Subgroups — C₅₃×#19630 #27227142

Order 13568.

858 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3226	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3152	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1722	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1713	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1706	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#137×C₅₃	—	2	4	no	—
10	C₅₃×#532	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#500	—	2	2	no	—
13	C₅₃×#500	C₂²	1	4	yes	—
14	C₅₃×#500	C₂²	1	2	yes	—
15	C₅₃×#500	C₂²	1	1	yes	characteristic
16	C₅₃×#500	C₂²	1	2	yes	—
17	C₅₃×#497	—	2	4	no	—
18	C₅₃×#494	C₂²	1	4	yes	—
19	C₅₃×#493	C₂²	1	1	yes	characteristic
20	C₅₃×#492	C₂²	1	4	yes	—
21	C₅₃×#486	C₂²	1	4	yes	—
22	C₅₃×#486	C₂²	1	2	yes	—
23	C₅₃×#480	—	2	4	no	—
24	C₅₃×#474	—	2	4	no	—
25	C₅₃×#466	—	2	2	no	—
26	C₄×D₈×C₅₃	—	2	2	no	—
27	C₅₃×#431	C₂²	1	1	yes	characteristic
28	C₂×#107×C₅₃	—	2	2	no	—
29	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#322	C₂²	1	2	yes	—
32	C₈²×C₅₃	C₂²	1	2	yes	—
33	C₂×#41×C₅₃	—	2	2	no	—
34	C₂²×Q₈×C₅₃	—	2	2	no	—
35	#137×C₅₃	—	4	16	no	—
36	C₂×Q₁₆×C₅₃	—	2	8	no	—
37	C₂×SD₁₆×C₅₃	—	2	8	no	—
38	C₂×D₈×C₅₃	—	2	2	no	—
39	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
40	C₂²×C₈×C₅₃	D₄	1	2	yes	—
41	C₄⋊Q₈×C₅₃	—	2	2	no	—
42	C₄⋊Q₈×C₅₃	—	2	2	no	—
43	C₄⋊Q₈×C₅₃	—	2	2	no	—
44	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
45	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
46	#127×C₅₃	C₂³	1	2	yes	—
47	#124×C₅₃	—	2	2	no	—
48	#121×C₅₃	—	2	2	no	—
49	#121×C₅₃	—	2	2	no	—
50	C₄×D₄×C₅₃	—	2	2	no	—
51	C₄×D₄×C₅₃	—	2	2	no	—
52	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	#107×C₅₃	—	2	4	no	—
56	#107×C₅₃	—	2	4	no	—
57	#107×C₅₃	—	2	2	no	—
58	#107×C₅₃	—	2	8	no	—
59	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
60	#102×C₅₃	—	2	4	no	—
61	#102×C₅₃	—	2	8	no	—
62	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
63	#97×C₅₃	D₄	1	2	yes	—
64	C₄×C₈×C₅₃	D₄	1	2	yes	—
65	C₄×C₈×C₅₃	C₂³	1	2	yes	—
66	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₅₃	C₂³	1	2	yes	—
68	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	#41×C₅₃	—	4	8	no	—
70	C₂×Q₈×C₅₃	—	2	2	no	—
71	C₂×Q₈×C₅₃	—	2	4	no	—
72	C₂×Q₈×C₅₃	—	4	2	no	—
73	C₂×Q₈×C₅₃	—	4	8	no	—
74	C₂×D₄×C₅₃	—	2	2	no	—
75	C₂×D₄×C₅₃	—	2	2	no	—
76	C₂×D₄×C₅₃	—	2	2	no	—
77	C₂×D₄×C₅₃	—	4	2	no	—
78	C₂²×C₄×C₅₃	—	4	2	no	—
79	C₂²×C₄×C₅₃	—	4	2	no	—
80	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
83	Q₁₆×C₅₃	—	4	16	no	—
84	SD₁₆×C₅₃	—	4	16	no	—
85	D₈×C₅₃	—	4	4	no	—
86	M₁₆×C₅₃	D₈	1	8	yes	complemented
87	C₂×C₈×C₅₃	—	2	4	no	—
88	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
90	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
91	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
92	C₄⋊C₄×C₅₃	—	2	4	no	—
93	C₄⋊C₄×C₅₃	—	2	2	no	—
94	C₄⋊C₄×C₅₃	—	2	4	no	—
95	C₄⋊C₄×C₅₃	—	2	2	no	—
96	C₄⋊C₄×C₅₃	—	4	2	no	—
97	C₄⋊C₄×C₅₃	—	4	2	no	—
98	C₂²⋊C₄×C₅₃	—	4	2	no	—
99	C₂²⋊C₄×C₅₃	—	4	2	no	—
100	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
104	C₂³×C₅₃	—	4	2	no	—
105	C₂³×C₅₃	#127	1	1	yes	characteristic
106	Q₈×C₅₃	—	4	8	no	—
107	Q₈×C₅₃	—	4	4	no	—
108	Q₈×C₅₃	—	8	4	no	—
109	D₄×C₅₃	—	4	4	no	—
110	D₄×C₅₃	—	4	4	no	—
111	D₄×C₅₃	—	4	4	no	—
112	D₄×C₅₃	—	8	4	no	—
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	4	2	no	—
116	C₂×C₄×C₅₃	—	4	4	no	—
117	C₂×C₄×C₅₃	—	8	4	no	—
118	C₂×C₄×C₅₃	—	8	4	no	—
119	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
120	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
123	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₅₃	—	2	2	no	—
127	C₈×C₅₃	—	2	4	no	—
128	C₈×C₅₃	C₂×D₈	1	8	yes	—
129	C₈×C₅₃	#137	1	4	yes	complemented
130	#19630	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₅₃	—	2	1	no	—
132	C₂²×C₅₃	—	2	1	no	—
133	C₂²×C₅₃	—	8	2	no	—
134	C₂²×C₅₃	—	8	2	no	—
135	C₂²×C₅₃	—	8	2	no	—
136	C₂²×C₅₃	#493	1	2	yes	—
137	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
138	C₄×C₅₃	—	2	1	no	—
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	—	8	2	no	—
141	C₄×C₅₃	—	8	4	no	—
142	C₄×C₅₃	#493	1	2	yes	—
143	C₄×C₅₃	C₂²×D₈	1	2	yes	—
144	C₄×C₅₃	C₂×#137	1	2	yes	—
145	C₄×C₅₃	#578	1	2	yes	—
146	#3226	C₂×C₅₃	1	2	yes	complemented
147	#3169	C₂×C₅₃	1	1	yes	characteristic
148	C₂×#500	C₂×C₅₃	1	1	yes	characteristic, complemented
149	#3152	C₂×C₅₃	1	2	yes	complemented
150	#1722	C₂×C₅₃	1	4	yes	complemented
151	#1713	C₂×C₅₃	1	2	yes	complemented
152	#1706	C₂×C₅₃	1	2	yes	complemented
153	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
154	C₂×C₅₃	—	2	2	no	—
155	C₂×C₅₃	—	16	2	no	—
156	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
157	C₂×C₅₃	#3152	1	1	yes	characteristic
158	C₂×C₅₃	#3156	1	1	yes	characteristic
159	C₂×#137	—	2	4	no	—
160	#532	C₂²×C₅₃	1	2	yes	—
161	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
162	#500	—	2	2	no	—
163	#500	C₂²×C₅₃	1	2	yes	—
164	#500	C₂²×C₅₃	1	2	yes	—
165	#500	C₂²×C₅₃	1	1	yes	characteristic
166	#500	C₂²×C₅₃	1	4	yes	—
167	#497	—	2	4	no	—
168	#494	C₂²×C₅₃	1	4	yes	—
169	#493	C₂²×C₅₃	1	1	yes	characteristic
170	#492	C₂²×C₅₃	1	4	yes	—
171	#486	C₂²×C₅₃	1	2	yes	—
172	#486	C₂²×C₅₃	1	4	yes	—
173	#480	—	2	4	no	—
174	#474	—	2	4	no	—
175	#466	—	2	2	no	—
176	C₄×D₈	—	2	2	no	—
177	#431	C₂²×C₅₃	1	1	yes	characteristic
178	C₂×#107	—	2	2	no	—
179	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
181	#322	C₂²×C₅₃	1	2	yes	—
182	C₈²	C₂²×C₅₃	1	2	yes	—
183	C₅₃	#19630	1	1	yes	characteristic, direct product factor, Sylow
184	C₂×#41	—	2	2	no	—
185	C₂²×Q₈	—	2	2	no	—
186	#137	—	4	16	no	—
187	C₂×Q₁₆	—	2	8	no	—
188	C₂×SD₁₆	—	2	8	no	—
189	C₂×D₈	—	2	2	no	—
190	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
191	C₂²×C₈	D₄×C₅₃	1	2	yes	—
192	C₄⋊Q₈	—	2	2	no	—
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
196	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
197	#127	C₂³×C₅₃	1	2	yes	—
198	#124	—	2	2	no	—
199	#121	—	2	2	no	—
200	#121	—	2	2	no	—
201	C₄×D₄	—	2	2	no	—
202	C₄×D₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
206	#107	—	2	4	no	—
207	#107	—	2	4	no	—
208	#107	—	2	2	no	—
209	#107	—	2	8	no	—
210	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
211	#102	—	2	4	no	—
212	#102	—	2	8	no	—
213	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
214	#97	D₄×C₅₃	1	2	yes	—
215	C₄×C₈	D₄×C₅₃	1	2	yes	—
216	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
217	C₄×C₈	C₂³×C₅₃	1	2	yes	—
218	C₄×C₈	C₂³×C₅₃	1	2	yes	—
219	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
220	#41	—	4	8	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	4	2	no	—
224	C₂×Q₈	—	4	8	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
234	Q₁₆	—	4	16	no	—
235	SD₁₆	—	4	16	no	—
236	D₈	—	4	4	no	—
237	M₁₆	D₈×C₅₃	1	8	yes	complemented
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
240	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
241	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
242	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	4	2	no	—
251	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
252	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
253	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
254	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
255	C₂³	—	4	2	no	—
256	C₂³	#127×C₅₃	1	1	yes	characteristic
257	Q₈	—	4	8	no	—
258	Q₈	—	4	4	no	—
259	Q₈	—	8	4	no	—
260	D₄	—	4	4	no	—
261	D₄	—	4	4	no	—
262	D₄	—	4	4	no	—
263	D₄	—	8	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	—	8	4	no	—
270	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
271	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
272	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
273	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
274	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	4	no	—
279	C₈	C₂×D₈×C₅₃	1	8	yes	—
280	C₈	#137×C₅₃	1	4	yes	complemented
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	8	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	C₅₃×#493	1	2	yes	—
287	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	8	2	no	—
291	C₄	—	8	4	no	—
292	C₄	C₅₃×#493	1	2	yes	—
293	C₄	C₂²×D₈×C₅₃	1	2	yes	—
294	C₄	C₂×#137×C₅₃	1	2	yes	—
295	C₄	C₅₃×#578	1	2	yes	—
296	C₂	—	2	2	no	—
297	C₂	—	16	2	no	—
298	C₂	C₂×C₅₃×#493	1	1	yes	characteristic
299	C₂	C₅₃×#3152	1	1	yes	characteristic
300	C₂	C₅₃×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3169	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
1. C₅₃×#3226	C₂	semidirect
3. C₂×C₅₃×#500	C₂	semidirect
4. C₅₃×#3152	C₂	semidirect
5. C₅₃×#1722	C₂	semidirect
6. C₅₃×#1713	C₂	semidirect
7. C₅₃×#1706	C₂	semidirect
8. C₈×M₁₆×C₅₃	C₂	semidirect
86. M₁₆×C₅₃	D₈	semidirect
129. C₈×C₅₃	#137	semidirect
130. #19630	C₅₃	direct (N × H)
146. #3226	C₂×C₅₃	semidirect
148. C₂×#500	C₂×C₅₃	semidirect
149. #3152	C₂×C₅₃	semidirect
150. #1722	C₂×C₅₃	semidirect
151. #1713	C₂×C₅₃	semidirect
152. #1706	C₂×C₅₃	semidirect
153. C₈×M₁₆	C₂×C₅₃	semidirect
183. C₅₃	#19630	direct (N × H)
237. M₁₆	D₈×C₅₃	semidirect
280. C₈	#137×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3226	2
C₅₃×#3169	2
C₂×C₅₃×#500	2
C₅₃×#3152	2
C₅₃×#1722	2
C₅₃×#1713	2
C₅₃×#1706	2
C₈×M₁₆×C₅₃	2
#19630	53

Subgroups — C53×#19630 #27227142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19630 #27227142