Subgroups — C₂×C₅₃×#1945 #27228673

Order 13568.

974 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#424	C₂	1	2	yes	maximal
4	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1945	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅₃×#339	C₂	1	2	yes	maximal
7	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×#117×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#108×C₅₃	—	2	4	no	—
13	C₂×#108×C₅₃	C₂²	1	4	yes	—
14	C₅₃×#427	C₄	1	4	yes	complemented
15	C₅₃×#427	C₂²	1	4	yes	—
16	C₂×#107×C₅₃	C₄	1	2	yes	complemented
17	C₂×#106×C₅₃	C₄	1	2	yes	complemented
18	C₅₃×#424	—	2	4	no	—
19	C₅₃×#424	C₂²	1	8	yes	—
20	C₂×C₄⋊C₈×C₅₃	—	2	4	no	—
21	C₂×C₄×C₈×C₅₃	—	2	4	no	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
23	C₅₃×#339	C₂²	1	8	yes	—
24	C₂³×C₄×C₅₃	—	2	1	no	—
25	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₅₃	—	2	2	no	—
27	C₂×M₁₆×C₅₃	—	2	4	no	—
28	C₂×M₁₆×C₅₃	—	2	4	no	—
29	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
30	C₂²×C₈×C₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	—	2	2	no	—
32	C₂²×C₈×C₅₃	—	2	2	no	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₅₃	D₄	1	2	yes	—
37	C₂²×C₈×C₅₃	D₄	1	4	yes	—
38	C₂²×C₈×C₅₃	D₄	1	2	yes	—
39	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
40	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
42	#117×C₅₃	—	2	2	no	—
43	#117×C₅₃	—	2	2	no	—
44	#117×C₅₃	C₂×C₄	1	4	yes	—
45	#117×C₅₃	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
47	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
48	C₂×C₄²×C₅₃	—	2	2	no	—
49	C₂×C₄²×C₅₃	—	2	2	no	—
50	C₂×C₄²×C₅₃	—	2	1	no	—
51	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
52	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
53	#108×C₅₃	—	2	8	no	—
54	#107×C₅₃	C₂×C₄	1	8	yes	complemented
55	#106×C₅₃	C₂×C₄	1	8	yes	complemented
56	C₄⋊C₈×C₅₃	—	2	16	no	—
57	C₄×C₈×C₅₃	—	2	8	no	—
58	C₂⁴×C₅₃	#41	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	—	2	8	no	—
60	C₂²×C₄×C₅₃	—	2	4	no	—
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	—	2	4	no	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	#41	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₅₃	—	2	4	no	—
74	M₁₆×C₅₃	—	2	4	no	—
75	M₁₆×C₅₃	—	4	4	no	—
76	C₂×C₈×C₅₃	—	2	4	no	—
77	C₂×C₈×C₅₃	—	2	4	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	—	2	4	no	—
81	C₂×C₈×C₅₃	—	4	4	no	—
82	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
83	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
84	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
85	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
88	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
89	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
90	C₄⋊C₄×C₅₃	—	2	4	no	—
91	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
92	C₂²⋊C₄×C₅₃	—	2	4	no	—
93	C₂²⋊C₄×C₅₃	—	2	4	no	—
94	C₄²×C₅₃	—	2	4	no	—
95	C₄²×C₅₃	—	2	4	no	—
96	C₄²×C₅₃	—	2	2	no	—
97	C₄²×C₅₃	—	2	4	no	—
98	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
99	C₂³×C₅₃	—	2	4	no	—
100	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
101	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
102	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
103	C₂³×C₅₃	#125	1	2	yes	—
104	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₅₃	—	2	8	no	—
106	C₂×C₄×C₅₃	—	2	16	no	—
107	C₂×C₄×C₅₃	—	2	4	no	—
108	C₂×C₄×C₅₃	—	2	8	no	—
109	C₂×C₄×C₅₃	—	4	8	no	—
110	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
116	C₂×C₄×C₅₃	#121	1	2	yes	—
117	C₂×C₄×C₅₃	#121	1	2	yes	—
118	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₅₃	#125	1	2	yes	—
120	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
121	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
122	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
128	C₈×C₅₃	—	2	4	no	—
129	C₈×C₅₃	—	4	4	no	—
130	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
131	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
132	C₂×#1945	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	—	2	4	no	—
136	C₂²×C₅₃	—	2	4	no	—
137	C₂²×C₅₃	#389	1	2	yes	—
138	C₂²×C₅₃	#389	1	2	yes	—
139	C₂²×C₅₃	#443	1	4	yes	—
140	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
142	C₂²×C₅₃	C₂×#125	1	1	yes	characteristic
143	C₄×C₅₃	—	2	8	no	—
144	C₄×C₅₃	—	4	8	no	—
145	C₄×C₅₃	#389	1	2	yes	—
146	C₄×C₅₃	#389	1	2	yes	—
147	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
148	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
150	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
151	C₂²×#108	C₂×C₅₃	1	1	yes	characteristic
152	C₂×#427	C₂×C₅₃	1	1	yes	characteristic
153	C₂×#424	C₂×C₅₃	1	2	yes	—
154	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
155	#1945	C₂×C₅₃	1	8	yes	direct product factor
156	C₂×#339	C₂×C₅₃	1	2	yes	—
157	C₂×C₅₃	—	2	1	no	—
158	C₂×C₅₃	—	2	1	no	—
159	C₂×C₅₃	—	2	2	no	—
160	C₂×C₅₃	#1945	1	4	yes	direct product factor
161	C₂×C₅₃	C₂×#389	1	1	yes	characteristic
162	C₂×C₅₃	C₂×#443	1	2	yes	—
163	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
164	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
165	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
166	C₂×#117	C₂²×C₅₃	1	2	yes	—
167	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
168	C₂×#108	—	2	4	no	—
169	C₂×#108	C₂²×C₅₃	1	4	yes	—
170	#427	C₄×C₅₃	1	4	yes	complemented
171	#427	C₂²×C₅₃	1	4	yes	—
172	C₂×#107	C₄×C₅₃	1	2	yes	complemented
173	C₂×#106	C₄×C₅₃	1	2	yes	complemented
174	#424	—	2	4	no	—
175	#424	C₂²×C₅₃	1	8	yes	—
176	C₂×C₄⋊C₈	—	2	4	no	—
177	C₂×C₄×C₈	—	2	4	no	—
178	C₂×C₄×C₈	C₂²×C₅₃	1	4	yes	—
179	#339	C₂²×C₅₃	1	8	yes	—
180	C₅₃	C₂×#1945	1	1	yes	characteristic, direct product factor, Sylow
181	C₂³×C₄	—	2	1	no	—
182	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	4	no	—
185	C₂×M₁₆	—	2	4	no	—
186	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
191	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
192	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
193	C₂²×C₈	D₄×C₅₃	1	2	yes	—
194	C₂²×C₈	D₄×C₅₃	1	2	yes	—
195	C₂²×C₈	D₄×C₅₃	1	4	yes	—
196	C₂²×C₈	Q₈×C₅₃	1	4	yes	—
197	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
198	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	C₂×C₄×C₅₃	1	4	yes	—
202	#117	C₂³×C₅₃	1	4	yes	—
203	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	Q₈×C₅₃	1	4	yes	—
209	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
210	#108	—	2	8	no	—
211	#107	C₂×C₄×C₅₃	1	8	yes	complemented
212	#106	C₂×C₄×C₅₃	1	8	yes	complemented
213	C₄⋊C₈	—	2	16	no	—
214	C₄×C₈	—	2	8	no	—
215	C₂⁴	#41×C₅₃	1	1	yes	characteristic
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
221	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
225	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
226	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
227	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
228	C₂²×C₄	#41×C₅₃	1	2	yes	—
229	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
230	M₁₆	—	2	4	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	4	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	4	4	no	—
239	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
240	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
241	C₂×C₈	C₄⋊C₄×C₅₃	1	8	yes	—
242	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
243	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
244	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
245	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
246	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	4	no	—
255	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
256	C₂³	—	2	4	no	—
257	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
258	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
259	C₂³	C₂²⋊Q₈×C₅₃	1	2	yes	—
260	C₂³	#125×C₅₃	1	2	yes	—
261	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	16	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	4	8	no	—
267	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
268	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
269	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
270	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
271	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
272	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
273	C₂×C₄	#121×C₅₃	1	2	yes	—
274	C₂×C₄	#121×C₅₃	1	2	yes	—
275	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
276	C₂×C₄	#125×C₅₃	1	2	yes	—
277	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
278	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
279	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
280	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
282	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
283	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
284	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
285	C₈	—	2	4	no	—
286	C₈	—	4	4	no	—
287	C₈	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
288	C₈	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	2	4	no	—
293	C₂²	C₅₃×#389	1	2	yes	—
294	C₂²	C₅₃×#389	1	2	yes	—
295	C₂²	C₅₃×#443	1	4	yes	—
296	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
297	C₂²	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
298	C₂²	C₂×#125×C₅₃	1	1	yes	characteristic
299	C₄	—	2	8	no	—
300	C₄	—	4	8	no	—
301	C₄	C₅₃×#389	1	2	yes	—
302	C₄	C₅₃×#389	1	2	yes	—
303	C₄	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
304	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
305	C₄	C₂×#121×C₅₃	1	1	yes	characteristic
306	C₄	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	2	2	no	—
310	C₂	C₅₃×#1945	1	4	yes	direct product factor
311	C₂	C₂×C₅₃×#389	1	1	yes	characteristic
312	C₂	C₂×C₅₃×#443	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂⁴×C₅₃	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅₃×#1945	C₂	direct (N × H)
14. C₅₃×#427	C₄	semidirect
16. C₂×#107×C₅₃	C₄	semidirect
17. C₂×#106×C₅₃	C₄	semidirect
54. #107×C₅₃	C₂×C₄	semidirect
55. #106×C₅₃	C₂×C₄	semidirect
132. C₂×#1945	C₅₃	direct (N × H)
155. #1945	C₂×C₅₃	direct (N × H)
160. C₂×C₅₃	#1945	direct (N × H)
170. #427	C₄×C₅₃	semidirect
172. C₂×#107	C₄×C₅₃	semidirect
173. C₂×#106	C₄×C₅₃	semidirect
180. C₅₃	C₂×#1945	direct (N × H)
211. #107	C₂×C₄×C₅₃	semidirect
212. #106	C₂×C₄×C₅₃	semidirect
310. C₂	C₅₃×#1945	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×#108×C₅₃	2
C₂×C₅₃×#427	2
C₂×C₅₃×#424	2
C₂²×C₄×C₈×C₅₃	2
C₅₃×#1945	2
C₂×C₅₃×#339	2
C₂×#1945	53

Subgroups — C2×C53×#1945 #27228673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1945 #27228673