Subgroups — C₅₃×#18086 #27225598

Order 13568.

572 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×#3170	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2986	C₂	1	2	yes	maximal
3	C₅₃×#1591	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1587	C₂	1	4	yes	maximal, complemented
5	C₈×Q₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1485	C₂	1	2	yes	maximal
8	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#533	C₂²	1	2	yes	—
10	C₅₃×#500	C₄	1	1	yes	characteristic
11	C₅₃×#494	C₄	1	1	yes	characteristic
12	C₅₃×#487	C₄	1	2	yes	—
13	C₅₃×#467	C₄	1	2	yes	—
14	C₅₃×#467	C₄	1	2	yes	—
15	C₅₃×#446	—	2	2	no	—
16	C₅₃×#446	C₂²	1	4	yes	—
17	C₈×Q₈×C₅₃	—	2	2	no	—
18	C₈×Q₈×C₅₃	C₂²	1	4	yes	—
19	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
20	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#433	—	2	4	no	—
22	C₅₃×#431	C₂²	1	2	yes	—
23	C₅₃×#431	C₂²	1	1	yes	characteristic
24	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#335	C₂²	1	1	yes	characteristic
27	C₅₃×#335	C₂²	1	1	yes	characteristic
28	C₅₃×#334	C₂²	1	2	yes	—
29	C₅₃×Q₈⋊C₈	C₂²	1	4	yes	—
30	C₅₃×Q₈⋊C₈	C₂²	1	4	yes	—
31	C₅₃×#322	C₂²	1	2	yes	—
32	C₈²×C₅₃	C₂²	1	2	yes	—
33	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×M₁₆×C₅₃	—	2	1	no	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
40	#125×C₅₃	—	2	2	no	—
41	C₂²⋊Q₈×C₅₃	—	2	2	no	—
42	C₂²⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
43	C₄×Q₈×C₅₃	—	2	2	no	—
44	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
45	#117×C₅₃	—	2	2	no	—
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	#107×C₅₃	C₂×C₄	1	2	yes	—
48	#107×C₅₃	C₂×C₄	1	2	yes	—
49	C₄⋊C₈×C₅₃	—	2	4	no	—
50	C₄⋊C₈×C₅₃	—	2	4	no	—
51	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
54	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
55	#103×C₅₃	C₂×C₄	1	4	yes	—
56	#103×C₅₃	C₂×C₄	1	4	yes	—
57	C₂²⋊C₈×C₅₃	—	2	2	no	—
58	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
59	#97×C₅₃	—	2	4	no	—
60	#97×C₅₃	—	2	1	no	—
61	C₄×C₈×C₅₃	—	2	4	no	—
62	C₄×C₈×C₅₃	—	2	2	no	—
63	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂³	1	2	yes	—
66	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₅₃	C₂³	1	2	yes	—
69	C₂×Q₈×C₅₃	—	2	2	no	—
70	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
74	Q₁₆×C₅₃	—	2	4	no	—
75	Q₁₆×C₅₃	—	4	2	no	—
76	M₁₆×C₅₃	—	2	4	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	—	2	2	no	—
81	C₂×C₈×C₅₃	—	4	1	no	—
82	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
88	C₄⋊C₄×C₅₃	—	2	2	no	—
89	C₄⋊C₄×C₅₃	—	2	2	no	—
90	C₄⋊C₄×C₅₃	—	2	2	no	—
91	C₄⋊C₄×C₅₃	—	2	4	no	—
92	C₄⋊C₄×C₅₃	—	2	4	no	—
93	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
94	C₂²⋊C₄×C₅₃	—	2	4	no	—
95	C₄²×C₅₃	—	2	4	no	—
96	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₅₃	#41	1	1	yes	characteristic
99	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
100	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
101	Q₈×C₅₃	—	2	4	no	—
102	Q₈×C₅₃	—	2	4	no	—
103	Q₈×C₅₃	—	4	4	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	4	no	—
110	C₂×C₄×C₅₃	—	2	4	no	—
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	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
117	C₈×C₅₃	—	2	4	no	—
118	C₈×C₅₃	—	2	1	no	—
119	C₈×C₅₃	—	2	1	no	—
120	C₈×C₅₃	—	2	1	no	—
121	C₈×C₅₃	—	2	2	no	—
122	C₈×C₅₃	—	2	2	no	—
123	C₈×C₅₃	—	4	2	no	—
124	C₈×C₅₃	—	4	2	no	—
125	C₈×C₅₃	#131	1	2	yes	—
126	#18086	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₅₃	—	2	1	no	—
128	C₂²×C₅₃	—	2	1	no	—
129	C₂²×C₅₃	—	2	1	no	—
130	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	4	4	no	—
135	C₄×C₅₃	—	4	4	no	—
136	C₄×C₅₃	#443	1	2	yes	—
137	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
138	C₄×C₅₃	#568	1	1	yes	characteristic
139	C₄×C₅₃	#578	1	1	yes	characteristic
140	C₄×C₅₃	#578	1	1	yes	characteristic
141	#3170	C₂×C₅₃	1	1	yes	characteristic
142	#2986	C₂×C₅₃	1	2	yes	—
143	#1591	C₂×C₅₃	1	2	yes	complemented
144	#1587	C₂×C₅₃	1	4	yes	complemented
145	C₈×Q₁₆	C₂×C₅₃	1	2	yes	complemented
146	#1564	C₂×C₅₃	1	1	yes	characteristic
147	#1485	C₂×C₅₃	1	2	yes	—
148	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic
149	C₂×C₅₃	—	4	1	no	—
150	C₂×C₅₃	#2935	1	1	yes	characteristic
151	C₂×C₅₃	#2950	1	1	yes	characteristic
152	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
153	#533	C₂²×C₅₃	1	2	yes	—
154	#500	C₄×C₅₃	1	1	yes	characteristic
155	#494	C₄×C₅₃	1	1	yes	characteristic
156	#487	C₄×C₅₃	1	2	yes	—
157	#467	C₄×C₅₃	1	2	yes	—
158	#467	C₄×C₅₃	1	2	yes	—
159	#446	—	2	2	no	—
160	#446	C₂²×C₅₃	1	4	yes	—
161	C₈×Q₈	—	2	2	no	—
162	C₈×Q₈	C₂²×C₅₃	1	4	yes	—
163	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	—
164	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	—
165	#433	—	2	4	no	—
166	#431	C₂²×C₅₃	1	2	yes	—
167	#431	C₂²×C₅₃	1	1	yes	characteristic
168	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
169	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
170	#335	C₂²×C₅₃	1	1	yes	characteristic
171	#335	C₂²×C₅₃	1	1	yes	characteristic
172	#334	C₂²×C₅₃	1	2	yes	—
173	Q₈⋊C₈	C₂²×C₅₃	1	4	yes	—
174	Q₈⋊C₈	C₂²×C₅₃	1	4	yes	—
175	#322	C₂²×C₅₃	1	2	yes	—
176	C₈²	C₂²×C₅₃	1	2	yes	—
177	C₅₃	#18086	1	1	yes	characteristic, direct product factor, Sylow
178	C₂×Q₁₆	C₂×C₄×C₅₃	1	2	yes	—
179	C₂×Q₁₆	C₂×C₄×C₅₃	1	2	yes	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
182	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
183	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
184	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
185	#125	—	2	2	no	—
186	C₂²⋊Q₈	—	2	2	no	—
187	C₂²⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
188	C₄×Q₈	—	2	2	no	—
189	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
190	#117	—	2	2	no	—
191	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
192	#107	C₂×C₄×C₅₃	1	2	yes	—
193	#107	C₂×C₄×C₅₃	1	2	yes	—
194	C₄⋊C₈	—	2	4	no	—
195	C₄⋊C₈	—	2	4	no	—
196	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
197	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
198	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
199	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
200	#103	C₂×C₄×C₅₃	1	4	yes	—
201	#103	C₂×C₄×C₅₃	1	4	yes	—
202	C₂²⋊C₈	—	2	2	no	—
203	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
204	#97	—	2	4	no	—
205	#97	—	2	1	no	—
206	C₄×C₈	—	2	4	no	—
207	C₄×C₈	—	2	2	no	—
208	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
209	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
210	C₄×C₈	C₂³×C₅₃	1	2	yes	—
211	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
212	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
213	C₄×C₈	C₂³×C₅₃	1	2	yes	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
216	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
218	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
219	Q₁₆	—	2	4	no	—
220	Q₁₆	—	4	2	no	—
221	M₁₆	—	2	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
228	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
229	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
230	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
231	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
232	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
239	C₂²⋊C₄	—	2	4	no	—
240	C₄²	—	2	4	no	—
241	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
242	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
243	C₄²	#41×C₅₃	1	1	yes	characteristic
244	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
245	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
246	Q₈	—	2	4	no	—
247	Q₈	—	2	4	no	—
248	Q₈	—	4	4	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
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	C₂³×C₄×C₅₃	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
261	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
262	C₈	—	2	4	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	2	no	—
267	C₈	—	2	2	no	—
268	C₈	—	4	2	no	—
269	C₈	—	4	2	no	—
270	C₈	#131×C₅₃	1	2	yes	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	4	4	no	—
279	C₄	—	4	4	no	—
280	C₄	C₅₃×#443	1	2	yes	—
281	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
282	C₄	C₅₃×#568	1	1	yes	characteristic
283	C₄	C₅₃×#578	1	1	yes	characteristic
284	C₄	C₅₃×#578	1	1	yes	characteristic
285	C₂	—	4	1	no	—
286	C₂	C₅₃×#2935	1	1	yes	characteristic
287	C₂	C₅₃×#2950	1	1	yes	characteristic
288	C₂	C₂×C₅₃×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₅₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1591	C₂	semidirect
4. C₅₃×#1587	C₂	semidirect
5. C₈×Q₁₆×C₅₃	C₂	semidirect
126. #18086	C₅₃	direct (N × H)
143. #1591	C₂×C₅₃	semidirect
144. #1587	C₂×C₅₃	semidirect
145. C₈×Q₁₆	C₂×C₅₃	semidirect
177. C₅₃	#18086	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×#3170	2
C₅₃×#2986	2
C₅₃×#1591	2
C₅₃×#1587	2
C₈×Q₁₆×C₅₃	2
C₅₃×#1564	2
C₅₃×#1485	2
C₈×M₁₆×C₅₃	2
#18086	53

Subgroups — C53×#18086 #27225598

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#18086 #27225598