Subgroups — C₅₃×#18701 #27226213

Order 13568.

924 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×#3548	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#3254	C₂	1	2	yes	maximal, complemented
3	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1654	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1647	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1561	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1492	C₂	1	2	yes	maximal, complemented
8	C₅₃×#562	—	2	2	no	—
9	C₅₃×#554	—	2	2	no	—
10	C₅₃×#550	—	2	2	no	—
11	C₅₃×#547	C₂²	1	4	yes	—
12	C₅₃×#540	—	2	2	no	—
13	C₅₃×#537	—	2	4	no	—
14	C₅₃×#533	C₂²	1	1	yes	characteristic
15	C₅₃×#533	C₂²	1	1	yes	characteristic
16	C₅₃×#532	C₂²	1	2	yes	—
17	C₅₃×#500	C₂²	1	2	yes	—
18	C₅₃×#499	—	2	2	no	—
19	C₅₃×#498	C₂²	1	2	yes	—
20	C₅₃×#477	—	2	2	no	—
21	C₅₃×#474	C₂²	1	4	yes	—
22	C₅₃×#468	—	2	4	no	—
23	C₅₃×#461	C₂²	1	4	yes	—
24	C₅₃×#460	—	2	2	no	—
25	C₅₃×#427	—	2	2	no	—
26	C₅₃×#423	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#332	C₂²	1	4	yes	—
29	C₅₃×D₄⋊C₈	C₂²	1	4	yes	—
30	C₅₃×D₄⋊C₈	C₂²	1	4	yes	—
31	C₂×#41×C₅₃	—	2	4	no	—
32	C₂×#41×C₅₃	—	2	2	no	—
33	C₂×SD₁₆×C₅₃	—	2	4	no	—
34	C₂×M₁₆×C₅₃	—	2	2	no	—
35	C₂²×C₈×C₅₃	—	2	1	no	—
36	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
37	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
38	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
39	#127×C₅₃	—	2	2	no	—
40	#127×C₅₃	D₄	1	2	yes	—
41	#125×C₅₃	—	2	2	no	—
42	#125×C₅₃	—	2	1	no	—
43	#125×C₅₃	D₄	1	2	yes	—
44	#124×C₅₃	—	2	2	no	—
45	#123×C₅₃	—	2	8	no	—
46	C₂²⋊Q₈×C₅₃	—	2	8	no	—
47	C₂²⋊Q₈×C₅₃	—	2	4	no	—
48	C₂²⋊Q₈×C₅₃	—	2	2	no	—
49	C₂²⋊Q₈×C₅₃	—	2	2	no	—
50	#121×C₅₃	—	2	8	no	—
51	#121×C₅₃	—	2	8	no	—
52	#121×C₅₃	—	2	4	no	—
53	#121×C₅₃	—	2	2	no	—
54	#121×C₅₃	D₄	1	4	yes	—
55	C₄×Q₈×C₅₃	—	2	2	no	—
56	C₄×Q₈×C₅₃	—	2	2	no	—
57	C₄×D₄×C₅₃	—	2	4	no	—
58	C₄×D₄×C₅₃	—	2	2	no	—
59	C₄×D₄×C₅₃	C₂³	1	4	yes	—
60	#117×C₅₃	—	2	1	no	—
61	#117×C₅₃	D₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
63	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
64	#107×C₅₃	—	2	4	no	—
65	#107×C₅₃	—	2	4	no	—
66	#106×C₅₃	—	2	4	no	—
67	C₄⋊C₈×C₅₃	—	2	2	no	—
68	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
69	#103×C₅₃	—	2	4	no	—
70	#102×C₅₃	—	2	4	no	—
71	C₄×C₈×C₅₃	—	2	1	no	—
72	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	#41×C₅₃	—	8	2	no	—
75	#41×C₅₃	—	8	2	no	—
76	#41×C₅₃	—	8	4	no	—
77	#41×C₅₃	—	8	4	no	—
78	C₂×Q₈×C₅₃	—	2	2	no	—
79	C₂×Q₈×C₅₃	—	2	2	no	—
80	C₂×Q₈×C₅₃	—	2	2	no	—
81	C₂×D₄×C₅₃	—	2	8	no	—
82	C₂×D₄×C₅₃	—	2	4	no	—
83	C₂×D₄×C₅₃	—	2	2	no	—
84	C₂×D₄×C₅₃	—	2	2	no	—
85	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₅₃	—	2	8	no	—
87	C₂²×C₄×C₅₃	—	2	4	no	—
88	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
90	SD₁₆×C₅₃	—	8	4	no	—
91	M₁₆×C₅₃	—	8	2	no	—
92	C₂×C₈×C₅₃	—	2	4	no	—
93	C₂×C₈×C₅₃	—	2	1	no	—
94	C₂×C₈×C₅₃	—	2	1	no	—
95	C₂×C₈×C₅₃	—	2	4	no	—
96	C₄⋊C₄×C₅₃	—	2	8	no	—
97	C₄⋊C₄×C₅₃	—	2	2	no	—
98	C₄⋊C₄×C₅₃	—	2	4	no	—
99	C₄⋊C₄×C₅₃	—	2	2	no	—
100	C₄⋊C₄×C₅₃	—	2	2	no	—
101	C₄⋊C₄×C₅₃	—	2	2	no	—
102	C₄⋊C₄×C₅₃	—	4	2	no	—
103	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
104	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
105	C₂²⋊C₄×C₅₃	—	2	8	no	—
106	C₂²⋊C₄×C₅₃	—	2	2	no	—
107	C₂²⋊C₄×C₅₃	—	2	4	no	—
108	C₂²⋊C₄×C₅₃	—	2	4	no	—
109	C₂²⋊C₄×C₅₃	—	4	1	no	—
110	C₄²×C₅₃	—	2	2	no	—
111	C₄²×C₅₃	—	4	1	no	—
112	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂×D₄	1	2	yes	—
114	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
115	C₂³×C₅₃	—	2	4	no	—
116	C₂³×C₅₃	#120	1	1	yes	characteristic
117	Q₈×C₅₃	—	4	2	no	—
118	Q₈×C₅₃	—	4	2	no	—
119	Q₈×C₅₃	—	4	2	no	—
120	Q₈×C₅₃	—	4	2	no	—
121	Q₈×C₅₃	—	8	2	no	—
122	D₄×C₅₃	—	2	4	no	—
123	D₄×C₅₃	—	2	4	no	—
124	D₄×C₅₃	—	4	4	no	—
125	D₄×C₅₃	—	8	2	no	—
126	D₄×C₅₃	—	8	4	no	—
127	D₄×C₅₃	—	8	8	no	—
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	—	2	4	no	—
132	C₂×C₄×C₅₃	—	2	4	no	—
133	C₂×C₄×C₅₃	—	4	4	no	—
134	C₂×C₄×C₅₃	—	4	4	no	—
135	C₂×C₄×C₅₃	—	4	2	no	—
136	C₂×C₄×C₅₃	—	8	8	no	—
137	C₂×C₄×C₅₃	#120	1	2	yes	—
138	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
141	C₈×C₅₃	—	2	2	no	—
142	C₈×C₅₃	—	2	2	no	—
143	C₈×C₅₃	—	4	4	no	—
144	#18701	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂²×C₅₃	—	2	2	no	—
146	C₂²×C₅₃	—	2	1	no	—
147	C₂²×C₅₃	—	4	4	no	—
148	C₂²×C₅₃	—	4	4	no	—
149	C₂²×C₅₃	—	4	4	no	—
150	C₂²×C₅₃	C₂×#120	1	1	yes	characteristic
151	C₄×C₅₃	—	2	2	no	—
152	C₄×C₅₃	—	2	1	no	—
153	C₄×C₅₃	—	4	4	no	—
154	C₄×C₅₃	—	4	4	no	—
155	C₄×C₅₃	—	8	2	no	—
156	C₄×C₅₃	#456	1	2	yes	—
157	C₄×C₅₃	#576	1	2	yes	—
158	C₄×C₅₃	#577	1	2	yes	—
159	#3548	C₂×C₅₃	1	1	yes	characteristic
160	#3254	C₂×C₅₃	1	2	yes	complemented
161	#3160	C₂×C₅₃	1	1	yes	characteristic, complemented
162	#1654	C₂×C₅₃	1	4	yes	complemented
163	#1647	C₂×C₅₃	1	4	yes	complemented
164	#1561	C₂×C₅₃	1	1	yes	characteristic, complemented
165	#1492	C₂×C₅₃	1	2	yes	complemented
166	C₂×C₅₃	—	4	1	no	—
167	C₂×C₅₃	—	8	4	no	—
168	C₂×C₅₃	#3012	1	2	yes	—
169	C₂×C₅₃	C₂×#456	1	1	yes	characteristic
170	#562	—	2	2	no	—
171	#554	—	2	2	no	—
172	#550	—	2	2	no	—
173	#547	C₂²×C₅₃	1	4	yes	—
174	#540	—	2	2	no	—
175	#537	—	2	4	no	—
176	#533	C₂²×C₅₃	1	1	yes	characteristic
177	#533	C₂²×C₅₃	1	1	yes	characteristic
178	#532	C₂²×C₅₃	1	2	yes	—
179	#500	C₂²×C₅₃	1	2	yes	—
180	#499	—	2	2	no	—
181	#498	C₂²×C₅₃	1	2	yes	—
182	#477	—	2	2	no	—
183	#474	C₂²×C₅₃	1	4	yes	—
184	#468	—	2	4	no	—
185	#461	C₂²×C₅₃	1	4	yes	—
186	#460	—	2	2	no	—
187	#427	—	2	2	no	—
188	#423	C₂²×C₅₃	1	2	yes	—
189	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
190	#332	C₂²×C₅₃	1	4	yes	—
191	D₄⋊C₈	C₂²×C₅₃	1	4	yes	—
192	D₄⋊C₈	C₂²×C₅₃	1	4	yes	—
193	C₅₃	#18701	1	1	yes	characteristic, direct product factor, Sylow
194	C₂×#41	—	2	4	no	—
195	C₂×#41	—	2	2	no	—
196	C₂×SD₁₆	—	2	4	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
200	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
201	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
202	#127	—	2	2	no	—
203	#127	D₄×C₅₃	1	2	yes	—
204	#125	—	2	2	no	—
205	#125	—	2	1	no	—
206	#125	D₄×C₅₃	1	2	yes	—
207	#124	—	2	2	no	—
208	#123	—	2	8	no	—
209	C₂²⋊Q₈	—	2	8	no	—
210	C₂²⋊Q₈	—	2	2	no	—
211	C₂²⋊Q₈	—	2	4	no	—
212	C₂²⋊Q₈	—	2	2	no	—
213	#121	—	2	8	no	—
214	#121	—	2	8	no	—
215	#121	—	2	4	no	—
216	#121	—	2	2	no	—
217	#121	D₄×C₅₃	1	4	yes	—
218	C₄×Q₈	—	2	2	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×D₄	—	2	4	no	—
221	C₄×D₄	—	2	2	no	—
222	C₄×D₄	C₂³×C₅₃	1	4	yes	—
223	#117	—	2	1	no	—
224	#117	D₄×C₅₃	1	2	yes	—
225	C₂×C₄⋊C₄	—	2	4	no	—
226	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
227	#107	—	2	4	no	—
228	#107	—	2	4	no	—
229	#106	—	2	4	no	—
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
232	#103	—	2	4	no	—
233	#102	—	2	4	no	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
237	#41	—	8	2	no	—
238	#41	—	8	2	no	—
239	#41	—	8	4	no	—
240	#41	—	8	4	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×D₄	—	2	8	no	—
245	C₂×D₄	—	2	4	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	C₂×D₄×C₅₃	1	4	yes	—
249	C₂²×C₄	—	2	8	no	—
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
253	SD₁₆	—	8	4	no	—
254	M₁₆	—	8	2	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	4	no	—
259	C₄⋊C₄	—	2	8	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
267	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
268	C₂²⋊C₄	—	2	8	no	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₂²⋊C₄	—	2	4	no	—
271	C₂²⋊C₄	—	2	4	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	4	1	no	—
275	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
276	C₄²	C₂×D₄×C₅₃	1	2	yes	—
277	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
278	C₂³	—	2	4	no	—
279	C₂³	#120×C₅₃	1	1	yes	characteristic
280	Q₈	—	4	2	no	—
281	Q₈	—	4	2	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	8	2	no	—
285	D₄	—	2	4	no	—
286	D₄	—	2	4	no	—
287	D₄	—	4	4	no	—
288	D₄	—	8	2	no	—
289	D₄	—	8	4	no	—
290	D₄	—	8	8	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	8	no	—
300	C₂×C₄	#120×C₅₃	1	2	yes	—
301	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
304	C₈	—	2	2	no	—
305	C₈	—	2	2	no	—
306	C₈	—	4	4	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	4	4	no	—
312	C₂²	C₂×#120×C₅₃	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	4	no	—
316	C₄	—	4	4	no	—
317	C₄	—	8	2	no	—
318	C₄	C₅₃×#456	1	2	yes	—
319	C₄	C₅₃×#576	1	2	yes	—
320	C₄	C₅₃×#577	1	2	yes	—
321	C₂	—	4	1	no	—
322	C₂	—	8	4	no	—
323	C₂	C₅₃×#3012	1	2	yes	—
324	C₂	C₂×C₅₃×#456	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3548	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#3254	C₂	semidirect
3. C₅₃×#3160	C₂	semidirect
4. C₅₃×#1654	C₂	semidirect
5. C₅₃×#1647	C₂	semidirect
6. C₅₃×#1561	C₂	semidirect
7. C₅₃×#1492	C₂	semidirect
144. #18701	C₅₃	direct (N × H)
160. #3254	C₂×C₅₃	semidirect
161. #3160	C₂×C₅₃	semidirect
162. #1654	C₂×C₅₃	semidirect
163. #1647	C₂×C₅₃	semidirect
164. #1561	C₂×C₅₃	semidirect
165. #1492	C₂×C₅₃	semidirect
193. C₅₃	#18701	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×#3548	2
C₅₃×#3254	2
C₅₃×#3160	2
C₅₃×#1654	2
C₅₃×#1647	2
C₅₃×#1561	2
C₅₃×#1492	2
#18701	53

Subgroups — C53×#18701 #27226213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18701 #27226213