Subgroups — C₂×C₅₃×#1384 #27220076

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#427	C₂	1	2	yes	maximal
3	C₅₃×#1384	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
5	C₂×#117×C₅₃	—	2	2	no	—
6	C₂×M₃₂×C₅₃	—	2	2	no	—
7	C₂×M₃₂×C₅₃	—	2	2	no	—
8	C₂×M₃₂×C₅₃	C₄	1	2	yes	complemented
9	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
10	C₂²×C₁₆×C₅₃	C₄	1	2	yes	complemented
11	C₅₃×#427	—	2	2	no	—
12	C₅₃×#427	—	2	2	no	—
13	C₅₃×#427	C₄	1	4	yes	complemented
14	C₅₃×#427	C₂²	1	4	yes	—
15	C₂×#107×C₅₃	—	2	2	no	—
16	C₂×#106×C₅₃	C₄	1	4	yes	complemented
17	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	D₄	1	2	yes	—
23	C₂²×C₈×C₅₃	D₄	1	2	yes	—
24	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	#117×C₅₃	—	2	4	no	—
29	#117×C₅₃	—	2	4	no	—
30	#117×C₅₃	—	4	2	no	—
31	#117×C₅₃	—	4	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
33	C₂×C₂²⋊C₄×C₅₃	—	4	2	no	—
34	C₂×C₄²×C₅₃	—	4	2	no	—
35	M₃₂×C₅₃	—	2	4	no	—
36	M₃₂×C₅₃	—	2	4	no	—
37	C₂×C₁₆×C₅₃	—	2	2	no	—
38	C₂×C₁₆×C₅₃	—	2	2	no	—
39	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
40	#107×C₅₃	—	2	4	no	—
41	#107×C₅₃	—	2	4	no	—
42	#106×C₅₃	—	2	4	no	—
43	#106×C₅₃	C₂×C₄	1	8	yes	complemented
44	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	—	4	4	no	—
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
52	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
53	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₄²	1	2	yes	—
58	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
74	C₄⋊C₄×C₅₃	—	2	8	no	—
75	C₄⋊C₄×C₅₃	—	4	4	no	—
76	C₂²⋊C₄×C₅₃	—	4	4	no	—
77	C₂²⋊C₄×C₅₃	—	4	4	no	—
78	C₄²×C₅₃	—	4	4	no	—
79	C₄²×C₅₃	—	4	4	no	—
80	C₁₆×C₅₃	—	2	4	no	—
81	C₂³×C₅₃	—	4	2	no	—
82	C₂³×C₅₃	#102	1	2	yes	—
83	C₂³×C₅₃	#102	1	2	yes	—
84	C₂³×C₅₃	#107	1	1	yes	characteristic
85	C₂³×C₅₃	#107	1	1	yes	characteristic
86	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	—	4	8	no	—
88	C₂×C₄×C₅₃	—	8	8	no	—
89	C₂×C₄×C₅₃	#95	1	2	yes	—
90	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#102	1	2	yes	—
93	C₂×C₄×C₅₃	#102	1	2	yes	—
94	C₂×C₄×C₅₃	#103	1	2	yes	—
95	C₂×C₄×C₅₃	#103	1	2	yes	—
96	C₂×C₄×C₅₃	#103	1	2	yes	—
97	C₂×C₄×C₅₃	#103	1	2	yes	—
98	C₂×C₄×C₅₃	#106	1	2	yes	—
99	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
107	C₈×C₅₃	#95	1	1	yes	characteristic
108	C₈×C₅₃	#95	1	2	yes	—
109	C₈×C₅₃	#95	1	1	yes	characteristic
110	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
111	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×#1384	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₅₃	—	2	4	no	—
115	C₂²×C₅₃	—	2	2	no	—
116	C₂²×C₅₃	—	4	2	no	—
117	C₂²×C₅₃	—	4	1	no	—
118	C₂²×C₅₃	—	4	1	no	—
119	C₂²×C₅₃	#340	1	2	yes	—
120	C₂²×C₅₃	#340	1	1	yes	characteristic
121	C₂²×C₅₃	#340	1	1	yes	characteristic
122	C₂²×C₅₃	C₂×#102	1	2	yes	—
123	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
124	C₄×C₅₃	—	8	8	no	—
125	C₄×C₅₃	#340	1	2	yes	—
126	C₄×C₅₃	#340	1	1	yes	characteristic
127	C₄×C₅₃	#340	1	1	yes	characteristic
128	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
129	C₄×C₅₃	C₂×#103	1	2	yes	—
130	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
131	C₂²×M₃₂	C₂×C₅₃	1	1	yes	characteristic
132	C₂×#427	C₂×C₅₃	1	2	yes	—
133	#1384	C₂×C₅₃	1	4	yes	direct product factor
134	C₂×C₅₃	—	2	2	no	—
135	C₂×C₅₃	—	2	2	no	—
136	C₂×C₅₃	—	2	1	no	—
137	C₂×C₅₃	—	2	1	no	—
138	C₂×C₅₃	#1384	1	2	yes	direct product factor
139	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
140	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
141	C₂×#117	—	2	2	no	—
142	C₂×M₃₂	—	2	2	no	—
143	C₂×M₃₂	—	2	2	no	—
144	C₂×M₃₂	C₄×C₅₃	1	2	yes	complemented
145	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
146	C₂²×C₁₆	C₄×C₅₃	1	2	yes	complemented
147	#427	—	2	2	no	—
148	#427	—	2	2	no	—
149	#427	C₄×C₅₃	1	4	yes	complemented
150	#427	C₂²×C₅₃	1	4	yes	—
151	C₂×#107	—	2	2	no	—
152	C₂×#106	C₄×C₅₃	1	4	yes	complemented
153	C₅₃	C₂×#1384	1	1	yes	characteristic, direct product factor, Sylow
154	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
155	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
156	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
157	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
158	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
159	C₂²×C₈	D₄×C₅₃	1	2	yes	—
160	C₂²×C₈	D₄×C₅₃	1	2	yes	—
161	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
162	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
163	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
165	#117	—	2	4	no	—
166	#117	—	2	4	no	—
167	#117	—	4	2	no	—
168	#117	—	4	2	no	—
169	C₂×C₄⋊C₄	—	2	4	no	—
170	C₂×C₂²⋊C₄	—	4	2	no	—
171	C₂×C₄²	—	4	2	no	—
172	M₃₂	—	2	4	no	—
173	M₃₂	—	2	4	no	—
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	complemented
177	#107	—	2	4	no	—
178	#107	—	2	4	no	—
179	#106	—	2	4	no	—
180	#106	C₂×C₄×C₅₃	1	8	yes	complemented
181	C₂⁴	D₈×C₅₃	1	1	yes	characteristic
182	C₂²×C₄	—	4	4	no	—
183	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
184	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
185	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
186	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
187	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
188	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
189	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
190	C₂²×C₄	Q₁₆×C₅₃	1	1	yes	characteristic
191	C₂²×C₄	Q₁₆×C₅₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
193	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
194	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
195	C₂×C₈	C₄²×C₅₃	1	2	yes	—
196	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
197	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
198	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
199	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
201	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
202	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
203	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
204	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
205	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
206	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
207	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
208	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
209	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
210	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
211	C₄⋊C₄	—	2	8	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₂²⋊C₄	—	4	4	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₄²	—	4	4	no	—
216	C₄²	—	4	4	no	—
217	C₁₆	—	2	4	no	—
218	C₂³	—	4	2	no	—
219	C₂³	#102×C₅₃	1	2	yes	—
220	C₂³	#102×C₅₃	1	2	yes	—
221	C₂³	#107×C₅₃	1	1	yes	characteristic
222	C₂³	#107×C₅₃	1	1	yes	characteristic
223	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	—	8	8	no	—
226	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
227	C₂×C₄	#95×C₅₃	1	2	yes	—
228	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
229	C₂×C₄	#102×C₅₃	1	2	yes	—
230	C₂×C₄	#102×C₅₃	1	2	yes	—
231	C₂×C₄	#103×C₅₃	1	2	yes	—
232	C₂×C₄	#103×C₅₃	1	2	yes	—
233	C₂×C₄	#103×C₅₃	1	2	yes	—
234	C₂×C₄	#103×C₅₃	1	2	yes	—
235	C₂×C₄	#106×C₅₃	1	2	yes	—
236	C₂×C₄	#106×C₅₃	1	1	yes	characteristic
237	C₂×C₄	#106×C₅₃	1	1	yes	characteristic
238	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
239	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
240	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
241	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
242	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
243	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
244	C₈	#95×C₅₃	1	1	yes	characteristic
245	C₈	#95×C₅₃	1	2	yes	—
246	C₈	#95×C₅₃	1	1	yes	characteristic
247	C₈	C₂×C₄²×C₅₃	1	1	yes	characteristic
248	C₈	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
249	C₈	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	1	no	—
254	C₂²	—	4	1	no	—
255	C₂²	C₅₃×#340	1	2	yes	—
256	C₂²	C₅₃×#340	1	1	yes	characteristic
257	C₂²	C₅₃×#340	1	1	yes	characteristic
258	C₂²	C₂×#102×C₅₃	1	2	yes	—
259	C₂²	C₂×#107×C₅₃	1	1	yes	characteristic
260	C₄	—	8	8	no	—
261	C₄	C₅₃×#340	1	1	yes	characteristic
262	C₄	C₅₃×#340	1	2	yes	—
263	C₄	C₅₃×#340	1	1	yes	characteristic
264	C₄	C₂×#95×C₅₃	1	1	yes	characteristic
265	C₄	C₂×#103×C₅₃	1	2	yes	—
266	C₄	C₂×#106×C₅₃	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	2	2	no	—
269	C₂	—	2	1	no	—
270	C₂	—	2	1	no	—
271	C₂	C₅₃×#1384	1	2	yes	direct product factor
272	C₂	C₂×C₅₃×#340	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#106
commutator	C₈	C₂×C₄²×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	#340
prime radical	C₂⁴×C₅₃	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1384	C₂	direct (N × H)
8. C₂×M₃₂×C₅₃	C₄	semidirect
10. C₂²×C₁₆×C₅₃	C₄	semidirect
13. C₅₃×#427	C₄	semidirect
16. C₂×#106×C₅₃	C₄	semidirect
39. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
43. #106×C₅₃	C₂×C₄	semidirect
113. C₂×#1384	C₅₃	direct (N × H)
133. #1384	C₂×C₅₃	direct (N × H)
138. C₂×C₅₃	#1384	direct (N × H)
144. C₂×M₃₂	C₄×C₅₃	semidirect
146. C₂²×C₁₆	C₄×C₅₃	semidirect
149. #427	C₄×C₅₃	semidirect
152. C₂×#106	C₄×C₅₃	semidirect
153. C₅₃	C₂×#1384	direct (N × H)
176. C₂×C₁₆	C₂×C₄×C₅₃	semidirect
180. #106	C₂×C₄×C₅₃	semidirect
271. C₂	C₅₃×#1384	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×M₃₂×C₅₃	2
C₂×C₅₃×#427	2
C₅₃×#1384	2
C₂×#1384	53

Subgroups — C2×C53×#1384 #27220076

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1384 #27220076