Subgroups — C₅₃×#8601 #27216113

Order 13568.

462 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	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1564	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1458	C₂	1	2	yes	maximal
4	C₅₃×#1275	C₂	1	2	yes	maximal, complemented
5	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#431	C₂²	1	4	yes	—
9	C₅₃×#405	—	2	2	no	—
10	C₅₃×#405	—	2	2	no	—
11	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
12	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#335	—	2	2	no	—
16	C₅₃×#334	—	2	2	no	—
17	C₅₃×#324	—	2	2	no	—
18	C₅₃×#323	—	2	2	no	—
19	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₅₃	—	2	2	no	—
21	C₂×M₁₆×C₅₃	—	2	2	no	—
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂²×C₈×C₅₃	—	2	1	no	—
24	C₂²×C₈×C₅₃	—	2	1	no	—
25	C₂²×C₈×C₅₃	—	2	2	no	—
26	C₂²×C₈×C₅₃	C₈	1	4	yes	complemented
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	#117×C₅₃	—	2	2	no	—
32	#117×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₅₃	D₄	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	—	2	4	no	—
40	C₂²⋊C₈×C₅₃	—	2	4	no	—
41	#97×C₅₃	—	2	2	no	—
42	#97×C₅₃	—	2	2	no	—
43	C₄×C₈×C₅₃	—	2	1	no	—
44	C₄×C₈×C₅₃	—	2	1	no	—
45	C₄×C₈×C₅₃	—	2	4	no	—
46	C₄×C₈×C₅₃	D₄	1	2	yes	—
47	C₄×C₈×C₅₃	D₄	1	2	yes	—
48	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	—	2	1	no	—
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
61	M₁₆×C₅₃	—	2	8	no	—
62	C₂×C₈×C₅₃	—	2	1	no	—
63	C₂×C₈×C₅₃	—	2	1	no	—
64	C₂×C₈×C₅₃	—	2	1	no	—
65	C₂×C₈×C₅₃	—	2	1	no	—
66	C₂×C₈×C₅₃	—	2	1	no	—
67	C₂×C₈×C₅₃	—	2	1	no	—
68	C₂×C₈×C₅₃	—	2	4	no	—
69	C₂×C₈×C₅₃	—	2	4	no	—
70	C₂×C₈×C₅₃	—	4	4	no	—
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
75	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
77	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
78	C₄⋊C₄×C₅₃	—	2	2	no	—
79	C₄⋊C₄×C₅₃	—	2	2	no	—
80	C₂²⋊C₄×C₅₃	—	2	2	no	—
81	C₂²⋊C₄×C₅₃	—	2	2	no	—
82	C₄²×C₅₃	—	2	1	no	—
83	C₄²×C₅₃	—	2	1	no	—
84	C₄²×C₅₃	C₂×D₄	1	2	yes	—
85	C₄²×C₅₃	#41	1	2	yes	—
86	C₂³×C₅₃	—	2	1	no	—
87	C₂³×C₅₃	—	2	1	no	—
88	C₂³×C₅₃	—	4	1	no	—
89	C₂³×C₅₃	—	4	1	no	—
90	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
91	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	4	1	no	—
101	C₂×C₄×C₅₃	—	4	1	no	—
102	C₂×C₄×C₅₃	—	4	1	no	—
103	C₂×C₄×C₅₃	—	4	1	no	—
104	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#121	1	2	yes	—
108	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
113	C₈×C₅₃	—	2	2	no	—
114	C₈×C₅₃	—	2	2	no	—
115	C₈×C₅₃	—	2	2	no	—
116	C₈×C₅₃	—	8	4	no	—
117	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
118	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
119	#8601	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₅₃	—	2	1	no	—
121	C₂²×C₅₃	—	2	1	no	—
122	C₂²×C₅₃	—	2	2	no	—
123	C₂²×C₅₃	—	4	2	no	—
124	C₂²×C₅₃	—	4	2	no	—
125	C₂²×C₅₃	—	4	1	no	—
126	C₂²×C₅₃	—	4	1	no	—
127	C₂²×C₅₃	#380	1	1	yes	characteristic
128	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
129	C₂²×C₅₃	#423	1	1	yes	characteristic
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	2	2	no	—
133	C₄×C₅₃	—	4	1	no	—
134	C₄×C₅₃	—	4	1	no	—
135	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
136	C₄×C₅₃	#407	1	1	yes	characteristic
137	C₄×C₅₃	#431	1	1	yes	characteristic
138	C₄×C₅₃	#432	1	1	yes	characteristic
139	C₂×#405	C₂×C₅₃	1	1	yes	characteristic
140	#1564	C₂×C₅₃	1	2	yes	complemented
141	#1458	C₂×C₅₃	1	2	yes	—
142	#1275	C₂×C₅₃	1	2	yes	complemented
143	C₂×C₅₃	—	2	1	no	—
144	C₂×C₅₃	—	2	1	no	—
145	C₂×C₅₃	—	4	2	no	—
146	C₂×C₅₃	#1797	1	1	yes	characteristic
147	C₂×C₅₃	#1812	1	1	yes	characteristic
148	C₂×C₅₃	#1813	1	1	yes	characteristic
149	C₂²×M₁₆	C₄×C₅₃	1	1	yes	characteristic
150	C₂³×C₈	C₄×C₅₃	1	1	yes	characteristic
151	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
152	#431	C₂²×C₅₃	1	4	yes	—
153	#405	—	2	2	no	—
154	#405	—	2	2	no	—
155	C₂×#97	C₄×C₅₃	1	1	yes	characteristic
156	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
157	C₂×C₄×C₈	C₄×C₅₃	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
159	#335	—	2	2	no	—
160	#334	—	2	2	no	—
161	#324	—	2	2	no	—
162	#323	—	2	2	no	—
163	C₅₃	#8601	1	1	yes	characteristic, direct product factor, Sylow
164	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂²×C₈	—	2	1	no	—
169	C₂²×C₈	—	2	1	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	C₈×C₅₃	1	4	yes	complemented
172	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
173	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
174	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
175	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
176	#117	—	2	2	no	—
177	#117	—	2	2	no	—
178	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
179	C₂×C₄⋊C₄	Q₈×C₅₃	1	1	yes	characteristic
180	C₂×C₂²⋊C₄	D₄×C₅₃	1	1	yes	characteristic
181	C₂×C₂²⋊C₄	Q₈×C₅₃	1	1	yes	characteristic
182	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
184	C₄⋊C₈	—	2	4	no	—
185	C₂²⋊C₈	—	2	4	no	—
186	#97	—	2	2	no	—
187	#97	—	2	2	no	—
188	C₄×C₈	—	2	1	no	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	—	2	4	no	—
191	C₄×C₈	D₄×C₅₃	1	2	yes	—
192	C₄×C₈	D₄×C₅₃	1	2	yes	—
193	C₂⁴	C₄⋊C₄×C₅₃	1	1	yes	characteristic
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
199	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
200	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
202	C₂²×C₄	M₁₆×C₅₃	1	2	yes	—
203	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
206	M₁₆	—	2	8	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	4	no	—
216	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
217	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
219	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
220	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
221	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
222	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	2	2	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	2	1	no	—
229	C₄²	C₂×D₄×C₅₃	1	2	yes	—
230	C₄²	#41×C₅₃	1	2	yes	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	4	1	no	—
234	C₂³	—	4	1	no	—
235	C₂³	C₄⋊C₈×C₅₃	1	2	yes	—
236	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
250	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
251	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
252	C₂×C₄	#121×C₅₃	1	2	yes	—
253	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
254	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
255	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
256	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
257	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	2	2	no	—
260	C₈	—	2	2	no	—
261	C₈	—	8	4	no	—
262	C₈	C₂²⋊C₈×C₅₃	1	2	yes	—
263	C₈	C₂²⋊C₈×C₅₃	1	2	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	1	no	—
271	C₂²	C₅₃×#380	1	1	yes	characteristic
272	C₂²	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
273	C₂²	C₅₃×#423	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
280	C₄	C₅₃×#407	1	1	yes	characteristic
281	C₄	C₅₃×#431	1	1	yes	characteristic
282	C₄	C₅₃×#432	1	1	yes	characteristic
283	C₂	—	2	1	no	—
284	C₂	—	2	1	no	—
285	C₂	—	4	2	no	—
286	C₂	C₅₃×#1797	1	1	yes	characteristic
287	C₂	C₅₃×#1812	1	1	yes	characteristic
288	C₂	C₅₃×#1813	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#117
commutator	C₂×C₄	C₂²×C₈×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#380
prime radical	C₂⁴×C₅₃	C₄⋊C₄
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×#1564	C₂	semidirect
4. C₅₃×#1275	C₂	semidirect
26. C₂²×C₈×C₅₃	C₈	semidirect
119. #8601	C₅₃	direct (N × H)
140. #1564	C₂×C₅₃	semidirect
142. #1275	C₂×C₅₃	semidirect
163. C₅₃	#8601	direct (N × H)
171. C₂²×C₈	C₈×C₅₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅₃×#405	2
C₅₃×#1564	2
C₅₃×#1458	2
C₅₃×#1275	2
#8601	53

Subgroups — C53×#8601 #27216113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8601 #27216113