Subgroups — C₅₃×#33907 #27241419

Order 13568.

480 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3342	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3341	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2237	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2234	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2227	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2224	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2217	C₂	1	2	yes	maximal, complemented
9	C₄×SD₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2158	C₂	1	2	yes	maximal, complemented
12	C₅₃×#556	—	2	1	no	—
13	C₅₃×#555	—	2	1	no	—
14	C₂×SD₃₂×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#500	C₂²	1	1	yes	characteristic
17	C₅₃×#500	C₂²	1	1	yes	characteristic
18	C₅₃×#485	—	2	2	no	—
19	C₅₃×#482	—	2	2	no	—
20	C₅₃×#476	—	2	1	no	—
21	C₅₃×#475	—	2	1	no	—
22	C₅₃×#467	C₂²	1	2	yes	—
23	C₅₃×#466	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#427	C₂²	1	2	yes	—
28	C₅₃×#427	C₂²	1	2	yes	—
29	C₅₃×#367	C₂²	1	1	yes	characteristic
30	C₅₃×#367	C₂²	1	2	yes	—
31	C₅₃×#367	C₂²	1	2	yes	—
32	C₅₃×#366	C₂²	1	2	yes	—
33	C₅₃×#366	C₂²	1	2	yes	—
34	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₅₃×#358	C₂²	1	1	yes	characteristic
36	C₅₃×#358	C₂²	1	2	yes	—
37	C₅₃×#357	C₂²	1	1	yes	characteristic
38	C₅₃×#357	C₂²	1	2	yes	—
39	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₅₃	—	2	1	no	—
45	C₄⋊Q₈×C₅₃	—	2	1	no	—
46	#126×C₅₃	—	4	2	no	—
47	#124×C₅₃	—	4	2	no	—
48	#124×C₅₃	—	4	1	no	—
49	C₂²⋊Q₈×C₅₃	—	2	2	no	—
50	#121×C₅₃	—	2	2	no	—
51	#121×C₅₃	—	4	1	no	—
52	C₄×Q₈×C₅₃	—	2	1	no	—
53	C₄×Q₈×C₅₃	—	4	1	no	—
54	C₄×D₄×C₅₃	—	2	1	no	—
55	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
56	#117×C₅₃	—	2	2	no	—
57	#117×C₅₃	—	2	2	no	—
58	#117×C₅₃	—	4	1	no	—
59	SD₃₂×C₅₃	—	2	2	no	—
60	C₂×C₁₆×C₅₃	—	2	4	no	—
61	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
62	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
64	#107×C₅₃	—	2	1	no	—
65	#107×C₅₃	—	2	1	no	—
66	#107×C₅₃	C₂³	1	2	yes	—
67	#107×C₅₃	C₂³	1	2	yes	—
68	#107×C₅₃	C₂³	1	1	yes	characteristic
69	#107×C₅₃	C₂³	1	1	yes	characteristic
70	#106×C₅₃	—	2	2	no	—
71	#106×C₅₃	—	2	2	no	—
72	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
73	#103×C₅₃	—	2	1	no	—
74	#103×C₅₃	—	2	2	no	—
75	#102×C₅₃	—	2	1	no	—
76	#102×C₅₃	—	2	2	no	—
77	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₅₃	—	2	1	no	—
80	C₂×Q₈×C₅₃	—	4	1	no	—
81	C₂×Q₈×C₅₃	—	4	1	no	—
82	C₂×D₄×C₅₃	—	2	1	no	—
83	C₂×D₄×C₅₃	—	4	2	no	—
84	C₂×D₄×C₅₃	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	—	4	1	no	—
86	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
87	Q₁₆×C₅₃	—	2	1	no	—
88	Q₁₆×C₅₃	—	2	1	no	—
89	D₈×C₅₃	—	2	1	no	—
90	D₈×C₅₃	—	2	1	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₅₃	—	2	1	no	—
97	C₄⋊C₄×C₅₃	—	2	2	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	1	no	—
102	C₄⋊C₄×C₅₃	—	4	2	no	—
103	C₄⋊C₄×C₅₃	—	4	1	no	—
104	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
105	C₂²⋊C₄×C₅₃	—	4	2	no	—
106	C₂²⋊C₄×C₅₃	—	4	2	no	—
107	C₂²⋊C₄×C₅₃	—	4	2	no	—
108	C₂²⋊C₄×C₅₃	—	4	2	no	—
109	C₂²⋊C₄×C₅₃	—	4	1	no	—
110	C₂²⋊C₄×C₅₃	D₈	1	2	yes	—
111	C₄²×C₅₃	—	4	2	no	—
112	C₄²×C₅₃	—	4	2	no	—
113	C₄²×C₅₃	—	4	1	no	—
114	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₅₃	—	2	2	no	—
116	C₁₆×C₅₃	—	2	1	no	—
117	C₁₆×C₅₃	#41	1	2	yes	—
118	C₂³×C₅₃	—	4	1	no	—
119	C₂³×C₅₃	C₂×D₈	1	2	yes	—
120	Q₈×C₅₃	—	4	1	no	—
121	Q₈×C₅₃	—	4	1	no	—
122	Q₈×C₅₃	—	8	2	no	—
123	Q₈×C₅₃	—	16	1	no	—
124	D₄×C₅₃	—	2	2	no	—
125	D₄×C₅₃	—	4	1	no	—
126	D₄×C₅₃	—	4	1	no	—
127	D₄×C₅₃	—	16	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	4	1	no	—
131	C₂×C₄×C₅₃	—	4	2	no	—
132	C₂×C₄×C₅₃	—	4	1	no	—
133	C₂×C₄×C₅₃	—	4	2	no	—
134	C₂×C₄×C₅₃	—	4	1	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	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₅₃	—	2	2	no	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
143	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
144	#33907	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂²×C₅₃	—	2	2	no	—
146	C₂²×C₅₃	—	2	2	no	—
147	C₂²×C₅₃	—	2	2	no	—
148	C₂²×C₅₃	—	8	1	no	—
149	C₂²×C₅₃	—	8	1	no	—
150	C₂²×C₅₃	—	8	1	no	—
151	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
152	C₄×C₅₃	—	2	2	no	—
153	C₄×C₅₃	—	2	1	no	—
154	C₄×C₅₃	—	8	1	no	—
155	C₄×C₅₃	—	8	1	no	—
156	C₄×C₅₃	—	8	2	no	—
157	C₄×C₅₃	—	8	1	no	—
158	C₄×C₅₃	—	8	2	no	—
159	C₄×C₅₃	#508	1	2	yes	—
160	C₄×C₅₃	#547	1	1	yes	characteristic
161	C₄×C₅₃	#578	1	1	yes	characteristic
162	#3342	C₂×C₅₃	1	1	yes	characteristic, complemented
163	#3341	C₂×C₅₃	1	1	yes	characteristic
164	#2254	C₂×C₅₃	1	1	yes	characteristic, complemented
165	#2237	C₂×C₅₃	1	2	yes	complemented
166	#2234	C₂×C₅₃	1	2	yes	complemented
167	#2227	C₂×C₅₃	1	1	yes	characteristic, complemented
168	#2224	C₂×C₅₃	1	1	yes	characteristic, complemented
169	#2217	C₂×C₅₃	1	2	yes	complemented
170	C₄×SD₃₂	C₂×C₅₃	1	1	yes	characteristic, complemented
171	D₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
172	#2158	C₂×C₅₃	1	2	yes	complemented
173	C₂×C₅₃	—	4	2	no	—
174	C₂×C₅₃	—	16	1	no	—
175	C₂×C₅₃	#3334	1	1	yes	characteristic
176	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
177	C₂×C₅₃	#3416	1	1	yes	characteristic
178	#556	—	2	1	no	—
179	#555	—	2	1	no	—
180	C₂×SD₃₂	C₂²×C₅₃	1	1	yes	characteristic
181	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	—
182	#500	C₂²×C₅₃	1	1	yes	characteristic
183	#500	C₂²×C₅₃	1	1	yes	characteristic
184	#485	—	2	2	no	—
185	#482	—	2	2	no	—
186	#476	—	2	1	no	—
187	#475	—	2	1	no	—
188	#467	C₂²×C₅₃	1	2	yes	—
189	#466	C₂²×C₅₃	1	2	yes	—
190	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
191	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
192	C₈×D₄	C₂²×C₅₃	1	1	yes	characteristic
193	#427	C₂²×C₅₃	1	2	yes	—
194	#427	C₂²×C₅₃	1	2	yes	—
195	#367	C₂²×C₅₃	1	1	yes	characteristic
196	#367	C₂²×C₅₃	1	2	yes	—
197	#367	C₂²×C₅₃	1	2	yes	—
198	#366	C₂²×C₅₃	1	2	yes	—
199	#366	C₂²×C₅₃	1	2	yes	—
200	C₄⋊C₁₆	C₂²×C₅₃	1	1	yes	characteristic
201	#358	C₂²×C₅₃	1	1	yes	characteristic
202	#358	C₂²×C₅₃	1	2	yes	—
203	#357	C₂²×C₅₃	1	1	yes	characteristic
204	#357	C₂²×C₅₃	1	2	yes	—
205	C₂²⋊C₁₆	C₂²×C₅₃	1	2	yes	—
206	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
207	C₅₃	#33907	1	1	yes	characteristic, direct product factor, Sylow
208	C₂×Q₁₆	C₂³×C₅₃	1	1	yes	characteristic
209	C₂×D₈	C₂³×C₅₃	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	#126	—	4	2	no	—
214	#124	—	4	1	no	—
215	#124	—	4	2	no	—
216	C₂²⋊Q₈	—	2	2	no	—
217	#121	—	2	2	no	—
218	#121	—	4	1	no	—
219	C₄×Q₈	—	2	1	no	—
220	C₄×Q₈	—	4	1	no	—
221	C₄×D₄	—	2	1	no	—
222	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
223	#117	—	2	2	no	—
224	#117	—	2	2	no	—
225	#117	—	4	1	no	—
226	SD₃₂	—	2	2	no	—
227	C₂×C₁₆	—	2	4	no	—
228	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
229	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
230	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
231	#107	—	2	1	no	—
232	#107	—	2	1	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	#107	C₂³×C₅₃	1	2	yes	—
237	#106	—	2	2	no	—
238	#106	—	2	2	no	—
239	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
240	#103	—	2	1	no	—
241	#103	—	2	2	no	—
242	#102	—	2	1	no	—
243	#102	—	2	2	no	—
244	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
245	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂×D₄	D₈×C₅₃	1	1	yes	characteristic
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
254	Q₁₆	—	2	1	no	—
255	Q₁₆	—	2	1	no	—
256	D₈	—	2	1	no	—
257	D₈	—	2	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
262	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	D₈×C₅₃	1	1	yes	characteristic
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	1	no	—
277	C₂²⋊C₄	D₈×C₅₃	1	2	yes	—
278	C₄²	—	4	2	no	—
279	C₄²	—	4	2	no	—
280	C₄²	—	4	1	no	—
281	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
282	C₁₆	—	2	2	no	—
283	C₁₆	—	2	1	no	—
284	C₁₆	#41×C₅₃	1	2	yes	—
285	C₂³	—	4	1	no	—
286	C₂³	C₂×D₈×C₅₃	1	2	yes	—
287	Q₈	—	4	1	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	8	2	no	—
290	Q₈	—	16	1	no	—
291	D₄	—	2	2	no	—
292	D₄	—	4	1	no	—
293	D₄	—	4	1	no	—
294	D₄	—	16	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	8	2	no	—
303	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
304	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
305	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
307	C₈	—	2	2	no	—
308	C₈	—	2	1	no	—
309	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
310	C₈	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	8	1	no	—
315	C₂²	—	8	1	no	—
316	C₂²	—	8	1	no	—
317	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	1	no	—
320	C₄	—	8	1	no	—
321	C₄	—	8	1	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	1	no	—
325	C₄	C₅₃×#508	1	2	yes	—
326	C₄	C₅₃×#547	1	1	yes	characteristic
327	C₄	C₅₃×#578	1	1	yes	characteristic
328	C₂	—	4	2	no	—
329	C₂	—	16	1	no	—
330	C₂	C₅₃×#3334	1	1	yes	characteristic
331	C₂	C₂×C₅₃×#508	1	1	yes	characteristic
332	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	C₅₃×#3341	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×D₄×C₅₃ ◁ C₈×D₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3342	C₂	semidirect
3. C₅₃×#2254	C₂	semidirect
4. C₅₃×#2237	C₂	semidirect
5. C₅₃×#2234	C₂	semidirect
6. C₅₃×#2227	C₂	semidirect
7. C₅₃×#2224	C₂	semidirect
8. C₅₃×#2217	C₂	semidirect
9. C₄×SD₃₂×C₅₃	C₂	semidirect
10. D₄×C₁₆×C₅₃	C₂	semidirect
11. C₅₃×#2158	C₂	semidirect
144. #33907	C₅₃	direct (N × H)
162. #3342	C₂×C₅₃	semidirect
164. #2254	C₂×C₅₃	semidirect
165. #2237	C₂×C₅₃	semidirect
166. #2234	C₂×C₅₃	semidirect
167. #2227	C₂×C₅₃	semidirect
168. #2224	C₂×C₅₃	semidirect
169. #2217	C₂×C₅₃	semidirect
170. C₄×SD₃₂	C₂×C₅₃	semidirect
171. D₄×C₁₆	C₂×C₅₃	semidirect
172. #2158	C₂×C₅₃	semidirect
207. C₅₃	#33907	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×#3342	2
C₅₃×#3341	2
C₅₃×#2254	2
C₅₃×#2237	2
C₅₃×#2234	2
C₅₃×#2227	2
C₅₃×#2224	2
C₅₃×#2217	2
C₄×SD₃₂×C₅₃	2
D₄×C₁₆×C₅₃	2
C₅₃×#2158	2
#33907	53

Subgroups — C53×#33907 #27241419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33907 #27241419