Subgroups — C₅₃×#3180 #20272117

Order 6784.

410 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	C₅₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#501	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#496	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#490	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#489	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#466	C₂	1	2	yes	maximal, complemented
8	C₅₃×#465	C₂	1	2	yes	maximal, complemented
9	C₅₃×#442	C₂	1	2	yes	maximal, complemented
10	C₅₃×#440	C₂	1	2	yes	maximal, complemented
11	C₂×#97×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₅₃	—	2	1	no	—
13	C₂×SD₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂×D₈×C₅₃	C₂²	1	2	yes	—
15	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
17	#127×C₅₃	C₂²	1	1	yes	characteristic
18	#126×C₅₃	—	2	1	no	—
19	#125×C₅₃	C₂²	1	1	yes	characteristic
20	#124×C₅₃	—	2	1	no	—
21	#124×C₅₃	C₂²	1	1	yes	characteristic
22	#123×C₅₃	—	2	1	no	—
23	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	#121×C₅₃	—	2	1	no	—
25	#121×C₅₃	C₂²	1	1	yes	characteristic
26	#121×C₅₃	C₂²	1	2	yes	—
27	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₅₃	—	2	1	no	—
29	C₄×D₄×C₅₃	—	2	1	no	—
30	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
31	C₄×D₄×C₅₃	C₂²	1	2	yes	—
32	#117×C₅₃	—	2	1	no	—
33	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
34	#107×C₅₃	C₂²	1	2	yes	—
35	#106×C₅₃	C₂²	1	2	yes	—
36	#103×C₅₃	C₂²	1	2	yes	—
37	#102×C₅₃	C₂²	1	2	yes	—
38	#102×C₅₃	C₂²	1	2	yes	—
39	#102×C₅₃	C₂²	1	2	yes	—
40	#97×C₅₃	C₂²	1	1	yes	characteristic
41	#97×C₅₃	C₂²	1	1	yes	characteristic
42	#97×C₅₃	C₂²	1	2	yes	—
43	#41×C₅₃	—	4	4	no	—
44	C₂×Q₈×C₅₃	—	2	1	no	—
45	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₅₃	—	2	1	no	—
47	C₂×D₄×C₅₃	—	2	2	no	—
48	C₂×D₄×C₅₃	—	2	1	no	—
49	C₂×D₄×C₅₃	—	2	1	no	—
50	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×D₄×C₅₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₅₃	—	2	1	no	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
57	SD₁₆×C₅₃	—	2	4	no	—
58	D₈×C₅₃	—	2	4	no	—
59	C₂×C₈×C₅₃	—	2	4	no	—
60	C₂×C₈×C₅₃	C₂³	1	2	yes	—
61	C₂×C₈×C₅₃	C₂³	1	2	yes	—
62	C₄⋊C₄×C₅₃	—	2	1	no	—
63	C₄⋊C₄×C₅₃	—	2	1	no	—
64	C₄⋊C₄×C₅₃	—	2	1	no	—
65	C₄⋊C₄×C₅₃	—	2	1	no	—
66	C₄⋊C₄×C₅₃	—	2	1	no	—
67	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
70	C₂²⋊C₄×C₅₃	—	2	1	no	—
71	C₂²⋊C₄×C₅₃	—	2	1	no	—
72	C₂²⋊C₄×C₅₃	—	2	1	no	—
73	C₂²⋊C₄×C₅₃	—	2	2	no	—
74	C₂²⋊C₄×C₅₃	—	2	1	no	—
75	C₂²⋊C₄×C₅₃	—	2	2	no	—
76	C₂²⋊C₄×C₅₃	—	2	1	no	—
77	C₄²×C₅₃	—	2	1	no	—
78	C₄²×C₅₃	D₄	1	1	yes	characteristic
79	C₄²×C₅₃	D₄	1	1	yes	characteristic
80	C₄²×C₅₃	C₂³	1	1	yes	characteristic
81	C₄²×C₅₃	C₂³	1	1	yes	characteristic
82	C₂³×C₅₃	—	2	2	no	—
83	C₂³×C₅₃	—	2	1	no	—
84	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
85	Q₈×C₅₃	—	2	2	no	—
86	Q₈×C₅₃	—	4	2	no	—
87	D₄×C₅₃	—	2	2	no	—
88	D₄×C₅₃	—	2	2	no	—
89	D₄×C₅₃	—	2	2	no	—
90	D₄×C₅₃	—	4	2	no	—
91	D₄×C₅₃	—	4	2	no	—
92	D₄×C₅₃	—	4	2	no	—
93	D₄×C₅₃	—	4	4	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	4	1	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	4	4	no	—
103	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
109	C₈×C₅₃	—	2	2	no	—
110	C₈×C₅₃	#41	1	4	yes	complemented
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	—	2	1	no	—
113	C₂²×C₅₃	—	4	1	no	—
114	C₂²×C₅₃	—	4	2	no	—
115	C₂²×C₅₃	—	4	2	no	—
116	C₂²×C₅₃	—	4	1	no	—
117	C₂²×C₅₃	—	4	1	no	—
118	C₂²×C₅₃	—	4	2	no	—
119	C₂²×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
120	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
121	C₄×C₅₃	—	2	1	no	—
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	—	2	1	no	—
125	C₄×C₅₃	—	2	1	no	—
126	C₄×C₅₃	—	4	1	no	—
127	C₄×C₅₃	—	4	2	no	—
128	C₄×C₅₃	—	4	1	no	—
129	C₄×C₅₃	—	4	1	no	—
130	C₄×C₅₃	C₂×#41	1	2	yes	—
131	#3180	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂×C₅₃	—	2	2	no	—
133	C₂×C₅₃	—	8	2	no	—
134	C₂×C₅₃	—	8	1	no	—
135	C₂×C₅₃	#532	1	1	yes	characteristic
136	C₂×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₅₃	#575	1	1	yes	characteristic
138	#532	C₂×C₅₃	1	1	yes	characteristic, complemented
139	#529	C₂×C₅₃	1	1	yes	characteristic, complemented
140	#501	C₂×C₅₃	1	1	yes	characteristic, complemented
141	#496	C₂×C₅₃	1	1	yes	characteristic, complemented
142	#490	C₂×C₅₃	1	1	yes	characteristic, complemented
143	#489	C₂×C₅₃	1	1	yes	characteristic, complemented
144	#466	C₂×C₅₃	1	2	yes	complemented
145	#465	C₂×C₅₃	1	2	yes	complemented
146	#442	C₂×C₅₃	1	2	yes	complemented
147	#440	C₂×C₅₃	1	2	yes	complemented
148	C₂×#97	C₂×C₅₃	1	1	yes	characteristic, complemented
149	C₅₃	#3180	1	1	yes	characteristic, direct product factor, Sylow
150	C₂×#41	—	2	1	no	—
151	C₂×SD₁₆	C₂²×C₅₃	1	2	yes	—
152	C₂×D₈	C₂²×C₅₃	1	2	yes	—
153	C₂²×C₈	C₂²×C₅₃	1	2	yes	—
154	C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
155	#127	C₂²×C₅₃	1	1	yes	characteristic
156	#126	—	2	1	no	—
157	#125	C₂²×C₅₃	1	1	yes	characteristic
158	#124	—	2	1	no	—
159	#124	C₂²×C₅₃	1	1	yes	characteristic
160	#123	—	2	1	no	—
161	C₂²⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
162	#121	—	2	1	no	—
163	#121	C₂²×C₅₃	1	1	yes	characteristic
164	#121	C₂²×C₅₃	1	2	yes	—
165	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
166	C₄×D₄	—	2	1	no	—
167	C₄×D₄	—	2	1	no	—
168	C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
169	C₄×D₄	C₂²×C₅₃	1	2	yes	—
170	#117	—	2	1	no	—
171	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
172	#107	C₂²×C₅₃	1	2	yes	—
173	#106	C₂²×C₅₃	1	2	yes	—
174	#103	C₂²×C₅₃	1	2	yes	—
175	#102	C₂²×C₅₃	1	2	yes	—
176	#102	C₂²×C₅₃	1	2	yes	—
177	#102	C₂²×C₅₃	1	2	yes	—
178	#97	C₂²×C₅₃	1	2	yes	—
179	#97	C₂²×C₅₃	1	1	yes	characteristic
180	#97	C₂²×C₅₃	1	1	yes	characteristic
181	#41	—	4	4	no	—
182	C₂×Q₈	—	2	1	no	—
183	C₂×Q₈	C₂³×C₅₃	1	1	yes	characteristic
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	C₂³×C₅₃	1	1	yes	characteristic
189	C₂×D₄	C₂³×C₅₃	1	2	yes	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	D₄×C₅₃	1	1	yes	characteristic
193	C₂²×C₄	D₄×C₅₃	1	1	yes	characteristic
194	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
195	SD₁₆	—	2	4	no	—
196	D₈	—	2	4	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	C₂³×C₅₃	1	2	yes	—
199	C₂×C₈	C₂³×C₅₃	1	2	yes	—
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
206	C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
207	C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
208	C₂²⋊C₄	—	2	1	no	—
209	C₂²⋊C₄	—	2	1	no	—
210	C₂²⋊C₄	—	2	1	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	2	1	no	—
215	C₄²	—	2	1	no	—
216	C₄²	D₄×C₅₃	1	1	yes	characteristic
217	C₄²	D₄×C₅₃	1	1	yes	characteristic
218	C₄²	C₂³×C₅₃	1	1	yes	characteristic
219	C₄²	C₂³×C₅₃	1	1	yes	characteristic
220	C₂³	—	2	1	no	—
221	C₂³	—	2	2	no	—
222	C₂³	C₂×D₄×C₅₃	1	1	yes	characteristic
223	Q₈	—	2	2	no	—
224	Q₈	—	4	2	no	—
225	D₄	—	2	2	no	—
226	D₄	—	2	2	no	—
227	D₄	—	2	2	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
242	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
243	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
245	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
246	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	#41×C₅₃	1	4	yes	complemented
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	4	2	no	—
257	C₂²	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
258	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	4	1	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	1	no	—
268	C₄	C₂×#41×C₅₃	1	2	yes	—
269	C₂	—	2	2	no	—
270	C₂	—	8	2	no	—
271	C₂	—	8	1	no	—
272	C₂	C₅₃×#532	1	1	yes	characteristic
273	C₂	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
274	C₂	C₅₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#532	C₂	semidirect
2. C₅₃×#529	C₂	semidirect
3. C₅₃×#501	C₂	semidirect
4. C₅₃×#496	C₂	semidirect
5. C₅₃×#490	C₂	semidirect
6. C₅₃×#489	C₂	semidirect
7. C₅₃×#466	C₂	semidirect
8. C₅₃×#465	C₂	semidirect
9. C₅₃×#442	C₂	semidirect
10. C₅₃×#440	C₂	semidirect
11. C₂×#97×C₅₃	C₂	semidirect
110. C₈×C₅₃	#41	semidirect
131. #3180	C₅₃	direct (N × H)
138. #532	C₂×C₅₃	semidirect
139. #529	C₂×C₅₃	semidirect
140. #501	C₂×C₅₃	semidirect
141. #496	C₂×C₅₃	semidirect
142. #490	C₂×C₅₃	semidirect
143. #489	C₂×C₅₃	semidirect
144. #466	C₂×C₅₃	semidirect
145. #465	C₂×C₅₃	semidirect
146. #442	C₂×C₅₃	semidirect
147. #440	C₂×C₅₃	semidirect
148. C₂×#97	C₂×C₅₃	semidirect
149. C₅₃	#3180	direct (N × H)
248. C₈	#41×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#532	2
C₅₃×#529	2
C₅₃×#501	2
C₅₃×#496	2
C₅₃×#490	2
C₅₃×#489	2
C₅₃×#466	2
C₅₃×#465	2
C₅₃×#442	2
C₅₃×#440	2
C₂×#97×C₅₃	2
#3180	53

Subgroups — C53×#3180 #20272117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#3180 #20272117