Subgroups — C₅₃×#12600 #27220112

Order 13568.

514 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×C₂²⋊C₁₆	C₂	1	2	yes	maximal
3	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1374	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
9	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
10	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#405	—	2	2	no	—
12	C₅₃×#405	—	2	2	no	—
13	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
14	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×C₂²⋊C₁₆	C₄	1	8	yes	complemented
18	C₅₃×#346	—	2	2	no	—
19	C₄×C₁₆×C₅₃	—	2	2	no	—
20	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₅₃	—	2	2	no	—
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂×M₁₆×C₅₃	C₈	1	4	yes	—
24	C₂²×C₈×C₅₃	—	2	1	no	—
25	C₂²×C₈×C₅₃	—	2	1	no	—
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	#117×C₅₃	—	2	2	no	—
33	#117×C₅₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
35	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₆×C₅₃	—	2	2	no	—
39	C₂×C₁₆×C₅₃	—	2	2	no	—
40	C₂×C₁₆×C₅₃	—	2	2	no	—
41	C₂×C₁₆×C₅₃	—	2	2	no	—
42	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
43	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
44	#97×C₅₃	—	2	2	no	—
45	#97×C₅₃	—	2	2	no	—
46	C₄×C₈×C₅₃	—	2	1	no	—
47	C₄×C₈×C₅₃	—	2	1	no	—
48	C₄×C₈×C₅₃	D₄	1	2	yes	—
49	C₄×C₈×C₅₃	D₄	1	2	yes	—
50	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	—	2	1	no	—
53	C₂²×C₄×C₅₃	—	2	1	no	—
54	C₂²×C₄×C₅₃	—	2	1	no	—
55	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	M₁₆×C₅₃	—	2	8	no	—
62	C₂×C₈×C₅₃	—	2	1	no	—
63	C₂×C₈×C₅₃	—	2	1	no	—
64	C₂×C₈×C₅₃	—	2	1	no	—
65	C₂×C₈×C₅₃	—	2	1	no	—
66	C₂×C₈×C₅₃	—	2	4	no	—
67	C₂×C₈×C₅₃	—	2	4	no	—
68	C₂×C₈×C₅₃	C₄²	1	2	yes	—
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
71	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	#41	1	2	yes	—
77	C₄⋊C₄×C₅₃	—	2	4	no	—
78	C₂²⋊C₄×C₅₃	—	2	4	no	—
79	C₄²×C₅₃	—	2	1	no	—
80	C₄²×C₅₃	—	2	1	no	—
81	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₅₃	M₁₆	1	2	yes	—
83	C₁₆×C₅₃	—	2	4	no	—
84	C₁₆×C₅₃	—	2	4	no	—
85	C₂³×C₅₃	—	2	1	no	—
86	C₂³×C₅₃	—	2	1	no	—
87	C₂³×C₅₃	—	2	4	no	—
88	C₂³×C₅₃	C₄×C₈	1	2	yes	—
89	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	4	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	4	no	—
99	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
100	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
101	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
107	C₈×C₅₃	—	2	2	no	—
108	C₈×C₅₃	—	2	2	no	—
109	C₈×C₅₃	C₂²⋊C₈	1	4	yes	—
110	C₈×C₅₃	C₄×D₄	1	2	yes	—
111	C₈×C₅₃	C₄×D₄	1	2	yes	—
112	#12600	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₅₃	—	2	4	no	—
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	—	4	2	no	—
116	C₂²×C₅₃	—	4	2	no	—
117	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
118	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₂²×C₅₃	#431	1	1	yes	characteristic
120	C₂²×C₅₃	#504	1	4	yes	—
121	C₄×C₅₃	—	2	4	no	—
122	C₄×C₅₃	—	2	2	no	—
123	C₄×C₅₃	—	4	2	no	—
124	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
125	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
126	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
127	C₄×C₅₃	#431	1	1	yes	characteristic
128	C₂×#405	C₂×C₅₃	1	1	yes	characteristic
129	C₂×C₂²⋊C₁₆	C₂×C₅₃	1	2	yes	—
130	C₂×#346	C₂×C₅₃	1	1	yes	characteristic, complemented
131	C₂×C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
132	#1374	C₂×C₅₃	1	2	yes	complemented
133	C₂×C₅₃	—	4	2	no	—
134	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
135	C₂×C₅₃	#2108	1	2	yes	—
136	C₂×C₅₃	#2113	1	2	yes	—
137	C₂×C₅₃	#2164	1	2	yes	—
138	C₂²×M₁₆	C₄×C₅₃	1	1	yes	characteristic
139	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
140	C₂×#117	C₄×C₅₃	1	1	yes	characteristic
141	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	—
142	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	—
143	#405	—	2	2	no	—
144	#405	—	2	2	no	—
145	C₂×#97	C₄×C₅₃	1	1	yes	characteristic
146	C₂×#97	C₄×C₅₃	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
149	C₂²⋊C₁₆	C₄×C₅₃	1	8	yes	complemented
150	#346	—	2	2	no	—
151	C₄×C₁₆	—	2	2	no	—
152	C₅₃	#12600	1	1	yes	characteristic, direct product factor, Sylow
153	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	C₈×C₅₃	1	4	yes	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
161	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
162	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
163	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
165	#117	—	2	2	no	—
166	#117	—	2	2	no	—
167	C₂×C₄⋊C₄	C₈×C₅₃	1	2	yes	—
168	C₂×C₂²⋊C₄	C₈×C₅₃	1	2	yes	—
169	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
170	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	—
176	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	—
177	#97	—	2	2	no	—
178	#97	—	2	2	no	—
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	D₄×C₅₃	1	2	yes	—
182	C₄×C₈	D₄×C₅₃	1	2	yes	—
183	C₂⁴	C₂×C₈×C₅₃	1	1	yes	characteristic
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
189	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
190	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
192	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
193	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
194	M₁₆	—	2	8	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	C₄²×C₅₃	1	2	yes	—
202	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
203	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
204	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
205	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
206	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
207	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
208	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
209	C₂×C₈	#41×C₅₃	1	2	yes	—
210	C₄⋊C₄	—	2	4	no	—
211	C₂²⋊C₄	—	2	4	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
215	C₄²	M₁₆×C₅₃	1	2	yes	—
216	C₁₆	—	2	4	no	—
217	C₁₆	—	2	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	1	no	—
221	C₂³	C₄×C₈×C₅₃	1	2	yes	—
222	C₂³	C₂²×C₈×C₅₃	1	1	yes	characteristic
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
233	C₂×C₄	C₂²⋊C₈×C₅₃	1	4	yes	—
234	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
236	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
237	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
238	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
239	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	C₂²⋊C₈×C₅₃	1	4	yes	—
243	C₈	C₄×D₄×C₅₃	1	2	yes	—
244	C₈	C₄×D₄×C₅₃	1	2	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
250	C₂²	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
251	C₂²	C₅₃×#431	1	1	yes	characteristic
252	C₂²	C₅₃×#504	1	4	yes	—
253	C₄	—	2	2	no	—
254	C₄	—	2	4	no	—
255	C₄	—	4	2	no	—
256	C₄	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
257	C₄	C₄×M₁₆×C₅₃	1	1	yes	characteristic
258	C₄	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
259	C₄	C₅₃×#431	1	1	yes	characteristic
260	C₂	—	4	2	no	—
261	C₂	C₄×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
262	C₂	C₅₃×#2108	1	2	yes	—
263	C₂	C₅₃×#2113	1	2	yes	—
264	C₂	C₅₃×#2164	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂×C₅₃×#346	C₂	semidirect
4. C₂×C₄×C₁₆×C₅₃	C₂	semidirect
5. C₅₃×#1374	C₂	semidirect
17. C₅₃×C₂²⋊C₁₆	C₄	semidirect
112. #12600	C₅₃	direct (N × H)
130. C₂×#346	C₂×C₅₃	semidirect
131. C₂×C₄×C₁₆	C₂×C₅₃	semidirect
132. #1374	C₂×C₅₃	semidirect
149. C₂²⋊C₁₆	C₄×C₅₃	semidirect
152. C₅₃	#12600	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×#405	2
C₂×C₅₃×C₂²⋊C₁₆	2
C₂×C₅₃×#346	2
C₂×C₄×C₁₆×C₅₃	2
C₅₃×#1374	2
#12600	53

Subgroups — C53×#12600 #27220112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12600 #27220112