₃	—	2	2	no	—
43	C₂²×Q₈×C₅₃	—	2	8	no	—
44	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
45	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
47	C₂²×D₄×C₅₃	—	2	1	no	—
48	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₅₃	—	2	1	no	—
50	C₂³×C₄×C₅₃	—	2	1	no	—
51	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×M₁₆×C₅₃	—	2	8	no	—
53	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
54	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
55	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
56	C₄⋊Q₈×C₅₃	—	2	8	no	—
57	C₄⋊Q₈×C₅₃	—	2	4	no	—
58	C₄⋊Q₈×C₅₃	C₂³	1	8	yes	—
59	C₄⋊Q₈×C₅₃	C₂³	1	8	yes	—
60	#125×C₅₃	—	2	8	no	—
61	#125×C₅₃	C₂³	1	8	yes	—
62	#124×C₅₃	—	2	4	no	—
63	#124×C₅₃	C₂³	1	8	yes	—
64	#123×C₅₃	—	2	16	no	—
65	C₂²⋊Q₈×C₅₃	—	2	16	no	—
66	C₂²⋊Q₈×C₅₃	—	2	16	no	—
67	C₂²⋊Q₈×C₅₃	—	2	8	no	—
68	C₄×Q₈×C₅₃	—	2	8	no	—
69	#117×C₅₃	—	2	2	no	—
70	#117×C₅₃	—	2	2	no	—
71	#117×C₅₃	D₄	1	4	yes	—
72	#117×C₅₃	C₂³	1	4	yes	—
73	#117×C₅₃	C₂³	1	4	yes	—
74	#117×C₅₃	C₂³	1	4	yes	—
75	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
76	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
77	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
78	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
80	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
81	C₂×C₄²×C₅₃	—	2	2	no	—
82	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
83	#103×C₅₃	C₂³	1	16	yes	—
84	#103×C₅₃	C₂³	1	32	yes	—
85	#102×C₅₃	C₂³	1	16	yes	—
86	#97×C₅₃	C₂³	1	8	yes	—
87	C₄×C₈×C₅₃	C₂³	1	8	yes	—
88	C₂⁴×C₅₃	—	2	1	no	—
89	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
90	#41×C₅₃	—	2	8	no	—
91	#41×C₅₃	—	2	8	no	—
92	#41×C₅₃	—	4	8	no	—
93	C₂×Q₈×C₅₃	—	2	4	no	—
94	C₂×Q₈×C₅₃	—	2	2	no	—
95	C₂×Q₈×C₅₃	—	2	1	no	—
96	C₂×Q₈×C₅₃	—	2	8	no	—
97	C₂×Q₈×C₅₃	—	2	8	no	—
98	C₂×Q₈×C₅₃	—	2	8	no	—
99	C₂×Q₈×C₅₃	—	2	8	no	—
100	C₂×Q₈×C₅₃	—	2	4	no	—
101	C₂×Q₈×C₅₃	—	4	4	no	—
102	C₂×Q₈×C₅₃	—	4	8	no	—
103	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
104	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
105	C₂×D₄×C₅₃	—	2	4	no	—
106	C₂×D₄×C₅₃	—	2	1	no	—
107	C₂×D₄×C₅₃	—	2	4	no	—
108	C₂×D₄×C₅₃	—	4	4	no	—
109	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
110	C₂²×C₄×C₅₃	—	2	2	no	—
111	C₂²×C₄×C₅₃	—	2	1	no	—
112	C₂²×C₄×C₅₃	—	2	4	no	—
113	C₂²×C₄×C₅₃	—	2	4	no	—
114	C₂²×C₄×C₅₃	—	2	4	no	—
115	C₂²×C₄×C₅₃	—	4	4	no	—
116	C₂²×C₄×C₅₃	—	4	4	no	—
117	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
118	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₅₃	#41	1	8	yes	—
121	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
122	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
123	M₁₆×C₅₃	—	2	8	no	—
124	M₁₆×C₅₃	—	2	8	no	—
125	C₂×C₈×C₅₃	—	2	8	no	—
126	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
127	C₄⋊C₄×C₅₃	—	2	16	no	—
128	C₄⋊C₄×C₅₃	—	2	4	no	—
129	C₄⋊C₄×C₅₃	—	2	8	no	—
130	C₄⋊C₄×C₅₃	—	2	16	no	—
131	C₄⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
132	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
133	C₂²⋊C₄×C₅₃	—	2	4	no	—
134	C₂²⋊C₄×C₅₃	—	2	4	no	—
135	C₂²⋊C₄×C₅₃	—	2	16	no	—
136	C₂²⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
137	C₄²×C₅₃	—	2	4	no	—
138	C₄²×C₅₃	—	2	4	no	—
139	C₄²×C₅₃	C₂×D₄	1	4	yes	—
140	C₄²×C₅₃	C₂⁴	1	4	yes	—
141	C₂³×C₅₃	—	2	2	no	—
142	C₂³×C₅₃	—	4	2	no	—
143	C₂³×C₅₃	—	4	4	no	—
144	C₂³×C₅₃	—	4	2	no	—
145	C₂³×C₅₃	#124	1	4	yes	—
146	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
147	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
148	Q₈×C₅₃	—	2	8	no	—
149	Q₈×C₅₃	—	2	4	no	—
150	Q₈×C₅₃	—	4	8	no	—
151	Q₈×C₅₃	—	4	4	no	—
152	Q₈×C₅₃	—	4	2	no	—
153	Q₈×C₅₃	—	8	8	no	—
154	D₄×C₅₃	—	2	4	no	—
155	D₄×C₅₃	—	4	8	no	—
156	D₄×C₅₃	—	4	2	no	—
157	C₂×C₄×C₅₃	—	2	2	no	—
158	C₂×C₄×C₅₃	—	2	4	no	—
159	C₂×C₄×C₅₃	—	2	8	no	—
160	C₂×C₄×C₅₃	—	4	4	no	—
161	C₂×C₄×C₅₃	—	4	2	no	—
162	C₂×C₄×C₅₃	—	4	4	no	—
163	C₂×C₄×C₅₃	—	4	8	no	—
164	C₂×C₄×C₅₃	—	4	8	no	—
165	C₂×C₄×C₅₃	—	8	4	no	—
166	C₂×C₄×C₅₃	#124	1	4	yes	—
167	C₂×C₄×C₅₃	#124	1	8	yes	—
168	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
169	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
170	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
172	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
173	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
174	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
175	C₈×C₅₃	—	2	8	no	—
176	#60620	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂²×C₅₃	—	2	2	no	—
178	C₂²×C₅₃	—	2	4	no	—
179	C₂²×C₅₃	—	4	4	no	—
180	C₂²×C₅₃	—	4	4	no	—
181	C₂²×C₅₃	—	4	2	no	—
182	C₂²×C₅₃	—	8	4	no	—
183	C₂²×C₅₃	C₂×#124	1	2	yes	—
184	C₂²×C₅₃	C₂×#124	1	4	yes	—
185	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
186	C₄×C₅₃	—	2	8	no	—
187	C₄×C₅₃	—	4	8	no	—
188	C₄×C₅₃	—	4	2	no	—
189	C₄×C₅₃	—	4	4	no	—
190	C₄×C₅₃	C₂×#124	1	4	yes	—
191	C₄×C₅₃	C₂×#124	1	2	yes	—
192	C₄×C₅₃	C₂²×#41	1	2	yes	—
193	#3458	C₂×C₅₃	1	1	yes	characteristic, complemented
194	C₂×#536	C₂×C₅₃	1	1	yes	characteristic
195	#3140	C₂×C₅₃	1	4	yes	complemented
196	#3139	C₂×C₅₃	1	4	yes	complemented
197	#3137	C₂×C₅₃	1	16	yes	complemented
198	#2896	C₂×C₅₃	1	2	yes	—
199	C₂×#419	C₂×C₅₃	1	2	yes	complemented
200	#2873	C₂×C₅₃	1	1	yes	characteristic, complemented
201	C₂×C₅₃	—	2	2	no	—
202	C₂×C₅₃	—	2	4	no	—
203	C₂×C₅₃	—	8	2	no	—
204	C₂×C₅₃	C₂²×#124	1	1	yes	characteristic
205	C₂×C₅₃	#3586	1	2	yes	—
206	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
207	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
208	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
209	#544	—	2	2	no	—
210	#544	C₂²×C₅₃	1	8	yes	complemented
211	#543	—	2	2	no	—
212	#536	—	2	4	no	—
213	#536	C₂²×C₅₃	1	8	yes	—
214	#533	—	2	4	no	—
215	#533	C₂²×C₅₃	1	4	yes	—
216	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
217	C₂×#124	C₂²×C₅₃	1	2	yes	—
218	C₂×#123	—	2	2	no	—
219	C₂×C₂²⋊Q₈	—	2	2	no	—
220	C₂×#117	C₂²×C₅₃	1	2	yes	—
221	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
222	#491	C₂²×C₅₃	1	16	yes	complemented
223	#489	C₂²×C₅₃	1	16	yes	complemented
224	#488	C₂²×C₅₃	1	16	yes	complemented
225	#487	C₂²×C₅₃	1	16	yes	complemented
226	#419	C₂²×C₅₃	1	4	yes	—
227	#419	C₂²×C₅₃	1	8	yes	—
228	#419	C₂²×C₅₃	1	8	yes	complemented
229	#418	C₂²×C₅₃	1	4	yes	—
230	#417	C₂²×C₅₃	1	8	yes	—
231	#416	C₂²×C₅₃	1	8	yes	—
232	C₂×#103	C₂²×C₅₃	1	8	yes	complemented
233	#405	C₂²×C₅₃	1	8	yes	complemented
234	C₄×M₁₆	C₂²×C₅₃	1	4	yes	—
235	C₅₃	#60620	1	1	yes	characteristic, direct product factor, Sylow
236	C₂×#41	—	2	2	no	—
237	C₂×#41	—	2	8	no	—
238	C₂×#41	C₂³×C₅₃	1	4	yes	—
239	C₂²×Q₈	—	2	1	no	—
240	C₂²×Q₈	—	2	2	no	—
241	C₂²×Q₈	—	2	8	no	—
242	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
243	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
244	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
245	C₂²×D₄	—	2	1	no	—
246	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
247	C₂³×C₄	—	2	1	no	—
248	C₂³×C₄	—	2	1	no	—
249	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
250	C₂×M₁₆	—	2	8	no	—
251	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
252	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
253	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
254	C₄⋊Q₈	—	2	8	no	—
255	C₄⋊Q₈	—	2	4	no	—
256	C₄⋊Q₈	C₂³×C₅₃	1	8	yes	—
257	C₄⋊Q₈	C₂³×C₅₃	1	8	yes	—
258	#125	—	2	8	no	—
259	#125	C₂³×C₅₃	1	8	yes	—
260	#124	—	2	4	no	—
261	#124	C₂³×C₅₃	1	8	yes	—
262	#123	—	2	16	no	—
263	C₂²⋊Q₈	—	2	8	no	—
264	C₂²⋊Q₈	—	2	16	no	—
265	C₂²⋊Q₈	—	2	16	no	—
266	C₄×Q₈	—	2	8	no	—
267	#117	—	2	2	no	—
268	#117	—	2	2	no	—
269	#117	D₄×C₅₃	1	4	yes	—
270	#117	C₂³×C₅₃	1	4	yes	—
271	#117	C₂³×C₅₃	1	4	yes	—
272	#117	C₂³×C₅₃	1	4	yes	—
273	C₂×C₄⋊C₄	—	2	4	no	—
274	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
275	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
276	C₂×C₂²⋊C₄	—	2	2	no	—
277	C₂×C₂²⋊C₄	—	2	4	no	—
278	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
279	C₂×C₄²	—	2	2	no	—
280	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
281	#103	C₂³×C₅₃	1	32	yes	—
282	#103	C₂³×C₅₃	1	16	yes	—
283	#102	C₂³×C₅₃	1	16	yes	—
284	#97	C₂³×C₅₃	1	8	yes	—
285	C₄×C₈	C₂³×C₅₃	1	8	yes	—
286	C₂⁴	—	2	1	no	—
287	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
288	#41	—	2	8	no	—
289	#41	—	2	8	no	—
290	#41	—	4	8	no	—
291	C₂×Q₈	—	2	4	no	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	—	2	1	no	—
294	C₂×Q₈	—	2	8	no	—
295	C₂×Q₈	—	2	8	no	—
296	C₂×Q₈	—	2	8	no	—
297	C₂×Q₈	—	2	8	no	—
298	C₂×Q₈	—	2	4	no	—
299	C₂×Q₈	—	4	4	no	—
300	C₂×Q₈	—	4	8	no	—
301	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
302	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
303	C₂×D₄	—	2	4	no	—
304	C₂×D₄	—	2	1	no	—
305	C₂×D₄	—	2	4	no	—
306	C₂×D₄	—	4	4	no	—
307	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	2	4	no	—
311	C₂²×C₄	—	2	4	no	—
312	C₂²×C₄	—	2	4	no	—
313	C₂²×C₄	—	4	4	no	—
314	C₂²×C₄	—	4	4	no	—
315	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
316	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
317	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
318	C₂²×C₄	#41×C₅₃	1	8	yes	—
319	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
320	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
321	M₁₆	—	2	8	no	—
322	M₁₆	—	2	8	no	—
323	C₂×C₈	—	2	8	no	—
324	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
325	C₄⋊C₄	—	2	16	no	—
326	C₄⋊C₄	—	2	4	no	—
327	C₄⋊C₄	—	2	8	no	—
328	C₄⋊C₄	—	2	16	no	—
329	C₄⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
330	C₄⋊C₄	C₂⁴×C₅₃	1	8	yes	—
331	C₂²⋊C₄	—	2	4	no	—
332	C₂²⋊C₄	—	2	4	no	—
333	C₂²⋊C₄	—	2	16	no	—
334	C₂²⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
335	C₄²	—	2	4	no	—
336	C₄²	—	2	4	no	—
337	C₄²	C₂×D₄×C₅₃	1	4	yes	—
338	C₄²	C₂⁴×C₅₃	1	4	yes	—
339	C₂³	—	2	2	no	—
340	C₂³	—	4	2	no	—
341	C₂³	—	4	4	no	—
342	C₂³	—	4	2	no	—
343	C₂³	#124×C₅₃	1	4	yes	—
344	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
345	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
346	Q₈	—	2	8	no	—
347	Q₈	—	2	4	no	—
348	Q₈	—	4	8	no	—
349	Q₈	—	4	4	no	—
350	Q₈	—	4	2	no	—
351	Q₈	—	8	8	no	—
352	D₄	—	2	4	no	—
353	D₄	—	4	8	no	—
354	D₄	—	4	2	no	—
355	C₂×C₄	—	2	8	no	—
356	C₂×C₄	—	2	4	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	4	4	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	4	no	—
361	C₂×C₄	—	4	8	no	—
362	C₂×C₄	—	4	8	no	—
363	C₂×C₄	—	8	4	no	—
364	C₂×C₄	#124×C₅₃	1	4	yes	—
365	C₂×C₄	#124×C₅₃	1	8	yes	—
366	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
367	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
368	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
369	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
370	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
371	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
372	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
373	C₈	—	2	8	no	—
374	C₂²	—	2	2	no	—
375	C₂²	—	2	4	no	—
376	C₂²	—	4	4	no	—
377	C₂²	—	4	4	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	8	4	no	—
380	C₂²	C₂×#124×C₅₃	1	2	yes	—
381	C₂²	C₂×#124×C₅₃	1	4	yes	—
382	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
383	C₄	—	2	8	no	—
384	C₄	—	4	8	no	—
385	C₄	—	4	2	no	—
386	C₄	—	4	4	no	—
387	C₄	C₂×#124×C₅₃	1	4	yes	—
388	C₄	C₂×#124×C₅₃	1	2	yes	—
389	C₄	C₂²×#41×C₅₃	1	2	yes	—
390	C₂	—	2	2	no	—
391	C₂	—	2	4	no	—
392	C₂	—	8	2	no	—
393	C₂	C₂²×#124×C₅₃	1	1	yes	characteristic
394	C₂	C₅₃×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₂²×#41×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×#117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3458	C₂	semidirect
3. C₅₃×#3140	C₂	semidirect
4. C₅₃×#3139	C₂	semidirect
5. C₅₃×#3137	C₂	semidirect
7. C₂×C₅₃×#419	C₂	semidirect
8. C₅₃×#2873	C₂	semidirect
13. C₅₃×#544	C₂²	semidirect
25. C₅₃×#491	C₂²	semidirect
26. C₅₃×#489	C₂²	semidirect
27. C₅₃×#488	C₂²	semidirect
28. C₅₃×#487	C₂²	semidirect
31. C₅₃×#419	C₂²	semidirect
35. C₂×#103×C₅₃	C₂²	semidirect
36. C₅₃×#405	C₂²	semidirect
176. #60620	C₅₃	direct (N × H)
193. #3458	C₂×C₅₃	semidirect
195. #3140	C₂×C₅₃	semidirect
196. #3139	C₂×C₅₃	semidirect
197. #3137	C₂×C₅₃	semidirect
199. C₂×#419	C₂×C₅₃	semidirect
200. #2873	C₂×C₅₃	semidirect
210. #544	C₂²×C₅₃	semidirect
222. #491	C₂²×C₅₃	semidirect
223. #489	C₂²×C₅₃	semidirect
224. #488	C₂²×C₅₃	semidirect
225. #487	C₂²×C₅₃	semidirect
228. #419	C₂²×C₅₃	semidirect
232. C₂×#103	C₂²×C₅₃	semidirect
233. #405	C₂²×C₅₃	semidirect
235. C₅₃	#60620	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×#3458	2
C₂×C₅₃×#536	2
C₅₃×#3140	2
C₅₃×#3139	2
C₅₃×#3137	2
C₅₃×#2896	2
C₂×C₅₃×#419	2
C₅₃×#2873	2
#60620	53

Subgroups — C53×#60620 #27268132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60620 #27268132