Subgroups — C₅₃×#22707 #27230219

Order 13568.

910 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2354	C₂	1	2	yes	maximal
3	C₅₃×#1947	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1854	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1834	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1808	C₂	1	2	yes	maximal, complemented
7	C₄×#106×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1735	C₂	1	2	yes	maximal, complemented
9	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#117×C₅₃	C₂²	1	4	yes	—
12	C₈×D₄×C₅₃	C₄	1	8	yes	complemented
13	C₂×#107×C₅₃	C₂²	1	4	yes	—
14	C₂×#106×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#390	—	2	2	no	—
19	C₅₃×#388	—	2	4	no	—
20	C₅₃×#384	C₂²	1	2	yes	—
21	C₅₃×#380	C₂²	1	4	yes	—
22	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
23	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₅₃×#340	C₂²	1	4	yes	—
25	C₅₃×#340	C₂²	1	1	yes	characteristic
26	C₅₃×#340	C₂²	1	2	yes	—
27	C₅₃×#340	C₂²	1	1	yes	characteristic
28	C₅₃×#340	C₂²	1	2	yes	—
29	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
31	C₂²×C₈×C₅₃	—	4	4	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
37	C₄×D₄×C₅₃	Q₈	1	4	yes	—
38	#117×C₅₃	—	2	8	no	—
39	#117×C₅₃	—	4	4	no	—
40	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
46	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
47	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
48	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
49	C₂×C₄²×C₅₃	—	2	4	no	—
50	C₂×C₄²×C₅₃	—	2	2	no	—
51	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	#107×C₅₃	—	4	4	no	—
53	#106×C₅₃	—	2	4	no	—
54	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
55	C₂²⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
56	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
57	#95×C₅₃	—	2	4	no	—
58	#95×C₅₃	—	2	2	no	—
59	C₂⁴×C₅₃	C₂×D₄	1	2	yes	—
60	C₂×D₄×C₅₃	—	2	4	no	—
61	C₂×D₄×C₅₃	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₅₃	—	2	4	no	—
63	C₂²×C₄×C₅₃	—	2	4	no	—
64	C₂²×C₄×C₅₃	—	2	2	no	—
65	C₂²×C₄×C₅₃	—	2	2	no	—
66	C₂²×C₄×C₅₃	—	2	4	no	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
72	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	4	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	4	8	no	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
80	C₄⋊C₄×C₅₃	—	2	2	no	—
81	C₄⋊C₄×C₅₃	—	2	2	no	—
82	C₄⋊C₄×C₅₃	—	2	4	no	—
83	C₄⋊C₄×C₅₃	—	2	2	no	—
84	C₄⋊C₄×C₅₃	—	2	4	no	—
85	C₄⋊C₄×C₅₃	—	4	4	no	—
86	C₄⋊C₄×C₅₃	—	8	2	no	—
87	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
88	C₄⋊C₄×C₅₃	#41	1	4	yes	—
89	C₂²⋊C₄×C₅₃	—	4	8	no	—
90	C₂²⋊C₄×C₅₃	—	8	4	no	—
91	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
92	C₄²×C₅₃	—	4	8	no	—
93	C₄²×C₅₃	—	4	4	no	—
94	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
95	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
96	C₂³×C₅₃	—	2	4	no	—
97	C₂³×C₅₃	—	2	4	no	—
98	C₂³×C₅₃	—	2	2	no	—
99	C₂³×C₅₃	—	2	2	no	—
100	C₂³×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
101	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
102	D₄×C₅₃	—	2	8	no	—
103	C₂×C₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	4	no	—
107	C₂×C₄×C₅₃	—	2	4	no	—
108	C₂×C₄×C₅₃	—	2	8	no	—
109	C₂×C₄×C₅₃	—	2	4	no	—
110	C₂×C₄×C₅₃	—	4	2	no	—
111	C₂×C₄×C₅₃	—	4	2	no	—
112	C₂×C₄×C₅₃	—	4	4	no	—
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	4	2	no	—
115	C₂×C₄×C₅₃	—	4	4	no	—
116	C₂×C₄×C₅₃	—	4	2	no	—
117	C₂×C₄×C₅₃	—	4	4	no	—
118	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
119	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₅₃	#135	1	4	yes	—
122	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
125	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
126	C₈×C₅₃	—	2	2	no	—
127	C₈×C₅₃	—	2	2	no	—
128	C₈×C₅₃	—	2	4	no	—
129	#22707	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₅₃	—	2	4	no	—
131	C₂²×C₅₃	—	2	4	no	—
132	C₂²×C₅₃	—	2	4	no	—
133	C₂²×C₅₃	—	4	4	no	—
134	C₂²×C₅₃	—	4	4	no	—
135	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₅₃	#546	1	2	yes	—
137	C₂²×C₅₃	C₂×#135	1	2	yes	—
138	C₂²×C₅₃	#576	1	1	yes	characteristic
139	C₂²×C₅₃	#578	1	1	yes	characteristic
140	C₄×C₅₃	—	2	4	no	—
141	C₄×C₅₃	—	2	2	no	—
142	C₄×C₅₃	—	4	4	no	—
143	C₄×C₅₃	—	8	4	no	—
144	C₄×C₅₃	—	8	2	no	—
145	C₄×C₅₃	#427	1	4	yes	—
146	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
147	C₄×C₅₃	#518	1	1	yes	characteristic
148	C₄×C₅₃	#554	1	2	yes	—
149	C₂×C₈×D₄	C₂×C₅₃	1	1	yes	characteristic
150	#2354	C₂×C₅₃	1	2	yes	—
151	#1947	C₂×C₅₃	1	2	yes	complemented
152	#1854	C₂×C₅₃	1	4	yes	complemented
153	#1834	C₂×C₅₃	1	1	yes	characteristic, complemented
154	#1808	C₂×C₅₃	1	2	yes	complemented
155	C₄×#106	C₂×C₅₃	1	1	yes	characteristic, complemented
156	#1735	C₂×C₅₃	1	2	yes	complemented
157	C₂×C₅₃	—	4	4	no	—
158	C₂×C₅₃	#2349	1	1	yes	characteristic
159	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
160	C₂×C₅₃	#2913	1	1	yes	characteristic
161	C₂×C₅₃	#3302	1	2	yes	—
162	C₂×C₅₃	#3305	1	2	yes	—
163	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
164	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
165	C₂×#117	C₂²×C₅₃	1	4	yes	—
166	C₈×D₄	C₄×C₅₃	1	8	yes	complemented
167	C₂×#107	C₂²×C₅₃	1	4	yes	—
168	C₂×#106	C₂²×C₅₃	1	2	yes	—
169	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
170	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
171	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
172	#390	—	2	2	no	—
173	#388	—	2	4	no	—
174	#384	C₂²×C₅₃	1	2	yes	—
175	#380	C₂²×C₅₃	1	4	yes	—
176	C₄×C₄⋊C₄	—	2	2	no	—
177	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
178	#340	C₂²×C₅₃	1	1	yes	characteristic
179	#340	C₂²×C₅₃	1	2	yes	—
180	#340	C₂²×C₅₃	1	2	yes	—
181	#340	C₂²×C₅₃	1	4	yes	—
182	#340	C₂²×C₅₃	1	1	yes	characteristic
183	C₅₃	#22707	1	1	yes	characteristic, direct product factor, Sylow
184	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
185	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
186	C₂²×C₈	—	4	4	no	—
187	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	—
188	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
190	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
191	C₄×D₄	C₂×C₄×C₅₃	1	4	yes	—
192	C₄×D₄	Q₈×C₅₃	1	4	yes	—
193	#117	—	2	8	no	—
194	#117	—	4	4	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
198	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
199	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
200	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
201	C₂×C₂²⋊C₄	—	2	4	no	—
202	C₂×C₂²⋊C₄	—	2	4	no	—
203	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
207	#107	—	4	4	no	—
208	#106	—	2	4	no	—
209	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
210	C₂²⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
211	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
212	#95	—	2	4	no	—
213	#95	—	2	2	no	—
214	C₂⁴	C₂×D₄×C₅₃	1	2	yes	—
215	C₂×D₄	—	2	4	no	—
216	C₂×D₄	C₄⋊C₄×C₅₃	1	4	yes	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
223	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
227	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	4	8	no	—
232	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
233	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
234	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	4	no	—
241	C₄⋊C₄	—	8	2	no	—
242	C₄⋊C₄	C₄⋊C₄×C₅₃	1	4	yes	—
243	C₄⋊C₄	#41×C₅₃	1	4	yes	—
244	C₂²⋊C₄	—	4	8	no	—
245	C₂²⋊C₄	—	8	4	no	—
246	C₂²⋊C₄	C₄⋊C₄×C₅₃	1	8	yes	—
247	C₄²	—	4	8	no	—
248	C₄²	—	4	4	no	—
249	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
250	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
251	C₂³	—	2	4	no	—
252	C₂³	—	2	4	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
256	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
257	D₄	—	2	8	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
274	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
275	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
276	C₂×C₄	#135×C₅₃	1	4	yes	—
277	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
278	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
279	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
280	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
281	C₈	—	2	2	no	—
282	C₈	—	2	2	no	—
283	C₈	—	2	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	4	no	—
289	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
290	C₂²	C₅₃×#546	1	2	yes	—
291	C₂²	C₂×#135×C₅₃	1	2	yes	—
292	C₂²	C₅₃×#576	1	1	yes	characteristic
293	C₂²	C₅₃×#578	1	1	yes	characteristic
294	C₄	—	2	4	no	—
295	C₄	—	2	2	no	—
296	C₄	—	4	4	no	—
297	C₄	—	8	4	no	—
298	C₄	—	8	2	no	—
299	C₄	C₅₃×#427	1	4	yes	—
300	C₄	C₄×#41×C₅₃	1	1	yes	characteristic
301	C₄	C₅₃×#518	1	1	yes	characteristic
302	C₄	C₅₃×#554	1	2	yes	—
303	C₂	—	4	4	no	—
304	C₂	C₅₃×#2349	1	1	yes	characteristic
305	C₂	C₂×C₅₃×#427	1	1	yes	characteristic
306	C₂	C₅₃×#2913	1	1	yes	characteristic
307	C₂	C₅₃×#3302	1	2	yes	—
308	C₂	C₅₃×#3305	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1947	C₂	semidirect
4. C₅₃×#1854	C₂	semidirect
5. C₅₃×#1834	C₂	semidirect
6. C₅₃×#1808	C₂	semidirect
7. C₄×#106×C₅₃	C₂	semidirect
8. C₅₃×#1735	C₂	semidirect
12. C₈×D₄×C₅₃	C₄	semidirect
129. #22707	C₅₃	direct (N × H)
151. #1947	C₂×C₅₃	semidirect
152. #1854	C₂×C₅₃	semidirect
153. #1834	C₂×C₅₃	semidirect
154. #1808	C₂×C₅₃	semidirect
155. C₄×#106	C₂×C₅₃	semidirect
156. #1735	C₂×C₅₃	semidirect
166. C₈×D₄	C₄×C₅₃	semidirect
183. C₅₃	#22707	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×C₈×D₄×C₅₃	2
C₅₃×#2354	2
C₅₃×#1947	2
C₅₃×#1854	2
C₅₃×#1834	2
C₅₃×#1808	2
C₄×#106×C₅₃	2
C₅₃×#1735	2
#22707	53

Subgroups — C53×#22707 #27230219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22707 #27230219