Subgroups — C₅₃×#19663 #27227175

Order 13568.

690 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3230	C₂	1	2	yes	maximal
2	C₅₃×#3209	C₂	1	2	yes	maximal
3	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#498	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1723	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1707	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#533	C₂²	1	1	yes	characteristic
9	C₅₃×#533	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#500	C₂²	1	4	yes	—
12	C₅₃×#500	C₂²	1	2	yes	—
13	C₅₃×#499	—	2	2	no	—
14	C₅₃×#498	—	2	2	no	—
15	C₅₃×#498	C₂²	1	2	yes	—
16	C₅₃×#498	C₂²	1	2	yes	—
17	C₅₃×#498	C₂²	1	2	yes	—
18	C₅₃×#494	C₂²	1	4	yes	—
19	C₅₃×#487	C₂²	1	4	yes	—
20	C₅₃×#487	C₂²	1	4	yes	—
21	C₅₃×#484	—	2	4	no	—
22	C₅₃×#477	—	2	4	no	—
23	C₅₃×#462	—	2	4	no	—
24	C₅₃×#451	—	2	4	no	—
25	C₅₃×#431	C₂²	1	2	yes	—
26	C₅₃×#427	—	2	2	no	—
27	C₂×#106×C₅₃	—	2	2	no	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#322	C₂²	1	2	yes	—
30	C₈²×C₅₃	C₂²	1	2	yes	—
31	C₂²×Q₈×C₅₃	—	2	2	no	—
32	C₂×Q₁₆×C₅₃	—	2	8	no	—
33	C₂²×C₈×C₅₃	D₄	1	2	yes	—
34	C₄⋊Q₈×C₅₃	—	2	2	no	—
35	C₄⋊Q₈×C₅₃	—	2	2	no	—
36	C₄⋊Q₈×C₅₃	—	2	2	no	—
37	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
38	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
39	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
40	#125×C₅₃	—	2	1	no	—
41	#125×C₅₃	—	2	1	no	—
42	C₂²⋊Q₈×C₅₃	—	2	2	no	—
43	C₂²⋊Q₈×C₅₃	—	2	2	no	—
44	C₄×Q₈×C₅₃	—	2	2	no	—
45	C₄×Q₈×C₅₃	—	2	2	no	—
46	#117×C₅₃	—	2	1	no	—
47	#117×C₅₃	—	2	1	no	—
48	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
50	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
51	#107×C₅₃	—	2	4	no	—
52	#107×C₅₃	—	2	8	no	—
53	#106×C₅₃	—	2	4	no	—
54	#106×C₅₃	—	2	4	no	—
55	#106×C₅₃	—	2	4	no	—
56	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
57	#103×C₅₃	—	2	8	no	—
58	#103×C₅₃	—	2	8	no	—
59	C₂²⋊C₈×C₅₃	D₄	1	4	yes	—
60	C₄×C₈×C₅₃	D₄	1	2	yes	—
61	C₄×C₈×C₅₃	C₂³	1	4	yes	—
62	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×Q₈×C₅₃	—	2	2	no	—
65	C₂×Q₈×C₅₃	—	2	4	no	—
66	C₂×Q₈×C₅₃	—	2	2	no	—
67	C₂×Q₈×C₅₃	—	4	2	no	—
68	C₂×Q₈×C₅₃	—	8	4	no	—
69	C₂²×C₄×C₅₃	—	4	2	no	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	Q₁₆×C₅₃	—	8	8	no	—
73	C₂×C₈×C₅₃	Q₁₆	1	8	yes	complemented
74	C₂×C₈×C₅₃	C₂×D₄	1	8	yes	—
75	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
77	C₄⋊C₄×C₅₃	—	2	2	no	—
78	C₄⋊C₄×C₅₃	—	2	4	no	—
79	C₄⋊C₄×C₅₃	—	2	4	no	—
80	C₄⋊C₄×C₅₃	—	2	2	no	—
81	C₄⋊C₄×C₅₃	—	2	2	no	—
82	C₄⋊C₄×C₅₃	—	2	2	no	—
83	C₄⋊C₄×C₅₃	—	4	2	no	—
84	C₄⋊C₄×C₅₃	—	4	2	no	—
85	C₄⋊C₄×C₅₃	—	4	2	no	—
86	C₄⋊C₄×C₅₃	—	4	2	no	—
87	C₂²⋊C₄×C₅₃	—	4	1	no	—
88	C₂²⋊C₄×C₅₃	—	4	1	no	—
89	C₄²×C₅₃	—	4	1	no	—
90	C₄²×C₅₃	—	4	1	no	—
91	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₅₃	C₂×D₄	1	2	yes	—
93	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
94	C₂³×C₅₃	#127	1	1	yes	characteristic
95	Q₈×C₅₃	—	4	4	no	—
96	Q₈×C₅₃	—	4	8	no	—
97	Q₈×C₅₃	—	4	4	no	—
98	Q₈×C₅₃	—	8	4	no	—
99	C₂×C₄×C₅₃	—	2	4	no	—
100	C₂×C₄×C₅₃	—	4	4	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	8	4	no	—
104	C₂×C₄×C₅₃	#127	1	2	yes	—
105	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
107	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
109	C₈×C₅₃	—	2	8	no	—
110	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
111	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
112	#19663	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₅₃	—	2	2	no	—
114	C₂²×C₅₃	#494	1	2	yes	—
115	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
116	C₄×C₅₃	—	2	2	no	—
117	C₄×C₅₃	—	8	4	no	—
118	C₄×C₅₃	—	8	2	no	—
119	C₄×C₅₃	—	8	2	no	—
120	C₄×C₅₃	#494	1	2	yes	—
121	C₄×C₅₃	C₂²×Q₁₆	1	2	yes	—
122	C₄×C₅₃	#577	1	4	yes	—
123	#3230	C₂×C₅₃	1	2	yes	—
124	#3209	C₂×C₅₃	1	2	yes	—
125	#3160	C₂×C₅₃	1	1	yes	characteristic
126	C₂×#498	C₂×C₅₃	1	1	yes	characteristic
127	#1723	C₂×C₅₃	1	4	yes	complemented
128	#1707	C₂×C₅₃	1	4	yes	complemented
129	#1450	C₂×C₅₃	1	1	yes	characteristic
130	C₂×C₅₃	—	2	2	no	—
131	C₂×C₅₃	C₂×#494	1	1	yes	characteristic
132	C₂×C₅₃	#3154	1	2	yes	—
133	#533	C₂²×C₅₃	1	1	yes	characteristic
134	#533	C₂²×C₅₃	1	1	yes	characteristic
135	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
136	#500	C₂²×C₅₃	1	2	yes	—
137	#500	C₂²×C₅₃	1	4	yes	—
138	#499	—	2	2	no	—
139	#498	—	2	2	no	—
140	#498	C₂²×C₅₃	1	2	yes	—
141	#498	C₂²×C₅₃	1	2	yes	—
142	#498	C₂²×C₅₃	1	2	yes	—
143	#494	C₂²×C₅₃	1	4	yes	—
144	#487	C₂²×C₅₃	1	4	yes	—
145	#487	C₂²×C₅₃	1	4	yes	—
146	#484	—	2	4	no	—
147	#477	—	2	4	no	—
148	#462	—	2	4	no	—
149	#451	—	2	4	no	—
150	#431	C₂²×C₅₃	1	2	yes	—
151	#427	—	2	2	no	—
152	C₂×#106	—	2	2	no	—
153	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
154	#322	C₂²×C₅₃	1	2	yes	—
155	C₈²	C₂²×C₅₃	1	2	yes	—
156	C₅₃	#19663	1	1	yes	characteristic, direct product factor, Sylow
157	C₂²×Q₈	—	2	2	no	—
158	C₂×Q₁₆	—	2	8	no	—
159	C₂²×C₈	D₄×C₅₃	1	2	yes	—
160	C₄⋊Q₈	—	2	2	no	—
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
164	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
165	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
166	#125	—	2	1	no	—
167	#125	—	2	1	no	—
168	C₂²⋊Q₈	—	2	2	no	—
169	C₂²⋊Q₈	—	2	2	no	—
170	C₄×Q₈	—	2	2	no	—
171	C₄×Q₈	—	2	2	no	—
172	#117	—	2	1	no	—
173	#117	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
177	#107	—	2	4	no	—
178	#107	—	2	8	no	—
179	#106	—	2	4	no	—
180	#106	—	2	4	no	—
181	#106	—	2	4	no	—
182	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
183	#103	—	2	8	no	—
184	#103	—	2	8	no	—
185	C₂²⋊C₈	D₄×C₅₃	1	4	yes	—
186	C₄×C₈	D₄×C₅₃	1	2	yes	—
187	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
188	C₄×C₈	C₂³×C₅₃	1	4	yes	—
189	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂×Q₈	—	8	4	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
198	Q₁₆	—	8	8	no	—
199	C₂×C₈	Q₁₆×C₅₃	1	8	yes	complemented
200	C₂×C₈	C₂×D₄×C₅₃	1	8	yes	—
201	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
202	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₂²⋊C₄	—	4	1	no	—
214	C₂²⋊C₄	—	4	1	no	—
215	C₄²	—	4	1	no	—
216	C₄²	—	4	1	no	—
217	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
218	C₄²	C₂×D₄×C₅₃	1	2	yes	—
219	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
220	C₂³	#127×C₅₃	1	1	yes	characteristic
221	Q₈	—	4	4	no	—
222	Q₈	—	4	8	no	—
223	Q₈	—	4	4	no	—
224	Q₈	—	8	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	8	4	no	—
230	C₂×C₄	#127×C₅₃	1	2	yes	—
231	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
232	C₂×C₄	C₂×Q₁₆×C₅₃	1	4	yes	—
233	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
235	C₈	—	2	8	no	—
236	C₈	C₂×Q₁₆×C₅₃	1	4	yes	—
237	C₈	C₂×Q₁₆×C₅₃	1	4	yes	—
238	C₂²	—	2	2	no	—
239	C₂²	C₅₃×#494	1	2	yes	—
240	C₂²	C₂×#127×C₅₃	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	8	4	no	—
243	C₄	—	8	2	no	—
244	C₄	—	8	2	no	—
245	C₄	C₅₃×#494	1	2	yes	—
246	C₄	C₂²×Q₁₆×C₅₃	1	2	yes	—
247	C₄	C₅₃×#577	1	4	yes	—
248	C₂	—	2	2	no	—
249	C₂	C₂×C₅₃×#494	1	1	yes	characteristic
250	C₂	C₅₃×#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₂³×C₅₃	#127
LCS	G ▷ C₄² ▷ C₂² ▷ 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
5. C₅₃×#1723	C₂	semidirect
6. C₅₃×#1707	C₂	semidirect
73. C₂×C₈×C₅₃	Q₁₆	semidirect
112. #19663	C₅₃	direct (N × H)
127. #1723	C₂×C₅₃	semidirect
128. #1707	C₂×C₅₃	semidirect
156. C₅₃	#19663	direct (N × H)
199. C₂×C₈	Q₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3230	2
C₅₃×#3209	2
C₅₃×#3160	2
C₂×C₅₃×#498	2
C₅₃×#1723	2
C₅₃×#1707	2
C₅₃×#1450	2
#19663	53

Subgroups — C53×#19663 #27227175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19663 #27227175