Subgroups — C₅₃×#12633 #27220145

Order 13568.

460 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×#2931	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2132	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2126	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1757	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1395	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₅₃	C₄	1	1	yes	characteristic, complemented
7	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#532	C₂²	1	1	yes	characteristic
9	C₅₃×#436	—	2	2	no	—
10	C₈×D₄×C₅₃	—	2	2	no	—
11	C₅₃×#423	C₄	1	1	yes	characteristic, complemented
12	C₅₃×#413	C₂²	1	2	yes	—
13	C₅₃×#410	C₂²	1	2	yes	—
14	C₅₃×#408	C₄	1	1	yes	characteristic, complemented
15	C₅₃×#408	C₂²	1	1	yes	characteristic
16	C₅₃×#405	—	2	2	no	—
17	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
18	C₅₃×#356	C₄	1	2	yes	complemented
19	C₅₃×#355	C₄	1	2	yes	complemented
20	C₅₃×#354	C₄	1	2	yes	—
21	C₅₃×#353	C₄	1	2	yes	—
22	C₅₃×#344	—	2	2	no	—
23	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
25	#131×C₅₃	—	4	1	no	—
26	#131×C₅₃	—	4	1	no	—
27	#131×C₅₃	—	4	1	no	—
28	#131×C₅₃	—	4	1	no	—
29	C₂×M₁₆×C₅₃	—	2	1	no	—
30	C₂×M₁₆×C₅₃	—	2	2	no	—
31	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
34	C₂²×C₈×C₅₃	—	2	1	no	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	D₄	1	2	yes	—
38	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
40	#124×C₅₃	C₂×C₄	1	1	yes	characteristic
41	#124×C₅₃	C₂×C₄	1	1	yes	characteristic
42	#121×C₅₃	—	2	2	no	—
43	C₄×D₄×C₅₃	—	2	2	no	—
44	#117×C₅₃	—	2	2	no	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₄⋊C₈×C₅₃	—	2	2	no	—
47	#101×C₅₃	C₂×C₄	1	4	yes	—
48	#100×C₅₃	—	2	2	no	—
49	#99×C₅₃	—	2	2	no	—
50	#99×C₅₃	C₂×C₄	1	4	yes	—
51	C₂²⋊C₈×C₅₃	—	2	2	no	—
52	C₂²⋊C₈×C₅₃	—	2	2	no	—
53	#97×C₅₃	—	4	2	no	—
54	C₄×C₈×C₅₃	—	2	1	no	—
55	C₄×C₈×C₅₃	—	2	1	no	—
56	C₄×C₈×C₅₃	—	4	2	no	—
57	#41×C₅₃	—	4	1	no	—
58	#41×C₅₃	—	4	1	no	—
59	#41×C₅₃	—	4	1	no	—
60	#41×C₅₃	—	4	1	no	—
61	C₂×Q₈×C₅₃	C₄²	1	1	yes	characteristic
62	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×D₄×C₅₃	—	2	1	no	—
64	C₂×D₄×C₅₃	—	2	2	no	—
65	C₂×D₄×C₅₃	C₄²	1	1	yes	characteristic
66	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×D₄×C₅₃	#41	1	2	yes	—
68	C₂²×C₄×C₅₃	—	2	1	no	—
69	C₂²×C₄×C₅₃	—	2	1	no	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
72	M₁₆×C₅₃	—	2	4	no	—
73	M₁₆×C₅₃	—	4	1	no	—
74	M₁₆×C₅₃	—	4	2	no	—
75	M₁₆×C₅₃	—	4	2	no	—
76	M₁₆×C₅₃	—	4	2	no	—
77	M₁₆×C₅₃	—	4	1	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	—	2	1	no	—
80	C₂×C₈×C₅₃	—	2	2	no	—
81	C₂×C₈×C₅₃	—	2	1	no	—
82	C₂×C₈×C₅₃	—	2	1	no	—
83	C₂×C₈×C₅₃	—	2	1	no	—
84	C₂×C₈×C₅₃	—	4	2	no	—
85	C₂×C₈×C₅₃	—	4	1	no	—
86	C₂×C₈×C₅₃	—	4	1	no	—
87	C₂×C₈×C₅₃	—	4	1	no	—
88	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
90	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₄⋊C₄×C₅₃	—	4	2	no	—
93	C₂²⋊C₄×C₅₃	—	2	2	no	—
94	C₂²⋊C₄×C₅₃	—	2	2	no	—
95	C₂²⋊C₄×C₅₃	—	2	4	no	—
96	C₄²×C₅₃	—	2	1	no	—
97	C₄²×C₅₃	—	4	2	no	—
98	C₄²×C₅₃	C₄²	1	1	yes	characteristic
99	C₄²×C₅₃	C₄²	1	1	yes	characteristic
100	C₂³×C₅₃	—	2	1	no	—
101	C₂³×C₅₃	#117	1	1	yes	characteristic
102	C₂³×C₅₃	C₄×D₄	1	2	yes	—
103	Q₈×C₅₃	—	4	1	no	—
104	Q₈×C₅₃	—	4	1	no	—
105	D₄×C₅₃	—	4	1	no	—
106	D₄×C₅₃	—	4	2	no	—
107	D₄×C₅₃	—	4	1	no	—
108	D₄×C₅₃	—	4	4	no	—
109	D₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	4	1	no	—
119	C₂×C₄×C₅₃	—	4	1	no	—
120	C₂×C₄×C₅₃	—	4	4	no	—
121	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
124	C₈×C₅₃	—	2	2	no	—
125	C₈×C₅₃	—	2	1	no	—
126	C₈×C₅₃	—	4	4	no	—
127	C₈×C₅₃	—	4	1	no	—
128	C₈×C₅₃	—	4	1	no	—
129	C₈×C₅₃	#99	1	1	yes	characteristic
130	C₈×C₅₃	#99	1	1	yes	characteristic
131	#12633	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂²×C₅₃	—	2	2	no	—
133	C₂²×C₅₃	—	2	1	no	—
134	C₂²×C₅₃	—	4	1	no	—
135	C₂²×C₅₃	—	4	2	no	—
136	C₂²×C₅₃	—	4	2	no	—
137	C₂²×C₅₃	—	4	1	no	—
138	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₅₃	—	2	2	no	—
140	C₄×C₅₃	—	2	2	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	4	1	no	—
143	C₄×C₅₃	—	4	1	no	—
144	C₄×C₅₃	—	8	4	no	—
145	C₄×C₅₃	C₂×#99	1	1	yes	characteristic
146	C₄×C₅₃	#410	1	1	yes	characteristic
147	#2931	C₂×C₅₃	1	1	yes	characteristic
148	#2132	C₂×C₅₃	1	1	yes	characteristic
149	#2126	C₂×C₅₃	1	1	yes	characteristic
150	#1757	C₂×C₅₃	1	2	yes	complemented
151	#1395	C₂×C₅₃	1	2	yes	complemented
152	C₂×C₅₃	—	2	1	no	—
153	C₂×C₅₃	—	4	2	no	—
154	C₂×C₅₃	—	4	1	no	—
155	C₂×C₅₃	—	8	1	no	—
156	C₂×C₅₃	C₄×#99	1	1	yes	characteristic
157	C₂×#131	C₄×C₅₃	1	1	yes	characteristic, complemented
158	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
159	#532	C₂²×C₅₃	1	1	yes	characteristic
160	#436	—	2	2	no	—
161	C₈×D₄	—	2	2	no	—
162	#423	C₄×C₅₃	1	1	yes	characteristic, complemented
163	#413	C₂²×C₅₃	1	2	yes	—
164	#410	C₂²×C₅₃	1	2	yes	—
165	#408	C₄×C₅₃	1	1	yes	characteristic, complemented
166	#408	C₂²×C₅₃	1	1	yes	characteristic
167	#405	—	2	2	no	—
168	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic, complemented
169	#356	C₄×C₅₃	1	2	yes	complemented
170	#355	C₄×C₅₃	1	2	yes	complemented
171	#354	C₄×C₅₃	1	2	yes	—
172	#353	C₄×C₅₃	1	2	yes	—
173	#344	—	2	2	no	—
174	C₅₃	#12633	1	1	yes	characteristic, direct product factor, Sylow
175	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
176	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
177	#131	—	4	1	no	—
178	#131	—	4	1	no	—
179	#131	—	4	1	no	—
180	#131	—	4	1	no	—
181	C₂×M₁₆	—	2	1	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
184	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
185	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
188	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
189	C₂²×C₈	D₄×C₅₃	1	2	yes	—
190	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
191	#127	C₂×C₄×C₅₃	1	1	yes	characteristic
192	#124	C₂×C₄×C₅₃	1	1	yes	characteristic
193	#124	C₂×C₄×C₅₃	1	1	yes	characteristic
194	#121	—	2	2	no	—
195	C₄×D₄	—	2	2	no	—
196	#117	—	2	2	no	—
197	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
198	C₄⋊C₈	—	2	2	no	—
199	#101	C₂×C₄×C₅₃	1	4	yes	—
200	#100	—	2	2	no	—
201	#99	—	2	2	no	—
202	#99	C₂×C₄×C₅₃	1	4	yes	—
203	C₂²⋊C₈	—	2	2	no	—
204	C₂²⋊C₈	—	2	2	no	—
205	#97	—	4	2	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	4	2	no	—
209	#41	—	4	1	no	—
210	#41	—	4	1	no	—
211	#41	—	4	1	no	—
212	#41	—	4	1	no	—
213	C₂×Q₈	C₄²×C₅₃	1	1	yes	characteristic
214	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	C₄²×C₅₃	1	1	yes	characteristic
218	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
219	C₂×D₄	#41×C₅₃	1	2	yes	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	4	1	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	4	1	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
241	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
242	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	—	2	2	no	—
246	C₂²⋊C₄	—	2	2	no	—
247	C₂²⋊C₄	—	2	4	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	4	2	no	—
250	C₄²	C₄²×C₅₃	1	1	yes	characteristic
251	C₄²	C₄²×C₅₃	1	1	yes	characteristic
252	C₂³	—	2	1	no	—
253	C₂³	#117×C₅₃	1	1	yes	characteristic
254	C₂³	C₄×D₄×C₅₃	1	2	yes	—
255	Q₈	—	4	1	no	—
256	Q₈	—	4	1	no	—
257	D₄	—	4	1	no	—
258	D₄	—	4	2	no	—
259	D₄	—	4	1	no	—
260	D₄	—	4	4	no	—
261	D₄	—	4	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
274	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
275	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
276	C₈	—	2	2	no	—
277	C₈	—	2	1	no	—
278	C₈	—	4	4	no	—
279	C₈	—	4	1	no	—
280	C₈	—	4	1	no	—
281	C₈	#99×C₅₃	1	1	yes	characteristic
282	C₈	#99×C₅₃	1	1	yes	characteristic
283	C₂²	—	2	2	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	1	no	—
289	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	8	4	no	—
296	C₄	C₂×#99×C₅₃	1	1	yes	characteristic
297	C₄	C₅₃×#410	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	4	2	no	—
300	C₂	—	4	1	no	—
301	C₂	—	8	1	no	—
302	C₂	C₄×#99×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₅₃	#99
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×#41	C₂³×C₅₃
socle	C₂×C₅₃	C₄×#99
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₅₃ ◁ C₂×C₈×C₅₃ ◁ C₂×#131×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1757	C₂	semidirect
5. C₅₃×#1395	C₂	semidirect
6. C₂×#131×C₅₃	C₄	semidirect
11. C₅₃×#423	C₄	semidirect
14. C₅₃×#408	C₄	semidirect
17. C₂×C₄×C₈×C₅₃	C₄	semidirect
18. C₅₃×#356	C₄	semidirect
19. C₅₃×#355	C₄	semidirect
131. #12633	C₅₃	direct (N × H)
150. #1757	C₂×C₅₃	semidirect
151. #1395	C₂×C₅₃	semidirect
157. C₂×#131	C₄×C₅₃	semidirect
162. #423	C₄×C₅₃	semidirect
165. #408	C₄×C₅₃	semidirect
168. C₂×C₄×C₈	C₄×C₅₃	semidirect
169. #356	C₄×C₅₃	semidirect
170. #355	C₄×C₅₃	semidirect
174. C₅₃	#12633	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×#2931	2
C₅₃×#2132	2
C₅₃×#2126	2
C₅₃×#1757	2
C₅₃×#1395	2
#12633	53

Subgroups — C53×#12633 #27220145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12633 #27220145