Subgroups — C₅₃×#10701 #27218213

Order 13568.

998 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#376	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#339	C₂	1	4	yes	maximal
5	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
6	C₂×#117×C₅₃	C₂²	1	2	yes	—
7	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₄×M₁₆×C₅₃	—	2	2	no	—
10	C₄×M₁₆×C₅₃	C₄	1	4	yes	complemented
11	C₂×#97×C₅₃	—	2	1	no	—
12	C₂×C₄×C₈×C₅₃	—	2	1	no	—
13	C₄×C₄⋊C₄×C₅₃	C₄	1	4	yes	complemented
14	C₄×C₂²⋊C₄×C₅₃	—	2	2	no	—
15	C₅₃×#376	—	2	2	no	—
16	C₅₃×#376	—	2	1	no	—
17	C₅₃×#376	C₄	1	4	yes	complemented
18	C₂×#95×C₅₃	—	2	2	no	—
19	C₄³×C₅₃	—	2	1	no	—
20	C₅₃×#339	C₄	1	16	yes	complemented
21	C₂³×C₄×C₅₃	—	2	2	no	—
22	C₂³×C₄×C₅₃	D₄	1	2	yes	—
23	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₅₃	—	2	2	no	—
25	C₂×M₁₆×C₅₃	—	2	4	no	—
26	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
27	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₅₃	—	2	2	no	—
29	#117×C₅₃	—	2	4	no	—
30	#117×C₅₃	C₂×C₄	1	8	yes	—
31	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	—	2	4	no	—
34	C₂×C₄²×C₅₃	—	2	2	no	—
35	C₂×C₄²×C₅₃	—	2	2	no	—
36	C₂×C₄²×C₅₃	—	2	4	no	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅₃	D₄	1	2	yes	—
43	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
46	#97×C₅₃	—	2	4	no	—
47	C₄×C₈×C₅₃	—	2	4	no	—
48	#95×C₅₃	—	2	8	no	—
49	#95×C₅₃	—	2	4	no	—
50	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	—	2	4	no	—
52	C₂²×C₄×C₅₃	—	2	4	no	—
53	C₂²×C₄×C₅₃	—	2	8	no	—
54	C₂²×C₄×C₅₃	—	4	4	no	—
55	C₂²×C₄×C₅₃	—	4	8	no	—
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
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	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	#41	1	4	yes	—
68	M₁₆×C₅₃	—	2	4	no	—
69	M₁₆×C₅₃	—	2	8	no	—
70	M₁₆×C₅₃	C₄²	1	8	yes	complemented
71	C₂×C₈×C₅₃	—	2	4	no	—
72	C₂×C₈×C₅₃	—	2	8	no	—
73	C₄⋊C₄×C₅₃	—	2	4	no	—
74	C₄⋊C₄×C₅₃	C₄²	1	8	yes	—
75	C₂²⋊C₄×C₅₃	—	2	8	no	—
76	C₄²×C₅₃	—	2	4	no	—
77	C₄²×C₅₃	—	2	8	no	—
78	C₄²×C₅₃	—	2	8	no	—
79	C₄²×C₅₃	—	4	8	no	—
80	C₄²×C₅₃	C₄²	1	4	yes	—
81	C₄²×C₅₃	C₄²	1	8	yes	complemented
82	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
85	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
86	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
87	C₂³×C₅₃	—	2	4	no	—
88	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂³×C₅₃	C₄×D₄	1	4	yes	—
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	—	4	4	no	—
97	C₂×C₄×C₅₃	—	4	4	no	—
98	C₂×C₄×C₅₃	—	4	8	no	—
99	C₂×C₄×C₅₃	—	4	16	no	—
100	C₂×C₄×C₅₃	—	4	8	no	—
101	C₂×C₄×C₅₃	#95	1	4	yes	—
102	C₂×C₄×C₅₃	#95	1	4	yes	—
103	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#117	1	2	yes	—
111	C₂×C₄×C₅₃	#117	1	2	yes	—
112	C₂×C₄×C₅₃	#117	1	2	yes	—
113	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
114	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
115	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
116	C₈×C₅₃	—	2	8	no	—
117	#10701	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	—	2	8	no	—
121	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
122	C₂²×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
123	C₂²×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
124	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
125	C₂²×C₅₃	#421	1	4	yes	—
126	C₄×C₅₃	—	2	4	no	—
127	C₄×C₅₃	—	2	4	no	—
128	C₄×C₅₃	—	4	8	no	—
129	C₄×C₅₃	—	4	8	no	—
130	C₄×C₅₃	C₄³	1	1	yes	characteristic
131	C₄×C₅₃	#376	1	1	yes	characteristic
132	C₄×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
133	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
134	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic
136	C₄×#117	C₂×C₅₃	1	1	yes	characteristic
137	C₂×#376	C₂×C₅₃	1	1	yes	characteristic
138	C₂×#339	C₂×C₅₃	1	4	yes	—
139	C₂×C₅₃	—	2	2	no	—
140	C₂×C₅₃	—	2	2	no	—
141	C₂×C₅₃	C₄×#95	1	1	yes	characteristic
142	C₂×C₅₃	#1733	1	2	yes	—
143	C₂×C₅₃	#1759	1	4	yes	—
144	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
145	C₂×#117	C₂²×C₅₃	1	2	yes	—
146	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
147	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
148	C₄×M₁₆	—	2	2	no	—
149	C₄×M₁₆	C₄×C₅₃	1	4	yes	complemented
150	C₂×#97	—	2	1	no	—
151	C₂×C₄×C₈	—	2	1	no	—
152	C₄×C₄⋊C₄	C₄×C₅₃	1	4	yes	complemented
153	C₄×C₂²⋊C₄	—	2	2	no	—
154	#376	—	2	2	no	—
155	#376	—	2	1	no	—
156	#376	C₄×C₅₃	1	4	yes	complemented
157	C₂×#95	—	2	2	no	—
158	C₄³	—	2	1	no	—
159	#339	C₄×C₅₃	1	16	yes	complemented
160	C₅₃	#10701	1	1	yes	characteristic, direct product factor, Sylow
161	C₂³×C₄	—	2	2	no	—
162	C₂³×C₄	D₄×C₅₃	1	2	yes	—
163	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	C₂×C₄×C₅₃	1	8	yes	—
167	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
168	C₂²×C₈	—	2	2	no	—
169	#117	—	2	4	no	—
170	#117	C₂×C₄×C₅₃	1	8	yes	—
171	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
172	C₂×C₂²⋊C₄	—	2	2	no	—
173	C₂×C₄²	—	2	2	no	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	—	2	4	no	—
176	C₂×C₄²	—	2	4	no	—
177	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	—
178	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
179	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
180	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
181	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
182	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
183	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
184	C₂×C₄²	D₄×C₅₃	1	2	yes	—
185	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
186	#97	—	2	4	no	—
187	C₄×C₈	—	2	4	no	—
188	#95	—	2	8	no	—
189	#95	—	2	4	no	—
190	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	4	4	no	—
195	C₂²×C₄	—	4	8	no	—
196	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
197	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
198	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
199	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
200	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
204	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
207	C₂²×C₄	#41×C₅₃	1	4	yes	—
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	8	no	—
210	M₁₆	C₄²×C₅₃	1	8	yes	complemented
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	8	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	C₄²×C₅₃	1	8	yes	—
215	C₂²⋊C₄	—	2	8	no	—
216	C₄²	—	2	4	no	—
217	C₄²	—	2	8	no	—
218	C₄²	—	2	8	no	—
219	C₄²	—	4	8	no	—
220	C₄²	C₄²×C₅₃	1	4	yes	—
221	C₄²	C₄²×C₅₃	1	8	yes	complemented
222	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
223	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
224	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
225	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
226	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
227	C₂³	—	2	4	no	—
228	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
229	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
230	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
231	C₂³	C₄×D₄×C₅₃	1	4	yes	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	—	4	16	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	#95×C₅₃	1	4	yes	—
242	C₂×C₄	#95×C₅₃	1	4	yes	—
243	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
244	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
245	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
246	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
247	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
249	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
250	C₂×C₄	#117×C₅₃	1	2	yes	—
251	C₂×C₄	#117×C₅₃	1	2	yes	—
252	C₂×C₄	#117×C₅₃	1	2	yes	—
253	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
254	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
255	C₂×C₄	C₄×Q₈×C₅₃	1	4	yes	—
256	C₈	—	2	8	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	8	no	—
260	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
261	C₂²	C₄×C₂²⋊C₄×C₅₃	1	2	yes	—
262	C₂²	C₄×C₂²⋊C₄×C₅₃	1	2	yes	—
263	C₂²	C₄×C₄⋊C₄×C₅₃	1	2	yes	—
264	C₂²	C₅₃×#421	1	4	yes	—
265	C₄	—	2	4	no	—
266	C₄	—	2	4	no	—
267	C₄	—	4	8	no	—
268	C₄	—	4	8	no	—
269	C₄	C₄³×C₅₃	1	1	yes	characteristic
270	C₄	C₅₃×#376	1	1	yes	characteristic
271	C₄	C₄×C₂²⋊C₄×C₅₃	1	2	yes	—
272	C₄	C₄×C₄⋊C₄×C₅₃	1	2	yes	—
273	C₄	C₄×C₄⋊C₄×C₅₃	1	2	yes	—
274	C₂	—	2	2	no	—
275	C₂	—	2	2	no	—
276	C₂	C₄×#95×C₅₃	1	1	yes	characteristic
277	C₂	C₅₃×#1733	1	2	yes	—
278	C₂	C₅₃×#1759	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₄	C₄³×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₄×M₁₆×C₅₃	C₄	semidirect
13. C₄×C₄⋊C₄×C₅₃	C₄	semidirect
17. C₅₃×#376	C₄	semidirect
20. C₅₃×#339	C₄	semidirect
70. M₁₆×C₅₃	C₄²	semidirect
81. C₄²×C₅₃	C₄²	semidirect
117. #10701	C₅₃	direct (N × H)
149. C₄×M₁₆	C₄×C₅₃	semidirect
152. C₄×C₄⋊C₄	C₄×C₅₃	semidirect
156. #376	C₄×C₅₃	semidirect
159. #339	C₄×C₅₃	semidirect
160. C₅₃	#10701	direct (N × H)
210. M₁₆	C₄²×C₅₃	semidirect
221. C₄²	C₄²×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×M₁₆×C₅₃	2
C₄×#117×C₅₃	2
C₂×C₅₃×#376	2
C₂×C₅₃×#339	2
#10701	53

Subgroups — C53×#10701 #27218213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10701 #27218213