Subgroups — C₅₃×#33492 #27241004

Order 13568.

564 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2929	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2169	C₂	1	4	yes	maximal, complemented
3	C₅₃×#2168	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2164	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2113	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2112	C₂	1	2	yes	maximal, complemented
7	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#525	C₄	1	1	yes	characteristic
10	C₂×M₃₂×C₅₃	C₂²	1	4	yes	—
11	C₂×M₃₂×C₅₃	C₂²	1	2	yes	complemented
12	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	complemented
13	C₅₃×#435	C₄	1	4	yes	—
14	C₈×D₄×C₅₃	C₂²	1	4	yes	—
15	C₅₃×#424	C₄	1	1	yes	characteristic
16	C₅₃×#408	C₄	1	1	yes	characteristic
17	C₅₃×#407	C₄	1	1	yes	characteristic
18	C₅₃×#405	C₂²	1	1	yes	characteristic
19	C₅₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
20	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
21	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
22	C₅₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
23	C₅₃×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
24	C₅₃×#346	C₂²	1	2	yes	complemented
25	C₄×C₁₆×C₅₃	C₂²	1	2	yes	complemented
26	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	#131×C₅₃	—	4	2	no	—
29	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	—	4	2	no	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	#123×C₅₃	C₈	1	2	yes	—
39	C₂²⋊Q₈×C₅₃	C₈	1	2	yes	—
40	#121×C₅₃	C₈	1	2	yes	—
41	#120×C₅₃	C₈	1	2	yes	—
42	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
43	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
44	M₃₂×C₅₃	—	4	4	no	—
45	M₃₂×C₅₃	—	4	2	no	—
46	C₂×C₁₆×C₅₃	—	4	2	no	—
47	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
48	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
49	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
50	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
51	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
52	#97×C₅₃	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₅₃	C₂³	1	2	yes	—
54	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
55	#41×C₅₃	—	4	1	no	—
56	#41×C₅₃	—	4	1	no	—
57	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
58	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
59	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	—	4	1	no	—
62	C₂²×C₄×C₅₃	—	4	1	no	—
63	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	M₁₆×C₅₃	—	4	1	no	—
69	M₁₆×C₅₃	—	4	2	no	—
70	M₁₆×C₅₃	—	4	1	no	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	—	2	4	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	4	1	no	—
76	C₂×C₈×C₅₃	—	4	2	no	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
83	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
84	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
85	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
86	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
87	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
88	C₁₆×C₅₃	—	2	4	no	—
89	C₁₆×C₅₃	—	2	4	no	—
90	C₂³×C₅₃	—	4	1	no	—
91	C₂³×C₅₃	—	4	2	no	—
92	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
93	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
94	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
95	Q₈×C₅₃	—	4	1	no	—
96	D₄×C₅₃	—	4	4	no	—
97	D₄×C₅₃	—	4	1	no	—
98	D₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	4	1	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
111	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
112	C₈×C₅₃	—	2	2	no	—
113	C₈×C₅₃	—	2	1	no	—
114	C₈×C₅₃	—	2	2	no	—
115	C₈×C₅₃	—	2	2	no	—
116	C₈×C₅₃	H(5,𝔽₂)	1	2	yes	—
117	#33492	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	—	2	4	no	—
120	C₂²×C₅₃	—	2	2	no	—
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	—	2	1	no	—
123	C₂²×C₅₃	—	2	2	no	—
124	C₂²×C₅₃	—	2	2	no	—
125	C₂²×C₅₃	—	4	4	no	—
126	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
127	C₄×C₅₃	—	2	4	no	—
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	#518	1	1	yes	characteristic
132	C₄×C₅₃	#568	1	1	yes	characteristic
133	#2929	C₂×C₅₃	1	1	yes	characteristic
134	#2169	C₂×C₅₃	1	4	yes	complemented
135	#2168	C₂×C₅₃	1	4	yes	complemented
136	#2164	C₂×C₅₃	1	2	yes	complemented
137	#2113	C₂×C₅₃	1	2	yes	complemented
138	#2112	C₂×C₅₃	1	2	yes	complemented
139	C₂×C₅₃	—	4	2	no	—
140	C₂×C₅₃	—	4	2	no	—
141	C₂×C₅₃	—	4	1	no	—
142	C₂×C₅₃	#2972	1	1	yes	characteristic
143	C₂×C₅₃	#3407	1	2	yes	—
144	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
145	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
146	#525	C₄×C₅₃	1	1	yes	characteristic
147	C₂×M₃₂	C₂²×C₅₃	1	4	yes	—
148	C₂×M₃₂	C₂²×C₅₃	1	2	yes	complemented
149	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	complemented
150	#435	C₄×C₅₃	1	4	yes	—
151	C₈×D₄	C₂²×C₅₃	1	4	yes	—
152	#424	C₄×C₅₃	1	1	yes	characteristic
153	#408	C₄×C₅₃	1	1	yes	characteristic
154	#407	C₄×C₅₃	1	1	yes	characteristic
155	#405	C₂²×C₅₃	1	1	yes	characteristic
156	C₄⋊C₁₆	C₂²×C₅₃	1	4	yes	complemented
157	C₂²⋊C₁₆	C₂²×C₅₃	1	4	yes	—
158	C₂²⋊C₁₆	C₂²×C₅₃	1	2	yes	—
159	C₂²⋊C₁₆	C₂²×C₅₃	1	2	yes	—
160	C₂²⋊C₁₆	C₂²×C₅₃	1	4	yes	complemented
161	#346	C₂²×C₅₃	1	2	yes	complemented
162	C₄×C₁₆	C₂²×C₅₃	1	2	yes	complemented
163	C₅₃	#33492	1	1	yes	characteristic, direct product factor, Sylow
164	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
165	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
166	#131	—	4	2	no	—
167	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
168	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
169	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
170	C₂²×C₈	—	4	2	no	—
171	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
172	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
173	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
175	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
176	#123	C₈×C₅₃	1	2	yes	—
177	C₂²⋊Q₈	C₈×C₅₃	1	2	yes	—
178	#121	C₈×C₅₃	1	2	yes	—
179	#120	C₈×C₅₃	1	2	yes	—
180	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
181	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
182	M₃₂	—	4	4	no	—
183	M₃₂	—	4	2	no	—
184	C₂×C₁₆	—	4	2	no	—
185	C₂×C₁₆	C₂³×C₅₃	1	4	yes	—
186	C₂×C₁₆	C₂³×C₅₃	1	4	yes	—
187	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
188	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
189	C₂²⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
190	#97	C₂×C₄×C₅₃	1	2	yes	—
191	C₄×C₈	C₂³×C₅₃	1	2	yes	—
192	C₂⁴	C₂×C₈×C₅₃	1	1	yes	characteristic
193	#41	—	4	1	no	—
194	#41	—	4	1	no	—
195	C₂×Q₈	C₂×C₈×C₅₃	1	1	yes	characteristic
196	C₂×D₄	C₂×C₈×C₅₃	1	4	yes	—
197	C₂×D₄	C₂×C₈×C₅₃	1	2	yes	—
198	C₂×D₄	C₂×C₈×C₅₃	1	1	yes	characteristic
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	—	4	1	no	—
201	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
202	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
205	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
206	M₁₆	—	4	2	no	—
207	M₁₆	—	4	1	no	—
208	M₁₆	—	4	1	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
216	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
217	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
218	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
219	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
220	C₄⋊C₄	C₂×C₈×C₅₃	1	2	yes	—
221	C₄⋊C₄	C₂×C₈×C₅₃	1	4	yes	—
222	C₂²⋊C₄	C₂×C₈×C₅₃	1	2	yes	—
223	C₂²⋊C₄	C₂×C₈×C₅₃	1	4	yes	—
224	C₂²⋊C₄	C₂×C₈×C₅₃	1	4	yes	—
225	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
226	C₁₆	—	2	4	no	—
227	C₁₆	—	2	4	no	—
228	C₂³	—	4	1	no	—
229	C₂³	—	4	2	no	—
230	C₂³	C₂²×C₈×C₅₃	1	1	yes	characteristic
231	C₂³	C₂²×C₈×C₅₃	1	2	yes	—
232	C₂³	C₂²×C₈×C₅₃	1	2	yes	—
233	Q₈	—	4	1	no	—
234	D₄	—	4	4	no	—
235	D₄	—	4	1	no	—
236	D₄	—	4	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	C₂²×C₈×C₅₃	1	4	yes	—
246	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
247	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
248	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
249	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	2	1	no	—
252	C₈	—	2	2	no	—
253	C₈	—	2	2	no	—
254	C₈	H(5,𝔽₂)×C₅₃	1	2	yes	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	4	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	4	4	no	—
263	C₂²	C₂³×C₈×C₅₃	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	4	no	—
268	C₄	C₅₃×#518	1	1	yes	characteristic
269	C₄	C₅₃×#568	1	1	yes	characteristic
270	C₂	—	4	1	no	—
271	C₂	—	4	2	no	—
272	C₂	—	4	2	no	—
273	C₂	C₅₃×#2972	1	1	yes	characteristic
274	C₂	C₅₃×#3407	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₅₃	C₂⁴
commutator	C₂²	C₂³×C₈×C₅₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂³×C₈
prime radical	C₅₃×#525	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2169	C₂	semidirect
3. C₅₃×#2168	C₂	semidirect
4. C₅₃×#2164	C₂	semidirect
5. C₅₃×#2113	C₂	semidirect
6. C₅₃×#2112	C₂	semidirect
11. C₂×M₃₂×C₅₃	C₂²	semidirect
12. C₂²×C₁₆×C₅₃	C₂²	semidirect
19. C₅₃×C₄⋊C₁₆	C₂²	semidirect
23. C₅₃×C₂²⋊C₁₆	C₂²	semidirect
24. C₅₃×#346	C₂²	semidirect
25. C₄×C₁₆×C₅₃	C₂²	semidirect
117. #33492	C₅₃	direct (N × H)
134. #2169	C₂×C₅₃	semidirect
135. #2168	C₂×C₅₃	semidirect
136. #2164	C₂×C₅₃	semidirect
137. #2113	C₂×C₅₃	semidirect
138. #2112	C₂×C₅₃	semidirect
148. C₂×M₃₂	C₂²×C₅₃	semidirect
149. C₂²×C₁₆	C₂²×C₅₃	semidirect
156. C₄⋊C₁₆	C₂²×C₅₃	semidirect
160. C₂²⋊C₁₆	C₂²×C₅₃	semidirect
161. #346	C₂²×C₅₃	semidirect
162. C₄×C₁₆	C₂²×C₅₃	semidirect
163. C₅₃	#33492	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#2929	2
C₅₃×#2169	2
C₅₃×#2168	2
C₅₃×#2164	2
C₅₃×#2113	2
C₅₃×#2112	2
#33492	53

Subgroups — C53×#33492 #27241004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33492 #27241004