Subgroups — C₅₃×#21353 #27228865

Order 13568.

866 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
2	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2283	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1940	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1850	C₂	1	4	yes	maximal
6	C₅₃×#1768	C₂	1	4	yes	maximal, complemented
7	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#431	—	2	4	no	—
10	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	8	yes	—
12	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×#97×C₅₃	—	2	4	no	—
14	C₂×#97×C₅₃	C₂²	1	2	yes	—
15	C₂×#97×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#383	C₄	1	2	yes	—
18	C₅₃×#382	C₄	1	2	yes	—
19	C₅₃×#381	C₄	1	2	yes	—
20	C₅₃×#380	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₅₃×#376	C₂²	1	2	yes	—
24	C₅₃×#336	C₂²	1	8	yes	—
25	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	—	2	8	no	—
29	C₂²×C₈×C₅₃	—	2	4	no	—
30	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
31	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
32	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
33	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	D₄	1	4	yes	—
37	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
38	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	—	2	16	no	—
42	C₄⋊C₈×C₅₃	—	2	4	no	—
43	C₂²⋊C₈×C₅₃	—	2	4	no	—
44	#97×C₅₃	—	2	4	no	—
45	#97×C₅₃	—	2	4	no	—
46	C₄×C₈×C₅₃	—	2	4	no	—
47	#95×C₅₃	C₂×C₄	1	4	yes	—
48	#95×C₅₃	C₂×C₄	1	4	yes	—
49	#95×C₅₃	C₂×C₄	1	4	yes	—
50	#95×C₅₃	C₂×C₄	1	4	yes	—
51	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	—	2	4	no	—
53	C₂²×C₄×C₅₃	—	2	4	no	—
54	C₂²×C₄×C₅₃	—	4	2	no	—
55	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₅₃	—	2	16	no	—
67	C₂×C₈×C₅₃	—	2	4	no	—
68	C₂×C₈×C₅₃	—	2	4	no	—
69	C₂×C₈×C₅₃	—	2	4	no	—
70	C₂×C₈×C₅₃	—	2	8	no	—
71	C₂×C₈×C₅₃	#41	1	8	yes	—
72	C₄⋊C₄×C₅₃	—	4	4	no	—
73	C₂²⋊C₄×C₅₃	—	2	8	no	—
74	C₄²×C₅₃	—	2	8	no	—
75	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
76	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
77	C₄²×C₅₃	C₂×D₄	1	2	yes	—
78	C₄²×C₅₃	C₂×D₄	1	2	yes	—
79	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
80	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
81	C₂³×C₅₃	—	2	2	no	—
82	C₂³×C₅₃	—	2	2	no	—
83	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
84	C₂³×C₅₃	#131	1	4	yes	—
85	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
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	8	no	—
92	C₂×C₄×C₅₃	—	2	4	no	—
93	C₂×C₄×C₅₃	—	2	4	no	—
94	C₂×C₄×C₅₃	—	2	8	no	—
95	C₂×C₄×C₅₃	—	2	8	no	—
96	C₂×C₄×C₅₃	—	2	8	no	—
97	C₂×C₄×C₅₃	—	4	4	no	—
98	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
106	C₈×C₅₃	—	2	8	no	—
107	C₈×C₅₃	—	4	4	no	—
108	#21353	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂²×C₅₃	—	2	2	no	—
110	C₂²×C₅₃	—	2	8	no	—
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	#424	1	4	yes	—
113	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₅₃	C₄×#41	1	2	yes	—
115	C₂²×C₅₃	C₂×#131	1	2	yes	—
116	C₂²×C₅₃	#568	1	2	yes	—
117	C₄×C₅₃	—	2	2	no	—
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	2	2	no	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	4	8	no	—
123	C₄×C₅₃	#525	1	2	yes	—
124	C₄×C₅₃	#533	1	2	yes	—
125	C₂×#431	C₂×C₅₃	1	1	yes	characteristic
126	C₂²×#97	C₂×C₅₃	1	1	yes	characteristic
127	#2283	C₂×C₅₃	1	1	yes	characteristic
128	#1940	C₂×C₅₃	1	4	yes	complemented
129	#1850	C₂×C₅₃	1	4	yes	—
130	#1768	C₂×C₅₃	1	4	yes	complemented
131	C₂×C₅₃	—	2	4	no	—
132	C₂×C₅₃	#2296	1	1	yes	characteristic
133	C₂×C₅₃	C₂×#424	1	1	yes	characteristic
134	C₂×C₅₃	#2905	1	1	yes	characteristic
135	C₂×C₅₃	#2968	1	4	yes	—
136	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
137	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
138	#431	—	2	4	no	—
139	C₂×C₄⋊C₈	C₂²×C₅₃	1	8	yes	—
140	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
141	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
142	C₂×#97	—	2	4	no	—
143	C₂×#97	C₂²×C₅₃	1	2	yes	—
144	C₂×#97	C₂²×C₅₃	1	2	yes	—
145	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
146	#383	C₄×C₅₃	1	2	yes	—
147	#382	C₄×C₅₃	1	2	yes	—
148	#381	C₄×C₅₃	1	2	yes	—
149	#380	C₄×C₅₃	1	2	yes	—
150	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
151	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
152	#376	C₂²×C₅₃	1	2	yes	—
153	#336	C₂²×C₅₃	1	8	yes	—
154	C₅₃	#21353	1	1	yes	characteristic, direct product factor, Sylow
155	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
156	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
157	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
158	C₂²×C₈	—	2	4	no	—
159	C₂²×C₈	—	2	8	no	—
160	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
161	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
162	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
163	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
164	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
165	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
166	C₂×C₄²	D₄×C₅₃	1	4	yes	—
167	C₂×C₄²	Q₈×C₅₃	1	4	yes	—
168	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
169	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
170	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
171	C₄⋊C₈	—	2	16	no	—
172	C₄⋊C₈	—	2	4	no	—
173	C₂²⋊C₈	—	2	4	no	—
174	#97	—	2	4	no	—
175	#97	—	2	4	no	—
176	C₄×C₈	—	2	4	no	—
177	#95	C₂×C₄×C₅₃	1	4	yes	—
178	#95	C₂×C₄×C₅₃	1	4	yes	—
179	#95	C₂×C₄×C₅₃	1	4	yes	—
180	#95	C₂×C₄×C₅₃	1	4	yes	—
181	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
186	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
187	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
188	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
189	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
190	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
192	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
194	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
195	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
196	C₂×C₈	—	2	16	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	8	no	—
201	C₂×C₈	#41×C₅₃	1	8	yes	—
202	C₄⋊C₄	—	4	4	no	—
203	C₂²⋊C₄	—	2	8	no	—
204	C₄²	—	2	8	no	—
205	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
206	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
207	C₄²	C₂×D₄×C₅₃	1	2	yes	—
208	C₄²	C₂×D₄×C₅₃	1	2	yes	—
209	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
210	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
214	C₂³	#131×C₅₃	1	4	yes	—
215	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
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	8	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
230	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
231	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
232	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
233	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
234	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
235	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
236	C₈	—	2	8	no	—
237	C₈	—	4	4	no	—
238	C₂²	—	2	8	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	C₅₃×#424	1	4	yes	—
242	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
243	C₂²	C₄×#41×C₅₃	1	2	yes	—
244	C₂²	C₂×#131×C₅₃	1	2	yes	—
245	C₂²	C₅₃×#568	1	2	yes	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	4	8	no	—
252	C₄	C₅₃×#525	1	2	yes	—
253	C₄	C₅₃×#533	1	2	yes	—
254	C₂	—	2	4	no	—
255	C₂	C₅₃×#2296	1	1	yes	characteristic
256	C₂	C₂×C₅₃×#424	1	1	yes	characteristic
257	C₂	C₅₃×#2905	1	1	yes	characteristic
258	C₂	C₅₃×#2968	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1940	C₂	semidirect
6. C₅₃×#1768	C₂	semidirect
108. #21353	C₅₃	direct (N × H)
128. #1940	C₂×C₅₃	semidirect
130. #1768	C₂×C₅₃	semidirect
154. C₅₃	#21353	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×#431	2
C₂²×#97×C₅₃	2
C₅₃×#2283	2
C₅₃×#1940	2
C₅₃×#1850	2
C₅₃×#1768	2
#21353	53

Subgroups — C53×#21353 #27228865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21353 #27228865