Subgroups — C₅₃×#2454 #20271391

Order 6784.

548 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#125×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#399	C₂	1	2	yes	maximal, complemented
4	C₅₃×#396	C₂	1	2	yes	maximal, complemented
5	C₅₃×#390	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#390	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#385	C₂	1	2	yes	maximal, complemented
10	C₅₃×#380	C₂	1	2	yes	maximal, complemented
11	C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
14	#125×C₅₃	—	2	4	no	—
15	C₄×D₄×C₅₃	—	2	4	no	—
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
33	#95×C₅₃	C₂²	1	1	yes	characteristic
34	#95×C₅₃	C₂²	1	1	yes	characteristic
35	#95×C₅₃	C₂²	1	2	yes	—
36	#95×C₅₃	C₂²	1	2	yes	—
37	C₂⁴×C₅₃	C₂³	1	2	yes	—
38	C₂×D₄×C₅₃	—	2	2	no	—
39	C₂×D₄×C₅₃	—	2	2	no	—
40	C₂×D₄×C₅₃	Q₈	1	4	yes	—
41	C₂²×C₄×C₅₃	—	2	4	no	—
42	C₂²×C₄×C₅₃	—	2	4	no	—
43	C₂²×C₄×C₅₃	—	4	2	no	—
44	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₅₃	—	2	4	no	—
55	C₄⋊C₄×C₅₃	—	2	4	no	—
56	C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₄⋊C₄×C₅₃	—	2	2	no	—
58	C₄⋊C₄×C₅₃	—	2	2	no	—
59	C₄⋊C₄×C₅₃	—	2	2	no	—
60	C₄⋊C₄×C₅₃	—	4	2	no	—
61	C₄⋊C₄×C₅₃	—	4	2	no	—
62	C₄⋊C₄×C₅₃	D₄	1	4	yes	—
63	C₂²⋊C₄×C₅₃	—	2	4	no	—
64	C₂²⋊C₄×C₅₃	—	2	4	no	—
65	C₂²⋊C₄×C₅₃	—	4	2	no	—
66	C₂²⋊C₄×C₅₃	—	4	2	no	—
67	C₂²⋊C₄×C₅₃	—	4	2	no	—
68	C₄²×C₅₃	—	2	2	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₂³×C₅₃	—	2	2	no	—
72	C₂³×C₅₃	—	2	2	no	—
73	C₂³×C₅₃	—	2	2	no	—
74	C₂³×C₅₃	—	2	2	no	—
75	C₂³×C₅₃	—	2	2	no	—
76	C₂³×C₅₃	—	2	2	no	—
77	C₂³×C₅₃	C₂×Q₈	1	4	yes	—
78	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
79	D₄×C₅₃	—	2	8	no	—
80	C₂×C₄×C₅₃	—	2	1	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	1	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	4	8	no	—
105	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
106	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
108	C₂×C₄×C₅₃	#41	1	2	yes	—
109	C₂×C₄×C₅₃	#41	1	2	yes	—
110	C₂×C₄×C₅₃	#41	1	2	yes	—
111	C₂²×C₅₃	—	2	4	no	—
112	C₂²×C₅₃	—	2	4	no	—
113	C₂²×C₅₃	—	2	4	no	—
114	C₂²×C₅₃	—	4	2	no	—
115	C₂²×C₅₃	—	4	2	no	—
116	C₂²×C₅₃	—	4	2	no	—
117	C₂²×C₅₃	—	4	2	no	—
118	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
119	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
121	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
122	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
123	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
124	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
125	C₄×C₅₃	—	2	2	no	—
126	C₄×C₅₃	—	2	2	no	—
127	C₄×C₅₃	—	2	4	no	—
128	C₄×C₅₃	—	4	1	no	—
129	C₄×C₅₃	—	4	2	no	—
130	C₄×C₅₃	—	4	1	no	—
131	C₄×C₅₃	—	4	1	no	—
132	C₄×C₅₃	—	4	1	no	—
133	C₄×C₅₃	—	4	2	no	—
134	C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
135	#2454	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₅₃	—	4	4	no	—
137	C₂×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₅₃	#529	1	1	yes	characteristic
139	C₂×C₅₃	#540	1	1	yes	characteristic
140	C₂×C₅₃	#546	1	1	yes	characteristic
141	C₂×C₅₃	#550	1	1	yes	characteristic
142	C₂×C₅₃	#554	1	1	yes	characteristic
143	C₂×C₅₃	#554	1	1	yes	characteristic
144	C₂×#125	C₂×C₅₃	1	1	yes	characteristic, complemented
145	C₂×C₄×D₄	C₂×C₅₃	1	1	yes	characteristic
146	#399	C₂×C₅₃	1	2	yes	complemented
147	#396	C₂×C₅₃	1	2	yes	complemented
148	#390	C₂×C₅₃	1	1	yes	characteristic
149	#390	C₂×C₅₃	1	1	yes	characteristic
150	#389	C₂×C₅₃	1	1	yes	characteristic, complemented
151	#389	C₂×C₅₃	1	1	yes	characteristic, complemented
152	#385	C₂×C₅₃	1	2	yes	complemented
153	#380	C₂×C₅₃	1	2	yes	complemented
154	C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
155	C₅₃	#2454	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
157	C₂³×C₄	C₂²×C₅₃	1	2	yes	—
158	#125	—	2	4	no	—
159	C₄×D₄	—	2	4	no	—
160	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
161	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
162	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
163	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
165	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
166	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
167	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
168	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
170	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
171	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
172	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
173	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
174	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
175	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
176	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
177	#95	C₂²×C₅₃	1	1	yes	characteristic
178	#95	C₂²×C₅₃	1	1	yes	characteristic
179	#95	C₂²×C₅₃	1	2	yes	—
180	#95	C₂²×C₅₃	1	2	yes	—
181	C₂⁴	C₂³×C₅₃	1	2	yes	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	Q₈×C₅₃	1	4	yes	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
190	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
192	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
193	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
194	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
195	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	D₄×C₅₃	1	4	yes	—
207	C₂²⋊C₄	—	2	4	no	—
208	C₂²⋊C₄	—	2	4	no	—
209	C₂²⋊C₄	—	4	2	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	4	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	C₂×Q₈×C₅₃	1	4	yes	—
222	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
223	D₄	—	2	8	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	8	no	—
249	C₂×C₄	C₂×D₄×C₅₃	1	4	yes	—
250	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
251	C₂×C₄	C₂×Q₈×C₅₃	1	2	yes	—
252	C₂×C₄	#41×C₅₃	1	2	yes	—
253	C₂×C₄	#41×C₅₃	1	2	yes	—
254	C₂×C₄	#41×C₅₃	1	2	yes	—
255	C₂²	—	2	4	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
263	C₂²	C₂²×Q₈×C₅₃	1	1	yes	characteristic
264	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
265	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
266	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
267	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
268	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	4	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	2	no	—
278	C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
279	C₂	—	4	4	no	—
280	C₂	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
281	C₂	C₅₃×#529	1	1	yes	characteristic
282	C₂	C₅₃×#540	1	1	yes	characteristic
283	C₂	C₅₃×#546	1	1	yes	characteristic
284	C₂	C₅₃×#550	1	1	yes	characteristic
285	C₂	C₅₃×#554	1	1	yes	characteristic
286	C₂	C₅₃×#554	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#125×C₅₃	C₂	semidirect
3. C₅₃×#399	C₂	semidirect
4. C₅₃×#396	C₂	semidirect
7. C₅₃×#389	C₂	semidirect
8. C₅₃×#389	C₂	semidirect
9. C₅₃×#385	C₂	semidirect
10. C₅₃×#380	C₂	semidirect
11. C₄×C₄⋊C₄×C₅₃	C₂	semidirect
135. #2454	C₅₃	direct (N × H)
144. C₂×#125	C₂×C₅₃	semidirect
146. #399	C₂×C₅₃	semidirect
147. #396	C₂×C₅₃	semidirect
150. #389	C₂×C₅₃	semidirect
151. #389	C₂×C₅₃	semidirect
152. #385	C₂×C₅₃	semidirect
153. #380	C₂×C₅₃	semidirect
154. C₄×C₄⋊C₄	C₂×C₅₃	semidirect
155. C₅₃	#2454	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×#125×C₅₃	2
C₂×C₄×D₄×C₅₃	2
C₅₃×#399	2
C₅₃×#396	2
C₅₃×#390	2
C₅₃×#389	2
C₅₃×#385	2
C₅₃×#380	2
C₄×C₄⋊C₄×C₅₃	2
#2454	53

Subgroups — C53×#2454 #20271391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2454 #20271391