Subgroups — C₅₃×#17979 #27225491

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3167	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1588	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1585	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1492	C₂	1	1	yes	maximal, characteristic
9	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#131×C₅₃	—	2	1	no	—
11	C₅₃×#533	C₂²	1	1	yes	characteristic
12	C₅₃×#532	C₂²	1	1	yes	characteristic
13	C₅₃×#498	C₄	1	1	yes	characteristic
14	C₅₃×#492	C₄	1	1	yes	characteristic
15	C₅₃×#487	C₄	1	1	yes	characteristic
16	C₅₃×#486	C₄	1	1	yes	characteristic
17	C₅₃×#465	C₄	1	4	yes	—
18	C₅₃×#446	—	2	1	no	—
19	C₅₃×#446	C₂²	1	2	yes	—
20	C₈×Q₈×C₅₃	—	2	1	no	—
21	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
22	C₄×SD₁₆×C₅₃	C₂²	1	4	yes	—
23	C₅₃×#436	—	2	1	no	—
24	C₅₃×#436	C₂²	1	2	yes	—
25	C₈×D₄×C₅₃	—	2	1	no	—
26	C₈×D₄×C₅₃	C₂²	1	2	yes	—
27	C₅₃×#424	—	2	1	no	—
28	C₅₃×#423	C₂²	1	1	yes	characteristic
29	C₅₃×#423	C₂²	1	1	yes	characteristic
30	C₅₃×#408	—	2	1	no	—
31	C₅₃×#405	—	2	1	no	—
32	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#335	C₂²	1	4	yes	—
36	C₅₃×Q₈⋊C₈	C₂²	1	4	yes	—
37	C₅₃×D₄⋊C₈	C₂²	1	4	yes	—
38	C₅₃×#322	C₂²	1	2	yes	—
39	C₅₃×#322	C₂²	1	2	yes	—
40	C₂×#41×C₅₃	—	2	1	no	—
41	C₂×SD₁₆×C₅₃	C₂×C₄	1	4	yes	—
42	#131×C₅₃	—	4	4	no	—
43	C₂×M₁₆×C₅₃	—	2	2	no	—
44	C₂×M₁₆×C₅₃	—	2	1	no	—
45	C₂×M₁₆×C₅₃	—	2	1	no	—
46	C₂²×C₈×C₅₃	—	2	2	no	—
47	C₂²×C₈×C₅₃	—	2	1	no	—
48	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
55	#125×C₅₃	—	2	1	no	—
56	#124×C₅₃	—	2	1	no	—
57	C₂²⋊Q₈×C₅₃	—	2	1	no	—
58	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
59	#121×C₅₃	—	2	1	no	—
60	#121×C₅₃	C₂×C₄	1	2	yes	—
61	C₄×Q₈×C₅₃	—	2	1	no	—
62	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
63	C₄×D₄×C₅₃	—	2	1	no	—
64	C₄×D₄×C₅₃	C₂³	1	2	yes	—
65	#117×C₅₃	—	2	1	no	—
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	#106×C₅₃	C₂×C₄	1	4	yes	—
68	C₄⋊C₈×C₅₃	—	2	2	no	—
69	C₄⋊C₈×C₅₃	—	2	2	no	—
70	C₄⋊C₈×C₅₃	—	2	1	no	—
71	C₄⋊C₈×C₅₃	—	2	1	no	—
72	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
73	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
74	#103×C₅₃	C₂×C₄	1	4	yes	—
75	#102×C₅₃	C₂×C₄	1	4	yes	—
76	C₂²⋊C₈×C₅₃	—	2	2	no	—
77	C₂²⋊C₈×C₅₃	—	2	2	no	—
78	#97×C₅₃	—	2	2	no	—
79	C₄×C₈×C₅₃	—	2	2	no	—
80	C₄×C₈×C₅₃	—	2	1	no	—
81	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
84	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂³	1	2	yes	—
86	C₄×C₈×C₅₃	C₂³	1	2	yes	—
87	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
89	#41×C₅₃	—	4	2	no	—
90	#41×C₅₃	—	4	2	no	—
91	C₂×Q₈×C₅₃	—	2	1	no	—
92	C₂×Q₈×C₅₃	—	2	1	no	—
93	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
94	C₂×D₄×C₅₃	—	2	2	no	—
95	C₂×D₄×C₅₃	—	2	1	no	—
96	C₂×D₄×C₅₃	—	2	1	no	—
97	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₅₃	—	2	2	no	—
99	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
102	SD₁₆×C₅₃	—	4	4	no	—
103	M₁₆×C₅₃	—	4	4	no	—
104	M₁₆×C₅₃	—	4	2	no	—
105	M₁₆×C₅₃	—	4	2	no	—
106	C₂×C₈×C₅₃	—	2	4	no	—
107	C₂×C₈×C₅₃	—	2	2	no	—
108	C₂×C₈×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	—	2	2	no	—
111	C₂×C₈×C₅₃	—	2	2	no	—
112	C₂×C₈×C₅₃	—	4	2	no	—
113	C₂×C₈×C₅₃	—	4	4	no	—
114	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
115	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
116	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₄⋊C₄×C₅₃	—	2	1	no	—
121	C₄⋊C₄×C₅₃	—	2	1	no	—
122	C₄⋊C₄×C₅₃	—	2	1	no	—
123	C₄⋊C₄×C₅₃	—	2	1	no	—
124	C₄⋊C₄×C₅₃	—	2	2	no	—
125	C₄⋊C₄×C₅₃	—	2	2	no	—
126	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
127	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
128	C₂²⋊C₄×C₅₃	—	2	2	no	—
129	C₂²⋊C₄×C₅₃	—	2	2	no	—
130	C₂²⋊C₄×C₅₃	—	2	2	no	—
131	C₄²×C₅₃	—	2	2	no	—
132	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₅₃	#41	1	1	yes	characteristic
135	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
136	C₂³×C₅₃	—	2	2	no	—
137	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
138	Q₈×C₅₃	—	2	4	no	—
139	Q₈×C₅₃	—	4	2	no	—
140	Q₈×C₅₃	—	4	2	no	—
141	D₄×C₅₃	—	2	4	no	—
142	D₄×C₅₃	—	4	4	no	—
143	D₄×C₅₃	—	4	2	no	—
144	D₄×C₅₃	—	4	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	—	4	2	no	—
151	C₂×C₄×C₅₃	—	4	2	no	—
152	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
156	C₂×C₄×C₅₃	#137	1	2	yes	—
157	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
160	C₈×C₅₃	—	2	4	no	—
161	C₈×C₅₃	—	2	2	no	—
162	C₈×C₅₃	—	2	2	no	—
163	C₈×C₅₃	—	2	1	no	—
164	C₈×C₅₃	—	2	1	no	—
165	C₈×C₅₃	—	2	1	no	—
166	C₈×C₅₃	—	2	1	no	—
167	C₈×C₅₃	—	4	2	no	—
168	C₈×C₅₃	—	4	2	no	—
169	#17979	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂²×C₅₃	—	2	1	no	—
171	C₂²×C₅₃	—	2	1	no	—
172	C₂²×C₅₃	—	4	2	no	—
173	C₂²×C₅₃	—	4	2	no	—
174	C₂²×C₅₃	—	4	2	no	—
175	C₂²×C₅₃	#440	1	2	yes	—
176	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
177	C₄×C₅₃	—	2	1	no	—
178	C₄×C₅₃	—	2	1	no	—
179	C₄×C₅₃	—	4	2	no	—
180	C₄×C₅₃	—	4	2	no	—
181	C₄×C₅₃	—	4	2	no	—
182	C₄×C₅₃	#440	1	2	yes	—
183	C₄×C₅₃	#443	1	2	yes	—
184	C₄×C₅₃	#568	1	2	yes	—
185	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
187	#3167	C₂×C₅₃	1	1	yes	characteristic
188	#2960	C₂×C₅₃	1	1	yes	characteristic, complemented
189	#2931	C₂×C₅₃	1	1	yes	characteristic
190	#1588	C₂×C₅₃	1	4	yes	complemented
191	#1585	C₂×C₅₃	1	4	yes	complemented
192	#1567	C₂×C₅₃	1	1	yes	characteristic, complemented
193	#1493	C₂×C₅₃	1	1	yes	characteristic, complemented
194	#1492	C₂×C₅₃	1	1	yes	characteristic
195	C₂×#322	C₂×C₅₃	1	1	yes	characteristic, complemented
196	C₂×C₅₃	—	2	2	no	—
197	C₂×C₅₃	—	8	2	no	—
198	C₂×C₅₃	#2932	1	1	yes	characteristic
199	C₂×C₅₃	C₂×#440	1	1	yes	characteristic
200	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
201	C₂×#131	—	2	1	no	—
202	#533	C₂²×C₅₃	1	1	yes	characteristic
203	#532	C₂²×C₅₃	1	1	yes	characteristic
204	#498	C₄×C₅₃	1	1	yes	characteristic
205	#492	C₄×C₅₃	1	1	yes	characteristic
206	#487	C₄×C₅₃	1	1	yes	characteristic
207	#486	C₄×C₅₃	1	1	yes	characteristic
208	#465	C₄×C₅₃	1	4	yes	—
209	#446	—	2	1	no	—
210	#446	C₂²×C₅₃	1	2	yes	—
211	C₈×Q₈	—	2	1	no	—
212	C₈×Q₈	C₂²×C₅₃	1	2	yes	—
213	C₄×SD₁₆	C₂²×C₅₃	1	4	yes	—
214	#436	—	2	1	no	—
215	#436	C₂²×C₅₃	1	2	yes	—
216	C₈×D₄	—	2	1	no	—
217	C₈×D₄	C₂²×C₅₃	1	2	yes	—
218	#424	—	2	1	no	—
219	#423	C₂²×C₅₃	1	1	yes	characteristic
220	#423	C₂²×C₅₃	1	1	yes	characteristic
221	#408	—	2	1	no	—
222	#405	—	2	1	no	—
223	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
224	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
225	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
226	#335	C₂²×C₅₃	1	4	yes	—
227	Q₈⋊C₈	C₂²×C₅₃	1	4	yes	—
228	D₄⋊C₈	C₂²×C₅₃	1	4	yes	—
229	#322	C₂²×C₅₃	1	2	yes	—
230	#322	C₂²×C₅₃	1	2	yes	—
231	C₅₃	#17979	1	1	yes	characteristic, direct product factor, Sylow
232	C₂×#41	—	2	1	no	—
233	C₂×SD₁₆	C₂×C₄×C₅₃	1	4	yes	—
234	#131	—	4	4	no	—
235	C₂×M₁₆	—	2	1	no	—
236	C₂×M₁₆	—	2	1	no	—
237	C₂×M₁₆	—	2	2	no	—
238	C₂²×C₈	—	2	1	no	—
239	C₂²×C₈	—	2	2	no	—
240	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
241	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
242	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
243	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
244	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
245	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
246	#127	C₂×C₄×C₅₃	1	1	yes	characteristic
247	#125	—	2	1	no	—
248	#124	—	2	1	no	—
249	C₂²⋊Q₈	—	2	1	no	—
250	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
251	#121	—	2	1	no	—
252	#121	C₂×C₄×C₅₃	1	2	yes	—
253	C₄×Q₈	—	2	1	no	—
254	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
255	C₄×D₄	—	2	1	no	—
256	C₄×D₄	C₂³×C₅₃	1	2	yes	—
257	#117	—	2	1	no	—
258	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
259	#106	C₂×C₄×C₅₃	1	4	yes	—
260	C₄⋊C₈	—	2	2	no	—
261	C₄⋊C₈	—	2	2	no	—
262	C₄⋊C₈	—	2	1	no	—
263	C₄⋊C₈	—	2	1	no	—
264	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
265	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
266	#103	C₂×C₄×C₅₃	1	4	yes	—
267	#102	C₂×C₄×C₅₃	1	4	yes	—
268	C₂²⋊C₈	—	2	2	no	—
269	C₂²⋊C₈	—	2	2	no	—
270	#97	—	2	2	no	—
271	C₄×C₈	—	2	2	no	—
272	C₄×C₈	—	2	1	no	—
273	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
274	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
275	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
276	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
277	C₄×C₈	C₂³×C₅₃	1	2	yes	—
278	C₄×C₈	C₂³×C₅₃	1	2	yes	—
279	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
280	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
281	#41	—	4	2	no	—
282	#41	—	4	2	no	—
283	C₂×Q₈	—	2	1	no	—
284	C₂×Q₈	—	2	1	no	—
285	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	—	2	1	no	—
288	C₂×D₄	—	2	1	no	—
289	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
292	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
293	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
294	SD₁₆	—	4	4	no	—
295	M₁₆	—	4	4	no	—
296	M₁₆	—	4	2	no	—
297	M₁₆	—	4	2	no	—
298	C₂×C₈	—	2	4	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	4	2	no	—
305	C₂×C₈	—	4	4	no	—
306	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
307	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
308	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
309	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
310	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
311	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	2	1	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
319	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
320	C₂²⋊C₄	—	2	2	no	—
321	C₂²⋊C₄	—	2	2	no	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
325	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
326	C₄²	#41×C₅₃	1	1	yes	characteristic
327	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
328	C₂³	—	2	2	no	—
329	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
330	Q₈	—	2	4	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	4	2	no	—
333	D₄	—	2	4	no	—
334	D₄	—	4	4	no	—
335	D₄	—	4	2	no	—
336	D₄	—	4	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
345	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
346	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
347	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
348	C₂×C₄	#137×C₅₃	1	2	yes	—
349	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
350	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
351	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
352	C₈	—	2	4	no	—
353	C₈	—	2	2	no	—
354	C₈	—	2	2	no	—
355	C₈	—	2	1	no	—
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	4	2	no	—
360	C₈	—	4	2	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	4	2	no	—
364	C₂²	—	4	2	no	—
365	C₂²	—	4	2	no	—
366	C₂²	C₅₃×#440	1	2	yes	—
367	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	2	no	—
372	C₄	—	4	2	no	—
373	C₄	C₅₃×#440	1	2	yes	—
374	C₄	C₅₃×#443	1	2	yes	—
375	C₄	C₅₃×#568	1	2	yes	—
376	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
377	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	8	2	no	—
380	C₂	C₅₃×#2932	1	1	yes	characteristic
381	C₂	C₂×C₅₃×#440	1	1	yes	characteristic
382	C₂	C₂×C₅₃×#443	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×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2960	C₂	semidirect
4. C₅₃×#1588	C₂	semidirect
5. C₅₃×#1585	C₂	semidirect
6. C₅₃×#1567	C₂	semidirect
7. C₅₃×#1493	C₂	semidirect
9. C₂×C₅₃×#322	C₂	semidirect
169. #17979	C₅₃	direct (N × H)
188. #2960	C₂×C₅₃	semidirect
190. #1588	C₂×C₅₃	semidirect
191. #1585	C₂×C₅₃	semidirect
192. #1567	C₂×C₅₃	semidirect
193. #1493	C₂×C₅₃	semidirect
195. C₂×#322	C₂×C₅₃	semidirect
231. C₅₃	#17979	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×#3167	2
C₅₃×#2960	2
C₅₃×#2931	2
C₅₃×#1588	2
C₅₃×#1585	2
C₅₃×#1567	2
C₅₃×#1493	2
C₅₃×#1492	2
C₂×C₅₃×#322	2
#17979	53

Subgroups — C53×#17979 #27225491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17979 #27225491