Subgroups — C₅₃×#18642 #27226154

Order 13568.

1002 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3208	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3038	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1627	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1626	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1549	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1476	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×C₅₃	—	2	1	no	—
11	Q₈²×C₅₃	—	2	1	no	—
12	C₅₃×#557	C₂²	1	2	yes	complemented
13	C₅₃×#554	—	2	1	no	—
14	C₅₃×#554	—	2	2	no	—
15	D₄×Q₈×C₅₃	—	2	2	no	—
16	D₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
17	C₅₃×#544	—	2	2	no	—
18	C₅₃×#544	—	2	1	no	—
19	C₅₃×#533	—	2	1	no	—
20	C₅₃×#533	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
22	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#520	—	2	1	no	—
25	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#492	—	2	1	no	—
27	C₅₃×#486	—	2	1	no	—
28	C₅₃×#486	—	2	2	no	—
29	C₅₃×#486	C₂²	1	4	yes	—
30	C₅₃×#460	—	2	1	no	—
31	C₅₃×#460	C₂²	1	4	yes	—
32	C₅₃×#460	C₂²	1	2	yes	—
33	C₅₃×#459	—	2	1	no	—
34	C₅₃×#459	C₂²	1	2	yes	—
35	C₅₃×#450	—	2	2	no	—
36	C₅₃×#431	C₂²	1	1	yes	characteristic
37	C₅₃×#423	C₂²	1	1	yes	characteristic
38	C₅₃×#418	—	2	1	no	—
39	C₂×#102×C₅₃	—	2	2	no	—
40	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#332	C₂²	1	4	yes	complemented
42	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
43	C₅₃×Q₈⋊C₈	C₂²	1	4	yes	complemented
44	C₅₃×D₄⋊C₈	C₂²	1	2	yes	complemented
45	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
46	C₂²×Q₈×C₅₃	—	2	2	no	—
47	C₂²×Q₈×C₅₃	—	2	1	no	—
48	C₂²×D₄×C₅₃	—	2	1	no	—
49	C₂²×D₄×C₅₃	—	2	1	no	—
50	C₂²×D₄×C₅₃	—	2	1	no	—
51	AGL(1,ℤ/8)×C₅₃	—	4	4	no	—
52	C₂×SD₁₆×C₅₃	—	2	2	no	—
53	C₂×SD₁₆×C₅₃	—	2	4	no	—
54	C₂×D₈×C₅₃	—	2	2	no	—
55	C₂×M₁₆×C₅₃	—	2	1	no	—
56	C₂²×C₈×C₅₃	—	2	1	no	—
57	C₄⋊Q₈×C₅₃	—	2	2	no	—
58	C₄⋊Q₈×C₅₃	—	2	2	no	—
59	C₄⋊Q₈×C₅₃	—	2	4	no	—
60	C₄⋊Q₈×C₅₃	—	2	2	no	—
61	C₄⋊Q₈×C₅₃	—	2	1	no	—
62	C₄⋊Q₈×C₅₃	—	2	1	no	—
63	C₄⋊Q₈×C₅₃	—	2	2	no	—
64	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
65	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
66	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
67	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
68	#127×C₅₃	—	2	1	no	—
69	#127×C₅₃	—	2	1	no	—
70	#127×C₅₃	—	2	1	no	—
71	#127×C₅₃	C₂³	1	2	yes	—
72	#125×C₅₃	—	2	2	no	—
73	#125×C₅₃	—	2	4	no	—
74	#125×C₅₃	—	2	2	no	—
75	#125×C₅₃	—	2	1	no	—
76	C₂²⋊Q₈×C₅₃	—	2	2	no	—
77	C₂²⋊Q₈×C₅₃	—	2	4	no	—
78	C₂²⋊Q₈×C₅₃	—	2	4	no	—
79	C₂²⋊Q₈×C₅₃	—	2	2	no	—
80	C₂²⋊Q₈×C₅₃	—	2	4	no	—
81	C₂²⋊Q₈×C₅₃	—	2	1	no	—
82	C₂²⋊Q₈×C₅₃	D₄	1	2	yes	—
83	C₄×Q₈×C₅₃	—	2	2	no	—
84	C₄×Q₈×C₅₃	—	2	2	no	—
85	C₄×Q₈×C₅₃	—	2	1	no	—
86	C₄×Q₈×C₅₃	—	2	2	no	—
87	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
88	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
89	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
90	C₄×D₄×C₅₃	—	2	4	no	—
91	C₄×D₄×C₅₃	—	2	2	no	—
92	C₄×D₄×C₅₃	—	2	1	no	—
93	C₄×D₄×C₅₃	—	2	1	no	—
94	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
95	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
96	#117×C₅₃	—	2	2	no	—
97	#117×C₅₃	—	2	1	no	—
98	#117×C₅₃	—	2	1	no	—
99	#117×C₅₃	D₄	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
101	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
102	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
103	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
104	C₂×C₄²×C₅₃	—	2	1	no	—
105	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
106	C₄⋊C₈×C₅₃	—	2	1	no	—
107	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
108	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
109	#102×C₅₃	—	2	8	no	—
110	#102×C₅₃	—	2	4	no	—
111	#102×C₅₃	—	2	2	no	—
112	#102×C₅₃	—	2	2	no	—
113	C₂²⋊C₈×C₅₃	—	2	1	no	—
114	C₄×C₈×C₅₃	—	2	1	no	—
115	C₄×C₈×C₅₃	—	2	1	no	—
116	C₄×C₈×C₅₃	C₂³	1	2	yes	—
117	C₂⁴×C₅₃	—	4	1	no	—
118	#41×C₅₃	—	2	2	no	—
119	#41×C₅₃	SD₁₆	1	4	yes	complemented
120	C₂×Q₈×C₅₃	—	2	4	no	—
121	C₂×Q₈×C₅₃	—	2	1	no	—
122	C₂×Q₈×C₅₃	—	2	2	no	—
123	C₂×Q₈×C₅₃	—	2	2	no	—
124	C₂×Q₈×C₅₃	—	8	2	no	—
125	C₂×Q₈×C₅₃	—	8	4	no	—
126	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
127	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
128	C₂×D₄×C₅₃	—	2	1	no	—
129	C₂×D₄×C₅₃	—	2	2	no	—
130	C₂×D₄×C₅₃	—	2	2	no	—
131	C₂×D₄×C₅₃	—	2	2	no	—
132	C₂×D₄×C₅₃	—	4	4	no	—
133	C₂×D₄×C₅₃	—	4	1	no	—
134	C₂×D₄×C₅₃	—	4	1	no	—
135	C₂×D₄×C₅₃	—	4	1	no	—
136	C₂×D₄×C₅₃	—	4	2	no	—
137	C₂×D₄×C₅₃	—	4	2	no	—
138	C₂×D₄×C₅₃	—	8	2	no	—
139	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₅₃	—	2	1	no	—
141	C₂²×C₄×C₅₃	—	2	4	no	—
142	C₂²×C₄×C₅₃	—	2	2	no	—
143	C₂²×C₄×C₅₃	—	2	1	no	—
144	C₂²×C₄×C₅₃	—	2	1	no	—
145	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
148	SD₁₆×C₅₃	—	4	4	no	—
149	SD₁₆×C₅₃	—	8	4	no	—
150	D₈×C₅₃	—	4	4	no	—
151	M₁₆×C₅₃	—	4	2	no	—
152	C₂×C₈×C₅₃	—	2	2	no	—
153	C₂×C₈×C₅₃	—	2	2	no	—
154	C₂×C₈×C₅₃	—	2	2	no	—
155	C₂×C₈×C₅₃	—	4	2	no	—
156	C₄⋊C₄×C₅₃	—	2	4	no	—
157	C₄⋊C₄×C₅₃	—	2	4	no	—
158	C₄⋊C₄×C₅₃	—	2	4	no	—
159	C₄⋊C₄×C₅₃	—	2	4	no	—
160	C₄⋊C₄×C₅₃	—	2	2	no	—
161	C₄⋊C₄×C₅₃	—	2	1	no	—
162	C₄⋊C₄×C₅₃	—	2	1	no	—
163	C₄⋊C₄×C₅₃	—	2	2	no	—
164	C₄⋊C₄×C₅₃	—	2	1	no	—
165	C₄⋊C₄×C₅₃	—	2	1	no	—
166	C₄⋊C₄×C₅₃	—	2	2	no	—
167	C₄⋊C₄×C₅₃	—	2	2	no	—
168	C₄⋊C₄×C₅₃	—	2	1	no	—
169	C₄⋊C₄×C₅₃	—	2	2	no	—
170	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
171	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
172	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
173	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
174	C₂²⋊C₄×C₅₃	—	2	4	no	—
175	C₂²⋊C₄×C₅₃	—	2	2	no	—
176	C₂²⋊C₄×C₅₃	—	2	1	no	—
177	C₂²⋊C₄×C₅₃	—	2	1	no	—
178	C₂²⋊C₄×C₅₃	—	2	1	no	—
179	C₄²×C₅₃	—	2	4	no	—
180	C₄²×C₅₃	—	2	1	no	—
181	C₄²×C₅₃	—	2	1	no	—
182	C₄²×C₅₃	—	2	1	no	—
183	C₄²×C₅₃	—	2	2	no	—
184	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
185	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
186	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
187	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
188	C₂³×C₅₃	—	2	1	no	—
189	C₂³×C₅₃	—	4	2	no	—
190	C₂³×C₅₃	—	8	2	no	—
191	C₂³×C₅₃	—	8	2	no	—
192	C₂³×C₅₃	—	8	2	no	—
193	C₂³×C₅₃	#120	1	1	yes	characteristic
194	Q₈×C₅₃	—	2	1	no	—
195	Q₈×C₅₃	—	2	4	no	—
196	Q₈×C₅₃	—	4	4	no	—
197	Q₈×C₅₃	—	4	2	no	—
198	Q₈×C₅₃	—	8	2	no	—
199	Q₈×C₅₃	—	8	4	no	—
200	Q₈×C₅₃	C₂×SD₁₆	1	2	yes	—
201	D₄×C₅₃	—	2	4	no	—
202	D₄×C₅₃	—	2	1	no	—
203	D₄×C₅₃	—	4	4	no	—
204	D₄×C₅₃	—	4	2	no	—
205	D₄×C₅₃	—	4	2	no	—
206	D₄×C₅₃	—	4	4	no	—
207	D₄×C₅₃	—	8	2	no	—
208	D₄×C₅₃	—	8	2	no	—
209	D₄×C₅₃	—	8	2	no	—
210	D₄×C₅₃	C₂×SD₁₆	1	2	yes	—
211	C₂×C₄×C₅₃	—	2	1	no	—
212	C₂×C₄×C₅₃	—	2	2	no	—
213	C₂×C₄×C₅₃	—	2	4	no	—
214	C₂×C₄×C₅₃	—	2	2	no	—
215	C₂×C₄×C₅₃	—	2	4	no	—
216	C₂×C₄×C₅₃	—	2	2	no	—
217	C₂×C₄×C₅₃	—	2	1	no	—
218	C₂×C₄×C₅₃	—	2	1	no	—
219	C₂×C₄×C₅₃	—	2	2	no	—
220	C₂×C₄×C₅₃	—	4	2	no	—
221	C₂×C₄×C₅₃	—	4	2	no	—
222	C₂×C₄×C₅₃	—	8	2	no	—
223	C₂×C₄×C₅₃	—	8	4	no	—
224	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
225	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
226	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
227	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
228	C₂×C₄×C₅₃	#137	1	2	yes	—
229	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
232	C₈×C₅₃	—	4	2	no	—
233	C₈×C₅₃	—	4	2	no	—
234	C₈×C₅₃	—	4	2	no	—
235	#18642	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
236	C₂²×C₅₃	—	2	1	no	—
237	C₂²×C₅₃	—	2	1	no	—
238	C₂²×C₅₃	—	2	2	no	—
239	C₂²×C₅₃	—	4	1	no	—
240	C₂²×C₅₃	—	4	1	no	—
241	C₂²×C₅₃	—	8	2	no	—
242	C₂²×C₅₃	—	8	2	no	—
243	C₂²×C₅₃	—	8	2	no	—
244	C₂²×C₅₃	—	16	4	no	—
245	C₂²×C₅₃	#448	1	2	yes	—
246	C₂²×C₅₃	C₂×#120	1	1	yes	characteristic
247	C₄×C₅₃	—	2	2	no	—
248	C₄×C₅₃	—	2	1	no	—
249	C₄×C₅₃	—	2	1	no	—
250	C₄×C₅₃	—	4	1	no	—
251	C₄×C₅₃	—	4	2	no	—
252	C₄×C₅₃	—	4	4	no	—
253	C₄×C₅₃	—	4	1	no	—
254	C₄×C₅₃	—	4	2	no	—
255	C₄×C₅₃	#448	1	2	yes	—
256	C₄×C₅₃	#454	1	2	yes	—
257	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
258	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
259	C₄×C₅₃	#576	1	2	yes	—
260	#3503	C₂×C₅₃	1	1	yes	characteristic, complemented
261	#3208	C₂×C₅₃	1	1	yes	characteristic, complemented
262	C₂×#486	C₂×C₅₃	1	1	yes	characteristic, complemented
263	#3038	C₂×C₅₃	1	1	yes	characteristic
264	#1627	C₂×C₅₃	1	4	yes	complemented
265	#1626	C₂×C₅₃	1	4	yes	complemented
266	#1549	C₂×C₅₃	1	1	yes	characteristic, complemented
267	#1493	C₂×C₅₃	1	1	yes	characteristic, complemented
268	#1476	C₂×C₅₃	1	1	yes	characteristic, complemented
269	C₂×C₅₃	—	2	2	no	—
270	C₂×C₅₃	—	4	2	no	—
271	C₂×C₅₃	—	16	2	no	—
272	C₂×C₅₃	C₂×#448	1	1	yes	characteristic
273	C₂×C₅₃	#3008	1	1	yes	characteristic
274	C₂×C₅₃	C₂×#454	1	1	yes	characteristic
275	C₂×AGL(1,ℤ/8)	—	2	1	no	—
276	Q₈²	—	2	1	no	—
277	#557	C₂²×C₅₃	1	2	yes	complemented
278	#554	—	2	1	no	—
279	#554	—	2	2	no	—
280	D₄×Q₈	—	2	2	no	—
281	D₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
282	#544	—	2	2	no	—
283	#544	—	2	1	no	—
284	#533	—	2	1	no	—
285	#533	C₂²×C₅₃	1	1	yes	characteristic
286	C₂×C₄⋊Q₈	—	2	1	no	—
287	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
288	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
289	#520	—	2	1	no	—
290	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
291	#492	—	2	1	no	—
292	#486	—	2	2	no	—
293	#486	—	2	1	no	—
294	#486	C₂²×C₅₃	1	4	yes	—
295	#460	—	2	1	no	—
296	#460	C₂²×C₅₃	1	2	yes	—
297	#460	C₂²×C₅₃	1	4	yes	—
298	#459	—	2	1	no	—
299	#459	C₂²×C₅₃	1	2	yes	—
300	#450	—	2	2	no	—
301	#431	C₂²×C₅₃	1	1	yes	characteristic
302	#423	C₂²×C₅₃	1	1	yes	characteristic
303	#418	—	2	1	no	—
304	C₂×#102	—	2	2	no	—
305	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
306	#332	C₂²×C₅₃	1	4	yes	complemented
307	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	complemented
308	Q₈⋊C₈	C₂²×C₅₃	1	4	yes	complemented
309	D₄⋊C₈	C₂²×C₅₃	1	2	yes	complemented
310	C₅₃	#18642	1	1	yes	characteristic, direct product factor, Sylow
311	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
312	C₂²×Q₈	—	2	2	no	—
313	C₂²×Q₈	—	2	1	no	—
314	C₂²×D₄	—	2	1	no	—
315	C₂²×D₄	—	2	1	no	—
316	C₂²×D₄	—	2	1	no	—
317	AGL(1,ℤ/8)	—	4	4	no	—
318	C₂×SD₁₆	—	2	2	no	—
319	C₂×SD₁₆	—	2	4	no	—
320	C₂×D₈	—	2	2	no	—
321	C₂×M₁₆	—	2	1	no	—
322	C₂²×C₈	—	2	1	no	—
323	C₄⋊Q₈	—	2	2	no	—
324	C₄⋊Q₈	—	2	2	no	—
325	C₄⋊Q₈	—	2	4	no	—
326	C₄⋊Q₈	—	2	2	no	—
327	C₄⋊Q₈	—	2	2	no	—
328	C₄⋊Q₈	—	2	1	no	—
329	C₄⋊Q₈	—	2	1	no	—
330	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
331	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
332	C₄⋊Q₈	D₄×C₅₃	1	1	yes	characteristic
333	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
334	#127	—	2	1	no	—
335	#127	—	2	1	no	—
336	#127	—	2	1	no	—
337	#127	C₂³×C₅₃	1	2	yes	—
338	#125	—	2	2	no	—
339	#125	—	2	4	no	—
340	#125	—	2	2	no	—
341	#125	—	2	1	no	—
342	C₂²⋊Q₈	—	2	2	no	—
343	C₂²⋊Q₈	—	2	4	no	—
344	C₂²⋊Q₈	—	2	4	no	—
345	C₂²⋊Q₈	—	2	1	no	—
346	C₂²⋊Q₈	—	2	2	no	—
347	C₂²⋊Q₈	—	2	4	no	—
348	C₂²⋊Q₈	D₄×C₅₃	1	2	yes	—
349	C₄×Q₈	—	2	2	no	—
350	C₄×Q₈	—	2	2	no	—
351	C₄×Q₈	—	2	2	no	—
352	C₄×Q₈	—	2	1	no	—
353	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
354	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
355	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
356	C₄×D₄	—	2	4	no	—
357	C₄×D₄	—	2	2	no	—
358	C₄×D₄	—	2	1	no	—
359	C₄×D₄	—	2	1	no	—
360	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
361	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
362	#117	—	2	2	no	—
363	#117	—	2	1	no	—
364	#117	—	2	1	no	—
365	#117	D₄×C₅₃	1	1	yes	characteristic
366	C₂×C₄⋊C₄	—	2	2	no	—
367	C₂×C₄⋊C₄	—	2	2	no	—
368	C₂×C₄⋊C₄	—	2	1	no	—
369	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
370	C₂×C₄²	—	2	1	no	—
371	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
372	C₄⋊C₈	—	2	1	no	—
373	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
374	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
375	#102	—	2	8	no	—
376	#102	—	2	2	no	—
377	#102	—	2	2	no	—
378	#102	—	2	4	no	—
379	C₂²⋊C₈	—	2	1	no	—
380	C₄×C₈	—	2	1	no	—
381	C₄×C₈	—	2	1	no	—
382	C₄×C₈	C₂³×C₅₃	1	2	yes	—
383	C₂⁴	—	4	1	no	—
384	#41	—	2	2	no	—
385	#41	SD₁₆×C₅₃	1	4	yes	complemented
386	C₂×Q₈	—	2	4	no	—
387	C₂×Q₈	—	2	1	no	—
388	C₂×Q₈	—	2	2	no	—
389	C₂×Q₈	—	2	2	no	—
390	C₂×Q₈	—	8	2	no	—
391	C₂×Q₈	—	8	4	no	—
392	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
393	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
394	C₂×D₄	—	2	1	no	—
395	C₂×D₄	—	2	2	no	—
396	C₂×D₄	—	2	2	no	—
397	C₂×D₄	—	2	2	no	—
398	C₂×D₄	—	4	4	no	—
399	C₂×D₄	—	4	1	no	—
400	C₂×D₄	—	4	1	no	—
401	C₂×D₄	—	4	1	no	—
402	C₂×D₄	—	4	2	no	—
403	C₂×D₄	—	4	2	no	—
404	C₂×D₄	—	8	2	no	—
405	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	2	4	no	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	1	no	—
410	C₂²×C₄	—	2	1	no	—
411	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
412	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
413	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
414	SD₁₆	—	4	4	no	—
415	SD₁₆	—	8	4	no	—
416	D₈	—	4	4	no	—
417	M₁₆	—	4	2	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	2	2	no	—
421	C₂×C₈	—	4	2	no	—
422	C₄⋊C₄	—	2	4	no	—
423	C₄⋊C₄	—	2	4	no	—
424	C₄⋊C₄	—	2	4	no	—
425	C₄⋊C₄	—	2	4	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	2	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	2	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	2	no	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	2	2	no	—
436	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
437	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
438	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
439	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
440	C₂²⋊C₄	—	2	4	no	—
441	C₂²⋊C₄	—	2	2	no	—
442	C₂²⋊C₄	—	2	1	no	—
443	C₂²⋊C₄	—	2	1	no	—
444	C₂²⋊C₄	—	2	1	no	—
445	C₄²	—	2	4	no	—
446	C₄²	—	2	1	no	—
447	C₄²	—	2	1	no	—
448	C₄²	—	2	1	no	—
449	C₄²	—	2	2	no	—
450	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
451	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
452	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
453	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
454	C₂³	—	2	1	no	—
455	C₂³	—	4	2	no	—
456	C₂³	—	8	2	no	—
457	C₂³	—	8	2	no	—
458	C₂³	—	8	2	no	—
459	C₂³	#120×C₅₃	1	1	yes	characteristic
460	Q₈	—	2	1	no	—
461	Q₈	—	2	4	no	—
462	Q₈	—	4	4	no	—
463	Q₈	—	4	2	no	—
464	Q₈	—	8	2	no	—
465	Q₈	—	8	4	no	—
466	Q₈	C₂×SD₁₆×C₅₃	1	2	yes	—
467	D₄	—	2	4	no	—
468	D₄	—	2	1	no	—
469	D₄	—	4	4	no	—
470	D₄	—	4	2	no	—
471	D₄	—	4	2	no	—
472	D₄	—	4	4	no	—
473	D₄	—	8	2	no	—
474	D₄	—	8	2	no	—
475	D₄	—	8	2	no	—
476	D₄	C₂×SD₁₆×C₅₃	1	2	yes	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	8	2	no	—
489	C₂×C₄	—	8	4	no	—
490	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
491	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
492	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
493	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
494	C₂×C₄	#137×C₅₃	1	2	yes	—
495	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
496	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
497	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
498	C₈	—	4	2	no	—
499	C₈	—	4	2	no	—
500	C₈	—	4	2	no	—
501	C₂²	—	2	1	no	—
502	C₂²	—	2	1	no	—
503	C₂²	—	2	2	no	—
504	C₂²	—	4	1	no	—
505	C₂²	—	4	1	no	—
506	C₂²	—	8	2	no	—
507	C₂²	—	8	2	no	—
508	C₂²	—	8	2	no	—
509	C₂²	—	16	4	no	—
510	C₂²	C₅₃×#448	1	2	yes	—
511	C₂²	C₂×#120×C₅₃	1	1	yes	characteristic
512	C₄	—	2	2	no	—
513	C₄	—	2	1	no	—
514	C₄	—	2	1	no	—
515	C₄	—	4	1	no	—
516	C₄	—	4	2	no	—
517	C₄	—	4	4	no	—
518	C₄	—	4	1	no	—
519	C₄	—	4	2	no	—
520	C₄	C₅₃×#448	1	2	yes	—
521	C₄	C₅₃×#454	1	2	yes	—
522	C₄	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
523	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
524	C₄	C₅₃×#576	1	2	yes	—
525	C₂	—	2	2	no	—
526	C₂	—	4	2	no	—
527	C₂	—	16	2	no	—
528	C₂	C₂×C₅₃×#448	1	1	yes	characteristic
529	C₂	C₅₃×#3008	1	1	yes	characteristic
530	C₂	C₂×C₅₃×#454	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3038	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3503	C₂	semidirect
2. C₅₃×#3208	C₂	semidirect
3. C₂×C₅₃×#486	C₂	semidirect
5. C₅₃×#1627	C₂	semidirect
6. C₅₃×#1626	C₂	semidirect
7. C₅₃×#1549	C₂	semidirect
8. C₅₃×#1493	C₂	semidirect
9. C₅₃×#1476	C₂	semidirect
12. C₅₃×#557	C₂²	semidirect
16. D₄×Q₈×C₅₃	C₂²	semidirect
41. C₅₃×#332	C₂²	semidirect
42. C₅₃×Q₈⋊C₈	C₂²	semidirect
43. C₅₃×Q₈⋊C₈	C₂²	semidirect
44. C₅₃×D₄⋊C₈	C₂²	semidirect
119. #41×C₅₃	SD₁₆	semidirect
235. #18642	C₅₃	direct (N × H)
260. #3503	C₂×C₅₃	semidirect
261. #3208	C₂×C₅₃	semidirect
262. C₂×#486	C₂×C₅₃	semidirect
264. #1627	C₂×C₅₃	semidirect
265. #1626	C₂×C₅₃	semidirect
266. #1549	C₂×C₅₃	semidirect
267. #1493	C₂×C₅₃	semidirect
268. #1476	C₂×C₅₃	semidirect
277. #557	C₂²×C₅₃	semidirect
281. D₄×Q₈	C₂²×C₅₃	semidirect
306. #332	C₂²×C₅₃	semidirect
307. Q₈⋊C₈	C₂²×C₅₃	semidirect
308. Q₈⋊C₈	C₂²×C₅₃	semidirect
309. D₄⋊C₈	C₂²×C₅₃	semidirect
310. C₅₃	#18642	direct (N × H)
385. #41	SD₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3503	2
C₅₃×#3208	2
C₂×C₅₃×#486	2
C₅₃×#3038	2
C₅₃×#1627	2
C₅₃×#1626	2
C₅₃×#1549	2
C₅₃×#1493	2
C₅₃×#1476	2
#18642	53

Subgroups — C53×#18642 #27226154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18642 #27226154