Subgroups — C₅₃×#11330 #27218842

Order 13568.

722 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#428	C₂	1	2	yes	maximal
2	C₂²×#106×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#343	C₂	1	2	yes	maximal
4	C₂×C₅₃×#341	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#428	—	2	4	no	—
12	C₅₃×#428	—	2	4	no	—
13	C₂×#107×C₅₃	—	2	2	no	—
14	C₂×#106×C₅₃	—	2	2	no	—
15	C₂×#106×C₅₃	—	2	4	no	—
16	C₂×#106×C₅₃	—	2	2	no	—
17	C₅₃×#343	—	2	4	no	—
18	C₅₃×#341	C₄	1	4	yes	complemented
19	C₅₃×#336	—	2	2	no	—
20	C₂³×C₄×C₅₃	—	2	2	no	—
21	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂×M₁₆×C₅₃	—	2	2	no	—
25	C₂×M₁₆×C₅₃	—	2	2	no	—
26	C₂×M₁₆×C₅₃	—	2	1	no	—
27	C₂×M₁₆×C₅₃	—	2	2	no	—
28	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
30	C₂×M₁₆×C₅₃	Q₈	1	2	yes	—
31	C₂×M₁₆×C₅₃	Q₈	1	4	yes	—
32	C₂²×C₈×C₅₃	—	2	2	no	—
33	C₂²×C₈×C₅₃	—	2	2	no	—
34	C₂²×C₈×C₅₃	—	2	1	no	—
35	C₂²×C₈×C₅₃	—	2	2	no	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₅₃	D₄	1	2	yes	—
38	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
39	#117×C₅₃	—	2	2	no	—
40	#117×C₅₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₅₃	D₄	1	4	yes	—
47	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
48	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
49	C₂×C₄²×C₅₃	—	2	1	no	—
50	#107×C₅₃	—	2	8	no	—
51	#106×C₅₃	—	2	8	no	—
52	#106×C₅₃	—	2	8	no	—
53	C₂⁴×C₅₃	#41	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	—	2	4	no	—
56	C₂²×C₄×C₅₃	—	4	2	no	—
57	C₂²×C₄×C₅₃	—	4	2	no	—
58	C₂²×C₄×C₅₃	—	8	4	no	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	#41	1	2	yes	—
68	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
69	M₁₆×C₅₃	—	2	4	no	—
70	M₁₆×C₅₃	—	2	4	no	—
71	M₁₆×C₅₃	—	2	4	no	—
72	M₁₆×C₅₃	—	2	4	no	—
73	M₁₆×C₅₃	—	4	2	no	—
74	M₁₆×C₅₃	C₄⋊C₄	1	8	yes	—
75	C₂×C₈×C₅₃	—	2	4	no	—
76	C₂×C₈×C₅₃	—	2	4	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	2	4	no	—
79	C₂×C₈×C₅₃	—	2	4	no	—
80	C₂×C₈×C₅₃	—	2	4	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	—	4	1	no	—
84	C₂×C₈×C₅₃	—	4	1	no	—
85	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
86	C₄⋊C₄×C₅₃	—	2	4	no	—
87	C₄⋊C₄×C₅₃	—	2	2	no	—
88	C₄⋊C₄×C₅₃	—	2	4	no	—
89	C₄⋊C₄×C₅₃	—	2	2	no	—
90	C₄⋊C₄×C₅₃	—	4	2	no	—
91	C₄⋊C₄×C₅₃	—	4	2	no	—
92	C₂²⋊C₄×C₅₃	—	4	2	no	—
93	C₄²×C₅₃	—	4	1	no	—
94	C₄²×C₅₃	—	4	1	no	—
95	C₂³×C₅₃	—	2	4	no	—
96	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
97	C₂³×C₅₃	#121	1	2	yes	—
98	C₂³×C₅₃	#121	1	1	yes	characteristic
99	C₂³×C₅₃	#125	1	2	yes	—
100	C₂³×C₅₃	#125	1	1	yes	characteristic
101	C₂×C₄×C₅₃	—	4	4	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	4	4	no	—
104	C₂×C₄×C₅₃	—	8	4	no	—
105	C₂×C₄×C₅₃	—	8	4	no	—
106	C₂×C₄×C₅₃	—	8	4	no	—
107	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#120	1	2	yes	—
113	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
119	C₂×C₄×C₅₃	#123	1	2	yes	—
120	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
122	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
124	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
125	C₈×C₅₃	—	2	4	no	—
126	C₈×C₅₃	—	2	8	no	—
127	C₈×C₅₃	—	4	2	no	—
128	#11330	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂²×C₅₃	—	2	4	no	—
130	C₂²×C₅₃	—	2	2	no	—
131	C₂²×C₅₃	—	2	2	no	—
132	C₂²×C₅₃	—	2	2	no	—
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	#389	1	2	yes	—
135	C₂²×C₅₃	#389	1	1	yes	characteristic
136	C₂²×C₅₃	#392	1	1	yes	characteristic
137	C₂²×C₅₃	#398	1	2	yes	—
138	C₂²×C₅₃	#398	1	1	yes	characteristic
139	C₂²×C₅₃	#471	1	4	yes	—
140	C₄×C₅₃	—	8	2	no	—
141	C₄×C₅₃	—	8	4	no	—
142	C₄×C₅₃	#384	1	1	yes	characteristic
143	C₄×C₅₃	#385	1	1	yes	characteristic
144	C₄×C₅₃	#385	1	2	yes	—
145	C₄×C₅₃	#395	1	1	yes	characteristic
146	C₄×C₅₃	#395	1	2	yes	—
147	C₄×C₅₃	#399	1	1	yes	characteristic
148	C₂×#428	C₂×C₅₃	1	2	yes	—
149	C₂²×#106	C₂×C₅₃	1	1	yes	characteristic
150	C₂×#343	C₂×C₅₃	1	2	yes	—
151	C₂×#341	C₂×C₅₃	1	1	yes	characteristic
152	#1729	C₂×C₅₃	1	1	yes	characteristic
153	C₂×C₅₃	—	2	2	no	—
154	C₂×C₅₃	—	2	2	no	—
155	C₂×C₅₃	#1444	1	1	yes	characteristic
156	C₂×C₅₃	#1951	1	2	yes	—
157	C₂×C₅₃	#2019	1	2	yes	—
158	C₂×C₅₃	#2090	1	2	yes	—
159	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
160	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
161	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
162	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
163	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
164	#428	—	2	4	no	—
165	#428	—	2	4	no	—
166	C₂×#107	—	2	2	no	—
167	C₂×#106	—	2	2	no	—
168	C₂×#106	—	2	4	no	—
169	C₂×#106	—	2	2	no	—
170	#343	—	2	4	no	—
171	#341	C₄×C₅₃	1	4	yes	complemented
172	#336	—	2	2	no	—
173	C₅₃	#11330	1	1	yes	characteristic, direct product factor, Sylow
174	C₂³×C₄	—	2	2	no	—
175	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
183	C₂×M₁₆	D₄×C₅₃	1	4	yes	—
184	C₂×M₁₆	Q₈×C₅₃	1	2	yes	—
185	C₂×M₁₆	Q₈×C₅₃	1	4	yes	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
191	C₂²×C₈	D₄×C₅₃	1	2	yes	—
192	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
193	#117	—	2	2	no	—
194	#117	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	D₄×C₅₃	1	4	yes	—
201	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
202	C₂×C₂²⋊C₄	—	2	1	no	—
203	C₂×C₄²	—	2	1	no	—
204	#107	—	2	8	no	—
205	#106	—	2	8	no	—
206	#106	—	2	8	no	—
207	C₂⁴	#41×C₅₃	1	1	yes	characteristic
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	—	8	4	no	—
213	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
218	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
220	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
221	C₂²×C₄	#41×C₅₃	1	2	yes	—
222	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	C₄⋊C₄×C₅₃	1	8	yes	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₄²	—	4	1	no	—
248	C₄²	—	4	1	no	—
249	C₂³	—	2	4	no	—
250	C₂³	C₄×Q₈×C₅₃	1	1	yes	characteristic
251	C₂³	#121×C₅₃	1	2	yes	—
252	C₂³	#121×C₅₃	1	1	yes	characteristic
253	C₂³	#125×C₅₃	1	1	yes	characteristic
254	C₂³	#125×C₅₃	1	2	yes	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	8	4	no	—
259	C₂×C₄	—	8	4	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
263	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
264	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
265	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
266	C₂×C₄	#120×C₅₃	1	2	yes	—
267	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
268	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
269	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
270	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
271	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
272	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
273	C₂×C₄	#123×C₅₃	1	2	yes	—
274	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
275	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
276	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
277	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
278	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
279	C₈	—	2	4	no	—
280	C₈	—	2	8	no	—
281	C₈	—	4	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	4	no	—
287	C₂²	C₅₃×#389	1	2	yes	—
288	C₂²	C₅₃×#389	1	1	yes	characteristic
289	C₂²	C₅₃×#392	1	1	yes	characteristic
290	C₂²	C₅₃×#398	1	2	yes	—
291	C₂²	C₅₃×#398	1	1	yes	characteristic
292	C₂²	C₅₃×#471	1	4	yes	—
293	C₄	—	8	2	no	—
294	C₄	—	8	4	no	—
295	C₄	C₅₃×#384	1	1	yes	characteristic
296	C₄	C₅₃×#385	1	1	yes	characteristic
297	C₄	C₅₃×#385	1	2	yes	—
298	C₄	C₅₃×#395	1	1	yes	characteristic
299	C₄	C₅₃×#395	1	2	yes	—
300	C₄	C₅₃×#399	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	2	2	no	—
303	C₂	C₅₃×#1444	1	1	yes	characteristic
304	C₂	C₅₃×#1951	1	2	yes	—
305	C₂	C₅₃×#2019	1	2	yes	—
306	C₂	C₅₃×#2090	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	#121
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
18. C₅₃×#341	C₄	semidirect
128. #11330	C₅₃	direct (N × H)
171. #341	C₄×C₅₃	semidirect
173. C₅₃	#11330	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#428	2
C₂²×#106×C₅₃	2
C₂×C₅₃×#343	2
C₂×C₅₃×#341	2
C₅₃×#1729	2
#11330	53

Subgroups — C53×#11330 #27218842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11330 #27218842