Subgroups — C₄×C₅₃×#500 #27229232

Order 13568.

878 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	C₄×C₄⋊Q₈×C₅₃	C₂	1	2	yes	maximal
3	C₅₃×#1990	C₂	1	2	yes	maximal
4	C₅₃×#1942	C₂	1	4	yes	maximal
5	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic
6	C₄×#107×C₅₃	C₂	1	4	yes	maximal
7	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₅₃	—	2	2	no	—
10	C₅₃×#500	C₄	1	8	yes	direct product factor
11	C₂×#107×C₅₃	C₂²	1	4	yes	—
12	C₂×#107×C₅₃	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#391	—	2	2	no	—
18	C₅₃×#389	—	2	2	no	—
19	C₅₃×#389	C₂²	1	4	yes	—
20	C₅₃×#384	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
23	C₄³×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#340	C₂²	1	8	yes	—
25	C₂²×Q₈×C₅₃	—	2	2	no	—
26	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
27	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
28	C₄⋊Q₈×C₅₃	—	2	4	no	—
29	C₄⋊Q₈×C₅₃	C₂×C₄	1	8	yes	—
30	C₄×Q₈×C₅₃	—	4	8	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
36	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
37	C₂×C₄²×C₅₃	—	2	4	no	—
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
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	8	no	—
46	#107×C₅₃	C₂×C₄	1	16	yes	—
47	C₄×C₈×C₅₃	—	2	4	no	—
48	C₄×C₈×C₅₃	—	2	2	no	—
49	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₅₃	Q₈	1	8	yes	complemented
51	#95×C₅₃	—	2	4	no	—
52	#95×C₅₃	—	2	4	no	—
53	C₂×Q₈×C₅₃	—	2	8	no	—
54	C₂×Q₈×C₅₃	—	4	4	no	—
55	C₂×Q₈×C₅₃	—	4	4	no	—
56	C₂²×C₄×C₅₃	—	2	4	no	—
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
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	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
69	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
70	C₂×C₈×C₅₃	#41	1	4	yes	—
71	C₄⋊C₄×C₅₃	—	2	8	no	—
72	C₄⋊C₄×C₅₃	—	2	4	no	—
73	C₄⋊C₄×C₅₃	—	2	8	no	—
74	C₄⋊C₄×C₅₃	—	4	8	no	—
75	C₄⋊C₄×C₅₃	—	4	4	no	—
76	C₄⋊C₄×C₅₃	—	4	4	no	—
77	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
78	C₄²×C₅₃	—	2	2	no	—
79	C₄²×C₅₃	—	2	2	no	—
80	C₄²×C₅₃	—	2	2	no	—
81	C₄²×C₅₃	—	4	8	no	—
82	C₄²×C₅₃	D₈	1	4	yes	—
83	C₄²×C₅₃	Q₁₆	1	4	yes	—
84	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
85	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
86	C₄²×C₅₃	#41	1	2	yes	—
87	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
88	Q₈×C₅₃	—	4	16	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	8	no	—
96	C₂×C₄×C₅₃	—	4	4	no	—
97	C₂×C₄×C₅₃	—	4	4	no	—
98	C₂×C₄×C₅₃	—	4	4	no	—
99	C₂×C₄×C₅₃	—	4	4	no	—
100	C₂×C₄×C₅₃	—	4	4	no	—
101	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
104	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
109	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
110	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
111	C₂×C₄×C₅₃	#135	1	2	yes	—
112	C₂×C₄×C₅₃	#135	1	2	yes	—
113	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
115	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
116	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
117	C₈×C₅₃	—	2	4	no	—
118	C₈×C₅₃	C₄×Q₈	1	8	yes	complemented
119	C₄×#500	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	C₂²×D₈	1	1	yes	characteristic
124	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
125	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
126	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
127	C₄×C₅₃	—	2	2	no	—
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	—	2	2	no	—
130	C₄×C₅₃	—	4	8	no	—
131	C₄×C₅₃	—	8	4	no	—
132	C₄×C₅₃	C₄×D₈	1	4	yes	—
133	C₄×C₅₃	C₄×Q₁₆	1	4	yes	—
134	C₄×C₅₃	#500	1	4	yes	direct product factor
135	C₄×C₅₃	C₂×C₄×Q₈	1	2	yes	—
136	C₄×C₅₃	#533	1	2	yes	—
137	C₂×#500	C₂×C₅₃	1	1	yes	characteristic
138	C₄×C₄⋊Q₈	C₂×C₅₃	1	2	yes	—
139	#1990	C₂×C₅₃	1	2	yes	—
140	#1942	C₂×C₅₃	1	4	yes	—
141	#1845	C₂×C₅₃	1	1	yes	characteristic
142	C₄×#107	C₂×C₅₃	1	4	yes	—
143	C₄²×C₈	C₂×C₅₃	1	1	yes	characteristic
144	C₂×C₅₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
145	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
146	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
147	C₂×C₅₃	C₂×#500	1	1	yes	characteristic
148	C₂×C₅₃	#3160	1	1	yes	characteristic
149	C₂×C₅₃	#3169	1	1	yes	characteristic
150	C₂×C₅₃	#3170	1	1	yes	characteristic
151	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
152	C₂×C₄×Q₈	—	2	2	no	—
153	#500	C₄×C₅₃	1	8	yes	direct product factor
154	C₂×#107	C₂²×C₅₃	1	4	yes	—
155	C₂×#107	C₂²×C₅₃	1	4	yes	—
156	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
157	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
160	#391	—	2	2	no	—
161	#389	—	2	2	no	—
162	#389	C₂²×C₅₃	1	4	yes	—
163	#384	C₂²×C₅₃	1	2	yes	—
164	C₄×C₄⋊C₄	—	2	2	no	—
165	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
166	C₄³	C₂²×C₅₃	1	1	yes	characteristic
167	#340	C₂²×C₅₃	1	8	yes	—
168	C₅₃	C₄×#500	1	1	yes	characteristic, direct product factor, Sylow
169	C₂²×Q₈	—	2	2	no	—
170	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
171	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
172	C₄⋊Q₈	—	2	4	no	—
173	C₄⋊Q₈	C₂×C₄×C₅₃	1	8	yes	—
174	C₄×Q₈	—	4	8	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	4	no	—
178	C₂×C₄⋊C₄	—	2	2	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	D₄×C₅₃	1	1	yes	characteristic
183	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
184	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
185	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
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	8	no	—
190	#107	C₂×C₄×C₅₃	1	16	yes	—
191	C₄×C₈	—	2	4	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
194	C₄×C₈	Q₈×C₅₃	1	8	yes	complemented
195	#95	—	2	4	no	—
196	#95	—	2	4	no	—
197	C₂×Q₈	—	2	8	no	—
198	C₂×Q₈	—	4	4	no	—
199	C₂×Q₈	—	4	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
208	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
212	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
213	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
214	C₂×C₈	#41×C₅₃	1	4	yes	—
215	C₄⋊C₄	—	2	8	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	8	no	—
218	C₄⋊C₄	—	4	8	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	4	8	no	—
226	C₄²	D₈×C₅₃	1	4	yes	—
227	C₄²	Q₁₆×C₅₃	1	4	yes	—
228	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
229	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
230	C₄²	#41×C₅₃	1	2	yes	—
231	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
232	Q₈	—	4	16	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
246	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
247	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
248	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
249	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
250	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
251	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
252	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
253	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
254	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
255	C₂×C₄	#135×C₅₃	1	2	yes	—
256	C₂×C₄	#135×C₅₃	1	2	yes	—
257	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
258	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
259	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
260	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
261	C₈	—	2	4	no	—
262	C₈	C₄×Q₈×C₅₃	1	8	yes	complemented
263	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
264	C₂²	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
265	C₂²	C₅₃×#532	1	1	yes	characteristic
266	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
267	C₂²	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
268	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
269	C₂²	C₂×#135×C₅₃	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	4	8	no	—
274	C₄	—	8	4	no	—
275	C₄	C₄×D₈×C₅₃	1	4	yes	—
276	C₄	C₄×Q₁₆×C₅₃	1	4	yes	—
277	C₄	C₅₃×#500	1	4	yes	direct product factor
278	C₄	C₂×C₄×Q₈×C₅₃	1	2	yes	—
279	C₄	C₅₃×#533	1	2	yes	—
280	C₂	C₄×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
281	C₂	C₂×C₄×D₈×C₅₃	1	1	yes	characteristic
282	C₂	C₂×C₄×Q₁₆×C₅₃	1	1	yes	characteristic
283	C₂	C₂×C₅₃×#500	1	1	yes	characteristic
284	C₂	C₅₃×#3160	1	1	yes	characteristic
285	C₂	C₅₃×#3169	1	1	yes	characteristic
286	C₂	C₅₃×#3170	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle, prime radical	C₂³×C₅₃	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₅₃×#500	C₄	direct (N × H)
50. C₄×C₈×C₅₃	Q₈	semidirect
118. C₈×C₅₃	C₄×Q₈	semidirect
119. C₄×#500	C₅₃	direct (N × H)
134. C₄×C₅₃	#500	direct (N × H)
153. #500	C₄×C₅₃	direct (N × H)
168. C₅₃	C₄×#500	direct (N × H)
194. C₄×C₈	Q₈×C₅₃	semidirect
262. C₈	C₄×Q₈×C₅₃	semidirect
277. C₄	C₅₃×#500	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×#500	2
C₄×C₄⋊Q₈×C₅₃	2
C₅₃×#1990	2
C₅₃×#1942	2
C₅₃×#1845	2
C₄×#107×C₅₃	2
C₄²×C₈×C₅₃	2
C₄×#500	53

Subgroups — C4×C53×#500 #27229232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅₃×#500 #27229232