Subgroups — C₅₃×#17989 #27225501

Order 13568.

624 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2981	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1586	C₂	1	4	yes	maximal, complemented
4	C₅₃×#1581	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1496	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#532	C₂²	1	2	yes	—
9	C₅₃×#500	C₄	1	1	yes	characteristic
10	C₅₃×#493	C₄	1	1	yes	characteristic
11	C₅₃×#486	C₄	1	2	yes	—
12	C₅₃×#466	C₄	1	2	yes	—
13	C₅₃×#466	C₄	1	2	yes	—
14	C₄×D₈×C₅₃	C₂²	1	2	yes	—
15	C₄×D₈×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#436	C₂²	1	4	yes	—
17	C₅₃×#435	—	2	4	no	—
18	C₈×D₄×C₅₃	C₂²	1	4	yes	—
19	C₅₃×#432	C₂²	1	2	yes	—
20	C₅₃×#431	C₂²	1	1	yes	characteristic
21	C₅₃×#431	C₂²	1	1	yes	characteristic
22	C₅₃×#408	—	2	4	no	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#335	C₂²	1	2	yes	—
25	C₅₃×#334	C₂²	1	2	yes	—
26	C₅₃×D₄⋊C₈	C₂²	1	8	yes	—
27	C₅₃×#322	C₂²	1	2	yes	—
28	C₅₃×#322	C₂²	1	2	yes	—
29	C₂×#41×C₅₃	—	2	2	no	—
30	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₅₃	—	2	4	no	—
33	C₂²×C₈×C₅₃	—	2	4	no	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
37	#127×C₅₃	C₂×C₄	1	2	yes	—
38	#124×C₅₃	—	2	2	no	—
39	#121×C₅₃	—	2	2	no	—
40	#121×C₅₃	C₂×C₄	1	4	yes	—
41	C₄×D₄×C₅₃	—	2	2	no	—
42	C₄×D₄×C₅₃	C₂³	1	4	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	#107×C₅₃	C₂×C₄	1	2	yes	—
45	#107×C₅₃	C₂×C₄	1	2	yes	—
46	C₄⋊C₈×C₅₃	—	2	1	no	—
47	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
48	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
49	#102×C₅₃	C₂×C₄	1	4	yes	—
50	#102×C₅₃	C₂×C₄	1	4	yes	—
51	C₂²⋊C₈×C₅₃	—	2	4	no	—
52	C₂²⋊C₈×C₅₃	—	2	4	no	—
53	C₂²⋊C₈×C₅₃	—	2	2	no	—
54	C₂²⋊C₈×C₅₃	—	2	1	no	—
55	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
56	#97×C₅₃	—	2	2	no	—
57	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₅₃	C₂³	1	2	yes	—
60	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₅₃	C₂³	1	2	yes	—
63	#41×C₅₃	—	8	4	no	—
64	C₂×Q₈×C₅₃	—	2	2	no	—
65	C₂×D₄×C₅₃	—	2	4	no	—
66	C₂×D₄×C₅₃	—	2	2	no	—
67	C₂×D₄×C₅₃	—	2	2	no	—
68	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₅₃	—	2	4	no	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
73	D₈×C₅₃	—	4	2	no	—
74	D₈×C₅₃	—	4	2	no	—
75	M₁₆×C₅₃	—	8	4	no	—
76	C₂×C₈×C₅₃	—	2	4	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	4	1	no	—
79	C₂×C₈×C₅₃	—	4	1	no	—
80	C₂×C₈×C₅₃	—	8	4	no	—
81	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	C₂×D₄	1	4	yes	—
86	C₄⋊C₄×C₅₃	—	2	2	no	—
87	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
88	C₂²⋊C₄×C₅₃	—	2	4	no	—
89	C₂²⋊C₄×C₅₃	—	2	4	no	—
90	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₅₃	#41	1	1	yes	characteristic
93	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
94	C₂³×C₅₃	—	2	4	no	—
95	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
96	Q₈×C₅₃	—	8	2	no	—
97	D₄×C₅₃	—	2	4	no	—
98	D₄×C₅₃	—	2	4	no	—
99	D₄×C₅₃	—	8	2	no	—
100	D₄×C₅₃	—	8	2	no	—
101	D₄×C₅₃	—	8	4	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	4	1	no	—
107	C₂×C₄×C₅₃	—	4	4	no	—
108	C₂×C₄×C₅₃	—	4	4	no	—
109	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
115	C₈×C₅₃	—	2	1	no	—
116	C₈×C₅₃	—	2	1	no	—
117	C₈×C₅₃	—	2	1	no	—
118	C₈×C₅₃	—	2	1	no	—
119	C₈×C₅₃	—	2	4	no	—
120	C₈×C₅₃	—	4	2	no	—
121	C₈×C₅₃	—	4	4	no	—
122	#17989	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₅₃	—	2	1	no	—
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	2	1	no	—
126	C₂²×C₅₃	—	4	4	no	—
127	C₂²×C₅₃	—	4	4	no	—
128	C₂²×C₅₃	—	4	4	no	—
129	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
130	C₄×C₅₃	—	2	1	no	—
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	#568	1	1	yes	characteristic
136	C₄×C₅₃	#568	1	1	yes	characteristic
137	C₄×C₅₃	#576	1	2	yes	—
138	#3169	C₂×C₅₃	1	1	yes	characteristic
139	#2981	C₂×C₅₃	1	2	yes	complemented
140	#1586	C₂×C₅₃	1	4	yes	complemented
141	#1581	C₂×C₅₃	1	4	yes	complemented
142	#1568	C₂×C₅₃	1	1	yes	characteristic, complemented
143	#1496	C₂×C₅₃	1	2	yes	complemented
144	#1452	C₂×C₅₃	1	1	yes	characteristic, complemented
145	C₂×C₅₃	—	4	1	no	—
146	C₂×C₅₃	—	8	4	no	—
147	C₂×C₅₃	#2932	1	1	yes	characteristic
148	C₂×C₅₃	#2948	1	1	yes	characteristic
149	C₂×C₅₃	#2957	1	1	yes	characteristic
150	#532	C₂²×C₅₃	1	2	yes	—
151	#500	C₄×C₅₃	1	1	yes	characteristic
152	#493	C₄×C₅₃	1	1	yes	characteristic
153	#486	C₄×C₅₃	1	2	yes	—
154	#466	C₄×C₅₃	1	2	yes	—
155	#466	C₄×C₅₃	1	2	yes	—
156	C₄×D₈	C₂²×C₅₃	1	2	yes	—
157	C₄×D₈	C₂²×C₅₃	1	2	yes	—
158	#436	C₂²×C₅₃	1	4	yes	—
159	#435	—	2	4	no	—
160	C₈×D₄	C₂²×C₅₃	1	4	yes	—
161	#432	C₂²×C₅₃	1	2	yes	—
162	#431	C₂²×C₅₃	1	1	yes	characteristic
163	#431	C₂²×C₅₃	1	1	yes	characteristic
164	#408	—	2	4	no	—
165	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
166	#335	C₂²×C₅₃	1	2	yes	—
167	#334	C₂²×C₅₃	1	2	yes	—
168	D₄⋊C₈	C₂²×C₅₃	1	8	yes	—
169	#322	C₂²×C₅₃	1	2	yes	—
170	#322	C₂²×C₅₃	1	2	yes	—
171	C₅₃	#17989	1	1	yes	characteristic, direct product factor, Sylow
172	C₂×#41	—	2	2	no	—
173	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
174	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
175	C₂×M₁₆	—	2	4	no	—
176	C₂²×C₈	—	2	4	no	—
177	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
178	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
179	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
180	#127	C₂×C₄×C₅₃	1	2	yes	—
181	#124	—	2	2	no	—
182	#121	—	2	2	no	—
183	#121	C₂×C₄×C₅₃	1	4	yes	—
184	C₄×D₄	—	2	2	no	—
185	C₄×D₄	C₂³×C₅₃	1	4	yes	—
186	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
187	#107	C₂×C₄×C₅₃	1	2	yes	—
188	#107	C₂×C₄×C₅₃	1	2	yes	—
189	C₄⋊C₈	—	2	1	no	—
190	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
191	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
192	#102	C₂×C₄×C₅₃	1	4	yes	—
193	#102	C₂×C₄×C₅₃	1	4	yes	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	—	2	1	no	—
196	C₂²⋊C₈	—	2	4	no	—
197	C₂²⋊C₈	—	2	4	no	—
198	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
199	#97	—	2	2	no	—
200	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
201	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
202	C₄×C₈	C₂³×C₅₃	1	2	yes	—
203	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
204	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
205	C₄×C₈	C₂³×C₅₃	1	2	yes	—
206	#41	—	8	4	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
215	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
216	D₈	—	4	2	no	—
217	D₈	—	4	2	no	—
218	M₁₆	—	8	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	8	4	no	—
224	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
225	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
226	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
227	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
234	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
235	C₄²	#41×C₅₃	1	1	yes	characteristic
236	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
237	C₂³	—	2	4	no	—
238	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
239	Q₈	—	8	2	no	—
240	D₄	—	2	4	no	—
241	D₄	—	2	4	no	—
242	D₄	—	8	2	no	—
243	D₄	—	8	2	no	—
244	D₄	—	8	4	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
253	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
254	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
255	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
256	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
257	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	4	no	—
263	C₈	—	4	2	no	—
264	C₈	—	4	4	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	4	no	—
271	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	4	no	—
277	C₄	C₅₃×#568	1	1	yes	characteristic
278	C₄	C₅₃×#568	1	1	yes	characteristic
279	C₄	C₅₃×#576	1	2	yes	—
280	C₂	—	4	1	no	—
281	C₂	—	8	4	no	—
282	C₂	C₅₃×#2932	1	1	yes	characteristic
283	C₂	C₅₃×#2948	1	1	yes	characteristic
284	C₂	C₅₃×#2957	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	C₅₃×#3169	C₂
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
2. C₅₃×#2981	C₂	semidirect
3. C₅₃×#1586	C₂	semidirect
4. C₅₃×#1581	C₂	semidirect
5. C₅₃×#1568	C₂	semidirect
6. C₅₃×#1496	C₂	semidirect
7. C₅₃×#1452	C₂	semidirect
122. #17989	C₅₃	direct (N × H)
139. #2981	C₂×C₅₃	semidirect
140. #1586	C₂×C₅₃	semidirect
141. #1581	C₂×C₅₃	semidirect
142. #1568	C₂×C₅₃	semidirect
143. #1496	C₂×C₅₃	semidirect
144. #1452	C₂×C₅₃	semidirect
171. C₅₃	#17989	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×#3169	2
C₅₃×#2981	2
C₅₃×#1586	2
C₅₃×#1581	2
C₅₃×#1568	2
C₅₃×#1496	2
C₅₃×#1452	2
#17989	53

Subgroups — C53×#17989 #27225501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17989 #27225501