Subgroups — C₅₃×#12623 #27220135

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2929	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2120	C₂	1	2	yes	maximal
3	C₅₃×#1757	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1392	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
8	C₅₃×#525	C₂²	1	1	yes	characteristic
9	C₅₃×#435	—	2	2	no	—
10	C₈×D₄×C₅₃	—	2	2	no	—
11	C₅₃×#424	C₄	1	1	yes	characteristic, complemented
12	C₅₃×#413	C₂²	1	2	yes	—
13	C₅₃×#410	C₂²	1	2	yes	—
14	C₅₃×#408	C₂²	1	1	yes	characteristic
15	C₅₃×#407	C₄	1	1	yes	characteristic, complemented
16	C₅₃×#405	—	2	2	no	—
17	C₅₃×#405	C₄	1	1	yes	characteristic, complemented
18	C₅₃×#352	C₄	1	4	yes	complemented
19	C₅₃×AΣL(1,𝔽₁₆)	C₄	1	4	yes	complemented
20	C₅₃×#344	—	2	2	no	—
21	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
22	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	#131×C₅₃	—	4	1	no	—
24	#131×C₅₃	—	4	1	no	—
25	C₂×M₁₆×C₅₃	—	2	2	no	—
26	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
29	C₂²×C₈×C₅₃	—	2	1	no	—
30	C₂²×C₈×C₅₃	—	2	1	no	—
31	C₂²×C₈×C₅₃	—	4	1	no	—
32	C₂²×C₈×C₅₃	—	4	1	no	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	D₄	1	2	yes	—
36	#123×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²⋊Q₈×C₅₃	—	2	1	no	—
38	#121×C₅₃	—	2	1	no	—
39	#120×C₅₃	C₂×C₄	1	2	yes	—
40	C₄×D₄×C₅₃	—	2	2	no	—
41	#117×C₅₃	—	2	2	no	—
42	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	#101×C₅₃	—	2	2	no	—
45	#100×C₅₃	C₂×C₄	1	4	yes	—
46	#99×C₅₃	—	2	2	no	—
47	#99×C₅₃	C₂×C₄	1	4	yes	—
48	C₂²⋊C₈×C₅₃	—	2	2	no	—
49	C₂²⋊C₈×C₅₃	—	2	2	no	—
50	#97×C₅₃	—	2	1	no	—
51	#97×C₅₃	—	4	2	no	—
52	C₄×C₈×C₅₃	—	2	1	no	—
53	C₄×C₈×C₅₃	—	4	2	no	—
54	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
55	#41×C₅₃	—	4	1	no	—
56	#41×C₅₃	—	4	1	no	—
57	C₂×Q₈×C₅₃	#41	1	1	yes	characteristic
58	C₂×D₄×C₅₃	—	2	2	no	—
59	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
60	C₂×D₄×C₅₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	—	2	1	no	—
62	C₂²×C₄×C₅₃	—	2	1	no	—
63	C₂²×C₄×C₅₃	—	4	1	no	—
64	C₂²×C₄×C₅₃	—	4	1	no	—
65	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
68	M₁₆×C₅₃	—	2	4	no	—
69	M₁₆×C₅₃	—	4	1	no	—
70	M₁₆×C₅₃	—	4	2	no	—
71	M₁₆×C₅₃	—	4	2	no	—
72	M₁₆×C₅₃	—	4	1	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	1	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	1	no	—
77	C₂×C₈×C₅₃	—	2	1	no	—
78	C₂×C₈×C₅₃	—	2	1	no	—
79	C₂×C₈×C₅₃	—	4	2	no	—
80	C₂×C₈×C₅₃	—	4	2	no	—
81	C₂×C₈×C₅₃	—	4	1	no	—
82	C₂×C₈×C₅₃	—	4	1	no	—
83	C₂×C₈×C₅₃	—	4	1	no	—
84	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₅₃	—	2	1	no	—
88	C₄⋊C₄×C₅₃	—	2	2	no	—
89	C₄⋊C₄×C₅₃	—	4	2	no	—
90	C₂²⋊C₄×C₅₃	—	2	2	no	—
91	C₂²⋊C₄×C₅₃	—	2	2	no	—
92	C₂²⋊C₄×C₅₃	—	2	4	no	—
93	C₂²⋊C₄×C₅₃	C₄²	1	2	yes	—
94	C₄²×C₅₃	—	2	1	no	—
95	C₄²×C₅₃	—	4	2	no	—
96	C₂³×C₅₃	—	2	1	no	—
97	C₂³×C₅₃	—	4	1	no	—
98	C₂³×C₅₃	—	4	2	no	—
99	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
100	C₂³×C₅₃	C₄×D₄	1	2	yes	—
101	Q₈×C₅₃	—	4	1	no	—
102	D₄×C₅₃	—	4	2	no	—
103	D₄×C₅₃	—	4	1	no	—
104	D₄×C₅₃	—	4	4	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	4	2	no	—
112	C₂×C₄×C₅₃	—	4	1	no	—
113	C₂×C₄×C₅₃	—	4	1	no	—
114	C₂×C₄×C₅₃	—	4	1	no	—
115	C₂×C₄×C₅₃	—	4	4	no	—
116	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
119	C₈×C₅₃	—	2	1	no	—
120	C₈×C₅₃	—	2	2	no	—
121	C₈×C₅₃	—	4	4	no	—
122	C₈×C₅₃	—	4	1	no	—
123	C₈×C₅₃	—	4	1	no	—
124	C₈×C₅₃	#99	1	1	yes	characteristic
125	C₈×C₅₃	#99	1	1	yes	characteristic
126	#12623	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₅₃	—	2	2	no	—
128	C₂²×C₅₃	—	2	2	no	—
129	C₂²×C₅₃	—	2	1	no	—
130	C₂²×C₅₃	—	4	2	no	—
131	C₂²×C₅₃	—	4	1	no	—
132	C₂²×C₅₃	—	4	1	no	—
133	C₂²×C₅₃	—	4	2	no	—
134	C₂²×C₅₃	—	4	2	no	—
135	C₂²×C₅₃	—	4	1	no	—
136	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₅₃	—	2	2	no	—
138	C₄×C₅₃	—	2	1	no	—
139	C₄×C₅₃	—	4	2	no	—
140	C₄×C₅₃	—	4	1	no	—
141	C₄×C₅₃	—	8	4	no	—
142	C₄×C₅₃	C₂×#99	1	1	yes	characteristic
143	C₄×C₅₃	#410	1	1	yes	characteristic
144	#2929	C₂×C₅₃	1	1	yes	characteristic
145	#2120	C₂×C₅₃	1	2	yes	—
146	#1757	C₂×C₅₃	1	2	yes	complemented
147	#1392	C₂×C₅₃	1	1	yes	characteristic, complemented
148	#1391	C₂×C₅₃	1	1	yes	characteristic, complemented
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	—	4	2	no	—
151	C₂×C₅₃	—	4	2	no	—
152	C₂×C₅₃	—	4	1	no	—
153	C₂×C₅₃	C₄×#99	1	1	yes	characteristic
154	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
155	C₂³×C₈	C₄×C₅₃	1	1	yes	characteristic, complemented
156	#525	C₂²×C₅₃	1	1	yes	characteristic
157	#435	—	2	2	no	—
158	C₈×D₄	—	2	2	no	—
159	#424	C₄×C₅₃	1	1	yes	characteristic, complemented
160	#413	C₂²×C₅₃	1	2	yes	—
161	#410	C₂²×C₅₃	1	2	yes	—
162	#408	C₂²×C₅₃	1	1	yes	characteristic
163	#407	C₄×C₅₃	1	1	yes	characteristic, complemented
164	#405	—	2	2	no	—
165	#405	C₄×C₅₃	1	1	yes	characteristic, complemented
166	#352	C₄×C₅₃	1	4	yes	complemented
167	AΣL(1,𝔽₁₆)	C₄×C₅₃	1	4	yes	complemented
168	#344	—	2	2	no	—
169	C₅₃	#12623	1	1	yes	characteristic, direct product factor, Sylow
170	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
171	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
172	#131	—	4	1	no	—
173	#131	—	4	1	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
176	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
177	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	4	1	no	—
181	C₂²×C₈	—	4	1	no	—
182	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
183	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
184	C₂²×C₈	D₄×C₅₃	1	2	yes	—
185	#123	C₂×C₄×C₅₃	1	2	yes	—
186	C₂²⋊Q₈	—	2	1	no	—
187	#121	—	2	1	no	—
188	#120	C₂×C₄×C₅₃	1	2	yes	—
189	C₄×D₄	—	2	2	no	—
190	#117	—	2	2	no	—
191	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
192	C₄⋊C₈	—	2	2	no	—
193	#101	—	2	2	no	—
194	#100	C₂×C₄×C₅₃	1	4	yes	—
195	#99	—	2	2	no	—
196	#99	C₂×C₄×C₅₃	1	4	yes	—
197	C₂²⋊C₈	—	2	2	no	—
198	C₂²⋊C₈	—	2	2	no	—
199	#97	—	2	1	no	—
200	#97	—	4	2	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	4	2	no	—
203	C₂⁴	C₄²×C₅₃	1	1	yes	characteristic
204	#41	—	4	1	no	—
205	#41	—	4	1	no	—
206	C₂×Q₈	#41×C₅₃	1	1	yes	characteristic
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
209	C₂×D₄	#41×C₅₃	1	1	yes	characteristic
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
215	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	4	1	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	1	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
234	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
235	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₂²⋊C₄	C₄²×C₅₃	1	2	yes	—
243	C₄²	—	2	1	no	—
244	C₄²	—	4	2	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	4	1	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
249	C₂³	C₄×D₄×C₅₃	1	2	yes	—
250	Q₈	—	4	1	no	—
251	D₄	—	4	2	no	—
252	D₄	—	4	1	no	—
253	D₄	—	4	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
266	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
267	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
268	C₈	—	2	1	no	—
269	C₈	—	2	2	no	—
270	C₈	—	4	4	no	—
271	C₈	—	4	1	no	—
272	C₈	—	4	1	no	—
273	C₈	#99×C₅₃	1	1	yes	characteristic
274	C₈	#99×C₅₃	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	1	no	—
289	C₄	—	8	4	no	—
290	C₄	C₂×#99×C₅₃	1	1	yes	characteristic
291	C₄	C₅₃×#410	1	1	yes	characteristic
292	C₂	—	2	1	no	—
293	C₂	—	4	2	no	—
294	C₂	—	4	2	no	—
295	C₂	—	4	1	no	—
296	C₂	C₄×#99×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₅₃	#99
commutator	C₂³	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×#41	C₂³×C₅₃
socle	C₂×C₅₃	C₄×#99
prime radical	C₅₃×#525	C₂²
LCS	G ▷ C₂³ ▷ 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
3. C₅₃×#1757	C₂	semidirect
4. C₅₃×#1392	C₂	semidirect
5. C₅₃×#1391	C₂	semidirect
7. C₂³×C₈×C₅₃	C₄	semidirect
11. C₅₃×#424	C₄	semidirect
15. C₅₃×#407	C₄	semidirect
17. C₅₃×#405	C₄	semidirect
18. C₅₃×#352	C₄	semidirect
19. C₅₃×AΣL(1,𝔽₁₆)	C₄	semidirect
126. #12623	C₅₃	direct (N × H)
146. #1757	C₂×C₅₃	semidirect
147. #1392	C₂×C₅₃	semidirect
148. #1391	C₂×C₅₃	semidirect
155. C₂³×C₈	C₄×C₅₃	semidirect
159. #424	C₄×C₅₃	semidirect
163. #407	C₄×C₅₃	semidirect
165. #405	C₄×C₅₃	semidirect
166. #352	C₄×C₅₃	semidirect
167. AΣL(1,𝔽₁₆)	C₄×C₅₃	semidirect
169. C₅₃	#12623	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×#2929	2
C₅₃×#2120	2
C₅₃×#1757	2
C₅₃×#1392	2
C₅₃×#1391	2
#12623	53

Subgroups — C53×#12623 #27220135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12623 #27220135