Subgroups — C₅₃×#33923 #27241435

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 326 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
2	C₅₃×#3338	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×#2235	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2227	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2225	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#2219	C₂	1	2	yes	maximal, complemented
9	C₄×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#2169	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#2159	C₂	1	2	yes	maximal, complemented
12	C₅₃×#556	—	2	1	no	—
13	C₅₃×#554	—	2	1	no	—
14	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×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×#484	—	2	2	no	—
20	C₅₃×#477	—	2	1	no	—
21	C₅₃×#475	—	2	1	no	—
22	C₅₃×#467	C₂²	1	2	yes	—
23	C₅₃×#467	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#427	C₂²	1	2	yes	—
28	C₂×#107×C₅₃	C₂²	1	2	yes	—
29	C₅₃×#367	C₂²	1	2	yes	—
30	C₅₃×#367	C₂²	1	2	yes	—
31	C₅₃×#367	C₂²	1	2	yes	—
32	C₅₃×#366	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
37	C₅₃×#358	C₂²	1	2	yes	—
38	C₅₃×#358	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×Q₁₆×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	#125×C₅₃	—	4	1	no	—
48	#124×C₅₃	—	4	1	no	—
49	C₂²⋊Q₈×C₅₃	—	2	2	no	—
50	C₂²⋊Q₈×C₅₃	—	2	2	no	—
51	C₂²⋊Q₈×C₅₃	—	4	2	no	—
52	C₄×Q₈×C₅₃	—	2	1	no	—
53	C₄×Q₈×C₅₃	—	2	1	no	—
54	C₄×Q₈×C₅₃	—	4	1	no	—
55	C₄×D₄×C₅₃	—	4	1	no	—
56	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
57	#117×C₅₃	—	2	2	no	—
58	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
59	Q₃₂×C₅₃	—	2	2	no	—
60	M₃₂×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	—	2	2	no	—
67	#107×C₅₃	C₂³	1	2	yes	—
68	#107×C₅₃	C₂³	1	2	yes	—
69	#107×C₅₃	C₂³	1	1	yes	characteristic
70	#107×C₅₃	C₂³	1	1	yes	characteristic
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	#103×C₅₃	—	2	1	no	—
76	#103×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	—	2	1	no	—
81	C₂×Q₈×C₅₃	—	4	1	no	—
82	C₂×Q₈×C₅₃	—	4	1	no	—
83	C₂×D₄×C₅₃	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	—	4	2	no	—
85	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
86	Q₁₆×C₅₃	—	2	1	no	—
87	Q₁₆×C₅₃	—	2	1	no	—
88	Q₁₆×C₅₃	—	2	1	no	—
89	Q₁₆×C₅₃	—	2	1	no	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₅₃	—	2	1	no	—
96	C₄⋊C₄×C₅₃	—	2	2	no	—
97	C₄⋊C₄×C₅₃	—	2	1	no	—
98	C₄⋊C₄×C₅₃	—	2	2	no	—
99	C₄⋊C₄×C₅₃	—	4	2	no	—
100	C₄⋊C₄×C₅₃	—	4	1	no	—
101	C₄⋊C₄×C₅₃	—	4	1	no	—
102	C₄⋊C₄×C₅₃	—	4	1	no	—
103	C₄⋊C₄×C₅₃	—	4	2	no	—
104	C₄⋊C₄×C₅₃	—	4	2	no	—
105	C₄⋊C₄×C₅₃	—	4	1	no	—
106	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
107	C₂²⋊C₄×C₅₃	—	4	2	no	—
108	C₂²⋊C₄×C₅₃	—	4	2	no	—
109	C₂²⋊C₄×C₅₃	—	4	2	no	—
110	C₂²⋊C₄×C₅₃	D₈	1	2	yes	—
111	C₄²×C₅₃	—	4	2	no	—
112	C₄²×C₅₃	—	4	1	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	C₂×D₈	1	2	yes	—
119	Q₈×C₅₃	—	4	1	no	—
120	Q₈×C₅₃	—	4	1	no	—
121	Q₈×C₅₃	—	4	1	no	—
122	Q₈×C₅₃	—	4	1	no	—
123	Q₈×C₅₃	—	8	2	no	—
124	Q₈×C₅₃	—	16	1	no	—
125	D₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	4	1	no	—
129	C₂×C₄×C₅₃	—	4	2	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	—	16	2	no	—
135	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₅₃	—	2	2	no	—
140	C₈×C₅₃	—	2	1	no	—
141	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
142	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
143	#33923	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₅₃	—	2	2	no	—
145	C₂²×C₅₃	—	2	2	no	—
146	C₂²×C₅₃	—	2	2	no	—
147	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
148	C₄×C₅₃	—	2	2	no	—
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	—	8	1	no	—
151	C₄×C₅₃	—	8	1	no	—
152	C₄×C₅₃	—	8	2	no	—
153	C₄×C₅₃	—	8	1	no	—
154	C₄×C₅₃	—	8	1	no	—
155	C₄×C₅₃	—	8	2	no	—
156	C₄×C₅₃	#510	1	2	yes	—
157	C₄×C₅₃	#547	1	1	yes	characteristic
158	C₄×C₅₃	#578	1	1	yes	characteristic
159	#3342	C₂×C₅₃	1	1	yes	characteristic
160	#3338	C₂×C₅₃	1	1	yes	characteristic
161	#2254	C₂×C₅₃	1	1	yes	characteristic, complemented
162	#2237	C₂×C₅₃	1	2	yes	complemented
163	#2235	C₂×C₅₃	1	2	yes	complemented
164	#2227	C₂×C₅₃	1	1	yes	characteristic, complemented
165	#2225	C₂×C₅₃	1	1	yes	characteristic, complemented
166	#2219	C₂×C₅₃	1	2	yes	complemented
167	C₄×Q₃₂	C₂×C₅₃	1	1	yes	characteristic, complemented
168	#2169	C₂×C₅₃	1	1	yes	characteristic
169	#2159	C₂×C₅₃	1	2	yes	complemented
170	C₂×C₅₃	—	4	2	no	—
171	C₂×C₅₃	#3334	1	1	yes	characteristic
172	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
173	C₂×C₅₃	#3416	1	1	yes	characteristic
174	#556	—	2	1	no	—
175	#554	—	2	1	no	—
176	C₂×Q₃₂	C₂²×C₅₃	1	1	yes	characteristic
177	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
178	#500	C₂²×C₅₃	1	1	yes	characteristic
179	#500	C₂²×C₅₃	1	1	yes	characteristic
180	#485	—	2	2	no	—
181	#484	—	2	2	no	—
182	#477	—	2	1	no	—
183	#475	—	2	1	no	—
184	#467	C₂²×C₅₃	1	2	yes	—
185	#467	C₂²×C₅₃	1	2	yes	—
186	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
187	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
188	C₈×D₄	C₂²×C₅₃	1	1	yes	characteristic
189	#427	C₂²×C₅₃	1	2	yes	—
190	C₂×#107	C₂²×C₅₃	1	2	yes	—
191	#367	C₂²×C₅₃	1	2	yes	—
192	#367	C₂²×C₅₃	1	2	yes	—
193	#367	C₂²×C₅₃	1	2	yes	—
194	#366	C₂²×C₅₃	1	1	yes	characteristic
195	#366	C₂²×C₅₃	1	2	yes	—
196	C₄⋊C₁₆	C₂²×C₅₃	1	1	yes	characteristic
197	#358	C₂²×C₅₃	1	1	yes	characteristic
198	#358	C₂²×C₅₃	1	1	yes	characteristic
199	#358	C₂²×C₅₃	1	2	yes	—
200	#358	C₂²×C₅₃	1	2	yes	—
201	C₂²⋊C₁₆	C₂²×C₅₃	1	2	yes	—
202	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
203	C₅₃	#33923	1	1	yes	characteristic, direct product factor, Sylow
204	C₂×Q₁₆	C₂³×C₅₃	1	1	yes	characteristic
205	C₂×Q₁₆	C₂³×C₅₃	1	1	yes	characteristic
206	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	#126	—	4	2	no	—
210	#125	—	4	1	no	—
211	#124	—	4	1	no	—
212	C₂²⋊Q₈	—	2	2	no	—
213	C₂²⋊Q₈	—	2	2	no	—
214	C₂²⋊Q₈	—	4	2	no	—
215	C₄×Q₈	—	2	1	no	—
216	C₄×Q₈	—	2	1	no	—
217	C₄×Q₈	—	4	1	no	—
218	C₄×D₄	—	4	1	no	—
219	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
220	#117	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	Q₃₂	—	2	2	no	—
223	M₃₂	—	2	4	no	—
224	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
225	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
226	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
227	#107	—	2	2	no	—
228	#107	—	2	1	no	—
229	#107	—	2	1	no	—
230	#107	C₂³×C₅₃	1	1	yes	characteristic
231	#107	C₂³×C₅₃	1	1	yes	characteristic
232	#107	C₂³×C₅₃	1	2	yes	—
233	#107	C₂³×C₅₃	1	2	yes	—
234	#106	—	2	2	no	—
235	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
236	#103	—	2	1	no	—
237	#103	—	2	1	no	—
238	#103	—	2	2	no	—
239	#103	—	2	2	no	—
240	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
241	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×D₄	D₈×C₅₃	1	1	yes	characteristic
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
249	Q₁₆	—	2	1	no	—
250	Q₁₆	—	2	1	no	—
251	Q₁₆	—	2	1	no	—
252	Q₁₆	—	2	1	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
256	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
257	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	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	D₈×C₅₃	1	1	yes	characteristic
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	D₈×C₅₃	1	2	yes	—
274	C₄²	—	4	2	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	4	1	no	—
277	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
278	C₁₆	—	2	2	no	—
279	C₁₆	—	2	1	no	—
280	C₁₆	#41×C₅₃	1	2	yes	—
281	C₂³	C₂×D₈×C₅₃	1	2	yes	—
282	Q₈	—	4	1	no	—
283	Q₈	—	4	1	no	—
284	Q₈	—	4	1	no	—
285	Q₈	—	4	1	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	16	1	no	—
288	D₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	16	2	no	—
298	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
299	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
300	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	2	1	no	—
304	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
305	C₈	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	8	1	no	—
313	C₄	—	8	1	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	1	no	—
316	C₄	—	8	2	no	—
317	C₄	—	8	1	no	—
318	C₄	C₅₃×#510	1	2	yes	—
319	C₄	C₅₃×#547	1	1	yes	characteristic
320	C₄	C₅₃×#578	1	1	yes	characteristic
321	C₂	—	4	2	no	—
322	C₂	C₅₃×#3334	1	1	yes	characteristic
323	C₂	C₂×C₅₃×#510	1	1	yes	characteristic
324	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×D₄×C₅₃	D₈
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
3. C₅₃×#2254	C₂	semidirect
4. C₅₃×#2237	C₂	semidirect
5. C₅₃×#2235	C₂	semidirect
6. C₅₃×#2227	C₂	semidirect
7. C₅₃×#2225	C₂	semidirect
8. C₅₃×#2219	C₂	semidirect
9. C₄×Q₃₂×C₅₃	C₂	semidirect
11. C₅₃×#2159	C₂	semidirect
143. #33923	C₅₃	direct (N × H)
161. #2254	C₂×C₅₃	semidirect
162. #2237	C₂×C₅₃	semidirect
163. #2235	C₂×C₅₃	semidirect
164. #2227	C₂×C₅₃	semidirect
165. #2225	C₂×C₅₃	semidirect
166. #2219	C₂×C₅₃	semidirect
167. C₄×Q₃₂	C₂×C₅₃	semidirect
169. #2159	C₂×C₅₃	semidirect
203. C₅₃	#33923	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×#3338	2
C₅₃×#2254	2
C₅₃×#2237	2
C₅₃×#2235	2
C₅₃×#2227	2
C₅₃×#2225	2
C₅₃×#2219	2
C₄×Q₃₂×C₅₃	2
C₅₃×#2169	2
C₅₃×#2159	2
#33923	53

Subgroups — C53×#33923 #27241435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33923 #27241435