Subgroups — C₂×C₅₃×#1944 #27228665

Order 13568.

1142 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#106×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#389	C₂	1	2	yes	maximal
5	C₅₃×#1944	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×#107×C₅₃	—	2	4	no	—
14	C₂×#107×C₅₃	C₂²	1	4	yes	—
15	C₂×#106×C₅₃	C₄	1	8	yes	complemented
16	C₂×#106×C₅₃	C₂²	1	4	yes	—
17	C₂×#97×C₅₃	—	2	4	no	—
18	C₂×#97×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	1	yes	characteristic
26	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₅₃	D₄	1	4	yes	—
32	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
33	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
40	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
41	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
42	C₂×C₄⋊C₄×C₅₃	—	4	8	no	—
43	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
44	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
45	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
46	C₂×C₄²×C₅₃	—	2	2	no	—
47	C₂×C₄²×C₅₃	—	2	2	no	—
48	C₂×C₄²×C₅₃	D₄	1	2	yes	—
49	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
50	#107×C₅₃	—	2	8	no	—
51	#106×C₅₃	C₂×C₄	1	16	yes	complemented
52	#97×C₅₃	—	2	8	no	—
53	#95×C₅₃	—	2	8	no	—
54	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	—	2	8	no	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	—	2	4	no	—
59	C₂²×C₄×C₅₃	—	2	4	no	—
60	C₂²×C₄×C₅₃	—	2	4	no	—
61	C₂²×C₄×C₅₃	—	2	4	no	—
62	C₂²×C₄×C₅₃	—	4	4	no	—
63	C₂²×C₄×C₅₃	—	4	4	no	—
64	C₂²×C₄×C₅₃	—	4	4	no	—
65	C₂²×C₄×C₅₃	—	4	4	no	—
66	C₂²×C₄×C₅₃	—	4	4	no	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₅₃	#41	1	2	yes	—
75	C₂²×C₄×C₅₃	#41	1	2	yes	—
76	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₅₃	—	2	8	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	C₄⋊C₄	1	16	yes	complemented
81	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
83	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
84	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
85	C₄⋊C₄×C₅₃	—	2	8	no	—
86	C₄⋊C₄×C₅₃	—	2	4	no	—
87	C₄⋊C₄×C₅₃	—	2	4	no	—
88	C₄⋊C₄×C₅₃	—	2	4	no	—
89	C₄⋊C₄×C₅₃	—	4	16	no	—
90	C₄⋊C₄×C₅₃	—	4	4	no	—
91	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
92	C₄²×C₅₃	—	2	4	no	—
93	C₄²×C₅₃	—	2	2	no	—
94	C₄²×C₅₃	—	2	2	no	—
95	C₂³×C₅₃	C₄×D₄	1	2	yes	—
96	C₂³×C₅₃	#121	1	2	yes	—
97	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
98	C₂³×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
99	C₂³×C₅₃	#137	1	4	yes	—
100	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	—	2	8	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	4	4	no	—
107	C₂×C₄×C₅₃	—	4	4	no	—
108	C₂×C₄×C₅₃	—	4	16	no	—
109	C₂×C₄×C₅₃	—	4	4	no	—
110	C₂×C₄×C₅₃	—	4	4	no	—
111	C₂×C₄×C₅₃	—	4	4	no	—
112	C₂×C₄×C₅₃	—	4	4	no	—
113	C₂×C₄×C₅₃	—	4	4	no	—
114	C₂×C₄×C₅₃	—	8	8	no	—
115	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
116	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
118	C₂×C₄×C₅₃	#121	1	2	yes	—
119	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
120	C₂×C₄×C₅₃	#125	1	4	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×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
128	C₈×C₅₃	—	2	4	no	—
129	C₈×C₅₃	C₂×C₄⋊C₄	1	8	yes	complemented
130	C₂×#1944	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₅₃	#389	1	4	yes	—
132	C₂²×C₅₃	#440	1	8	yes	—
133	C₂²×C₅₃	#469	1	4	yes	—
134	C₂²×C₅₃	#470	1	4	yes	—
135	C₂²×C₅₃	#501	1	8	yes	—
136	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
138	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
139	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
140	C₂²×C₅₃	C₂×#137	1	2	yes	—
141	C₄×C₅₃	—	4	8	no	—
142	C₄×C₅₃	—	4	2	no	—
143	C₄×C₅₃	—	4	2	no	—
144	C₄×C₅₃	—	8	4	no	—
145	C₄×C₅₃	#389	1	4	yes	—
146	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
148	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
149	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
150	C₂²×#107	C₂×C₅₃	1	1	yes	characteristic
151	C₂²×#106	C₂×C₅₃	1	1	yes	characteristic
152	C₂²×#97	C₂×C₅₃	1	1	yes	characteristic
153	C₂×#389	C₂×C₅₃	1	2	yes	—
154	#1944	C₂×C₅₃	1	8	yes	direct product factor
155	C₂×#340	C₂×C₅₃	1	2	yes	—
156	C₂×C₅₃	#1944	1	8	yes	direct product factor
157	C₂×C₅₃	C₂×#389	1	1	yes	characteristic
158	C₂×C₅₃	C₂×#440	1	2	yes	—
159	C₂×C₅₃	C₂×#469	1	1	yes	characteristic
160	C₂×C₅₃	C₂×#470	1	1	yes	characteristic
161	C₂×C₅₃	C₂×#501	1	2	yes	—
162	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
163	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
164	C₂²×C₄⋊C₄	—	2	2	no	—
165	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
166	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
167	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
168	C₂×#107	—	2	4	no	—
169	C₂×#107	C₂²×C₅₃	1	4	yes	—
170	C₂×#106	C₄×C₅₃	1	8	yes	complemented
171	C₂×#106	C₂²×C₅₃	1	4	yes	—
172	C₂×#97	—	2	4	no	—
173	C₂×#97	C₂²×C₅₃	1	4	yes	—
174	#389	—	2	4	no	—
175	#389	C₂²×C₅₃	1	8	yes	—
176	C₂×#95	—	2	2	no	—
177	#340	C₂²×C₅₃	1	8	yes	—
178	C₅₃	C₂×#1944	1	1	yes	characteristic, direct product factor, Sylow
179	C₂³×C₄	—	2	2	no	—
180	C₂³×C₄	—	2	2	no	—
181	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
182	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
183	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
187	C₂²×C₈	D₄×C₅₃	1	4	yes	—
188	C₂²×C₈	Q₈×C₅₃	1	4	yes	—
189	C₂²×C₈	Q₈×C₅₃	1	4	yes	—
190	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
191	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	—	2	4	no	—
196	C₂×C₄⋊C₄	—	2	8	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	4	8	no	—
199	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
200	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
201	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	D₄×C₅₃	1	2	yes	—
205	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
206	#107	—	2	8	no	—
207	#106	C₂×C₄×C₅₃	1	16	yes	complemented
208	#97	—	2	8	no	—
209	#95	—	2	8	no	—
210	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	4	4	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
227	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
229	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
230	C₂²×C₄	#41×C₅₃	1	2	yes	—
231	C₂²×C₄	#41×C₅₃	1	2	yes	—
232	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
233	C₂×C₈	—	2	8	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	C₄⋊C₄×C₅₃	1	16	yes	complemented
237	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
238	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
239	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
240	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
241	C₄⋊C₄	—	2	8	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	4	16	no	—
246	C₄⋊C₄	—	4	4	no	—
247	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
248	C₄²	—	2	4	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₂³	C₄×D₄×C₅₃	1	2	yes	—
252	C₂³	#121×C₅₃	1	2	yes	—
253	C₂³	C₄⋊Q₈×C₅₃	1	2	yes	—
254	C₂³	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
255	C₂³	#137×C₅₃	1	4	yes	—
256	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	16	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	8	8	no	—
271	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
272	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
273	C₂×C₄	C₄×Q₈×C₅₃	1	4	yes	—
274	C₂×C₄	#121×C₅₃	1	2	yes	—
275	C₂×C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
276	C₂×C₄	#125×C₅₃	1	4	yes	—
277	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
278	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
279	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
280	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
281	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
282	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
283	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
284	C₈	—	2	4	no	—
285	C₈	C₂×C₄⋊C₄×C₅₃	1	8	yes	complemented
286	C₂²	C₅₃×#389	1	4	yes	—
287	C₂²	C₅₃×#440	1	8	yes	—
288	C₂²	C₅₃×#469	1	4	yes	—
289	C₂²	C₅₃×#470	1	4	yes	—
290	C₂²	C₅₃×#501	1	8	yes	—
291	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
292	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
293	C₂²	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
294	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
295	C₂²	C₂×#137×C₅₃	1	2	yes	—
296	C₄	—	4	8	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	2	no	—
299	C₄	—	8	4	no	—
300	C₄	C₅₃×#389	1	4	yes	—
301	C₄	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
302	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
303	C₄	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
304	C₄	C₂×#125×C₅₃	1	1	yes	characteristic
305	C₂	C₅₃×#1944	1	8	yes	direct product factor
306	C₂	C₂×C₅₃×#389	1	1	yes	characteristic
307	C₂	C₂×C₅₃×#440	1	2	yes	—
308	C₂	C₂×C₅₃×#469	1	1	yes	characteristic
309	C₂	C₂×C₅₃×#470	1	1	yes	characteristic
310	C₂	C₂×C₅₃×#501	1	2	yes	—

	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×#1944	C₂	direct (N × H)
15. C₂×#106×C₅₃	C₄	semidirect
51. #106×C₅₃	C₂×C₄	semidirect
80. C₂×C₈×C₅₃	C₄⋊C₄	semidirect
129. C₈×C₅₃	C₂×C₄⋊C₄	semidirect
130. C₂×#1944	C₅₃	direct (N × H)
154. #1944	C₂×C₅₃	direct (N × H)
156. C₂×C₅₃	#1944	direct (N × H)
170. C₂×#106	C₄×C₅₃	semidirect
178. C₅₃	C₂×#1944	direct (N × H)
207. #106	C₂×C₄×C₅₃	semidirect
236. C₂×C₈	C₄⋊C₄×C₅₃	semidirect
285. C₈	C₂×C₄⋊C₄×C₅₃	semidirect
305. C₂	C₅₃×#1944	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×#107×C₅₃	2
C₂²×#106×C₅₃	2
C₂²×#97×C₅₃	2
C₂×C₅₃×#389	2
C₅₃×#1944	2
C₂×C₅₃×#340	2
C₂×#1944	53

Subgroups — C2×C53×#1944 #27228665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1944 #27228665