Subgroups — C₅₃×#17813 #27225325

Order 13568.

730 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#423	C₂	1	2	yes	maximal
3	C₅₃×#1570	C₂	1	8	yes	maximal, complemented
4	C₅₃×#1566	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₅₃	—	2	2	no	—
6	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#431	C₄	1	4	yes	—
8	C₅₃×#431	C₂²	1	4	yes	—
9	C₅₃×#423	—	2	2	no	—
10	C₅₃×#423	—	2	2	no	—
11	C₅₃×#423	C₂²	1	4	yes	—
12	C₅₃×#423	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
14	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
16	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#334	C₂²	1	16	yes	complemented
19	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
22	C₂×M₁₆×C₅₃	—	2	4	no	—
23	C₂×M₁₆×C₅₃	—	2	4	no	—
24	C₂×M₁₆×C₅₃	—	4	8	no	—
25	C₂²×C₈×C₅₃	—	2	4	no	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₃	D₄	1	2	yes	—
30	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
35	C₄⋊C₈×C₅₃	—	2	4	no	—
36	C₄⋊C₈×C₅₃	—	2	4	no	—
37	C₄⋊C₈×C₅₃	—	2	4	no	—
38	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
39	C₄⋊C₈×C₅₃	C₂³	1	8	yes	—
40	C₂²⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₅₃	C₂³	1	4	yes	—
43	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	—	4	2	no	—
45	C₂²×C₄×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	SD₁₆	1	8	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
59	M₁₆×C₅₃	—	4	8	no	—
60	M₁₆×C₅₃	—	4	8	no	—
61	C₂×C₈×C₅₃	—	2	8	no	—
62	C₂×C₈×C₅₃	—	4	4	no	—
63	C₂×C₈×C₅₃	—	4	8	no	—
64	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
65	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₅₃	C₂×D₄	1	2	yes	—
71	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
73	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
74	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
75	C₂³×C₅₃	—	4	1	no	—
76	C₂³×C₅₃	—	4	1	no	—
77	C₂³×C₅₃	#106	1	4	yes	—
78	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
79	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	4	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	4	1	no	—
85	C₂×C₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	—	4	1	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	#106	1	4	yes	—
89	C₂×C₄×C₅₃	#106	1	8	yes	—
90	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
97	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
98	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
99	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
102	C₈×C₅₃	—	2	8	no	—
103	C₈×C₅₃	—	4	8	no	—
104	#17813	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×C₅₃	—	2	2	no	—
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	—	2	2	no	—
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	—	4	4	no	—
110	C₂²×C₅₃	C₂×#106	1	4	yes	—
111	C₂²×C₅₃	C₂×#106	1	2	yes	—
112	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₄×C₅₃	—	2	2	no	—
114	C₄×C₅₃	—	2	1	no	—
115	C₄×C₅₃	—	2	2	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	C₂×#106	1	4	yes	—
118	C₄×C₅₃	C₂×#106	1	2	yes	—
119	C₄×C₅₃	#430	1	2	yes	—
120	C₄×C₅₃	#568	1	2	yes	—
121	C₄×C₅₃	C₂²×SD₁₆	1	2	yes	—
122	C₂×#431	C₂×C₅₃	1	1	yes	characteristic
123	C₂×#423	C₂×C₅₃	1	2	yes	—
124	#1570	C₂×C₅₃	1	8	yes	complemented
125	#1566	C₂×C₅₃	1	4	yes	complemented
126	C₂×C₅₃	—	2	4	no	—
127	C₂×C₅₃	—	2	2	no	—
128	C₂×C₅₃	#2905	1	1	yes	characteristic
129	C₂×C₅₃	C₂²×#106	1	1	yes	characteristic
130	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
131	C₂²×M₁₆	—	2	2	no	—
132	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
133	#431	C₄×C₅₃	1	4	yes	—
134	#431	C₂²×C₅₃	1	4	yes	—
135	#423	—	2	2	no	—
136	#423	—	2	2	no	—
137	#423	C₂²×C₅₃	1	4	yes	—
138	#423	C₂²×C₅₃	1	4	yes	—
139	C₂×C₄⋊C₈	—	2	2	no	—
140	C₂×C₄⋊C₈	C₄×C₅₃	1	2	yes	—
141	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
142	C₂×C₂²⋊C₈	C₄×C₅₃	1	2	yes	—
143	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
144	#334	C₂²×C₅₃	1	16	yes	complemented
145	C₅₃	#17813	1	1	yes	characteristic, direct product factor, Sylow
146	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
147	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
148	C₂³×C₄	Q₈×C₅₃	1	1	yes	characteristic
149	C₂×M₁₆	—	2	4	no	—
150	C₂×M₁₆	—	2	4	no	—
151	C₂×M₁₆	—	4	8	no	—
152	C₂²×C₈	—	2	4	no	—
153	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
154	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
155	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
156	C₂×C₄²	D₄×C₅₃	1	2	yes	—
157	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
158	C₂×C₄²	Q₈×C₅₃	1	1	yes	characteristic
159	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
160	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
161	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
162	C₄⋊C₈	—	2	4	no	—
163	C₄⋊C₈	—	2	4	no	—
164	C₄⋊C₈	—	2	4	no	—
165	C₄⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
166	C₄⋊C₈	C₂³×C₅₃	1	8	yes	—
167	C₂²⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
168	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
169	C₄×C₈	C₂³×C₅₃	1	4	yes	—
170	C₂⁴	C₄⋊C₄×C₅₃	1	1	yes	characteristic
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
174	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
175	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
176	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
177	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
178	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
179	C₂²×C₄	SD₁₆×C₅₃	1	8	yes	—
180	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
181	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
183	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
184	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
186	M₁₆	—	4	8	no	—
187	M₁₆	—	4	8	no	—
188	C₂×C₈	—	2	8	no	—
189	C₂×C₈	—	4	4	no	—
190	C₂×C₈	—	4	8	no	—
191	C₂×C₈	C₂²×C₄×C₅₃	1	8	yes	—
192	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
193	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
194	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
195	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
196	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
197	C₄²	C₂×D₄×C₅₃	1	2	yes	—
198	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
199	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
200	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
201	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
202	C₂³	—	4	1	no	—
203	C₂³	—	4	1	no	—
204	C₂³	#106×C₅₃	1	4	yes	—
205	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
206	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	#106×C₅₃	1	4	yes	—
216	C₂×C₄	#106×C₅₃	1	8	yes	—
217	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
218	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
219	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
220	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
222	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
223	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
224	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
225	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
226	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
227	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
228	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
229	C₈	—	2	8	no	—
230	C₈	—	4	8	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	4	4	no	—
236	C₂²	C₂×#106×C₅₃	1	4	yes	—
237	C₂²	C₂×#106×C₅₃	1	2	yes	—
238	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
239	C₄	—	2	1	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	1	no	—
243	C₄	C₂×#106×C₅₃	1	4	yes	—
244	C₄	C₂×#106×C₅₃	1	2	yes	—
245	C₄	C₅₃×#430	1	2	yes	—
246	C₄	C₅₃×#568	1	2	yes	—
247	C₄	C₂²×SD₁₆×C₅₃	1	2	yes	—
248	C₂	—	2	2	no	—
249	C₂	—	2	4	no	—
250	C₂	C₅₃×#2905	1	1	yes	characteristic
251	C₂	C₂²×#106×C₅₃	1	1	yes	characteristic
252	C₂	C₂×C₅₃×#430	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	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1570	C₂	semidirect
4. C₅₃×#1566	C₂	semidirect
18. C₅₃×#334	C₂²	semidirect
104. #17813	C₅₃	direct (N × H)
124. #1570	C₂×C₅₃	semidirect
125. #1566	C₂×C₅₃	semidirect
144. #334	C₂²×C₅₃	semidirect
145. C₅₃	#17813	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×#431	2
C₂×C₅₃×#423	2
C₅₃×#1570	2
C₅₃×#1566	2
#17813	53

Subgroups — C53×#17813 #27225325

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17813 #27225325