Subgroups — C₂×C₅₃×#2084 #27231167

Order 13568.

1086 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#106×C₅₃	C₂	1	2	yes	maximal
2	C₂×C₅₃×#389	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#384	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2084	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅₃×#340	C₂	1	2	yes	maximal
6	C₂×C₅₃×#336	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
9	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
10	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
11	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#106×C₅₃	—	2	8	no	—
16	C₂×#106×C₅₃	C₂²	1	8	yes	—
17	C₅₃×#389	—	2	2	no	—
18	C₅₃×#389	C₂²	1	4	yes	—
19	C₅₃×#384	—	2	2	no	—
20	C₅₃×#384	C₂²	1	4	yes	—
21	C₂×#95×C₅₃	—	2	1	no	—
22	C₅₃×#340	C₂²	1	8	yes	—
23	C₅₃×#336	C₂²	1	4	yes	—
24	C₂³×C₄×C₅₃	—	2	1	no	—
25	C₂³×C₄×C₅₃	—	2	2	no	—
26	C₂³×C₄×C₅₃	—	2	1	no	—
27	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	—	2	4	no	—
31	C₂²×C₈×C₅₃	—	2	4	no	—
32	C₂²×C₈×C₅₃	Q₈	1	8	yes	—
33	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
34	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
44	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
45	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
46	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
47	C₂×C₄⋊C₄×C₅₃	D₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
50	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
51	C₂×C₄²×C₅₃	—	2	4	no	—
52	C₂×C₄²×C₅₃	D₄	1	2	yes	—
53	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
54	#106×C₅₃	—	2	16	no	—
55	#95×C₅₃	—	2	4	no	—
56	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	—	2	2	no	—
59	C₂²×C₄×C₅₃	—	2	2	no	—
60	C₂²×C₄×C₅₃	—	2	2	no	—
61	C₂²×C₄×C₅₃	—	2	4	no	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	—	2	2	no	—
64	C₂²×C₄×C₅₃	—	2	4	no	—
65	C₂²×C₄×C₅₃	—	4	2	no	—
66	C₂²×C₄×C₅₃	—	4	2	no	—
67	C₂²×C₄×C₅₃	—	4	4	no	—
68	C₂²×C₄×C₅₃	—	4	2	no	—
69	C₂²×C₄×C₅₃	—	4	4	no	—
70	C₂²×C₄×C₅₃	—	4	2	no	—
71	C₂²×C₄×C₅₃	—	4	4	no	—
72	C₂²×C₄×C₅₃	—	4	2	no	—
73	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
74	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
75	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
80	C₂²×C₄×C₅₃	#41	1	2	yes	—
81	C₂²×C₄×C₅₃	#41	1	4	yes	—
82	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
83	C₂×C₈×C₅₃	—	2	16	no	—
84	C₂×C₈×C₅₃	—	2	4	no	—
85	C₂×C₈×C₅₃	—	2	4	no	—
86	C₂×C₈×C₅₃	C₂×Q₈	1	8	yes	—
87	C₄⋊C₄×C₅₃	—	2	4	no	—
88	C₄⋊C₄×C₅₃	—	2	2	no	—
89	C₄⋊C₄×C₅₃	—	2	4	no	—
90	C₄⋊C₄×C₅₃	—	2	4	no	—
91	C₄⋊C₄×C₅₃	—	2	4	no	—
92	C₄⋊C₄×C₅₃	—	4	8	no	—
93	C₄⋊C₄×C₅₃	—	4	4	no	—
94	C₄⋊C₄×C₅₃	—	4	2	no	—
95	C₄⋊C₄×C₅₃	—	4	8	no	—
96	C₄⋊C₄×C₅₃	—	4	8	no	—
97	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
98	C₄²×C₅₃	—	2	4	no	—
99	C₄²×C₅₃	—	2	4	no	—
100	C₂³×C₅₃	#121	1	2	yes	—
101	C₂³×C₅₃	#123	1	2	yes	—
102	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
103	C₂³×C₅₃	C₂×SD₁₆	1	2	yes	—
104	C₂³×C₅₃	C₂×SD₁₆	1	2	yes	—
105	C₂³×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
106	C₂³×C₅₃	#137	1	2	yes	—
107	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
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	8	no	—
113	C₂×C₄×C₅₃	—	2	8	no	—
114	C₂×C₄×C₅₃	—	2	4	no	—
115	C₂×C₄×C₅₃	—	4	2	no	—
116	C₂×C₄×C₅₃	—	4	2	no	—
117	C₂×C₄×C₅₃	—	4	8	no	—
118	C₂×C₄×C₅₃	—	4	4	no	—
119	C₂×C₄×C₅₃	—	4	2	no	—
120	C₂×C₄×C₅₃	—	4	2	no	—
121	C₂×C₄×C₅₃	—	4	4	no	—
122	C₂×C₄×C₅₃	—	4	2	no	—
123	C₂×C₄×C₅₃	—	4	4	no	—
124	C₂×C₄×C₅₃	—	8	8	no	—
125	C₂×C₄×C₅₃	—	8	4	no	—
126	C₂×C₄×C₅₃	#121	1	2	yes	—
127	C₂×C₄×C₅₃	C₂²⋊Q₈	1	8	yes	—
128	C₂×C₄×C₅₃	#123	1	2	yes	—
129	C₂×C₄×C₅₃	#125	1	8	yes	—
130	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
131	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
132	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
133	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
135	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
136	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
137	C₈×C₅₃	—	2	8	no	—
138	C₂×#2084	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂²×C₅₃	#398	1	4	yes	—
140	C₂²×C₅₃	#460	1	4	yes	—
141	C₂²×C₅₃	#461	1	4	yes	—
142	C₂²×C₅₃	#481	1	4	yes	—
143	C₂²×C₅₃	#483	1	4	yes	—
144	C₂²×C₅₃	#498	1	4	yes	—
145	C₂²×C₅₃	#501	1	4	yes	—
146	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
147	C₂²×C₅₃	C₂×#123	1	1	yes	characteristic
148	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
149	C₂²×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
150	C₂²×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
151	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂²×C₅₃	C₂×#137	1	1	yes	characteristic
153	C₄×C₅₃	—	2	4	no	—
154	C₄×C₅₃	—	2	4	no	—
155	C₄×C₅₃	—	4	4	no	—
156	C₄×C₅₃	—	8	2	no	—
157	C₄×C₅₃	—	8	4	no	—
158	C₄×C₅₃	#398	1	4	yes	—
159	C₄×C₅₃	C₂×C₂²⋊Q₈	1	2	yes	—
160	C₄×C₅₃	C₂×#125	1	2	yes	—
161	C₂²×#106	C₂×C₅₃	1	2	yes	—
162	C₂×#389	C₂×C₅₃	1	1	yes	characteristic
163	C₂×#384	C₂×C₅₃	1	1	yes	characteristic
164	#2084	C₂×C₅₃	1	8	yes	direct product factor
165	C₂×#340	C₂×C₅₃	1	2	yes	—
166	C₂×#336	C₂×C₅₃	1	1	yes	characteristic
167	C₂×C₅₃	#2084	1	8	yes	direct product factor
168	C₂×C₅₃	C₂×#398	1	1	yes	characteristic
169	C₂×C₅₃	C₂×#461	1	1	yes	characteristic
170	C₂×C₅₃	C₂×#460	1	1	yes	characteristic
171	C₂×C₅₃	C₂×#483	1	1	yes	characteristic
172	C₂×C₅₃	C₂×#481	1	1	yes	characteristic
173	C₂×C₅₃	C₂×#498	1	1	yes	characteristic
174	C₂×C₅₃	C₂×#501	1	1	yes	characteristic
175	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
176	C₂²×C₄⋊C₄	—	2	1	no	—
177	C₂²×C₄⋊C₄	—	2	1	no	—
178	C₂²×C₄⋊C₄	—	2	1	no	—
179	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
180	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
181	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
182	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
183	C₂×#106	—	2	8	no	—
184	C₂×#106	C₂²×C₅₃	1	8	yes	—
185	#389	—	2	2	no	—
186	#389	C₂²×C₅₃	1	4	yes	—
187	#384	—	2	2	no	—
188	#384	C₂²×C₅₃	1	4	yes	—
189	C₂×#95	—	2	1	no	—
190	#340	C₂²×C₅₃	1	8	yes	—
191	#336	C₂²×C₅₃	1	4	yes	—
192	C₅₃	C₂×#2084	1	1	yes	characteristic, direct product factor, Sylow
193	C₂³×C₄	—	2	1	no	—
194	C₂³×C₄	—	2	2	no	—
195	C₂³×C₄	—	2	1	no	—
196	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
197	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
198	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
199	C₂²×C₈	—	2	4	no	—
200	C₂²×C₈	—	2	4	no	—
201	C₂²×C₈	Q₈×C₅₃	1	8	yes	—
202	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	—	2	4	no	—
209	C₂×C₄⋊C₄	—	2	4	no	—
210	C₂×C₄⋊C₄	—	2	4	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	4	4	no	—
214	C₂×C₄⋊C₄	—	4	4	no	—
215	C₂×C₄⋊C₄	—	4	4	no	—
216	C₂×C₄⋊C₄	D₄×C₅₃	1	4	yes	—
217	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
218	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
219	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
220	C₂×C₄²	—	2	4	no	—
221	C₂×C₄²	D₄×C₅₃	1	2	yes	—
222	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
223	#106	—	2	16	no	—
224	#95	—	2	4	no	—
225	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	4	4	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	4	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	SD₁₆×C₅₃	1	4	yes	—
243	C₂²×C₄	SD₁₆×C₅₃	1	4	yes	—
244	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
245	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
248	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
249	C₂²×C₄	#41×C₅₃	1	2	yes	—
250	C₂²×C₄	#41×C₅₃	1	4	yes	—
251	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
252	C₂×C₈	—	2	16	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	C₂×Q₈×C₅₃	1	8	yes	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	4	8	no	—
262	C₄⋊C₄	—	4	4	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	8	no	—
265	C₄⋊C₄	—	4	8	no	—
266	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
267	C₄²	—	2	4	no	—
268	C₄²	—	2	4	no	—
269	C₂³	#121×C₅₃	1	2	yes	—
270	C₂³	#123×C₅₃	1	2	yes	—
271	C₂³	C₄⋊Q₈×C₅₃	1	2	yes	—
272	C₂³	C₂×SD₁₆×C₅₃	1	2	yes	—
273	C₂³	C₂×SD₁₆×C₅₃	1	2	yes	—
274	C₂³	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
275	C₂³	#137×C₅₃	1	2	yes	—
276	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	8	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	8	8	no	—
294	C₂×C₄	—	8	4	no	—
295	C₂×C₄	#121×C₅₃	1	2	yes	—
296	C₂×C₄	C₂²⋊Q₈×C₅₃	1	8	yes	—
297	C₂×C₄	#123×C₅₃	1	2	yes	—
298	C₂×C₄	#125×C₅₃	1	8	yes	—
299	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
300	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
301	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
302	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
303	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
304	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
305	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
306	C₈	—	2	8	no	—
307	C₂²	C₅₃×#398	1	4	yes	—
308	C₂²	C₅₃×#460	1	4	yes	—
309	C₂²	C₅₃×#461	1	4	yes	—
310	C₂²	C₅₃×#481	1	4	yes	—
311	C₂²	C₅₃×#483	1	4	yes	—
312	C₂²	C₅₃×#498	1	4	yes	—
313	C₂²	C₅₃×#501	1	4	yes	—
314	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
315	C₂²	C₂×#123×C₅₃	1	1	yes	characteristic
316	C₂²	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
317	C₂²	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
318	C₂²	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
319	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
320	C₂²	C₂×#137×C₅₃	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	2	4	no	—
323	C₄	—	4	4	no	—
324	C₄	—	8	4	no	—
325	C₄	—	8	2	no	—
326	C₄	C₅₃×#398	1	4	yes	—
327	C₄	C₂×C₂²⋊Q₈×C₅₃	1	2	yes	—
328	C₄	C₂×#125×C₅₃	1	2	yes	—
329	C₂	C₅₃×#2084	1	8	yes	direct product factor
330	C₂	C₂×C₅₃×#398	1	1	yes	characteristic
331	C₂	C₂×C₅₃×#461	1	1	yes	characteristic
332	C₂	C₂×C₅₃×#460	1	1	yes	characteristic
333	C₂	C₂×C₅₃×#483	1	1	yes	characteristic
334	C₂	C₂×C₅₃×#481	1	1	yes	characteristic
335	C₂	C₂×C₅₃×#498	1	1	yes	characteristic
336	C₂	C₂×C₅₃×#501	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅₃	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#2084	C₂	direct (N × H)
138. C₂×#2084	C₅₃	direct (N × H)
164. #2084	C₂×C₅₃	direct (N × H)
167. C₂×C₅₃	#2084	direct (N × H)
192. C₅₃	C₂×#2084	direct (N × H)
329. C₂	C₅₃×#2084	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#106×C₅₃	2
C₂×C₅₃×#389	2
C₂×C₅₃×#384	2
C₅₃×#2084	2
C₂×C₅₃×#340	2
C₂×C₅₃×#336	2
C₂×#2084	53

Subgroups — C2×C53×#2084 #27231167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2084 #27231167