Subgroups — C₅₃×#33936 #27241448

Order 13568.

474 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3393	C₂	1	1	yes	maximal, characteristic
2	Q₈×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2243	C₂	1	2	yes	maximal
4	C₅₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2226	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#2225	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2211	C₂	1	2	yes	maximal
8	C₅₃×#2209	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#2176	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
12	Q₈²×C₅₃	—	2	1	no	—
13	C₅₃×#556	—	2	1	no	—
14	C₂×Q₃₂×C₅₃	C₂²	1	2	yes	—
15	C₂×SD₃₂×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#500	C₂²	1	2	yes	—
17	C₅₃×#500	C₂²	1	1	yes	characteristic
18	C₅₃×#495	C₂²	1	2	yes	—
19	C₅₃×#494	C₂²	1	1	yes	characteristic
20	C₅₃×#477	—	2	1	no	—
21	C₅₃×#477	—	2	2	no	—
22	C₅₃×#464	—	2	2	no	—
23	C₅₃×#461	—	2	1	no	—
24	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
26	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
27	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
28	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#367	C₂²	1	1	yes	characteristic
30	C₅₃×#367	C₂²	1	2	yes	—
31	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
33	C₅₃×#358	C₂²	1	1	yes	characteristic
34	C₅₃×#358	C₂²	1	2	yes	—
35	C₅₃×#358	C₂²	1	2	yes	—
36	C₅₃×#358	C₂²	1	2	yes	—
37	C₅₃×#357	C₂²	1	1	yes	characteristic
38	C₅₃×#357	C₂²	1	2	yes	—
39	C₅₃×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₅₃	—	2	1	no	—
42	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
43	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×SD₁₆×C₅₃	—	2	2	no	—
45	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₄⋊Q₈×C₅₃	—	2	1	no	—
47	C₄⋊Q₈×C₅₃	—	2	1	no	—
48	C₄⋊Q₈×C₅₃	—	2	2	no	—
49	C₄⋊Q₈×C₅₃	—	4	1	no	—
50	C₄⋊Q₈×C₅₃	—	4	2	no	—
51	#126×C₅₃	—	4	2	no	—
52	#124×C₅₃	—	2	2	no	—
53	C₂²⋊Q₈×C₅₃	—	4	1	no	—
54	C₄×Q₈×C₅₃	—	2	2	no	—
55	C₄×Q₈×C₅₃	—	2	1	no	—
56	C₄×Q₈×C₅₃	—	2	2	no	—
57	C₄×Q₈×C₅₃	—	4	1	no	—
58	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
59	C₄×D₄×C₅₃	—	2	1	no	—
60	#117×C₅₃	—	4	1	no	—
61	Q₃₂×C₅₃	—	2	4	no	—
62	SD₃₂×C₅₃	—	2	2	no	—
63	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
65	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
66	#107×C₅₃	—	2	1	no	—
67	#107×C₅₃	—	2	2	no	—
68	#107×C₅₃	C₂³	1	1	yes	characteristic
69	#107×C₅₃	C₂³	1	2	yes	—
70	#107×C₅₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
72	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
73	#103×C₅₃	—	2	2	no	—
74	#103×C₅₃	—	2	1	no	—
75	#103×C₅₃	—	2	2	no	—
76	#102×C₅₃	—	2	1	no	—
77	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₅₃	C₂³	1	2	yes	—
79	C₂×Q₈×C₅₃	—	2	1	no	—
80	C₂×Q₈×C₅₃	—	2	2	no	—
81	C₂×Q₈×C₅₃	—	4	2	no	—
82	C₂×Q₈×C₅₃	—	4	1	no	—
83	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
84	C₂×D₄×C₅₃	—	2	1	no	—
85	C₂²×C₄×C₅₃	—	4	1	no	—
86	Q₁₆×C₅₃	—	2	2	no	—
87	Q₁₆×C₅₃	—	2	1	no	—
88	Q₁₆×C₅₃	—	2	2	no	—
89	Q₁₆×C₅₃	—	8	1	no	—
90	Q₁₆×C₅₃	#41	1	2	yes	—
91	SD₁₆×C₅₃	—	8	2	no	—
92	D₈×C₅₃	—	2	1	no	—
93	D₈×C₅₃	—	2	1	no	—
94	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₅₃	—	2	1	no	—
98	C₄⋊C₄×C₅₃	—	2	1	no	—
99	C₄⋊C₄×C₅₃	—	2	2	no	—
100	C₄⋊C₄×C₅₃	—	4	1	no	—
101	C₄⋊C₄×C₅₃	—	4	2	no	—
102	C₄⋊C₄×C₅₃	—	4	2	no	—
103	C₄⋊C₄×C₅₃	—	4	1	no	—
104	C₄⋊C₄×C₅₃	—	4	2	no	—
105	C₄⋊C₄×C₅₃	—	4	1	no	—
106	C₄⋊C₄×C₅₃	—	4	1	no	—
107	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
108	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
109	C₂²⋊C₄×C₅₃	—	4	2	no	—
110	C₂²⋊C₄×C₅₃	—	4	1	no	—
111	C₂²⋊C₄×C₅₃	—	4	2	no	—
112	C₄²×C₅₃	—	4	2	no	—
113	C₄²×C₅₃	—	4	2	no	—
114	C₄²×C₅₃	—	4	1	no	—
115	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅₃	C₂×D₄	1	2	yes	—
117	C₁₆×C₅₃	—	2	1	no	—
118	C₁₆×C₅₃	—	2	2	no	—
119	C₁₆×C₅₃	—	2	1	no	—
120	C₂³×C₅₃	—	4	1	no	—
121	Q₈×C₅₃	—	2	2	no	—
122	Q₈×C₅₃	—	2	2	no	—
123	Q₈×C₅₃	—	4	1	no	—
124	Q₈×C₅₃	—	4	2	no	—
125	Q₈×C₅₃	—	4	2	no	—
126	Q₈×C₅₃	—	16	2	no	—
127	Q₈×C₅₃	—	16	1	no	—
128	D₄×C₅₃	—	4	1	no	—
129	D₄×C₅₃	—	4	1	no	—
130	C₂×C₄×C₅₃	—	4	1	no	—
131	C₂×C₄×C₅₃	—	4	1	no	—
132	C₂×C₄×C₅₃	—	4	2	no	—
133	C₂×C₄×C₅₃	—	4	1	no	—
134	C₂×C₄×C₅₃	—	4	2	no	—
135	C₂×C₄×C₅₃	—	8	2	no	—
136	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₅₃	—	2	2	no	—
142	C₈×C₅₃	—	2	1	no	—
143	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
144	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
145	#33936	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂²×C₅₃	—	8	1	no	—
147	C₂²×C₅₃	—	8	1	no	—
148	C₂²×C₅₃	—	8	1	no	—
149	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
150	C₄×C₅₃	—	2	2	no	—
151	C₄×C₅₃	—	2	2	no	—
152	C₄×C₅₃	—	2	1	no	—
153	C₄×C₅₃	—	4	2	no	—
154	C₄×C₅₃	—	8	2	no	—
155	C₄×C₅₃	—	8	1	no	—
156	C₄×C₅₃	—	8	1	no	—
157	C₄×C₅₃	—	8	2	no	—
158	C₄×C₅₃	#510	1	2	yes	—
159	C₄×C₅₃	#548	1	1	yes	characteristic
160	C₄×C₅₃	#577	1	1	yes	characteristic
161	#3393	C₂×C₅₃	1	1	yes	characteristic
162	Q₈×Q₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
163	#2243	C₂×C₅₃	1	2	yes	—
164	#2241	C₂×C₅₃	1	1	yes	characteristic, complemented
165	#2226	C₂×C₅₃	1	1	yes	characteristic
166	#2225	C₂×C₅₃	1	2	yes	complemented
167	#2211	C₂×C₅₃	1	2	yes	—
168	#2209	C₂×C₅₃	1	1	yes	characteristic, complemented
169	#2183	C₂×C₅₃	1	1	yes	characteristic, complemented
170	#2176	C₂×C₅₃	1	2	yes	complemented
171	C₄×SD₃₂	C₂×C₅₃	1	1	yes	characteristic
172	C₂×C₅₃	—	16	1	no	—
173	C₂×C₅₃	#3358	1	1	yes	characteristic
174	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
175	C₂×C₅₃	#3416	1	1	yes	characteristic
176	Q₈²	—	2	1	no	—
177	#556	—	2	1	no	—
178	C₂×Q₃₂	C₂²×C₅₃	1	2	yes	—
179	C₂×SD₃₂	C₂²×C₅₃	1	1	yes	characteristic
180	#500	C₂²×C₅₃	1	2	yes	—
181	#500	C₂²×C₅₃	1	1	yes	characteristic
182	#495	C₂²×C₅₃	1	2	yes	—
183	#494	C₂²×C₅₃	1	1	yes	characteristic
184	#477	—	2	1	no	—
185	#477	—	2	2	no	—
186	#464	—	2	2	no	—
187	#461	—	2	1	no	—
188	C₈×Q₈	C₂²×C₅₃	1	1	yes	characteristic
189	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
190	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	—
191	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	—
192	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
193	#367	C₂²×C₅₃	1	1	yes	characteristic
194	#367	C₂²×C₅₃	1	2	yes	—
195	C₄⋊C₁₆	C₂²×C₅₃	1	1	yes	characteristic
196	C₄⋊C₁₆	C₂²×C₅₃	1	2	yes	—
197	#358	C₂²×C₅₃	1	1	yes	characteristic
198	#358	C₂²×C₅₃	1	2	yes	—
199	#358	C₂²×C₅₃	1	2	yes	—
200	#358	C₂²×C₅₃	1	2	yes	—
201	#357	C₂²×C₅₃	1	1	yes	characteristic
202	#357	C₂²×C₅₃	1	2	yes	—
203	#346	C₂²×C₅₃	1	2	yes	—
204	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
205	C₅₃	#33936	1	1	yes	characteristic, direct product factor, Sylow
206	C₂×Q₁₆	—	2	1	no	—
207	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
208	C₂×Q₁₆	C₂³×C₅₃	1	1	yes	characteristic
209	C₂×SD₁₆	—	2	2	no	—
210	C₂×D₈	C₂³×C₅₃	1	1	yes	characteristic
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	2	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	4	1	no	—
215	C₄⋊Q₈	—	4	2	no	—
216	#126	—	4	2	no	—
217	#124	—	2	2	no	—
218	C₂²⋊Q₈	—	4	1	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×Q₈	—	4	1	no	—
223	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
224	C₄×D₄	—	2	1	no	—
225	#117	—	4	1	no	—
226	Q₃₂	—	2	4	no	—
227	SD₃₂	—	2	2	no	—
228	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
229	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
230	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
231	#107	—	2	1	no	—
232	#107	—	2	2	no	—
233	#107	C₂³×C₅₃	1	1	yes	characteristic
234	#107	C₂³×C₅₃	1	2	yes	—
235	#107	C₂³×C₅₃	1	1	yes	characteristic
236	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
237	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
238	#103	—	2	2	no	—
239	#103	—	2	1	no	—
240	#103	—	2	2	no	—
241	#102	—	2	1	no	—
242	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₅₃	1	2	yes	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	D₈×C₅₃	1	1	yes	characteristic
249	C₂×D₄	—	2	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	Q₁₆	—	2	2	no	—
252	Q₁₆	—	2	1	no	—
253	Q₁₆	—	2	2	no	—
254	Q₁₆	—	8	1	no	—
255	Q₁₆	#41×C₅₃	1	2	yes	—
256	SD₁₆	—	8	2	no	—
257	D₈	—	2	1	no	—
258	D₈	—	2	1	no	—
259	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
261	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	D₈×C₅₃	1	1	yes	characteristic
273	C₄⋊C₄	D₈×C₅₃	1	2	yes	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	1	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	—	4	2	no	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₅₃	1	2	yes	—
282	C₁₆	—	2	1	no	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	2	1	no	—
285	C₂³	—	4	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	16	2	no	—
292	Q₈	—	16	1	no	—
293	D₄	—	4	1	no	—
294	D₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
302	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
303	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
304	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
309	C₈	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
310	C₂²	—	8	1	no	—
311	C₂²	—	8	1	no	—
312	C₂²	—	8	1	no	—
313	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	1	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	1	no	—
322	C₄	C₅₃×#510	1	2	yes	—
323	C₄	C₅₃×#548	1	1	yes	characteristic
324	C₄	C₅₃×#577	1	1	yes	characteristic
325	C₂	—	16	1	no	—
326	C₂	C₅₃×#3358	1	1	yes	characteristic
327	C₂	C₂×C₅₃×#510	1	1	yes	characteristic
328	C₂	C₅₃×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
prime radical	C₂×D₈×C₅₃	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×Q₈×C₅₃ ◁ C₈×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×Q₁₆×C₅₃	C₂	semidirect
4. C₅₃×#2241	C₂	semidirect
6. C₅₃×#2225	C₂	semidirect
8. C₅₃×#2209	C₂	semidirect
9. C₅₃×#2183	C₂	semidirect
10. C₅₃×#2176	C₂	semidirect
145. #33936	C₅₃	direct (N × H)
162. Q₈×Q₁₆	C₂×C₅₃	semidirect
164. #2241	C₂×C₅₃	semidirect
166. #2225	C₂×C₅₃	semidirect
168. #2209	C₂×C₅₃	semidirect
169. #2183	C₂×C₅₃	semidirect
170. #2176	C₂×C₅₃	semidirect
205. C₅₃	#33936	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3393	2
Q₈×Q₁₆×C₅₃	2
C₅₃×#2243	2
C₅₃×#2241	2
C₅₃×#2226	2
C₅₃×#2225	2
C₅₃×#2211	2
C₅₃×#2209	2
C₅₃×#2183	2
C₅₃×#2176	2
C₄×SD₃₂×C₅₃	2
#33936	53

Subgroups — C53×#33936 #27241448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33936 #27241448