Subgroups — C₅₃×#11393 #27218905

Order 13568.

850 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3458	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1740	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1735	C₂	1	2	yes	maximal
4	C₅₃×#1733	C₂	1	2	yes	maximal
5	C₅₃×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#544	—	2	4	no	—
9	C₅₃×#544	C₄	1	4	yes	complemented
10	C₅₃×#543	—	2	2	no	—
11	C₅₃×#533	—	2	4	no	—
12	C₅₃×#533	—	2	2	no	—
13	C₂×#117×C₅₃	C₂²	1	2	yes	—
14	C₂×#117×C₅₃	C₂²	1	2	yes	—
15	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#342	—	2	2	no	—
17	C₅₃×#340	—	2	4	no	—
18	C₅₃×#339	—	2	4	no	—
19	C₅₃×#336	—	2	2	no	—
20	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₅₃	—	2	2	no	—
22	C₂×M₁₆×C₅₃	—	2	1	no	—
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂×M₁₆×C₅₃	Q₈	1	2	yes	complemented
25	C₂²×C₈×C₅₃	—	2	1	no	—
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	C₂²×C₈×C₅₃	—	2	1	no	—
28	C₂²×C₈×C₅₃	—	2	1	no	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	Q₈	1	2	yes	complemented
31	C₄⋊Q₈×C₅₃	—	2	4	no	—
32	C₄⋊Q₈×C₅₃	—	2	8	no	—
33	#125×C₅₃	—	2	4	no	—
34	#125×C₅₃	—	2	8	no	—
35	C₂²⋊Q₈×C₅₃	—	2	16	no	—
36	C₂²⋊Q₈×C₅₃	—	2	8	no	—
37	C₄×Q₈×C₅₃	—	2	8	no	—
38	#117×C₅₃	—	2	2	no	—
39	#117×C₅₃	—	2	2	no	—
40	#117×C₅₃	—	2	2	no	—
41	#117×C₅₃	—	2	2	no	—
42	#117×C₅₃	—	2	2	no	—
43	#117×C₅₃	—	2	2	no	—
44	#117×C₅₃	—	2	2	no	—
45	#117×C₅₃	C₂×C₄	1	4	yes	—
46	#117×C₅₃	D₄	1	4	yes	—
47	#117×C₅₃	Q₈	1	4	yes	complemented
48	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₅₃	D₄	1	4	yes	—
51	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
54	C₂×C₄²×C₅₃	—	2	2	no	—
55	C₂×C₄²×C₅₃	—	2	2	no	—
56	C₂×C₄²×C₅₃	D₄	1	2	yes	—
57	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×Q₈×C₅₃	—	2	8	no	—
59	C₂²×C₄×C₅₃	—	2	4	no	—
60	C₂²×C₄×C₅₃	—	4	2	no	—
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	#41	1	2	yes	—
72	M₁₆×C₅₃	—	2	4	no	—
73	M₁₆×C₅₃	—	4	2	no	—
74	C₂×C₈×C₅₃	—	2	4	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	4	1	no	—
78	C₂×C₈×C₅₃	—	4	1	no	—
79	C₂×C₈×C₅₃	—	4	2	no	—
80	C₂×C₈×C₅₃	—	4	1	no	—
81	C₂×C₈×C₅₃	—	4	1	no	—
82	C₄⋊C₄×C₅₃	—	2	16	no	—
83	C₄⋊C₄×C₅₃	—	2	8	no	—
84	C₄⋊C₄×C₅₃	—	2	4	no	—
85	C₄⋊C₄×C₅₃	—	2	4	no	—
86	C₄⋊C₄×C₅₃	—	2	4	no	—
87	C₄⋊C₄×C₅₃	—	4	2	no	—
88	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	8	yes	—
89	C₂²⋊C₄×C₅₃	—	2	4	no	—
90	C₂²⋊C₄×C₅₃	—	2	4	no	—
91	C₂²⋊C₄×C₅₃	—	2	4	no	—
92	C₂²⋊C₄×C₅₃	—	2	4	no	—
93	C₂²⋊C₄×C₅₃	—	4	2	no	—
94	C₄²×C₅₃	—	2	4	no	—
95	C₄²×C₅₃	—	2	2	no	—
96	C₄²×C₅₃	—	2	4	no	—
97	C₄²×C₅₃	—	2	2	no	—
98	C₄²×C₅₃	—	2	4	no	—
99	C₄²×C₅₃	—	4	2	no	—
100	C₂³×C₅₃	—	4	2	no	—
101	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂³×C₅₃	#120	1	2	yes	—
103	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
104	C₂³×C₅₃	#124	1	1	yes	characteristic
105	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
106	Q₈×C₅₃	—	8	8	no	—
107	C₂×C₄×C₅₃	—	2	8	no	—
108	C₂×C₄×C₅₃	—	2	4	no	—
109	C₂×C₄×C₅₃	—	4	4	no	—
110	C₂×C₄×C₅₃	—	4	4	no	—
111	C₂×C₄×C₅₃	—	8	4	no	—
112	C₂×C₄×C₅₃	—	8	4	no	—
113	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
114	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
117	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
122	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₅₃	#124	1	2	yes	—
126	C₂×C₄×C₅₃	#124	1	2	yes	—
127	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
129	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
130	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
131	C₈×C₅₃	—	4	2	no	—
132	C₈×C₅₃	—	4	2	no	—
133	#11393	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	—	2	4	no	—
136	C₂²×C₅₃	—	4	2	no	—
137	C₂²×C₅₃	—	4	1	no	—
138	C₂²×C₅₃	—	4	1	no	—
139	C₂²×C₅₃	#379	1	1	yes	characteristic
140	C₂²×C₅₃	#391	1	1	yes	characteristic
141	C₂²×C₅₃	#391	1	1	yes	characteristic
142	C₂²×C₅₃	#393	1	2	yes	—
143	C₂²×C₅₃	#395	1	2	yes	—
144	C₄×C₅₃	—	4	4	no	—
145	C₄×C₅₃	—	4	8	no	—
146	C₄×C₅₃	—	8	4	no	—
147	C₄×C₅₃	#391	1	2	yes	—
148	C₄×C₅₃	#391	1	2	yes	—
149	C₄×C₅₃	#393	1	1	yes	characteristic
150	C₄×C₅₃	#393	1	1	yes	characteristic
151	C₄×C₅₃	#395	1	1	yes	characteristic
152	C₄×C₅₃	#395	1	1	yes	characteristic
153	#3458	C₂×C₅₃	1	1	yes	characteristic
154	#1740	C₂×C₅₃	1	1	yes	characteristic
155	#1735	C₂×C₅₃	1	2	yes	—
156	#1733	C₂×C₅₃	1	2	yes	—
157	#1729	C₂×C₅₃	1	1	yes	characteristic
158	C₂×C₅₃	—	2	2	no	—
159	C₂×C₅₃	—	2	1	no	—
160	C₂×C₅₃	—	2	1	no	—
161	C₂×C₅₃	—	2	2	no	—
162	C₂×C₅₃	#1446	1	1	yes	characteristic
163	C₂×C₅₃	#1794	1	2	yes	—
164	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
165	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
166	#544	—	2	4	no	—
167	#544	C₄×C₅₃	1	4	yes	complemented
168	#543	—	2	2	no	—
169	#533	—	2	4	no	—
170	#533	—	2	2	no	—
171	C₂×#117	C₂²×C₅₃	1	2	yes	—
172	C₂×#117	C₂²×C₅₃	1	2	yes	—
173	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
174	#342	—	2	2	no	—
175	#340	—	2	4	no	—
176	#339	—	2	4	no	—
177	#336	—	2	2	no	—
178	C₅₃	#11393	1	1	yes	characteristic, direct product factor, Sylow
179	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
180	C₂×M₁₆	—	2	1	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	Q₈×C₅₃	1	2	yes	complemented
184	C₂²×C₈	—	2	1	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	Q₈×C₅₃	1	2	yes	complemented
190	C₄⋊Q₈	—	2	4	no	—
191	C₄⋊Q₈	—	2	8	no	—
192	#125	—	2	4	no	—
193	#125	—	2	8	no	—
194	C₂²⋊Q₈	—	2	16	no	—
195	C₂²⋊Q₈	—	2	8	no	—
196	C₄×Q₈	—	2	8	no	—
197	#117	—	2	2	no	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	#117	C₂×C₄×C₅₃	1	4	yes	—
205	#117	D₄×C₅₃	1	4	yes	—
206	#117	Q₈×C₅₃	1	4	yes	complemented
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
209	C₂×C₄⋊C₄	D₄×C₅₃	1	4	yes	—
210	C₂×C₂²⋊C₄	—	2	2	no	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	D₄×C₅₃	1	2	yes	—
216	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
217	C₂×Q₈	—	2	8	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
221	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
225	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
228	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
229	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
230	C₂²×C₄	#41×C₅₃	1	2	yes	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	4	2	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	—	4	1	no	—
241	C₄⋊C₄	—	2	16	no	—
242	C₄⋊C₄	—	2	8	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	8	yes	—
248	C₂²⋊C₄	—	2	4	no	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	4	2	no	—
259	C₂³	—	4	2	no	—
260	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
261	C₂³	#120×C₅₃	1	2	yes	—
262	C₂³	C₂²⋊Q₈×C₅₃	1	2	yes	—
263	C₂³	#124×C₅₃	1	1	yes	characteristic
264	C₂³	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
265	Q₈	—	8	8	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	8	4	no	—
271	C₂×C₄	—	8	4	no	—
272	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
273	C₂×C₄	C₄×Q₈×C₅₃	1	1	yes	characteristic
274	C₂×C₄	C₄×Q₈×C₅₃	1	1	yes	characteristic
275	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
276	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
277	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
278	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
279	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
280	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
281	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
282	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
283	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
284	C₂×C₄	#124×C₅₃	1	2	yes	—
285	C₂×C₄	#124×C₅₃	1	2	yes	—
286	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
287	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
288	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
289	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
290	C₈	—	4	2	no	—
291	C₈	—	4	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₅₃×#379	1	1	yes	characteristic
298	C₂²	C₅₃×#391	1	1	yes	characteristic
299	C₂²	C₅₃×#391	1	1	yes	characteristic
300	C₂²	C₅₃×#393	1	2	yes	—
301	C₂²	C₅₃×#395	1	2	yes	—
302	C₄	—	4	4	no	—
303	C₄	—	4	8	no	—
304	C₄	—	8	4	no	—
305	C₄	C₅₃×#391	1	2	yes	—
306	C₄	C₅₃×#391	1	2	yes	—
307	C₄	C₅₃×#393	1	1	yes	characteristic
308	C₄	C₅₃×#393	1	1	yes	characteristic
309	C₄	C₅₃×#395	1	1	yes	characteristic
310	C₄	C₅₃×#395	1	1	yes	characteristic
311	C₂	—	2	2	no	—
312	C₂	—	2	1	no	—
313	C₂	—	2	1	no	—
314	C₂	—	2	2	no	—
315	C₂	C₅₃×#1446	1	1	yes	characteristic
316	C₂	C₅₃×#1794	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	#120
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂²×C₅₃	#379
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₅₃×#544	C₄	semidirect
24. C₂×M₁₆×C₅₃	Q₈	semidirect
30. C₂²×C₈×C₅₃	Q₈	semidirect
47. #117×C₅₃	Q₈	semidirect
133. #11393	C₅₃	direct (N × H)
167. #544	C₄×C₅₃	semidirect
178. C₅₃	#11393	direct (N × H)
183. C₂×M₁₆	Q₈×C₅₃	semidirect
189. C₂²×C₈	Q₈×C₅₃	semidirect
206. #117	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3458	2
C₅₃×#1740	2
C₅₃×#1735	2
C₅₃×#1733	2
C₅₃×#1729	2
#11393	53

Subgroups — C53×#11393 #27218905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11393 #27218905