Subgroups — C₅₃×#17795 #27225307

Order 13568.

710 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2920	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1569	C₂	1	8	yes	maximal, complemented
4	C₅₃×#1564	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₅₃	C₄	1	2	yes	—
6	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#432	—	2	2	no	—
8	C₅₃×#432	C₂²	1	4	yes	—
9	C₅₃×#431	—	2	2	no	—
10	C₅₃×#431	C₂²	1	4	yes	—
11	C₅₃×#423	C₂²	1	4	yes	—
12	C₅₃×#407	—	2	2	no	—
13	C₄×M₁₆×C₅₃	—	2	2	no	—
14	C₄×M₁₆×C₅₃	C₄	1	4	yes	—
15	C₄×M₁₆×C₅₃	C₂²	1	4	yes	—
16	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
17	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
19	C₅₃×#335	C₂²	1	8	yes	complemented
20	C₅₃×#334	C₂²	1	8	yes	complemented
21	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
24	C₂×M₁₆×C₅₃	—	2	8	no	—
25	C₂×M₁₆×C₅₃	—	2	4	no	—
26	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	D₄	1	2	yes	—
33	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	—	2	4	no	—
38	C₄⋊C₈×C₅₃	C₂³	1	8	yes	—
39	C₂²⋊C₈×C₅₃	—	2	4	no	—
40	C₂²⋊C₈×C₅₃	—	2	4	no	—
41	#97×C₅₃	—	2	4	no	—
42	#97×C₅₃	C₂×C₄	1	4	yes	—
43	#97×C₅₃	C₂×C₄	1	4	yes	—
44	C₄×C₈×C₅₃	—	2	4	no	—
45	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
46	C₄×C₈×C₅₃	C₂³	1	4	yes	—
47	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	—	2	4	no	—
49	C₂²×C₄×C₅₃	—	4	2	no	—
50	C₂²×C₄×C₅₃	—	4	2	no	—
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂²×C₄	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×D₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
63	M₁₆×C₅₃	—	2	16	no	—
64	M₁₆×C₅₃	—	4	8	no	—
65	C₂×C₈×C₅₃	—	2	4	no	—
66	C₂×C₈×C₅₃	—	2	8	no	—
67	C₂×C₈×C₅₃	M₁₆	1	8	yes	complemented
68	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
70	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
71	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
72	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
73	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
75	C₄²×C₅₃	C₂×D₄	1	2	yes	—
76	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
78	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
79	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
80	C₂³×C₅₃	—	2	2	no	—
81	C₂³×C₅₃	—	4	1	no	—
82	C₂³×C₅₃	—	4	1	no	—
83	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
84	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	4	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	2	4	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	4	1	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	4	1	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
103	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
106	C₈×C₅₃	—	2	4	no	—
107	C₈×C₅₃	—	4	8	no	—
108	C₈×C₅₃	C₂×M₁₆	1	8	yes	—
109	#17795	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	—	2	2	no	—
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	—	4	4	no	—
116	C₂²×C₅₃	#423	1	2	yes	—
117	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	#423	1	2	yes	—
124	C₄×C₅₃	#430	1	2	yes	—
125	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
126	C₄×C₅₃	#575	1	2	yes	—
127	#2920	C₂×C₅₃	1	2	yes	—
128	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic
129	#1569	C₂×C₅₃	1	8	yes	complemented
130	#1564	C₂×C₅₃	1	4	yes	complemented
131	C₂×C₅₃	—	2	2	no	—
132	C₂×C₅₃	—	4	2	no	—
133	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
134	C₂×C₅₃	#2911	1	1	yes	characteristic
135	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
136	C₂²×M₁₆	C₄×C₅₃	1	2	yes	—
137	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
138	#432	—	2	2	no	—
139	#432	C₂²×C₅₃	1	4	yes	—
140	#431	—	2	2	no	—
141	#431	C₂²×C₅₃	1	4	yes	—
142	#423	C₂²×C₅₃	1	4	yes	—
143	#407	—	2	2	no	—
144	C₄×M₁₆	—	2	2	no	—
145	C₄×M₁₆	C₄×C₅₃	1	4	yes	—
146	C₄×M₁₆	C₂²×C₅₃	1	4	yes	—
147	C₂×#97	C₄×C₅₃	1	1	yes	characteristic
148	C₂×#97	C₄×C₅₃	1	1	yes	characteristic
149	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
150	#335	C₂²×C₅₃	1	8	yes	complemented
151	#334	C₂²×C₅₃	1	8	yes	complemented
152	C₅₃	#17795	1	1	yes	characteristic, direct product factor, Sylow
153	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
154	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
155	C₂³×C₄	Q₈×C₅₃	1	1	yes	characteristic
156	C₂×M₁₆	—	2	4	no	—
157	C₂×M₁₆	—	2	8	no	—
158	C₂×M₁₆	C₂×C₄×C₅₃	1	8	yes	—
159	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
160	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
161	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
162	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
163	C₂×C₄²	D₄×C₅₃	1	2	yes	—
164	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
165	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
166	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
167	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
168	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
169	C₄⋊C₈	—	2	4	no	—
170	C₄⋊C₈	C₂³×C₅₃	1	8	yes	—
171	C₂²⋊C₈	—	2	4	no	—
172	C₂²⋊C₈	—	2	4	no	—
173	#97	—	2	4	no	—
174	#97	C₂×C₄×C₅₃	1	4	yes	—
175	#97	C₂×C₄×C₅₃	1	4	yes	—
176	C₄×C₈	—	2	4	no	—
177	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
178	C₄×C₈	C₂³×C₅₃	1	4	yes	—
179	C₂⁴	C₄⋊C₄×C₅₃	1	1	yes	characteristic
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
184	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
186	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
187	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
188	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
189	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
190	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
193	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
195	M₁₆	—	2	16	no	—
196	M₁₆	—	4	8	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	8	no	—
199	C₂×C₈	M₁₆×C₅₃	1	8	yes	complemented
200	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
201	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
202	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
203	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
204	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
205	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
206	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
207	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
208	C₄²	C₂×D₄×C₅₃	1	2	yes	—
209	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
210	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
211	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
212	C₂³	—	2	2	no	—
213	C₂³	—	4	1	no	—
214	C₂³	—	4	1	no	—
215	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
216	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
230	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
231	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
232	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
234	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
235	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
237	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₈	—	4	8	no	—
240	C₈	C₂×M₁₆×C₅₃	1	8	yes	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	4	no	—
247	C₂²	C₅₃×#423	1	2	yes	—
248	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	C₅₃×#423	1	2	yes	—
255	C₄	C₅₃×#430	1	2	yes	—
256	C₄	C₂²×M₁₆×C₅₃	1	2	yes	—
257	C₄	C₅₃×#575	1	2	yes	—
258	C₂	—	2	2	no	—
259	C₂	—	4	2	no	—
260	C₂	C₂×C₅₃×#423	1	1	yes	characteristic
261	C₂	C₅₃×#2911	1	1	yes	characteristic
262	C₂	C₂×C₅₃×#430	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1569	C₂	semidirect
4. C₅₃×#1564	C₂	semidirect
19. C₅₃×#335	C₂²	semidirect
20. C₅₃×#334	C₂²	semidirect
67. C₂×C₈×C₅₃	M₁₆	semidirect
109. #17795	C₅₃	direct (N × H)
129. #1569	C₂×C₅₃	semidirect
130. #1564	C₂×C₅₃	semidirect
150. #335	C₂²×C₅₃	semidirect
151. #334	C₂²×C₅₃	semidirect
152. C₅₃	#17795	direct (N × H)
199. C₂×C₈	M₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2920	2
C₂×C₄×M₁₆×C₅₃	2
C₅₃×#1569	2
C₅₃×#1564	2
#17795	53

Subgroups — C53×#17795 #27225307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17795 #27225307