Subgroups — C₅₃×#2993 #20271930

Order 6784.

354 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#529	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#436	C₂	1	2	yes	maximal, complemented
6	C₅₃×#435	C₂	1	2	yes	maximal, complemented
7	C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#433	C₂	1	2	yes	maximal, complemented
9	C₅₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#424	C₂	1	2	yes	maximal, complemented
11	C₅₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
13	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
15	#126×C₅₃	C₄	1	2	yes	—
16	#125×C₅₃	C₄	1	1	yes	characteristic
17	#124×C₅₃	C₄	1	1	yes	characteristic
18	#123×C₅₃	C₄	1	2	yes	—
19	C₂²⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
20	#121×C₅₃	C₄	1	1	yes	characteristic
21	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₅₃	C₂²	1	2	yes	—
24	#117×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
26	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
27	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
31	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
32	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂²⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
36	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
37	#97×C₅₃	C₂²	1	2	yes	—
38	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₅₃	C₂²	1	2	yes	—
41	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
48	M₁₆×C₅₃	—	4	2	no	—
49	M₁₆×C₅₃	—	4	2	no	—
50	C₂×C₈×C₅₃	—	4	2	no	—
51	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂³	1	2	yes	—
56	C₂×C₈×C₅₃	C₂³	1	2	yes	—
57	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
58	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
59	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
61	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
63	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
64	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
65	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
66	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
67	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
68	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₅₃	C₂³	1	1	yes	characteristic
71	C₄²×C₅₃	C₂³	1	2	yes	—
72	C₄²×C₅₃	C₂³	1	1	yes	characteristic
73	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
75	Q₈×C₅₃	—	2	2	no	—
76	D₄×C₅₃	—	2	2	no	—
77	D₄×C₅₃	—	4	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	4	1	no	—
84	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
93	C₈×C₅₃	—	2	1	no	—
94	C₈×C₅₃	—	2	2	no	—
95	C₈×C₅₃	—	2	1	no	—
96	C₈×C₅₃	—	2	2	no	—
97	C₈×C₅₃	—	2	1	no	—
98	C₈×C₅₃	—	2	1	no	—
99	C₂²×C₅₃	—	2	2	no	—
100	C₂²×C₅₃	—	2	1	no	—
101	C₂²×C₅₃	—	2	1	no	—
102	C₂²×C₅₃	—	2	2	no	—
103	C₂²×C₅₃	—	2	1	no	—
104	C₂²×C₅₃	—	2	2	no	—
105	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
106	C₄×C₅₃	—	2	2	no	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	2	2	no	—
113	C₄×C₅₃	#131	1	2	yes	—
114	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
115	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
116	#2993	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂×C₅₃	—	4	1	no	—
118	C₂×C₅₃	—	4	2	no	—
119	C₂×C₅₃	#520	1	1	yes	characteristic
120	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
121	C₂×C₅₃	#568	1	1	yes	characteristic
122	#529	C₂×C₅₃	1	1	yes	characteristic
123	#446	C₂×C₅₃	1	1	yes	characteristic, complemented
124	C₈×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
125	#436	C₂×C₅₃	1	1	yes	characteristic, complemented
126	#436	C₂×C₅₃	1	2	yes	complemented
127	#435	C₂×C₅₃	1	2	yes	complemented
128	C₈×D₄	C₂×C₅₃	1	1	yes	characteristic, complemented
129	#433	C₂×C₅₃	1	2	yes	complemented
130	#431	C₂×C₅₃	1	1	yes	characteristic, complemented
131	#424	C₂×C₅₃	1	2	yes	complemented
132	#423	C₂×C₅₃	1	1	yes	characteristic, complemented
133	C₅₃	#2993	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×M₁₆	C₂²×C₅₃	1	2	yes	—
135	C₂×M₁₆	C₂²×C₅₃	1	2	yes	—
136	C₂²×C₈	C₂²×C₅₃	1	2	yes	—
137	#126	C₄×C₅₃	1	2	yes	—
138	#125	C₄×C₅₃	1	1	yes	characteristic
139	#124	C₄×C₅₃	1	1	yes	characteristic
140	#123	C₄×C₅₃	1	2	yes	—
141	C₂²⋊Q₈	C₄×C₅₃	1	1	yes	characteristic
142	#121	C₄×C₅₃	1	1	yes	characteristic
143	C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
144	C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
145	C₄×D₄	C₂²×C₅₃	1	2	yes	—
146	#117	C₂²×C₅₃	1	2	yes	—
147	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
148	C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
149	C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
150	C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
151	C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
152	C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
153	C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
154	C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
155	C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
156	C₂²⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
157	C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
158	C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
159	#97	C₂²×C₅₃	1	2	yes	—
160	C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
161	C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
162	C₄×C₈	C₂²×C₅₃	1	2	yes	—
163	C₂×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
164	C₂×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
165	C₂×D₄	C₂×C₄×C₅₃	1	2	yes	—
166	C₂²×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
170	M₁₆	—	4	2	no	—
171	M₁₆	—	4	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
174	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
175	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
176	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
177	C₂×C₈	C₂³×C₅₃	1	2	yes	—
178	C₂×C₈	C₂³×C₅₃	1	2	yes	—
179	C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
180	C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
181	C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
182	C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
183	C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
184	C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
185	C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
186	C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
187	C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
188	C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
189	C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
190	C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
191	C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
192	C₄²	C₂³×C₅₃	1	1	yes	characteristic
193	C₄²	C₂³×C₅₃	1	2	yes	—
194	C₄²	C₂³×C₅₃	1	1	yes	characteristic
195	C₂³	C₂²×C₄×C₅₃	1	2	yes	—
196	C₂³	C₂²×C₄×C₅₃	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	D₄	—	2	2	no	—
199	D₄	—	4	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
207	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
208	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
209	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
210	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
211	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
212	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
213	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
214	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	2	no	—
217	C₈	—	2	1	no	—
218	C₈	—	2	2	no	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	2	no	—
227	C₂²	C₂³×C₄×C₅₃	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	2	no	—
235	C₄	#131×C₅₃	1	2	yes	—
236	C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
237	C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
238	C₂	—	4	1	no	—
239	C₂	—	4	2	no	—
240	C₂	C₅₃×#520	1	1	yes	characteristic
241	C₂	C₂×#131×C₅₃	1	1	yes	characteristic
242	C₂	C₅₃×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₅₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂³×C₄
prime radical	#121×C₅₃	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#446	C₂	semidirect
3. C₈×Q₈×C₅₃	C₂	semidirect
4. C₅₃×#436	C₂	semidirect
5. C₅₃×#436	C₂	semidirect
6. C₅₃×#435	C₂	semidirect
7. C₈×D₄×C₅₃	C₂	semidirect
8. C₅₃×#433	C₂	semidirect
9. C₅₃×#431	C₂	semidirect
10. C₅₃×#424	C₂	semidirect
11. C₅₃×#423	C₂	semidirect
116. #2993	C₅₃	direct (N × H)
123. #446	C₂×C₅₃	semidirect
124. C₈×Q₈	C₂×C₅₃	semidirect
125. #436	C₂×C₅₃	semidirect
126. #436	C₂×C₅₃	semidirect
127. #435	C₂×C₅₃	semidirect
128. C₈×D₄	C₂×C₅₃	semidirect
129. #433	C₂×C₅₃	semidirect
130. #431	C₂×C₅₃	semidirect
131. #424	C₂×C₅₃	semidirect
132. #423	C₂×C₅₃	semidirect
133. C₅₃	#2993	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#529	2
C₅₃×#446	2
C₈×Q₈×C₅₃	2
C₅₃×#436	2
C₅₃×#435	2
C₈×D₄×C₅₃	2
C₅₃×#433	2
C₅₃×#431	2
C₅₃×#424	2
C₅₃×#423	2
#2993	53

Subgroups — C53×#2993 #20271930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2993 #20271930