Subgroups — C₅₃×#21156 #27228668

Order 13568.

982 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic
2	C₂²×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2317	C₂	1	2	yes	maximal
5	C₅₃×#1944	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1942	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1736	C₂	1	2	yes	maximal
8	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×#117×C₅₃	C₂²	1	2	yes	—
11	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#427	C₄	1	8	yes	complemented
14	C₂×#107×C₅₃	—	2	4	no	—
15	C₂×#107×C₅₃	C₂²	1	2	yes	—
16	C₂×#107×C₅₃	C₂²	1	2	yes	—
17	C₂×#107×C₅₃	C₂²	1	2	yes	—
18	C₂×#106×C₅₃	C₂²	1	2	yes	—
19	C₅₃×#405	—	2	4	no	—
20	C₂×#97×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
22	C₅₃×#389	C₂²	1	8	yes	—
23	C₅₃×#385	—	2	4	no	—
24	C₅₃×#384	—	2	2	no	—
25	C₅₃×#382	—	2	2	no	—
26	C₅₃×#340	C₂²	1	8	yes	—
27	C₂³×C₄×C₅₃	—	2	2	no	—
28	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₅₃	—	2	2	no	—
30	C₂×M₁₆×C₅₃	—	2	2	no	—
31	C₂×M₁₆×C₅₃	Q₈	1	4	yes	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₅₃	D₄	1	4	yes	—
34	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
35	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	#117×C₅₃	—	2	2	no	—
39	#117×C₅₃	—	2	4	no	—
40	#117×C₅₃	—	2	2	no	—
41	#117×C₅₃	C₂×C₄	1	8	yes	—
42	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
46	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
49	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
50	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
52	C₂×C₄²×C₅₃	—	2	2	no	—
53	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
54	#107×C₅₃	—	2	4	no	—
55	#107×C₅₃	—	2	4	no	—
56	#107×C₅₃	C₂×C₄	1	8	yes	complemented
57	#106×C₅₃	C₂×C₄	1	8	yes	complemented
58	#97×C₅₃	—	2	4	no	—
59	C₄×C₈×C₅₃	—	2	4	no	—
60	#95×C₅₃	—	2	8	no	—
61	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	—	2	4	no	—
63	C₂²×C₄×C₅₃	—	2	4	no	—
64	C₂²×C₄×C₅₃	—	4	4	no	—
65	C₂²×C₄×C₅₃	—	8	4	no	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₅₃	#41	1	2	yes	—
74	C₂²×C₄×C₅₃	#41	1	2	yes	—
75	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
76	M₁₆×C₅₃	—	2	8	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	C₄⋊C₄	1	16	yes	complemented
80	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
85	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
86	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
87	C₄⋊C₄×C₅₃	—	2	2	no	—
88	C₄⋊C₄×C₅₃	—	2	2	no	—
89	C₄⋊C₄×C₅₃	—	2	4	no	—
90	C₄⋊C₄×C₅₃	—	2	4	no	—
91	C₄⋊C₄×C₅₃	—	2	4	no	—
92	C₄⋊C₄×C₅₃	—	4	4	no	—
93	C₄⋊C₄×C₅₃	—	8	8	no	—
94	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
95	C₂²⋊C₄×C₅₃	—	2	8	no	—
96	C₂²⋊C₄×C₅₃	—	2	4	no	—
97	C₄²×C₅₃	—	2	8	no	—
98	C₄²×C₅₃	—	2	2	no	—
99	C₄²×C₅₃	—	2	2	no	—
100	C₂³×C₅₃	—	2	2	no	—
101	C₂³×C₅₃	—	2	2	no	—
102	C₂³×C₅₃	C₄×D₄	1	2	yes	—
103	C₂³×C₅₃	#121	1	2	yes	—
104	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
105	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	—	2	8	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	4	no	—
110	C₂×C₄×C₅₃	—	2	4	no	—
111	C₂×C₄×C₅₃	—	4	4	no	—
112	C₂×C₄×C₅₃	—	4	4	no	—
113	C₂×C₄×C₅₃	—	4	4	no	—
114	C₂×C₄×C₅₃	—	4	8	no	—
115	C₂×C₄×C₅₃	—	8	8	no	—
116	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
119	C₂×C₄×C₅₃	#121	1	2	yes	—
120	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
121	C₂×C₄×C₅₃	#125	1	4	yes	—
122	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
123	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
129	C₈×C₅₃	—	2	4	no	—
130	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
131	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
132	#21156	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₅₃	—	2	4	no	—
134	C₂²×C₅₃	—	2	4	no	—
135	C₂²×C₅₃	—	2	2	no	—
136	C₂²×C₅₃	—	2	2	no	—
137	C₂²×C₅₃	#389	1	4	yes	—
138	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
140	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
141	C₂²×C₅₃	#576	1	2	yes	—
142	C₂²×C₅₃	#578	1	2	yes	—
143	C₄×C₅₃	—	4	8	no	—
144	C₄×C₅₃	—	4	2	no	—
145	C₄×C₅₃	—	4	2	no	—
146	C₄×C₅₃	—	8	4	no	—
147	C₄×C₅₃	#389	1	4	yes	—
148	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
150	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
151	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
152	C₂×#427	C₂×C₅₃	1	1	yes	characteristic
153	C₂²×#107	C₂×C₅₃	1	1	yes	characteristic
154	C₂×#405	C₂×C₅₃	1	1	yes	characteristic
155	#2317	C₂×C₅₃	1	2	yes	—
156	#1944	C₂×C₅₃	1	4	yes	complemented
157	#1942	C₂×C₅₃	1	4	yes	complemented
158	#1736	C₂×C₅₃	1	2	yes	—
159	C₂×C₅₃	—	2	4	no	—
160	C₂×C₅₃	C₂×#389	1	1	yes	characteristic
161	C₂×C₅₃	#2951	1	2	yes	—
162	C₂×C₅₃	#3058	1	1	yes	characteristic
163	C₂×C₅₃	#3059	1	1	yes	characteristic
164	C₂×C₅₃	#3164	1	2	yes	—
165	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
166	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
167	C₂×#117	C₂²×C₅₃	1	2	yes	—
168	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
169	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
170	#427	C₄×C₅₃	1	8	yes	complemented
171	C₂×#107	—	2	4	no	—
172	C₂×#107	C₂²×C₅₃	1	2	yes	—
173	C₂×#107	C₂²×C₅₃	1	2	yes	—
174	C₂×#107	C₂²×C₅₃	1	2	yes	—
175	C₂×#106	C₂²×C₅₃	1	2	yes	—
176	#405	—	2	4	no	—
177	C₂×#97	C₂²×C₅₃	1	2	yes	—
178	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
179	#389	C₂²×C₅₃	1	8	yes	—
180	#385	—	2	4	no	—
181	#384	—	2	2	no	—
182	#382	—	2	2	no	—
183	#340	C₂²×C₅₃	1	8	yes	—
184	C₅₃	#21156	1	1	yes	characteristic, direct product factor, Sylow
185	C₂³×C₄	—	2	2	no	—
186	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	Q₈×C₅₃	1	4	yes	—
190	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
191	C₂²×C₈	D₄×C₅₃	1	4	yes	—
192	C₂²×C₈	Q₈×C₅₃	1	4	yes	—
193	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
194	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
196	#117	—	2	2	no	—
197	#117	—	2	4	no	—
198	#117	—	2	2	no	—
199	#117	C₂×C₄×C₅₃	1	8	yes	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	8	no	—
204	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
205	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
206	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
207	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
212	#107	—	2	4	no	—
213	#107	—	2	4	no	—
214	#107	C₂×C₄×C₅₃	1	8	yes	complemented
215	#106	C₂×C₄×C₅₃	1	8	yes	complemented
216	#97	—	2	4	no	—
217	C₄×C₈	—	2	4	no	—
218	#95	—	2	8	no	—
219	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	8	4	no	—
224	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
227	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
228	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
230	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
231	C₂²×C₄	#41×C₅₃	1	2	yes	—
232	C₂²×C₄	#41×C₅₃	1	2	yes	—
233	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
234	M₁₆	—	2	8	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	C₄⋊C₄×C₅₃	1	16	yes	complemented
238	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
239	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
240	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
241	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
242	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
243	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
244	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	—	8	8	no	—
252	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
253	C₂²⋊C₄	—	2	8	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₄²	—	2	8	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₄×D₄×C₅₃	1	2	yes	—
261	C₂³	#121×C₅₃	1	2	yes	—
262	C₂³	C₄⋊Q₈×C₅₃	1	2	yes	—
263	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	8	no	—
273	C₂×C₄	—	8	8	no	—
274	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
275	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
276	C₂×C₄	C₄×Q₈×C₅₃	1	4	yes	—
277	C₂×C₄	#121×C₅₃	1	2	yes	—
278	C₂×C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
279	C₂×C₄	#125×C₅₃	1	4	yes	—
280	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
281	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
283	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
284	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
285	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
286	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
287	C₈	—	2	4	no	—
288	C₈	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
289	C₈	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	C₅₃×#389	1	4	yes	—
295	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
296	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
297	C₂²	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
298	C₂²	C₅₃×#576	1	2	yes	—
299	C₂²	C₅₃×#578	1	2	yes	—
300	C₄	—	4	8	no	—
301	C₄	—	4	2	no	—
302	C₄	—	4	2	no	—
303	C₄	—	8	4	no	—
304	C₄	C₅₃×#389	1	4	yes	—
305	C₄	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
306	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
307	C₄	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
308	C₄	C₂×#125×C₅₃	1	1	yes	characteristic
309	C₂	—	2	4	no	—
310	C₂	C₂×C₅₃×#389	1	1	yes	characteristic
311	C₂	C₅₃×#2951	1	2	yes	—
312	C₂	C₅₃×#3058	1	1	yes	characteristic
313	C₂	C₅₃×#3059	1	1	yes	characteristic
314	C₂	C₅₃×#3164	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×#117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅₃×#1944	C₂	semidirect
6. C₅₃×#1942	C₂	semidirect
13. C₅₃×#427	C₄	semidirect
56. #107×C₅₃	C₂×C₄	semidirect
57. #106×C₅₃	C₂×C₄	semidirect
79. C₂×C₈×C₅₃	C₄⋊C₄	semidirect
132. #21156	C₅₃	direct (N × H)
156. #1944	C₂×C₅₃	semidirect
157. #1942	C₂×C₅₃	semidirect
170. #427	C₄×C₅₃	semidirect
184. C₅₃	#21156	direct (N × H)
214. #107	C₂×C₄×C₅₃	semidirect
215. #106	C₂×C₄×C₅₃	semidirect
237. C₂×C₈	C₄⋊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×#427	2
C₂²×#107×C₅₃	2
C₂×C₅₃×#405	2
C₅₃×#2317	2
C₅₃×#1944	2
C₅₃×#1942	2
C₅₃×#1736	2
#21156	53

Subgroups — C53×#21156 #27228668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21156 #27228668