Subgroups — C₅₃×#60089 #27267601

Order 13568.

4902 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#585	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#568	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#520	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2898	C₂	1	8	yes	maximal, complemented
5	C₅₃×#2897	C₂	1	8	yes	maximal, complemented
6	C₅₃×#2896	C₂	1	6	yes	maximal, complemented
7	C₂×C₅₃×#417	C₂	1	6	yes	maximal, complemented
8	C₅₃×#585	—	2	8	no	—
9	C₅₃×#585	C₄	1	16	yes	complemented
10	C₂×2₋¹⁺⁴×C₅₃	—	2	1	no	—
11	C₂×2₋¹⁺⁴×C₅₃	C₂²	1	4	yes	—
12	C₂×H(5,𝔽₂)×C₅₃	—	2	3	no	—
13	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	4	yes	—
14	C₂²×#41×C₅₃	—	2	4	no	—
15	C₂²×#41×C₅₃	C₂²	1	6	yes	—
16	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#568	—	2	8	no	—
18	C₂×#131×C₅₃	C₂²	1	8	yes	complemented
19	C₂²×M₁₆×C₅₃	C₂²	1	6	yes	complemented
20	C₅₃×#520	—	2	4	no	—
21	C₅₃×#520	C₂²	1	8	yes	complemented
22	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
23	C₂×C₄×D₄×C₅₃	—	2	3	no	—
24	C₂×#117×C₅₃	C₂²	1	3	yes	complemented
25	C₂²×C₄⋊C₄×C₅₃	C₂²	1	3	yes	complemented
26	C₅₃×#419	C₂²	1	12	yes	complemented
27	C₅₃×#418	C₂²	1	12	yes	complemented
28	C₅₃×#417	C₂²	1	12	yes	complemented
29	C₅₃×#417	C₂²	1	24	yes	complemented
30	C₅₃×#417	C₂²	1	4	yes	complemented
31	C₅₃×#416	C₂²	1	24	yes	complemented
32	C₂×#103×C₅₃	C₂²	1	12	yes	complemented
33	C₂×#102×C₅₃	C₂²	1	12	yes	complemented
34	2₋¹⁺⁴×C₅₃	—	2	16	no	—
35	2₋¹⁺⁴×C₅₃	—	2	16	no	—
36	2₋¹⁺⁴×C₅₃	C₂×C₄	1	32	yes	complemented
37	H(5,𝔽₂)×C₅₃	—	2	16	no	—
38	H(5,𝔽₂)×C₅₃	—	2	48	no	—
39	H(5,𝔽₂)×C₅₃	C₂×C₄	1	32	yes	complemented
40	C₂×#41×C₅₃	—	2	64	no	—
41	C₂×#41×C₅₃	—	2	4	no	—
42	C₂×#41×C₅₃	—	2	12	no	—
43	C₂×#41×C₅₃	—	2	6	no	—
44	C₂×#41×C₅₃	—	2	6	no	—
45	C₂×#41×C₅₃	—	2	24	no	—
46	C₂×#41×C₅₃	C₂×C₄	1	8	yes	—
47	C₂×#41×C₅₃	C₂×C₄	1	48	yes	complemented
48	C₂×#41×C₅₃	D₄	1	8	yes	—
49	C₂×#41×C₅₃	D₄	1	4	yes	—
50	C₂×#41×C₅₃	C₂³	1	4	yes	—
51	C₂×#41×C₅₃	C₂³	1	12	yes	—
52	C₂×#41×C₅₃	C₂³	1	4	yes	—
53	C₂²×Q₈×C₅₃	—	2	4	no	—
54	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
55	C₂²×Q₈×C₅₃	C₂³	1	6	yes	—
56	C₂²×D₄×C₅₃	—	2	6	no	—
57	C₂²×D₄×C₅₃	—	2	12	no	—
58	C₂²×D₄×C₅₃	D₄	1	3	yes	—
59	C₂²×D₄×C₅₃	C₂³	1	6	yes	—
60	C₂³×C₄×C₅₃	—	2	6	no	—
61	C₂³×C₄×C₅₃	—	2	3	no	—
62	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
63	#131×C₅₃	—	2	32	no	—
64	C₂×M₁₆×C₅₃	—	2	24	no	—
65	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
66	C₂×M₁₆×C₅₃	C₂³	1	12	yes	—
67	C₂²×C₈×C₅₃	C₂³	1	12	yes	—
68	C₄×Q₈×C₅₃	—	2	8	no	—
69	C₄×D₄×C₅₃	—	2	24	no	—
70	#117×C₅₃	—	2	6	no	—
71	#117×C₅₃	C₂³	1	12	yes	—
72	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
73	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
74	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
75	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
76	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
77	C₂×C₂²⋊C₄×C₅₃	—	2	3	no	—
78	C₂×C₄²×C₅₃	—	2	3	no	—
79	#103×C₅₃	C₂³	1	24	yes	—
80	#103×C₅₃	C₂³	1	8	yes	—
81	#102×C₅₃	C₂³	1	24	yes	—
82	#102×C₅₃	C₂³	1	8	yes	—
83	C₂⁴×C₅₃	—	2	6	no	—
84	C₂⁴×C₅₃	C₂×D₄	1	3	yes	—
85	#41×C₅₃	—	2	32	no	—
86	#41×C₅₃	—	2	96	no	—
87	#41×C₅₃	—	2	48	no	—
88	#41×C₅₃	—	2	48	no	—
89	#41×C₅₃	—	2	8	no	—
90	#41×C₅₃	—	2	24	no	—
91	#41×C₅₃	C₂²⋊C₄	1	32	yes	—
92	#41×C₅₃	C₂²×C₄	1	16	yes	—
93	#41×C₅₃	C₂²×C₄	1	16	yes	—
94	#41×C₅₃	C₂²×C₄	1	48	yes	complemented
95	#41×C₅₃	C₂×D₄	1	16	yes	—
96	C₂×Q₈×C₅₃	—	2	32	no	—
97	C₂×Q₈×C₅₃	—	2	6	no	—
98	C₂×Q₈×C₅₃	—	2	12	no	—
99	C₂×Q₈×C₅₃	C₂²⋊C₄	1	8	yes	—
100	C₂×Q₈×C₅₃	C₂²×C₄	1	24	yes	complemented
101	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
102	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
103	C₂×D₄×C₅₃	—	2	48	no	—
104	C₂×D₄×C₅₃	—	2	4	no	—
105	C₂×D₄×C₅₃	—	2	12	no	—
106	C₂×D₄×C₅₃	—	2	96	no	—
107	C₂×D₄×C₅₃	—	2	6	no	—
108	C₂×D₄×C₅₃	—	2	12	no	—
109	C₂×D₄×C₅₃	C₂²⋊C₄	1	24	yes	—
110	C₂×D₄×C₅₃	C₂²×C₄	1	24	yes	complemented
111	C₂×D₄×C₅₃	C₂×D₄	1	12	yes	—
112	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
113	C₂²×C₄×C₅₃	—	2	6	no	—
114	C₂²×C₄×C₅₃	—	2	48	no	—
115	C₂²×C₄×C₅₃	—	2	4	no	—
116	C₂²×C₄×C₅₃	—	2	12	no	—
117	C₂²×C₄×C₅₃	—	2	1	no	—
118	C₂²×C₄×C₅₃	—	2	3	no	—
119	C₂²×C₄×C₅₃	—	2	12	no	—
120	C₂²×C₄×C₅₃	—	8	6	no	—
121	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
122	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
123	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
124	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
125	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
127	M₁₆×C₅₃	—	2	8	no	—
128	M₁₆×C₅₃	—	2	24	no	—
129	C₂×C₈×C₅₃	—	2	24	no	—
130	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
131	C₄⋊C₄×C₅₃	—	2	3	no	—
132	C₄⋊C₄×C₅₃	—	2	12	no	—
133	C₄⋊C₄×C₅₃	—	2	1	no	—
134	C₄⋊C₄×C₅₃	C₂⁴	1	6	yes	—
135	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
136	C₂²⋊C₄×C₅₃	—	2	12	no	—
137	C₄²×C₅₃	—	2	12	no	—
138	C₂³×C₅₃	—	2	4	no	—
139	C₂³×C₅₃	—	2	24	no	—
140	C₂³×C₅₃	—	4	3	no	—
141	C₂³×C₅₃	—	4	3	no	—
142	C₂³×C₅₃	—	8	6	no	—
143	C₂³×C₅₃	—	8	6	no	—
144	C₂³×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
145	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
146	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
147	Q₈×C₅₃	—	2	24	no	—
148	Q₈×C₅₃	—	2	4	no	—
149	Q₈×C₅₃	C₂×C₂²⋊C₄	1	16	yes	—
150	Q₈×C₅₃	C₂³×C₄	1	8	yes	—
151	D₄×C₅₃	—	2	16	no	—
152	D₄×C₅₃	—	2	48	no	—
153	D₄×C₅₃	—	2	24	no	—
154	D₄×C₅₃	—	2	4	no	—
155	D₄×C₅₃	C₂×C₂²⋊C₄	1	48	yes	—
156	D₄×C₅₃	C₂³×C₄	1	8	yes	—
157	C₂×C₄×C₅₃	—	2	24	no	—
158	C₂×C₄×C₅₃	—	2	4	no	—
159	C₂×C₄×C₅₃	—	2	16	no	—
160	C₂×C₄×C₅₃	—	2	48	no	—
161	C₂×C₄×C₅₃	—	2	6	no	—
162	C₂×C₄×C₅₃	—	2	2	no	—
163	C₂×C₄×C₅₃	—	4	2	no	—
164	C₂×C₄×C₅₃	—	4	6	no	—
165	C₂×C₄×C₅₃	—	8	12	no	—
166	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	24	yes	—
167	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
168	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
169	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
170	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
172	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
173	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
174	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
175	C₈×C₅₃	—	2	8	no	—
176	#60089	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂²×C₅₃	—	2	1	no	—
178	C₂²×C₅₃	—	2	6	no	—
179	C₂²×C₅₃	—	2	1	no	—
180	C₂²×C₅₃	—	2	6	no	—
181	C₂²×C₅₃	—	2	8	no	—
182	C₂²×C₅₃	—	4	12	no	—
183	C₂²×C₅₃	—	4	4	no	—
184	C₂²×C₅₃	—	4	4	no	—
185	C₂²×C₅₃	—	8	24	no	—
186	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	12	yes	—
187	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	2	yes	—
188	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
189	C₄×C₅₃	—	2	8	no	—
190	C₄×C₅₃	—	4	6	no	—
191	C₄×C₅₃	—	4	2	no	—
192	C₄×C₅₃	C₂²×C₂²⋊C₄	1	2	yes	—
193	C₄×C₅₃	C₂²×C₂²⋊C₄	1	12	yes	—
194	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
195	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
196	C₂×#585	C₂×C₅₃	1	1	yes	characteristic
197	C₂×#568	C₂×C₅₃	1	1	yes	characteristic, complemented
198	C₂×#520	C₂×C₅₃	1	1	yes	characteristic, complemented
199	#2898	C₂×C₅₃	1	8	yes	complemented
200	#2897	C₂×C₅₃	1	8	yes	complemented
201	#2896	C₂×C₅₃	1	6	yes	complemented
202	C₂×#417	C₂×C₅₃	1	6	yes	complemented
203	C₂×C₅₃	—	2	2	no	—
204	C₂×C₅₃	—	2	12	no	—
205	C₂×C₅₃	—	4	8	no	—
206	C₂×C₅₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
207	C₂×C₅₃	#3585	1	1	yes	characteristic
208	C₂×C₅₃	#3586	1	1	yes	characteristic
209	#585	—	2	8	no	—
210	#585	C₄×C₅₃	1	16	yes	complemented
211	C₂×2₋¹⁺⁴	—	2	1	no	—
212	C₂×2₋¹⁺⁴	C₂²×C₅₃	1	4	yes	—
213	C₂×H(5,𝔽₂)	—	2	3	no	—
214	C₂×H(5,𝔽₂)	C₂²×C₅₃	1	4	yes	—
215	C₂²×#41	—	2	4	no	—
216	C₂²×#41	C₂²×C₅₃	1	6	yes	—
217	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
218	#568	—	2	8	no	—
219	C₂×#131	C₂²×C₅₃	1	8	yes	complemented
220	C₂²×M₁₆	C₂²×C₅₃	1	6	yes	complemented
221	#520	—	2	4	no	—
222	#520	C₂²×C₅₃	1	8	yes	complemented
223	C₂×C₄×Q₈	—	2	1	no	—
224	C₂×C₄×D₄	—	2	3	no	—
225	C₂×#117	C₂²×C₅₃	1	3	yes	complemented
226	C₂²×C₄⋊C₄	C₂²×C₅₃	1	3	yes	complemented
227	#419	C₂²×C₅₃	1	12	yes	complemented
228	#418	C₂²×C₅₃	1	12	yes	complemented
229	#417	C₂²×C₅₃	1	12	yes	complemented
230	#417	C₂²×C₅₃	1	24	yes	complemented
231	#417	C₂²×C₅₃	1	4	yes	complemented
232	#416	C₂²×C₅₃	1	24	yes	complemented
233	C₂×#103	C₂²×C₅₃	1	12	yes	complemented
234	C₂×#102	C₂²×C₅₃	1	12	yes	complemented
235	C₅₃	#60089	1	1	yes	characteristic, direct product factor, Sylow
236	2₋¹⁺⁴	—	2	16	no	—
237	2₋¹⁺⁴	—	2	16	no	—
238	2₋¹⁺⁴	C₂×C₄×C₅₃	1	32	yes	complemented
239	H(5,𝔽₂)	—	2	16	no	—
240	H(5,𝔽₂)	—	2	48	no	—
241	H(5,𝔽₂)	C₂×C₄×C₅₃	1	32	yes	complemented
242	C₂×#41	—	2	64	no	—
243	C₂×#41	—	2	4	no	—
244	C₂×#41	—	2	12	no	—
245	C₂×#41	—	2	6	no	—
246	C₂×#41	—	2	6	no	—
247	C₂×#41	—	2	24	no	—
248	C₂×#41	C₂×C₄×C₅₃	1	8	yes	—
249	C₂×#41	C₂×C₄×C₅₃	1	48	yes	complemented
250	C₂×#41	D₄×C₅₃	1	4	yes	—
251	C₂×#41	D₄×C₅₃	1	8	yes	—
252	C₂×#41	C₂³×C₅₃	1	4	yes	—
253	C₂×#41	C₂³×C₅₃	1	12	yes	—
254	C₂×#41	C₂³×C₅₃	1	4	yes	—
255	C₂²×Q₈	—	2	4	no	—
256	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
257	C₂²×Q₈	C₂³×C₅₃	1	6	yes	—
258	C₂²×D₄	—	2	6	no	—
259	C₂²×D₄	—	2	12	no	—
260	C₂²×D₄	D₄×C₅₃	1	3	yes	—
261	C₂²×D₄	C₂³×C₅₃	1	6	yes	—
262	C₂³×C₄	—	2	6	no	—
263	C₂³×C₄	—	2	3	no	—
264	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
265	#131	—	2	32	no	—
266	C₂×M₁₆	—	2	24	no	—
267	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
268	C₂×M₁₆	C₂³×C₅₃	1	12	yes	—
269	C₂²×C₈	C₂³×C₅₃	1	12	yes	—
270	C₄×Q₈	—	2	8	no	—
271	C₄×D₄	—	2	24	no	—
272	#117	—	2	6	no	—
273	#117	C₂³×C₅₃	1	12	yes	—
274	C₂×C₄⋊C₄	—	2	3	no	—
275	C₂×C₄⋊C₄	—	2	6	no	—
276	C₂×C₄⋊C₄	C₂³×C₅₃	1	3	yes	—
277	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
278	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
279	C₂×C₂²⋊C₄	—	2	3	no	—
280	C₂×C₄²	—	2	3	no	—
281	#103	C₂³×C₅₃	1	24	yes	—
282	#103	C₂³×C₅₃	1	8	yes	—
283	#102	C₂³×C₅₃	1	24	yes	—
284	#102	C₂³×C₅₃	1	8	yes	—
285	C₂⁴	—	2	6	no	—
286	C₂⁴	C₂×D₄×C₅₃	1	3	yes	—
287	#41	—	2	32	no	—
288	#41	—	2	96	no	—
289	#41	—	2	48	no	—
290	#41	—	2	48	no	—
291	#41	—	2	8	no	—
292	#41	—	2	24	no	—
293	#41	C₂²⋊C₄×C₅₃	1	32	yes	—
294	#41	C₂²×C₄×C₅₃	1	16	yes	—
295	#41	C₂²×C₄×C₅₃	1	16	yes	—
296	#41	C₂²×C₄×C₅₃	1	48	yes	complemented
297	#41	C₂×D₄×C₅₃	1	16	yes	—
298	C₂×Q₈	—	2	32	no	—
299	C₂×Q₈	—	2	6	no	—
300	C₂×Q₈	—	2	12	no	—
301	C₂×Q₈	C₂²⋊C₄×C₅₃	1	8	yes	—
302	C₂×Q₈	C₂²×C₄×C₅₃	1	24	yes	complemented
303	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
304	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
305	C₂×D₄	—	2	48	no	—
306	C₂×D₄	—	2	4	no	—
307	C₂×D₄	—	2	12	no	—
308	C₂×D₄	—	2	96	no	—
309	C₂×D₄	—	2	6	no	—
310	C₂×D₄	—	2	12	no	—
311	C₂×D₄	C₂²⋊C₄×C₅₃	1	24	yes	—
312	C₂×D₄	C₂²×C₄×C₅₃	1	24	yes	complemented
313	C₂×D₄	C₂×D₄×C₅₃	1	12	yes	—
314	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
315	C₂²×C₄	—	2	6	no	—
316	C₂²×C₄	—	2	48	no	—
317	C₂²×C₄	—	2	4	no	—
318	C₂²×C₄	—	2	12	no	—
319	C₂²×C₄	—	2	1	no	—
320	C₂²×C₄	—	2	3	no	—
321	C₂²×C₄	—	2	12	no	—
322	C₂²×C₄	—	8	6	no	—
323	C₂²×C₄	C₂²×C₄×C₅₃	1	12	yes	—
324	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
325	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
326	C₂²×C₄	C₂×D₄×C₅₃	1	12	yes	—
327	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
328	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
329	M₁₆	—	2	8	no	—
330	M₁₆	—	2	24	no	—
331	C₂×C₈	—	2	24	no	—
332	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
333	C₄⋊C₄	—	2	3	no	—
334	C₄⋊C₄	—	2	12	no	—
335	C₄⋊C₄	—	2	1	no	—
336	C₄⋊C₄	C₂⁴×C₅₃	1	6	yes	—
337	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
338	C₂²⋊C₄	—	2	12	no	—
339	C₄²	—	2	12	no	—
340	C₂³	—	2	4	no	—
341	C₂³	—	2	24	no	—
342	C₂³	—	4	3	no	—
343	C₂³	—	4	3	no	—
344	C₂³	—	8	6	no	—
345	C₂³	—	8	6	no	—
346	C₂³	C₂×C₂²⋊C₄×C₅₃	1	12	yes	—
347	C₂³	C₂²×D₄×C₅₃	1	6	yes	—
348	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
349	Q₈	—	2	24	no	—
350	Q₈	—	2	4	no	—
351	Q₈	C₂×C₂²⋊C₄×C₅₃	1	16	yes	—
352	Q₈	C₂³×C₄×C₅₃	1	8	yes	—
353	D₄	—	2	16	no	—
354	D₄	—	2	48	no	—
355	D₄	—	2	24	no	—
356	D₄	—	2	4	no	—
357	D₄	C₂×C₂²⋊C₄×C₅₃	1	48	yes	—
358	D₄	C₂³×C₄×C₅₃	1	8	yes	—
359	C₂×C₄	—	2	24	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	16	no	—
362	C₂×C₄	—	2	48	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	6	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	6	no	—
367	C₂×C₄	—	8	12	no	—
368	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	24	yes	—
369	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	12	yes	—
370	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
371	C₂×C₄	C₂³×C₄×C₅₃	1	12	yes	—
372	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
373	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
374	C₂×C₄	C₂²×D₄×C₅₃	1	12	yes	—
375	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
376	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
377	C₈	—	2	8	no	—
378	C₂²	—	2	1	no	—
379	C₂²	—	2	6	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	6	no	—
382	C₂²	—	2	8	no	—
383	C₂²	—	4	12	no	—
384	C₂²	—	4	4	no	—
385	C₂²	—	4	4	no	—
386	C₂²	—	8	24	no	—
387	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	12	yes	—
388	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	2	yes	—
389	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
390	C₄	—	2	8	no	—
391	C₄	—	4	6	no	—
392	C₄	—	4	2	no	—
393	C₄	C₂²×C₂²⋊C₄×C₅₃	1	2	yes	—
394	C₄	C₂²×C₂²⋊C₄×C₅₃	1	12	yes	—
395	C₄	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
396	C₄	C₂³×D₄×C₅₃	1	1	yes	characteristic
397	C₂	—	2	2	no	—
398	C₂	—	2	12	no	—
399	C₂	—	4	8	no	—
400	C₂	C₂³×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
401	C₂	C₅₃×#3585	1	1	yes	characteristic
402	C₂	C₅₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator	C₄	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₂×C₅₃×#585	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#568	C₂	semidirect
3. C₂×C₅₃×#520	C₂	semidirect
4. C₅₃×#2898	C₂	semidirect
5. C₅₃×#2897	C₂	semidirect
6. C₅₃×#2896	C₂	semidirect
7. C₂×C₅₃×#417	C₂	semidirect
9. C₅₃×#585	C₄	semidirect
18. C₂×#131×C₅₃	C₂²	semidirect
19. C₂²×M₁₆×C₅₃	C₂²	semidirect
21. C₅₃×#520	C₂²	semidirect
24. C₂×#117×C₅₃	C₂²	semidirect
25. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
26. C₅₃×#419	C₂²	semidirect
27. C₅₃×#418	C₂²	semidirect
28. C₅₃×#417	C₂²	semidirect
29. C₅₃×#417	C₂²	semidirect
30. C₅₃×#417	C₂²	semidirect
31. C₅₃×#416	C₂²	semidirect
32. C₂×#103×C₅₃	C₂²	semidirect
33. C₂×#102×C₅₃	C₂²	semidirect
36. 2₋¹⁺⁴×C₅₃	C₂×C₄	semidirect
39. H(5,𝔽₂)×C₅₃	C₂×C₄	semidirect
47. C₂×#41×C₅₃	C₂×C₄	semidirect
94. #41×C₅₃	C₂²×C₄	semidirect
100. C₂×Q₈×C₅₃	C₂²×C₄	semidirect
110. C₂×D₄×C₅₃	C₂²×C₄	semidirect
176. #60089	C₅₃	direct (N × H)
197. C₂×#568	C₂×C₅₃	semidirect
198. C₂×#520	C₂×C₅₃	semidirect
199. #2898	C₂×C₅₃	semidirect
200. #2897	C₂×C₅₃	semidirect
201. #2896	C₂×C₅₃	semidirect
202. C₂×#417	C₂×C₅₃	semidirect
210. #585	C₄×C₅₃	semidirect
219. C₂×#131	C₂²×C₅₃	semidirect
220. C₂²×M₁₆	C₂²×C₅₃	semidirect
222. #520	C₂²×C₅₃	semidirect
225. C₂×#117	C₂²×C₅₃	semidirect
226. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
227. #419	C₂²×C₅₃	semidirect
228. #418	C₂²×C₅₃	semidirect
229. #417	C₂²×C₅₃	semidirect
230. #417	C₂²×C₅₃	semidirect
231. #417	C₂²×C₅₃	semidirect
232. #416	C₂²×C₅₃	semidirect
233. C₂×#103	C₂²×C₅₃	semidirect
234. C₂×#102	C₂²×C₅₃	semidirect
235. C₅₃	#60089	direct (N × H)
238. 2₋¹⁺⁴	C₂×C₄×C₅₃	semidirect
241. H(5,𝔽₂)	C₂×C₄×C₅₃	semidirect
249. C₂×#41	C₂×C₄×C₅₃	semidirect
296. #41	C₂²×C₄×C₅₃	semidirect
302. C₂×Q₈	C₂²×C₄×C₅₃	semidirect
312. C₂×D₄	C₂²×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#585	2
C₂×C₅₃×#568	2
C₂×C₅₃×#520	2
C₅₃×#2898	2
C₅₃×#2897	2
C₅₃×#2896	2
C₂×C₅₃×#417	2
#60089	53

Subgroups — C53×#60089 #27267601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60089 #27267601