Subgroups — C₅₃×#12646 #27220158

Order 13568.

428 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#443	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2148	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2145	C₂	1	1	yes	maximal, characteristic
4	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1772	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1388	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#131×C₅₃	—	2	1	no	—
10	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#443	—	2	1	no	—
13	C₅₃×#443	—	2	1	no	—
14	C₅₃×#443	C₄	1	2	yes	complemented
15	C₅₃×#443	C₄	1	2	yes	complemented
16	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#427	C₂²	1	1	yes	characteristic
19	C₂×#104×C₅₃	—	2	1	no	—
20	C₅₃×#405	—	2	1	no	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#362	C₄	1	2	yes	complemented
23	C₅₃×#361	C₄	1	2	yes	complemented
24	C₅₃×#360	C₄	1	4	yes	complemented
25	C₅₃×#346	—	2	1	no	—
26	C₄×C₁₆×C₅₃	—	2	1	no	—
27	C₅₃×#339	—	2	1	no	—
28	C₂×#41×C₅₃	—	2	1	no	—
29	#135×C₅₃	—	2	1	no	—
30	#135×C₅₃	—	2	1	no	—
31	#135×C₅₃	C₂×C₄	1	2	yes	—
32	#135×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×SD₁₆×C₅₃	—	2	1	no	—
35	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	#131×C₅₃	—	2	2	no	—
37	#131×C₅₃	—	2	2	no	—
38	#131×C₅₃	—	4	1	no	—
39	#131×C₅₃	—	4	1	no	—
40	C₂×M₁₆×C₅₃	—	2	1	no	—
41	C₂×M₁₆×C₅₃	—	2	1	no	—
42	C₂×M₁₆×C₅₃	—	4	1	no	—
43	C₂²×C₈×C₅₃	—	4	1	no	—
44	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	#117×C₅₃	—	2	1	no	—
47	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
48	M₃₂×C₅₃	—	2	1	no	—
49	M₃₂×C₅₃	—	2	1	no	—
50	M₃₂×C₅₃	C₂×C₄	1	2	yes	—
51	M₃₂×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₁₆×C₅₃	—	2	1	no	—
53	C₂×C₁₆×C₅₃	—	2	1	no	—
54	#108×C₅₃	—	2	1	no	—
55	#108×C₅₃	—	2	1	no	—
56	#108×C₅₃	C₂×C₄	1	2	yes	—
57	#108×C₅₃	C₂×C₄	1	2	yes	—
58	#107×C₅₃	—	2	1	no	—
59	#106×C₅₃	C₂×C₄	1	2	yes	—
60	#104×C₅₃	—	2	2	no	—
61	#104×C₅₃	—	2	2	no	—
62	#97×C₅₃	—	4	1	no	—
63	C₄×C₈×C₅₃	—	4	1	no	—
64	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
67	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
68	#41×C₅₃	—	2	2	no	—
69	#41×C₅₃	—	2	2	no	—
70	#41×C₅₃	—	4	1	no	—
71	#41×C₅₃	—	4	1	no	—
72	C₂×Q₈×C₅₃	—	2	1	no	—
73	C₂×D₄×C₅₃	—	2	1	no	—
74	C₂×D₄×C₅₃	—	4	1	no	—
75	C₂²×C₄×C₅₃	—	2	1	no	—
76	C₂²×C₄×C₅₃	—	4	1	no	—
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	Q₁₆×C₅₃	—	2	1	no	—
79	Q₁₆×C₅₃	C₄²	1	2	yes	—
80	SD₁₆×C₅₃	—	2	2	no	—
81	SD₁₆×C₅₃	—	2	2	no	—
82	D₈×C₅₃	—	2	1	no	—
83	D₈×C₅₃	C₄²	1	2	yes	—
84	M₁₆×C₅₃	—	2	2	no	—
85	M₁₆×C₅₃	—	2	2	no	—
86	M₁₆×C₅₃	—	4	1	no	—
87	M₁₆×C₅₃	—	4	1	no	—
88	M₁₆×C₅₃	—	4	2	no	—
89	M₁₆×C₅₃	—	4	2	no	—
90	C₂×C₈×C₅₃	—	4	1	no	—
91	C₂×C₈×C₅₃	—	4	1	no	—
92	C₂×C₈×C₅₃	—	4	2	no	—
93	C₂×C₈×C₅₃	—	4	2	no	—
94	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
97	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
98	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
99	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
100	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
105	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
106	C₄⋊C₄×C₅₃	—	2	2	no	—
107	C₂²⋊C₄×C₅₃	—	4	1	no	—
108	C₄²×C₅₃	—	2	1	no	—
109	C₄²×C₅₃	—	4	1	no	—
110	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
112	C₁₆×C₅₃	—	2	2	no	—
113	C₁₆×C₅₃	—	2	2	no	—
114	C₂³×C₅₃	—	4	1	no	—
115	C₂³×C₅₃	#135	1	1	yes	characteristic
116	Q₈×C₅₃	—	2	2	no	—
117	Q₈×C₅₃	—	4	1	no	—
118	D₄×C₅₃	—	2	2	no	—
119	D₄×C₅₃	—	4	2	no	—
120	D₄×C₅₃	—	4	2	no	—
121	D₄×C₅₃	—	4	1	no	—
122	C₂×C₄×C₅₃	—	2	1	no	—
123	C₂×C₄×C₅₃	—	2	1	no	—
124	C₂×C₄×C₅₃	—	4	2	no	—
125	C₂×C₄×C₅₃	—	4	1	no	—
126	C₂×C₄×C₅₃	—	4	2	no	—
127	C₂×C₄×C₅₃	—	4	2	no	—
128	C₂×C₄×C₅₃	—	4	2	no	—
129	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
135	C₈×C₅₃	—	4	2	no	—
136	C₈×C₅₃	—	4	2	no	—
137	C₈×C₅₃	#102	1	1	yes	characteristic
138	C₈×C₅₃	#102	1	1	yes	characteristic
139	C₈×C₅₃	#102	1	1	yes	characteristic
140	C₈×C₅₃	#102	1	1	yes	characteristic
141	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
142	C₈×C₅₃	#117	1	1	yes	characteristic
143	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
144	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
145	#12646	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂²×C₅₃	—	4	1	no	—
147	C₂²×C₅₃	—	4	2	no	—
148	C₂²×C₅₃	—	8	2	no	—
149	C₂²×C₅₃	#416	1	1	yes	characteristic
150	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
151	C₂²×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
152	C₄×C₅₃	—	2	2	no	—
153	C₄×C₅₃	—	4	1	no	—
154	C₄×C₅₃	—	4	2	no	—
155	C₄×C₅₃	—	8	2	no	—
156	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
158	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
159	C₄×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
160	C₂×#443	C₂×C₅₃	1	1	yes	characteristic
161	#2148	C₂×C₅₃	1	1	yes	characteristic
162	#2145	C₂×C₅₃	1	1	yes	characteristic
163	C₄×M₃₂	C₂×C₅₃	1	1	yes	characteristic, complemented
164	#1772	C₂×C₅₃	1	1	yes	characteristic, complemented
165	#1389	C₂×C₅₃	1	1	yes	characteristic, complemented
166	#1388	C₂×C₅₃	1	1	yes	characteristic, complemented
167	C₂×C₅₃	—	2	1	no	—
168	C₂×C₅₃	—	2	1	no	—
169	C₂×C₅₃	—	2	1	no	—
170	C₂×C₅₃	—	8	2	no	—
171	C₂×C₅₃	C₄×#102	1	1	yes	characteristic
172	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
173	C₂×#131	—	2	1	no	—
174	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
175	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
176	#443	—	2	1	no	—
177	#443	—	2	1	no	—
178	#443	C₄×C₅₃	1	2	yes	complemented
179	#443	C₄×C₅₃	1	2	yes	complemented
180	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
181	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
182	#427	C₂²×C₅₃	1	1	yes	characteristic
183	C₂×#104	—	2	1	no	—
184	#405	—	2	1	no	—
185	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
186	#362	C₄×C₅₃	1	2	yes	complemented
187	#361	C₄×C₅₃	1	2	yes	complemented
188	#360	C₄×C₅₃	1	4	yes	complemented
189	#346	—	2	1	no	—
190	C₄×C₁₆	—	2	1	no	—
191	#339	—	2	1	no	—
192	C₅₃	#12646	1	1	yes	characteristic, direct product factor, Sylow
193	C₂×#41	—	2	1	no	—
194	#135	—	2	1	no	—
195	#135	—	2	1	no	—
196	#135	C₂×C₄×C₅₃	1	2	yes	—
197	#135	C₂×C₄×C₅₃	1	2	yes	—
198	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
199	C₂×SD₁₆	—	2	1	no	—
200	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic
201	#131	—	2	2	no	—
202	#131	—	2	2	no	—
203	#131	—	4	1	no	—
204	#131	—	4	1	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	2	1	no	—
207	C₂×M₁₆	—	4	1	no	—
208	C₂²×C₈	—	4	1	no	—
209	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
211	#117	—	2	1	no	—
212	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
213	M₃₂	—	2	1	no	—
214	M₃₂	—	2	1	no	—
215	M₃₂	C₂×C₄×C₅₃	1	2	yes	—
216	M₃₂	C₂×C₄×C₅₃	1	2	yes	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	2	1	no	—
219	#108	—	2	1	no	—
220	#108	—	2	1	no	—
221	#108	C₂×C₄×C₅₃	1	2	yes	—
222	#108	C₂×C₄×C₅₃	1	2	yes	—
223	#107	—	2	1	no	—
224	#106	C₂×C₄×C₅₃	1	2	yes	—
225	#104	—	2	2	no	—
226	#104	—	2	2	no	—
227	#97	—	4	1	no	—
228	C₄×C₈	—	4	1	no	—
229	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
230	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
231	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
232	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
233	#41	—	2	2	no	—
234	#41	—	2	2	no	—
235	#41	—	4	1	no	—
236	#41	—	4	1	no	—
237	C₂×Q₈	—	2	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
243	Q₁₆	—	2	1	no	—
244	Q₁₆	C₄²×C₅₃	1	2	yes	—
245	SD₁₆	—	2	2	no	—
246	SD₁₆	—	2	2	no	—
247	D₈	—	2	1	no	—
248	D₈	C₄²×C₅₃	1	2	yes	—
249	M₁₆	—	2	2	no	—
250	M₁₆	—	2	2	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	M₁₆	—	4	2	no	—
254	M₁₆	—	4	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	—	4	2	no	—
259	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
260	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
261	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
262	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
263	C₂×C₈	SD₁₆×C₅₃	1	1	yes	characteristic
264	C₂×C₈	SD₁₆×C₅₃	1	1	yes	characteristic
265	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
266	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
267	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
268	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
269	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
270	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	4	1	no	—
275	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
276	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
277	C₁₆	—	2	2	no	—
278	C₁₆	—	2	2	no	—
279	C₂³	—	4	1	no	—
280	C₂³	#135×C₅₃	1	1	yes	characteristic
281	Q₈	—	2	2	no	—
282	Q₈	—	4	1	no	—
283	D₄	—	2	2	no	—
284	D₄	—	4	2	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
295	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
296	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
297	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
298	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
299	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
300	C₈	—	4	2	no	—
301	C₈	—	4	2	no	—
302	C₈	#102×C₅₃	1	1	yes	characteristic
303	C₈	#102×C₅₃	1	1	yes	characteristic
304	C₈	#102×C₅₃	1	1	yes	characteristic
305	C₈	#102×C₅₃	1	1	yes	characteristic
306	C₈	C₂×C₄²×C₅₃	1	1	yes	characteristic
307	C₈	#117×C₅₃	1	1	yes	characteristic
308	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
309	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
310	C₂²	—	4	1	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	C₅₃×#416	1	1	yes	characteristic
314	C₂²	C₄×D₈×C₅₃	1	1	yes	characteristic
315	C₂²	C₄×SD₁₆×C₅₃	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	2	no	—
319	C₄	—	8	2	no	—
320	C₄	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
321	C₄	C₂×#102×C₅₃	1	1	yes	characteristic
322	C₄	C₄×D₈×C₅₃	1	1	yes	characteristic
323	C₄	C₄×SD₁₆×C₅₃	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	8	2	no	—
328	C₂	C₄×#102×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₅₃	#102
commutator	C₈	C₂×C₄²×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	C₄×#102
prime radical	C₂×#135×C₅₃	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₅₃ ◁ C₂²×C₈×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄×M₃₂×C₅₃	C₂	semidirect
5. C₅₃×#1772	C₂	semidirect
6. C₅₃×#1389	C₂	semidirect
7. C₅₃×#1388	C₂	semidirect
14. C₅₃×#443	C₄	semidirect
15. C₅₃×#443	C₄	semidirect
22. C₅₃×#362	C₄	semidirect
23. C₅₃×#361	C₄	semidirect
24. C₅₃×#360	C₄	semidirect
145. #12646	C₅₃	direct (N × H)
163. C₄×M₃₂	C₂×C₅₃	semidirect
164. #1772	C₂×C₅₃	semidirect
165. #1389	C₂×C₅₃	semidirect
166. #1388	C₂×C₅₃	semidirect
178. #443	C₄×C₅₃	semidirect
179. #443	C₄×C₅₃	semidirect
186. #362	C₄×C₅₃	semidirect
187. #361	C₄×C₅₃	semidirect
188. #360	C₄×C₅₃	semidirect
192. C₅₃	#12646	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×#443	2
C₅₃×#2148	2
C₅₃×#2145	2
C₄×M₃₂×C₅₃	2
C₅₃×#1772	2
C₅₃×#1389	2
C₅₃×#1388	2
#12646	53

Subgroups — C53×#12646 #27220158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12646 #27220158