Subgroups — D₅₃×#1845 #27384491

Order 13568.

1358 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#107×D₅₃	C₂	1	4	yes	maximal
2	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#384×D₅₃	C₂	1	2	yes	maximal
4	#20274875	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274874	C₂	1	2	yes	maximal, complemented
6	#20274873	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₅₃	C₂²	1	2	yes	—
9	C₂×C₄⋊C₄×D₅₃	—	2	4	no	—
10	C₂×C₄⋊C₄×D₅₃	C₂²	1	4	yes	—
11	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
12	C₂×#13133151	C₂²	1	4	yes	—
13	C₂×#13133150	C₂²	1	8	yes	—
14	#107×D₅₃	—	2	16	no	—
15	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
17	C₄×C₈×D₅₃	C₄	1	8	yes	complemented
18	#16810493	C₂²	1	2	yes	—
19	#16810492	—	2	4	no	—
20	#16810492	C₂²	1	4	yes	—
21	C₂×#107×C₅₃	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#384	C₂²	1	2	yes	—
24	C₂²×C₄×D₅₃	—	2	4	no	—
25	C₂²×C₄×D₅₃	D₄	1	2	yes	—
26	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
27	C₂²×#88322	C₂³	1	2	yes	—
28	C₂×C₈×D₅₃	C₂×C₄	1	8	yes	—
29	C₂×C₈×D₅₃	D₄	1	8	yes	—
30	C₂×C₈×D₅₃	Q₈	1	8	yes	—
31	C₂×C₄⋊Dic₅₃	—	2	4	no	—
32	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
33	C₂×#11719919	—	2	8	no	—
34	C₂×#11719919	—	2	4	no	—
35	C₄⋊C₄×D₅₃	—	2	16	no	—
36	C₄⋊C₄×D₅₃	—	4	16	no	—
37	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
38	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
39	C₄²×D₅₃	C₂×C₄	1	4	yes	—
40	C₄²×D₅₃	Q₈	1	4	yes	—
41	#13133151	—	2	8	no	—
42	#13133150	—	2	16	no	—
43	C₄×#88322	C₂×C₄	1	4	yes	complemented
44	C₈×Dic₅₃	C₂×C₄	1	8	yes	complemented
45	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
47	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
48	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	#107×C₅₃	—	2	8	no	—
50	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
51	C₂³×D₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	—	2	4	no	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
55	C₂²×Dic₅₃	—	2	4	no	—
56	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×D₅₃	—	4	8	no	—
59	C₂×C₄×D₅₃	—	4	16	no	—
60	C₂×C₄×D₅₃	C₄⋊C₄	1	8	yes	—
61	C₂×C₄×D₅₃	D₈	1	8	yes	—
62	C₂×C₄×D₅₃	Q₁₆	1	8	yes	—
63	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
64	C₂×C₄×D₅₃	C₂×D₄	1	4	yes	—
65	C₂×C₄×D₅₃	C₂×Q₈	1	4	yes	—
66	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
69	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
70	C₄⋊C₄×C₅₃	—	2	8	no	—
71	C₄⋊C₄×C₅₃	—	4	8	no	—
72	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
74	C₄⋊Dic₅₃	—	2	8	no	—
75	C₄⋊Dic₅₃	—	4	8	no	—
76	#11719919	—	4	8	no	—
77	#11719919	—	4	16	no	—
78	C₄×Dic₅₃	C₄⋊C₄	1	8	yes	—
79	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
80	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
81	C₂×#88322	C₂²×C₄	1	4	yes	—
82	C₂×#88322	C₂×D₄	1	4	yes	—
83	C₂×#88322	C₂×Q₈	1	4	yes	—
84	C₈×D₅₃	C₄⋊C₄	1	32	yes	complemented
85	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
86	C₂²×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
87	C₂²×D₅₃	#127	1	2	yes	—
88	C₂²×D₅₃	C₄⋊Q₈	1	2	yes	—
89	C₂²×D₅₃	C₂×D₈	1	4	yes	—
90	C₂²×D₅₃	C₂×Q₁₆	1	4	yes	—
91	C₂×C₄×C₅₃	—	4	4	no	—
92	C₂×C₄×C₅₃	—	4	8	no	—
93	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
94	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
95	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
96	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
98	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
99	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
100	C₂×Dic₅₃	—	4	4	no	—
101	C₂×Dic₅₃	—	4	8	no	—
102	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×Dic₅₃	#127	1	2	yes	—
105	C₂×Dic₅₃	C₄⋊Q₈	1	2	yes	—
106	C₂×Dic₅₃	C₂×D₈	1	4	yes	—
107	C₂×Dic₅₃	C₂×Q₁₆	1	4	yes	—
108	C₄×D₅₃	—	8	16	no	—
109	C₄×D₅₃	#107	1	32	yes	complemented
110	C₄×D₅₃	C₂×C₄⋊C₄	1	8	yes	—
111	C₄×D₅₃	C₄⋊Q₈	1	8	yes	—
112	C₈×C₅₃	C₂×C₄⋊C₄	1	8	yes	complemented
113	#88322	C₂×C₄⋊C₄	1	8	yes	complemented
114	C₂×#1845	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
117	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
118	C₂²×C₅₃	C₂²×D₈	1	2	yes	—
119	C₂²×C₅₃	C₂²×Q₁₆	1	2	yes	—
120	C₂×D₅₃	#384	1	4	yes	—
121	C₂×D₅₃	C₂×#107	1	8	yes	—
122	C₂×D₅₃	#493	1	4	yes	—
123	C₂×D₅₃	#494	1	4	yes	—
124	C₂×D₅₃	#500	1	8	yes	—
125	C₄×C₅₃	—	8	4	no	—
126	C₄×C₅₃	C₂×#107	1	8	yes	complemented
127	C₄×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
128	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
129	Dic₅₃	—	8	4	no	—
130	Dic₅₃	#384	1	4	yes	—
131	Dic₅₃	C₂×#107	1	8	yes	complemented
132	C₂²×#107	—	53	4	no	—
133	C₂²×C₄×C₈	—	53	1	no	—
134	C₂×#384	—	53	2	no	—
135	#1845	—	53	4	no	—
136	#1845	—	53	1	no	—
137	#1845	—	53	2	no	—
138	#1845	D₅₃	1	1	yes	characteristic, direct product factor
139	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
140	C₂×C₅₃	C₂²×#107	1	2	yes	—
141	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
142	C₂×C₅₃	C₂×#494	1	1	yes	characteristic
143	C₂×C₅₃	C₂×#500	1	2	yes	—
144	D₅₃	#1845	1	8	yes	direct product factor
145	C₂³×C₈	—	53	2	no	—
146	C₂²×C₄⋊C₄	—	53	4	no	—
147	C₂²×C₄⋊C₄	—	106	4	no	—
148	C₂²×C₄²	—	53	1	no	—
149	C₂×#107	—	53	4	no	—
150	C₂×#107	—	53	8	no	—
151	C₂×#107	—	106	16	no	—
152	C₂×#107	C₂×D₅₃	1	4	yes	—
153	C₂×C₄×C₈	—	53	2	no	—
154	C₂×C₄×C₈	—	53	1	no	—
155	C₂×C₄×C₈	—	53	8	no	—
156	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
157	#384	—	53	4	no	—
158	#384	—	53	2	no	—
159	#384	—	106	4	no	—
160	#384	C₂×D₅₃	1	2	yes	—
161	C₅₃	C₂×#1845	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₄	—	53	2	no	—
163	C₂³×C₄	—	53	1	no	—
164	C₂³×C₄	—	106	4	no	—
165	C₂²×C₈	—	53	2	no	—
166	C₂²×C₈	—	53	8	no	—
167	C₂²×C₈	—	53	8	no	—
168	C₂²×C₈	—	53	8	no	—
169	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
170	C₂×C₄⋊C₄	—	2	4	no	—
171	C₂×C₄⋊C₄	—	53	4	no	—
172	C₂×C₄⋊C₄	—	106	16	no	—
173	C₂×C₄⋊C₄	—	106	4	no	—
174	C₂×C₄⋊C₄	—	106	8	no	—
175	C₂×C₄⋊C₄	—	106	4	no	—
176	C₂×C₄⋊C₄	—	212	16	no	—
177	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
178	C₂×C₄²	—	53	2	no	—
179	C₂×C₄²	—	53	1	no	—
180	C₂×C₄²	—	53	4	no	—
181	C₂×C₄²	—	53	4	no	—
182	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
183	#107	—	2	8	no	—
184	#107	—	106	16	no	—
185	#107	—	106	8	no	—
186	C₄×C₈	—	53	8	no	—
187	C₄×C₈	—	53	4	no	—
188	C₄×C₈	C₄×D₅₃	1	4	yes	complemented
189	C₂⁴	—	53	1	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	53	2	no	—
192	C₂²×C₄	—	53	1	no	—
193	C₂²×C₄	—	53	8	no	—
194	C₂²×C₄	—	53	8	no	—
195	C₂²×C₄	—	53	8	no	—
196	C₂²×C₄	—	53	2	no	—
197	C₂²×C₄	—	53	4	no	—
198	C₂²×C₄	—	53	2	no	—
199	C₂²×C₄	—	53	4	no	—
200	C₂²×C₄	—	106	4	no	—
201	C₂²×C₄	—	212	8	no	—
202	C₂²×C₄	—	212	16	no	—
203	C₂²×C₄	D₄×D₅₃	1	2	yes	—
204	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
205	C₂×C₈	—	53	4	no	—
206	C₂×C₈	—	53	4	no	—
207	C₂×C₈	—	53	4	no	—
208	C₂×C₈	—	53	32	no	—
209	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
210	C₂×C₈	D₄×D₅₃	1	4	yes	—
211	C₂×C₈	Q₈×D₅₃	1	4	yes	—
212	C₄⋊C₄	—	2	8	no	—
213	C₄⋊C₄	—	4	8	no	—
214	C₄⋊C₄	—	106	8	no	—
215	C₄⋊C₄	—	212	8	no	—
216	C₄⋊C₄	—	212	16	no	—
217	C₄⋊C₄	—	212	8	no	—
218	C₄²	—	53	8	no	—
219	C₄²	—	53	2	no	—
220	C₄²	—	53	2	no	—
221	C₄²	C₂×C₄×D₅₃	1	2	yes	—
222	C₄²	Q₈×D₅₃	1	2	yes	—
223	C₂³	—	53	2	no	—
224	C₂³	—	53	4	no	—
225	C₂³	—	53	4	no	—
226	C₂³	—	53	2	no	—
227	C₂³	—	53	2	no	—
228	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
229	C₂×C₄	—	4	8	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	53	2	no	—
232	C₂×C₄	—	53	4	no	—
233	C₂×C₄	—	53	4	no	—
234	C₂×C₄	—	53	4	no	—
235	C₂×C₄	—	53	2	no	—
236	C₂×C₄	—	53	2	no	—
237	C₂×C₄	—	53	8	no	—
238	C₂×C₄	—	53	8	no	—
239	C₂×C₄	—	53	32	no	—
240	C₂×C₄	—	424	16	no	—
241	C₂×C₄	—	212	8	no	—
242	C₂×C₄	—	212	4	no	—
243	C₂×C₄	C₄⋊C₄×D₅₃	1	4	yes	—
244	C₂×C₄	D₈×D₅₃	1	4	yes	—
245	C₂×C₄	Q₁₆×D₅₃	1	4	yes	—
246	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
247	C₂×C₄	C₂×D₄×D₅₃	1	2	yes	—
248	C₂×C₄	C₂×Q₈×D₅₃	1	2	yes	—
249	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
250	C₈	—	53	8	no	—
251	C₈	C₄⋊C₄×D₅₃	1	8	yes	complemented
252	C₂²	—	53	8	no	—
253	C₂²	—	53	4	no	—
254	C₂²	—	53	8	no	—
255	C₂²	—	53	4	no	—
256	C₂²	—	53	4	no	—
257	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
258	C₂²	#127×D₅₃	1	1	yes	characteristic
259	C₂²	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
260	C₂²	C₂×D₈×D₅₃	1	2	yes	—
261	C₂²	C₂×Q₁₆×D₅₃	1	2	yes	—
262	C₄	—	8	4	no	—
263	C₄	—	53	4	no	—
264	C₄	—	53	8	no	—
265	C₄	—	424	4	no	—
266	C₄	#107×D₅₃	1	8	yes	complemented
267	C₄	C₂×C₄⋊C₄×D₅₃	1	2	yes	—
268	C₄	C₄⋊Q₈×D₅₃	1	2	yes	—
269	C₂	—	53	8	no	—
270	C₂	#384×D₅₃	1	1	yes	characteristic
271	C₂	C₂×#107×D₅₃	1	2	yes	—
272	C₂	#493×D₅₃	1	1	yes	characteristic
273	C₂	#494×D₅₃	1	1	yes	characteristic
274	C₂	#500×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1845	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂³×D₅₃	C₂×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1845	D₅₃
hypocenter	C₅₃	C₂×#1845
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
4. #20274875	C₂	semidirect
5. #20274874	C₂	semidirect
6. #20274873	C₂	semidirect
7. C₅₃×#1845	C₂	semidirect
17. C₄×C₈×D₅₃	C₄	semidirect
43. C₄×#88322	C₂×C₄	semidirect
44. C₈×Dic₅₃	C₂×C₄	semidirect
50. C₄×C₈×C₅₃	C₂×C₄	semidirect
84. C₈×D₅₃	C₄⋊C₄	semidirect
109. C₄×D₅₃	#107	semidirect
112. C₈×C₅₃	C₂×C₄⋊C₄	semidirect
113. #88322	C₂×C₄⋊C₄	semidirect
126. C₄×C₅₃	C₂×#107	semidirect
131. Dic₅₃	C₂×#107	semidirect
138. #1845	D₅₃	direct (N × H)
144. D₅₃	#1845	direct (N × H)
161. C₅₃	C₂×#1845	semidirect
188. C₄×C₈	C₄×D₅₃	semidirect
251. C₈	C₄⋊C₄×D₅₃	semidirect
266. C₄	#107×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#107×D₅₃	2
C₂×C₄×C₈×D₅₃	2
#384×D₅₃	2
#20274875	2
#20274874	2
#20274873	2
C₅₃×#1845	2
C₂×#1845	53

Subgroups — D53×#1845 #27384491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1845 #27384491