Subgroups — C₅₃×#2638 #20271575

Order 6784.

520 conjugacy classes of subgroups.

The subgroup lattice has 278 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	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#398	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#397	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×#387	C₂	1	2	yes	maximal, complemented
10	C₅₃×#385	C₂	1	2	yes	maximal, complemented
11	C₅₃×#381	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	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²⋊Q₈×C₅₃	—	2	4	no	—
17	C₄×D₄×C₅₃	—	2	4	no	—
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
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	1	yes	characteristic
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	2	yes	—
27	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
30	#95×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	—	2	2	no	—
43	C₂×D₄×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	—	2	2	no	—
45	C₂²×C₄×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	—	4	2	no	—
47	C₂²×C₄×C₅₃	D₄	1	4	yes	—
48	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	—	2	2	no	—
58	C₄⋊C₄×C₅₃	—	2	4	no	—
59	C₄⋊C₄×C₅₃	—	2	2	no	—
60	C₄⋊C₄×C₅₃	—	2	4	no	—
61	C₄⋊C₄×C₅₃	—	4	2	no	—
62	C₂²⋊C₄×C₅₃	—	2	2	no	—
63	C₂²⋊C₄×C₅₃	—	2	4	no	—
64	C₂²⋊C₄×C₅₃	—	2	4	no	—
65	C₂²⋊C₄×C₅₃	—	2	2	no	—
66	C₂²⋊C₄×C₅₃	—	4	2	no	—
67	C₂²⋊C₄×C₅₃	—	4	2	no	—
68	C₄²×C₅₃	—	2	2	no	—
69	C₂³×C₅₃	—	2	1	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	2	no	—
76	C₂³×C₅₃	—	2	2	no	—
77	C₂³×C₅₃	C₂×D₄	1	2	yes	—
78	C₂³×C₅₃	#41	1	2	yes	—
79	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
80	Q₈×C₅₃	—	4	4	no	—
81	D₄×C₅₃	—	2	8	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	2	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	2	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	4	4	no	—
103	C₂×C₄×C₅₃	—	4	4	no	—
104	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₅₃	#41	1	2	yes	—
107	C₂²×C₅₃	—	2	2	no	—
108	C₂²×C₅₃	—	2	2	no	—
109	C₂²×C₅₃	—	2	2	no	—
110	C₂²×C₅₃	—	2	2	no	—
111	C₂²×C₅₃	—	2	4	no	—
112	C₂²×C₅₃	—	4	1	no	—
113	C₂²×C₅₃	—	4	1	no	—
114	C₂²×C₅₃	—	4	2	no	—
115	C₂²×C₅₃	—	4	2	no	—
116	C₂²×C₅₃	—	4	2	no	—
117	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
118	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
119	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
120	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
121	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
122	C₄×C₅₃	—	2	2	no	—
123	C₄×C₅₃	—	2	2	no	—
124	C₄×C₅₃	—	2	2	no	—
125	C₄×C₅₃	—	4	2	no	—
126	C₄×C₅₃	—	4	1	no	—
127	C₄×C₅₃	—	4	2	no	—
128	C₄×C₅₃	—	4	1	no	—
129	C₄×C₅₃	—	4	2	no	—
130	C₄×C₅₃	—	4	2	no	—
131	#2638	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂×C₅₃	—	4	2	no	—
133	C₂×C₅₃	—	4	2	no	—
134	C₂×C₅₃	#525	1	1	yes	characteristic
135	C₂×C₅₃	#534	1	1	yes	characteristic
136	C₂×C₅₃	#536	1	1	yes	characteristic
137	C₂×C₅₃	#539	1	2	yes	—
138	C₂×C₅₃	#542	1	2	yes	—
139	C₂×C₂²⋊Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
140	C₂×C₄×D₄	C₂×C₅₃	1	1	yes	characteristic
141	#398	C₂×C₅₃	1	1	yes	characteristic, complemented
142	#397	C₂×C₅₃	1	1	yes	characteristic, complemented
143	#397	C₂×C₅₃	1	2	yes	complemented
144	#395	C₂×C₅₃	1	1	yes	characteristic, complemented
145	#394	C₂×C₅₃	1	2	yes	—
146	#393	C₂×C₅₃	1	1	yes	characteristic, complemented
147	#387	C₂×C₅₃	1	2	yes	complemented
148	#385	C₂×C₅₃	1	2	yes	complemented
149	#381	C₂×C₅₃	1	1	yes	characteristic, complemented
150	C₅₃	#2638	1	1	yes	characteristic, direct product factor, Sylow
151	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
152	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
153	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
154	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
155	C₂²⋊Q₈	—	2	4	no	—
156	C₄×D₄	—	2	4	no	—
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	1	yes	characteristic
160	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
161	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
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×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
168	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
169	#95	C₂²×C₅₃	1	2	yes	—
170	#95	C₂²×C₅₃	1	1	yes	characteristic
171	#95	C₂²×C₅₃	1	1	yes	characteristic
172	#95	C₂²×C₅₃	1	2	yes	—
173	#95	C₂²×C₅₃	1	2	yes	—
174	#95	C₂²×C₅₃	1	2	yes	—
175	#95	C₂²×C₅₃	1	2	yes	—
176	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
177	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	D₄×C₅₃	1	4	yes	—
187	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
188	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
189	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
190	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
191	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
192	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
193	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
194	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	4	2	no	—
206	C₂²⋊C₄	—	4	2	no	—
207	C₄²	—	2	2	no	—
208	C₂³	—	2	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	C₂×D₄×C₅₃	1	2	yes	—
217	C₂³	#41×C₅₃	1	2	yes	—
218	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
219	Q₈	—	4	4	no	—
220	D₄	—	2	8	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	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	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
244	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
245	C₂×C₄	#41×C₅₃	1	2	yes	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	4	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
257	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
258	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
259	C₂²	H(5,𝔽₂)×C₅₃	1	2	yes	—
260	C₂²	2₋¹⁺⁴×C₅₃	1	2	yes	—
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	2	no	—
265	C₄	—	4	1	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	1	no	—
268	C₄	—	4	2	no	—
269	C₄	—	4	2	no	—
270	C₂	—	4	2	no	—
271	C₂	—	4	2	no	—
272	C₂	C₅₃×#525	1	1	yes	characteristic
273	C₂	C₅₃×#534	1	1	yes	characteristic
274	C₂	C₅₃×#536	1	1	yes	characteristic
275	C₂	C₅₃×#539	1	2	yes	—
276	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
1. C₂×C₂²⋊Q₈×C₅₃	C₂	semidirect
3. C₅₃×#398	C₂	semidirect
4. C₅₃×#397	C₂	semidirect
5. C₅₃×#397	C₂	semidirect
6. C₅₃×#395	C₂	semidirect
8. C₅₃×#393	C₂	semidirect
9. C₅₃×#387	C₂	semidirect
10. C₅₃×#385	C₂	semidirect
11. C₅₃×#381	C₂	semidirect
131. #2638	C₅₃	direct (N × H)
139. C₂×C₂²⋊Q₈	C₂×C₅₃	semidirect
141. #398	C₂×C₅₃	semidirect
142. #397	C₂×C₅₃	semidirect
143. #397	C₂×C₅₃	semidirect
144. #395	C₂×C₅₃	semidirect
146. #393	C₂×C₅₃	semidirect
147. #387	C₂×C₅₃	semidirect
148. #385	C₂×C₅₃	semidirect
149. #381	C₂×C₅₃	semidirect
150. C₅₃	#2638	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×C₂²⋊Q₈×C₅₃	2
C₂×C₄×D₄×C₅₃	2
C₅₃×#398	2
C₅₃×#397	2
C₅₃×#395	2
C₅₃×#394	2
C₅₃×#393	2
C₅₃×#387	2
C₅₃×#385	2
C₅₃×#381	2
#2638	53

Subgroups — C53×#2638 #20271575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2638 #20271575