Subgroups — C₅₃×#19927 #27227439

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3384	C₂	1	2	yes	maximal
2	C₅₃×#3299	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1707	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1692	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1683	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1649	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1632	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1591	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
10	C₅₃×#554	C₂²	1	2	yes	—
11	D₄×Q₈×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#501	—	2	2	no	—
13	C₅₃×#500	C₂²	1	1	yes	characteristic
14	C₅₃×#498	C₂²	1	2	yes	—
15	C₅₃×#494	C₂²	1	1	yes	characteristic
16	C₅₃×#487	C₂²	1	2	yes	—
17	C₅₃×#484	—	2	2	no	—
18	C₅₃×#478	—	2	2	no	—
19	C₅₃×#477	C₂²	1	2	yes	—
20	C₅₃×#476	—	2	2	no	—
21	C₅₃×#476	C₂²	1	2	yes	—
22	C₅₃×#474	C₂²	1	2	yes	—
23	C₅₃×#467	—	2	2	no	—
24	C₅₃×#462	C₂²	1	2	yes	—
25	C₅₃×#451	—	2	2	no	—
26	C₅₃×#446	C₂²	1	2	yes	—
27	C₅₃×#442	—	2	2	no	—
28	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
29	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₈×D₄×C₅₃	C₂²	1	2	yes	—
31	C₂×#103×C₅₃	—	2	2	no	—
32	C₅₃×#335	C₂²	1	1	yes	characteristic
33	C₅₃×#335	C₂²	1	1	yes	characteristic
34	C₅₃×#333	C₂²	1	2	yes	—
35	C₅₃×#332	C₂²	1	2	yes	—
36	C₅₃×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₅₃	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₅₃	—	2	2	no	—
40	C₂×Q₁₆×C₅₃	—	2	2	no	—
41	C₂×Q₁₆×C₅₃	—	4	1	no	—
42	C₂×Q₁₆×C₅₃	D₄	1	1	yes	characteristic
43	C₂×D₈×C₅₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	—	2	2	no	—
45	C₄⋊Q₈×C₅₃	—	2	2	no	—
46	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
48	#125×C₅₃	—	2	2	no	—
49	C₂²⋊Q₈×C₅₃	—	2	2	no	—
50	C₂²⋊Q₈×C₅₃	—	2	2	no	—
51	C₂²⋊Q₈×C₅₃	—	4	2	no	—
52	C₂²⋊Q₈×C₅₃	—	4	2	no	—
53	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
54	C₄×D₄×C₅₃	—	2	2	no	—
55	C₄×D₄×C₅₃	—	2	2	no	—
56	C₄×D₄×C₅₃	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
58	#107×C₅₃	—	2	2	no	—
59	#107×C₅₃	—	2	2	no	—
60	#107×C₅₃	—	4	1	no	—
61	#107×C₅₃	D₄	1	1	yes	characteristic
62	#107×C₅₃	D₄	1	1	yes	characteristic
63	#106×C₅₃	—	2	2	no	—
64	#106×C₅₃	—	2	2	no	—
65	#106×C₅₃	—	2	2	no	—
66	C₄⋊C₈×C₅₃	—	2	2	no	—
67	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
68	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
69	#103×C₅₃	—	2	2	no	—
70	#103×C₅₃	—	2	2	no	—
71	#103×C₅₃	—	2	2	no	—
72	#103×C₅₃	—	2	2	no	—
73	#103×C₅₃	D₄	1	2	yes	—
74	#102×C₅₃	—	2	2	no	—
75	#102×C₅₃	—	2	2	no	—
76	#102×C₅₃	D₄	1	2	yes	—
77	C₂²⋊C₈×C₅₃	—	2	2	no	—
78	#97×C₅₃	—	2	2	no	—
79	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₅₃	—	2	2	no	—
83	C₂×Q₈×C₅₃	—	2	2	no	—
84	C₂×Q₈×C₅₃	—	4	2	no	—
85	C₂×Q₈×C₅₃	—	4	4	no	—
86	C₂×Q₈×C₅₃	—	8	2	no	—
87	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
88	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₅₃	—	2	2	no	—
90	C₂²×C₄×C₅₃	—	4	2	no	—
91	C₂²×C₄×C₅₃	—	4	2	no	—
92	Q₁₆×C₅₃	—	4	1	no	—
93	Q₁₆×C₅₃	—	8	2	no	—
94	Q₁₆×C₅₃	—	8	2	no	—
95	D₈×C₅₃	Q₁₆	1	4	yes	complemented
96	C₂×C₈×C₅₃	—	2	1	no	—
97	C₂×C₈×C₅₃	—	2	2	no	—
98	C₂×C₈×C₅₃	—	2	2	no	—
99	C₂×C₈×C₅₃	—	8	2	no	—
100	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₄⋊C₄×C₅₃	—	2	2	no	—
105	C₄⋊C₄×C₅₃	—	2	2	no	—
106	C₄⋊C₄×C₅₃	—	2	2	no	—
107	C₄⋊C₄×C₅₃	—	2	2	no	—
108	C₄⋊C₄×C₅₃	—	2	2	no	—
109	C₄⋊C₄×C₅₃	—	4	2	no	—
110	C₄⋊C₄×C₅₃	—	4	2	no	—
111	C₄⋊C₄×C₅₃	—	4	2	no	—
112	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
113	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
114	C₂²⋊C₄×C₅₃	—	2	2	no	—
115	C₂²⋊C₄×C₅₃	—	4	2	no	—
116	C₂²⋊C₄×C₅₃	—	4	2	no	—
117	C₄²×C₅₃	—	2	2	no	—
118	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
119	C₂³×C₅₃	—	2	2	no	—
120	Q₈×C₅₃	—	2	2	no	—
121	Q₈×C₅₃	—	2	2	no	—
122	Q₈×C₅₃	—	4	2	no	—
123	Q₈×C₅₃	—	4	2	no	—
124	Q₈×C₅₃	—	8	2	no	—
125	Q₈×C₅₃	—	8	4	no	—
126	D₄×C₅₃	—	2	2	no	—
127	D₄×C₅₃	C₂×Q₁₆	1	4	yes	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	4	2	no	—
132	C₂×C₄×C₅₃	—	4	2	no	—
133	C₂×C₄×C₅₃	—	4	2	no	—
134	C₂×C₄×C₅₃	—	4	2	no	—
135	C₂×C₄×C₅₃	—	8	4	no	—
136	C₂×C₄×C₅₃	—	16	2	no	—
137	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
140	C₈×C₅₃	—	2	2	no	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	—	2	1	no	—
143	C₈×C₅₃	—	2	1	no	—
144	C₈×C₅₃	—	4	2	no	—
145	C₈×C₅₃	—	4	2	no	—
146	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
147	#19927	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂²×C₅₃	—	2	4	no	—
149	C₂²×C₅₃	—	4	2	no	—
150	C₂²×C₅₃	—	4	2	no	—
151	C₂²×C₅₃	D₄²	1	1	yes	characteristic
152	C₄×C₅₃	—	2	1	no	—
153	C₄×C₅₃	—	4	2	no	—
154	C₄×C₅₃	—	4	2	no	—
155	C₄×C₅₃	—	4	2	no	—
156	C₄×C₅₃	—	8	2	no	—
157	C₄×C₅₃	—	8	2	no	—
158	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
159	C₄×C₅₃	#575	1	1	yes	characteristic
160	C₄×C₅₃	#576	1	1	yes	characteristic
161	C₄×C₅₃	#578	1	1	yes	characteristic
162	#3384	C₂×C₅₃	1	2	yes	—
163	#3299	C₂×C₅₃	1	2	yes	complemented
164	#1707	C₂×C₅₃	1	1	yes	characteristic, complemented
165	#1692	C₂×C₅₃	1	2	yes	complemented
166	#1683	C₂×C₅₃	1	2	yes	complemented
167	#1649	C₂×C₅₃	1	2	yes	complemented
168	#1632	C₂×C₅₃	1	2	yes	complemented
169	#1591	C₂×C₅₃	1	1	yes	characteristic, complemented
170	C₈×D₈	C₂×C₅₃	1	1	yes	characteristic
171	C₂×C₅₃	—	4	4	no	—
172	C₂×C₅₃	#3273	1	1	yes	characteristic
173	C₂×C₅₃	D₄×Q₁₆	1	1	yes	characteristic
174	C₂×C₅₃	#3291	1	1	yes	characteristic
175	#554	C₂²×C₅₃	1	2	yes	—
176	D₄×Q₈	C₂²×C₅₃	1	2	yes	—
177	#501	—	2	2	no	—
178	#500	C₂²×C₅₃	1	1	yes	characteristic
179	#498	C₂²×C₅₃	1	2	yes	—
180	#494	C₂²×C₅₃	1	1	yes	characteristic
181	#487	C₂²×C₅₃	1	2	yes	—
182	#484	—	2	2	no	—
183	#478	—	2	2	no	—
184	#477	C₂²×C₅₃	1	2	yes	—
185	#476	—	2	2	no	—
186	#476	C₂²×C₅₃	1	2	yes	—
187	#474	C₂²×C₅₃	1	2	yes	—
188	#467	—	2	2	no	—
189	#462	C₂²×C₅₃	1	2	yes	—
190	#451	—	2	2	no	—
191	#446	C₂²×C₅₃	1	2	yes	—
192	#442	—	2	2	no	—
193	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
194	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
195	C₈×D₄	C₂²×C₅₃	1	2	yes	—
196	C₂×#103	—	2	2	no	—
197	#335	C₂²×C₅₃	1	1	yes	characteristic
198	#335	C₂²×C₅₃	1	1	yes	characteristic
199	#333	C₂²×C₅₃	1	2	yes	—
200	#332	C₂²×C₅₃	1	2	yes	—
201	Q₈⋊C₈	C₂²×C₅₃	1	2	yes	—
202	D₄⋊C₈	C₂²×C₅₃	1	2	yes	—
203	C₈²	C₂²×C₅₃	1	1	yes	characteristic
204	C₅₃	#19927	1	1	yes	characteristic, direct product factor, Sylow
205	C₂²×Q₈	—	2	2	no	—
206	C₂×Q₁₆	—	2	2	no	—
207	C₂×Q₁₆	—	4	1	no	—
208	C₂×Q₁₆	D₄×C₅₃	1	1	yes	characteristic
209	C₂×D₈	D₄×C₅₃	1	1	yes	characteristic
210	C₂²×C₈	—	2	2	no	—
211	C₄⋊Q₈	—	2	2	no	—
212	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
213	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
214	#125	—	2	2	no	—
215	C₂²⋊Q₈	—	2	2	no	—
216	C₂²⋊Q₈	—	2	2	no	—
217	C₂²⋊Q₈	—	4	2	no	—
218	C₂²⋊Q₈	—	4	2	no	—
219	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
220	C₄×D₄	—	2	2	no	—
221	C₄×D₄	—	2	2	no	—
222	C₄×D₄	C₂³×C₅₃	1	2	yes	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	#107	—	2	2	no	—
225	#107	—	2	2	no	—
226	#107	—	4	1	no	—
227	#107	D₄×C₅₃	1	1	yes	characteristic
228	#107	D₄×C₅₃	1	1	yes	characteristic
229	#106	—	2	2	no	—
230	#106	—	2	2	no	—
231	#106	—	2	2	no	—
232	C₄⋊C₈	—	2	2	no	—
233	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
234	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
235	#103	—	2	2	no	—
236	#103	—	2	2	no	—
237	#103	—	2	2	no	—
238	#103	—	2	2	no	—
239	#103	D₄×C₅₃	1	2	yes	—
240	#102	—	2	2	no	—
241	#102	—	2	2	no	—
242	#102	D₄×C₅₃	1	2	yes	—
243	C₂²⋊C₈	—	2	2	no	—
244	#97	—	2	2	no	—
245	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
246	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
247	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	4	2	no	—
251	C₂×Q₈	—	4	4	no	—
252	C₂×Q₈	—	8	2	no	—
253	C₂×Q₈	C₂×D₄×C₅₃	1	2	yes	—
254	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	—	4	2	no	—
258	Q₁₆	—	4	1	no	—
259	Q₁₆	—	8	2	no	—
260	Q₁₆	—	8	2	no	—
261	D₈	Q₁₆×C₅₃	1	4	yes	complemented
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	8	2	no	—
266	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
267	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
268	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
269	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
279	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
285	C₂³	—	2	2	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	8	2	no	—
291	Q₈	—	8	4	no	—
292	D₄	—	2	2	no	—
293	D₄	C₂×Q₁₆×C₅₃	1	4	yes	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	—	16	2	no	—
303	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
305	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	4	2	no	—
311	C₈	—	4	2	no	—
312	C₈	C₂×Q₁₆×C₅₃	1	2	yes	—
313	C₂²	—	2	4	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	D₄²×C₅₃	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	2	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
324	C₄	C₅₃×#575	1	1	yes	characteristic
325	C₄	C₅₃×#576	1	1	yes	characteristic
326	C₄	C₅₃×#578	1	1	yes	characteristic
327	C₂	—	4	4	no	—
328	C₂	C₅₃×#3273	1	1	yes	characteristic
329	C₂	D₄×Q₁₆×C₅₃	1	1	yes	characteristic
330	C₂	C₅₃×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₂×D₈×C₅₃	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#3299	C₂	semidirect
3. C₅₃×#1707	C₂	semidirect
4. C₅₃×#1692	C₂	semidirect
5. C₅₃×#1683	C₂	semidirect
6. C₅₃×#1649	C₂	semidirect
7. C₅₃×#1632	C₂	semidirect
8. C₅₃×#1591	C₂	semidirect
95. D₈×C₅₃	Q₁₆	semidirect
147. #19927	C₅₃	direct (N × H)
163. #3299	C₂×C₅₃	semidirect
164. #1707	C₂×C₅₃	semidirect
165. #1692	C₂×C₅₃	semidirect
166. #1683	C₂×C₅₃	semidirect
167. #1649	C₂×C₅₃	semidirect
168. #1632	C₂×C₅₃	semidirect
169. #1591	C₂×C₅₃	semidirect
204. C₅₃	#19927	direct (N × H)
261. D₈	Q₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3384	2
C₅₃×#3299	2
C₅₃×#1707	2
C₅₃×#1692	2
C₅₃×#1683	2
C₅₃×#1649	2
C₅₃×#1632	2
C₅₃×#1591	2
C₈×D₈×C₅₃	2
#19927	53

Subgroups — C53×#19927 #27227439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19927 #27227439