Subgroups — C₂×C₅₃×#2140 #27240889

Order 13568.

1406 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2140	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅₃×#357	C₂	1	4	yes	maximal, complemented
6	C₂³×D₄×C₅₃	—	2	1	no	—
7	C₂²×D₈×C₅₃	—	2	4	no	—
8	C₂²×D₈×C₅₃	—	2	1	no	—
9	C₂²×D₈×C₅₃	—	2	2	no	—
10	C₂²×D₈×C₅₃	C₄	1	8	yes	complemented
11	C₂²×D₈×C₅₃	C₂²	1	2	yes	—
12	C₂²×D₈×C₅₃	C₂²	1	4	yes	—
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#117×C₅₃	—	2	1	no	—
15	C₂×M₃₂×C₅₃	—	2	4	no	—
16	C₂×M₃₂×C₅₃	C₂²	1	4	yes	complemented
17	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	complemented
18	C₅₃×#427	—	2	2	no	—
19	C₅₃×#427	C₂²	1	4	yes	complemented
20	C₂×#107×C₅₃	C₂²	1	2	yes	complemented
21	C₂×#106×C₅₃	—	2	1	no	—
22	C₅₃×#357	C₂²	1	16	yes	complemented
23	C₂⁵×C₅₃	—	4	1	no	—
24	C₂²×D₄×C₅₃	—	2	8	no	—
25	C₂²×D₄×C₅₃	—	2	4	no	—
26	C₂²×D₄×C₅₃	—	2	2	no	—
27	C₂²×D₄×C₅₃	—	4	2	no	—
28	C₂²×D₄×C₅₃	—	4	1	no	—
29	C₂²×D₄×C₅₃	—	4	4	no	—
30	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
31	C₂×D₈×C₅₃	—	2	8	no	—
32	C₂×D₈×C₅₃	—	2	16	no	—
33	C₂×D₈×C₅₃	—	2	4	no	—
34	C₂×D₈×C₅₃	—	2	8	no	—
35	C₂×D₈×C₅₃	—	2	4	no	—
36	C₂×D₈×C₅₃	—	2	2	no	—
37	C₂×D₈×C₅₃	C₂×C₄	1	8	yes	—
38	C₂×D₈×C₅₃	C₂×C₄	1	16	yes	complemented
39	C₂×D₈×C₅₃	C₂³	1	4	yes	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₅₃	D₄	1	4	yes	—
42	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	D₄	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
46	#117×C₅₃	—	2	4	no	—
47	#117×C₅₃	—	4	2	no	—
48	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
50	C₂×C₄²×C₅₃	—	4	1	no	—
51	M₃₂×C₅₃	—	2	8	no	—
52	C₂×C₁₆×C₅₃	—	2	4	no	—
53	C₂×C₁₆×C₅₃	C₂³	1	4	yes	complemented
54	#107×C₅₃	—	2	2	no	—
55	#107×C₅₃	C₂³	1	4	yes	complemented
56	#106×C₅₃	—	2	4	no	—
57	C₂⁴×C₅₃	—	4	8	no	—
58	C₂⁴×C₅₃	—	4	4	no	—
59	C₂⁴×C₅₃	—	4	2	no	—
60	C₂⁴×C₅₃	—	8	8	no	—
61	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
62	C₂×D₄×C₅₃	—	2	8	no	—
63	C₂×D₄×C₅₃	—	2	16	no	—
64	C₂×D₄×C₅₃	—	2	4	no	—
65	C₂×D₄×C₅₃	—	4	16	no	—
66	C₂×D₄×C₅₃	—	4	8	no	—
67	C₂×D₄×C₅₃	—	4	4	no	—
68	C₂×D₄×C₅₃	—	4	2	no	—
69	C₂×D₄×C₅₃	—	4	8	no	—
70	C₂×D₄×C₅₃	—	4	4	no	—
71	C₂²×C₄×C₅₃	—	4	2	no	—
72	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
73	C₂²×C₄×C₅₃	D₈	1	2	yes	—
74	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
76	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	D₈×C₅₃	—	2	8	no	—
79	D₈×C₅₃	—	2	16	no	—
80	D₈×C₅₃	—	2	4	no	—
81	D₈×C₅₃	C₂²×C₄	1	8	yes	—
82	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
83	C₂×C₈×C₅₃	C₂²⋊C₄	1	8	yes	—
84	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
90	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₅₃	—	2	4	no	—
92	C₄⋊C₄×C₅₃	—	4	2	no	—
93	C₂²⋊C₄×C₅₃	—	4	4	no	—
94	C₄²×C₅₃	—	4	4	no	—
95	C₁₆×C₅₃	—	2	4	no	—
96	C₂³×C₅₃	—	2	4	no	—
97	C₂³×C₅₃	—	4	4	no	—
98	C₂³×C₅₃	—	4	16	no	—
99	C₂³×C₅₃	—	4	8	no	—
100	C₂³×C₅₃	—	8	16	no	—
101	C₂³×C₅₃	—	8	32	no	—
102	C₂³×C₅₃	—	8	8	no	—
103	C₂³×C₅₃	#102	1	4	yes	—
104	C₂³×C₅₃	C₂×D₈	1	2	yes	—
105	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
106	D₄×C₅₃	—	2	8	no	—
107	D₄×C₅₃	—	4	4	no	—
108	D₄×C₅₃	—	4	8	no	—
109	D₄×C₅₃	—	4	16	no	—
110	C₂×C₄×C₅₃	—	4	4	no	—
111	C₂×C₄×C₅₃	—	8	4	no	—
112	C₂×C₄×C₅₃	#102	1	8	yes	—
113	C₂×C₄×C₅₃	#102	1	4	yes	—
114	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
115	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
116	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
119	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
120	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
121	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
122	C₈×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
123	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
124	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×#2140	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₅₃	—	2	4	no	—
127	C₂²×C₅₃	—	2	8	no	—
128	C₂²×C₅₃	—	4	8	no	—
129	C₂²×C₅₃	—	8	32	no	—
130	C₂²×C₅₃	—	8	8	no	—
131	C₂²×C₅₃	—	8	16	no	—
132	C₂²×C₅₃	C₂×#102	1	4	yes	—
133	C₂²×C₅₃	C₂×#102	1	2	yes	—
134	C₂²×C₅₃	#509	1	4	yes	—
135	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
136	C₄×C₅₃	—	8	4	no	—
137	C₄×C₅₃	C₂×#102	1	4	yes	—
138	C₄×C₅₃	C₂×#102	1	2	yes	—
139	C₄×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
140	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
141	C₂³×D₈	C₂×C₅₃	1	1	yes	characteristic
142	C₂²×M₃₂	C₂×C₅₃	1	1	yes	characteristic, complemented
143	C₂×#427	C₂×C₅₃	1	1	yes	characteristic, complemented
144	#2140	C₂×C₅₃	1	8	yes	direct product factor
145	C₂×#357	C₂×C₅₃	1	4	yes	complemented
146	C₂×C₅₃	—	2	4	no	—
147	C₂×C₅₃	—	8	8	no	—
148	C₂×C₅₃	#2140	1	4	yes	direct product factor
149	C₂×C₅₃	C₂²×#102	1	1	yes	characteristic
150	C₂×C₅₃	C₂×#509	1	2	yes	—
151	C₂³×D₄	—	2	1	no	—
152	C₂²×D₈	—	2	4	no	—
153	C₂²×D₈	—	2	1	no	—
154	C₂²×D₈	—	2	2	no	—
155	C₂²×D₈	C₄×C₅₃	1	8	yes	complemented
156	C₂²×D₈	C₂²×C₅₃	1	2	yes	—
157	C₂²×D₈	C₂²×C₅₃	1	4	yes	—
158	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
159	C₂×#117	—	2	1	no	—
160	C₂×M₃₂	—	2	4	no	—
161	C₂×M₃₂	C₂²×C₅₃	1	4	yes	complemented
162	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	complemented
163	#427	—	2	2	no	—
164	#427	C₂²×C₅₃	1	4	yes	complemented
165	C₂×#107	C₂²×C₅₃	1	2	yes	complemented
166	C₂×#106	—	2	1	no	—
167	#357	C₂²×C₅₃	1	16	yes	complemented
168	C₅₃	C₂×#2140	1	1	yes	characteristic, direct product factor, Sylow
169	C₂⁵	—	4	1	no	—
170	C₂²×D₄	—	2	2	no	—
171	C₂²×D₄	—	2	4	no	—
172	C₂²×D₄	—	2	8	no	—
173	C₂²×D₄	—	4	1	no	—
174	C₂²×D₄	—	4	2	no	—
175	C₂²×D₄	—	4	4	no	—
176	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
177	C₂×D₈	—	2	4	no	—
178	C₂×D₈	—	2	2	no	—
179	C₂×D₈	—	2	4	no	—
180	C₂×D₈	—	2	8	no	—
181	C₂×D₈	—	2	16	no	—
182	C₂×D₈	—	2	8	no	—
183	C₂×D₈	C₂×C₄×C₅₃	1	8	yes	—
184	C₂×D₈	C₂×C₄×C₅₃	1	16	yes	complemented
185	C₂×D₈	C₂³×C₅₃	1	4	yes	—
186	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
187	C₂²×C₈	D₄×C₅₃	1	4	yes	—
188	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
189	C₂²×C₈	D₄×C₅₃	1	2	yes	—
190	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
191	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
192	#117	—	2	4	no	—
193	#117	—	4	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₂²⋊C₄	—	4	1	no	—
196	C₂×C₄²	—	4	1	no	—
197	M₃₂	—	2	8	no	—
198	C₂×C₁₆	—	2	4	no	—
199	C₂×C₁₆	C₂³×C₅₃	1	4	yes	complemented
200	#107	—	2	2	no	—
201	#107	C₂³×C₅₃	1	4	yes	complemented
202	#106	—	2	4	no	—
203	C₂⁴	—	4	8	no	—
204	C₂⁴	—	4	4	no	—
205	C₂⁴	—	4	2	no	—
206	C₂⁴	—	8	8	no	—
207	C₂⁴	D₈×C₅₃	1	1	yes	characteristic
208	C₂×D₄	—	2	8	no	—
209	C₂×D₄	—	2	16	no	—
210	C₂×D₄	—	2	4	no	—
211	C₂×D₄	—	4	16	no	—
212	C₂×D₄	—	4	8	no	—
213	C₂×D₄	—	4	4	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	8	no	—
216	C₂×D₄	—	4	4	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
219	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
220	C₂²×C₄	D₈×C₅₃	1	2	yes	—
221	C₂²×C₄	SD₁₆×C₅₃	1	4	yes	—
222	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
224	D₈	—	2	8	no	—
225	D₈	—	2	16	no	—
226	D₈	—	2	4	no	—
227	D₈	C₂²×C₄×C₅₃	1	8	yes	—
228	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
229	C₂×C₈	C₂²⋊C₄×C₅₃	1	8	yes	—
230	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
231	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
232	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
233	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
234	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
235	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
236	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	4	no	—
240	C₄²	—	4	4	no	—
241	C₁₆	—	2	4	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	4	4	no	—
244	C₂³	—	4	16	no	—
245	C₂³	—	4	8	no	—
246	C₂³	—	8	16	no	—
247	C₂³	—	8	32	no	—
248	C₂³	—	8	8	no	—
249	C₂³	#102×C₅₃	1	4	yes	—
250	C₂³	C₂×D₈×C₅₃	1	2	yes	—
251	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
252	D₄	—	2	8	no	—
253	D₄	—	4	4	no	—
254	D₄	—	4	8	no	—
255	D₄	—	4	16	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	#102×C₅₃	1	8	yes	—
259	C₂×C₄	#102×C₅₃	1	4	yes	—
260	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
261	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
262	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
263	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
264	C₂×C₄	C₂×SD₁₆×C₅₃	1	4	yes	—
265	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
266	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
267	C₈	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
268	C₈	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
269	C₈	C₂³×C₄×C₅₃	1	1	yes	characteristic
270	C₈	C₂²×D₄×C₅₃	1	1	yes	characteristic
271	C₂²	—	2	4	no	—
272	C₂²	—	2	8	no	—
273	C₂²	—	4	8	no	—
274	C₂²	—	8	32	no	—
275	C₂²	—	8	8	no	—
276	C₂²	—	8	16	no	—
277	C₂²	C₂×#102×C₅₃	1	4	yes	—
278	C₂²	C₂×#102×C₅₃	1	2	yes	—
279	C₂²	C₅₃×#509	1	4	yes	—
280	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
281	C₄	—	8	4	no	—
282	C₄	C₂×#102×C₅₃	1	4	yes	—
283	C₄	C₂×#102×C₅₃	1	2	yes	—
284	C₄	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
285	C₄	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
286	C₂	—	2	4	no	—
287	C₂	—	8	8	no	—
288	C₂	C₅₃×#2140	1	4	yes	direct product factor
289	C₂	C₂²×#102×C₅₃	1	1	yes	characteristic
290	C₂	C₂×C₅₃×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂×D₈
commutator	C₈	C₂³×C₄×C₅₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
prime radical	C₂³×D₈×C₅₃	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×M₃₂×C₅₃	C₂	semidirect
3. C₂×C₅₃×#427	C₂	semidirect
4. C₅₃×#2140	C₂	direct (N × H)
5. C₂×C₅₃×#357	C₂	semidirect
10. C₂²×D₈×C₅₃	C₄	semidirect
16. C₂×M₃₂×C₅₃	C₂²	semidirect
17. C₂²×C₁₆×C₅₃	C₂²	semidirect
19. C₅₃×#427	C₂²	semidirect
20. C₂×#107×C₅₃	C₂²	semidirect
22. C₅₃×#357	C₂²	semidirect
38. C₂×D₈×C₅₃	C₂×C₄	semidirect
53. C₂×C₁₆×C₅₃	C₂³	semidirect
55. #107×C₅₃	C₂³	semidirect
125. C₂×#2140	C₅₃	direct (N × H)
142. C₂²×M₃₂	C₂×C₅₃	semidirect
143. C₂×#427	C₂×C₅₃	semidirect
144. #2140	C₂×C₅₃	direct (N × H)
145. C₂×#357	C₂×C₅₃	semidirect
148. C₂×C₅₃	#2140	direct (N × H)
155. C₂²×D₈	C₄×C₅₃	semidirect
161. C₂×M₃₂	C₂²×C₅₃	semidirect
162. C₂²×C₁₆	C₂²×C₅₃	semidirect
164. #427	C₂²×C₅₃	semidirect
165. C₂×#107	C₂²×C₅₃	semidirect
167. #357	C₂²×C₅₃	semidirect
168. C₅₃	C₂×#2140	direct (N × H)
184. C₂×D₈	C₂×C₄×C₅₃	semidirect
199. C₂×C₁₆	C₂³×C₅₃	semidirect
201. #107	C₂³×C₅₃	semidirect
288. C₂	C₅₃×#2140	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×D₈×C₅₃	2
C₂²×M₃₂×C₅₃	2
C₂×C₅₃×#427	2
C₅₃×#2140	2
C₂×C₅₃×#357	2
C₂×#2140	53

Subgroups — C2×C53×#2140 #27240889

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2140 #27240889