Subgroups — C₂×C₅₃×#1941 #27228661

Order 13568.

1182 conjugacy classes of subgroups.

The subgroup lattice has 300 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	1	yes	maximal, characteristic
2	C₂²×#106×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#389	C₂	1	2	yes	maximal
5	C₅₃×#1941	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅₃×#340	C₂	1	2	yes	maximal
7	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊C₄×C₅₃	—	2	2	no	—
10	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
12	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#106×C₅₃	—	2	4	no	—
14	C₂×#106×C₅₃	C₄	1	8	yes	complemented
15	C₂×#106×C₅₃	C₂²	1	4	yes	—
16	C₂×#106×C₅₃	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₅₃	—	2	4	no	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
19	C₅₃×#389	—	2	4	no	—
20	C₅₃×#389	C₂²	1	8	yes	—
21	C₂×#95×C₅₃	—	2	2	no	—
22	C₅₃×#340	C₂²	1	8	yes	—
23	C₂³×C₄×C₅₃	—	2	2	no	—
24	C₂³×C₄×C₅₃	—	2	2	no	—
25	C₂³×C₄×C₅₃	D₄	1	2	yes	—
26	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₅₃	D₄	1	4	yes	—
31	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
32	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
33	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
40	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
41	C₂×C₄⋊C₄×C₅₃	—	4	8	no	—
42	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
43	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
45	C₂×C₄²×C₅₃	—	2	2	no	—
46	C₂×C₄²×C₅₃	—	2	2	no	—
47	C₂×C₄²×C₅₃	D₄	1	2	yes	—
48	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
49	#106×C₅₃	—	2	8	no	—
50	#106×C₅₃	C₂×C₄	1	16	yes	complemented
51	C₄×C₈×C₅₃	—	2	8	no	—
52	#95×C₅₃	—	2	8	no	—
53	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	—	2	4	no	—
55	C₂²×C₄×C₅₃	—	2	4	no	—
56	C₂²×C₄×C₅₃	—	2	8	no	—
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	—	2	4	no	—
59	C₂²×C₄×C₅₃	—	4	4	no	—
60	C₂²×C₄×C₅₃	—	4	4	no	—
61	C₂²×C₄×C₅₃	—	4	4	no	—
62	C₂²×C₄×C₅₃	—	4	4	no	—
63	C₂²×C₄×C₅₃	—	4	4	no	—
64	C₂²×C₄×C₅₃	SD₁₆	1	8	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₅₃	#41	1	2	yes	—
72	C₂²×C₄×C₅₃	#41	1	2	yes	—
73	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₅₃	—	2	8	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	C₄⋊C₄	1	16	yes	complemented
78	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
80	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
81	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
82	C₄⋊C₄×C₅₃	—	2	8	no	—
83	C₄⋊C₄×C₅₃	—	2	4	no	—
84	C₄⋊C₄×C₅₃	—	2	4	no	—
85	C₄⋊C₄×C₅₃	—	2	4	no	—
86	C₄⋊C₄×C₅₃	—	4	4	no	—
87	C₄⋊C₄×C₅₃	—	4	16	no	—
88	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
89	C₄²×C₅₃	—	2	4	no	—
90	C₄²×C₅₃	—	2	2	no	—
91	C₄²×C₅₃	—	2	2	no	—
92	C₂³×C₅₃	C₄×D₄	1	2	yes	—
93	C₂³×C₅₃	#121	1	2	yes	—
94	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
95	C₂³×C₅₃	C₂×SD₁₆	1	4	yes	—
96	C₂³×C₅₃	#135	1	4	yes	—
97	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	—	2	4	no	—
99	C₂×C₄×C₅₃	—	2	4	no	—
100	C₂×C₄×C₅₃	—	2	16	no	—
101	C₂×C₄×C₅₃	—	2	8	no	—
102	C₂×C₄×C₅₃	—	4	4	no	—
103	C₂×C₄×C₅₃	—	4	4	no	—
104	C₂×C₄×C₅₃	—	4	16	no	—
105	C₂×C₄×C₅₃	—	4	4	no	—
106	C₂×C₄×C₅₃	—	4	4	no	—
107	C₂×C₄×C₅₃	—	4	4	no	—
108	C₂×C₄×C₅₃	—	8	8	no	—
109	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
110	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
112	C₂×C₄×C₅₃	#121	1	2	yes	—
113	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
114	C₂×C₄×C₅₃	#125	1	4	yes	—
115	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₅₃	C₂×SD₁₆	1	8	yes	—
117	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
123	C₈×C₅₃	—	2	4	no	—
124	C₈×C₅₃	C₂×C₄⋊C₄	1	8	yes	complemented
125	C₂×#1941	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₅₃	#389	1	4	yes	—
127	C₂²×C₅₃	C₄×SD₁₆	1	8	yes	—
128	C₂²×C₅₃	#465	1	8	yes	—
129	C₂²×C₅₃	#498	1	4	yes	—
130	C₂²×C₅₃	#499	1	4	yes	—
131	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
133	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
134	C₂²×C₅₃	C₂²×SD₁₆	1	2	yes	—
135	C₂²×C₅₃	C₂×#135	1	2	yes	—
136	C₄×C₅₃	—	2	8	no	—
137	C₄×C₅₃	—	4	8	no	—
138	C₄×C₅₃	—	8	4	no	—
139	C₄×C₅₃	#389	1	4	yes	—
140	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
142	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
143	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
144	C₂²×#106	C₂×C₅₃	1	1	yes	characteristic
145	C₂²×#106	C₂×C₅₃	1	1	yes	characteristic
146	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
147	C₂×#389	C₂×C₅₃	1	2	yes	—
148	#1941	C₂×C₅₃	1	8	yes	direct product factor
149	C₂×#340	C₂×C₅₃	1	2	yes	—
150	C₂×C₅₃	#1941	1	8	yes	direct product factor
151	C₂×C₅₃	C₂×#389	1	1	yes	characteristic
152	C₂×C₅₃	C₂×C₄×SD₁₆	1	2	yes	—
153	C₂×C₅₃	C₂×#465	1	2	yes	—
154	C₂×C₅₃	C₂×#498	1	1	yes	characteristic
155	C₂×C₅₃	C₂×#499	1	1	yes	characteristic
156	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
157	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
158	C₂²×C₄⋊C₄	—	2	2	no	—
159	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
160	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
161	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
162	C₂×#106	—	2	4	no	—
163	C₂×#106	C₄×C₅₃	1	8	yes	complemented
164	C₂×#106	C₂²×C₅₃	1	4	yes	—
165	C₂×#106	C₂²×C₅₃	1	4	yes	—
166	C₂×C₄×C₈	—	2	4	no	—
167	C₂×C₄×C₈	C₂²×C₅₃	1	4	yes	—
168	#389	—	2	4	no	—
169	#389	C₂²×C₅₃	1	8	yes	—
170	C₂×#95	—	2	2	no	—
171	#340	C₂²×C₅₃	1	8	yes	—
172	C₅₃	C₂×#1941	1	1	yes	characteristic, direct product factor, Sylow
173	C₂³×C₄	—	2	2	no	—
174	C₂³×C₄	—	2	2	no	—
175	C₂³×C₄	D₄×C₅₃	1	2	yes	—
176	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
180	C₂²×C₈	D₄×C₅₃	1	4	yes	—
181	C₂²×C₈	Q₈×C₅₃	1	4	yes	—
182	C₂²×C₈	Q₈×C₅₃	1	4	yes	—
183	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
184	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	4	no	—
187	C₂×C₄⋊C₄	—	2	4	no	—
188	C₂×C₄⋊C₄	—	2	4	no	—
189	C₂×C₄⋊C₄	—	2	8	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	4	8	no	—
192	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
193	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
194	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	D₄×C₅₃	1	2	yes	—
198	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
199	#106	—	2	8	no	—
200	#106	C₂×C₄×C₅₃	1	16	yes	complemented
201	C₄×C₈	—	2	8	no	—
202	#95	—	2	8	no	—
203	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	4	4	no	—
214	C₂²×C₄	SD₁₆×C₅₃	1	8	yes	—
215	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
217	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
218	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
220	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
221	C₂²×C₄	#41×C₅₃	1	2	yes	—
222	C₂²×C₄	#41×C₅₃	1	2	yes	—
223	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	C₄⋊C₄×C₅₃	1	16	yes	complemented
228	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
229	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
230	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
231	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄⋊C₄	—	4	16	no	—
238	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
239	C₄²	—	2	4	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₂³	C₄×D₄×C₅₃	1	2	yes	—
243	C₂³	#121×C₅₃	1	2	yes	—
244	C₂³	C₄⋊Q₈×C₅₃	1	2	yes	—
245	C₂³	C₂×SD₁₆×C₅₃	1	4	yes	—
246	C₂³	#135×C₅₃	1	4	yes	—
247	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	16	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	16	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	8	8	no	—
259	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
260	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
261	C₂×C₄	C₄×Q₈×C₅₃	1	4	yes	—
262	C₂×C₄	#121×C₅₃	1	2	yes	—
263	C₂×C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
264	C₂×C₄	#125×C₅₃	1	4	yes	—
265	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
266	C₂×C₄	C₂×SD₁₆×C₅₃	1	8	yes	—
267	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
268	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
269	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
270	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
271	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
272	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
273	C₈	—	2	4	no	—
274	C₈	C₂×C₄⋊C₄×C₅₃	1	8	yes	complemented
275	C₂²	C₅₃×#389	1	4	yes	—
276	C₂²	C₄×SD₁₆×C₅₃	1	8	yes	—
277	C₂²	C₅₃×#465	1	8	yes	—
278	C₂²	C₅₃×#498	1	4	yes	—
279	C₂²	C₅₃×#499	1	4	yes	—
280	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
281	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
282	C₂²	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
283	C₂²	C₂²×SD₁₆×C₅₃	1	2	yes	—
284	C₂²	C₂×#135×C₅₃	1	2	yes	—
285	C₄	—	2	8	no	—
286	C₄	—	4	8	no	—
287	C₄	—	8	4	no	—
288	C₄	C₅₃×#389	1	4	yes	—
289	C₄	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
290	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
291	C₄	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
292	C₄	C₂×#125×C₅₃	1	1	yes	characteristic
293	C₂	C₅₃×#1941	1	8	yes	direct product factor
294	C₂	C₂×C₅₃×#389	1	1	yes	characteristic
295	C₂	C₂×C₄×SD₁₆×C₅₃	1	2	yes	—
296	C₂	C₂×C₅₃×#465	1	2	yes	—
297	C₂	C₂×C₅₃×#498	1	1	yes	characteristic
298	C₂	C₂×C₅₃×#499	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂×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
5. C₅₃×#1941	C₂	direct (N × H)
14. C₂×#106×C₅₃	C₄	semidirect
50. #106×C₅₃	C₂×C₄	semidirect
77. C₂×C₈×C₅₃	C₄⋊C₄	semidirect
124. C₈×C₅₃	C₂×C₄⋊C₄	semidirect
125. C₂×#1941	C₅₃	direct (N × H)
148. #1941	C₂×C₅₃	direct (N × H)
150. C₂×C₅₃	#1941	direct (N × H)
163. C₂×#106	C₄×C₅₃	semidirect
172. C₅₃	C₂×#1941	direct (N × H)
200. #106	C₂×C₄×C₅₃	semidirect
227. C₂×C₈	C₄⋊C₄×C₅₃	semidirect
274. C₈	C₂×C₄⋊C₄×C₅₃	semidirect
293. C₂	C₅₃×#1941	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
C₂²×C₄×C₈×C₅₃	2
C₂×C₅₃×#389	2
C₅₃×#1941	2
C₂×C₅₃×#340	2
C₂×#1941	53

Subgroups — C2×C53×#1941 #27228661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1941 #27228661