Subgroups — C₂×C₅₃×#1961 #27228710

Order 13568.

2286 conjugacy classes of subgroups.

The subgroup lattice has 298 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×#97×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1961	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	C₂²	1	2	yes	—
14	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#102×C₅₃	—	2	16	no	—
16	C₂×#102×C₅₃	C₂²	1	16	yes	complemented
17	C₂×#97×C₅₃	—	2	4	no	—
18	C₂×#97×C₅₃	C₂²	1	4	yes	complemented
19	C₅₃×#382	—	2	2	no	—
20	C₅₃×#382	C₂²	1	4	yes	complemented
21	C₂×#95×C₅₃	—	2	2	no	—
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	16	no	—
30	C₂²×D₄×C₅₃	C₂×C₄	1	8	yes	—
31	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
32	C₂³×C₄×C₅₃	—	2	2	no	—
33	C₂³×C₄×C₅₃	D₄	1	2	yes	—
34	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	—	2	4	no	—
36	C₂²×C₈×C₅₃	—	2	4	no	—
37	C₂²×C₈×C₅₃	D₄	1	8	yes	complemented
38	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
39	#127×C₅₃	—	2	16	no	—
40	#127×C₅₃	—	2	8	no	—
41	#127×C₅₃	C₂×C₄	1	16	yes	complemented
42	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
44	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₅₃	D₄	1	4	yes	—
47	C₂×C₄²×C₅₃	D₄	1	2	yes	—
48	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
49	#102×C₅₃	—	2	32	no	—
50	#97×C₅₃	—	2	8	no	—
51	#95×C₅₃	—	2	8	no	—
52	C₂⁴×C₅₃	—	2	8	no	—
53	C₂⁴×C₅₃	—	2	4	no	—
54	C₂⁴×C₅₃	—	4	4	no	—
55	C₂⁴×C₅₃	—	4	4	no	—
56	C₂⁴×C₅₃	—	4	16	no	—
57	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×D₄×C₅₃	—	2	8	no	—
59	C₂×D₄×C₅₃	—	2	32	no	—
60	C₂×D₄×C₅₃	—	2	32	no	—
61	C₂×D₄×C₅₃	—	2	8	no	—
62	C₂×D₄×C₅₃	—	2	4	no	—
63	C₂×D₄×C₅₃	—	2	8	no	—
64	C₂×D₄×C₅₃	—	4	32	no	—
65	C₂×D₄×C₅₃	—	4	64	no	—
66	C₂×D₄×C₅₃	—	4	16	no	—
67	C₂×D₄×C₅₃	—	4	4	no	—
68	C₂×D₄×C₅₃	—	4	4	no	—
69	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
70	C₂²×C₄×C₅₃	—	2	8	no	—
71	C₂²×C₄×C₅₃	—	2	4	no	—
72	C₂²×C₄×C₅₃	—	4	4	no	—
73	C₂²×C₄×C₅₃	—	4	8	no	—
74	C₂²×C₄×C₅₃	C₂²⋊C₄	1	8	yes	—
75	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₅₃	C₂×D₄	1	4	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×D₄	1	4	yes	—
80	C₂²×C₄×C₅₃	#41	1	4	yes	—
81	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₅₃	—	2	16	no	—
83	C₂×C₈×C₅₃	—	2	4	no	—
84	C₂×C₈×C₅₃	—	2	4	no	—
85	C₂×C₈×C₅₃	C₂×D₄	1	8	yes	complemented
86	C₄⋊C₄×C₅₃	—	2	8	no	—
87	C₄⋊C₄×C₅₃	—	4	4	no	—
88	C₄²×C₅₃	C₂²⋊C₄	1	8	yes	—
89	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
90	C₄²×C₅₃	C₂×D₄	1	4	yes	—
91	C₂³×C₅₃	—	2	8	no	—
92	C₂³×C₅₃	—	4	4	no	—
93	C₂³×C₅₃	—	4	4	no	—
94	C₂³×C₅₃	—	4	16	no	—
95	C₂³×C₅₃	—	4	16	no	—
96	C₂³×C₅₃	—	4	32	no	—
97	C₂³×C₅₃	—	8	16	no	—
98	C₂³×C₅₃	—	8	16	no	—
99	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂³×C₅₃	#124	1	2	yes	—
101	C₂³×C₅₃	#127	1	2	yes	—
102	C₂³×C₅₃	AGL(1,ℤ/8)	1	8	yes	—
103	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
104	D₄×C₅₃	—	2	16	no	—
105	D₄×C₅₃	—	4	64	no	—
106	D₄×C₅₃	—	4	8	no	—
107	D₄×C₅₃	—	4	16	no	—
108	C₂×C₄×C₅₃	—	2	8	no	—
109	C₂×C₄×C₅₃	—	2	16	no	—
110	C₂×C₄×C₅₃	—	4	4	no	—
111	C₂×C₄×C₅₃	—	4	8	no	—
112	C₂×C₄×C₅₃	—	8	8	no	—
113	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	8	yes	—
114	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
115	C₂×C₄×C₅₃	C₄×D₄	1	8	yes	—
116	C₂×C₄×C₅₃	#121	1	8	yes	—
117	C₂×C₄×C₅₃	#124	1	2	yes	—
118	C₂×C₄×C₅₃	#127	1	2	yes	—
119	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
121	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
123	C₈×C₅₃	—	2	8	no	—
124	C₂×#1961	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×C₅₃	—	4	16	no	—
126	C₂²×C₅₃	—	4	8	no	—
127	C₂²×C₅₃	—	8	8	no	—
128	C₂²×C₅₃	—	8	8	no	—
129	C₂²×C₅₃	—	8	32	no	—
130	C₂²×C₅₃	#390	1	4	yes	—
131	C₂²×C₅₃	#418	1	8	yes	—
132	C₂²×C₅₃	#490	1	8	yes	—
133	C₂²×C₅₃	#496	1	8	yes	—
134	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₅₃	C₂×#124	1	1	yes	characteristic
136	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
137	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
138	C₄×C₅₃	—	2	8	no	—
139	C₄×C₅₃	—	8	4	no	—
140	C₄×C₅₃	#390	1	4	yes	—
141	C₄×C₅₃	C₂×C₄×D₄	1	2	yes	—
142	C₄×C₅₃	C₂×#121	1	2	yes	—
143	C₂²×#127	C₂×C₅₃	1	1	yes	characteristic
144	C₂²×#102	C₂×C₅₃	1	4	yes	complemented
145	C₂²×#97	C₂×C₅₃	1	1	yes	characteristic, complemented
146	C₂×#382	C₂×C₅₃	1	1	yes	characteristic, complemented
147	#1961	C₂×C₅₃	1	8	yes	direct product factor
148	C₂×C₅₃	—	8	8	no	—
149	C₂×C₅₃	#1961	1	8	yes	direct product factor
150	C₂×C₅₃	C₂×#390	1	1	yes	characteristic
151	C₂×C₅₃	C₂×#418	1	2	yes	—
152	C₂×C₅₃	C₂×#490	1	2	yes	—
153	C₂×C₅₃	C₂×#496	1	2	yes	—
154	C₂³×D₄	—	2	2	no	—
155	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
156	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
157	C₂×#127	—	2	4	no	—
158	C₂×#127	—	2	2	no	—
159	C₂×#127	C₄×C₅₃	1	8	yes	complemented
160	C₂×#127	C₂²×C₅₃	1	4	yes	—
161	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
162	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
163	C₂×#102	—	2	16	no	—
164	C₂×#102	C₂²×C₅₃	1	16	yes	complemented
165	C₂×#97	—	2	4	no	—
166	C₂×#97	C₂²×C₅₃	1	4	yes	complemented
167	#382	—	2	2	no	—
168	#382	C₂²×C₅₃	1	4	yes	complemented
169	C₂×#95	—	2	2	no	—
170	C₅₃	C₂×#1961	1	1	yes	characteristic, direct product factor, Sylow
171	C₂⁵	—	2	2	no	—
172	C₂²×D₄	—	2	8	no	—
173	C₂²×D₄	—	2	2	no	—
174	C₂²×D₄	—	2	4	no	—
175	C₂²×D₄	—	2	8	no	—
176	C₂²×D₄	—	2	8	no	—
177	C₂²×D₄	—	2	16	no	—
178	C₂²×D₄	—	4	16	no	—
179	C₂²×D₄	C₂×C₄×C₅₃	1	8	yes	—
180	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
181	C₂³×C₄	—	2	2	no	—
182	C₂³×C₄	D₄×C₅₃	1	2	yes	—
183	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
184	C₂²×C₈	—	2	4	no	—
185	C₂²×C₈	—	2	4	no	—
186	C₂²×C₈	D₄×C₅₃	1	8	yes	complemented
187	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
188	#127	—	2	8	no	—
189	#127	—	2	16	no	—
190	#127	C₂×C₄×C₅₃	1	16	yes	complemented
191	C₂×C₄⋊C₄	—	2	8	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
194	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
195	C₂×C₄²	D₄×C₅₃	1	4	yes	—
196	C₂×C₄²	D₄×C₅₃	1	2	yes	—
197	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
198	#102	—	2	32	no	—
199	#97	—	2	8	no	—
200	#95	—	2	8	no	—
201	C₂⁴	—	2	8	no	—
202	C₂⁴	—	2	4	no	—
203	C₂⁴	—	4	4	no	—
204	C₂⁴	—	4	4	no	—
205	C₂⁴	—	4	16	no	—
206	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
207	C₂×D₄	—	2	8	no	—
208	C₂×D₄	—	2	32	no	—
209	C₂×D₄	—	2	32	no	—
210	C₂×D₄	—	2	8	no	—
211	C₂×D₄	—	2	4	no	—
212	C₂×D₄	—	2	8	no	—
213	C₂×D₄	—	4	32	no	—
214	C₂×D₄	—	4	64	no	—
215	C₂×D₄	—	4	16	no	—
216	C₂×D₄	—	4	4	no	—
217	C₂×D₄	—	4	4	no	—
218	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	4	8	no	—
223	C₂²×C₄	C₂²⋊C₄×C₅₃	1	8	yes	—
224	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
226	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
228	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
229	C₂²×C₄	#41×C₅₃	1	4	yes	—
230	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
231	C₂×C₈	—	2	16	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	C₂×D₄×C₅₃	1	8	yes	complemented
235	C₄⋊C₄	—	2	8	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄²	C₂²⋊C₄×C₅₃	1	8	yes	—
238	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
239	C₄²	C₂×D₄×C₅₃	1	4	yes	—
240	C₂³	—	2	8	no	—
241	C₂³	—	4	4	no	—
242	C₂³	—	4	4	no	—
243	C₂³	—	4	16	no	—
244	C₂³	—	4	16	no	—
245	C₂³	—	4	32	no	—
246	C₂³	—	8	16	no	—
247	C₂³	—	8	16	no	—
248	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
249	C₂³	#124×C₅₃	1	2	yes	—
250	C₂³	#127×C₅₃	1	2	yes	—
251	C₂³	AGL(1,ℤ/8)×C₅₃	1	8	yes	—
252	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
253	D₄	—	2	16	no	—
254	D₄	—	4	64	no	—
255	D₄	—	4	8	no	—
256	D₄	—	4	16	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	16	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	8	no	—
261	C₂×C₄	—	8	8	no	—
262	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	8	yes	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
264	C₂×C₄	C₄×D₄×C₅₃	1	8	yes	—
265	C₂×C₄	#121×C₅₃	1	8	yes	—
266	C₂×C₄	#124×C₅₃	1	2	yes	—
267	C₂×C₄	#127×C₅₃	1	2	yes	—
268	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
269	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
270	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
271	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
272	C₈	—	2	8	no	—
273	C₂²	—	4	8	no	—
274	C₂²	—	4	16	no	—
275	C₂²	—	8	32	no	—
276	C₂²	—	8	8	no	—
277	C₂²	—	8	8	no	—
278	C₂²	C₅₃×#390	1	4	yes	—
279	C₂²	C₅₃×#418	1	8	yes	—
280	C₂²	C₅₃×#490	1	8	yes	—
281	C₂²	C₅₃×#496	1	8	yes	—
282	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
283	C₂²	C₂×#124×C₅₃	1	1	yes	characteristic
284	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
285	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	4	yes	—
286	C₄	—	2	8	no	—
287	C₄	—	8	4	no	—
288	C₄	C₅₃×#390	1	4	yes	—
289	C₄	C₂×C₄×D₄×C₅₃	1	2	yes	—
290	C₄	C₂×#121×C₅₃	1	2	yes	—
291	C₂	—	8	8	no	—
292	C₂	C₅₃×#1961	1	8	yes	direct product factor
293	C₂	C₂×C₅₃×#390	1	1	yes	characteristic
294	C₂	C₂×C₅₃×#418	1	2	yes	—
295	C₂	C₂×C₅₃×#490	1	2	yes	—
296	C₂	C₂×C₅₃×#496	1	2	yes	—

	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×#97×C₅₃	C₂	semidirect
4. C₂×C₅₃×#382	C₂	semidirect
5. C₅₃×#1961	C₂	direct (N × H)
11. C₂×#127×C₅₃	C₄	semidirect
16. C₂×#102×C₅₃	C₂²	semidirect
18. C₂×#97×C₅₃	C₂²	semidirect
20. C₅₃×#382	C₂²	semidirect
37. C₂²×C₈×C₅₃	D₄	semidirect
41. #127×C₅₃	C₂×C₄	semidirect
85. C₂×C₈×C₅₃	C₂×D₄	semidirect
124. C₂×#1961	C₅₃	direct (N × H)
144. C₂²×#102	C₂×C₅₃	semidirect
145. C₂²×#97	C₂×C₅₃	semidirect
146. C₂×#382	C₂×C₅₃	semidirect
147. #1961	C₂×C₅₃	direct (N × H)
149. C₂×C₅₃	#1961	direct (N × H)
159. C₂×#127	C₄×C₅₃	semidirect
164. C₂×#102	C₂²×C₅₃	semidirect
166. C₂×#97	C₂²×C₅₃	semidirect
168. #382	C₂²×C₅₃	semidirect
170. C₅₃	C₂×#1961	direct (N × H)
186. C₂²×C₈	D₄×C₅₃	semidirect
190. #127	C₂×C₄×C₅₃	semidirect
234. C₂×C₈	C₂×D₄×C₅₃	semidirect
292. C₂	C₅₃×#1961	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×#97×C₅₃	2
C₂×C₅₃×#382	2
C₅₃×#1961	2
C₂×#1961	53

Subgroups — C2×C53×#1961 #27228710

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1961 #27228710