Subgroups — C₅₃×#21307 #27228819

Order 13568.

1106 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3458	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2873	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2296	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1986	C₂	1	8	yes	maximal, complemented
5	C₅₃×#1733	C₂	1	2	yes	maximal
6	C₂×C₅₃×#339	C₂	1	2	yes	maximal
7	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
8	C₅₃×#544	—	2	2	no	—
9	C₅₃×#544	C₄	1	8	yes	complemented
10	C₅₃×#543	—	2	2	no	—
11	C₅₃×#533	—	2	4	no	—
12	C₅₃×#533	C₂²	1	4	yes	—
13	C₂×#117×C₅₃	C₂²	1	2	yes	—
14	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#405	—	2	4	no	—
18	C₄×M₁₆×C₅₃	C₂²	1	4	yes	—
19	C₅₃×#389	—	2	2	no	—
20	C₅₃×#385	—	2	2	no	—
21	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
22	C₅₃×#339	C₂²	1	8	yes	—
23	C₅₃×#339	C₂²	1	8	yes	—
24	C₂³×C₄×C₅₃	—	2	1	no	—
25	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₅₃	—	2	2	no	—
27	C₂×M₁₆×C₅₃	—	2	4	no	—
28	C₂×M₁₆×C₅₃	Q₈	1	8	yes	complemented
29	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
30	C₂²×C₈×C₅₃	—	2	2	no	—
31	C₄⋊Q₈×C₅₃	—	2	8	no	—
32	C₄⋊Q₈×C₅₃	C₂×C₄	1	8	yes	complemented
33	#125×C₅₃	—	2	8	no	—
34	#125×C₅₃	C₂×C₄	1	8	yes	complemented
35	C₂²⋊Q₈×C₅₃	—	2	16	no	—
36	C₂²⋊Q₈×C₅₃	—	2	8	no	—
37	C₄×Q₈×C₅₃	—	2	8	no	—
38	#117×C₅₃	—	2	2	no	—
39	#117×C₅₃	—	2	2	no	—
40	#117×C₅₃	—	2	4	no	—
41	#117×C₅₃	C₂×C₄	1	4	yes	—
42	#117×C₅₃	C₂×C₄	1	4	yes	—
43	#117×C₅₃	D₄	1	4	yes	—
44	#117×C₅₃	C₂³	1	4	yes	—
45	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
46	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
47	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
49	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
51	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
52	C₂×C₄²×C₅₃	—	2	2	no	—
53	C₂×C₄²×C₅₃	—	2	1	no	—
54	C₂×C₄²×C₅₃	—	2	4	no	—
55	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
56	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
57	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
58	#97×C₅₃	—	2	4	no	—
59	C₄×C₈×C₅₃	—	2	4	no	—
60	#95×C₅₃	—	2	4	no	—
61	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×Q₈×C₅₃	—	2	8	no	—
63	C₂²×C₄×C₅₃	—	2	2	no	—
64	C₂²×C₄×C₅₃	—	2	4	no	—
65	C₂²×C₄×C₅₃	—	2	2	no	—
66	C₂²×C₄×C₅₃	—	4	4	no	—
67	C₂²×C₄×C₅₃	—	4	2	no	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
73	C₂²×C₄×C₅₃	#41	1	4	yes	—
74	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
75	M₁₆×C₅₃	—	2	8	no	—
76	M₁₆×C₅₃	—	2	4	no	—
77	M₁₆×C₅₃	C₂×Q₈	1	8	yes	—
78	C₂×C₈×C₅₃	—	2	8	no	—
79	C₂×C₈×C₅₃	—	2	4	no	—
80	C₄⋊C₄×C₅₃	—	2	4	no	—
81	C₄⋊C₄×C₅₃	—	2	8	no	—
82	C₄⋊C₄×C₅₃	—	2	16	no	—
83	C₄⋊C₄×C₅₃	—	4	8	no	—
84	C₄⋊C₄×C₅₃	—	4	2	no	—
85	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	8	yes	—
86	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
87	C₂²⋊C₄×C₅₃	—	2	4	no	—
88	C₂²⋊C₄×C₅₃	—	2	4	no	—
89	C₂²⋊C₄×C₅₃	—	4	2	no	—
90	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	8	yes	—
91	C₄²×C₅₃	—	2	4	no	—
92	C₄²×C₅₃	—	2	4	no	—
93	C₄²×C₅₃	—	2	4	no	—
94	C₄²×C₅₃	—	2	4	no	—
95	C₄²×C₅₃	—	4	2	no	—
96	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
97	C₄²×C₅₃	C₂×D₄	1	4	yes	—
98	C₂³×C₅₃	—	4	2	no	—
99	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂³×C₅₃	#124	1	2	yes	—
101	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
102	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
103	Q₈×C₅₃	—	8	8	no	—
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	—	2	8	no	—
106	C₂×C₄×C₅₃	—	4	4	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	—	4	8	no	—
109	C₂×C₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	—	4	4	no	—
111	C₂×C₄×C₅₃	—	4	2	no	—
112	C₂×C₄×C₅₃	—	8	4	no	—
113	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	8	yes	—
114	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
115	C₂×C₄×C₅₃	C₄×Q₈	1	8	yes	—
116	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
117	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
118	C₂×C₄×C₅₃	#124	1	2	yes	—
119	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
120	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
123	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
124	C₈×C₅₃	—	2	8	no	—
125	#21307	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	—	2	4	no	—
128	C₂²×C₅₃	—	4	2	no	—
129	C₂²×C₅₃	—	4	2	no	—
130	C₂²×C₅₃	#391	1	4	yes	—
131	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₅₃	C₂×#124	1	1	yes	characteristic
133	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
134	C₄×C₅₃	—	2	8	no	—
135	C₄×C₅₃	—	4	4	no	—
136	C₄×C₅₃	—	4	8	no	—
137	C₄×C₅₃	—	8	2	no	—
138	C₄×C₅₃	#391	1	4	yes	—
139	C₄×C₅₃	C₂×C₄×Q₈	1	2	yes	—
140	C₄×C₅₃	C₂×C₂²⋊Q₈	1	2	yes	—
141	#3458	C₂×C₅₃	1	1	yes	characteristic
142	#2873	C₂×C₅₃	1	1	yes	characteristic
143	#2296	C₂×C₅₃	1	1	yes	characteristic
144	#1986	C₂×C₅₃	1	8	yes	complemented
145	#1733	C₂×C₅₃	1	2	yes	—
146	C₂×#339	C₂×C₅₃	1	2	yes	—
147	C₂×C₅₃	—	2	1	no	—
148	C₂×C₅₃	—	2	1	no	—
149	C₂×C₅₃	—	2	4	no	—
150	C₂×C₅₃	C₂×#391	1	1	yes	characteristic
151	C₂×C₅₃	#2901	1	2	yes	—
152	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
153	#544	—	2	2	no	—
154	#544	C₄×C₅₃	1	8	yes	complemented
155	#543	—	2	2	no	—
156	#533	—	2	4	no	—
157	#533	C₂²×C₅₃	1	4	yes	—
158	C₂×#117	C₂²×C₅₃	1	2	yes	—
159	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
160	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
161	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
162	#405	—	2	4	no	—
163	C₄×M₁₆	C₂²×C₅₃	1	4	yes	—
164	#389	—	2	2	no	—
165	#385	—	2	2	no	—
166	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
167	#339	C₂²×C₅₃	1	8	yes	—
168	#339	C₂²×C₅₃	1	8	yes	—
169	C₅₃	#21307	1	1	yes	characteristic, direct product factor, Sylow
170	C₂³×C₄	—	2	1	no	—
171	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	4	no	—
174	C₂×M₁₆	Q₈×C₅₃	1	8	yes	complemented
175	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₄⋊Q₈	—	2	8	no	—
178	C₄⋊Q₈	C₂×C₄×C₅₃	1	8	yes	complemented
179	#125	—	2	8	no	—
180	#125	C₂×C₄×C₅₃	1	8	yes	complemented
181	C₂²⋊Q₈	—	2	8	no	—
182	C₂²⋊Q₈	—	2	16	no	—
183	C₄×Q₈	—	2	8	no	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	—	2	4	no	—
187	#117	C₂×C₄×C₅₃	1	4	yes	—
188	#117	C₂×C₄×C₅₃	1	4	yes	—
189	#117	D₄×C₅₃	1	4	yes	—
190	#117	C₂³×C₅₃	1	4	yes	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
194	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₂²⋊C₄	—	2	1	no	—
197	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	—	2	4	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
202	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
203	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
204	#97	—	2	4	no	—
205	C₄×C₈	—	2	4	no	—
206	#95	—	2	4	no	—
207	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
208	C₂×Q₈	—	2	8	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
216	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
218	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
219	C₂²×C₄	#41×C₅₃	1	4	yes	—
220	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
221	M₁₆	—	2	8	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	C₂×Q₈×C₅₃	1	8	yes	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	2	4	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	8	no	—
228	C₄⋊C₄	—	2	16	no	—
229	C₄⋊C₄	—	4	8	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	8	yes	—
232	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	8	yes	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	2	4	no	—
241	C₄²	—	4	2	no	—
242	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
243	C₄²	C₂×D₄×C₅₃	1	4	yes	—
244	C₂³	—	4	2	no	—
245	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
246	C₂³	#124×C₅₃	1	2	yes	—
247	C₂³	C₄⋊Q₈×C₅₃	1	2	yes	—
248	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
249	Q₈	—	8	8	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	8	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	8	4	no	—
259	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	8	yes	—
260	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
261	C₂×C₄	C₄×Q₈×C₅₃	1	8	yes	—
262	C₂×C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
263	C₂×C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
264	C₂×C₄	#124×C₅₃	1	2	yes	—
265	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
266	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
267	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
268	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
269	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
270	C₈	—	2	8	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	2	no	—
275	C₂²	C₅₃×#391	1	4	yes	—
276	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
277	C₂²	C₂×#124×C₅₃	1	1	yes	characteristic
278	C₂²	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
279	C₄	—	2	8	no	—
280	C₄	—	4	4	no	—
281	C₄	—	4	8	no	—
282	C₄	—	8	2	no	—
283	C₄	C₅₃×#391	1	4	yes	—
284	C₄	C₂×C₄×Q₈×C₅₃	1	2	yes	—
285	C₄	C₂×C₂²⋊Q₈×C₅₃	1	2	yes	—
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	2	4	no	—
289	C₂	C₂×C₅₃×#391	1	1	yes	characteristic
290	C₂	C₅₃×#2901	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	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×#117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1986	C₂	semidirect
9. C₅₃×#544	C₄	semidirect
28. C₂×M₁₆×C₅₃	Q₈	semidirect
32. C₄⋊Q₈×C₅₃	C₂×C₄	semidirect
34. #125×C₅₃	C₂×C₄	semidirect
125. #21307	C₅₃	direct (N × H)
144. #1986	C₂×C₅₃	semidirect
154. #544	C₄×C₅₃	semidirect
169. C₅₃	#21307	direct (N × H)
174. C₂×M₁₆	Q₈×C₅₃	semidirect
178. C₄⋊Q₈	C₂×C₄×C₅₃	semidirect
180. #125	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×#3458	2
C₅₃×#2873	2
C₅₃×#2296	2
C₅₃×#1986	2
C₅₃×#1733	2
C₂×C₅₃×#339	2
#21307	53

Subgroups — C53×#21307 #27228819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21307 #27228819