Subgroups — C₂×C₅₃×#1957 #27228705

Order 13568.

2414 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#102×C₅₃	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1957	C₂	1	8	yes	maximal, direct product factor
6	C₂³×D₄×C₅₃	—	2	2	no	—
7	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
8	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
9	C₂×#127×C₅₃	—	2	4	no	—
10	C₂×#127×C₅₃	—	2	2	no	—
11	C₂×#127×C₅₃	C₄	1	8	yes	complemented
12	C₂×#127×C₅₃	C₂²	1	4	yes	—
13	C₂²×C₄⋊C₄×C₅₃	—	2	2	no	—
14	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#102×C₅₃	—	2	16	no	—
17	C₂×#102×C₅₃	C₂²	1	16	yes	complemented
18	C₂×C₄×C₈×C₅₃	—	2	4	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	complemented
20	C₅₃×#384	—	2	2	no	—
21	C₅₃×#384	C₂²	1	4	yes	complemented
22	C₂⁵×C₅₃	—	2	2	no	—
23	C₂²×D₄×C₅₃	—	2	16	no	—
24	C₂²×D₄×C₅₃	—	2	8	no	—
25	C₂²×D₄×C₅₃	—	2	2	no	—
26	C₂²×D₄×C₅₃	—	2	4	no	—
27	C₂²×D₄×C₅₃	—	2	8	no	—
28	C₂²×D₄×C₅₃	—	2	8	no	—
29	C₂²×D₄×C₅₃	—	4	8	no	—
30	C₂²×D₄×C₅₃	—	4	8	no	—
31	C₂²×D₄×C₅₃	C₂×C₄	1	8	yes	—
32	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
33	C₂³×C₄×C₅₃	—	2	2	no	—
34	C₂³×C₄×C₅₃	D₄	1	2	yes	—
35	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	—	2	4	no	—
37	C₂²×C₈×C₅₃	—	2	4	no	—
38	C₂²×C₈×C₅₃	D₄	1	8	yes	complemented
39	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
40	#127×C₅₃	—	2	16	no	—
41	#127×C₅₃	—	2	8	no	—
42	#127×C₅₃	C₂×C₄	1	16	yes	complemented
43	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
46	C₂×C₄⋊C₄×C₅₃	—	4	8	no	—
47	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
48	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₅₃	D₄	1	4	yes	—
50	C₂×C₄²×C₅₃	D₄	1	2	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
52	#102×C₅₃	—	2	32	no	—
53	C₄×C₈×C₅₃	—	2	8	no	—
54	C₂⁴×C₅₃	—	2	8	no	—
55	C₂⁴×C₅₃	—	2	4	no	—
56	C₂⁴×C₅₃	—	4	4	no	—
57	C₂⁴×C₅₃	—	4	4	no	—
58	C₂⁴×C₅₃	—	4	8	no	—
59	C₂⁴×C₅₃	—	4	8	no	—
60	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂×D₄×C₅₃	—	2	8	no	—
62	C₂×D₄×C₅₃	—	2	32	no	—
63	C₂×D₄×C₅₃	—	2	32	no	—
64	C₂×D₄×C₅₃	—	2	8	no	—
65	C₂×D₄×C₅₃	—	2	4	no	—
66	C₂×D₄×C₅₃	—	2	8	no	—
67	C₂×D₄×C₅₃	—	4	64	no	—
68	C₂×D₄×C₅₃	—	4	16	no	—
69	C₂×D₄×C₅₃	—	4	4	no	—
70	C₂×D₄×C₅₃	—	4	4	no	—
71	C₂×D₄×C₅₃	—	4	16	no	—
72	C₂×D₄×C₅₃	—	4	16	no	—
73	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
74	C₂²×C₄×C₅₃	—	2	4	no	—
75	C₂²×C₄×C₅₃	—	4	4	no	—
76	C₂²×C₄×C₅₃	—	4	8	no	—
77	C₂²×C₄×C₅₃	C₂²⋊C₄	1	8	yes	—
78	C₂²×C₄×C₅₃	D₈	1	8	yes	—
79	C₂²×C₄×C₅₃	SD₁₆	1	8	yes	—
80	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₅₃	#41	1	4	yes	—
86	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
87	C₂×C₈×C₅₃	—	2	16	no	—
88	C₂×C₈×C₅₃	—	2	4	no	—
89	C₂×C₈×C₅₃	—	2	4	no	—
90	C₂×C₈×C₅₃	C₂×D₄	1	8	yes	complemented
91	C₄⋊C₄×C₅₃	—	2	8	no	—
92	C₄⋊C₄×C₅₃	—	4	4	no	—
93	C₄⋊C₄×C₅₃	—	4	16	no	—
94	C₄²×C₅₃	C₂²⋊C₄	1	8	yes	—
95	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
96	C₄²×C₅₃	C₂×D₄	1	4	yes	—
97	C₂³×C₅₃	—	2	8	no	—
98	C₂³×C₅₃	—	4	4	no	—
99	C₂³×C₅₃	—	4	4	no	—
100	C₂³×C₅₃	—	4	8	no	—
101	C₂³×C₅₃	—	4	8	no	—
102	C₂³×C₅₃	—	4	16	no	—
103	C₂³×C₅₃	—	4	32	no	—
104	C₂³×C₅₃	—	8	16	no	—
105	C₂³×C₅₃	—	8	8	no	—
106	C₂³×C₅₃	—	8	8	no	—
107	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂³×C₅₃	#124	1	2	yes	—
109	C₂³×C₅₃	#127	1	2	yes	—
110	C₂³×C₅₃	C₂×D₈	1	4	yes	—
111	C₂³×C₅₃	C₂×SD₁₆	1	4	yes	—
112	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
113	D₄×C₅₃	—	2	16	no	—
114	D₄×C₅₃	—	4	8	no	—
115	D₄×C₅₃	—	4	16	no	—
116	D₄×C₅₃	—	4	64	no	—
117	C₂×C₄×C₅₃	—	4	4	no	—
118	C₂×C₄×C₅₃	—	4	8	no	—
119	C₂×C₄×C₅₃	—	8	8	no	—
120	C₂×C₄×C₅₃	#102	1	32	yes	complemented
121	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
122	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	8	yes	—
123	C₂×C₄×C₅₃	C₄×D₄	1	8	yes	—
124	C₂×C₄×C₅₃	#121	1	8	yes	—
125	C₂×C₄×C₅₃	#124	1	2	yes	—
126	C₂×C₄×C₅₃	#127	1	2	yes	—
127	C₂×C₄×C₅₃	C₂×D₈	1	8	yes	—
128	C₂×C₄×C₅₃	C₂×SD₁₆	1	8	yes	—
129	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
131	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
133	C₈×C₅₃	—	2	8	no	—
134	C₂×#1957	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂²×C₅₃	—	4	16	no	—
136	C₂²×C₅₃	—	4	8	no	—
137	C₂²×C₅₃	—	8	8	no	—
138	C₂²×C₅₃	—	8	4	no	—
139	C₂²×C₅₃	—	8	4	no	—
140	C₂²×C₅₃	—	8	32	no	—
141	C₂²×C₅₃	#390	1	4	yes	—
142	C₂²×C₅₃	C₂×#102	1	8	yes	—
143	C₂²×C₅₃	#486	1	8	yes	—
144	C₂²×C₅₃	#492	1	4	yes	—
145	C₂²×C₅₃	#493	1	4	yes	—
146	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₅₃	C₂×#124	1	1	yes	characteristic
148	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
149	C₂²×C₅₃	C₂²×D₈	1	2	yes	—
150	C₂²×C₅₃	C₂²×SD₁₆	1	2	yes	—
151	C₄×C₅₃	—	8	4	no	—
152	C₄×C₅₃	#390	1	4	yes	—
153	C₄×C₅₃	C₂×#102	1	16	yes	complemented
154	C₄×C₅₃	C₂×C₄×D₄	1	2	yes	—
155	C₄×C₅₃	C₂×#121	1	2	yes	—
156	C₂²×#127	C₂×C₅₃	1	1	yes	characteristic
157	C₂²×#102	C₂×C₅₃	1	4	yes	complemented
158	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
159	C₂×#384	C₂×C₅₃	1	1	yes	characteristic, complemented
160	#1957	C₂×C₅₃	1	8	yes	direct product factor
161	C₂×C₅₃	—	8	8	no	—
162	C₂×C₅₃	#1957	1	8	yes	direct product factor
163	C₂×C₅₃	C₂×#390	1	1	yes	characteristic
164	C₂×C₅₃	C₂²×#102	1	2	yes	—
165	C₂×C₅₃	C₂×#486	1	2	yes	—
166	C₂×C₅₃	C₂×#492	1	1	yes	characteristic
167	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
168	C₂³×D₄	—	2	2	no	—
169	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
170	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
171	C₂×#127	—	2	4	no	—
172	C₂×#127	—	2	2	no	—
173	C₂×#127	C₄×C₅₃	1	8	yes	complemented
174	C₂×#127	C₂²×C₅₃	1	4	yes	—
175	C₂²×C₄⋊C₄	—	2	2	no	—
176	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
177	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
178	C₂×#102	—	2	16	no	—
179	C₂×#102	C₂²×C₅₃	1	16	yes	complemented
180	C₂×C₄×C₈	—	2	4	no	—
181	C₂×C₄×C₈	C₂²×C₅₃	1	4	yes	complemented
182	#384	—	2	2	no	—
183	#384	C₂²×C₅₃	1	4	yes	complemented
184	C₅₃	C₂×#1957	1	1	yes	characteristic, direct product factor, Sylow
185	C₂⁵	—	2	2	no	—
186	C₂²×D₄	—	2	8	no	—
187	C₂²×D₄	—	2	2	no	—
188	C₂²×D₄	—	2	4	no	—
189	C₂²×D₄	—	2	8	no	—
190	C₂²×D₄	—	2	8	no	—
191	C₂²×D₄	—	2	16	no	—
192	C₂²×D₄	—	4	8	no	—
193	C₂²×D₄	—	4	8	no	—
194	C₂²×D₄	C₂×C₄×C₅₃	1	8	yes	—
195	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
196	C₂³×C₄	—	2	2	no	—
197	C₂³×C₄	D₄×C₅₃	1	2	yes	—
198	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
199	C₂²×C₈	—	2	4	no	—
200	C₂²×C₈	—	2	4	no	—
201	C₂²×C₈	D₄×C₅₃	1	8	yes	complemented
202	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
203	#127	—	2	8	no	—
204	#127	—	2	16	no	—
205	#127	C₂×C₄×C₅₃	1	16	yes	complemented
206	C₂×C₄⋊C₄	—	2	8	no	—
207	C₂×C₄⋊C₄	—	2	8	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	4	8	no	—
210	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
211	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
212	C₂×C₄²	D₄×C₅₃	1	4	yes	—
213	C₂×C₄²	D₄×C₅₃	1	2	yes	—
214	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
215	#102	—	2	32	no	—
216	C₄×C₈	—	2	8	no	—
217	C₂⁴	—	2	8	no	—
218	C₂⁴	—	2	4	no	—
219	C₂⁴	—	4	4	no	—
220	C₂⁴	—	4	4	no	—
221	C₂⁴	—	4	8	no	—
222	C₂⁴	—	4	8	no	—
223	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
224	C₂×D₄	—	2	8	no	—
225	C₂×D₄	—	2	32	no	—
226	C₂×D₄	—	2	32	no	—
227	C₂×D₄	—	2	8	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	8	no	—
230	C₂×D₄	—	4	64	no	—
231	C₂×D₄	—	4	16	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	—	4	4	no	—
234	C₂×D₄	—	4	16	no	—
235	C₂×D₄	—	4	16	no	—
236	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	—	4	8	no	—
240	C₂²×C₄	C₂²⋊C₄×C₅₃	1	8	yes	—
241	C₂²×C₄	D₈×C₅₃	1	8	yes	—
242	C₂²×C₄	SD₁₆×C₅₃	1	8	yes	—
243	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
244	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
246	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
247	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
248	C₂²×C₄	#41×C₅₃	1	4	yes	—
249	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
250	C₂×C₈	—	2	16	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	C₂×D₄×C₅₃	1	8	yes	complemented
254	C₄⋊C₄	—	2	8	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	—	4	16	no	—
257	C₄²	C₂²⋊C₄×C₅₃	1	8	yes	—
258	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
259	C₄²	C₂×D₄×C₅₃	1	4	yes	—
260	C₂³	—	2	8	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	4	4	no	—
263	C₂³	—	4	8	no	—
264	C₂³	—	4	8	no	—
265	C₂³	—	4	16	no	—
266	C₂³	—	4	32	no	—
267	C₂³	—	8	16	no	—
268	C₂³	—	8	8	no	—
269	C₂³	—	8	8	no	—
270	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
271	C₂³	#124×C₅₃	1	2	yes	—
272	C₂³	#127×C₅₃	1	2	yes	—
273	C₂³	C₂×D₈×C₅₃	1	4	yes	—
274	C₂³	C₂×SD₁₆×C₅₃	1	4	yes	—
275	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
276	D₄	—	2	16	no	—
277	D₄	—	4	8	no	—
278	D₄	—	4	16	no	—
279	D₄	—	4	64	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	8	no	—
282	C₂×C₄	—	8	8	no	—
283	C₂×C₄	#102×C₅₃	1	32	yes	complemented
284	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
285	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	8	yes	—
286	C₂×C₄	C₄×D₄×C₅₃	1	8	yes	—
287	C₂×C₄	#121×C₅₃	1	8	yes	—
288	C₂×C₄	#124×C₅₃	1	2	yes	—
289	C₂×C₄	#127×C₅₃	1	2	yes	—
290	C₂×C₄	C₂×D₈×C₅₃	1	8	yes	—
291	C₂×C₄	C₂×SD₁₆×C₅₃	1	8	yes	—
292	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
294	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
295	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
296	C₈	—	2	8	no	—
297	C₂²	—	4	8	no	—
298	C₂²	—	4	16	no	—
299	C₂²	—	8	32	no	—
300	C₂²	—	8	8	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	8	4	no	—
303	C₂²	C₅₃×#390	1	4	yes	—
304	C₂²	C₂×#102×C₅₃	1	8	yes	—
305	C₂²	C₅₃×#486	1	8	yes	—
306	C₂²	C₅₃×#492	1	4	yes	—
307	C₂²	C₅₃×#493	1	4	yes	—
308	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
309	C₂²	C₂×#124×C₅₃	1	1	yes	characteristic
310	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
311	C₂²	C₂²×D₈×C₅₃	1	2	yes	—
312	C₂²	C₂²×SD₁₆×C₅₃	1	2	yes	—
313	C₄	—	8	4	no	—
314	C₄	C₅₃×#390	1	4	yes	—
315	C₄	C₂×#102×C₅₃	1	16	yes	complemented
316	C₄	C₂×C₄×D₄×C₅₃	1	2	yes	—
317	C₄	C₂×#121×C₅₃	1	2	yes	—
318	C₂	—	8	8	no	—
319	C₂	C₅₃×#1957	1	8	yes	direct product factor
320	C₂	C₂×C₅₃×#390	1	1	yes	characteristic
321	C₂	C₂²×#102×C₅₃	1	2	yes	—
322	C₂	C₂×C₅₃×#486	1	2	yes	—
323	C₂	C₂×C₅₃×#492	1	1	yes	characteristic
324	C₂	C₂×C₅₃×#493	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂²×#127×C₅₃	C₂
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
2. C₂²×#102×C₅₃	C₂	semidirect
3. C₂²×C₄×C₈×C₅₃	C₂	semidirect
4. C₂×C₅₃×#384	C₂	semidirect
5. C₅₃×#1957	C₂	direct (N × H)
11. C₂×#127×C₅₃	C₄	semidirect
17. C₂×#102×C₅₃	C₂²	semidirect
19. C₂×C₄×C₈×C₅₃	C₂²	semidirect
21. C₅₃×#384	C₂²	semidirect
38. C₂²×C₈×C₅₃	D₄	semidirect
42. #127×C₅₃	C₂×C₄	semidirect
90. C₂×C₈×C₅₃	C₂×D₄	semidirect
120. C₂×C₄×C₅₃	#102	semidirect
134. C₂×#1957	C₅₃	direct (N × H)
153. C₄×C₅₃	C₂×#102	semidirect
157. C₂²×#102	C₂×C₅₃	semidirect
158. C₂²×C₄×C₈	C₂×C₅₃	semidirect
159. C₂×#384	C₂×C₅₃	semidirect
160. #1957	C₂×C₅₃	direct (N × H)
162. C₂×C₅₃	#1957	direct (N × H)
173. C₂×#127	C₄×C₅₃	semidirect
179. C₂×#102	C₂²×C₅₃	semidirect
181. C₂×C₄×C₈	C₂²×C₅₃	semidirect
183. #384	C₂²×C₅₃	semidirect
184. C₅₃	C₂×#1957	direct (N × H)
201. C₂²×C₈	D₄×C₅₃	semidirect
205. #127	C₂×C₄×C₅₃	semidirect
253. C₂×C₈	C₂×D₄×C₅₃	semidirect
283. C₂×C₄	#102×C₅₃	semidirect
315. C₄	C₂×#102×C₅₃	semidirect
319. C₂	C₅₃×#1957	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×#127×C₅₃	2
C₂²×#102×C₅₃	2
C₂²×C₄×C₈×C₅₃	2
C₂×C₅₃×#384	2
C₅₃×#1957	2
C₂×#1957	53

Subgroups — C2×C53×#1957 #27228705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1957 #27228705