Subgroups — C₅₃×#22237 #27229749

Order 13568.

694 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3135	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2961	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2932	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1994	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1962	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1929	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1866	C₂	1	4	yes	maximal, complemented
8	C₅₃×#1849	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1745	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#131×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#533	C₂²	1	1	yes	characteristic
12	C₅₃×#532	C₂²	1	1	yes	characteristic
13	C₅₃×#491	C₄	1	2	yes	complemented
14	C₅₃×#490	C₄	1	2	yes	complemented
15	C₅₃×#489	C₄	1	4	yes	complemented
16	C₅₃×#446	—	2	4	no	—
17	C₅₃×#435	—	2	4	no	—
18	C₂×#108×C₅₃	C₂²	1	4	yes	—
19	C₅₃×#424	C₂²	1	2	yes	—
20	C₅₃×#423	C₂²	1	1	yes	characteristic
21	C₅₃×#423	C₂²	1	1	yes	characteristic
22	C₂×#104×C₅₃	C₂²	1	4	yes	—
23	C₅₃×#416	C₂²	1	4	yes	—
24	C₅₃×#408	C₂²	1	2	yes	—
25	C₅₃×#405	C₂²	1	2	yes	—
26	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
28	C₄³×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#339	C₂²	1	4	yes	—
30	C₅₃×#336	C₂²	1	2	yes	—
31	C₅₃×#336	C₂²	1	2	yes	—
32	C₂×#41×C₅₃	C₂³	1	2	yes	—
33	#131×C₅₃	—	4	4	no	—
34	C₂×M₁₆×C₅₃	—	2	2	no	—
35	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
36	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₅₃	—	2	2	no	—
38	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
44	#125×C₅₃	C₂×C₄	1	2	yes	—
45	#124×C₅₃	C₂×C₄	1	2	yes	—
46	C₂²⋊Q₈×C₅₃	—	2	2	no	—
47	#121×C₅₃	—	2	2	no	—
48	C₄×Q₈×C₅₃	—	2	2	no	—
49	C₄×D₄×C₅₃	—	2	2	no	—
50	#117×C₅₃	C₂³	1	2	yes	—
51	C₂×C₄²×C₅₃	—	2	2	no	—
52	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	#108×C₅₃	—	4	4	no	—
55	C₄⋊C₈×C₅₃	—	2	4	no	—
56	C₄⋊C₈×C₅₃	—	2	2	no	—
57	C₄⋊C₈×C₅₃	—	2	2	no	—
58	#104×C₅₃	—	4	4	no	—
59	#103×C₅₃	C₂×C₄	1	8	yes	—
60	#102×C₅₃	C₂×C₄	1	8	yes	—
61	C₂²⋊C₈×C₅₃	—	2	4	no	—
62	#97×C₅₃	—	2	2	no	—
63	#97×C₅₃	C₂×C₄	1	2	yes	—
64	#97×C₅₃	C₂×C₄	1	2	yes	—
65	#97×C₅₃	D₄	1	4	yes	—
66	C₄×C₈×C₅₃	—	2	2	no	—
67	#41×C₅₃	—	2	4	no	—
68	#41×C₅₃	—	2	4	no	—
69	C₂×Q₈×C₅₃	—	2	2	no	—
70	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
71	C₂×D₄×C₅₃	—	2	2	no	—
72	C₂×D₄×C₅₃	—	2	2	no	—
73	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₅₃	—	2	2	no	—
75	C₂²×C₄×C₅₃	—	2	2	no	—
76	C₂²×C₄×C₅₃	#41	1	2	yes	—
77	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
78	M₁₆×C₅₃	—	2	4	no	—
79	M₁₆×C₅₃	—	2	4	no	—
80	M₁₆×C₅₃	—	4	4	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	—	4	4	no	—
88	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
91	C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₄⋊C₄×C₅₃	—	2	4	no	—
93	C₄⋊C₄×C₅₃	—	2	2	no	—
94	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
95	C₂²⋊C₄×C₅₃	—	2	2	no	—
96	C₂²⋊C₄×C₅₃	—	2	4	no	—
97	C₄²×C₅₃	—	2	2	no	—
98	C₄²×C₅₃	—	2	2	no	—
99	C₄²×C₅₃	—	2	2	no	—
100	C₄²×C₅₃	—	4	4	no	—
101	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₂×D₄	1	2	yes	—
104	C₂³×C₅₃	—	2	2	no	—
105	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
106	Q₈×C₅₃	—	2	4	no	—
107	Q₈×C₅₃	—	8	2	no	—
108	D₄×C₅₃	—	2	4	no	—
109	D₄×C₅₃	—	4	4	no	—
110	D₄×C₅₃	—	8	2	no	—
111	C₂×C₄×C₅₃	—	2	4	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	4	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	—	4	2	no	—
119	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
124	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
125	C₈×C₅₃	—	2	4	no	—
126	C₈×C₅₃	—	2	2	no	—
127	C₈×C₅₃	—	2	2	no	—
128	C₈×C₅₃	—	4	2	no	—
129	C₈×C₅₃	—	4	2	no	—
130	#22237	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₅₃	—	2	2	no	—
132	C₂²×C₅₃	—	4	2	no	—
133	C₂²×C₅₃	—	4	4	no	—
134	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂²×C₅₃	#542	1	2	yes	—
136	C₄×C₅₃	—	2	2	no	—
137	C₄×C₅₃	—	2	2	no	—
138	C₄×C₅₃	—	4	4	no	—
139	C₄×C₅₃	—	4	2	no	—
140	C₄×C₅₃	—	4	4	no	—
141	C₄×C₅₃	#520	1	2	yes	—
142	C₄×C₅₃	#535	1	1	yes	characteristic
143	C₄×C₅₃	#536	1	1	yes	characteristic
144	#3135	C₂×C₅₃	1	1	yes	characteristic
145	#2961	C₂×C₅₃	1	1	yes	characteristic, complemented
146	#2932	C₂×C₅₃	1	1	yes	characteristic
147	#1994	C₂×C₅₃	1	1	yes	characteristic, complemented
148	#1962	C₂×C₅₃	1	1	yes	characteristic
149	#1929	C₂×C₅₃	1	4	yes	complemented
150	#1866	C₂×C₅₃	1	4	yes	complemented
151	#1849	C₂×C₅₃	1	1	yes	characteristic, complemented
152	#1745	C₂×C₅₃	1	1	yes	characteristic, complemented
153	C₂×C₅₃	—	2	2	no	—
154	C₂×C₅₃	—	8	2	no	—
155	C₂×C₅₃	#2326	1	1	yes	characteristic
156	C₂×C₅₃	#2956	1	2	yes	—
157	C₂×#131	C₂²×C₅₃	1	2	yes	—
158	#533	C₂²×C₅₃	1	1	yes	characteristic
159	#532	C₂²×C₅₃	1	1	yes	characteristic
160	#491	C₄×C₅₃	1	2	yes	complemented
161	#490	C₄×C₅₃	1	2	yes	complemented
162	#489	C₄×C₅₃	1	4	yes	complemented
163	#446	—	2	4	no	—
164	#435	—	2	4	no	—
165	C₂×#108	C₂²×C₅₃	1	4	yes	—
166	#424	C₂²×C₅₃	1	2	yes	—
167	#423	C₂²×C₅₃	1	1	yes	characteristic
168	#423	C₂²×C₅₃	1	1	yes	characteristic
169	C₂×#104	C₂²×C₅₃	1	4	yes	—
170	#416	C₂²×C₅₃	1	4	yes	—
171	#408	C₂²×C₅₃	1	2	yes	—
172	#405	C₂²×C₅₃	1	2	yes	—
173	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
174	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
175	C₄³	C₂²×C₅₃	1	1	yes	characteristic
176	#339	C₂²×C₅₃	1	4	yes	—
177	#336	C₂²×C₅₃	1	2	yes	—
178	#336	C₂²×C₅₃	1	2	yes	—
179	C₅₃	#22237	1	1	yes	characteristic, direct product factor, Sylow
180	C₂×#41	C₂³×C₅₃	1	2	yes	—
181	#131	—	4	4	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
184	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
187	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
189	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
190	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
191	#127	C₂×C₄×C₅₃	1	1	yes	characteristic
192	#125	C₂×C₄×C₅₃	1	2	yes	—
193	#124	C₂×C₄×C₅₃	1	2	yes	—
194	C₂²⋊Q₈	—	2	2	no	—
195	#121	—	2	2	no	—
196	C₄×Q₈	—	2	2	no	—
197	C₄×D₄	—	2	2	no	—
198	#117	C₂³×C₅₃	1	2	yes	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
201	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
202	#108	—	4	4	no	—
203	C₄⋊C₈	—	2	4	no	—
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	2	2	no	—
206	#104	—	4	4	no	—
207	#103	C₂×C₄×C₅₃	1	8	yes	—
208	#102	C₂×C₄×C₅₃	1	8	yes	—
209	C₂²⋊C₈	—	2	4	no	—
210	#97	—	2	2	no	—
211	#97	C₂×C₄×C₅₃	1	2	yes	—
212	#97	C₂×C₄×C₅₃	1	2	yes	—
213	#97	D₄×C₅₃	1	4	yes	—
214	C₄×C₈	—	2	2	no	—
215	#41	—	2	4	no	—
216	#41	—	2	4	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	#41×C₅₃	1	2	yes	—
225	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	4	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	4	4	no	—
236	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
237	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
238	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	—	2	4	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	4	4	no	—
249	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
250	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
251	C₄²	C₂×D₄×C₅₃	1	2	yes	—
252	C₂³	—	2	2	no	—
253	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
254	Q₈	—	2	4	no	—
255	Q₈	—	8	2	no	—
256	D₄	—	2	4	no	—
257	D₄	—	4	4	no	—
258	D₄	—	8	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
268	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
269	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
270	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
271	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
272	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
273	C₈	—	2	4	no	—
274	C₈	—	2	2	no	—
275	C₈	—	2	2	no	—
276	C₈	—	4	2	no	—
277	C₈	—	4	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	4	4	no	—
280	C₂²	—	4	2	no	—
281	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
282	C₂²	C₅₃×#542	1	2	yes	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	4	4	no	—
286	C₄	—	4	4	no	—
287	C₄	—	4	2	no	—
288	C₄	C₅₃×#520	1	2	yes	—
289	C₄	C₅₃×#535	1	1	yes	characteristic
290	C₄	C₅₃×#536	1	1	yes	characteristic
291	C₂	—	2	2	no	—
292	C₂	—	8	2	no	—
293	C₂	C₅₃×#2326	1	1	yes	characteristic
294	C₂	C₅₃×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2961	C₂	semidirect
4. C₅₃×#1994	C₂	semidirect
6. C₅₃×#1929	C₂	semidirect
7. C₅₃×#1866	C₂	semidirect
8. C₅₃×#1849	C₂	semidirect
9. C₅₃×#1745	C₂	semidirect
13. C₅₃×#491	C₄	semidirect
14. C₅₃×#490	C₄	semidirect
15. C₅₃×#489	C₄	semidirect
130. #22237	C₅₃	direct (N × H)
145. #2961	C₂×C₅₃	semidirect
147. #1994	C₂×C₅₃	semidirect
149. #1929	C₂×C₅₃	semidirect
150. #1866	C₂×C₅₃	semidirect
151. #1849	C₂×C₅₃	semidirect
152. #1745	C₂×C₅₃	semidirect
160. #491	C₄×C₅₃	semidirect
161. #490	C₄×C₅₃	semidirect
162. #489	C₄×C₅₃	semidirect
179. C₅₃	#22237	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×#3135	2
C₅₃×#2961	2
C₅₃×#2932	2
C₅₃×#1994	2
C₅₃×#1962	2
C₅₃×#1929	2
C₅₃×#1866	2
C₅₃×#1849	2
C₅₃×#1745	2
#22237	53

Subgroups — C53×#22237 #27229749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22237 #27229749