Subgroups — C₅₃×#2463 #20271400

Order 6784.

668 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊Q₈×C₅₃	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₄×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₅₃×#394	C₂	1	2	yes	maximal, complemented
4	C₅₃×#393	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#392	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#388	C₂	1	2	yes	maximal, complemented
7	C₅₃×#387	C₂	1	2	yes	maximal, complemented
8	C₅₃×#386	C₂	1	2	yes	maximal, complemented
9	C₅₃×#385	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₅₃	C₂²	1	2	yes	—
12	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
15	C₂²⋊Q₈×C₅₃	—	2	8	no	—
16	C₄×D₄×C₅₃	—	2	8	no	—
17	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊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	2	yes	—
26	C₂×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
28	#95×C₅₃	C₂²	1	2	yes	—
29	#95×C₅₃	C₂²	1	2	yes	—
30	#95×C₅₃	C₂²	1	1	yes	characteristic
31	#95×C₅₃	C₂²	1	2	yes	—
32	#95×C₅₃	C₂²	1	1	yes	characteristic
33	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
34	C₂⁴×C₅₃	C₂³	1	2	yes	—
35	C₂×Q₈×C₅₃	—	2	4	no	—
36	C₂×Q₈×C₅₃	—	4	1	no	—
37	C₂×D₄×C₅₃	—	2	4	no	—
38	C₂×D₄×C₅₃	—	2	4	no	—
39	C₂×D₄×C₅₃	—	2	4	no	—
40	C₂×D₄×C₅₃	—	2	4	no	—
41	C₂×D₄×C₅₃	—	4	1	no	—
42	C₂²×C₄×C₅₃	—	2	4	no	—
43	C₂²×C₄×C₅₃	—	2	4	no	—
44	C₂²×C₄×C₅₃	—	2	4	no	—
45	C₂²×C₄×C₅₃	—	4	1	no	—
46	C₂²×C₄×C₅₃	—	4	2	no	—
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	2	yes	—
52	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
54	C₄⋊C₄×C₅₃	—	2	4	no	—
55	C₄⋊C₄×C₅₃	—	2	4	no	—
56	C₄⋊C₄×C₅₃	D₄	1	8	yes	complemented
57	C₂²⋊C₄×C₅₃	—	2	2	no	—
58	C₂²⋊C₄×C₅₃	—	2	4	no	—
59	C₂²⋊C₄×C₅₃	—	2	4	no	—
60	C₂²⋊C₄×C₅₃	—	2	4	no	—
61	C₂²⋊C₄×C₅₃	—	2	4	no	—
62	C₂²⋊C₄×C₅₃	—	2	4	no	—
63	C₂²⋊C₄×C₅₃	—	4	1	no	—
64	C₄²×C₅₃	—	2	4	no	—
65	C₂³×C₅₃	—	2	2	no	—
66	C₂³×C₅₃	—	2	2	no	—
67	C₂³×C₅₃	—	2	2	no	—
68	C₂³×C₅₃	—	2	2	no	—
69	C₂³×C₅₃	—	2	2	no	—
70	C₂³×C₅₃	—	2	2	no	—
71	C₂³×C₅₃	—	2	2	no	—
72	C₂³×C₅₃	—	2	1	no	—
73	C₂³×C₅₃	—	2	1	no	—
74	C₂³×C₅₃	—	2	2	no	—
75	C₂³×C₅₃	#41	1	2	yes	—
76	C₂³×C₅₃	#41	1	4	yes	—
77	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
78	Q₈×C₅₃	—	4	4	no	—
79	D₄×C₅₃	—	2	16	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	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	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	1	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	4	4	no	—
98	C₂×C₄×C₅₃	—	4	8	no	—
99	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
100	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
101	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
102	C₂²×C₅₃	—	2	4	no	—
103	C₂²×C₅₃	—	2	4	no	—
104	C₂²×C₅₃	—	2	2	no	—
105	C₂²×C₅₃	—	2	4	no	—
106	C₂²×C₅₃	—	2	4	no	—
107	C₂²×C₅₃	—	4	2	no	—
108	C₂²×C₅₃	—	4	2	no	—
109	C₂²×C₅₃	—	4	1	no	—
110	C₂²×C₅₃	—	4	2	no	—
111	C₂²×C₅₃	—	4	2	no	—
112	C₂²×C₅₃	—	4	2	no	—
113	C₂²×C₅₃	—	4	1	no	—
114	C₂²×C₅₃	C₂²×D₄	1	2	yes	—
115	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
116	C₂²×C₅₃	C₂×#41	1	2	yes	—
117	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
119	C₄×C₅₃	—	2	4	no	—
120	C₄×C₅₃	—	2	4	no	—
121	C₄×C₅₃	—	2	4	no	—
122	C₄×C₅₃	—	4	2	no	—
123	C₄×C₅₃	—	4	1	no	—
124	C₄×C₅₃	—	4	2	no	—
125	C₄×C₅₃	—	4	1	no	—
126	#2463	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂×C₅₃	—	4	2	no	—
128	C₂×C₅₃	—	4	4	no	—
129	C₂×C₅₃	#525	1	2	yes	—
130	C₂×C₅₃	#539	1	1	yes	characteristic
131	C₂×C₅₃	D₄²	1	1	yes	characteristic
132	C₂×C₅₃	#548	1	2	yes	—
133	C₂×C₅₃	#551	1	1	yes	characteristic
134	C₂×C₂²⋊Q₈	C₂×C₅₃	1	2	yes	complemented
135	C₂×C₄×D₄	C₂×C₅₃	1	2	yes	complemented
136	#394	C₂×C₅₃	1	2	yes	complemented
137	#393	C₂×C₅₃	1	1	yes	characteristic, complemented
138	#392	C₂×C₅₃	1	1	yes	characteristic
139	#388	C₂×C₅₃	1	2	yes	complemented
140	#387	C₂×C₅₃	1	2	yes	complemented
141	#386	C₂×C₅₃	1	2	yes	complemented
142	#385	C₂×C₅₃	1	1	yes	characteristic, complemented
143	C₅₃	#2463	1	1	yes	characteristic, direct product factor, Sylow
144	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
145	C₂²×D₄	C₂²×C₅₃	1	2	yes	—
146	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
147	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
148	C₂³×C₄	C₂²×C₅₃	1	2	yes	—
149	C₂²⋊Q₈	—	2	8	no	—
150	C₄×D₄	—	2	8	no	—
151	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
152	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
153	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
154	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
155	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
156	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
157	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
158	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
159	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
160	C₂×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
161	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
162	#95	C₂²×C₅₃	1	2	yes	—
163	#95	C₂²×C₅₃	1	2	yes	—
164	#95	C₂²×C₅₃	1	1	yes	characteristic
165	#95	C₂²×C₅₃	1	2	yes	—
166	#95	C₂²×C₅₃	1	1	yes	characteristic
167	C₂⁴	C₂³×C₅₃	1	2	yes	—
168	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	—	4	1	no	—
171	C₂×D₄	—	2	4	no	—
172	C₂×D₄	—	2	4	no	—
173	C₂×D₄	—	2	4	no	—
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	—	4	1	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
183	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
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	2	yes	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	D₄×C₅₃	1	8	yes	complemented
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₂²⋊C₄	—	2	4	no	—
194	C₂²⋊C₄	—	2	4	no	—
195	C₂²⋊C₄	—	2	4	no	—
196	C₂²⋊C₄	—	2	4	no	—
197	C₂²⋊C₄	—	4	1	no	—
198	C₄²	—	2	4	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	—	2	1	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	2	no	—
209	C₂³	#41×C₅₃	1	2	yes	—
210	C₂³	#41×C₅₃	1	4	yes	—
211	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
212	Q₈	—	4	4	no	—
213	D₄	—	2	16	no	—
214	D₄	—	4	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	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	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	8	no	—
233	C₂×C₄	C₂×D₄×C₅₃	1	4	yes	—
234	C₂×C₄	C₂×D₄×C₅₃	1	4	yes	—
235	C₂×C₄	C₂×D₄×C₅₃	1	4	yes	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	1	no	—
248	C₂²	C₂²×D₄×C₅₃	1	2	yes	—
249	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
250	C₂²	C₂×#41×C₅₃	1	2	yes	—
251	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
252	C₂²	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
253	C₄	—	2	4	no	—
254	C₄	—	2	4	no	—
255	C₄	—	2	4	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	1	no	—
258	C₄	—	4	2	no	—
259	C₄	—	4	1	no	—
260	C₂	—	4	2	no	—
261	C₂	—	4	4	no	—
262	C₂	C₅₃×#525	1	2	yes	—
263	C₂	C₅₃×#539	1	1	yes	characteristic
264	C₂	D₄²×C₅₃	1	1	yes	characteristic
265	C₂	C₅₃×#548	1	2	yes	—
266	C₂	C₅₃×#551	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₃
prime radical	C₅₃×#392	C₂
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
1. C₂×C₂²⋊Q₈×C₅₃	C₂	semidirect
2. C₂×C₄×D₄×C₅₃	C₂	semidirect
3. C₅₃×#394	C₂	semidirect
4. C₅₃×#393	C₂	semidirect
6. C₅₃×#388	C₂	semidirect
7. C₅₃×#387	C₂	semidirect
8. C₅₃×#386	C₂	semidirect
9. C₅₃×#385	C₂	semidirect
56. C₄⋊C₄×C₅₃	D₄	semidirect
126. #2463	C₅₃	direct (N × H)
134. C₂×C₂²⋊Q₈	C₂×C₅₃	semidirect
135. C₂×C₄×D₄	C₂×C₅₃	semidirect
136. #394	C₂×C₅₃	semidirect
137. #393	C₂×C₅₃	semidirect
139. #388	C₂×C₅₃	semidirect
140. #387	C₂×C₅₃	semidirect
141. #386	C₂×C₅₃	semidirect
142. #385	C₂×C₅₃	semidirect
143. C₅₃	#2463	direct (N × H)
190. C₄⋊C₄	D₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊Q₈×C₅₃	2
C₂×C₄×D₄×C₅₃	2
C₅₃×#394	2
C₅₃×#393	2
C₅₃×#392	2
C₅₃×#388	2
C₅₃×#387	2
C₅₃×#386	2
C₅₃×#385	2
#2463	53

Subgroups — C53×#2463 #20271400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2463 #20271400