Subgroups — C₅₃×#2663 #20271600

Order 6784.

498 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	C₂×#123×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#400	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#399	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#397	C₂	1	2	yes	maximal, complemented
6	C₅₃×#395	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#394	C₂	1	2	yes	maximal
8	C₅₃×#393	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#389	C₂	1	2	yes	maximal, complemented
10	C₅₃×#388	C₂	1	2	yes	maximal, complemented
11	C₄×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
15	#123×C₅₃	—	2	4	no	—
16	C₂²⋊Q₈×C₅₃	—	2	4	no	—
17	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
31	#95×C₅₃	C₂²	1	1	yes	characteristic
32	#95×C₅₃	C₂²	1	1	yes	characteristic
33	#95×C₅₃	C₂²	1	2	yes	—
34	#95×C₅₃	C₂²	1	2	yes	—
35	#95×C₅₃	C₂²	1	2	yes	—
36	#95×C₅₃	C₂²	1	2	yes	—
37	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
38	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₅₃	—	2	2	no	—
40	C₂×Q₈×C₅₃	—	4	2	no	—
41	C₂×D₄×C₅₃	—	2	2	no	—
42	C₂×D₄×C₅₃	—	4	2	no	—
43	C₂²×C₄×C₅₃	—	4	2	no	—
44	C₂²×C₄×C₅₃	D₄	1	4	yes	—
45	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
54	C₄⋊C₄×C₅₃	—	2	4	no	—
55	C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₄⋊C₄×C₅₃	—	2	2	no	—
58	C₄⋊C₄×C₅₃	—	2	4	no	—
59	C₄⋊C₄×C₅₃	—	4	2	no	—
60	C₂²⋊C₄×C₅₃	—	2	4	no	—
61	C₂²⋊C₄×C₅₃	—	2	2	no	—
62	C₂²⋊C₄×C₅₃	—	2	2	no	—
63	C₂²⋊C₄×C₅₃	—	2	2	no	—
64	C₂²⋊C₄×C₅₃	—	2	4	no	—
65	C₂²⋊C₄×C₅₃	—	2	4	no	—
66	C₂²⋊C₄×C₅₃	—	4	2	no	—
67	C₄²×C₅₃	—	4	2	no	—
68	C₂³×C₅₃	—	2	2	no	—
69	C₂³×C₅₃	—	2	2	no	—
70	C₂³×C₅₃	—	2	1	no	—
71	C₂³×C₅₃	—	2	1	no	—
72	C₂³×C₅₃	—	2	1	no	—
73	C₂³×C₅₃	—	2	2	no	—
74	C₂³×C₅₃	—	2	2	no	—
75	C₂³×C₅₃	—	2	1	no	—
76	C₂³×C₅₃	—	2	1	no	—
77	C₂³×C₅₃	C₂×D₄	1	2	yes	—
78	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
79	Q₈×C₅₃	—	4	4	no	—
80	D₄×C₅₃	—	4	4	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	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	2	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	4	4	no	—
102	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₅₃	#41	1	4	yes	—
105	C₂²×C₅₃	—	2	2	no	—
106	C₂²×C₅₃	—	2	2	no	—
107	C₂²×C₅₃	—	2	2	no	—
108	C₂²×C₅₃	—	4	2	no	—
109	C₂²×C₅₃	—	4	2	no	—
110	C₂²×C₅₃	—	4	1	no	—
111	C₂²×C₅₃	—	4	1	no	—
112	C₂²×C₅₃	—	4	1	no	—
113	C₂²×C₅₃	—	4	2	no	—
114	C₂²×C₅₃	—	4	2	no	—
115	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
116	C₂²×C₅₃	C₂×#41	1	2	yes	—
117	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
118	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
119	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	2	4	no	—
123	C₄×C₅₃	—	4	1	no	—
124	C₄×C₅₃	—	4	2	no	—
125	C₄×C₅₃	—	4	1	no	—
126	C₄×C₅₃	—	4	2	no	—
127	C₄×C₅₃	—	4	1	no	—
128	C₄×C₅₃	—	4	2	no	—
129	#2663	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₅₃	—	4	2	no	—
131	C₂×C₅₃	—	8	1	no	—
132	C₂×C₅₃	#532	1	1	yes	characteristic
133	C₂×C₅₃	#534	1	1	yes	characteristic
134	C₂×C₅₃	#536	1	1	yes	characteristic
135	C₂×C₅₃	#539	1	2	yes	—
136	C₂×C₅₃	#542	1	2	yes	—
137	C₂×#123	C₂×C₅₃	1	1	yes	characteristic
138	C₂×C₂²⋊Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
139	#400	C₂×C₅₃	1	1	yes	characteristic, complemented
140	#399	C₂×C₅₃	1	1	yes	characteristic, complemented
141	#397	C₂×C₅₃	1	2	yes	complemented
142	#395	C₂×C₅₃	1	1	yes	characteristic, complemented
143	#394	C₂×C₅₃	1	2	yes	—
144	#393	C₂×C₅₃	1	1	yes	characteristic, complemented
145	#389	C₂×C₅₃	1	2	yes	complemented
146	#388	C₂×C₅₃	1	2	yes	complemented
147	C₄×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
148	C₅₃	#2663	1	1	yes	characteristic, direct product factor, Sylow
149	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
150	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
151	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
152	#123	—	2	4	no	—
153	C₂²⋊Q₈	—	2	4	no	—
154	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
155	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
156	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
157	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
158	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
159	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
160	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
161	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
162	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
163	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
164	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
165	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
166	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
167	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
168	#95	C₂²×C₅₃	1	1	yes	characteristic
169	#95	C₂²×C₅₃	1	1	yes	characteristic
170	#95	C₂²×C₅₃	1	2	yes	—
171	#95	C₂²×C₅₃	1	2	yes	—
172	#95	C₂²×C₅₃	1	2	yes	—
173	#95	C₂²×C₅₃	1	2	yes	—
174	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
175	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	D₄×C₅₃	1	4	yes	—
182	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
183	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
184	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
186	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
187	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
188	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
190	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₂²⋊C₄	—	2	4	no	—
198	C₂²⋊C₄	—	2	2	no	—
199	C₂²⋊C₄	—	2	2	no	—
200	C₂²⋊C₄	—	2	2	no	—
201	C₂²⋊C₄	—	2	4	no	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₄²	—	4	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	C₂×D₄×C₅₃	1	2	yes	—
215	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
216	Q₈	—	4	4	no	—
217	D₄	—	4	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
240	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
241	C₂×C₄	#41×C₅₃	1	4	yes	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
253	C₂²	C₂×#41×C₅₃	1	2	yes	—
254	C₂²	H(5,𝔽₂)×C₅₃	1	2	yes	—
255	C₂²	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
256	C₂²	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	4	1	no	—
261	C₄	—	4	2	no	—
262	C₄	—	4	1	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	4	2	no	—
266	C₂	—	4	2	no	—
267	C₂	—	8	1	no	—
268	C₂	C₅₃×#532	1	1	yes	characteristic
269	C₂	C₅₃×#534	1	1	yes	characteristic
270	C₂	C₅₃×#536	1	1	yes	characteristic
271	C₂	C₅₃×#539	1	2	yes	—
272	C₂	C₅₃×#542	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂²⋊Q₈×C₅₃	C₂	semidirect
3. C₅₃×#400	C₂	semidirect
4. C₅₃×#399	C₂	semidirect
5. C₅₃×#397	C₂	semidirect
6. C₅₃×#395	C₂	semidirect
8. C₅₃×#393	C₂	semidirect
9. C₅₃×#389	C₂	semidirect
10. C₅₃×#388	C₂	semidirect
11. C₄×C₂²⋊C₄×C₅₃	C₂	semidirect
129. #2663	C₅₃	direct (N × H)
138. C₂×C₂²⋊Q₈	C₂×C₅₃	semidirect
139. #400	C₂×C₅₃	semidirect
140. #399	C₂×C₅₃	semidirect
141. #397	C₂×C₅₃	semidirect
142. #395	C₂×C₅₃	semidirect
144. #393	C₂×C₅₃	semidirect
145. #389	C₂×C₅₃	semidirect
146. #388	C₂×C₅₃	semidirect
147. C₄×C₂²⋊C₄	C₂×C₅₃	semidirect
148. C₅₃	#2663	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#123×C₅₃	2
C₂×C₂²⋊Q₈×C₅₃	2
C₅₃×#400	2
C₅₃×#399	2
C₅₃×#397	2
C₅₃×#395	2
C₅₃×#394	2
C₅₃×#393	2
C₅₃×#389	2
C₅₃×#388	2
C₄×C₂²⋊C₄×C₅₃	2
#2663	53

Subgroups — C53×#2663 #20271600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2663 #20271600