Subgroups — C₅₃×#61411 #27268923

Order 13568.

1838 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×#3509	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3398	C₂	1	8	yes	maximal, complemented
3	C₅₃×#3395	C₂	1	4	yes	maximal, complemented
4	C₅₃×#3393	C₂	1	4	yes	maximal, complemented
5	C₅₃×#3156	C₂	1	2	yes	maximal
6	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#3044	C₂	1	4	yes	maximal, complemented
8	C₅₃×#3039	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#2950	C₂	1	2	yes	maximal, complemented
11	C₄×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂×#137×C₅₃	C₂²	1	4	yes	—
13	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#557	C₂²	1	8	yes	—
17	C₅₃×#556	C₂²	1	8	yes	—
18	C₅₃×#552	—	2	8	no	—
19	C₅₃×#541	—	2	4	no	—
20	C₅₃×#536	C₂²	1	4	yes	—
21	C₅₃×#533	—	2	4	no	—
22	C₅₃×#533	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
24	C₅₃×#519	C₂²	1	4	yes	—
25	C₄×#41×C₅₃	C₂²	1	2	yes	—
26	C₅₃×#497	C₂²	1	8	yes	—
27	C₅₃×#495	C₂²	1	4	yes	—
28	C₅₃×#494	C₂²	1	4	yes	—
29	C₅₃×#494	C₂²	1	4	yes	—
30	C₅₃×#494	C₂²	1	4	yes	—
31	C₅₃×#464	C₂²	1	16	yes	—
32	C₅₃×#462	C₂²	1	8	yes	—
33	C₅₃×#462	C₂²	1	16	yes	—
34	C₅₃×#461	C₂²	1	8	yes	—
35	C₅₃×#446	C₂²	1	4	yes	—
36	C₈×Q₈×C₅₃	C₂²	1	4	yes	—
37	C₅₃×#441	C₂²	1	8	yes	—
38	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
39	C₄×Q₁₆×C₅₃	C₂²	1	8	yes	—
40	C₄×SD₁₆×C₅₃	C₂²	1	4	yes	—
41	C₄×D₈×C₅₃	C₂²	1	2	yes	—
42	C₅₃×#427	C₂²	1	1	yes	characteristic
43	C₅₃×#424	C₂²	1	2	yes	—
44	C₅₃×#423	C₂²	1	1	yes	characteristic
45	C₅₃×#419	C₂²	1	4	yes	—
46	C₅₃×#416	C₂²	1	2	yes	—
47	C₅₃×#405	C₂²	1	2	yes	—
48	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
49	C₂×#41×C₅₃	C₂³	1	2	yes	—
50	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
51	#137×C₅₃	—	4	8	no	—
52	#135×C₅₃	—	2	4	no	—
53	C₂×Q₁₆×C₅₃	—	2	8	no	—
54	C₂×Q₁₆×C₅₃	—	4	4	no	—
55	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×Q₁₆×C₅₃	C₂³	1	4	yes	—
57	C₂×Q₁₆×C₅₃	C₂³	1	4	yes	—
58	C₂×Q₁₆×C₅₃	C₂³	1	8	yes	—
59	C₂×Q₁₆×C₅₃	C₂³	1	8	yes	—
60	C₂×SD₁₆×C₅₃	C₂³	1	2	yes	—
61	C₂×SD₁₆×C₅₃	C₂³	1	8	yes	—
62	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
63	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
64	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
66	C₄⋊Q₈×C₅₃	—	2	2	no	—
67	C₄⋊Q₈×C₅₃	—	2	2	no	—
68	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
69	C₄⋊Q₈×C₅₃	C₂³	1	8	yes	—
70	C₄⋊Q₈×C₅₃	C₂³	1	8	yes	—
71	#126×C₅₃	—	2	16	no	—
72	#125×C₅₃	—	2	8	no	—
73	#125×C₅₃	—	2	16	no	—
74	#125×C₅₃	D₄	1	2	yes	—
75	#124×C₅₃	C₂³	1	8	yes	—
76	#123×C₅₃	—	2	8	no	—
77	C₂²⋊Q₈×C₅₃	—	2	8	no	—
78	C₂²⋊Q₈×C₅₃	—	2	8	no	—
79	C₂²⋊Q₈×C₅₃	D₄	1	4	yes	—
80	C₄×Q₈×C₅₃	—	2	8	no	—
81	C₄×Q₈×C₅₃	—	2	8	no	—
82	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
83	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
84	C₄×Q₈×C₅₃	C₂³	1	16	yes	—
85	C₄×D₄×C₅₃	—	2	4	no	—
86	C₄×D₄×C₅₃	C₂³	1	4	yes	—
87	#117×C₅₃	—	2	8	no	—
88	#117×C₅₃	—	2	8	no	—
89	#117×C₅₃	—	2	2	no	—
90	#117×C₅₃	C₂³	1	2	yes	—
91	#117×C₅₃	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
93	C₂×C₄²×C₅₃	—	2	2	no	—
94	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
95	#107×C₅₃	C₂³	1	2	yes	—
96	#106×C₅₃	C₂³	1	2	yes	—
97	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
98	C₄⋊C₈×C₅₃	C₂³	1	8	yes	—
99	#103×C₅₃	C₂³	1	4	yes	—
100	#103×C₅₃	C₂³	1	16	yes	—
101	#102×C₅₃	C₂³	1	4	yes	—
102	#97×C₅₃	C₂³	1	4	yes	—
103	C₄×C₈×C₅₃	C₂³	1	2	yes	—
104	C₄×C₈×C₅₃	C₂³	1	2	yes	—
105	C₄×C₈×C₅₃	C₂³	1	4	yes	—
106	#41×C₅₃	—	2	4	no	—
107	#41×C₅₃	—	2	4	no	—
108	C₂×Q₈×C₅₃	—	2	4	no	—
109	C₂×Q₈×C₅₃	—	2	8	no	—
110	C₂×Q₈×C₅₃	—	2	8	no	—
111	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
112	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
113	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
114	C₂×D₄×C₅₃	—	2	2	no	—
115	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
116	C₂²×C₄×C₅₃	—	2	4	no	—
117	C₂²×C₄×C₅₃	—	2	4	no	—
118	C₂²×C₄×C₅₃	—	2	2	no	—
119	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
121	Q₁₆×C₅₃	—	2	8	no	—
122	Q₁₆×C₅₃	—	2	8	no	—
123	Q₁₆×C₅₃	—	2	2	no	—
124	Q₁₆×C₅₃	—	4	8	no	—
125	Q₁₆×C₅₃	—	4	8	no	—
126	SD₁₆×C₅₃	—	2	4	no	—
127	SD₁₆×C₅₃	—	4	8	no	—
128	D₈×C₅₃	—	2	2	no	—
129	M₁₆×C₅₃	—	2	4	no	—
130	C₂×C₈×C₅₃	—	2	2	no	—
131	C₂×C₈×C₅₃	#41	1	4	yes	—
132	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
133	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
134	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
135	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
136	C₄⋊C₄×C₅₃	—	2	8	no	—
137	C₄⋊C₄×C₅₃	—	2	8	no	—
138	C₄⋊C₄×C₅₃	—	2	4	no	—
139	C₄⋊C₄×C₅₃	—	2	16	no	—
140	C₄⋊C₄×C₅₃	—	2	16	no	—
141	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
142	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
143	C₄⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
144	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
145	C₂²⋊C₄×C₅₃	—	2	2	no	—
146	C₂²⋊C₄×C₅₃	—	2	4	no	—
147	C₂²⋊C₄×C₅₃	—	2	16	no	—
148	C₂²⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
149	C₄²×C₅₃	—	2	2	no	—
150	C₄²×C₅₃	—	2	4	no	—
151	C₄²×C₅₃	—	2	16	no	—
152	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₅₃	C₂⁴	1	2	yes	—
155	C₄²×C₅₃	C₂⁴	1	4	yes	—
156	C₂³×C₅₃	—	2	2	no	—
157	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
158	Q₈×C₅₃	—	2	8	no	—
159	Q₈×C₅₃	—	2	8	no	—
160	Q₈×C₅₃	—	2	2	no	—
161	Q₈×C₅₃	—	2	8	no	—
162	Q₈×C₅₃	—	2	2	no	—
163	Q₈×C₅₃	—	4	8	no	—
164	D₄×C₅₃	—	2	2	no	—
165	D₄×C₅₃	—	2	2	no	—
166	D₄×C₅₃	—	4	4	no	—
167	C₂×C₄×C₅₃	—	2	4	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	2	8	no	—
170	C₂×C₄×C₅₃	—	2	4	no	—
171	C₂×C₄×C₅₃	—	4	8	no	—
172	C₂×C₄×C₅₃	—	4	4	no	—
173	C₂×C₄×C₅₃	—	4	8	no	—
174	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
175	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
176	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
178	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
179	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
180	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
181	C₈×C₅₃	—	2	2	no	—
182	C₈×C₅₃	—	2	4	no	—
183	C₈×C₅₃	C₂×#41	1	2	yes	—
184	C₈×C₅₃	C₂×#41	1	2	yes	—
185	#61411	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
186	C₂²×C₅₃	—	2	2	no	—
187	C₂²×C₅₃	—	4	2	no	—
188	C₂²×C₅₃	—	4	4	no	—
189	C₂²×C₅₃	#550	1	2	yes	—
190	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
191	C₄×C₅₃	—	2	2	no	—
192	C₄×C₅₃	—	2	8	no	—
193	C₄×C₅₃	—	4	2	no	—
194	C₄×C₅₃	—	4	8	no	—
195	C₄×C₅₃	—	4	4	no	—
196	C₄×C₅₃	#550	1	2	yes	—
197	C₄×C₅₃	#578	1	4	yes	—
198	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
199	C₄×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
200	#3509	C₂×C₅₃	1	2	yes	complemented
201	#3398	C₂×C₅₃	1	8	yes	complemented
202	#3395	C₂×C₅₃	1	4	yes	complemented
203	#3393	C₂×C₅₃	1	4	yes	complemented
204	#3156	C₂×C₅₃	1	2	yes	—
205	C₂×#494	C₂×C₅₃	1	1	yes	characteristic, complemented
206	#3044	C₂×C₅₃	1	4	yes	complemented
207	#3039	C₂×C₅₃	1	2	yes	complemented
208	#2960	C₂×C₅₃	1	1	yes	characteristic, complemented
209	#2950	C₂×C₅₃	1	2	yes	complemented
210	C₄×#135	C₂×C₅₃	1	1	yes	characteristic
211	C₂×C₅₃	—	2	2	no	—
212	C₂×C₅₃	—	8	2	no	—
213	C₂×C₅₃	C₂×#550	1	1	yes	characteristic
214	C₂×C₅₃	C₂×#578	1	2	yes	—
215	C₂×#137	C₂²×C₅₃	1	4	yes	—
216	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
217	C₂²×Q₁₆	C₂²×C₅₃	1	2	yes	—
218	C₂²×Q₁₆	C₂²×C₅₃	1	2	yes	—
219	#557	C₂²×C₅₃	1	8	yes	—
220	#556	C₂²×C₅₃	1	8	yes	—
221	#552	—	2	8	no	—
222	#541	—	2	4	no	—
223	#536	C₂²×C₅₃	1	4	yes	—
224	#533	—	2	4	no	—
225	#533	C₂²×C₅₃	1	1	yes	characteristic
226	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
227	#519	C₂²×C₅₃	1	4	yes	—
228	C₄×#41	C₂²×C₅₃	1	2	yes	—
229	#497	C₂²×C₅₃	1	8	yes	—
230	#495	C₂²×C₅₃	1	4	yes	—
231	#494	C₂²×C₅₃	1	4	yes	—
232	#494	C₂²×C₅₃	1	4	yes	—
233	#494	C₂²×C₅₃	1	4	yes	—
234	#464	C₂²×C₅₃	1	16	yes	—
235	#462	C₂²×C₅₃	1	8	yes	—
236	#462	C₂²×C₅₃	1	16	yes	—
237	#461	C₂²×C₅₃	1	8	yes	—
238	#446	C₂²×C₅₃	1	4	yes	—
239	C₈×Q₈	C₂²×C₅₃	1	4	yes	—
240	#441	C₂²×C₅₃	1	8	yes	—
241	C₄×Q₁₆	C₂²×C₅₃	1	8	yes	—
242	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	—
243	C₄×SD₁₆	C₂²×C₅₃	1	4	yes	—
244	C₄×D₈	C₂²×C₅₃	1	2	yes	—
245	#427	C₂²×C₅₃	1	1	yes	characteristic
246	#424	C₂²×C₅₃	1	2	yes	—
247	#423	C₂²×C₅₃	1	1	yes	characteristic
248	#419	C₂²×C₅₃	1	4	yes	—
249	#416	C₂²×C₅₃	1	2	yes	—
250	#405	C₂²×C₅₃	1	2	yes	—
251	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
252	C₅₃	#61411	1	1	yes	characteristic, direct product factor, Sylow
253	C₂×#41	C₂³×C₅₃	1	2	yes	—
254	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
255	#137	—	4	8	no	—
256	#135	—	2	4	no	—
257	C₂×Q₁₆	—	2	8	no	—
258	C₂×Q₁₆	—	4	4	no	—
259	C₂×Q₁₆	C₂³×C₅₃	1	4	yes	—
260	C₂×Q₁₆	C₂³×C₅₃	1	4	yes	—
261	C₂×Q₁₆	C₂³×C₅₃	1	1	yes	characteristic
262	C₂×Q₁₆	C₂³×C₅₃	1	8	yes	—
263	C₂×Q₁₆	C₂³×C₅₃	1	8	yes	—
264	C₂×SD₁₆	C₂³×C₅₃	1	2	yes	—
265	C₂×SD₁₆	C₂³×C₅₃	1	8	yes	—
266	C₂×D₈	C₂³×C₅₃	1	1	yes	characteristic
267	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
268	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
269	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
270	C₄⋊Q₈	—	2	2	no	—
271	C₄⋊Q₈	—	2	2	no	—
272	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
273	C₄⋊Q₈	C₂³×C₅₃	1	8	yes	—
274	C₄⋊Q₈	C₂³×C₅₃	1	8	yes	—
275	#126	—	2	16	no	—
276	#125	—	2	8	no	—
277	#125	—	2	16	no	—
278	#125	D₄×C₅₃	1	2	yes	—
279	#124	C₂³×C₅₃	1	8	yes	—
280	#123	—	2	8	no	—
281	C₂²⋊Q₈	—	2	8	no	—
282	C₂²⋊Q₈	—	2	8	no	—
283	C₂²⋊Q₈	D₄×C₅₃	1	4	yes	—
284	C₄×Q₈	—	2	8	no	—
285	C₄×Q₈	—	2	8	no	—
286	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
287	C₄×Q₈	C₂³×C₅₃	1	16	yes	—
288	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
289	C₄×D₄	—	2	4	no	—
290	C₄×D₄	C₂³×C₅₃	1	4	yes	—
291	#117	—	2	8	no	—
292	#117	—	2	8	no	—
293	#117	—	2	2	no	—
294	#117	C₂³×C₅₃	1	2	yes	—
295	#117	C₂³×C₅₃	1	2	yes	—
296	C₂×C₄⋊C₄	—	2	4	no	—
297	C₂×C₄²	—	2	2	no	—
298	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
299	#107	C₂³×C₅₃	1	2	yes	—
300	#106	C₂³×C₅₃	1	2	yes	—
301	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
302	C₄⋊C₈	C₂³×C₅₃	1	8	yes	—
303	#103	C₂³×C₅₃	1	4	yes	—
304	#103	C₂³×C₅₃	1	16	yes	—
305	#102	C₂³×C₅₃	1	4	yes	—
306	#97	C₂³×C₅₃	1	4	yes	—
307	C₄×C₈	C₂³×C₅₃	1	2	yes	—
308	C₄×C₈	C₂³×C₅₃	1	2	yes	—
309	C₄×C₈	C₂³×C₅₃	1	4	yes	—
310	#41	—	2	4	no	—
311	#41	—	2	4	no	—
312	C₂×Q₈	—	2	4	no	—
313	C₂×Q₈	—	2	8	no	—
314	C₂×Q₈	—	2	8	no	—
315	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
316	C₂×Q₈	C₂⁴×C₅₃	1	8	yes	—
317	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
318	C₂×D₄	—	2	2	no	—
319	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
320	C₂²×C₄	—	2	4	no	—
321	C₂²×C₄	—	2	4	no	—
322	C₂²×C₄	—	2	2	no	—
323	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
324	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
325	Q₁₆	—	2	8	no	—
326	Q₁₆	—	2	8	no	—
327	Q₁₆	—	2	2	no	—
328	Q₁₆	—	4	8	no	—
329	Q₁₆	—	4	8	no	—
330	SD₁₆	—	2	4	no	—
331	SD₁₆	—	4	8	no	—
332	D₈	—	2	2	no	—
333	M₁₆	—	2	4	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	#41×C₅₃	1	4	yes	—
336	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
337	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
338	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
339	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
340	C₄⋊C₄	—	2	8	no	—
341	C₄⋊C₄	—	2	8	no	—
342	C₄⋊C₄	—	2	4	no	—
343	C₄⋊C₄	—	2	16	no	—
344	C₄⋊C₄	—	2	16	no	—
345	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
346	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
347	C₄⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
348	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
349	C₂²⋊C₄	—	2	2	no	—
350	C₂²⋊C₄	—	2	4	no	—
351	C₂²⋊C₄	—	2	16	no	—
352	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
353	C₄²	—	2	2	no	—
354	C₄²	—	2	4	no	—
355	C₄²	—	2	16	no	—
356	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
357	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
358	C₄²	C₂⁴×C₅₃	1	2	yes	—
359	C₄²	C₂⁴×C₅₃	1	4	yes	—
360	C₂³	—	2	2	no	—
361	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
362	Q₈	—	2	8	no	—
363	Q₈	—	2	2	no	—
364	Q₈	—	2	8	no	—
365	Q₈	—	2	8	no	—
366	Q₈	—	2	2	no	—
367	Q₈	—	4	8	no	—
368	D₄	—	2	2	no	—
369	D₄	—	2	2	no	—
370	D₄	—	4	4	no	—
371	C₂×C₄	—	2	4	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	8	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	4	8	no	—
376	C₂×C₄	—	4	4	no	—
377	C₂×C₄	—	4	8	no	—
378	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
379	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
380	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
381	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
382	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
383	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
384	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
385	C₈	—	2	2	no	—
386	C₈	—	2	4	no	—
387	C₈	C₂×#41×C₅₃	1	2	yes	—
388	C₈	C₂×#41×C₅₃	1	2	yes	—
389	C₂²	—	2	2	no	—
390	C₂²	—	4	2	no	—
391	C₂²	—	4	4	no	—
392	C₂²	C₅₃×#550	1	2	yes	—
393	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
394	C₄	—	2	2	no	—
395	C₄	—	2	8	no	—
396	C₄	—	4	2	no	—
397	C₄	—	4	8	no	—
398	C₄	—	4	4	no	—
399	C₄	C₅₃×#550	1	2	yes	—
400	C₄	C₅₃×#578	1	4	yes	—
401	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
402	C₄	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
403	C₂	—	2	2	no	—
404	C₂	—	8	2	no	—
405	C₂	C₂×C₅₃×#550	1	1	yes	characteristic
406	C₂	C₂×C₅₃×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₂×#135×C₅₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₅₃×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3509	C₂	semidirect
2. C₅₃×#3398	C₂	semidirect
3. C₅₃×#3395	C₂	semidirect
4. C₅₃×#3393	C₂	semidirect
6. C₂×C₅₃×#494	C₂	semidirect
7. C₅₃×#3044	C₂	semidirect
8. C₅₃×#3039	C₂	semidirect
9. C₅₃×#2960	C₂	semidirect
10. C₅₃×#2950	C₂	semidirect
185. #61411	C₅₃	direct (N × H)
200. #3509	C₂×C₅₃	semidirect
201. #3398	C₂×C₅₃	semidirect
202. #3395	C₂×C₅₃	semidirect
203. #3393	C₂×C₅₃	semidirect
205. C₂×#494	C₂×C₅₃	semidirect
206. #3044	C₂×C₅₃	semidirect
207. #3039	C₂×C₅₃	semidirect
208. #2960	C₂×C₅₃	semidirect
209. #2950	C₂×C₅₃	semidirect
252. C₅₃	#61411	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×#3509	2
C₅₃×#3398	2
C₅₃×#3395	2
C₅₃×#3393	2
C₅₃×#3156	2
C₂×C₅₃×#494	2
C₅₃×#3044	2
C₅₃×#3039	2
C₅₃×#2960	2
C₅₃×#2950	2
C₄×#135×C₅₃	2
#61411	53

Subgroups — C53×#61411 #27268923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#61411 #27268923