Subgroups — C₅₃×#21723 #27229235

Order 13568.

734 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#499	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2330	C₂	1	2	yes	maximal
3	C₅₃×#1992	C₂	1	2	yes	maximal
4	C₅₃×#1944	C₂	1	2	yes	maximal
5	C₅₃×#1942	C₂	1	2	yes	maximal
6	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1776	C₂	1	2	yes	maximal
8	C₄×#107×C₅₃	C₂	1	2	yes	maximal
9	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
10	C₂×#125×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#499	C₄	1	8	yes	complemented
12	C₂×#107×C₅₃	C₂²	1	2	yes	—
13	C₂×#107×C₅₃	C₂²	1	2	yes	—
14	C₂×#107×C₅₃	C₂²	1	2	yes	—
15	C₂×#106×C₅₃	C₂²	1	2	yes	—
16	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#389	—	2	4	no	—
23	C₅₃×#389	C₂²	1	4	yes	—
24	C₅₃×#387	—	2	4	no	—
25	C₅₃×#384	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
27	C₅₃×#376	C₂²	1	1	yes	characteristic
28	C₅₃×#340	C₂²	1	4	yes	—
29	C₅₃×#340	C₂²	1	4	yes	—
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	#125×C₅₃	—	2	4	no	—
35	#125×C₅₃	C₂×C₄	1	8	yes	—
36	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
41	C₂×C₄²×C₅₃	—	2	4	no	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	#107×C₅₃	—	2	4	no	—
46	#107×C₅₃	—	2	4	no	—
47	#107×C₅₃	C₂×C₄	1	8	yes	—
48	#106×C₅₃	C₂×C₄	1	8	yes	—
49	#97×C₅₃	—	2	2	no	—
50	#97×C₅₃	—	2	2	no	—
51	#97×C₅₃	Q₈	1	4	yes	—
52	C₄×C₈×C₅₃	—	2	2	no	—
53	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
54	C₄×C₈×C₅₃	Q₈	1	4	yes	—
55	#95×C₅₃	—	2	4	no	—
56	#95×C₅₃	—	2	4	no	—
57	#95×C₅₃	D₄	1	2	yes	—
58	#95×C₅₃	D₄	1	2	yes	—
59	C₂²×C₄×C₅₃	—	2	4	no	—
60	C₂²×C₄×C₅₃	—	2	4	no	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₅₃	—	2	1	no	—
67	C₂×C₈×C₅₃	—	2	1	no	—
68	C₂×C₈×C₅₃	—	2	1	no	—
69	C₂×C₈×C₅₃	—	2	1	no	—
70	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₅₃	#41	1	2	yes	—
77	C₂×C₈×C₅₃	#41	1	2	yes	—
78	C₄⋊C₄×C₅₃	—	2	16	no	—
79	C₄⋊C₄×C₅₃	—	2	4	no	—
80	C₄⋊C₄×C₅₃	—	2	4	no	—
81	C₄⋊C₄×C₅₃	—	2	4	no	—
82	C₄⋊C₄×C₅₃	—	8	4	no	—
83	C₄⋊C₄×C₅₃	—	8	4	no	—
84	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
85	C₄²×C₅₃	—	2	1	no	—
86	C₄²×C₅₃	—	2	1	no	—
87	C₄²×C₅₃	—	8	4	no	—
88	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
90	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
91	C₄²×C₅₃	#41	1	2	yes	—
92	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
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	8	no	—
98	C₂×C₄×C₅₃	—	2	1	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	—	4	4	no	—
103	C₂×C₄×C₅₃	—	4	4	no	—
104	C₂×C₄×C₅₃	—	4	4	no	—
105	C₂×C₄×C₅₃	—	4	4	no	—
106	C₂×C₄×C₅₃	—	4	4	no	—
107	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
108	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
109	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
115	C₈×C₅₃	—	2	2	no	—
116	C₈×C₅₃	—	2	2	no	—
117	C₈×C₅₃	C₄×Q₈	1	4	yes	—
118	C₈×C₅₃	C₄×Q₈	1	4	yes	—
119	#21723	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
121	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₂²×C₅₃	#532	1	1	yes	characteristic
123	C₂²×C₅₃	#576	1	2	yes	—
124	C₂²×C₅₃	#578	1	2	yes	—
125	C₄×C₅₃	—	2	4	no	—
126	C₄×C₅₃	—	2	4	no	—
127	C₄×C₅₃	—	4	1	no	—
128	C₄×C₅₃	—	4	1	no	—
129	C₄×C₅₃	—	4	8	no	—
130	C₄×C₅₃	—	8	4	no	—
131	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
132	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
133	C₄×C₅₃	#533	1	1	yes	characteristic
134	C₄×C₅₃	#533	1	1	yes	characteristic
135	C₂×#499	C₂×C₅₃	1	1	yes	characteristic
136	#2330	C₂×C₅₃	1	2	yes	—
137	#1992	C₂×C₅₃	1	2	yes	—
138	#1944	C₂×C₅₃	1	2	yes	—
139	#1942	C₂×C₅₃	1	2	yes	—
140	#1845	C₂×C₅₃	1	1	yes	characteristic
141	#1776	C₂×C₅₃	1	2	yes	—
142	C₄×#107	C₂×C₅₃	1	2	yes	—
143	#1726	C₂×C₅₃	1	1	yes	characteristic
144	C₂×C₅₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
145	C₂×C₅₃	#2951	1	2	yes	—
146	C₂×C₅₃	#3164	1	2	yes	—
147	C₂×C₅₃	#3176	1	1	yes	characteristic
148	C₂×C₅₃	#3177	1	1	yes	characteristic
149	C₂×#125	C₂²×C₅₃	1	2	yes	—
150	#499	C₄×C₅₃	1	8	yes	complemented
151	C₂×#107	C₂²×C₅₃	1	2	yes	—
152	C₂×#107	C₂²×C₅₃	1	2	yes	—
153	C₂×#107	C₂²×C₅₃	1	2	yes	—
154	C₂×#106	C₂²×C₅₃	1	2	yes	—
155	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
156	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
157	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
161	#389	—	2	4	no	—
162	#389	C₂²×C₅₃	1	4	yes	—
163	#387	—	2	4	no	—
164	#384	C₂²×C₅₃	1	2	yes	—
165	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
166	#376	C₂²×C₅₃	1	1	yes	characteristic
167	#340	C₂²×C₅₃	1	4	yes	—
168	#340	C₂²×C₅₃	1	4	yes	—
169	C₅₃	#21723	1	1	yes	characteristic, direct product factor, Sylow
170	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
174	#125	—	2	4	no	—
175	#125	C₂×C₄×C₅₃	1	8	yes	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	—	2	4	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
180	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
181	C₂×C₄²	—	2	4	no	—
182	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
185	#107	—	2	4	no	—
186	#107	—	2	4	no	—
187	#107	C₂×C₄×C₅₃	1	8	yes	—
188	#106	C₂×C₄×C₅₃	1	8	yes	—
189	#97	—	2	2	no	—
190	#97	—	2	2	no	—
191	#97	Q₈×C₅₃	1	4	yes	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
194	C₄×C₈	Q₈×C₅₃	1	4	yes	—
195	#95	—	2	4	no	—
196	#95	—	2	4	no	—
197	#95	D₄×C₅₃	1	2	yes	—
198	#95	D₄×C₅₃	1	2	yes	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
205	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
211	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
212	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
213	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
214	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
215	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
216	C₂×C₈	#41×C₅₃	1	2	yes	—
217	C₂×C₈	#41×C₅₃	1	2	yes	—
218	C₄⋊C₄	—	2	16	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	8	4	no	—
223	C₄⋊C₄	—	8	4	no	—
224	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	8	4	no	—
228	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
229	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
230	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
231	C₄²	#41×C₅₃	1	2	yes	—
232	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
248	C₂×C₄	C₄⋊Q₈×C₅₃	1	4	yes	—
249	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
250	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
251	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
252	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
253	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
254	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	2	2	no	—
257	C₈	C₄×Q₈×C₅₃	1	4	yes	—
258	C₈	C₄×Q₈×C₅₃	1	4	yes	—
259	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
260	C₂²	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
261	C₂²	C₅₃×#532	1	1	yes	characteristic
262	C₂²	C₅₃×#576	1	2	yes	—
263	C₂²	C₅₃×#578	1	2	yes	—
264	C₄	—	2	4	no	—
265	C₄	—	2	4	no	—
266	C₄	—	4	8	no	—
267	C₄	—	4	1	no	—
268	C₄	—	4	1	no	—
269	C₄	—	8	4	no	—
270	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
271	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
272	C₄	C₅₃×#533	1	1	yes	characteristic
273	C₄	C₅₃×#533	1	1	yes	characteristic
274	C₂	C₄×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
275	C₂	C₅₃×#2951	1	2	yes	—
276	C₂	C₅₃×#3164	1	2	yes	—
277	C₂	C₅₃×#3176	1	1	yes	characteristic
278	C₂	C₅₃×#3177	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₅₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
11. C₅₃×#499	C₄	semidirect
119. #21723	C₅₃	direct (N × H)
150. #499	C₄×C₅₃	semidirect
169. C₅₃	#21723	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×#499	2
C₅₃×#2330	2
C₅₃×#1992	2
C₅₃×#1944	2
C₅₃×#1942	2
C₅₃×#1845	2
C₅₃×#1776	2
C₄×#107×C₅₃	2
C₅₃×#1726	2
#21723	53

Subgroups — C53×#21723 #27229235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21723 #27229235