Subgroups — C₅₃×#11440 #27218952

Order 13568.

450 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2112	C₂	1	2	yes	maximal
3	C₅₃×#1322	C₂	1	4	yes	maximal, complemented
4	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₅₃	—	2	2	no	—
8	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
9	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
10	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
12	C₅₃×C₂²⋊C₁₆	—	2	2	no	—
13	C₅₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
14	C₂³×C₄×C₅₃	C₈	1	1	yes	characteristic
15	C₂³×C₄×C₅₃	C₈	1	1	yes	characteristic
16	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	—	2	1	no	—
18	C₂²×C₈×C₅₃	—	2	1	no	—
19	C₂²×C₈×C₅₃	—	4	2	no	—
20	C₂²×C₈×C₅₃	—	4	2	no	—
21	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	D₄	1	2	yes	—
24	C₂²×C₈×C₅₃	D₄	1	2	yes	—
25	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	—	2	1	no	—
27	C₂×C₄²×C₅₃	—	2	1	no	—
28	C₂×C₄²×C₅₃	—	2	1	no	—
29	C₂×C₄²×C₅₃	—	2	1	no	—
30	C₂×C₄²×C₅₃	C₈	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
32	M₃₂×C₅₃	—	4	4	no	—
33	C₂×C₁₆×C₅₃	—	2	4	no	—
34	C₄⋊C₈×C₅₃	—	2	4	no	—
35	C₄⋊C₈×C₅₃	—	2	4	no	—
36	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	—	2	1	no	—
38	C₂²×C₄×C₅₃	—	2	1	no	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	—	2	4	no	—
42	C₂²×C₄×C₅₃	—	2	1	no	—
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂²×C₄×C₅₃	—	4	2	no	—
45	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	—	2	4	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	—	4	1	no	—
56	C₂×C₈×C₅₃	—	4	2	no	—
57	C₂×C₈×C₅₃	—	4	1	no	—
58	C₂×C₈×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	—	4	2	no	—
60	C₂×C₈×C₅₃	—	4	2	no	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
63	C₄²×C₅₃	—	2	2	no	—
64	C₄²×C₅₃	—	2	2	no	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₁₆×C₅₃	—	4	4	no	—
68	C₂³×C₅₃	—	2	2	no	—
69	C₂³×C₅₃	—	2	2	no	—
70	C₂³×C₅₃	—	2	2	no	—
71	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
72	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
73	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	—	2	4	no	—
75	C₂×C₄×C₅₃	—	2	4	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#99	1	2	yes	—
92	C₂×C₄×C₅₃	#101	1	2	yes	—
93	C₂×C₄×C₅₃	M₃₂	1	4	yes	complemented
94	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
96	C₈×C₅₃	—	2	2	no	—
97	C₈×C₅₃	—	2	2	no	—
98	C₈×C₅₃	—	4	2	no	—
99	#11440	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂²×C₅₃	—	2	8	no	—
101	C₂²×C₅₃	—	2	1	no	—
102	C₂²×C₅₃	—	2	1	no	—
103	C₂²×C₅₃	—	2	1	no	—
104	C₂²×C₅₃	—	2	1	no	—
105	C₂²×C₅₃	—	2	1	no	—
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	#324	1	2	yes	—
108	C₂²×C₅₃	#349	1	2	yes	—
109	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
110	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
111	C₄×C₅₃	—	2	4	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	2	1	no	—
114	C₄×C₅₃	—	4	2	no	—
115	C₄×C₅₃	#324	1	2	yes	—
116	C₄×C₅₃	C₂×#99	1	1	yes	characteristic
117	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
118	C₂²×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
119	#2112	C₂×C₅₃	1	2	yes	—
120	#1322	C₂×C₅₃	1	4	yes	complemented
121	C₂×C₅₃	—	2	2	no	—
122	C₂×C₅₃	—	2	2	no	—
123	C₂×C₅₃	—	2	2	no	—
124	C₂×C₅₃	C₂×#324	1	1	yes	characteristic
125	C₂×C₅₃	#2112	1	1	yes	characteristic
126	C₂×C₅₃	C₂×#349	1	1	yes	characteristic
127	C₂³×C₈	C₄×C₅₃	1	1	yes	characteristic
128	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
129	C₂²×C₄²	C₄×C₅₃	1	1	yes	characteristic
130	C₂×M₃₂	—	2	2	no	—
131	C₂×C₄⋊C₈	—	2	2	no	—
132	C₂×C₄⋊C₈	—	2	2	no	—
133	C₂×C₄⋊C₈	C₄×C₅₃	1	2	yes	—
134	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
135	C₂²⋊C₁₆	—	2	2	no	—
136	C₂²⋊C₁₆	C₂²×C₅₃	1	4	yes	—
137	C₅₃	#11440	1	1	yes	characteristic, direct product factor, Sylow
138	C₂³×C₄	C₈×C₅₃	1	1	yes	characteristic
139	C₂³×C₄	C₈×C₅₃	1	1	yes	characteristic
140	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	—	2	1	no	—
143	C₂²×C₈	—	4	2	no	—
144	C₂²×C₈	—	4	2	no	—
145	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
146	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
147	C₂²×C₈	D₄×C₅₃	1	2	yes	—
148	C₂²×C₈	D₄×C₅₃	1	2	yes	—
149	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
150	C₂×C₄²	—	2	1	no	—
151	C₂×C₄²	—	2	1	no	—
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	—	2	1	no	—
154	C₂×C₄²	C₈×C₅₃	1	2	yes	—
155	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
156	M₃₂	—	4	4	no	—
157	C₂×C₁₆	—	2	4	no	—
158	C₄⋊C₈	—	2	4	no	—
159	C₄⋊C₈	—	2	4	no	—
160	C₂⁴	C₂×C₈×C₅₃	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
170	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
173	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
174	C₂²×C₄	M₁₆×C₅₃	1	1	yes	characteristic
175	C₂²×C₄	M₁₆×C₅₃	1	1	yes	characteristic
176	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
186	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	2	no	—
191	C₁₆	—	4	4	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	2	2	no	—
195	C₂³	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
196	C₂³	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
197	C₂³	C₂²×C₈×C₅₃	1	1	yes	characteristic
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
214	C₂×C₄	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
215	C₂×C₄	#99×C₅₃	1	2	yes	—
216	C₂×C₄	#101×C₅₃	1	2	yes	—
217	C₂×C₄	M₃₂×C₅₃	1	4	yes	complemented
218	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
219	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	4	2	no	—
223	C₂²	—	2	8	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	C₅₃×#324	1	2	yes	—
231	C₂²	C₅₃×#349	1	2	yes	—
232	C₂²	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
233	C₂²	C₂×M₃₂×C₅₃	1	2	yes	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	4	no	—
237	C₄	—	4	2	no	—
238	C₄	C₅₃×#324	1	2	yes	—
239	C₄	C₂×#99×C₅₃	1	1	yes	characteristic
240	C₄	C₂×#101×C₅₃	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	2	2	no	—
243	C₂	—	2	2	no	—
244	C₂	C₂×C₅₃×#324	1	1	yes	characteristic
245	C₂	C₅₃×#2112	1	1	yes	characteristic
246	C₂	C₂×C₅₃×#349	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1322	C₂	semidirect
93. C₂×C₄×C₅₃	M₃₂	semidirect
99. #11440	C₅₃	direct (N × H)
120. #1322	C₂×C₅₃	semidirect
137. C₅₃	#11440	direct (N × H)
217. C₂×C₄	M₃₂×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊C₈×C₅₃	2
C₅₃×#2112	2
C₅₃×#1322	2
#11440	53

Subgroups — C53×#11440 #27218952

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11440 #27218952