Subgroups — C₅₃×#21725 #27229237

Order 13568.

750 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×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2331	C₂	1	2	yes	maximal
3	C₅₃×#1993	C₂	1	2	yes	maximal
4	C₅₃×#1943	C₂	1	2	yes	maximal
5	C₅₃×#1942	C₂	1	2	yes	maximal
6	C₅₃×#1846	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×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#500	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×#391	—	2	4	no	—
23	C₅₃×#389	C₂²	1	4	yes	—
24	C₅₃×#387	—	2	4	no	—
25	C₅₃×#382	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×Q₈×C₅₃	—	2	2	no	—
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	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₅₃	—	2	4	no	—
36	C₄⋊Q₈×C₅₃	C₂×C₄	1	8	yes	—
37	C₂×C₄⋊C₄×C₅₃	—	2	2	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	C₂×C₄	1	8	yes	—
47	#107×C₅₃	C₂×C₄	1	8	yes	—
48	#106×C₅₃	—	2	4	no	—
49	#97×C₅₃	—	2	2	no	—
50	#97×C₅₃	—	2	2	no	—
51	#97×C₅₃	Q₈	1	4	yes	complemented
52	C₄×C₈×C₅₃	—	2	2	no	—
53	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
54	C₄×C₈×C₅₃	Q₈	1	4	yes	complemented
55	#95×C₅₃	—	2	4	no	—
56	#95×C₅₃	—	2	2	no	—
57	#95×C₅₃	—	2	2	no	—
58	#95×C₅₃	—	2	4	no	—
59	#95×C₅₃	D₄	1	2	yes	—
60	#95×C₅₃	D₄	1	2	yes	—
61	C₂×Q₈×C₅₃	—	2	8	no	—
62	C₂×Q₈×C₅₃	—	8	4	no	—
63	C₂²×C₄×C₅₃	—	2	4	no	—
64	C₂²×C₄×C₅₃	—	2	4	no	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₅₃	—	2	1	no	—
71	C₂×C₈×C₅₃	—	2	1	no	—
72	C₂×C₈×C₅₃	—	2	1	no	—
73	C₂×C₈×C₅₃	—	2	1	no	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
77	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
78	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
79	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
80	C₂×C₈×C₅₃	#41	1	2	yes	—
81	C₂×C₈×C₅₃	#41	1	2	yes	—
82	C₄⋊C₄×C₅₃	—	2	8	no	—
83	C₄⋊C₄×C₅₃	—	2	4	no	—
84	C₄⋊C₄×C₅₃	—	2	4	no	—
85	C₄⋊C₄×C₅₃	—	2	4	no	—
86	C₄⋊C₄×C₅₃	—	8	4	no	—
87	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
88	C₄²×C₅₃	—	2	1	no	—
89	C₄²×C₅₃	—	2	1	no	—
90	C₄²×C₅₃	—	8	4	no	—
91	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
92	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
93	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
94	C₄²×C₅₃	#41	1	2	yes	—
95	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₅₃	—	8	8	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	8	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	4	4	no	—
106	C₂×C₄×C₅₃	—	4	4	no	—
107	C₂×C₄×C₅₃	—	4	4	no	—
108	C₂×C₄×C₅₃	—	4	4	no	—
109	C₂×C₄×C₅₃	—	4	4	no	—
110	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
113	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
119	C₈×C₅₃	—	2	2	no	—
120	C₈×C₅₃	—	2	2	no	—
121	C₈×C₅₃	C₄×Q₈	1	4	yes	—
122	C₈×C₅₃	C₄×Q₈	1	4	yes	—
123	#21725	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
125	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
126	C₂²×C₅₃	#532	1	1	yes	characteristic
127	C₂²×C₅₃	#576	1	2	yes	—
128	C₂²×C₅₃	#578	1	2	yes	—
129	C₄×C₅₃	—	2	4	no	—
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	4	2	no	—
133	C₄×C₅₃	—	4	8	no	—
134	C₄×C₅₃	—	8	4	no	—
135	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
136	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
137	C₄×C₅₃	#533	1	1	yes	characteristic
138	C₄×C₅₃	#533	1	1	yes	characteristic
139	C₂×#500	C₂×C₅₃	1	1	yes	characteristic
140	#2331	C₂×C₅₃	1	2	yes	—
141	#1993	C₂×C₅₃	1	2	yes	—
142	#1943	C₂×C₅₃	1	2	yes	—
143	#1942	C₂×C₅₃	1	2	yes	—
144	#1846	C₂×C₅₃	1	1	yes	characteristic
145	#1776	C₂×C₅₃	1	2	yes	—
146	C₄×#107	C₂×C₅₃	1	2	yes	—
147	#1726	C₂×C₅₃	1	1	yes	characteristic
148	C₂×C₅₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
149	C₂×C₅₃	#2948	1	1	yes	characteristic
150	C₂×C₅₃	#2950	1	1	yes	characteristic
151	C₂×C₅₃	#3164	1	2	yes	—
152	C₂×C₅₃	#3175	1	2	yes	—
153	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
154	#500	C₄×C₅₃	1	8	yes	complemented
155	C₂×#107	C₂²×C₅₃	1	2	yes	—
156	C₂×#107	C₂²×C₅₃	1	2	yes	—
157	C₂×#107	C₂²×C₅₃	1	2	yes	—
158	C₂×#106	C₂²×C₅₃	1	2	yes	—
159	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
160	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
161	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
165	#391	—	2	4	no	—
166	#389	C₂²×C₅₃	1	4	yes	—
167	#387	—	2	4	no	—
168	#382	C₂²×C₅₃	1	2	yes	—
169	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
170	#376	C₂²×C₅₃	1	1	yes	characteristic
171	#340	C₂²×C₅₃	1	4	yes	—
172	#340	C₂²×C₅₃	1	4	yes	—
173	C₅₃	#21725	1	1	yes	characteristic, direct product factor, Sylow
174	C₂²×Q₈	—	2	2	no	—
175	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
177	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
179	C₄⋊Q₈	—	2	4	no	—
180	C₄⋊Q₈	C₂×C₄×C₅₃	1	8	yes	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	4	no	—
183	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
184	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
185	C₂×C₄²	—	2	4	no	—
186	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
187	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
188	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
189	#107	—	2	4	no	—
190	#107	C₂×C₄×C₅₃	1	8	yes	—
191	#107	C₂×C₄×C₅₃	1	8	yes	—
192	#106	—	2	4	no	—
193	#97	—	2	2	no	—
194	#97	—	2	2	no	—
195	#97	Q₈×C₅₃	1	4	yes	complemented
196	C₄×C₈	—	2	2	no	—
197	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
198	C₄×C₈	Q₈×C₅₃	1	4	yes	complemented
199	#95	—	2	4	no	—
200	#95	—	2	2	no	—
201	#95	—	2	2	no	—
202	#95	—	2	4	no	—
203	#95	D₄×C₅₃	1	2	yes	—
204	#95	D₄×C₅₃	1	2	yes	—
205	C₂×Q₈	—	2	8	no	—
206	C₂×Q₈	—	8	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
210	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
213	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
219	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
220	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
221	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
222	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
223	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
224	C₂×C₈	#41×C₅₃	1	2	yes	—
225	C₂×C₈	#41×C₅₃	1	2	yes	—
226	C₄⋊C₄	—	2	8	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	8	4	no	—
231	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	8	4	no	—
235	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
236	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
237	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
238	C₄²	#41×C₅₃	1	2	yes	—
239	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
240	Q₈	—	8	8	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
255	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
256	C₂×C₄	C₄⋊Q₈×C₅₃	1	4	yes	—
257	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
258	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
260	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
261	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
262	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	2	2	no	—
265	C₈	C₄×Q₈×C₅₃	1	4	yes	—
266	C₈	C₄×Q₈×C₅₃	1	4	yes	—
267	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
268	C₂²	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
269	C₂²	C₅₃×#532	1	1	yes	characteristic
270	C₂²	C₅₃×#576	1	2	yes	—
271	C₂²	C₅₃×#578	1	2	yes	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	4	no	—
275	C₄	—	4	8	no	—
276	C₄	—	4	2	no	—
277	C₄	—	8	4	no	—
278	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
279	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
280	C₄	C₅₃×#533	1	1	yes	characteristic
281	C₄	C₅₃×#533	1	1	yes	characteristic
282	C₂	C₄×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
283	C₂	C₅₃×#2948	1	1	yes	characteristic
284	C₂	C₅₃×#2950	1	1	yes	characteristic
285	C₂	C₅₃×#3164	1	2	yes	—
286	C₂	C₅₃×#3175	1	2	yes	—

	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×#500	C₄	semidirect
51. #97×C₅₃	Q₈	semidirect
54. C₄×C₈×C₅₃	Q₈	semidirect
123. #21725	C₅₃	direct (N × H)
154. #500	C₄×C₅₃	semidirect
173. C₅₃	#21725	direct (N × H)
195. #97	Q₈×C₅₃	semidirect
198. C₄×C₈	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#500	2
C₅₃×#2331	2
C₅₃×#1993	2
C₅₃×#1943	2
C₅₃×#1942	2
C₅₃×#1846	2
C₅₃×#1776	2
C₄×#107×C₅₃	2
C₅₃×#1726	2
#21725	53

Subgroups — C53×#21725 #27229237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21725 #27229237