Subgroups — C₅₃×#17758 #27225270

Order 13568.

1240 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3498	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2989	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2925	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1557	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1555	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1553	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1547	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1545	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1545	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1540	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×#331	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₅₃	—	2	1	no	—
13	C₅₃×#550	—	2	1	no	—
14	C₅₃×#550	C₄	1	2	yes	—
15	D₄²×C₅₃	—	2	2	no	—
16	D₄²×C₅₃	—	2	1	no	—
17	D₄²×C₅₃	C₄	1	2	yes	—
18	D₄²×C₅₃	C₄	1	4	yes	—
19	C₅₃×#535	—	2	1	no	—
20	C₅₃×#535	—	2	2	no	—
21	C₅₃×#532	C₂²	1	1	yes	characteristic
22	C₅₃×#532	C₂²	1	2	yes	—
23	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#127×C₅₃	C₂²	1	2	yes	—
25	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#446	—	2	1	no	—
27	C₅₃×#436	—	2	1	no	—
28	C₅₃×#435	—	2	2	no	—
29	C₅₃×#433	—	2	2	no	—
30	C₅₃×#432	C₂²	1	2	yes	complemented
31	C₅₃×#432	C₂²	1	2	yes	complemented
32	C₅₃×#431	C₂²	1	1	yes	characteristic, complemented
33	C₅₃×#431	C₂²	1	2	yes	complemented
34	C₅₃×#423	C₂²	1	2	yes	complemented
35	C₅₃×#423	C₂²	1	1	yes	characteristic, complemented
36	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
37	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
38	C₅₃×#408	—	2	1	no	—
39	C₅₃×#405	—	2	1	no	—
40	C₅₃×#332	C₂²	1	1	yes	characteristic, complemented
41	C₅₃×#332	C₂²	1	1	yes	characteristic, complemented
42	C₅₃×#332	C₂²	1	2	yes	complemented
43	C₅₃×#331	C₂²	1	2	yes	complemented
44	C₅₃×#331	C₂²	1	2	yes	complemented
45	C₅₃×#331	C₂²	1	1	yes	characteristic, complemented
46	C₅₃×#331	C₂²	1	1	yes	characteristic, complemented
47	C₅₃×#331	C₂²	1	2	yes	complemented
48	C₅₃×#331	C₂²	1	2	yes	complemented
49	C₅₃×#331	C₂²	1	2	yes	complemented
50	H(5,𝔽₂)×C₅₃	—	4	4	no	—
51	H(5,𝔽₂)×C₅₃	—	4	4	no	—
52	C₂×#41×C₅₃	—	2	2	no	—
53	C₂×#41×C₅₃	—	2	2	no	—
54	C₂×#41×C₅₃	—	2	1	no	—
55	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
56	C₂²×D₄×C₅₃	—	2	4	no	—
57	C₂²×D₄×C₅₃	—	2	2	no	—
58	C₂²×D₄×C₅₃	—	2	2	no	—
59	C₂²×D₄×C₅₃	—	2	1	no	—
60	C₂²×D₄×C₅₃	—	2	1	no	—
61	C₂²×D₄×C₅₃	—	2	2	no	—
62	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
63	C₂²×D₄×C₅₃	C₂×C₄	1	4	yes	—
64	C₂×M₁₆×C₅₃	—	2	1	no	—
65	C₂×M₁₆×C₅₃	—	2	2	no	—
66	C₂²×C₈×C₅₃	—	2	1	no	—
67	C₂²×C₈×C₅₃	—	2	1	no	—
68	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
69	#127×C₅₃	—	2	2	no	—
70	#127×C₅₃	—	2	1	no	—
71	#127×C₅₃	—	2	1	no	—
72	#127×C₅₃	—	2	2	no	—
73	#127×C₅₃	C₂×C₄	1	2	yes	—
74	#127×C₅₃	C₂×C₄	1	4	yes	—
75	#127×C₅₃	C₂³	1	2	yes	—
76	#127×C₅₃	C₂³	1	1	yes	characteristic
77	#124×C₅₃	—	2	2	no	—
78	#124×C₅₃	—	2	1	no	—
79	#121×C₅₃	—	2	4	no	—
80	#121×C₅₃	—	2	2	no	—
81	#121×C₅₃	—	2	4	no	—
82	#121×C₅₃	—	2	2	no	—
83	#121×C₅₃	—	2	1	no	—
84	#121×C₅₃	—	2	2	no	—
85	#121×C₅₃	C₂×C₄	1	2	yes	—
86	#121×C₅₃	C₂×C₄	1	4	yes	—
87	#120×C₅₃	—	2	4	no	—
88	#120×C₅₃	—	2	4	no	—
89	#120×C₅₃	—	2	4	no	—
90	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
91	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
92	C₄×D₄×C₅₃	—	2	1	no	—
93	C₄×D₄×C₅₃	—	2	2	no	—
94	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
95	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
96	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
97	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
98	C₄×D₄×C₅₃	D₄	1	2	yes	—
99	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
100	#117×C₅₃	D₄	1	1	yes	characteristic
101	#117×C₅₃	D₄	1	2	yes	—
102	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
104	C₄⋊C₈×C₅₃	—	2	1	no	—
105	C₄⋊C₈×C₅₃	—	2	1	no	—
106	C₄⋊C₈×C₅₃	—	2	1	no	—
107	C₄⋊C₈×C₅₃	—	2	2	no	—
108	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
109	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
110	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
111	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
112	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
113	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
114	C₂²⋊C₈×C₅₃	—	2	2	no	—
115	C₂²⋊C₈×C₅₃	—	2	2	no	—
116	C₂²⋊C₈×C₅₃	—	2	1	no	—
117	C₂²⋊C₈×C₅₃	—	2	2	no	—
118	#97×C₅₃	—	2	2	no	—
119	#97×C₅₃	—	2	2	no	—
120	C₄×C₈×C₅₃	—	2	2	no	—
121	C₄×C₈×C₅₃	—	2	1	no	—
122	C₂⁴×C₅₃	—	2	4	no	—
123	C₂⁴×C₅₃	—	2	2	no	—
124	#41×C₅₃	—	2	2	no	—
125	#41×C₅₃	—	2	2	no	—
126	#41×C₅₃	—	4	4	no	—
127	#41×C₅₃	—	4	4	no	—
128	#41×C₅₃	—	4	8	no	—
129	#41×C₅₃	—	4	2	no	—
130	#41×C₅₃	—	4	2	no	—
131	C₂×Q₈×C₅₃	—	2	1	no	—
132	C₂×Q₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
133	C₂×D₄×C₅₃	—	2	4	no	—
134	C₂×D₄×C₅₃	—	2	2	no	—
135	C₂×D₄×C₅₃	—	2	4	no	—
136	C₂×D₄×C₅₃	—	2	2	no	—
137	C₂×D₄×C₅₃	—	2	4	no	—
138	C₂×D₄×C₅₃	—	2	2	no	—
139	C₂×D₄×C₅₃	—	2	2	no	—
140	C₂×D₄×C₅₃	—	2	4	no	—
141	C₂×D₄×C₅₃	—	2	1	no	—
142	C₂×D₄×C₅₃	—	2	2	no	—
143	C₂×D₄×C₅₃	—	2	1	no	—
144	C₂×D₄×C₅₃	—	2	2	no	—
145	C₂×D₄×C₅₃	—	4	4	no	—
146	C₂×D₄×C₅₃	—	4	4	no	—
147	C₂×D₄×C₅₃	—	4	16	no	—
148	C₂×D₄×C₅₃	—	4	8	no	—
149	C₂×D₄×C₅₃	—	4	2	no	—
150	C₂×D₄×C₅₃	—	4	2	no	—
151	C₂×D₄×C₅₃	—	8	1	no	—
152	C₂×D₄×C₅₃	—	8	2	no	—
153	C₂×D₄×C₅₃	—	8	1	no	—
154	C₂×D₄×C₅₃	—	8	2	no	—
155	C₂×D₄×C₅₃	—	8	2	no	—
156	C₂×D₄×C₅₃	—	8	4	no	—
157	C₂×D₄×C₅₃	C₂²⋊C₄	1	2	yes	—
158	C₂×D₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
159	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
160	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
161	C₂²×C₄×C₅₃	—	2	4	no	—
162	C₂²×C₄×C₅₃	—	2	2	no	—
163	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
164	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
165	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
169	M₁₆×C₅₃	—	4	2	no	—
170	M₁₆×C₅₃	—	8	2	no	—
171	C₂×C₈×C₅₃	—	2	1	no	—
172	C₂×C₈×C₅₃	—	2	1	no	—
173	C₂×C₈×C₅₃	—	2	2	no	—
174	C₂×C₈×C₅₃	—	2	1	no	—
175	C₂×C₈×C₅₃	—	2	1	no	—
176	C₂×C₈×C₅₃	—	2	2	no	—
177	C₂×C₈×C₅₃	—	4	2	no	—
178	C₂×C₈×C₅₃	—	8	1	no	—
179	C₄⋊C₄×C₅₃	—	2	1	no	—
180	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
181	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
182	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
183	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
184	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
185	C₂²⋊C₄×C₅₃	—	2	4	no	—
186	C₂²⋊C₄×C₅₃	—	2	2	no	—
187	C₂²⋊C₄×C₅₃	—	2	2	no	—
188	C₂²⋊C₄×C₅₃	—	2	4	no	—
189	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
190	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
191	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
192	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
193	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
194	C₄²×C₅₃	C₂×D₄	1	2	yes	—
195	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
196	C₄²×C₅₃	C₂×D₄	1	2	yes	—
197	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
198	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
199	C₂³×C₅₃	—	2	4	no	—
200	C₂³×C₅₃	—	2	2	no	—
201	C₂³×C₅₃	—	4	8	no	—
202	C₂³×C₅₃	—	4	4	no	—
203	C₂³×C₅₃	—	8	2	no	—
204	C₂³×C₅₃	—	8	4	no	—
205	C₂³×C₅₃	—	8	2	no	—
206	C₂³×C₅₃	—	8	4	no	—
207	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
208	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
209	Q₈×C₅₃	—	2	2	no	—
210	Q₈×C₅₃	—	4	1	no	—
211	Q₈×C₅₃	—	4	1	no	—
212	D₄×C₅₃	—	2	4	no	—
213	D₄×C₅₃	—	2	2	no	—
214	D₄×C₅₃	—	4	4	no	—
215	D₄×C₅₃	—	4	2	no	—
216	D₄×C₅₃	—	4	8	no	—
217	D₄×C₅₃	—	4	8	no	—
218	D₄×C₅₃	—	4	4	no	—
219	D₄×C₅₃	—	4	1	no	—
220	D₄×C₅₃	—	4	1	no	—
221	D₄×C₅₃	—	4	2	no	—
222	D₄×C₅₃	—	4	2	no	—
223	D₄×C₅₃	—	4	2	no	—
224	D₄×C₅₃	—	4	1	no	—
225	D₄×C₅₃	—	4	2	no	—
226	D₄×C₅₃	—	4	1	no	—
227	D₄×C₅₃	—	4	4	no	—
228	D₄×C₅₃	—	8	2	no	—
229	D₄×C₅₃	—	8	4	no	—
230	D₄×C₅₃	—	8	2	no	—
231	D₄×C₅₃	—	8	4	no	—
232	C₂×C₄×C₅₃	—	2	2	no	—
233	C₂×C₄×C₅₃	—	2	2	no	—
234	C₂×C₄×C₅₃	—	2	1	no	—
235	C₂×C₄×C₅₃	—	2	2	no	—
236	C₂×C₄×C₅₃	—	2	1	no	—
237	C₂×C₄×C₅₃	—	2	2	no	—
238	C₂×C₄×C₅₃	—	2	2	no	—
239	C₂×C₄×C₅₃	—	4	1	no	—
240	C₂×C₄×C₅₃	—	4	2	no	—
241	C₂×C₄×C₅₃	—	4	8	no	—
242	C₂×C₄×C₅₃	—	4	4	no	—
243	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
244	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
245	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
246	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
247	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
248	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
249	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
250	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
251	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
252	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
254	C₈×C₅₃	—	4	1	no	—
255	C₈×C₅₃	—	4	1	no	—
256	C₈×C₅₃	—	4	2	no	—
257	C₈×C₅₃	—	4	2	no	—
258	C₈×C₅₃	—	4	1	no	—
259	C₈×C₅₃	—	4	1	no	—
260	#17758	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
261	C₂²×C₅₃	—	2	1	no	—
262	C₂²×C₅₃	—	2	1	no	—
263	C₂²×C₅₃	—	2	2	no	—
264	C₂²×C₅₃	—	2	2	no	—
265	C₂²×C₅₃	—	2	2	no	—
266	C₂²×C₅₃	—	4	4	no	—
267	C₂²×C₅₃	—	4	2	no	—
268	C₂²×C₅₃	—	4	4	no	—
269	C₂²×C₅₃	—	4	2	no	—
270	C₂²×C₅₃	—	4	4	no	—
271	C₂²×C₅₃	—	4	2	no	—
272	C₂²×C₅₃	—	16	4	no	—
273	C₂²×C₅₃	—	16	8	no	—
274	C₂²×C₅₃	#418	1	2	yes	—
275	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
276	C₄×C₅₃	—	2	1	no	—
277	C₄×C₅₃	—	2	1	no	—
278	C₄×C₅₃	—	2	2	no	—
279	C₄×C₅₃	—	2	2	no	—
280	C₄×C₅₃	—	2	1	no	—
281	C₄×C₅₃	—	2	2	no	—
282	C₄×C₅₃	—	4	2	no	—
283	C₄×C₅₃	#418	1	2	yes	—
284	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
285	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
286	C₄×C₅₃	#576	1	1	yes	characteristic
287	C₄×C₅₃	#577	1	1	yes	characteristic
288	#3498	C₂×C₅₃	1	1	yes	characteristic
289	#2989	C₂×C₅₃	1	1	yes	characteristic, complemented
290	#2925	C₂×C₅₃	1	1	yes	characteristic, complemented
291	#1557	C₂×C₅₃	1	2	yes	complemented
292	#1555	C₂×C₅₃	1	2	yes	complemented
293	#1553	C₂×C₅₃	1	2	yes	complemented
294	#1547	C₂×C₅₃	1	1	yes	characteristic, complemented
295	#1545	C₂×C₅₃	1	1	yes	characteristic, complemented
296	#1545	C₂×C₅₃	1	2	yes	complemented
297	#1540	C₂×C₅₃	1	1	yes	characteristic, complemented
298	C₂×#331	C₂×C₅₃	1	1	yes	characteristic, complemented
299	C₂×C₅₃	—	2	2	no	—
300	C₂×C₅₃	—	4	2	no	—
301	C₂×C₅₃	—	8	4	no	—
302	C₂×C₅₃	—	8	2	no	—
303	C₂×C₅₃	#2888	1	1	yes	characteristic
304	C₂×C₅₃	C₂×#418	1	1	yes	characteristic
305	C₂×C₅₃	#2897	1	1	yes	characteristic
306	C₂×H(5,𝔽₂)	—	2	1	no	—
307	#550	—	2	1	no	—
308	#550	C₄×C₅₃	1	2	yes	—
309	D₄²	—	2	2	no	—
310	D₄²	—	2	1	no	—
311	D₄²	C₄×C₅₃	1	2	yes	—
312	D₄²	C₄×C₅₃	1	4	yes	—
313	#535	—	2	1	no	—
314	#535	—	2	2	no	—
315	#532	C₂²×C₅₃	1	1	yes	characteristic
316	#532	C₂²×C₅₃	1	2	yes	—
317	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
318	C₂×#127	C₂²×C₅₃	1	2	yes	—
319	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
320	#446	—	2	1	no	—
321	#436	—	2	1	no	—
322	#435	—	2	2	no	—
323	#433	—	2	2	no	—
324	#432	C₂²×C₅₃	1	2	yes	complemented
325	#432	C₂²×C₅₃	1	2	yes	complemented
326	#431	C₂²×C₅₃	1	1	yes	characteristic, complemented
327	#431	C₂²×C₅₃	1	2	yes	complemented
328	#423	C₂²×C₅₃	1	1	yes	characteristic, complemented
329	#423	C₂²×C₅₃	1	2	yes	complemented
330	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
331	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
332	#408	—	2	1	no	—
333	#405	—	2	1	no	—
334	#332	C₂²×C₅₃	1	1	yes	characteristic, complemented
335	#332	C₂²×C₅₃	1	1	yes	characteristic, complemented
336	#332	C₂²×C₅₃	1	2	yes	complemented
337	#331	C₂²×C₅₃	1	2	yes	complemented
338	#331	C₂²×C₅₃	1	2	yes	complemented
339	#331	C₂²×C₅₃	1	1	yes	characteristic, complemented
340	#331	C₂²×C₅₃	1	1	yes	characteristic, complemented
341	#331	C₂²×C₅₃	1	2	yes	complemented
342	#331	C₂²×C₅₃	1	2	yes	complemented
343	#331	C₂²×C₅₃	1	2	yes	complemented
344	C₅₃	#17758	1	1	yes	characteristic, direct product factor, Sylow
345	H(5,𝔽₂)	—	4	4	no	—
346	H(5,𝔽₂)	—	4	4	no	—
347	C₂×#41	—	2	2	no	—
348	C₂×#41	—	2	2	no	—
349	C₂×#41	—	2	1	no	—
350	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
351	C₂²×D₄	—	2	4	no	—
352	C₂²×D₄	—	2	2	no	—
353	C₂²×D₄	—	2	2	no	—
354	C₂²×D₄	—	2	1	no	—
355	C₂²×D₄	—	2	1	no	—
356	C₂²×D₄	—	2	2	no	—
357	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
358	C₂²×D₄	C₂×C₄×C₅₃	1	4	yes	—
359	C₂×M₁₆	—	2	1	no	—
360	C₂×M₁₆	—	2	2	no	—
361	C₂²×C₈	—	2	1	no	—
362	C₂²×C₈	—	2	1	no	—
363	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
364	#127	—	2	2	no	—
365	#127	—	2	1	no	—
366	#127	—	2	1	no	—
367	#127	—	2	2	no	—
368	#127	C₂×C₄×C₅₃	1	2	yes	—
369	#127	C₂×C₄×C₅₃	1	4	yes	—
370	#127	C₂³×C₅₃	1	2	yes	—
371	#127	C₂³×C₅₃	1	1	yes	characteristic
372	#124	—	2	2	no	—
373	#124	—	2	1	no	—
374	#121	—	2	1	no	—
375	#121	—	2	2	no	—
376	#121	—	2	4	no	—
377	#121	—	2	2	no	—
378	#121	—	2	4	no	—
379	#121	—	2	2	no	—
380	#121	C₂×C₄×C₅₃	1	2	yes	—
381	#121	C₂×C₄×C₅₃	1	4	yes	—
382	#120	—	2	4	no	—
383	#120	—	2	4	no	—
384	#120	—	2	4	no	—
385	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
386	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
387	C₄×D₄	—	2	1	no	—
388	C₄×D₄	—	2	2	no	—
389	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
390	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
391	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
392	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
393	C₄×D₄	D₄×C₅₃	1	2	yes	—
394	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
395	#117	D₄×C₅₃	1	1	yes	characteristic
396	#117	D₄×C₅₃	1	2	yes	—
397	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
398	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
399	C₄⋊C₈	—	2	1	no	—
400	C₄⋊C₈	—	2	1	no	—
401	C₄⋊C₈	—	2	1	no	—
402	C₄⋊C₈	—	2	2	no	—
403	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
404	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
405	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
406	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
407	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
408	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
409	C₂²⋊C₈	—	2	2	no	—
410	C₂²⋊C₈	—	2	2	no	—
411	C₂²⋊C₈	—	2	1	no	—
412	C₂²⋊C₈	—	2	2	no	—
413	#97	—	2	2	no	—
414	#97	—	2	2	no	—
415	C₄×C₈	—	2	2	no	—
416	C₄×C₈	—	2	1	no	—
417	C₂⁴	—	2	4	no	—
418	C₂⁴	—	2	2	no	—
419	#41	—	2	2	no	—
420	#41	—	2	2	no	—
421	#41	—	4	4	no	—
422	#41	—	4	4	no	—
423	#41	—	4	8	no	—
424	#41	—	4	2	no	—
425	#41	—	4	2	no	—
426	C₂×Q₈	—	2	1	no	—
427	C₂×Q₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
428	C₂×D₄	—	2	4	no	—
429	C₂×D₄	—	2	2	no	—
430	C₂×D₄	—	2	4	no	—
431	C₂×D₄	—	2	2	no	—
432	C₂×D₄	—	2	4	no	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	—	2	4	no	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	2	1	no	—
437	C₂×D₄	—	2	2	no	—
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	2	no	—
440	C₂×D₄	—	4	4	no	—
441	C₂×D₄	—	4	4	no	—
442	C₂×D₄	—	4	16	no	—
443	C₂×D₄	—	4	8	no	—
444	C₂×D₄	—	4	2	no	—
445	C₂×D₄	—	4	2	no	—
446	C₂×D₄	—	8	1	no	—
447	C₂×D₄	—	8	2	no	—
448	C₂×D₄	—	8	1	no	—
449	C₂×D₄	—	8	2	no	—
450	C₂×D₄	—	8	2	no	—
451	C₂×D₄	—	8	4	no	—
452	C₂×D₄	C₂²⋊C₄×C₅₃	1	2	yes	—
453	C₂×D₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
454	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
455	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
456	C₂²×C₄	—	2	4	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
459	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
460	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
461	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
462	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
463	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
464	M₁₆	—	4	2	no	—
465	M₁₆	—	8	2	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	2	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	2	no	—
472	C₂×C₈	—	4	2	no	—
473	C₂×C₈	—	8	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
476	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
477	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
478	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
479	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
480	C₂²⋊C₄	—	2	4	no	—
481	C₂²⋊C₄	—	2	2	no	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	2	no	—
484	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
485	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
486	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
487	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
488	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
489	C₄²	C₂×D₄×C₅₃	1	2	yes	—
490	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
491	C₄²	C₂×D₄×C₅₃	1	2	yes	—
492	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
493	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
494	C₂³	—	2	4	no	—
495	C₂³	—	2	2	no	—
496	C₂³	—	4	8	no	—
497	C₂³	—	4	4	no	—
498	C₂³	—	8	2	no	—
499	C₂³	—	8	4	no	—
500	C₂³	—	8	2	no	—
501	C₂³	—	8	4	no	—
502	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
503	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
504	Q₈	—	2	2	no	—
505	Q₈	—	4	1	no	—
506	Q₈	—	4	1	no	—
507	D₄	—	2	4	no	—
508	D₄	—	2	2	no	—
509	D₄	—	4	4	no	—
510	D₄	—	4	2	no	—
511	D₄	—	4	8	no	—
512	D₄	—	4	8	no	—
513	D₄	—	4	4	no	—
514	D₄	—	4	1	no	—
515	D₄	—	4	1	no	—
516	D₄	—	4	2	no	—
517	D₄	—	4	2	no	—
518	D₄	—	4	2	no	—
519	D₄	—	4	1	no	—
520	D₄	—	4	2	no	—
521	D₄	—	4	1	no	—
522	D₄	—	4	4	no	—
523	D₄	—	8	2	no	—
524	D₄	—	8	4	no	—
525	D₄	—	8	2	no	—
526	D₄	—	8	4	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	2	no	—
536	C₂×C₄	—	4	8	no	—
537	C₂×C₄	—	4	4	no	—
538	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
539	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
540	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
541	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
542	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
543	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
544	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
545	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
546	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
547	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
548	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
549	C₈	—	4	1	no	—
550	C₈	—	4	1	no	—
551	C₈	—	4	2	no	—
552	C₈	—	4	2	no	—
553	C₈	—	4	1	no	—
554	C₈	—	4	1	no	—
555	C₂²	—	2	1	no	—
556	C₂²	—	2	1	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	2	no	—
559	C₂²	—	2	2	no	—
560	C₂²	—	4	4	no	—
561	C₂²	—	4	2	no	—
562	C₂²	—	4	4	no	—
563	C₂²	—	4	2	no	—
564	C₂²	—	4	4	no	—
565	C₂²	—	4	2	no	—
566	C₂²	—	16	4	no	—
567	C₂²	—	16	8	no	—
568	C₂²	C₅₃×#418	1	2	yes	—
569	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
570	C₄	—	2	1	no	—
571	C₄	—	2	1	no	—
572	C₄	—	2	2	no	—
573	C₄	—	2	2	no	—
574	C₄	—	2	1	no	—
575	C₄	—	2	2	no	—
576	C₄	—	4	2	no	—
577	C₄	C₅₃×#418	1	2	yes	—
578	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
579	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
580	C₄	C₅₃×#576	1	1	yes	characteristic
581	C₄	C₅₃×#577	1	1	yes	characteristic
582	C₂	—	2	2	no	—
583	C₂	—	4	2	no	—
584	C₂	—	8	4	no	—
585	C₂	—	8	2	no	—
586	C₂	C₅₃×#2888	1	1	yes	characteristic
587	C₂	C₂×C₅₃×#418	1	1	yes	characteristic
588	C₂	C₅₃×#2897	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3498	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2989	C₂	semidirect
3. C₅₃×#2925	C₂	semidirect
4. C₅₃×#1557	C₂	semidirect
5. C₅₃×#1555	C₂	semidirect
6. C₅₃×#1553	C₂	semidirect
7. C₅₃×#1547	C₂	semidirect
8. C₅₃×#1545	C₂	semidirect
9. C₅₃×#1545	C₂	semidirect
10. C₅₃×#1540	C₂	semidirect
11. C₂×C₅₃×#331	C₂	semidirect
30. C₅₃×#432	C₂²	semidirect
31. C₅₃×#432	C₂²	semidirect
32. C₅₃×#431	C₂²	semidirect
33. C₅₃×#431	C₂²	semidirect
34. C₅₃×#423	C₂²	semidirect
35. C₅₃×#423	C₂²	semidirect
36. C₂×C₄⋊C₈×C₅₃	C₂²	semidirect
37. C₂×C₄⋊C₈×C₅₃	C₂²	semidirect
40. C₅₃×#332	C₂²	semidirect
41. C₅₃×#332	C₂²	semidirect
42. C₅₃×#332	C₂²	semidirect
43. C₅₃×#331	C₂²	semidirect
44. C₅₃×#331	C₂²	semidirect
45. C₅₃×#331	C₂²	semidirect
46. C₅₃×#331	C₂²	semidirect
47. C₅₃×#331	C₂²	semidirect
48. C₅₃×#331	C₂²	semidirect
49. C₅₃×#331	C₂²	semidirect
260. #17758	C₅₃	direct (N × H)
289. #2989	C₂×C₅₃	semidirect
290. #2925	C₂×C₅₃	semidirect
291. #1557	C₂×C₅₃	semidirect
292. #1555	C₂×C₅₃	semidirect
293. #1553	C₂×C₅₃	semidirect
294. #1547	C₂×C₅₃	semidirect
295. #1545	C₂×C₅₃	semidirect
296. #1545	C₂×C₅₃	semidirect
297. #1540	C₂×C₅₃	semidirect
298. C₂×#331	C₂×C₅₃	semidirect
324. #432	C₂²×C₅₃	semidirect
325. #432	C₂²×C₅₃	semidirect
326. #431	C₂²×C₅₃	semidirect
327. #431	C₂²×C₅₃	semidirect
328. #423	C₂²×C₅₃	semidirect
329. #423	C₂²×C₅₃	semidirect
330. C₂×C₄⋊C₈	C₂²×C₅₃	semidirect
331. C₂×C₄⋊C₈	C₂²×C₅₃	semidirect
334. #332	C₂²×C₅₃	semidirect
335. #332	C₂²×C₅₃	semidirect
336. #332	C₂²×C₅₃	semidirect
337. #331	C₂²×C₅₃	semidirect
338. #331	C₂²×C₅₃	semidirect
339. #331	C₂²×C₅₃	semidirect
340. #331	C₂²×C₅₃	semidirect
341. #331	C₂²×C₅₃	semidirect
342. #331	C₂²×C₅₃	semidirect
343. #331	C₂²×C₅₃	semidirect
344. C₅₃	#17758	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×#3498	2
C₅₃×#2989	2
C₅₃×#2925	2
C₅₃×#1557	2
C₅₃×#1555	2
C₅₃×#1553	2
C₅₃×#1547	2
C₅₃×#1545	2
C₅₃×#1540	2
C₂×C₅₃×#331	2
#17758	53

Subgroups — C53×#17758 #27225270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17758 #27225270