Subgroups — C₅₃×#9670 #27217182

Order 13568.

498 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1478	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1460	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
8	C₄×D₈×C₅₃	—	2	1	no	—
9	C₄×D₈×C₅₃	—	2	1	no	—
10	C₄×D₈×C₅₃	—	2	1	no	—
11	C₄×D₈×C₅₃	—	2	1	no	—
12	C₅₃×#431	C₂²	1	2	yes	—
13	C₅₃×#431	C₂²	1	1	yes	characteristic
14	C₅₃×#431	C₂²	1	1	yes	characteristic
15	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
16	C₂×#102×C₅₃	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#335	—	2	1	no	—
19	C₅₃×#334	—	2	1	no	—
20	C₅₃×D₄⋊C₈	—	2	4	no	—
21	C₅₃×#323	—	2	4	no	—
22	C₅₃×#322	—	2	1	no	—
23	C₅₃×#322	—	2	1	no	—
24	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₅₃	—	2	2	no	—
26	C₂×D₈×C₅₃	—	2	1	no	—
27	C₂×D₈×C₅₃	—	2	1	no	—
28	C₂×D₈×C₅₃	—	2	4	no	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×D₄×C₅₃	—	2	2	no	—
32	C₄×D₄×C₅₃	—	2	2	no	—
33	C₄×D₄×C₅₃	—	2	2	no	—
34	C₄×D₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	#107×C₅₃	—	2	1	no	—
39	#107×C₅₃	—	2	1	no	—
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₄⋊C₈×C₅₃	—	2	1	no	—
42	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
43	#102×C₅₃	—	2	2	no	—
44	#102×C₅₃	—	2	2	no	—
45	C₂²⋊C₈×C₅₃	—	2	2	no	—
46	C₂²⋊C₈×C₅₃	—	2	1	no	—
47	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
48	C₄×C₈×C₅₃	—	2	1	no	—
49	C₄×C₈×C₅₃	—	2	1	no	—
50	C₄×C₈×C₅₃	—	2	2	no	—
51	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
55	C₂⁴×C₅₃	—	2	2	no	—
56	C₂×D₄×C₅₃	—	2	2	no	—
57	C₂×D₄×C₅₃	—	2	2	no	—
58	C₂×D₄×C₅₃	—	2	2	no	—
59	C₂×D₄×C₅₃	—	2	2	no	—
60	C₂×D₄×C₅₃	—	2	2	no	—
61	C₂×D₄×C₅₃	M₁₆	1	4	yes	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	—	4	2	no	—
64	C₂²×C₄×C₅₃	—	4	2	no	—
65	C₂²×C₄×C₅₃	—	8	4	no	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
69	D₈×C₅₃	—	2	4	no	—
70	D₈×C₅₃	—	2	4	no	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	4	2	no	—
76	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₄⋊C₄×C₅₃	—	2	2	no	—
81	C₄⋊C₄×C₅₃	—	2	2	no	—
82	C₂²⋊C₄×C₅₃	—	4	2	no	—
83	C₂²⋊C₄×C₅₃	—	4	2	no	—
84	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₃	#41	1	1	yes	characteristic
88	C₂³×C₅₃	—	2	4	no	—
89	C₂³×C₅₃	—	4	2	no	—
90	C₂³×C₅₃	—	4	2	no	—
91	C₂³×C₅₃	—	8	2	no	—
92	C₂³×C₅₃	—	8	2	no	—
93	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
94	D₄×C₅₃	—	2	4	no	—
95	D₄×C₅₃	—	2	4	no	—
96	D₄×C₅₃	—	4	2	no	—
97	D₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	4	1	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	—	8	4	no	—
102	C₂×C₄×C₅₃	—	8	4	no	—
103	C₂×C₄×C₅₃	—	8	2	no	—
104	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₅₃	#131	1	2	yes	—
112	C₈×C₅₃	—	2	1	no	—
113	C₈×C₅₃	—	2	1	no	—
114	C₈×C₅₃	—	2	1	no	—
115	C₈×C₅₃	—	2	1	no	—
116	C₈×C₅₃	—	4	2	no	—
117	C₈×C₅₃	—	4	2	no	—
118	C₈×C₅₃	—	8	2	no	—
119	#9670	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	—	4	4	no	—
123	C₂²×C₅₃	—	8	4	no	—
124	C₂²×C₅₃	—	8	4	no	—
125	C₂²×C₅₃	—	8	2	no	—
126	C₂²×C₅₃	—	8	2	no	—
127	C₂²×C₅₃	#390	1	1	yes	characteristic
128	C₂²×C₅₃	#435	1	2	yes	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	8	2	no	—
133	C₄×C₅₃	#407	1	1	yes	characteristic
134	C₄×C₅₃	#408	1	1	yes	characteristic
135	C₄×C₅₃	#435	1	2	yes	—
136	C₄×C₅₃	#472	1	2	yes	—
137	C₂×C₄×D₈	C₂×C₅₃	1	1	yes	characteristic
138	#1564	C₂×C₅₃	1	1	yes	characteristic, complemented
139	#1478	C₂×C₅₃	1	2	yes	complemented
140	#1460	C₂×C₅₃	1	2	yes	complemented
141	#1452	C₂×C₅₃	1	1	yes	characteristic, complemented
142	C₂×C₅₃	—	2	2	no	—
143	C₂×C₅₃	—	8	4	no	—
144	C₂×C₅₃	#1955	1	1	yes	characteristic
145	C₂×C₅₃	#1976	1	1	yes	characteristic
146	C₂×C₅₃	#1978	1	1	yes	characteristic
147	C₂²×D₈	C₄×C₅₃	1	1	yes	characteristic
148	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
149	C₄×D₈	—	2	1	no	—
150	C₄×D₈	—	2	1	no	—
151	C₄×D₈	—	2	1	no	—
152	C₄×D₈	—	2	1	no	—
153	#431	C₂²×C₅₃	1	2	yes	—
154	#431	C₂²×C₅₃	1	1	yes	characteristic
155	#431	C₂²×C₅₃	1	1	yes	characteristic
156	C₂×#107	C₄×C₅₃	1	1	yes	characteristic
157	C₂×#102	C₄×C₅₃	1	2	yes	—
158	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
159	#335	—	2	1	no	—
160	#334	—	2	1	no	—
161	D₄⋊C₈	—	2	4	no	—
162	#323	—	2	4	no	—
163	#322	—	2	1	no	—
164	#322	—	2	1	no	—
165	C₅₃	#9670	1	1	yes	characteristic, direct product factor, Sylow
166	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
167	C₂³×C₄	—	2	2	no	—
168	C₂×D₈	—	2	4	no	—
169	C₂×D₈	—	2	1	no	—
170	C₂×D₈	—	2	1	no	—
171	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
172	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
173	C₄×D₄	—	2	2	no	—
174	C₄×D₄	—	2	2	no	—
175	C₄×D₄	—	2	2	no	—
176	C₄×D₄	—	2	2	no	—
177	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
178	C₂×C₂²⋊C₄	—	2	2	no	—
179	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
180	#107	—	2	1	no	—
181	#107	—	2	1	no	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	—	2	1	no	—
184	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
185	#102	—	2	2	no	—
186	#102	—	2	2	no	—
187	C₂²⋊C₈	—	2	1	no	—
188	C₂²⋊C₈	—	2	2	no	—
189	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
190	C₄×C₈	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
194	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
195	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
196	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
197	C₂⁴	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	M₁₆×C₅₃	1	4	yes	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	8	4	no	—
208	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
210	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
211	D₈	—	2	4	no	—
212	D₈	—	2	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
219	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
220	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
221	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	4	2	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
227	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
228	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
229	C₄²	#41×C₅₃	1	1	yes	characteristic
230	C₂³	—	2	4	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	2	no	—
233	C₂³	—	8	2	no	—
234	C₂³	—	8	2	no	—
235	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
236	D₄	—	2	4	no	—
237	D₄	—	2	4	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	8	4	no	—
245	C₂×C₄	—	8	2	no	—
246	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
247	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
248	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
249	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
250	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
251	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
252	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
253	C₂×C₄	#131×C₅₃	1	2	yes	—
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	4	2	no	—
259	C₈	—	4	2	no	—
260	C₈	—	8	2	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	4	4	no	—
264	C₂²	—	8	4	no	—
265	C₂²	—	8	4	no	—
266	C₂²	—	8	2	no	—
267	C₂²	—	8	2	no	—
268	C₂²	C₅₃×#390	1	1	yes	characteristic
269	C₂²	C₅₃×#435	1	2	yes	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	8	2	no	—
274	C₄	C₅₃×#407	1	1	yes	characteristic
275	C₄	C₅₃×#408	1	1	yes	characteristic
276	C₄	C₅₃×#435	1	2	yes	—
277	C₄	C₅₃×#472	1	2	yes	—
278	C₂	—	2	2	no	—
279	C₂	—	8	4	no	—
280	C₂	C₅₃×#1955	1	1	yes	characteristic
281	C₂	C₅₃×#1976	1	1	yes	characteristic
282	C₂	C₅₃×#1978	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#121
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#390
prime radical	C₂²×D₈×C₅₃	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1564	C₂	semidirect
3. C₅₃×#1478	C₂	semidirect
4. C₅₃×#1460	C₂	semidirect
5. C₅₃×#1452	C₂	semidirect
119. #9670	C₅₃	direct (N × H)
138. #1564	C₂×C₅₃	semidirect
139. #1478	C₂×C₅₃	semidirect
140. #1460	C₂×C₅₃	semidirect
141. #1452	C₂×C₅₃	semidirect
165. C₅₃	#9670	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₅₃	2
C₅₃×#1564	2
C₅₃×#1478	2
C₅₃×#1460	2
C₅₃×#1452	2
#9670	53

Subgroups — C53×#9670 #27217182

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9670 #27217182