Subgroups — C₅₃×#11371 #27218883

Order 13568.

886 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×C₅₃×#384	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#340	C₂	1	4	yes	maximal
3	C₂×C₅₃×#336	C₂	1	1	yes	maximal, characteristic
4	C₅₃×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
6	C₂²×C₄⋊C₄×C₅₃	—	2	2	no	—
7	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
9	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#384	—	2	2	no	—
11	C₅₃×#384	C₄	1	4	yes	complemented
12	C₂×#95×C₅₃	—	2	2	no	—
13	C₅₃×#340	—	2	8	no	—
14	C₅₃×#336	—	2	2	no	—
15	C₂³×C₄×C₅₃	—	2	2	no	—
16	C₂³×C₄×C₅₃	—	2	2	no	—
17	C₂³×C₄×C₅₃	D₄	1	2	yes	—
18	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	—	2	2	no	—
20	C₂²×C₈×C₅₃	—	2	4	no	—
21	C₂²×C₈×C₅₃	—	2	2	no	—
22	C₂²×C₈×C₅₃	—	2	4	no	—
23	C₂²×C₈×C₅₃	D₄	1	4	yes	—
24	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
25	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
31	C₂×C₄⋊C₄×C₅₃	—	4	8	no	—
32	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₅₃	Q₈	1	4	yes	—
34	C₂×C₄²×C₅₃	—	2	4	no	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₃	D₄	1	2	yes	—
37	#95×C₅₃	—	4	4	no	—
38	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	—	2	4	no	—
40	C₂²×C₄×C₅₃	—	2	4	no	—
41	C₂²×C₄×C₅₃	—	2	4	no	—
42	C₂²×C₄×C₅₃	—	2	4	no	—
43	C₂²×C₄×C₅₃	—	4	2	no	—
44	C₂²×C₄×C₅₃	—	4	2	no	—
45	C₂²×C₄×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	—	4	2	no	—
47	C₂²×C₄×C₅₃	—	4	4	no	—
48	C₂²×C₄×C₅₃	—	4	8	no	—
49	C₂²×C₄×C₅₃	—	4	4	no	—
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
51	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
52	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₅₃	#41	1	4	yes	—
58	C₂²×C₄×C₅₃	#41	1	2	yes	—
59	C₂²×C₄×C₅₃	#41	1	2	yes	—
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	—	2	8	no	—
62	C₂×C₈×C₅₃	—	4	2	no	—
63	C₂×C₈×C₅₃	—	4	2	no	—
64	C₂×C₈×C₅₃	—	4	4	no	—
65	C₄⋊C₄×C₅₃	—	2	8	no	—
66	C₄⋊C₄×C₅₃	—	2	8	no	—
67	C₄⋊C₄×C₅₃	—	4	16	no	—
68	C₄⋊C₄×C₅₃	—	4	4	no	—
69	C₄⋊C₄×C₅₃	—	4	4	no	—
70	C₄²×C₅₃	—	2	4	no	—
71	C₄²×C₅₃	—	2	4	no	—
72	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
74	C₂³×C₅₃	#123	1	2	yes	—
75	C₂³×C₅₃	#124	1	1	yes	characteristic
76	C₂³×C₅₃	#127	1	1	yes	characteristic
77	C₂³×C₅₃	C₂×SD₁₆	1	2	yes	—
78	C₂³×C₅₃	C₂×Q₁₆	1	2	yes	—
79	C₂³×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	4	no	—
82	C₂×C₄×C₅₃	—	2	8	no	—
83	C₂×C₄×C₅₃	—	2	8	no	—
84	C₂×C₄×C₅₃	—	4	4	no	—
85	C₂×C₄×C₅₃	—	4	4	no	—
86	C₂×C₄×C₅₃	—	4	8	no	—
87	C₂×C₄×C₅₃	—	4	8	no	—
88	C₂×C₄×C₅₃	—	8	4	no	—
89	C₂×C₄×C₅₃	—	8	4	no	—
90	C₂×C₄×C₅₃	—	8	2	no	—
91	C₂×C₄×C₅₃	—	8	2	no	—
92	C₂×C₄×C₅₃	—	8	4	no	—
93	C₂×C₄×C₅₃	#103	1	8	yes	—
94	C₂×C₄×C₅₃	#117	1	2	yes	—
95	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
97	C₂×C₄×C₅₃	#121	1	2	yes	—
98	C₂×C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
99	C₂×C₄×C₅₃	#123	1	4	yes	—
100	C₂×C₄×C₅₃	#123	1	2	yes	—
101	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#125	1	4	yes	—
103	C₂×C₄×C₅₃	#126	1	4	yes	—
104	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
105	C₈×C₅₃	—	4	4	no	—
106	#11371	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂²×C₅₃	#381	1	1	yes	characteristic
108	C₂²×C₅₃	#390	1	1	yes	characteristic
109	C₂²×C₅₃	#391	1	1	yes	characteristic
110	C₂²×C₅₃	#399	1	2	yes	—
111	C₂²×C₅₃	#400	1	2	yes	—
112	C₂²×C₅₃	C₂×#103	1	2	yes	—
113	C₂²×C₅₃	#418	1	2	yes	—
114	C₂²×C₅₃	#476	1	4	yes	—
115	C₂²×C₅₃	#477	1	4	yes	—
116	C₂²×C₅₃	#481	1	4	yes	—
117	C₂²×C₅₃	#484	1	4	yes	—
118	C₂²×C₅₃	#487	1	2	yes	—
119	C₂²×C₅₃	#490	1	2	yes	—
120	C₂²×C₅₃	#492	1	1	yes	characteristic
121	C₂²×C₅₃	#494	1	1	yes	characteristic
122	C₂²×C₅₃	#496	1	2	yes	—
123	C₄×C₅₃	—	2	8	no	—
124	C₄×C₅₃	—	8	4	no	—
125	C₄×C₅₃	—	8	4	no	—
126	C₄×C₅₃	#387	1	4	yes	—
127	C₄×C₅₃	#397	1	2	yes	—
128	C₄×C₅₃	#398	1	2	yes	—
129	C₂×#384	C₂×C₅₃	1	1	yes	characteristic
130	C₂×#340	C₂×C₅₃	1	4	yes	—
131	C₂×#336	C₂×C₅₃	1	1	yes	characteristic
132	C₄⋊#95	C₂×C₅₃	1	1	yes	characteristic
133	C₂×C₅₃	#1445	1	1	yes	characteristic
134	C₂×C₅₃	#1824	1	2	yes	—
135	C₂×C₅₃	#1956	1	1	yes	characteristic
136	C₂×C₅₃	#1961	1	1	yes	characteristic
137	C₂×C₅₃	#1987	1	2	yes	—
138	C₂×C₅₃	#2094	1	2	yes	—
139	C₂×C₅₃	#2095	1	2	yes	—
140	C₂×C₅₃	#2100	1	4	yes	—
141	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
142	C₂²×C₄⋊C₄	—	2	2	no	—
143	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
144	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
145	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
146	#384	—	2	2	no	—
147	#384	C₄×C₅₃	1	4	yes	complemented
148	C₂×#95	—	2	2	no	—
149	#340	—	2	8	no	—
150	#336	—	2	2	no	—
151	C₅₃	#11371	1	1	yes	characteristic, direct product factor, Sylow
152	C₂³×C₄	—	2	2	no	—
153	C₂³×C₄	—	2	2	no	—
154	C₂³×C₄	D₄×C₅₃	1	2	yes	—
155	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	2	4	no	—
159	C₂²×C₈	—	2	4	no	—
160	C₂²×C₈	D₄×C₅₃	1	4	yes	—
161	C₂×C₄⋊C₄	—	2	4	no	—
162	C₂×C₄⋊C₄	—	2	4	no	—
163	C₂×C₄⋊C₄	—	2	4	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	2	8	no	—
168	C₂×C₄⋊C₄	—	4	8	no	—
169	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
170	C₂×C₄⋊C₄	Q₈×C₅₃	1	4	yes	—
171	C₂×C₄²	—	2	4	no	—
172	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
173	C₂×C₄²	D₄×C₅₃	1	2	yes	—
174	#95	—	4	4	no	—
175	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	4	4	no	—
185	C₂²×C₄	—	4	8	no	—
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
188	C₂²×C₄	SD₁₆×C₅₃	1	4	yes	—
189	C₂²×C₄	Q₁₆×C₅₃	1	4	yes	—
190	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
192	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
194	C₂²×C₄	#41×C₅₃	1	4	yes	—
195	C₂²×C₄	#41×C₅₃	1	2	yes	—
196	C₂²×C₄	#41×C₅₃	1	2	yes	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	8	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	4	no	—
202	C₄⋊C₄	—	2	8	no	—
203	C₄⋊C₄	—	2	8	no	—
204	C₄⋊C₄	—	4	16	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄²	—	2	4	no	—
208	C₄²	—	2	4	no	—
209	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
210	C₂³	C₂²⋊Q₈×C₅₃	1	2	yes	—
211	C₂³	#123×C₅₃	1	2	yes	—
212	C₂³	#124×C₅₃	1	1	yes	characteristic
213	C₂³	#127×C₅₃	1	1	yes	characteristic
214	C₂³	C₂×SD₁₆×C₅₃	1	2	yes	—
215	C₂³	C₂×Q₁₆×C₅₃	1	2	yes	—
216	C₂³	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	2	8	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	8	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	—	8	4	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	—	8	4	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	4	no	—
230	C₂×C₄	#103×C₅₃	1	8	yes	—
231	C₂×C₄	#117×C₅₃	1	2	yes	—
232	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
233	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
234	C₂×C₄	#121×C₅₃	1	2	yes	—
235	C₂×C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
236	C₂×C₄	#123×C₅₃	1	4	yes	—
237	C₂×C₄	#123×C₅₃	1	2	yes	—
238	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
239	C₂×C₄	#125×C₅₃	1	4	yes	—
240	C₂×C₄	#126×C₅₃	1	4	yes	—
241	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
242	C₈	—	4	4	no	—
243	C₂²	C₅₃×#381	1	1	yes	characteristic
244	C₂²	C₅₃×#390	1	1	yes	characteristic
245	C₂²	C₅₃×#391	1	1	yes	characteristic
246	C₂²	C₅₃×#399	1	2	yes	—
247	C₂²	C₅₃×#400	1	2	yes	—
248	C₂²	C₂×#103×C₅₃	1	2	yes	—
249	C₂²	C₅₃×#418	1	2	yes	—
250	C₂²	C₅₃×#476	1	4	yes	—
251	C₂²	C₅₃×#477	1	4	yes	—
252	C₂²	C₅₃×#481	1	4	yes	—
253	C₂²	C₅₃×#484	1	4	yes	—
254	C₂²	C₅₃×#487	1	2	yes	—
255	C₂²	C₅₃×#490	1	2	yes	—
256	C₂²	C₅₃×#492	1	1	yes	characteristic
257	C₂²	C₅₃×#494	1	1	yes	characteristic
258	C₂²	C₅₃×#496	1	2	yes	—
259	C₄	—	2	8	no	—
260	C₄	—	8	4	no	—
261	C₄	—	8	4	no	—
262	C₄	C₅₃×#387	1	4	yes	—
263	C₄	C₅₃×#397	1	2	yes	—
264	C₄	C₅₃×#398	1	2	yes	—
265	C₂	C₅₃×#1445	1	1	yes	characteristic
266	C₂	C₅₃×#1824	1	2	yes	—
267	C₂	C₅₃×#1956	1	1	yes	characteristic
268	C₂	C₅₃×#1961	1	1	yes	characteristic
269	C₂	C₅₃×#1987	1	2	yes	—
270	C₂	C₅₃×#2094	1	2	yes	—
271	C₂	C₅₃×#2095	1	2	yes	—
272	C₂	C₅₃×#2100	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₅₃×#384	C₄	semidirect
106. #11371	C₅₃	direct (N × H)
147. #384	C₄×C₅₃	semidirect
151. C₅₃	#11371	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×C₅₃×#384	2
C₂×C₅₃×#340	2
C₂×C₅₃×#336	2
C₅₃×C₄⋊#95	2
#11371	53

Subgroups — C53×#11371 #27218883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11371 #27218883