Subgroups — C₅₃×#21346 #27228858

Order 13568.

746 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	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1985	C₂	1	2	yes	maximal
3	C₅₃×#1940	C₂	1	2	yes	maximal
4	C₅₃×#1916	C₂	1	4	yes	maximal
5	C₅₃×#1767	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1767	C₂	1	2	yes	maximal
7	C₅₃×#1745	C₂	1	2	yes	maximal
8	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₅₃	C₄	1	2	yes	—
10	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×#97×C₅₃	C₂²	1	2	yes	—
14	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#97×C₅₃	C₂²	1	2	yes	—
16	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
19	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
20	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₄³×C₅₃	C₂²	1	1	yes	characteristic
24	C₄³×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#336	C₂²	1	2	yes	—
26	C₅₃×#336	C₂²	1	4	yes	—
27	C₅₃×#336	C₂²	1	4	yes	—
28	C₅₃×#336	C₂²	1	4	yes	—
29	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
31	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
32	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
33	C₄×Q₈×C₅₃	—	2	8	no	—
34	C₄×Q₈×C₅₃	—	2	4	no	—
35	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
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	2	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
49	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₅₃	—	2	8	no	—
52	C₄⋊C₈×C₅₃	—	2	4	no	—
53	#97×C₅₃	—	2	4	no	—
54	#97×C₅₃	—	2	2	no	—
55	#97×C₅₃	—	2	4	no	—
56	#97×C₅₃	Q₈	1	4	yes	complemented
57	C₄×C₈×C₅₃	—	2	4	no	—
58	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
59	#95×C₅₃	C₂×C₄	1	2	yes	—
60	#95×C₅₃	C₂×C₄	1	2	yes	—
61	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×Q₈×C₅₃	—	2	2	no	—
63	C₂×Q₈×C₅₃	—	2	4	no	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	4	no	—
78	C₂×C₈×C₅₃	—	2	4	no	—
79	C₂×C₈×C₅₃	—	2	4	no	—
80	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
81	C₂×C₈×C₅₃	#41	1	4	yes	—
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	—	2	4	no	—
86	C₄⋊C₄×C₅₃	M₁₆	1	8	yes	—
87	C₄²×C₅₃	—	2	4	no	—
88	C₄²×C₅₃	—	2	4	no	—
89	C₄²×C₅₃	—	2	4	no	—
90	C₄²×C₅₃	—	2	2	no	—
91	C₄²×C₅₃	—	2	4	no	—
92	C₄²×C₅₃	—	2	4	no	—
93	C₄²×C₅₃	—	4	2	no	—
94	C₄²×C₅₃	—	4	2	no	—
95	C₄²×C₅₃	—	4	2	no	—
96	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
97	C₄²×C₅₃	#41	1	4	yes	—
98	C₄²×C₅₃	#41	1	2	yes	—
99	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
100	Q₈×C₅₃	—	2	8	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	2	2	no	—
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	#117	1	2	yes	—
118	C₂×C₄×C₅₃	#117	1	2	yes	—
119	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
120	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
121	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
122	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
123	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
126	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
127	C₈×C₅₃	—	2	4	no	—
128	C₈×C₅₃	—	2	4	no	—
129	C₈×C₅₃	—	4	4	no	—
130	#21346	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
132	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
133	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
134	C₂²×C₅₃	C₂²×M₁₆	1	2	yes	—
135	C₂²×C₅₃	#568	1	2	yes	—
136	C₄×C₅₃	—	2	2	no	—
137	C₄×C₅₃	—	2	4	no	—
138	C₄×C₅₃	—	2	2	no	—
139	C₄×C₅₃	—	2	4	no	—
140	C₄×C₅₃	—	2	4	no	—
141	C₄×C₅₃	—	2	4	no	—
142	C₄×C₅₃	—	4	2	no	—
143	C₄×C₅₃	—	4	2	no	—
144	C₄×C₅₃	#529	1	2	yes	—
145	C₄×C₅₃	#533	1	2	yes	—
146	C₄²×Q₈	C₂×C₅₃	1	1	yes	characteristic
147	#1985	C₂×C₅₃	1	2	yes	—
148	#1940	C₂×C₅₃	1	2	yes	—
149	#1916	C₂×C₅₃	1	4	yes	—
150	#1767	C₂×C₅₃	1	1	yes	characteristic
151	#1767	C₂×C₅₃	1	2	yes	—
152	#1745	C₂×C₅₃	1	2	yes	—
153	C₄×#97	C₂×C₅₃	1	1	yes	characteristic
154	C₂×C₅₃	#2295	1	1	yes	characteristic
155	C₂×C₅₃	#2920	1	1	yes	characteristic
156	C₂×C₅₃	#2923	1	1	yes	characteristic
157	C₂×C₅₃	Q₈×M₁₆	1	2	yes	—
158	C₂×C₅₃	#2970	1	2	yes	—
159	C₂×C₄×Q₈	C₄×C₅₃	1	2	yes	—
160	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
161	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
162	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
163	C₂×#97	C₂²×C₅₃	1	2	yes	—
164	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
165	C₂×#97	C₂²×C₅₃	1	2	yes	—
166	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
168	C₄×C₄⋊C₄	C₄×C₅₃	1	2	yes	—
169	C₄×C₄⋊C₄	C₄×C₅₃	1	2	yes	—
170	C₄×C₄⋊C₄	C₄×C₅₃	1	2	yes	—
171	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
172	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
173	C₄³	C₂²×C₅₃	1	1	yes	characteristic
174	C₄³	C₂²×C₅₃	1	2	yes	—
175	#336	C₂²×C₅₃	1	2	yes	—
176	#336	C₂²×C₅₃	1	4	yes	—
177	#336	C₂²×C₅₃	1	4	yes	—
178	#336	C₂²×C₅₃	1	4	yes	—
179	C₅₃	#21346	1	1	yes	characteristic, direct product factor, Sylow
180	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
182	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
183	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
184	C₄×Q₈	—	2	8	no	—
185	C₄×Q₈	—	2	4	no	—
186	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
187	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
188	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
189	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
190	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
191	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
192	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
193	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
194	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
195	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
196	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
197	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
198	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
200	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
202	C₄⋊C₈	—	2	8	no	—
203	C₄⋊C₈	—	2	4	no	—
204	#97	—	2	4	no	—
205	#97	—	2	2	no	—
206	#97	—	2	4	no	—
207	#97	Q₈×C₅₃	1	4	yes	complemented
208	C₄×C₈	—	2	4	no	—
209	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
210	#95	C₂×C₄×C₅₃	1	2	yes	—
211	#95	C₂×C₄×C₅₃	1	2	yes	—
212	#95	C₂×C₄×C₅₃	1	1	yes	characteristic
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
216	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
217	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
218	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
220	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
221	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
222	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
223	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
232	C₂×C₈	#41×C₅₃	1	4	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	M₁₆×C₅₃	1	8	yes	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	2	4	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	4	2	no	—
245	C₄²	—	4	2	no	—
246	C₄²	—	4	2	no	—
247	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
248	C₄²	#41×C₅₃	1	4	yes	—
249	C₄²	#41×C₅₃	1	2	yes	—
250	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
251	Q₈	—	2	8	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	#117×C₅₃	1	2	yes	—
269	C₂×C₄	#117×C₅₃	1	2	yes	—
270	C₂×C₄	C₄×Q₈×C₅₃	1	4	yes	—
271	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
272	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
273	C₂×C₄	C₂×M₁₆×C₅₃	1	4	yes	—
274	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
275	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
276	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
277	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
278	C₈	—	2	4	no	—
279	C₈	—	2	4	no	—
280	C₈	—	4	4	no	—
281	C₂²	C₂×#117×C₅₃	1	1	yes	characteristic
282	C₂²	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
283	C₂²	C₄×#41×C₅₃	1	1	yes	characteristic
284	C₂²	C₂²×M₁₆×C₅₃	1	2	yes	—
285	C₂²	C₅₃×#568	1	2	yes	—
286	C₄	—	2	4	no	—
287	C₄	—	2	4	no	—
288	C₄	—	2	4	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	4	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	2	no	—
293	C₄	—	4	2	no	—
294	C₄	C₅₃×#529	1	2	yes	—
295	C₄	C₅₃×#533	1	2	yes	—
296	C₂	C₅₃×#2295	1	1	yes	characteristic
297	C₂	C₅₃×#2920	1	1	yes	characteristic
298	C₂	C₅₃×#2923	1	1	yes	characteristic
299	C₂	Q₈×M₁₆×C₅₃	1	2	yes	—
300	C₂	C₅₃×#2970	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
56. #97×C₅₃	Q₈	semidirect
130. #21346	C₅₃	direct (N × H)
179. C₅₃	#21346	direct (N × H)
207. #97	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄²×Q₈×C₅₃	2
C₅₃×#1985	2
C₅₃×#1940	2
C₅₃×#1916	2
C₅₃×#1767	2
C₅₃×#1745	2
C₄×#97×C₅₃	2
#21346	53

Subgroups — C53×#21346 #27228858

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21346 #27228858