Subgroups — C₅₃×#8624 #27216136

Order 13568.

418 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#436	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1466	C₂	1	2	yes	maximal
5	C₅₃×#1290	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₅₃	C₄	1	2	yes	—
7	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#436	—	2	1	no	—
9	C₅₃×#436	—	2	1	no	—
10	C₅₃×#436	—	2	1	no	—
11	C₅₃×#436	—	2	1	no	—
12	C₅₃×#432	C₂²	1	2	yes	—
13	C₅₃×#432	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#335	—	2	1	no	—
18	C₅₃×#335	—	2	1	no	—
19	C₅₃×#334	—	2	1	no	—
20	C₅₃×#334	—	2	1	no	—
21	C₅₃×#323	—	2	4	no	—
22	C₂²×D₄×C₅₃	Q₈	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×M₁₆×C₅₃	—	2	2	no	—
25	C₂×M₁₆×C₅₃	—	2	2	no	—
26	C₂×M₁₆×C₅₃	—	4	4	no	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×D₄×C₅₃	—	2	1	no	—
31	C₄×D₄×C₅₃	—	2	1	no	—
32	C₄×D₄×C₅₃	—	2	1	no	—
33	C₄×D₄×C₅₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	—	2	1	no	—
38	C₄⋊C₈×C₅₃	—	2	1	no	—
39	C₄⋊C₈×C₅₃	—	2	2	no	—
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₂²⋊C₈×C₅₃	—	2	2	no	—
42	C₂²⋊C₈×C₅₃	—	2	2	no	—
43	C₂²⋊C₈×C₅₃	—	2	2	no	—
44	C₂²⋊C₈×C₅₃	—	2	2	no	—
45	#97×C₅₃	—	2	2	no	—
46	#97×C₅₃	—	2	2	no	—
47	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₂⁴×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₂×D₄×C₅₃	—	2	1	no	—
53	C₂×D₄×C₅₃	—	2	1	no	—
54	C₂×D₄×C₅₃	—	4	2	no	—
55	C₂²×C₄×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	—	4	2	no	—
58	C₂²×C₄×C₅₃	—	4	2	no	—
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
63	M₁₆×C₅₃	—	4	4	no	—
64	M₁₆×C₅₃	—	4	4	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	—	4	1	no	—
68	C₂×C₈×C₅₃	—	4	2	no	—
69	C₂×C₈×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	—	4	2	no	—
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₄⋊C₄×C₅₃	—	2	1	no	—
76	C₄⋊C₄×C₅₃	—	2	1	no	—
77	C₂²⋊C₄×C₅₃	—	2	2	no	—
78	C₂²⋊C₄×C₅₃	—	2	2	no	—
79	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₅₃	#41	1	1	yes	characteristic
82	C₄²×C₅₃	#41	1	1	yes	characteristic
83	C₂³×C₅₃	—	2	2	no	—
84	C₂³×C₅₃	—	2	4	no	—
85	C₂³×C₅₃	—	4	2	no	—
86	C₂³×C₅₃	—	4	2	no	—
87	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	D₄×C₅₃	—	4	2	no	—
89	D₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	4	1	no	—
95	C₂×C₄×C₅₃	—	4	4	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
105	C₈×C₅₃	—	2	1	no	—
106	C₈×C₅₃	—	2	1	no	—
107	C₈×C₅₃	—	2	1	no	—
108	C₈×C₅₃	—	4	2	no	—
109	C₈×C₅₃	—	8	2	no	—
110	C₈×C₅₃	—	8	2	no	—
111	C₈×C₅₃	#100	1	2	yes	—
112	#8624	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	—	2	1	no	—
115	C₂²×C₅₃	—	4	2	no	—
116	C₂²×C₅₃	—	4	2	no	—
117	C₂²×C₅₃	—	4	2	no	—
118	C₂²×C₅₃	—	8	4	no	—
119	C₂²×C₅₃	#380	1	1	yes	characteristic
120	C₂²×C₅₃	#430	1	2	yes	—
121	C₄×C₅₃	—	2	1	no	—
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	—	2	1	no	—
125	C₄×C₅₃	—	2	1	no	—
126	C₄×C₅₃	—	4	2	no	—
127	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
128	C₄×C₅₃	#413	1	1	yes	characteristic
129	C₂×#436	C₂×C₅₃	1	1	yes	characteristic
130	#1568	C₂×C₅₃	1	1	yes	characteristic, complemented
131	#1568	C₂×C₅₃	1	1	yes	characteristic, complemented
132	#1466	C₂×C₅₃	1	2	yes	—
133	#1290	C₂×C₅₃	1	2	yes	complemented
134	C₂×C₅₃	—	2	2	no	—
135	C₂×C₅₃	—	8	2	no	—
136	C₂×C₅₃	#1800	1	1	yes	characteristic
137	C₂×C₅₃	#1811	1	1	yes	characteristic
138	C₂×C₅₃	#1812	1	1	yes	characteristic
139	C₂²×M₁₆	C₄×C₅₃	1	2	yes	—
140	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
141	#436	—	2	1	no	—
142	#436	—	2	1	no	—
143	#436	—	2	1	no	—
144	#436	—	2	1	no	—
145	#432	C₂²×C₅₃	1	2	yes	—
146	#432	C₂²×C₅₃	1	2	yes	—
147	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
148	C₂×C₂²⋊C₈	C₄×C₅₃	1	2	yes	—
149	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
150	#335	—	2	1	no	—
151	#335	—	2	1	no	—
152	#334	—	2	1	no	—
153	#334	—	2	1	no	—
154	#323	—	2	4	no	—
155	C₅₃	#8624	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²×D₄	Q₈×C₅₃	1	1	yes	characteristic
157	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	4	4	no	—
161	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
162	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
163	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
164	C₄×D₄	—	2	1	no	—
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	—	2	1	no	—
167	C₄×D₄	—	2	1	no	—
168	C₂×C₄⋊C₄	Q₈×C₅₃	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
170	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	2	1	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₂²⋊C₈	—	2	2	no	—
177	C₂²⋊C₈	—	2	2	no	—
178	C₂²⋊C₈	—	2	2	no	—
179	#97	—	2	2	no	—
180	#97	—	2	2	no	—
181	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
182	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
183	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
184	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
185	C₂⁴	C₄⋊C₄×C₅₃	1	2	yes	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
194	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
197	M₁₆	—	4	4	no	—
198	M₁₆	—	4	4	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
207	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
214	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
215	C₄²	#41×C₅₃	1	1	yes	characteristic
216	C₄²	#41×C₅₃	1	1	yes	characteristic
217	C₂³	—	2	2	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	4	2	no	—
220	C₂³	—	4	2	no	—
221	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
234	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
235	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
236	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
237	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
238	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	4	2	no	—
243	C₈	—	8	2	no	—
244	C₈	—	8	2	no	—
245	C₈	#100×C₅₃	1	2	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	8	4	no	—
252	C₂²	C₅₃×#380	1	1	yes	characteristic
253	C₂²	C₅₃×#430	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	4	2	no	—
260	C₄	C₂×#100×C₅₃	1	1	yes	characteristic
261	C₄	C₅₃×#413	1	1	yes	characteristic
262	C₂	—	2	2	no	—
263	C₂	—	8	2	no	—
264	C₂	C₅₃×#1800	1	1	yes	characteristic
265	C₂	C₅₃×#1811	1	1	yes	characteristic
266	C₂	C₅₃×#1812	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#380
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂²×D₄×C₅₃	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#1568	C₂	semidirect
3. C₅₃×#1568	C₂	semidirect
5. C₅₃×#1290	C₂	semidirect
112. #8624	C₅₃	direct (N × H)
130. #1568	C₂×C₅₃	semidirect
131. #1568	C₂×C₅₃	semidirect
133. #1290	C₂×C₅₃	semidirect
155. C₅₃	#8624	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×C₅₃×#436	2
C₅₃×#1568	2
C₅₃×#1466	2
C₅₃×#1290	2
#8624	53

Subgroups — C53×#8624 #27216136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8624 #27216136