Subgroups — C₅₃×#17438 #27224950

Order 13568.

1290 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	C₄×H(5,𝔽₂)×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2988	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1478	C₂	1	3	yes	maximal, complemented
5	C₅₃×#1477	C₂	1	3	yes	maximal, complemented
6	C₅₃×#1476	C₂	1	3	yes	maximal, complemented
7	C₂×C₅₃×D₄⋊C₈	C₂	1	3	yes	maximal, complemented
8	C₂×H(5,𝔽₂)×C₅₃	C₄	1	1	yes	characteristic
9	C₅₃×#520	—	2	1	no	—
10	C₅₃×#520	C₄	1	3	yes	—
11	C₅₃×#520	C₄	1	1	yes	characteristic
12	C₅₃×#518	—	2	3	no	—
13	C₅₃×#518	C₄	1	3	yes	—
14	C₄×#41×C₅₃	—	2	1	no	—
15	C₄×#41×C₅₃	C₂²	1	3	yes	—
16	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₅₃	—	2	3	no	—
18	C₂×C₄×D₄×C₅₃	C₂²	1	3	yes	—
19	C₈×Q₈×C₅₃	—	2	1	no	—
20	C₈×Q₈×C₅₃	—	2	1	no	—
21	C₈×D₄×C₅₃	—	2	3	no	—
22	C₈×D₄×C₅₃	—	2	3	no	—
23	C₅₃×#431	C₂²	1	3	yes	complemented
24	C₅₃×#431	C₂²	1	3	yes	complemented
25	C₂×C₄⋊C₈×C₅₃	C₂²	1	3	yes	complemented
26	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	complemented
27	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
28	C₅₃×Q₈⋊C₈	C₂²	1	3	yes	complemented
29	C₅₃×D₄⋊C₈	C₂²	1	3	yes	complemented
30	C₅₃×D₄⋊C₈	C₂²	1	6	yes	complemented
31	C₅₃×D₄⋊C₈	C₂²	1	3	yes	complemented
32	H(5,𝔽₂)×C₅₃	—	2	4	no	—
33	H(5,𝔽₂)×C₅₃	C₈	1	8	yes	complemented
34	C₂×#41×C₅₃	—	2	1	no	—
35	C₂×#41×C₅₃	C₂×C₄	1	3	yes	—
36	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×D₄×C₅₃	—	2	3	no	—
38	C₂²×D₄×C₅₃	C₂×C₄	1	3	yes	—
39	C₂³×C₄×C₅₃	—	2	3	no	—
40	C₂²×C₈×C₅₃	—	2	3	no	—
41	C₂²×C₈×C₅₃	—	2	3	no	—
42	C₄×Q₈×C₅₃	—	2	1	no	—
43	C₄×Q₈×C₅₃	C₂×C₄	1	3	yes	—
44	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
45	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
46	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₄×D₄×C₅₃	—	2	3	no	—
48	C₄×D₄×C₅₃	—	2	3	no	—
49	C₄×D₄×C₅₃	—	2	6	no	—
50	C₄×D₄×C₅₃	—	2	3	no	—
51	C₄×D₄×C₅₃	—	2	3	no	—
52	C₄×D₄×C₅₃	—	2	3	no	—
53	C₄×D₄×C₅₃	—	2	3	no	—
54	C₄×D₄×C₅₃	—	2	3	no	—
55	C₄×D₄×C₅₃	C₂×C₄	1	6	yes	—
56	C₄×D₄×C₅₃	C₂×C₄	1	3	yes	—
57	C₄×D₄×C₅₃	D₄	1	3	yes	—
58	C₄×D₄×C₅₃	D₄	1	3	yes	—
59	C₄×D₄×C₅₃	C₂³	1	3	yes	—
60	#117×C₅₃	—	2	3	no	—
61	#117×C₅₃	—	2	3	no	—
62	#117×C₅₃	C₂×C₄	1	3	yes	—
63	#117×C₅₃	C₂×C₄	1	3	yes	—
64	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
65	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
66	C₂×C₂²⋊C₄×C₅₃	—	2	6	no	—
67	C₂×C₂²⋊C₄×C₅₃	—	2	3	no	—
68	C₂×C₄²×C₅₃	—	2	3	no	—
69	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
70	C₄⋊C₈×C₅₃	—	2	3	no	—
71	C₄⋊C₈×C₅₃	—	2	3	no	—
72	C₄⋊C₈×C₅₃	C₂³	1	3	yes	—
73	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₂²⋊C₈×C₅₃	—	2	3	no	—
75	C₂²⋊C₈×C₅₃	—	2	3	no	—
76	C₄×C₈×C₅₃	—	2	3	no	—
77	C₄×C₈×C₅₃	—	2	3	no	—
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₅₃	C₂³	1	3	yes	—
80	C₂⁴×C₅₃	—	2	3	no	—
81	#41×C₅₃	—	2	8	no	—
82	#41×C₅₃	—	2	6	no	—
83	#41×C₅₃	C₂×C₈	1	4	yes	—
84	#41×C₅₃	C₂×C₈	1	12	yes	complemented
85	#41×C₅₃	M₁₆	1	4	yes	—
86	C₂×Q₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×D₄×C₅₃	—	2	3	no	—
89	C₂×D₄×C₅₃	—	2	24	no	—
90	C₂×D₄×C₅₃	—	2	3	no	—
91	C₂×D₄×C₅₃	—	2	6	no	—
92	C₂×D₄×C₅₃	C₂²⋊C₄	1	3	yes	—
93	C₂×D₄×C₅₃	C₂×C₈	1	12	yes	complemented
94	C₂×D₄×C₅₃	C₂²×C₄	1	3	yes	—
95	C₂²×C₄×C₅₃	—	2	3	no	—
96	C₂²×C₄×C₅₃	—	2	3	no	—
97	C₂²×C₄×C₅₃	—	2	3	no	—
98	C₂²×C₄×C₅₃	—	2	3	no	—
99	C₂²×C₄×C₅₃	—	2	6	no	—
100	C₂²×C₄×C₅₃	—	8	3	no	—
101	C₂²×C₄×C₅₃	—	8	3	no	—
102	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
103	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
104	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
105	C₂×C₈×C₅₃	—	2	1	no	—
106	C₂×C₈×C₅₃	—	2	3	no	—
107	C₂×C₈×C₅₃	—	2	1	no	—
108	C₂×C₈×C₅₃	—	2	3	no	—
109	C₂×C₈×C₅₃	—	4	6	no	—
110	C₂×C₈×C₅₃	—	8	3	no	—
111	C₄⋊C₄×C₅₃	—	2	3	no	—
112	C₄⋊C₄×C₅₃	—	2	3	no	—
113	C₄⋊C₄×C₅₃	—	2	1	no	—
114	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	3	yes	—
115	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	3	yes	—
116	C₄⋊C₄×C₅₃	C₂²×C₄	1	3	yes	—
117	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂²⋊C₄×C₅₃	—	2	3	no	—
119	C₂²⋊C₄×C₅₃	—	2	6	no	—
120	C₂²⋊C₄×C₅₃	—	2	3	no	—
121	C₂²⋊C₄×C₅₃	—	2	3	no	—
122	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	6	yes	—
123	C₄²×C₅₃	—	2	3	no	—
124	C₄²×C₅₃	—	2	3	no	—
125	C₄²×C₅₃	—	2	1	no	—
126	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
127	C₄²×C₅₃	C₂×D₄	1	3	yes	—
128	C₄²×C₅₃	C₂×D₄	1	3	yes	—
129	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
130	C₂³×C₅₃	—	2	3	no	—
131	C₂³×C₅₃	—	2	12	no	—
132	C₂³×C₅₃	—	8	3	no	—
133	C₂³×C₅₃	—	8	3	no	—
134	C₂³×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
135	Q₈×C₅₃	—	2	1	no	—
136	Q₈×C₅₃	C₂²⋊C₈	1	4	yes	—
137	Q₈×C₅₃	C₂²×C₈	1	2	yes	—
138	D₄×C₅₃	—	2	12	no	—
139	D₄×C₅₃	—	2	3	no	—
140	D₄×C₅₃	—	2	12	no	—
141	D₄×C₅₃	C₂²⋊C₈	1	12	yes	—
142	D₄×C₅₃	C₂²×C₈	1	6	yes	—
143	C₂×C₄×C₅₃	—	2	1	no	—
144	C₂×C₄×C₅₃	—	2	3	no	—
145	C₂×C₄×C₅₃	—	2	3	no	—
146	C₂×C₄×C₅₃	—	2	12	no	—
147	C₂×C₄×C₅₃	—	2	3	no	—
148	C₂×C₄×C₅₃	—	2	1	no	—
149	C₂×C₄×C₅₃	—	4	3	no	—
150	C₂×C₄×C₅₃	—	4	3	no	—
151	C₂×C₄×C₅₃	—	4	3	no	—
152	C₂×C₄×C₅₃	—	8	3	no	—
153	C₂×C₄×C₅₃	—	8	6	no	—
154	C₂×C₄×C₅₃	—	8	3	no	—
155	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
156	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
157	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
158	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
159	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
160	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
163	C₈×C₅₃	—	2	2	no	—
164	C₈×C₅₃	—	4	3	no	—
165	C₈×C₅₃	—	4	3	no	—
166	C₈×C₅₃	—	4	1	no	—
167	#17438	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂²×C₅₃	—	2	3	no	—
169	C₂²×C₅₃	—	2	3	no	—
170	C₂²×C₅₃	—	2	6	no	—
171	C₂²×C₅₃	—	4	3	no	—
172	C₂²×C₅₃	—	4	3	no	—
173	C₂²×C₅₃	—	8	12	no	—
174	C₂²×C₅₃	C₂×C₂²⋊C₈	1	6	yes	—
175	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
176	C₄×C₅₃	—	2	2	no	—
177	C₄×C₅₃	—	2	3	no	—
178	C₄×C₅₃	—	2	1	no	—
179	C₄×C₅₃	—	2	3	no	—
180	C₄×C₅₃	—	4	3	no	—
181	C₄×C₅₃	—	4	1	no	—
182	C₄×C₅₃	C₂×C₂²⋊C₈	1	6	yes	—
183	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
184	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
185	C₄×C₅₃	#576	1	1	yes	characteristic
186	C₄×C₅₃	#577	1	1	yes	characteristic
187	C₄×H(5,𝔽₂)	C₂×C₅₃	1	1	yes	characteristic
188	#2988	C₂×C₅₃	1	1	yes	characteristic, complemented
189	C₈×#41	C₂×C₅₃	1	1	yes	characteristic, complemented
190	#1478	C₂×C₅₃	1	3	yes	complemented
191	#1477	C₂×C₅₃	1	3	yes	complemented
192	#1476	C₂×C₅₃	1	3	yes	complemented
193	C₂×D₄⋊C₈	C₂×C₅₃	1	3	yes	complemented
194	C₂×C₅₃	—	2	6	no	—
195	C₂×C₅₃	—	4	6	no	—
196	C₂×C₅₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
197	C₂×C₅₃	#2897	1	1	yes	characteristic
198	C₂×C₅₃	#2901	1	1	yes	characteristic
199	C₂×H(5,𝔽₂)	C₄×C₅₃	1	1	yes	characteristic
200	#520	—	2	1	no	—
201	#520	C₄×C₅₃	1	3	yes	—
202	#520	C₄×C₅₃	1	1	yes	characteristic
203	#518	—	2	3	no	—
204	#518	C₄×C₅₃	1	3	yes	—
205	C₄×#41	—	2	1	no	—
206	C₄×#41	C₂²×C₅₃	1	3	yes	—
207	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
208	C₂×C₄×D₄	—	2	3	no	—
209	C₂×C₄×D₄	C₂²×C₅₃	1	3	yes	—
210	C₈×Q₈	—	2	1	no	—
211	C₈×Q₈	—	2	1	no	—
212	C₈×D₄	—	2	3	no	—
213	C₈×D₄	—	2	3	no	—
214	#431	C₂²×C₅₃	1	3	yes	complemented
215	#431	C₂²×C₅₃	1	3	yes	complemented
216	C₂×C₄⋊C₈	C₂²×C₅₃	1	3	yes	complemented
217	C₂×C₄×C₈	C₂²×C₅₃	1	3	yes	complemented
218	Q₈⋊C₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
219	Q₈⋊C₈	C₂²×C₅₃	1	3	yes	complemented
220	D₄⋊C₈	C₂²×C₅₃	1	3	yes	complemented
221	D₄⋊C₈	C₂²×C₅₃	1	6	yes	complemented
222	D₄⋊C₈	C₂²×C₅₃	1	3	yes	complemented
223	C₅₃	#17438	1	1	yes	characteristic, direct product factor, Sylow
224	H(5,𝔽₂)	—	2	4	no	—
225	H(5,𝔽₂)	C₈×C₅₃	1	8	yes	complemented
226	C₂×#41	—	2	1	no	—
227	C₂×#41	C₂×C₄×C₅₃	1	3	yes	—
228	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
229	C₂²×D₄	—	2	3	no	—
230	C₂²×D₄	C₂×C₄×C₅₃	1	3	yes	—
231	C₂³×C₄	—	2	3	no	—
232	C₂²×C₈	—	2	3	no	—
233	C₂²×C₈	—	2	3	no	—
234	C₄×Q₈	—	2	1	no	—
235	C₄×Q₈	C₂×C₄×C₅₃	1	3	yes	—
236	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
237	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
238	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
239	C₄×D₄	—	2	3	no	—
240	C₄×D₄	—	2	3	no	—
241	C₄×D₄	—	2	3	no	—
242	C₄×D₄	—	2	3	no	—
243	C₄×D₄	—	2	3	no	—
244	C₄×D₄	—	2	6	no	—
245	C₄×D₄	—	2	3	no	—
246	C₄×D₄	—	2	3	no	—
247	C₄×D₄	C₂×C₄×C₅₃	1	6	yes	—
248	C₄×D₄	C₂×C₄×C₅₃	1	3	yes	—
249	C₄×D₄	D₄×C₅₃	1	3	yes	—
250	C₄×D₄	D₄×C₅₃	1	3	yes	—
251	C₄×D₄	C₂³×C₅₃	1	3	yes	—
252	#117	—	2	3	no	—
253	#117	—	2	3	no	—
254	#117	C₂×C₄×C₅₃	1	3	yes	—
255	#117	C₂×C₄×C₅₃	1	3	yes	—
256	C₂×C₄⋊C₄	—	2	3	no	—
257	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	3	yes	—
258	C₂×C₂²⋊C₄	—	2	3	no	—
259	C₂×C₂²⋊C₄	—	2	6	no	—
260	C₂×C₄²	—	2	3	no	—
261	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
262	C₄⋊C₈	—	2	3	no	—
263	C₄⋊C₈	—	2	3	no	—
264	C₄⋊C₈	C₂³×C₅₃	1	3	yes	—
265	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
266	C₂²⋊C₈	—	2	3	no	—
267	C₂²⋊C₈	—	2	3	no	—
268	C₄×C₈	—	2	3	no	—
269	C₄×C₈	—	2	3	no	—
270	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
271	C₄×C₈	C₂³×C₅₃	1	3	yes	—
272	C₂⁴	—	2	3	no	—
273	#41	—	2	8	no	—
274	#41	—	2	6	no	—
275	#41	C₂×C₈×C₅₃	1	4	yes	—
276	#41	C₂×C₈×C₅₃	1	12	yes	complemented
277	#41	M₁₆×C₅₃	1	4	yes	—
278	C₂×Q₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
279	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
280	C₂×D₄	—	2	3	no	—
281	C₂×D₄	—	2	24	no	—
282	C₂×D₄	—	2	3	no	—
283	C₂×D₄	—	2	6	no	—
284	C₂×D₄	C₂²⋊C₄×C₅₃	1	3	yes	—
285	C₂×D₄	C₂×C₈×C₅₃	1	12	yes	complemented
286	C₂×D₄	C₂²×C₄×C₅₃	1	3	yes	—
287	C₂²×C₄	—	2	3	no	—
288	C₂²×C₄	—	2	3	no	—
289	C₂²×C₄	—	2	3	no	—
290	C₂²×C₄	—	2	3	no	—
291	C₂²×C₄	—	2	6	no	—
292	C₂²×C₄	—	8	3	no	—
293	C₂²×C₄	—	8	3	no	—
294	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
295	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
296	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	3	no	—
300	C₂×C₈	—	2	3	no	—
301	C₂×C₈	—	4	6	no	—
302	C₂×C₈	—	8	3	no	—
303	C₄⋊C₄	—	2	3	no	—
304	C₄⋊C₄	—	2	3	no	—
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	3	yes	—
307	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	3	yes	—
308	C₄⋊C₄	C₂²×C₄×C₅₃	1	3	yes	—
309	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
310	C₂²⋊C₄	—	2	3	no	—
311	C₂²⋊C₄	—	2	3	no	—
312	C₂²⋊C₄	—	2	6	no	—
313	C₂²⋊C₄	—	2	3	no	—
314	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	6	yes	—
315	C₄²	—	2	3	no	—
316	C₄²	—	2	3	no	—
317	C₄²	—	2	1	no	—
318	C₄²	C₂²×C₄×C₅₃	1	3	yes	—
319	C₄²	C₂×D₄×C₅₃	1	3	yes	—
320	C₄²	C₂×D₄×C₅₃	1	3	yes	—
321	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
322	C₂³	—	2	3	no	—
323	C₂³	—	2	12	no	—
324	C₂³	—	8	3	no	—
325	C₂³	—	8	3	no	—
326	C₂³	C₂×C₂²⋊C₄×C₅₃	1	3	yes	—
327	Q₈	—	2	1	no	—
328	Q₈	C₂²⋊C₈×C₅₃	1	4	yes	—
329	Q₈	C₂²×C₈×C₅₃	1	2	yes	—
330	D₄	—	2	12	no	—
331	D₄	—	2	3	no	—
332	D₄	—	2	12	no	—
333	D₄	C₂²⋊C₈×C₅₃	1	12	yes	—
334	D₄	C₂²×C₈×C₅₃	1	6	yes	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	3	no	—
337	C₂×C₄	—	2	3	no	—
338	C₂×C₄	—	2	12	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	3	no	—
341	C₂×C₄	—	4	3	no	—
342	C₂×C₄	—	4	3	no	—
343	C₂×C₄	—	4	3	no	—
344	C₂×C₄	—	8	3	no	—
345	C₂×C₄	—	8	6	no	—
346	C₂×C₄	—	8	3	no	—
347	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	3	yes	—
348	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	3	yes	—
349	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	3	yes	—
350	C₂×C₄	C₂²×C₈×C₅₃	1	6	yes	—
351	C₂×C₄	C₂×M₁₆×C₅₃	1	6	yes	—
352	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
353	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
354	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
355	C₈	—	2	2	no	—
356	C₈	—	4	3	no	—
357	C₈	—	4	3	no	—
358	C₈	—	4	1	no	—
359	C₂²	—	2	3	no	—
360	C₂²	—	2	3	no	—
361	C₂²	—	2	6	no	—
362	C₂²	—	4	3	no	—
363	C₂²	—	4	3	no	—
364	C₂²	—	8	12	no	—
365	C₂²	C₂×C₂²⋊C₈×C₅₃	1	6	yes	—
366	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
367	C₄	—	2	2	no	—
368	C₄	—	2	3	no	—
369	C₄	—	2	1	no	—
370	C₄	—	2	3	no	—
371	C₄	—	4	3	no	—
372	C₄	—	4	1	no	—
373	C₄	C₂×C₂²⋊C₈×C₅₃	1	6	yes	—
374	C₄	C₂³×C₈×C₅₃	1	1	yes	characteristic
375	C₄	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
376	C₄	C₅₃×#576	1	1	yes	characteristic
377	C₄	C₅₃×#577	1	1	yes	characteristic
378	C₂	—	2	6	no	—
379	C₂	—	4	6	no	—
380	C₂	C₂²×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
381	C₂	C₅₃×#2897	1	1	yes	characteristic
382	C₂	C₅₃×#2901	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₄	C₂³×C₈×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂×H(5,𝔽₂)×C₅₃	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2988	C₂	semidirect
3. C₈×#41×C₅₃	C₂	semidirect
4. C₅₃×#1478	C₂	semidirect
5. C₅₃×#1477	C₂	semidirect
6. C₅₃×#1476	C₂	semidirect
7. C₂×C₅₃×D₄⋊C₈	C₂	semidirect
23. C₅₃×#431	C₂²	semidirect
24. C₅₃×#431	C₂²	semidirect
25. C₂×C₄⋊C₈×C₅₃	C₂²	semidirect
26. C₂×C₄×C₈×C₅₃	C₂²	semidirect
27. C₅₃×Q₈⋊C₈	C₂²	semidirect
28. C₅₃×Q₈⋊C₈	C₂²	semidirect
29. C₅₃×D₄⋊C₈	C₂²	semidirect
30. C₅₃×D₄⋊C₈	C₂²	semidirect
31. C₅₃×D₄⋊C₈	C₂²	semidirect
33. H(5,𝔽₂)×C₅₃	C₈	semidirect
84. #41×C₅₃	C₂×C₈	semidirect
93. C₂×D₄×C₅₃	C₂×C₈	semidirect
167. #17438	C₅₃	direct (N × H)
188. #2988	C₂×C₅₃	semidirect
189. C₈×#41	C₂×C₅₃	semidirect
190. #1478	C₂×C₅₃	semidirect
191. #1477	C₂×C₅₃	semidirect
192. #1476	C₂×C₅₃	semidirect
193. C₂×D₄⋊C₈	C₂×C₅₃	semidirect
214. #431	C₂²×C₅₃	semidirect
215. #431	C₂²×C₅₃	semidirect
216. C₂×C₄⋊C₈	C₂²×C₅₃	semidirect
217. C₂×C₄×C₈	C₂²×C₅₃	semidirect
218. Q₈⋊C₈	C₂²×C₅₃	semidirect
219. Q₈⋊C₈	C₂²×C₅₃	semidirect
220. D₄⋊C₈	C₂²×C₅₃	semidirect
221. D₄⋊C₈	C₂²×C₅₃	semidirect
222. D₄⋊C₈	C₂²×C₅₃	semidirect
223. C₅₃	#17438	direct (N × H)
225. H(5,𝔽₂)	C₈×C₅₃	semidirect
276. #41	C₂×C₈×C₅₃	semidirect
285. C₂×D₄	C₂×C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×H(5,𝔽₂)×C₅₃	2
C₅₃×#2988	2
C₈×#41×C₅₃	2
C₅₃×#1478	2
C₅₃×#1477	2
C₅₃×#1476	2
C₂×C₅₃×D₄⋊C₈	2
#17438	53

Subgroups — C53×#17438 #27224950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17438 #27224950