Subgroups — C₅₃×#60645 #27268157

Order 13568.

3158 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#575	C₂	1	2	yes	maximal, complemented
2	C₂²×#137×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₅₃×#3449	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#536	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#3156	C₂	1	8	yes	maximal, complemented
6	C₅₃×#3154	C₂	1	8	yes	maximal, complemented
7	C₅₃×#3153	C₂	1	4	yes	maximal, complemented
8	C₅₃×#3152	C₂	1	4	yes	maximal, complemented
9	C₅₃×#2873	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×#41×C₅₃	C₂²	1	2	yes	—
12	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#575	—	2	16	no	—
14	C₂×#137×C₅₃	—	2	16	no	—
15	C₂×#137×C₅₃	C₂²	1	8	yes	complemented
16	C₂×#137×C₅₃	C₂²	1	8	yes	complemented
17	C₂×#137×C₅₃	C₂²	1	4	yes	complemented
18	C₂×#137×C₅₃	C₂²	1	4	yes	complemented
19	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	4	yes	complemented
20	C₂×AGL(1,ℤ/8)×C₅₃	C₂²	1	4	yes	complemented
21	C₂×#135×C₅₃	C₂²	1	8	yes	—
22	C₂²×Q₁₆×C₅₃	C₂²	1	4	yes	complemented
23	C₂²×SD₁₆×C₅₃	C₂²	1	4	yes	complemented
24	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#537	—	2	2	no	—
26	C₅₃×#536	—	2	4	no	—
27	C₅₃×#536	C₂²	1	8	yes	complemented
28	C₅₃×#536	C₂²	1	4	yes	complemented
29	C₅₃×#535	C₂²	1	4	yes	complemented
30	C₅₃×#534	—	2	2	no	—
31	C₅₃×#532	C₂²	1	4	yes	—
32	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
33	C₂×#124×C₅₃	C₂²	1	2	yes	complemented
34	C₅₃×#525	—	2	4	no	—
35	C₂×#123×C₅₃	—	2	2	no	—
36	C₂×C₂²⋊Q₈×C₅₃	—	2	2	no	—
37	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#497	C₂²	1	16	yes	complemented
39	C₅₃×#497	C₂²	1	8	yes	complemented
40	C₅₃×#496	C₂²	1	8	yes	complemented
41	C₅₃×#495	C₂²	1	16	yes	complemented
42	C₅₃×#494	C₂²	1	8	yes	complemented
43	C₅₃×#492	C₂²	1	8	yes	complemented
44	C₅₃×#405	C₂²	1	8	yes	complemented
45	C₄×M₁₆×C₅₃	C₂²	1	4	yes	complemented
46	C₂×#41×C₅₃	—	2	2	no	—
47	C₂×#41×C₅₃	—	2	2	no	—
48	C₂×#41×C₅₃	—	2	2	no	—
49	C₂×#41×C₅₃	—	2	8	no	—
50	C₂×#41×C₅₃	—	2	8	no	—
51	C₂×#41×C₅₃	—	2	8	no	—
52	C₂×#41×C₅₃	C₂³	1	4	yes	—
53	C₂×#41×C₅₃	C₂³	1	4	yes	—
54	C₂×#41×C₅₃	C₂³	1	4	yes	—
55	C₂²×Q₈×C₅₃	—	2	1	no	—
56	C₂²×Q₈×C₅₃	—	2	2	no	—
57	C₂²×Q₈×C₅₃	—	2	8	no	—
58	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
59	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
61	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
62	C₂²×D₄×C₅₃	—	2	1	no	—
63	C₂²×D₄×C₅₃	—	2	2	no	—
64	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
66	C₂³×C₄×C₅₃	—	2	1	no	—
67	C₂³×C₄×C₅₃	—	2	1	no	—
68	C₂³×C₄×C₅₃	—	2	2	no	—
69	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
70	#137×C₅₃	—	2	32	no	—
71	#137×C₅₃	—	2	32	no	—
72	#137×C₅₃	—	2	16	no	—
73	#137×C₅₃	—	2	16	no	—
74	AGL(1,ℤ/8)×C₅₃	—	2	16	no	—
75	AGL(1,ℤ/8)×C₅₃	—	2	16	no	—
76	#135×C₅₃	—	2	32	no	—
77	C₂×Q₁₆×C₅₃	—	2	16	no	—
78	C₂×Q₁₆×C₅₃	C₂³	1	8	yes	—
79	C₂×Q₁₆×C₅₃	C₂³	1	16	yes	—
80	C₂×SD₁₆×C₅₃	—	2	16	no	—
81	C₂×SD₁₆×C₅₃	C₂³	1	8	yes	—
82	C₂×SD₁₆×C₅₃	C₂³	1	16	yes	—
83	C₂×SD₁₆×C₅₃	C₂³	1	8	yes	—
84	C₂×D₈×C₅₃	C₂³	1	8	yes	—
85	C₂×M₁₆×C₅₃	D₄	1	16	yes	complemented
86	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
87	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
88	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
89	C₄⋊Q₈×C₅₃	—	2	4	no	—
90	C₄⋊Q₈×C₅₃	C₂³	1	8	yes	—
91	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
92	#127×C₅₃	C₂³	1	4	yes	—
93	#124×C₅₃	—	2	4	no	—
94	#124×C₅₃	C₂³	1	8	yes	—
95	#124×C₅₃	C₂³	1	8	yes	—
96	#123×C₅₃	—	2	16	no	—
97	#123×C₅₃	—	2	8	no	—
98	C₂²⋊Q₈×C₅₃	—	2	16	no	—
99	C₂²⋊Q₈×C₅₃	—	2	8	no	—
100	#121×C₅₃	—	2	8	no	—
101	#121×C₅₃	—	2	8	no	—
102	#120×C₅₃	—	2	8	no	—
103	C₄×D₄×C₅₃	—	2	8	no	—
104	#117×C₅₃	D₄	1	4	yes	—
105	#117×C₅₃	C₂³	1	4	yes	—
106	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
107	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
108	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
109	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
110	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
111	#97×C₅₃	C₂³	1	8	yes	—
112	C₄×C₈×C₅₃	C₂³	1	8	yes	—
113	C₂⁴×C₅₃	—	2	1	no	—
114	C₂⁴×C₅₃	—	2	2	no	—
115	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
116	#41×C₅₃	—	2	8	no	—
117	#41×C₅₃	—	2	8	no	—
118	#41×C₅₃	—	2	8	no	—
119	#41×C₅₃	—	2	8	no	—
120	#41×C₅₃	—	2	8	no	—
121	#41×C₅₃	—	2	8	no	—
122	#41×C₅₃	—	4	8	no	—
123	#41×C₅₃	—	4	8	no	—
124	#41×C₅₃	—	4	8	no	—
125	C₂×Q₈×C₅₃	—	2	4	no	—
126	C₂×Q₈×C₅₃	—	2	2	no	—
127	C₂×Q₈×C₅₃	—	2	2	no	—
128	C₂×Q₈×C₅₃	—	2	1	no	—
129	C₂×Q₈×C₅₃	—	2	8	no	—
130	C₂×Q₈×C₅₃	—	2	4	no	—
131	C₂×Q₈×C₅₃	—	2	8	no	—
132	C₂×Q₈×C₅₃	—	2	8	no	—
133	C₂×Q₈×C₅₃	—	2	4	no	—
134	C₂×Q₈×C₅₃	—	2	4	no	—
135	C₂×Q₈×C₅₃	—	4	4	no	—
136	C₂×Q₈×C₅₃	—	4	8	no	—
137	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
138	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
139	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
140	C₂×D₄×C₅₃	—	2	4	no	—
141	C₂×D₄×C₅₃	—	2	4	no	—
142	C₂×D₄×C₅₃	—	2	4	no	—
143	C₂×D₄×C₅₃	—	2	4	no	—
144	C₂×D₄×C₅₃	—	2	4	no	—
145	C₂×D₄×C₅₃	—	2	4	no	—
146	C₂×D₄×C₅₃	—	2	2	no	—
147	C₂×D₄×C₅₃	—	2	1	no	—
148	C₂×D₄×C₅₃	—	2	8	no	—
149	C₂×D₄×C₅₃	—	4	4	no	—
150	C₂×D₄×C₅₃	—	4	8	no	—
151	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
152	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₅₃	—	2	2	no	—
154	C₂²×C₄×C₅₃	—	2	1	no	—
155	C₂²×C₄×C₅₃	—	2	2	no	—
156	C₂²×C₄×C₅₃	—	2	4	no	—
157	C₂²×C₄×C₅₃	—	2	4	no	—
158	C₂²×C₄×C₅₃	—	2	4	no	—
159	C₂²×C₄×C₅₃	—	2	4	no	—
160	C₂²×C₄×C₅₃	—	4	4	no	—
161	C₂²×C₄×C₅₃	—	4	4	no	—
162	C₂²×C₄×C₅₃	—	4	8	no	—
163	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
164	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
167	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
169	Q₁₆×C₅₃	—	2	32	no	—
170	Q₁₆×C₅₃	—	2	16	no	—
171	SD₁₆×C₅₃	—	2	32	no	—
172	SD₁₆×C₅₃	—	2	16	no	—
173	SD₁₆×C₅₃	—	2	16	no	—
174	D₈×C₅₃	—	2	16	no	—
175	M₁₆×C₅₃	C₂×D₄	1	16	yes	—
176	M₁₆×C₅₃	C₂×D₄	1	16	yes	—
177	C₂×C₈×C₅₃	C₂×D₄	1	16	yes	—
178	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
179	C₄⋊C₄×C₅₃	—	2	16	no	—
180	C₄⋊C₄×C₅₃	—	2	8	no	—
181	C₄⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
182	C₂²⋊C₄×C₅₃	—	2	16	no	—
183	C₂²⋊C₄×C₅₃	—	2	16	no	—
184	C₂²⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
185	C₄²×C₅₃	C₂×D₄	1	4	yes	—
186	C₄²×C₅₃	C₂⁴	1	4	yes	—
187	C₂³×C₅₃	—	2	4	no	—
188	C₂³×C₅₃	—	2	2	no	—
189	C₂³×C₅₃	—	4	2	no	—
190	C₂³×C₅₃	—	4	4	no	—
191	C₂³×C₅₃	—	4	2	no	—
192	C₂³×C₅₃	—	4	4	no	—
193	C₂³×C₅₃	—	8	4	no	—
194	C₂³×C₅₃	#127	1	4	yes	—
195	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
196	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
197	Q₈×C₅₃	—	2	8	no	—
198	Q₈×C₅₃	—	2	4	no	—
199	Q₈×C₅₃	—	2	4	no	—
200	Q₈×C₅₃	—	2	4	no	—
201	Q₈×C₅₃	—	4	8	no	—
202	Q₈×C₅₃	—	4	4	no	—
203	Q₈×C₅₃	—	4	4	no	—
204	Q₈×C₅₃	—	4	2	no	—
205	D₄×C₅₃	—	2	4	no	—
206	D₄×C₅₃	—	2	4	no	—
207	D₄×C₅₃	—	2	4	no	—
208	D₄×C₅₃	—	4	8	no	—
209	D₄×C₅₃	—	4	8	no	—
210	D₄×C₅₃	—	4	8	no	—
211	D₄×C₅₃	—	4	4	no	—
212	D₄×C₅₃	—	4	2	no	—
213	D₄×C₅₃	—	8	8	no	—
214	C₂×C₄×C₅₃	—	2	4	no	—
215	C₂×C₄×C₅₃	—	2	2	no	—
216	C₂×C₄×C₅₃	—	2	4	no	—
217	C₂×C₄×C₅₃	—	4	4	no	—
218	C₂×C₄×C₅₃	—	4	2	no	—
219	C₂×C₄×C₅₃	—	4	4	no	—
220	C₂×C₄×C₅₃	—	4	4	no	—
221	C₂×C₄×C₅₃	—	4	8	no	—
222	C₂×C₄×C₅₃	—	4	8	no	—
223	C₂×C₄×C₅₃	—	4	8	no	—
224	C₂×C₄×C₅₃	—	4	8	no	—
225	C₂×C₄×C₅₃	#127	1	4	yes	—
226	C₂×C₄×C₅₃	#127	1	8	yes	—
227	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
228	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
230	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
231	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
232	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
233	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
234	C₈×C₅₃	C₂²×D₄	1	16	yes	—
235	#60645	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
236	C₂²×C₅₃	—	2	2	no	—
237	C₂²×C₅₃	—	2	4	no	—
238	C₂²×C₅₃	—	4	4	no	—
239	C₂²×C₅₃	—	4	8	no	—
240	C₂²×C₅₃	—	4	4	no	—
241	C₂²×C₅₃	—	4	2	no	—
242	C₂²×C₅₃	—	4	4	no	—
243	C₂²×C₅₃	—	8	4	no	—
244	C₂²×C₅₃	—	8	8	no	—
245	C₂²×C₅₃	C₂×#127	1	4	yes	—
246	C₂²×C₅₃	C₂×#127	1	2	yes	—
247	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
248	C₄×C₅₃	—	2	8	no	—
249	C₄×C₅₃	—	4	4	no	—
250	C₄×C₅₃	—	4	2	no	—
251	C₄×C₅₃	—	4	4	no	—
252	C₄×C₅₃	C₂×#127	1	2	yes	—
253	C₄×C₅₃	C₂×#127	1	4	yes	—
254	C₄×C₅₃	C₂³×D₄	1	2	yes	—
255	C₂×#575	C₂×C₅₃	1	2	yes	complemented
256	C₂²×#137	C₂×C₅₃	1	2	yes	complemented
257	#3449	C₂×C₅₃	1	1	yes	characteristic, complemented
258	C₂×#536	C₂×C₅₃	1	1	yes	characteristic, complemented
259	#3156	C₂×C₅₃	1	8	yes	complemented
260	#3154	C₂×C₅₃	1	8	yes	complemented
261	#3153	C₂×C₅₃	1	4	yes	complemented
262	#3152	C₂×C₅₃	1	4	yes	complemented
263	#2873	C₂×C₅₃	1	1	yes	characteristic, complemented
264	C₂×C₅₃	—	2	2	no	—
265	C₂×C₅₃	—	2	4	no	—
266	C₂×C₅₃	—	8	4	no	—
267	C₂×C₅₃	—	8	2	no	—
268	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
269	C₂×C₅₃	#3586	1	2	yes	—
270	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
271	C₂²×#41	C₂²×C₅₃	1	2	yes	—
272	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
273	#575	—	2	16	no	—
274	C₂×#137	—	2	16	no	—
275	C₂×#137	C₂²×C₅₃	1	4	yes	complemented
276	C₂×#137	C₂²×C₅₃	1	8	yes	complemented
277	C₂×#137	C₂²×C₅₃	1	8	yes	complemented
278	C₂×#137	C₂²×C₅₃	1	4	yes	complemented
279	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	4	yes	complemented
280	C₂×AGL(1,ℤ/8)	C₂²×C₅₃	1	4	yes	complemented
281	C₂×#135	C₂²×C₅₃	1	8	yes	—
282	C₂²×Q₁₆	C₂²×C₅₃	1	4	yes	complemented
283	C₂²×SD₁₆	C₂²×C₅₃	1	4	yes	complemented
284	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
285	#537	—	2	2	no	—
286	#536	—	2	4	no	—
287	#536	C₂²×C₅₃	1	8	yes	complemented
288	#536	C₂²×C₅₃	1	4	yes	complemented
289	#535	C₂²×C₅₃	1	4	yes	complemented
290	#534	—	2	2	no	—
291	#532	C₂²×C₅₃	1	4	yes	—
292	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	complemented
293	C₂×#124	C₂²×C₅₃	1	2	yes	complemented
294	#525	—	2	4	no	—
295	C₂×#123	—	2	2	no	—
296	C₂×C₂²⋊Q₈	—	2	2	no	—
297	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
298	#497	C₂²×C₅₃	1	16	yes	complemented
299	#497	C₂²×C₅₃	1	8	yes	complemented
300	#496	C₂²×C₅₃	1	8	yes	complemented
301	#495	C₂²×C₅₃	1	16	yes	complemented
302	#494	C₂²×C₅₃	1	8	yes	complemented
303	#492	C₂²×C₅₃	1	8	yes	complemented
304	#405	C₂²×C₅₃	1	8	yes	complemented
305	C₄×M₁₆	C₂²×C₅₃	1	4	yes	complemented
306	C₅₃	#60645	1	1	yes	characteristic, direct product factor, Sylow
307	C₂×#41	—	2	2	no	—
308	C₂×#41	—	2	2	no	—
309	C₂×#41	—	2	2	no	—
310	C₂×#41	—	2	8	no	—
311	C₂×#41	—	2	8	no	—
312	C₂×#41	—	2	8	no	—
313	C₂×#41	C₂³×C₅₃	1	4	yes	—
314	C₂×#41	C₂³×C₅₃	1	4	yes	—
315	C₂×#41	C₂³×C₅₃	1	4	yes	—
316	C₂²×Q₈	—	2	1	no	—
317	C₂²×Q₈	—	2	2	no	—
318	C₂²×Q₈	—	2	8	no	—
319	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
320	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
321	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
322	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
323	C₂²×D₄	—	2	1	no	—
324	C₂²×D₄	—	2	2	no	—
325	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
326	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
327	C₂³×C₄	—	2	1	no	—
328	C₂³×C₄	—	2	1	no	—
329	C₂³×C₄	—	2	2	no	—
330	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
331	#137	—	2	32	no	—
332	#137	—	2	32	no	—
333	#137	—	2	16	no	—
334	#137	—	2	16	no	—
335	AGL(1,ℤ/8)	—	2	16	no	—
336	AGL(1,ℤ/8)	—	2	16	no	—
337	#135	—	2	32	no	—
338	C₂×Q₁₆	—	2	16	no	—
339	C₂×Q₁₆	C₂³×C₅₃	1	16	yes	—
340	C₂×Q₁₆	C₂³×C₅₃	1	8	yes	—
341	C₂×SD₁₆	—	2	16	no	—
342	C₂×SD₁₆	C₂³×C₅₃	1	16	yes	—
343	C₂×SD₁₆	C₂³×C₅₃	1	8	yes	—
344	C₂×SD₁₆	C₂³×C₅₃	1	8	yes	—
345	C₂×D₈	C₂³×C₅₃	1	8	yes	—
346	C₂×M₁₆	D₄×C₅₃	1	16	yes	complemented
347	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
348	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
349	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
350	C₄⋊Q₈	—	2	4	no	—
351	C₄⋊Q₈	C₂³×C₅₃	1	8	yes	—
352	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
353	#127	C₂³×C₅₃	1	4	yes	—
354	#124	—	2	4	no	—
355	#124	C₂³×C₅₃	1	8	yes	—
356	#124	C₂³×C₅₃	1	8	yes	—
357	#123	—	2	16	no	—
358	#123	—	2	8	no	—
359	C₂²⋊Q₈	—	2	16	no	—
360	C₂²⋊Q₈	—	2	8	no	—
361	#121	—	2	8	no	—
362	#121	—	2	8	no	—
363	#120	—	2	8	no	—
364	C₄×D₄	—	2	8	no	—
365	#117	D₄×C₅₃	1	4	yes	—
366	#117	C₂³×C₅₃	1	4	yes	—
367	C₂×C₄⋊C₄	—	2	4	no	—
368	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
369	C₂×C₂²⋊C₄	—	2	4	no	—
370	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
371	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
372	#97	C₂³×C₅₃	1	8	yes	—
373	C₄×C₈	C₂³×C₅₃	1	8	yes	—
374	C₂⁴	—	2	1	no	—
375	C₂⁴	—	2	2	no	—
376	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
377	#41	—	2	8	no	—
378	#41	—	2	8	no	—
379	#41	—	2	8	no	—
380	#41	—	2	8	no	—
381	#41	—	2	8	no	—
382	#41	—	2	8	no	—
383	#41	—	4	8	no	—
384	#41	—	4	8	no	—
385	#41	—	4	8	no	—
386	C₂×Q₈	—	2	4	no	—
387	C₂×Q₈	—	2	2	no	—
388	C₂×Q₈	—	2	2	no	—
389	C₂×Q₈	—	2	1	no	—
390	C₂×Q₈	—	2	8	no	—
391	C₂×Q₈	—	2	4	no	—
392	C₂×Q₈	—	2	8	no	—
393	C₂×Q₈	—	2	8	no	—
394	C₂×Q₈	—	2	4	no	—
395	C₂×Q₈	—	2	4	no	—
396	C₂×Q₈	—	4	4	no	—
397	C₂×Q₈	—	4	8	no	—
398	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
399	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
400	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	4	no	—
403	C₂×D₄	—	2	4	no	—
404	C₂×D₄	—	2	4	no	—
405	C₂×D₄	—	2	4	no	—
406	C₂×D₄	—	2	4	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	2	1	no	—
409	C₂×D₄	—	2	8	no	—
410	C₂×D₄	—	4	4	no	—
411	C₂×D₄	—	4	8	no	—
412	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
413	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	4	4	no	—
422	C₂²×C₄	—	4	4	no	—
423	C₂²×C₄	—	4	8	no	—
424	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
425	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
426	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
427	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
428	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
429	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
430	Q₁₆	—	2	32	no	—
431	Q₁₆	—	2	16	no	—
432	SD₁₆	—	2	32	no	—
433	SD₁₆	—	2	16	no	—
434	SD₁₆	—	2	16	no	—
435	D₈	—	2	16	no	—
436	M₁₆	C₂×D₄×C₅₃	1	16	yes	—
437	M₁₆	C₂×D₄×C₅₃	1	16	yes	—
438	C₂×C₈	C₂×D₄×C₅₃	1	16	yes	—
439	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
440	C₄⋊C₄	—	2	16	no	—
441	C₄⋊C₄	—	2	8	no	—
442	C₄⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
443	C₂²⋊C₄	—	2	16	no	—
444	C₂²⋊C₄	—	2	16	no	—
445	C₂²⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
446	C₄²	C₂×D₄×C₅₃	1	4	yes	—
447	C₄²	C₂⁴×C₅₃	1	4	yes	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	4	no	—
450	C₂³	—	4	2	no	—
451	C₂³	—	4	4	no	—
452	C₂³	—	4	2	no	—
453	C₂³	—	4	4	no	—
454	C₂³	—	8	4	no	—
455	C₂³	#127×C₅₃	1	4	yes	—
456	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
457	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
458	Q₈	—	2	8	no	—
459	Q₈	—	2	4	no	—
460	Q₈	—	2	4	no	—
461	Q₈	—	2	4	no	—
462	Q₈	—	4	8	no	—
463	Q₈	—	4	4	no	—
464	Q₈	—	4	4	no	—
465	Q₈	—	4	2	no	—
466	D₄	—	2	4	no	—
467	D₄	—	2	4	no	—
468	D₄	—	2	4	no	—
469	D₄	—	4	8	no	—
470	D₄	—	4	8	no	—
471	D₄	—	4	8	no	—
472	D₄	—	4	4	no	—
473	D₄	—	4	2	no	—
474	D₄	—	8	8	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	4	4	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	4	no	—
481	C₂×C₄	—	4	4	no	—
482	C₂×C₄	—	4	8	no	—
483	C₂×C₄	—	4	8	no	—
484	C₂×C₄	—	4	8	no	—
485	C₂×C₄	—	4	8	no	—
486	C₂×C₄	#127×C₅₃	1	4	yes	—
487	C₂×C₄	#127×C₅₃	1	8	yes	—
488	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
489	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
490	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
491	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
492	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
493	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
494	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
495	C₈	C₂²×D₄×C₅₃	1	16	yes	—
496	C₂²	—	2	2	no	—
497	C₂²	—	2	4	no	—
498	C₂²	—	4	4	no	—
499	C₂²	—	4	8	no	—
500	C₂²	—	4	4	no	—
501	C₂²	—	4	2	no	—
502	C₂²	—	4	4	no	—
503	C₂²	—	8	4	no	—
504	C₂²	—	8	8	no	—
505	C₂²	C₂×#127×C₅₃	1	4	yes	—
506	C₂²	C₂×#127×C₅₃	1	2	yes	—
507	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
508	C₄	—	2	8	no	—
509	C₄	—	4	4	no	—
510	C₄	—	4	2	no	—
511	C₄	—	4	4	no	—
512	C₄	C₂×#127×C₅₃	1	2	yes	—
513	C₄	C₂×#127×C₅₃	1	4	yes	—
514	C₄	C₂³×D₄×C₅₃	1	2	yes	—
515	C₂	—	2	2	no	—
516	C₂	—	2	4	no	—
517	C₂	—	8	4	no	—
518	C₂	—	8	2	no	—
519	C₂	C₂²×#127×C₅₃	1	1	yes	characteristic
520	C₂	C₅₃×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×#117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#575	C₂	semidirect
2. C₂²×#137×C₅₃	C₂	semidirect
3. C₅₃×#3449	C₂	semidirect
4. C₂×C₅₃×#536	C₂	semidirect
5. C₅₃×#3156	C₂	semidirect
6. C₅₃×#3154	C₂	semidirect
7. C₅₃×#3153	C₂	semidirect
8. C₅₃×#3152	C₂	semidirect
9. C₅₃×#2873	C₂	semidirect
15. C₂×#137×C₅₃	C₂²	semidirect
16. C₂×#137×C₅₃	C₂²	semidirect
17. C₂×#137×C₅₃	C₂²	semidirect
18. C₂×#137×C₅₃	C₂²	semidirect
19. C₂×AGL(1,ℤ/8)×C₅₃	C₂²	semidirect
20. C₂×AGL(1,ℤ/8)×C₅₃	C₂²	semidirect
22. C₂²×Q₁₆×C₅₃	C₂²	semidirect
23. C₂²×SD₁₆×C₅₃	C₂²	semidirect
27. C₅₃×#536	C₂²	semidirect
28. C₅₃×#536	C₂²	semidirect
29. C₅₃×#535	C₂²	semidirect
32. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
33. C₂×#124×C₅₃	C₂²	semidirect
38. C₅₃×#497	C₂²	semidirect
39. C₅₃×#497	C₂²	semidirect
40. C₅₃×#496	C₂²	semidirect
41. C₅₃×#495	C₂²	semidirect
42. C₅₃×#494	C₂²	semidirect
43. C₅₃×#492	C₂²	semidirect
44. C₅₃×#405	C₂²	semidirect
45. C₄×M₁₆×C₅₃	C₂²	semidirect
85. C₂×M₁₆×C₅₃	D₄	semidirect
235. #60645	C₅₃	direct (N × H)
255. C₂×#575	C₂×C₅₃	semidirect
256. C₂²×#137	C₂×C₅₃	semidirect
257. #3449	C₂×C₅₃	semidirect
258. C₂×#536	C₂×C₅₃	semidirect
259. #3156	C₂×C₅₃	semidirect
260. #3154	C₂×C₅₃	semidirect
261. #3153	C₂×C₅₃	semidirect
262. #3152	C₂×C₅₃	semidirect
263. #2873	C₂×C₅₃	semidirect
275. C₂×#137	C₂²×C₅₃	semidirect
276. C₂×#137	C₂²×C₅₃	semidirect
277. C₂×#137	C₂²×C₅₃	semidirect
278. C₂×#137	C₂²×C₅₃	semidirect
279. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
280. C₂×AGL(1,ℤ/8)	C₂²×C₅₃	semidirect
282. C₂²×Q₁₆	C₂²×C₅₃	semidirect
283. C₂²×SD₁₆	C₂²×C₅₃	semidirect
287. #536	C₂²×C₅₃	semidirect
288. #536	C₂²×C₅₃	semidirect
289. #535	C₂²×C₅₃	semidirect
292. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
293. C₂×#124	C₂²×C₅₃	semidirect
298. #497	C₂²×C₅₃	semidirect
299. #497	C₂²×C₅₃	semidirect
300. #496	C₂²×C₅₃	semidirect
301. #495	C₂²×C₅₃	semidirect
302. #494	C₂²×C₅₃	semidirect
303. #492	C₂²×C₅₃	semidirect
304. #405	C₂²×C₅₃	semidirect
305. C₄×M₁₆	C₂²×C₅₃	semidirect
306. C₅₃	#60645	direct (N × H)
346. C₂×M₁₆	D₄×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×#575	2
C₂²×#137×C₅₃	2
C₅₃×#3449	2
C₂×C₅₃×#536	2
C₅₃×#3156	2
C₅₃×#3154	2
C₅₃×#3153	2
C₅₃×#3152	2
C₅₃×#2873	2
#60645	53

Subgroups — C53×#60645 #27268157

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60645 #27268157