Subgroups — C₅₃×#21202 #27228714

Order 13568.

1558 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#419	C₂	1	4	yes	maximal
3	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2285	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1960	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1956	C₂	1	4	yes	maximal, complemented
7	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₅₃	—	2	4	no	—
10	C₂×C₄⋊Q₈×C₅₃	—	2	2	no	—
11	C₂×C₄⋊Q₈×C₅₃	C₄	1	8	yes	complemented
12	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
14	C₂×#117×C₅₃	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₅₃	—	2	2	no	—
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#419	—	2	16	no	—
18	C₂×#103×C₅₃	C₂²	1	16	yes	—
19	C₄×M₁₆×C₅₃	—	2	4	no	—
20	C₂×#97×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
22	C₅₃×#384	C₂²	1	2	yes	—
23	C₅₃×#382	C₂²	1	2	yes	—
24	C₅₃×#380	—	2	2	no	—
25	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₂²×Q₈×C₅₃	—	2	4	no	—
27	C₂²×Q₈×C₅₃	—	2	2	no	—
28	C₂²×Q₈×C₅₃	—	2	8	no	—
29	C₂²×Q₈×C₅₃	—	2	8	no	—
30	C₂²×Q₈×C₅₃	C₂×C₄	1	8	yes	—
31	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
32	C₂³×C₄×C₅₃	—	2	2	no	—
33	C₂³×C₄×C₅₃	D₄	1	2	yes	—
34	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₅₃	—	2	4	no	—
36	C₂×M₁₆×C₅₃	—	2	4	no	—
37	C₂×M₁₆×C₅₃	D₄	1	8	yes	—
38	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
39	C₄⋊Q₈×C₅₃	—	2	16	no	—
40	C₄⋊Q₈×C₅₃	—	2	4	no	—
41	C₄⋊Q₈×C₅₃	—	2	4	no	—
42	C₄⋊Q₈×C₅₃	C₂×C₄	1	8	yes	—
43	C₄⋊Q₈×C₅₃	C₂×C₄	1	8	yes	—
44	#117×C₅₃	—	2	8	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
46	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
47	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
48	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
49	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
50	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
51	C₂×C₄²×C₅₃	—	2	2	no	—
52	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₅₃	D₄	1	4	yes	—
54	C₂×C₄²×C₅₃	D₄	1	2	yes	—
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	#103×C₅₃	—	2	32	no	—
58	#97×C₅₃	—	2	4	no	—
59	C₄×C₈×C₅₃	—	2	4	no	—
60	#95×C₅₃	—	2	4	no	—
61	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×Q₈×C₅₃	—	2	8	no	—
63	C₂×Q₈×C₅₃	—	2	4	no	—
64	C₂×Q₈×C₅₃	—	2	32	no	—
65	C₂×Q₈×C₅₃	—	2	8	no	—
66	C₂×Q₈×C₅₃	—	2	8	no	—
67	C₂×Q₈×C₅₃	—	4	16	no	—
68	C₂×Q₈×C₅₃	—	4	4	no	—
69	C₂×Q₈×C₅₃	—	4	4	no	—
70	C₂×Q₈×C₅₃	C₂²×C₄	1	8	yes	—
71	C₂²×C₄×C₅₃	—	2	8	no	—
72	C₂²×C₄×C₅₃	—	2	8	no	—
73	C₂²×C₄×C₅₃	—	2	4	no	—
74	C₂²×C₄×C₅₃	—	2	4	no	—
75	C₂²×C₄×C₅₃	—	4	4	no	—
76	C₂²×C₄×C₅₃	—	4	4	no	—
77	C₂²×C₄×C₅₃	C₂²⋊C₄	1	8	yes	—
78	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
84	C₂²×C₄×C₅₃	#41	1	4	yes	—
85	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
86	M₁₆×C₅₃	—	2	16	no	—
87	C₂×C₈×C₅₃	—	2	4	no	—
88	C₂×C₈×C₅₃	—	2	4	no	—
89	C₂×C₈×C₅₃	C₂×D₄	1	8	yes	—
90	C₄⋊C₄×C₅₃	—	2	16	no	—
91	C₄⋊C₄×C₅₃	—	2	16	no	—
92	C₄⋊C₄×C₅₃	—	2	8	no	—
93	C₄⋊C₄×C₅₃	—	4	8	no	—
94	C₂²⋊C₄×C₅₃	—	4	4	no	—
95	C₄²×C₅₃	—	4	4	no	—
96	C₄²×C₅₃	C₂²⋊C₄	1	8	yes	—
97	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
98	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
99	C₄²×C₅₃	C₂×D₄	1	2	yes	—
100	C₄²×C₅₃	C₂×D₄	1	2	yes	—
101	C₂³×C₅₃	—	2	4	no	—
102	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂³×C₅₃	#124	1	2	yes	—
104	C₂³×C₅₃	#127	1	2	yes	—
105	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
106	Q₈×C₅₃	—	2	16	no	—
107	Q₈×C₅₃	—	4	8	no	—
108	Q₈×C₅₃	—	4	16	no	—
109	C₂×C₄×C₅₃	—	2	4	no	—
110	C₂×C₄×C₅₃	—	2	16	no	—
111	C₂×C₄×C₅₃	—	2	8	no	—
112	C₂×C₄×C₅₃	—	4	4	no	—
113	C₂×C₄×C₅₃	—	4	4	no	—
114	C₂×C₄×C₅₃	—	4	16	no	—
115	C₂×C₄×C₅₃	—	4	4	no	—
116	C₂×C₄×C₅₃	—	4	8	no	—
117	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
118	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
119	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
120	C₂×C₄×C₅₃	C₄×D₄	1	8	yes	—
121	C₂×C₄×C₅₃	#121	1	8	yes	—
122	C₂×C₄×C₅₃	#124	1	2	yes	—
123	C₂×C₄×C₅₃	#127	1	2	yes	—
124	C₂×C₄×C₅₃	#137	1	8	yes	—
125	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
127	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
129	C₈×C₅₃	—	2	8	no	—
130	#21202	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₅₃	—	2	4	no	—
132	C₂²×C₅₃	—	2	4	no	—
133	C₂²×C₅₃	—	2	4	no	—
134	C₂²×C₅₃	#390	1	4	yes	—
135	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
136	C₂²×C₅₃	C₂×#124	1	1	yes	characteristic
137	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
138	C₂²×C₅₃	C₂×#137	1	4	yes	—
139	C₄×C₅₃	—	2	4	no	—
140	C₄×C₅₃	—	4	8	no	—
141	C₄×C₅₃	—	8	4	no	—
142	C₄×C₅₃	#390	1	4	yes	—
143	C₄×C₅₃	#419	1	8	yes	complemented
144	C₄×C₅₃	C₂×C₄×D₄	1	2	yes	—
145	C₄×C₅₃	C₂×#121	1	2	yes	—
146	C₂²×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
147	C₂×#419	C₂×C₅₃	1	4	yes	—
148	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic
149	#2285	C₂×C₅₃	1	1	yes	characteristic
150	#1960	C₂×C₅₃	1	4	yes	complemented
151	#1956	C₂×C₅₃	1	4	yes	complemented
152	C₂×C₅₃	—	2	4	no	—
153	C₂×C₅₃	C₂×#390	1	1	yes	characteristic
154	C₂×C₅₃	C₂×#419	1	2	yes	—
155	C₂×C₅₃	#3139	1	2	yes	—
156	C₂×C₅₃	#3152	1	2	yes	—
157	C₂³×Q₈	C₂²×C₅₃	1	2	yes	—
158	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
159	C₂×C₄⋊Q₈	—	2	4	no	—
160	C₂×C₄⋊Q₈	—	2	2	no	—
161	C₂×C₄⋊Q₈	C₄×C₅₃	1	8	yes	complemented
162	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
163	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
164	C₂×#117	C₂²×C₅₃	1	2	yes	—
165	C₂²×C₄⋊C₄	—	2	2	no	—
166	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
167	#419	—	2	16	no	—
168	C₂×#103	C₂²×C₅₃	1	16	yes	—
169	C₄×M₁₆	—	2	4	no	—
170	C₂×#97	C₂²×C₅₃	1	2	yes	—
171	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
172	#384	C₂²×C₅₃	1	2	yes	—
173	#382	C₂²×C₅₃	1	2	yes	—
174	#380	—	2	2	no	—
175	C₄×C₄⋊C₄	—	2	2	no	—
176	C₅₃	#21202	1	1	yes	characteristic, direct product factor, Sylow
177	C₂²×Q₈	—	2	4	no	—
178	C₂²×Q₈	—	2	2	no	—
179	C₂²×Q₈	—	2	8	no	—
180	C₂²×Q₈	—	2	8	no	—
181	C₂²×Q₈	C₂×C₄×C₅₃	1	8	yes	—
182	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
183	C₂³×C₄	—	2	2	no	—
184	C₂³×C₄	D₄×C₅₃	1	2	yes	—
185	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
186	C₂×M₁₆	—	2	4	no	—
187	C₂×M₁₆	—	2	4	no	—
188	C₂×M₁₆	D₄×C₅₃	1	8	yes	—
189	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
190	C₄⋊Q₈	—	2	4	no	—
191	C₄⋊Q₈	—	2	4	no	—
192	C₄⋊Q₈	—	2	16	no	—
193	C₄⋊Q₈	C₂×C₄×C₅₃	1	8	yes	—
194	C₄⋊Q₈	C₂×C₄×C₅₃	1	8	yes	—
195	#117	—	2	8	no	—
196	C₂×C₄⋊C₄	—	2	4	no	—
197	C₂×C₄⋊C₄	—	2	4	no	—
198	C₂×C₄⋊C₄	—	2	16	no	—
199	C₂×C₄⋊C₄	—	2	4	no	—
200	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
201	C₂×C₂²⋊C₄	—	2	2	no	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
204	C₂×C₄²	D₄×C₅₃	1	4	yes	—
205	C₂×C₄²	D₄×C₅₃	1	2	yes	—
206	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
208	#103	—	2	32	no	—
209	#97	—	2	4	no	—
210	C₄×C₈	—	2	4	no	—
211	#95	—	2	4	no	—
212	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
213	C₂×Q₈	—	2	8	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	2	32	no	—
216	C₂×Q₈	—	2	8	no	—
217	C₂×Q₈	—	2	8	no	—
218	C₂×Q₈	—	4	16	no	—
219	C₂×Q₈	—	4	4	no	—
220	C₂×Q₈	—	4	4	no	—
221	C₂×Q₈	C₂²×C₄×C₅₃	1	8	yes	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	2	8	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	4	4	no	—
228	C₂²×C₄	C₂²⋊C₄×C₅₃	1	8	yes	—
229	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
230	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
235	C₂²×C₄	#41×C₅₃	1	4	yes	—
236	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
237	M₁₆	—	2	16	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	C₂×D₄×C₅₃	1	8	yes	—
241	C₄⋊C₄	—	2	16	no	—
242	C₄⋊C₄	—	2	16	no	—
243	C₄⋊C₄	—	2	8	no	—
244	C₄⋊C₄	—	4	8	no	—
245	C₂²⋊C₄	—	4	4	no	—
246	C₄²	—	4	4	no	—
247	C₄²	C₂²⋊C₄×C₅₃	1	8	yes	—
248	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
249	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
250	C₄²	C₂×D₄×C₅₃	1	2	yes	—
251	C₄²	C₂×D₄×C₅₃	1	2	yes	—
252	C₂³	—	2	4	no	—
253	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
254	C₂³	#124×C₅₃	1	2	yes	—
255	C₂³	#127×C₅₃	1	2	yes	—
256	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
257	Q₈	—	2	16	no	—
258	Q₈	—	4	8	no	—
259	Q₈	—	4	16	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	16	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	16	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	8	no	—
268	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
269	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
270	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
271	C₂×C₄	C₄×D₄×C₅₃	1	8	yes	—
272	C₂×C₄	#121×C₅₃	1	8	yes	—
273	C₂×C₄	#124×C₅₃	1	2	yes	—
274	C₂×C₄	#127×C₅₃	1	2	yes	—
275	C₂×C₄	#137×C₅₃	1	8	yes	—
276	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
278	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
279	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
280	C₈	—	2	8	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	C₅₃×#390	1	4	yes	—
285	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
286	C₂²	C₂×#124×C₅₃	1	1	yes	characteristic
287	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
288	C₂²	C₂×#137×C₅₃	1	4	yes	—
289	C₄	—	2	4	no	—
290	C₄	—	4	8	no	—
291	C₄	—	8	4	no	—
292	C₄	C₅₃×#390	1	4	yes	—
293	C₄	C₅₃×#419	1	8	yes	complemented
294	C₄	C₂×C₄×D₄×C₅₃	1	2	yes	—
295	C₄	C₂×#121×C₅₃	1	2	yes	—
296	C₂	—	2	4	no	—
297	C₂	C₂×C₅₃×#390	1	1	yes	characteristic
298	C₂	C₂×C₅₃×#419	1	2	yes	—
299	C₂	C₅₃×#3139	1	2	yes	—
300	C₂	C₅₃×#3152	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	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅₃×#1960	C₂	semidirect
6. C₅₃×#1956	C₂	semidirect
11. C₂×C₄⋊Q₈×C₅₃	C₄	semidirect
130. #21202	C₅₃	direct (N × H)
143. C₄×C₅₃	#419	semidirect
150. #1960	C₂×C₅₃	semidirect
151. #1956	C₂×C₅₃	semidirect
161. C₂×C₄⋊Q₈	C₄×C₅₃	semidirect
176. C₅₃	#21202	direct (N × H)
293. C₄	C₅₃×#419	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₅₃	2
C₂×C₅₃×#419	2
C₂×C₄×M₁₆×C₅₃	2
C₅₃×#2285	2
C₅₃×#1960	2
C₅₃×#1956	2
#21202	53

Subgroups — C53×#21202 #27228714

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21202 #27228714