Subgroups — C₅₃×#12649 #27220161

Order 13568.

428 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2943	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2145	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2143	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#347	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1773	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
9	C₄×#41×C₅₃	—	2	1	no	—
10	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#442	C₄	1	2	yes	complemented
13	C₅₃×#441	C₄	1	2	yes	complemented
14	C₅₃×#440	—	2	2	no	—
15	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#427	C₂²	1	1	yes	characteristic
17	C₅₃×#427	C₂²	1	1	yes	characteristic
18	C₅₃×#416	—	2	1	no	—
19	C₅₃×#405	—	2	1	no	—
20	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#360	C₄	1	4	yes	complemented
22	C₅₃×#359	C₄	1	4	yes	complemented
23	C₅₃×#347	—	2	2	no	—
24	C₅₃×#339	—	2	1	no	—
25	C₂×#41×C₅₃	—	2	1	no	—
26	#135×C₅₃	—	2	1	no	—
27	#135×C₅₃	—	2	1	no	—
28	#135×C₅₃	C₂×C₄	1	2	yes	—
29	#135×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×SD₁₆×C₅₃	—	2	1	no	—
32	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₅₃	—	2	1	no	—
34	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₅₃	—	2	2	no	—
37	C₄×D₄×C₅₃	—	2	2	no	—
38	C₄×D₄×C₅₃	—	4	2	no	—
39	#117×C₅₃	—	2	1	no	—
40	#117×C₅₃	—	2	1	no	—
41	#117×C₅₃	—	4	1	no	—
42	C₂×C₄²×C₅₃	—	4	1	no	—
43	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
44	M₃₂×C₅₃	—	2	1	no	—
45	M₃₂×C₅₃	—	2	1	no	—
46	M₃₂×C₅₃	—	2	1	no	—
47	M₃₂×C₅₃	C₂×C₄	1	2	yes	complemented
48	C₂×C₁₆×C₅₃	—	2	1	no	—
49	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
50	#108×C₅₃	—	2	1	no	—
51	#108×C₅₃	C₂×C₄	1	2	yes	complemented
52	#107×C₅₃	—	2	1	no	—
53	#107×C₅₃	—	2	1	no	—
54	#106×C₅₃	C₂×C₄	1	2	yes	—
55	#106×C₅₃	C₂×C₄	1	2	yes	complemented
56	#103×C₅₃	—	2	2	no	—
57	#102×C₅₃	—	2	2	no	—
58	#97×C₅₃	—	4	1	no	—
59	#97×C₅₃	C₂×C₄	1	2	yes	—
60	#97×C₅₃	D₄	1	2	yes	—
61	C₄×C₈×C₅₃	—	4	1	no	—
62	#41×C₅₃	—	2	2	no	—
63	#41×C₅₃	—	2	2	no	—
64	#41×C₅₃	—	4	1	no	—
65	#41×C₅₃	—	4	1	no	—
66	C₂×Q₈×C₅₃	—	2	1	no	—
67	C₂×D₄×C₅₃	—	2	1	no	—
68	C₂×D₄×C₅₃	—	4	1	no	—
69	C₂²×C₄×C₅₃	—	2	1	no	—
70	C₂²×C₄×C₅₃	—	4	2	no	—
71	C₂²×C₄×C₅₃	—	4	1	no	—
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	Q₁₆×C₅₃	—	2	1	no	—
74	Q₁₆×C₅₃	C₄²	1	2	yes	—
75	SD₁₆×C₅₃	—	2	2	no	—
76	SD₁₆×C₅₃	—	2	2	no	—
77	D₈×C₅₃	—	2	1	no	—
78	D₈×C₅₃	C₄²	1	2	yes	—
79	M₁₆×C₅₃	—	2	2	no	—
80	M₁₆×C₅₃	—	4	1	no	—
81	C₂×C₈×C₅₃	—	2	1	no	—
82	C₂×C₈×C₅₃	—	2	1	no	—
83	C₂×C₈×C₅₃	—	4	1	no	—
84	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
90	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
91	C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₄⋊C₄×C₅₃	—	2	2	no	—
93	C₄⋊C₄×C₅₃	—	4	2	no	—
94	C₂²⋊C₄×C₅₃	—	4	2	no	—
95	C₂²⋊C₄×C₅₃	—	4	1	no	—
96	C₂²⋊C₄×C₅₃	—	4	1	no	—
97	C₄²×C₅₃	—	2	1	no	—
98	C₄²×C₅₃	—	4	2	no	—
99	C₄²×C₅₃	—	4	1	no	—
100	C₄²×C₅₃	—	4	1	no	—
101	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
102	C₁₆×C₅₃	—	2	2	no	—
103	C₁₆×C₅₃	—	2	2	no	—
104	C₂³×C₅₃	—	4	1	no	—
105	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
106	Q₈×C₅₃	—	2	2	no	—
107	Q₈×C₅₃	—	4	1	no	—
108	D₄×C₅₃	—	2	2	no	—
109	D₄×C₅₃	—	4	2	no	—
110	D₄×C₅₃	—	4	2	no	—
111	D₄×C₅₃	—	4	1	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	4	1	no	—
114	C₂×C₄×C₅₃	—	4	2	no	—
115	C₂×C₄×C₅₃	—	4	2	no	—
116	C₂×C₄×C₅₃	—	4	2	no	—
117	C₂×C₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	8	1	no	—
119	C₂×C₄×C₅₃	—	8	4	no	—
120	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
126	C₈×C₅₃	—	2	2	no	—
127	C₈×C₅₃	—	4	2	no	—
128	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
129	C₈×C₅₃	#117	1	1	yes	characteristic
130	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
131	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
132	#12649	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₅₃	—	4	1	no	—
134	C₂²×C₅₃	—	4	2	no	—
135	C₂²×C₅₃	—	8	2	no	—
136	C₂²×C₅₃	#418	1	1	yes	characteristic
137	C₂²×C₅₃	#440	1	1	yes	characteristic
138	C₂²×C₅₃	#442	1	1	yes	characteristic
139	C₄×C₅₃	—	4	2	no	—
140	C₄×C₅₃	—	4	2	no	—
141	C₄×C₅₃	—	8	2	no	—
142	C₄×C₅₃	—	8	2	no	—
143	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₄×C₅₃	#417	1	1	yes	characteristic
145	C₄×C₅₃	#440	1	1	yes	characteristic
146	C₄×C₅₃	#442	1	1	yes	characteristic
147	#2943	C₂×C₅₃	1	1	yes	characteristic
148	#2145	C₂×C₅₃	1	1	yes	characteristic
149	#2143	C₂×C₅₃	1	1	yes	characteristic
150	C₂×#347	C₂×C₅₃	1	1	yes	characteristic, complemented
151	#1773	C₂×C₅₃	1	1	yes	characteristic, complemented
152	#1389	C₂×C₅₃	1	1	yes	characteristic, complemented
153	#1384	C₂×C₅₃	1	1	yes	characteristic, complemented
154	C₂×C₅₃	—	2	1	no	—
155	C₂×C₅₃	—	2	1	no	—
156	C₂×C₅₃	—	2	1	no	—
157	C₂×C₅₃	—	8	2	no	—
158	C₂×C₅₃	#1762	1	1	yes	characteristic
159	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
160	C₄×#41	—	2	1	no	—
161	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
162	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
163	#442	C₄×C₅₃	1	2	yes	complemented
164	#441	C₄×C₅₃	1	2	yes	complemented
165	#440	—	2	2	no	—
166	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
167	#427	C₂²×C₅₃	1	1	yes	characteristic
168	#427	C₂²×C₅₃	1	1	yes	characteristic
169	#416	—	2	1	no	—
170	#405	—	2	1	no	—
171	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
172	#360	C₄×C₅₃	1	4	yes	complemented
173	#359	C₄×C₅₃	1	4	yes	complemented
174	#347	—	2	2	no	—
175	#339	—	2	1	no	—
176	C₅₃	#12649	1	1	yes	characteristic, direct product factor, Sylow
177	C₂×#41	—	2	1	no	—
178	#135	—	2	1	no	—
179	#135	—	2	1	no	—
180	#135	C₂×C₄×C₅₃	1	2	yes	—
181	#135	C₂×C₄×C₅₃	1	2	yes	—
182	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
183	C₂×SD₁₆	—	2	1	no	—
184	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic
185	C₂×M₁₆	—	2	1	no	—
186	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
188	C₄×Q₈	—	2	2	no	—
189	C₄×D₄	—	2	2	no	—
190	C₄×D₄	—	4	2	no	—
191	#117	—	2	1	no	—
192	#117	—	2	1	no	—
193	#117	—	4	1	no	—
194	C₂×C₄²	—	4	1	no	—
195	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
196	M₃₂	—	2	1	no	—
197	M₃₂	—	2	1	no	—
198	M₃₂	—	2	1	no	—
199	M₃₂	C₂×C₄×C₅₃	1	2	yes	complemented
200	C₂×C₁₆	—	2	1	no	—
201	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	complemented
202	#108	—	2	1	no	—
203	#108	C₂×C₄×C₅₃	1	2	yes	complemented
204	#107	—	2	1	no	—
205	#107	—	2	1	no	—
206	#106	C₂×C₄×C₅₃	1	2	yes	—
207	#106	C₂×C₄×C₅₃	1	2	yes	complemented
208	#103	—	2	2	no	—
209	#102	—	2	2	no	—
210	#97	—	4	1	no	—
211	#97	C₂×C₄×C₅₃	1	2	yes	—
212	#97	D₄×C₅₃	1	2	yes	—
213	C₄×C₈	—	4	1	no	—
214	#41	—	2	2	no	—
215	#41	—	2	2	no	—
216	#41	—	4	1	no	—
217	#41	—	4	1	no	—
218	C₂×Q₈	—	2	1	no	—
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	—	4	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
225	Q₁₆	—	2	1	no	—
226	Q₁₆	C₄²×C₅₃	1	2	yes	—
227	SD₁₆	—	2	2	no	—
228	SD₁₆	—	2	2	no	—
229	D₈	—	2	1	no	—
230	D₈	C₄²×C₅₃	1	2	yes	—
231	M₁₆	—	2	2	no	—
232	M₁₆	—	4	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
237	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
238	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
239	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
240	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
241	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
242	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	4	1	no	—
248	C₂²⋊C₄	—	4	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	4	2	no	—
251	C₄²	—	4	1	no	—
252	C₄²	—	4	1	no	—
253	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
254	C₁₆	—	2	2	no	—
255	C₁₆	—	2	2	no	—
256	C₂³	—	4	1	no	—
257	C₂³	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
258	Q₈	—	2	2	no	—
259	Q₈	—	4	1	no	—
260	D₄	—	2	2	no	—
261	D₄	—	4	2	no	—
262	D₄	—	4	2	no	—
263	D₄	—	4	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	4	no	—
272	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
273	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
274	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
275	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
276	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
277	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	4	2	no	—
280	C₈	C₂×C₄²×C₅₃	1	1	yes	characteristic
281	C₈	#117×C₅₃	1	1	yes	characteristic
282	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
283	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
284	C₂²	—	4	1	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	8	2	no	—
287	C₂²	C₅₃×#418	1	1	yes	characteristic
288	C₂²	C₅₃×#440	1	1	yes	characteristic
289	C₂²	C₅₃×#442	1	1	yes	characteristic
290	C₄	—	4	2	no	—
291	C₄	—	4	2	no	—
292	C₄	—	8	2	no	—
293	C₄	—	8	2	no	—
294	C₄	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
295	C₄	C₅₃×#417	1	1	yes	characteristic
296	C₄	C₅₃×#440	1	1	yes	characteristic
297	C₄	C₅₃×#442	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	8	2	no	—
302	C₂	C₅₃×#1762	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	#442
commutator	C₈	C₂×C₄²×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	#1762
prime radical	C₂×#135×C₅₃	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₅₃×#347	C₂	semidirect
5. C₅₃×#1773	C₂	semidirect
6. C₅₃×#1389	C₂	semidirect
7. C₅₃×#1384	C₂	semidirect
12. C₅₃×#442	C₄	semidirect
13. C₅₃×#441	C₄	semidirect
21. C₅₃×#360	C₄	semidirect
22. C₅₃×#359	C₄	semidirect
47. M₃₂×C₅₃	C₂×C₄	semidirect
49. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
51. #108×C₅₃	C₂×C₄	semidirect
55. #106×C₅₃	C₂×C₄	semidirect
132. #12649	C₅₃	direct (N × H)
150. C₂×#347	C₂×C₅₃	semidirect
151. #1773	C₂×C₅₃	semidirect
152. #1389	C₂×C₅₃	semidirect
153. #1384	C₂×C₅₃	semidirect
163. #442	C₄×C₅₃	semidirect
164. #441	C₄×C₅₃	semidirect
172. #360	C₄×C₅₃	semidirect
173. #359	C₄×C₅₃	semidirect
176. C₅₃	#12649	direct (N × H)
199. M₃₂	C₂×C₄×C₅₃	semidirect
201. C₂×C₁₆	C₂×C₄×C₅₃	semidirect
203. #108	C₂×C₄×C₅₃	semidirect
207. #106	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2943	2
C₅₃×#2145	2
C₅₃×#2143	2
C₂×C₅₃×#347	2
C₅₃×#1773	2
C₅₃×#1389	2
C₅₃×#1384	2
#12649	53

Subgroups — C53×#12649 #27220161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12649 #27220161