Subgroups — C₅₃×#21415 #27228927

Order 13568.

1430 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3444	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1994	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1962	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1896	C₂	1	4	yes	maximal, complemented
7	C₄×#104×C₅₃	C₂	1	4	yes	maximal, complemented
8	C₅₃×#533	C₄	1	2	yes	complemented
9	C₅₃×#533	C₂²	1	1	yes	characteristic
10	C₅₃×#532	C₄	1	2	yes	complemented
11	C₅₃×#532	C₂²	1	1	yes	characteristic
12	C₅₃×#529	—	2	2	no	—
13	C₅₃×#529	—	2	2	no	—
14	C₅₃×#529	C₄	1	4	yes	complemented
15	C₅₃×#525	—	2	2	no	—
16	C₅₃×#525	C₂²	1	2	yes	—
17	C₄×#41×C₅₃	—	2	2	no	—
18	C₄×#41×C₅₃	C₂²	1	2	yes	—
19	C₂²×C₄²×C₅₃	—	2	2	no	—
20	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
21	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#432	—	2	2	no	—
23	C₅₃×#431	—	2	2	no	—
24	C₅₃×#423	C₂²	1	2	yes	—
25	C₂×#104×C₅₃	C₂²	1	8	yes	—
26	C₅₃×#407	C₂²	1	2	yes	—
27	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
28	C₄³×C₅₃	—	2	2	no	—
29	C₄³×C₅₃	C₂²	1	2	yes	—
30	C₄³×C₅₃	C₂²	1	2	yes	—
31	C₅₃×#339	C₂²	1	8	yes	—
32	C₂×#41×C₅₃	—	2	2	no	—
33	C₂×#41×C₅₃	C₂³	1	2	yes	—
34	C₂³×C₄×C₅₃	—	2	2	no	—
35	C₂³×C₄×C₅₃	D₄	1	2	yes	—
36	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
38	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	complemented
39	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
40	#127×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
41	#126×C₅₃	—	2	4	no	—
42	#125×C₅₃	—	2	2	no	—
43	#125×C₅₃	C₂×C₄	1	2	yes	complemented
44	#124×C₅₃	—	2	2	no	—
45	#124×C₅₃	C₂×C₄	1	2	yes	complemented
46	#123×C₅₃	—	2	4	no	—
47	#123×C₅₃	—	2	2	no	—
48	C₂²⋊Q₈×C₅₃	—	2	2	no	—
49	C₂²⋊Q₈×C₅₃	—	2	2	no	—
50	C₂²⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
51	#121×C₅₃	—	2	2	no	—
52	#121×C₅₃	—	2	2	no	—
53	#121×C₅₃	C₂×C₄	1	4	yes	—
54	#120×C₅₃	—	2	2	no	—
55	C₄×Q₈×C₅₃	—	2	2	no	—
56	C₄×Q₈×C₅₃	—	2	2	no	—
57	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	complemented
58	C₄×D₄×C₅₃	—	2	2	no	—
59	C₄×D₄×C₅₃	—	2	2	no	—
60	C₄×D₄×C₅₃	—	2	4	no	—
61	C₄×D₄×C₅₃	—	2	4	no	—
62	C₄×D₄×C₅₃	—	2	4	no	—
63	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	complemented
64	#117×C₅₃	—	2	4	no	—
65	#117×C₅₃	—	2	2	no	—
66	#117×C₅₃	—	2	2	no	—
67	#117×C₅₃	C₂³	1	2	yes	—
68	C₂×C₄²×C₅₃	—	2	16	no	—
69	C₂×C₄²×C₅₃	—	2	2	no	—
70	C₂×C₄²×C₅₃	—	2	8	no	—
71	C₂×C₄²×C₅₃	—	2	4	no	—
72	C₂×C₄²×C₅₃	—	2	4	no	—
73	C₂×C₄²×C₅₃	—	2	2	no	—
74	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
75	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
76	C₂×C₄²×C₅₃	D₄	1	2	yes	—
77	C₂×C₄²×C₅₃	D₄	1	2	yes	—
78	C₂×C₄²×C₅₃	D₄	1	2	yes	—
79	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
80	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₅₃	—	2	4	no	—
83	#104×C₅₃	—	2	16	no	—
84	C₂²⋊C₈×C₅₃	—	2	4	no	—
85	#97×C₅₃	—	2	2	no	—
86	C₄×C₈×C₅₃	—	2	2	no	—
87	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
88	#41×C₅₃	—	2	8	no	—
89	#41×C₅₃	—	4	8	no	—
90	C₂×Q₈×C₅₃	—	2	2	no	—
91	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
92	C₂×D₄×C₅₃	—	2	4	no	—
93	C₂×D₄×C₅₃	—	2	2	no	—
94	C₂×D₄×C₅₃	—	2	2	no	—
95	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₅₃	—	2	16	no	—
97	C₂²×C₄×C₅₃	—	2	8	no	—
98	C₂²×C₄×C₅₃	—	2	4	no	—
99	C₂²×C₄×C₅₃	—	2	8	no	—
100	C₂²×C₄×C₅₃	—	2	4	no	—
101	C₂²×C₄×C₅₃	—	2	4	no	—
102	C₂²×C₄×C₅₃	—	2	2	no	—
103	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
104	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
105	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
106	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
107	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
113	M₁₆×C₅₃	—	2	8	no	—
114	C₂×C₈×C₅₃	—	2	4	no	—
115	C₄⋊C₄×C₅₃	—	2	2	no	—
116	C₄⋊C₄×C₅₃	—	2	2	no	—
117	C₄⋊C₄×C₅₃	—	2	4	no	—
118	C₄⋊C₄×C₅₃	—	2	4	no	—
119	C₄⋊C₄×C₅₃	—	2	4	no	—
120	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
121	C₂²⋊C₄×C₅₃	—	2	4	no	—
122	C₂²⋊C₄×C₅₃	—	2	2	no	—
123	C₂²⋊C₄×C₅₃	—	2	4	no	—
124	C₂²⋊C₄×C₅₃	—	2	4	no	—
125	C₂²⋊C₄×C₅₃	—	2	4	no	—
126	C₄²×C₅₃	—	2	16	no	—
127	C₄²×C₅₃	—	2	16	no	—
128	C₄²×C₅₃	—	2	8	no	—
129	C₄²×C₅₃	—	2	4	no	—
130	C₄²×C₅₃	—	2	2	no	—
131	C₄²×C₅₃	—	2	4	no	—
132	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
133	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
134	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
135	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
136	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
137	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
138	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₅₃	C₂×D₄	1	2	yes	—
141	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
142	C₄²×C₅₃	#41	1	8	yes	complemented
143	C₂³×C₅₃	—	2	2	no	—
144	C₂³×C₅₃	—	2	4	no	—
145	C₂³×C₅₃	—	2	2	no	—
146	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
147	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
148	Q₈×C₅₃	—	2	4	no	—
149	Q₈×C₅₃	—	4	4	no	—
150	D₄×C₅₃	—	2	4	no	—
151	D₄×C₅₃	—	4	4	no	—
152	D₄×C₅₃	—	4	8	no	—
153	D₄×C₅₃	—	4	4	no	—
154	C₂×C₄×C₅₃	—	2	32	no	—
155	C₂×C₄×C₅₃	—	2	16	no	—
156	C₂×C₄×C₅₃	—	2	8	no	—
157	C₂×C₄×C₅₃	—	2	2	no	—
158	C₂×C₄×C₅₃	—	2	4	no	—
159	C₂×C₄×C₅₃	—	2	4	no	—
160	C₂×C₄×C₅₃	—	2	4	no	—
161	C₂×C₄×C₅₃	—	2	4	no	—
162	C₂×C₄×C₅₃	—	2	4	no	—
163	C₂×C₄×C₅₃	—	2	4	no	—
164	C₂×C₄×C₅₃	—	2	4	no	—
165	C₂×C₄×C₅₃	—	2	4	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	4	4	no	—
168	C₂×C₄×C₅₃	—	4	8	no	—
169	C₂×C₄×C₅₃	#104	1	8	yes	complemented
170	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
171	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
172	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
173	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
174	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
175	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
178	C₈×C₅₃	—	4	4	no	—
179	#21415	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
180	C₂²×C₅₃	—	2	8	no	—
181	C₂²×C₅₃	—	2	2	no	—
182	C₂²×C₅₃	—	2	2	no	—
183	C₂²×C₅₃	—	2	2	no	—
184	C₂²×C₅₃	—	4	4	no	—
185	C₂²×C₅₃	—	4	2	no	—
186	C₂²×C₅₃	C₂×#104	1	4	yes	—
187	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
188	C₂²×C₅₃	#525	1	2	yes	—
189	C₄×C₅₃	—	2	16	no	—
190	C₄×C₅₃	—	2	2	no	—
191	C₄×C₅₃	—	2	2	no	—
192	C₄×C₅₃	—	2	2	no	—
193	C₄×C₅₃	—	4	4	no	—
194	C₄×C₅₃	—	4	2	no	—
195	C₄×C₅₃	C₂×#104	1	4	yes	—
196	C₄×C₅₃	C₂×#104	1	4	yes	—
197	C₄×C₅₃	C₄×#41	1	2	yes	—
198	C₄×C₅₃	#532	1	2	yes	—
199	#3444	C₂×C₅₃	1	1	yes	characteristic
200	#2920	C₂×C₅₃	1	1	yes	characteristic, complemented
201	C₂×C₄³	C₂×C₅₃	1	1	yes	characteristic, complemented
202	#1994	C₂×C₅₃	1	2	yes	complemented
203	#1962	C₂×C₅₃	1	2	yes	complemented
204	#1896	C₂×C₅₃	1	4	yes	complemented
205	C₄×#104	C₂×C₅₃	1	4	yes	complemented
206	C₂×C₅₃	—	2	2	no	—
207	C₂×C₅₃	—	2	4	no	—
208	C₂×C₅₃	—	8	2	no	—
209	C₂×C₅₃	#2297	1	1	yes	characteristic
210	C₂×C₅₃	C₂²×#104	1	2	yes	—
211	#533	C₄×C₅₃	1	2	yes	complemented
212	#533	C₂²×C₅₃	1	1	yes	characteristic
213	#532	C₄×C₅₃	1	2	yes	complemented
214	#532	C₂²×C₅₃	1	1	yes	characteristic
215	#529	—	2	2	no	—
216	#529	—	2	2	no	—
217	#529	C₄×C₅₃	1	4	yes	complemented
218	#525	—	2	2	no	—
219	#525	C₂²×C₅₃	1	2	yes	—
220	C₄×#41	—	2	2	no	—
221	C₄×#41	C₂²×C₅₃	1	2	yes	—
222	C₂²×C₄²	—	2	2	no	—
223	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
224	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
225	#432	—	2	2	no	—
226	#431	—	2	2	no	—
227	#423	C₂²×C₅₃	1	2	yes	—
228	C₂×#104	C₂²×C₅₃	1	8	yes	—
229	#407	C₂²×C₅₃	1	2	yes	—
230	C₄×M₁₆	C₂²×C₅₃	1	2	yes	—
231	C₄³	—	2	2	no	—
232	C₄³	C₂²×C₅₃	1	2	yes	—
233	C₄³	C₂²×C₅₃	1	2	yes	—
234	#339	C₂²×C₅₃	1	8	yes	—
235	C₅₃	#21415	1	1	yes	characteristic, direct product factor, Sylow
236	C₂×#41	—	2	2	no	—
237	C₂×#41	C₂³×C₅₃	1	2	yes	—
238	C₂³×C₄	—	2	2	no	—
239	C₂³×C₄	D₄×C₅₃	1	2	yes	—
240	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
241	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
242	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic, complemented
243	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	complemented
244	#127	C₂×C₄×C₅₃	1	1	yes	characteristic, complemented
245	#126	—	2	4	no	—
246	#125	—	2	2	no	—
247	#125	C₂×C₄×C₅₃	1	2	yes	complemented
248	#124	—	2	2	no	—
249	#124	C₂×C₄×C₅₃	1	2	yes	complemented
250	#123	—	2	2	no	—
251	#123	—	2	4	no	—
252	C₂²⋊Q₈	—	2	2	no	—
253	C₂²⋊Q₈	—	2	2	no	—
254	C₂²⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
255	#121	—	2	2	no	—
256	#121	—	2	2	no	—
257	#121	C₂×C₄×C₅₃	1	4	yes	—
258	#120	—	2	2	no	—
259	C₄×Q₈	—	2	2	no	—
260	C₄×Q₈	—	2	2	no	—
261	C₄×Q₈	C₂×C₄×C₅₃	1	4	yes	complemented
262	C₄×D₄	—	2	4	no	—
263	C₄×D₄	—	2	4	no	—
264	C₄×D₄	—	2	4	no	—
265	C₄×D₄	—	2	2	no	—
266	C₄×D₄	—	2	2	no	—
267	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	complemented
268	#117	—	2	2	no	—
269	#117	—	2	2	no	—
270	#117	—	2	4	no	—
271	#117	C₂³×C₅₃	1	2	yes	—
272	C₂×C₄²	—	2	16	no	—
273	C₂×C₄²	—	2	2	no	—
274	C₂×C₄²	—	2	4	no	—
275	C₂×C₄²	—	2	4	no	—
276	C₂×C₄²	—	2	2	no	—
277	C₂×C₄²	—	2	8	no	—
278	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
279	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
280	C₂×C₄²	D₄×C₅₃	1	2	yes	—
281	C₂×C₄²	D₄×C₅₃	1	2	yes	—
282	C₂×C₄²	D₄×C₅₃	1	2	yes	—
283	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
284	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
285	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
286	C₄⋊C₈	—	2	4	no	—
287	#104	—	2	16	no	—
288	C₂²⋊C₈	—	2	4	no	—
289	#97	—	2	2	no	—
290	C₄×C₈	—	2	2	no	—
291	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
292	#41	—	2	8	no	—
293	#41	—	4	8	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
296	C₂×D₄	—	2	4	no	—
297	C₂×D₄	—	2	2	no	—
298	C₂×D₄	—	2	2	no	—
299	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
300	C₂²×C₄	—	2	16	no	—
301	C₂²×C₄	—	2	8	no	—
302	C₂²×C₄	—	2	4	no	—
303	C₂²×C₄	—	2	8	no	—
304	C₂²×C₄	—	2	4	no	—
305	C₂²×C₄	—	2	4	no	—
306	C₂²×C₄	—	2	2	no	—
307	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
308	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
309	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
310	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
311	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
312	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
313	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
314	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
315	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
316	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
317	M₁₆	—	2	8	no	—
318	C₂×C₈	—	2	4	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	4	no	—
322	C₄⋊C₄	—	2	4	no	—
323	C₄⋊C₄	—	2	4	no	—
324	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
325	C₂²⋊C₄	—	2	4	no	—
326	C₂²⋊C₄	—	2	2	no	—
327	C₂²⋊C₄	—	2	4	no	—
328	C₂²⋊C₄	—	2	4	no	—
329	C₂²⋊C₄	—	2	4	no	—
330	C₄²	—	2	16	no	—
331	C₄²	—	2	16	no	—
332	C₄²	—	2	8	no	—
333	C₄²	—	2	4	no	—
334	C₄²	—	2	2	no	—
335	C₄²	—	2	4	no	—
336	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
337	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
338	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
339	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
340	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
341	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
342	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
343	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
344	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
345	C₄²	C₂×D₄×C₅₃	1	2	yes	—
346	C₄²	#41×C₅₃	1	8	yes	complemented
347	C₂³	—	2	2	no	—
348	C₂³	—	2	4	no	—
349	C₂³	—	2	2	no	—
350	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
351	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
352	Q₈	—	2	4	no	—
353	Q₈	—	4	4	no	—
354	D₄	—	2	4	no	—
355	D₄	—	4	4	no	—
356	D₄	—	4	8	no	—
357	D₄	—	4	4	no	—
358	C₂×C₄	—	2	16	no	—
359	C₂×C₄	—	2	32	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	8	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	4	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	4	no	—
368	C₂×C₄	—	2	4	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	4	no	—
371	C₂×C₄	—	4	4	no	—
372	C₂×C₄	—	4	8	no	—
373	C₂×C₄	#104×C₅₃	1	8	yes	complemented
374	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
375	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
376	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
377	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
378	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
379	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
380	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
381	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
382	C₈	—	4	4	no	—
383	C₂²	—	2	8	no	—
384	C₂²	—	2	2	no	—
385	C₂²	—	2	2	no	—
386	C₂²	—	2	2	no	—
387	C₂²	—	4	2	no	—
388	C₂²	—	4	4	no	—
389	C₂²	C₂×#104×C₅₃	1	4	yes	—
390	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
391	C₂²	C₅₃×#525	1	2	yes	—
392	C₄	—	2	16	no	—
393	C₄	—	2	2	no	—
394	C₄	—	2	2	no	—
395	C₄	—	2	2	no	—
396	C₄	—	4	2	no	—
397	C₄	—	4	4	no	—
398	C₄	C₂×#104×C₅₃	1	4	yes	—
399	C₄	C₂×#104×C₅₃	1	4	yes	—
400	C₄	C₄×#41×C₅₃	1	2	yes	—
401	C₄	C₅₃×#532	1	2	yes	—
402	C₂	—	2	2	no	—
403	C₂	—	2	4	no	—
404	C₂	—	8	2	no	—
405	C₂	C₅₃×#2297	1	1	yes	characteristic
406	C₂	C₂²×#104×C₅₃	1	2	yes	—

	subgroup	G/H
center	C₄²×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₅₃×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2920	C₂	semidirect
3. C₂×C₄³×C₅₃	C₂	semidirect
4. C₅₃×#1994	C₂	semidirect
5. C₅₃×#1962	C₂	semidirect
6. C₅₃×#1896	C₂	semidirect
7. C₄×#104×C₅₃	C₂	semidirect
8. C₅₃×#533	C₄	semidirect
10. C₅₃×#532	C₄	semidirect
14. C₅₃×#529	C₄	semidirect
38. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
39. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
40. #127×C₅₃	C₂×C₄	semidirect
43. #125×C₅₃	C₂×C₄	semidirect
45. #124×C₅₃	C₂×C₄	semidirect
57. C₄×Q₈×C₅₃	C₂×C₄	semidirect
63. C₄×D₄×C₅₃	C₂×C₄	semidirect
142. C₄²×C₅₃	#41	semidirect
169. C₂×C₄×C₅₃	#104	semidirect
179. #21415	C₅₃	direct (N × H)
200. #2920	C₂×C₅₃	semidirect
201. C₂×C₄³	C₂×C₅₃	semidirect
202. #1994	C₂×C₅₃	semidirect
203. #1962	C₂×C₅₃	semidirect
204. #1896	C₂×C₅₃	semidirect
205. C₄×#104	C₂×C₅₃	semidirect
211. #533	C₄×C₅₃	semidirect
213. #532	C₄×C₅₃	semidirect
217. #529	C₄×C₅₃	semidirect
235. C₅₃	#21415	direct (N × H)
242. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
243. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
244. #127	C₂×C₄×C₅₃	semidirect
247. #125	C₂×C₄×C₅₃	semidirect
249. #124	C₂×C₄×C₅₃	semidirect
261. C₄×Q₈	C₂×C₄×C₅₃	semidirect
267. C₄×D₄	C₂×C₄×C₅₃	semidirect
346. C₄²	#41×C₅₃	semidirect
373. C₂×C₄	#104×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3444	2
C₅₃×#2920	2
C₂×C₄³×C₅₃	2
C₅₃×#1994	2
C₅₃×#1962	2
C₅₃×#1896	2
C₄×#104×C₅₃	2
#21415	53

Subgroups — C53×#21415 #27228927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21415 #27228927