Subgroups — C₅₃×#8189 #27215701

Order 13568.

420 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2922	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1476	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1476	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#433	—	2	2	no	—
10	C₅₃×#431	—	2	1	no	—
11	C₅₃×#431	—	2	1	no	—
12	C₅₃×#431	C₂²	1	1	yes	characteristic
13	C₅₃×#431	C₂²	1	1	yes	characteristic
14	C₅₃×#424	C₄	1	2	yes	—
15	C₅₃×#408	C₄	1	2	yes	—
16	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×Q₈⋊C₈	C₄	1	2	yes	—
21	C₅₃×Q₈⋊C₈	C₄	1	2	yes	—
22	C₅₃×D₄⋊C₈	C₄	1	2	yes	—
23	C₅₃×D₄⋊C₈	C₄	1	2	yes	—
24	C₅₃×#322	—	2	2	no	—
25	C₅₃×#322	—	2	1	no	—
26	C₈²×C₅₃	—	2	1	no	—
27	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₅₃	—	2	1	no	—
30	C₂²×C₈×C₅₃	—	2	1	no	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×D₄×C₅₃	—	2	1	no	—
35	C₄×D₄×C₅₃	—	2	1	no	—
36	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	#117×C₅₃	—	2	1	no	—
39	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	—	2	1	no	—
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₂²⋊C₈×C₅₃	—	2	2	no	—
47	C₂²⋊C₈×C₅₃	—	2	2	no	—
48	C₂²⋊C₈×C₅₃	—	2	1	no	—
49	C₂²⋊C₈×C₅₃	—	2	1	no	—
50	#97×C₅₃	—	2	1	no	—
51	C₄×C₈×C₅₃	—	2	1	no	—
52	C₄×C₈×C₅₃	—	2	1	no	—
53	C₄×C₈×C₅₃	—	2	1	no	—
54	C₄×C₈×C₅₃	—	2	1	no	—
55	C₄×C₈×C₅₃	—	2	1	no	—
56	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₅₃	D₄	1	2	yes	—
59	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
61	#41×C₅₃	—	2	4	no	—
62	C₂×Q₈×C₅₃	C₄²	1	1	yes	characteristic
63	C₂×D₄×C₅₃	—	2	1	no	—
64	C₂×D₄×C₅₃	C₄²	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	—	2	1	no	—
66	C₂²×C₄×C₅₃	—	2	1	no	—
67	C₂²×C₄×C₅₃	—	2	1	no	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
71	M₁₆×C₅₃	—	2	4	no	—
72	C₂×C₈×C₅₃	—	2	4	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	4	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	—	4	2	no	—
80	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
82	C₄⋊C₄×C₅₃	—	2	1	no	—
83	C₄⋊C₄×C₅₃	—	2	1	no	—
84	C₄⋊C₄×C₅₃	C₄²	1	1	yes	characteristic
85	C₄⋊C₄×C₅₃	C₄²	1	1	yes	characteristic
86	C₂²⋊C₄×C₅₃	—	2	1	no	—
87	C₂²⋊C₄×C₅₃	—	2	1	no	—
88	C₂²⋊C₄×C₅₃	—	2	1	no	—
89	C₄²×C₅₃	—	2	1	no	—
90	C₄²×C₅₃	—	2	1	no	—
91	C₄²×C₅₃	—	2	1	no	—
92	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂³×C₅₃	—	2	1	no	—
97	C₂³×C₅₃	#117	1	1	yes	characteristic
98	Q₈×C₅₃	—	2	2	no	—
99	D₄×C₅₃	—	2	2	no	—
100	D₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#131	1	4	yes	—
116	C₈×C₅₃	—	2	4	no	—
117	C₈×C₅₃	—	2	2	no	—
118	C₈×C₅₃	—	2	2	no	—
119	C₈×C₅₃	—	4	2	no	—
120	C₈×C₅₃	—	4	2	no	—
121	C₈×C₅₃	—	4	2	no	—
122	C₈×C₅₃	—	4	2	no	—
123	#8189	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	2	1	no	—
126	C₂²×C₅₃	—	4	1	no	—
127	C₂²×C₅₃	—	4	1	no	—
128	C₂²×C₅₃	—	4	1	no	—
129	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₅₃	#433	1	2	yes	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	4	1	no	—
134	C₄×C₅₃	—	4	1	no	—
135	C₄×C₅₃	—	4	1	no	—
136	C₄×C₅₃	#405	1	2	yes	—
137	C₄×C₅₃	#408	1	2	yes	—
138	C₄×C₅₃	#443	1	2	yes	—
139	C₄×C₅₃	#444	1	2	yes	—
140	#2922	C₂×C₅₃	1	1	yes	characteristic
141	#1476	C₂×C₅₃	1	1	yes	characteristic
142	#1476	C₂×C₅₃	1	1	yes	characteristic
143	#1450	C₂×C₅₃	1	1	yes	characteristic, complemented
144	C₂×#322	C₂×C₅₃	1	1	yes	characteristic, complemented
145	#1278	C₂×C₅₃	1	1	yes	characteristic, complemented
146	#1278	C₂×C₅₃	1	1	yes	characteristic, complemented
147	C₂×C₅₃	—	2	2	no	—
148	C₂×C₅₃	—	8	1	no	—
149	C₂×C₅₃	#1750	1	1	yes	characteristic
150	C₂×C₅₃	#1764	1	1	yes	characteristic
151	C₂×C₅₃	#1765	1	1	yes	characteristic
152	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
153	#433	—	2	2	no	—
154	#431	—	2	1	no	—
155	#431	—	2	1	no	—
156	#431	C₂²×C₅₃	1	1	yes	characteristic
157	#431	C₂²×C₅₃	1	1	yes	characteristic
158	#424	C₄×C₅₃	1	2	yes	—
159	#408	C₄×C₅₃	1	2	yes	—
160	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
161	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
164	Q₈⋊C₈	C₄×C₅₃	1	2	yes	—
165	Q₈⋊C₈	C₄×C₅₃	1	2	yes	—
166	D₄⋊C₈	C₄×C₅₃	1	2	yes	—
167	D₄⋊C₈	C₄×C₅₃	1	2	yes	—
168	#322	—	2	2	no	—
169	#322	—	2	1	no	—
170	C₈²	—	2	1	no	—
171	C₅₃	#8189	1	1	yes	characteristic, direct product factor, Sylow
172	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
173	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
177	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
178	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
179	C₄×D₄	—	2	1	no	—
180	C₄×D₄	—	2	1	no	—
181	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
182	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
183	#117	—	2	1	no	—
184	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
185	C₂×C₄²	—	2	1	no	—
186	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	—	2	2	no	—
189	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
190	C₄⋊C₈	C₂×C₄×C₅₃	1	2	yes	—
191	C₂²⋊C₈	—	2	1	no	—
192	C₂²⋊C₈	—	2	1	no	—
193	C₂²⋊C₈	—	2	2	no	—
194	C₂²⋊C₈	—	2	2	no	—
195	#97	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	—	2	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	2	1	no	—
201	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
202	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
203	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
204	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
205	C₄×C₈	D₄×C₅₃	1	2	yes	—
206	#41	—	2	4	no	—
207	C₂×Q₈	C₄²×C₅₃	1	1	yes	characteristic
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	C₄²×C₅₃	1	1	yes	characteristic
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
214	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
215	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
216	M₁₆	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
226	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	C₄²×C₅₃	1	1	yes	characteristic
230	C₄⋊C₄	C₄²×C₅₃	1	1	yes	characteristic
231	C₂²⋊C₄	—	2	1	no	—
232	C₂²⋊C₄	—	2	1	no	—
233	C₂²⋊C₄	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	2	1	no	—
237	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
238	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
239	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
240	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
241	C₂³	—	2	1	no	—
242	C₂³	#117×C₅₃	1	1	yes	characteristic
243	Q₈	—	2	2	no	—
244	D₄	—	2	2	no	—
245	D₄	—	4	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
255	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
256	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
257	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
258	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
259	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
260	C₂×C₄	#131×C₅₃	1	4	yes	—
261	C₈	—	2	4	no	—
262	C₈	—	2	2	no	—
263	C₈	—	2	2	no	—
264	C₈	—	4	2	no	—
265	C₈	—	4	2	no	—
266	C₈	—	4	2	no	—
267	C₈	—	4	2	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
274	C₂²	C₅₃×#433	1	2	yes	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	C₅₃×#405	1	2	yes	—
281	C₄	C₅₃×#408	1	2	yes	—
282	C₄	C₅₃×#443	1	2	yes	—
283	C₄	C₅₃×#444	1	2	yes	—
284	C₂	—	2	2	no	—
285	C₂	—	8	1	no	—
286	C₂	C₅₃×#1750	1	1	yes	characteristic
287	C₂	C₅₃×#1764	1	1	yes	characteristic
288	C₂	C₅₃×#1765	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₅₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1450	C₂	semidirect
5. C₂×C₅₃×#322	C₂	semidirect
6. C₅₃×#1278	C₂	semidirect
7. C₅₃×#1278	C₂	semidirect
123. #8189	C₅₃	direct (N × H)
143. #1450	C₂×C₅₃	semidirect
144. C₂×#322	C₂×C₅₃	semidirect
145. #1278	C₂×C₅₃	semidirect
146. #1278	C₂×C₅₃	semidirect
171. C₅₃	#8189	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×#2922	2
C₅₃×#1476	2
C₅₃×#1450	2
C₂×C₅₃×#322	2
C₅₃×#1278	2
#8189	53

Subgroups — C53×#8189 #27215701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8189 #27215701