Subgroups — C₅₃×#8625 #27216137

Order 13568.

342 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#446	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1467	C₂	1	2	yes	maximal
5	C₅₃×#1290	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#446	—	2	1	no	—
8	C₅₃×#446	—	2	1	no	—
9	C₅₃×#446	—	2	1	no	—
10	C₅₃×#446	—	2	1	no	—
11	C₅₃×#432	C₂²	1	2	yes	—
12	C₅₃×#432	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#97×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×#324	—	2	4	no	—
22	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×Q₈×C₅₃	—	2	1	no	—
27	C₄×Q₈×C₅₃	—	2	1	no	—
28	C₄×Q₈×C₅₃	—	2	1	no	—
29	C₄×Q₈×C₅₃	—	2	1	no	—
30	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	C₄⋊C₈×C₅₃	—	2	1	no	—
37	C₄⋊C₈×C₅₃	—	2	1	no	—
38	C₄⋊C₈×C₅₃	—	2	2	no	—
39	C₄⋊C₈×C₅₃	—	2	2	no	—
40	C₂²⋊C₈×C₅₃	—	2	2	no	—
41	C₂²⋊C₈×C₅₃	—	2	2	no	—
42	#97×C₅₃	—	2	2	no	—
43	#97×C₅₃	—	2	2	no	—
44	#97×C₅₃	—	2	2	no	—
45	#97×C₅₃	—	2	2	no	—
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
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	C₂×Q₈×C₅₃	—	2	1	no	—
51	C₂×Q₈×C₅₃	—	2	1	no	—
52	C₂×Q₈×C₅₃	—	4	2	no	—
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	4	1	no	—
61	C₂×C₈×C₅₃	—	4	2	no	—
62	C₂×C₈×C₅₃	—	4	2	no	—
63	C₂×C₈×C₅₃	—	4	2	no	—
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₄⋊C₄×C₅₃	—	2	2	no	—
69	C₄⋊C₄×C₅₃	—	2	2	no	—
70	C₄⋊C₄×C₅₃	—	2	1	no	—
71	C₄⋊C₄×C₅₃	—	2	1	no	—
72	C₄²×C₅₃	—	2	2	no	—
73	C₄²×C₅₃	—	2	2	no	—
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₅₃	#41	1	1	yes	characteristic
77	C₄²×C₅₃	#41	1	1	yes	characteristic
78	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	Q₈×C₅₃	—	4	2	no	—
80	Q₈×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	4	no	—
87	C₂×C₄×C₅₃	—	4	1	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
95	C₈×C₅₃	—	2	1	no	—
96	C₈×C₅₃	—	2	1	no	—
97	C₈×C₅₃	—	2	1	no	—
98	C₈×C₅₃	—	4	2	no	—
99	C₈×C₅₃	—	8	2	no	—
100	C₈×C₅₃	—	8	2	no	—
101	C₈×C₅₃	#101	1	2	yes	—
102	#8625	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂²×C₅₃	—	2	1	no	—
104	C₂²×C₅₃	—	2	1	no	—
105	C₂²×C₅₃	#380	1	1	yes	characteristic
106	C₂²×C₅₃	#430	1	2	yes	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	4	2	no	—
113	C₄×C₅₃	—	4	2	no	—
114	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
115	C₄×C₅₃	#413	1	1	yes	characteristic
116	C₂×#446	C₂×C₅₃	1	1	yes	characteristic
117	#1568	C₂×C₅₃	1	1	yes	characteristic
118	#1568	C₂×C₅₃	1	1	yes	characteristic
119	#1467	C₂×C₅₃	1	2	yes	—
120	#1290	C₂×C₅₃	1	2	yes	—
121	C₂×C₅₃	—	2	2	no	—
122	C₂×C₅₃	#1801	1	1	yes	characteristic
123	C₂×C₅₃	#1811	1	1	yes	characteristic
124	C₂×C₅₃	#1812	1	1	yes	characteristic
125	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
126	#446	—	2	1	no	—
127	#446	—	2	1	no	—
128	#446	—	2	1	no	—
129	#446	—	2	1	no	—
130	#432	C₂²×C₅₃	1	2	yes	—
131	#432	C₂²×C₅₃	1	2	yes	—
132	C₂×C₄⋊C₈	C₄×C₅₃	1	2	yes	—
133	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
134	C₂×#97	C₄×C₅₃	1	2	yes	—
135	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
136	#335	—	2	1	no	—
137	#335	—	2	1	no	—
138	#334	—	2	1	no	—
139	#334	—	2	1	no	—
140	#324	—	2	4	no	—
141	C₅₃	#8625	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²×Q₈	Q₈×C₅₃	1	1	yes	characteristic
143	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
144	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
145	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
146	C₄×Q₈	—	2	1	no	—
147	C₄×Q₈	—	2	1	no	—
148	C₄×Q₈	—	2	1	no	—
149	C₄×Q₈	—	2	1	no	—
150	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
151	C₂×C₄⋊C₄	Q₈×C₅₃	1	1	yes	characteristic
152	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
153	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	1	no	—
157	C₄⋊C₈	—	2	1	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₂²⋊C₈	—	2	2	no	—
161	C₂²⋊C₈	—	2	2	no	—
162	#97	—	2	2	no	—
163	#97	—	2	2	no	—
164	#97	—	2	2	no	—
165	#97	—	2	2	no	—
166	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
167	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
168	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
169	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
170	C₂×Q₈	—	2	1	no	—
171	C₂×Q₈	—	2	1	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
174	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
175	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
176	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
177	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
185	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
186	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
187	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
195	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
196	C₄²	#41×C₅₃	1	1	yes	characteristic
197	C₄²	#41×C₅₃	1	1	yes	characteristic
198	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
199	Q₈	—	4	2	no	—
200	Q₈	—	4	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
210	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
211	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
212	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
213	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
214	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	4	2	no	—
219	C₈	—	8	2	no	—
220	C₈	—	8	2	no	—
221	C₈	#101×C₅₃	1	2	yes	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	C₅₃×#380	1	1	yes	characteristic
225	C₂²	C₅₃×#430	1	2	yes	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	2	no	—
233	C₄	C₂×#101×C₅₃	1	1	yes	characteristic
234	C₄	C₅₃×#413	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	C₅₃×#1801	1	1	yes	characteristic
237	C₂	C₅₃×#1811	1	1	yes	characteristic
238	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	C₂³×C₅₃	C₂×C₄⋊C₄
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
102. #8625	C₅₃	direct (N × H)
141. C₅₃	#8625	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×#446	2
C₅₃×#1568	2
C₅₃×#1467	2
C₅₃×#1290	2
#8625	53

Subgroups — C53×#8625 #27216137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8625 #27216137