Subgroups — C₅₃×#22307 #27229819

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1984	C₂	1	4	yes	maximal
3	C₅₃×#1939	C₂	1	2	yes	maximal
4	C₅₃×#1916	C₂	1	4	yes	maximal
5	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
6	C₈×C₄⋊C₄×C₅₃	C₂	1	2	yes	maximal
7	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#391	C₄	1	2	yes	—
16	C₅₃×#389	C₄	1	2	yes	—
17	C₅₃×#389	C₄	1	2	yes	—
18	C₅₃×#384	C₄	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₄³×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#336	C₂²	1	2	yes	—
23	C₅₃×#336	C₂²	1	2	yes	—
24	C₅₃×#336	C₂²	1	4	yes	—
25	C₅₃×#336	C₂²	1	4	yes	—
26	C₅₃×#336	C₂²	1	2	yes	—
27	C₅₃×#336	C₂²	1	2	yes	—
28	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
31	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
32	C₄⋊Q₈×C₅₃	C₈	1	8	yes	complemented
33	C₄×Q₈×C₅₃	—	4	4	no	—
34	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₅₃	—	2	4	no	—
49	C₄⋊C₈×C₅₃	—	4	4	no	—
50	C₄×C₈×C₅₃	—	2	2	no	—
51	C₄×C₈×C₅₃	—	4	4	no	—
52	C₄×C₈×C₅₃	Q₈	1	4	yes	complemented
53	#95×C₅₃	C₂×C₄	1	4	yes	—
54	#95×C₅₃	C₂×C₄	1	4	yes	—
55	C₂×Q₈×C₅₃	—	4	2	no	—
56	C₂×Q₈×C₅₃	—	4	2	no	—
57	C₂×Q₈×C₅₃	C₂×C₈	1	8	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	—	2	2	no	—
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	—	2	4	no	—
71	C₂×C₈×C₅₃	—	2	4	no	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	4	no	—
75	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
76	C₄⋊C₄×C₅₃	—	2	4	no	—
77	C₄⋊C₄×C₅₃	—	4	2	no	—
78	C₄⋊C₄×C₅₃	—	4	2	no	—
79	C₄⋊C₄×C₅₃	—	4	4	no	—
80	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
81	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
82	C₄²×C₅₃	—	2	2	no	—
83	C₄²×C₅₃	—	2	4	no	—
84	C₄²×C₅₃	—	2	4	no	—
85	C₄²×C₅₃	—	2	2	no	—
86	C₄²×C₅₃	—	2	2	no	—
87	C₄²×C₅₃	—	4	1	no	—
88	C₄²×C₅₃	—	4	1	no	—
89	C₄²×C₅₃	C₂×C₈	1	4	yes	—
90	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
91	C₄²×C₅₃	#41	1	2	yes	—
92	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
93	Q₈×C₅₃	—	4	8	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	4	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	4	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	4	no	—
111	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
115	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₅₃	#131	1	2	yes	—
118	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
121	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
122	C₈×C₅₃	—	2	4	no	—
123	C₈×C₅₃	—	4	2	no	—
124	C₈×C₅₃	—	4	4	no	—
125	#22307	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
127	C₂²×C₅₃	#518	1	1	yes	characteristic
128	C₂²×C₅₃	#519	1	1	yes	characteristic
129	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
130	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
131	C₂²×C₅₃	#568	1	1	yes	characteristic
132	C₂²×C₅₃	#568	1	1	yes	characteristic
133	C₄×C₅₃	—	2	2	no	—
134	C₄×C₅₃	—	2	8	no	—
135	C₄×C₅₃	—	2	2	no	—
136	C₄×C₅₃	—	2	2	no	—
137	C₄×C₅₃	—	4	4	no	—
138	C₄×C₅₃	—	4	1	no	—
139	C₄×C₅₃	—	4	1	no	—
140	C₄×C₅₃	C₈×Q₈	1	4	yes	—
141	C₄×C₅₃	#542	1	2	yes	—
142	C₄×C₅₃	#544	1	2	yes	—
143	C₄×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
144	#1984	C₂×C₅₃	1	4	yes	—
145	#1939	C₂×C₅₃	1	2	yes	—
146	#1916	C₂×C₅₃	1	4	yes	—
147	#1831	C₂×C₅₃	1	1	yes	characteristic
148	C₈×C₄⋊C₄	C₂×C₅₃	1	2	yes	—
149	#1744	C₂×C₅₃	1	1	yes	characteristic
150	C₂×C₅₃	#2331	1	1	yes	characteristic
151	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
152	C₂×C₅₃	#2961	1	1	yes	characteristic
153	C₂×C₅₃	#2972	1	1	yes	characteristic
154	C₂×C₅₃	#2981	1	1	yes	characteristic
155	C₂×C₅₃	#2982	1	1	yes	characteristic
156	C₂×C₅₃	#2985	1	1	yes	characteristic
157	C₂×C₄⋊Q₈	C₄×C₅₃	1	1	yes	characteristic
158	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
159	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
160	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
161	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂²×C₅₃	1	4	yes	—
163	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
164	#391	C₄×C₅₃	1	2	yes	—
165	#389	C₄×C₅₃	1	2	yes	—
166	#389	C₄×C₅₃	1	2	yes	—
167	#384	C₄×C₅₃	1	1	yes	characteristic
168	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
169	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
170	C₄³	C₂²×C₅₃	1	1	yes	characteristic
171	#336	C₂²×C₅₃	1	2	yes	—
172	#336	C₂²×C₅₃	1	2	yes	—
173	#336	C₂²×C₅₃	1	2	yes	—
174	#336	C₂²×C₅₃	1	4	yes	—
175	#336	C₂²×C₅₃	1	4	yes	—
176	#336	C₂²×C₅₃	1	2	yes	—
177	C₅₃	#22307	1	1	yes	characteristic, direct product factor, Sylow
178	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
179	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
180	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
181	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
182	C₄⋊Q₈	C₈×C₅₃	1	8	yes	complemented
183	C₄×Q₈	—	4	4	no	—
184	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
185	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
186	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
187	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
188	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
189	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
190	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
191	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
192	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
193	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
195	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
196	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
197	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
198	C₄⋊C₈	—	2	4	no	—
199	C₄⋊C₈	—	4	4	no	—
200	C₄×C₈	—	2	2	no	—
201	C₄×C₈	—	4	4	no	—
202	C₄×C₈	Q₈×C₅₃	1	4	yes	complemented
203	#95	C₂×C₄×C₅₃	1	4	yes	—
204	#95	C₂×C₄×C₅₃	1	4	yes	—
205	C₂×Q₈	—	4	2	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂×Q₈	C₂×C₈×C₅₃	1	8	yes	—
208	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
209	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
210	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
212	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
213	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
215	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
216	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	C₂×C₈×C₅₃	1	8	yes	—
231	C₄⋊C₄	C₂×C₈×C₅₃	1	8	yes	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	C₂×C₈×C₅₃	1	4	yes	—
240	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
241	C₄²	#41×C₅₃	1	2	yes	—
242	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
243	Q₈	—	4	8	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
262	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
263	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
264	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
265	C₂×C₄	C₂²×C₈×C₅₃	1	8	yes	—
266	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
267	C₂×C₄	#131×C₅₃	1	2	yes	—
268	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
269	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
270	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
271	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
272	C₈	—	2	4	no	—
273	C₈	—	4	2	no	—
274	C₈	—	4	4	no	—
275	C₂²	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
276	C₂²	C₅₃×#518	1	1	yes	characteristic
277	C₂²	C₅₃×#519	1	1	yes	characteristic
278	C₂²	C₂³×C₈×C₅₃	1	1	yes	characteristic
279	C₂²	C₂×#131×C₅₃	1	1	yes	characteristic
280	C₂²	C₅₃×#568	1	1	yes	characteristic
281	C₂²	C₅₃×#568	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	8	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	4	1	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	1	no	—
289	C₄	C₈×Q₈×C₅₃	1	4	yes	—
290	C₄	C₅₃×#542	1	2	yes	—
291	C₄	C₅₃×#544	1	2	yes	—
292	C₂	C₅₃×#2331	1	1	yes	characteristic
293	C₂	C₂×C₈×Q₈×C₅₃	1	1	yes	characteristic
294	C₂	C₅₃×#2961	1	1	yes	characteristic
295	C₂	C₅₃×#2972	1	1	yes	characteristic
296	C₂	C₅₃×#2981	1	1	yes	characteristic
297	C₂	C₅₃×#2982	1	1	yes	characteristic
298	C₂	C₅₃×#2985	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
32. C₄⋊Q₈×C₅₃	C₈	semidirect
52. C₄×C₈×C₅₃	Q₈	semidirect
125. #22307	C₅₃	direct (N × H)
177. C₅₃	#22307	direct (N × H)
182. C₄⋊Q₈	C₈×C₅₃	semidirect
202. 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
C₄×C₄⋊Q₈×C₅₃	2
C₅₃×#1984	2
C₅₃×#1939	2
C₅₃×#1916	2
C₅₃×#1831	2
C₈×C₄⋊C₄×C₅₃	2
C₅₃×#1744	2
#22307	53

Subgroups — C53×#22307 #27229819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22307 #27229819