Subgroups — C₅₃×#22809 #27230321

Order 13568.

766 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2356	C₂	1	2	yes	maximal
3	C₅₃×#1969	C₂	1	2	yes	maximal
4	C₅₃×#1927	C₂	1	4	yes	maximal
5	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#1805	C₂	1	2	yes	maximal
7	C₄×#107×C₅₃	C₂	1	2	yes	maximal
8	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
11	C₄×Q₁₆×C₅₃	C₄	1	8	yes	complemented
12	C₂×#107×C₅₃	C₂²	1	2	yes	—
13	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#106×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#103×C₅₃	C₂²	1	4	yes	—
17	C₂×#103×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#391	C₂²	1	4	yes	—
21	C₅₃×#387	—	2	4	no	—
22	C₅₃×#383	—	2	2	no	—
23	C₅₃×#382	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
25	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#376	—	2	2	no	—
27	C₅₃×#340	C₂²	1	4	yes	—
28	C₅₃×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
30	C₂×Q₁₆×C₅₃	—	2	2	no	—
31	C₂×Q₁₆×C₅₃	—	2	2	no	—
32	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₅₃	—	2	4	no	—
37	C₄×Q₈×C₅₃	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
42	C₂×C₄²×C₅₃	—	2	2	no	—
43	C₂×C₄²×C₅₃	—	2	4	no	—
44	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	#107×C₅₃	—	2	4	no	—
47	#107×C₅₃	C₂×C₄	1	4	yes	—
48	#106×C₅₃	—	4	2	no	—
49	C₄⋊C₈×C₅₃	—	2	2	no	—
50	#103×C₅₃	—	4	4	no	—
51	#103×C₅₃	C₂×C₄	1	8	yes	—
52	C₄×C₈×C₅₃	—	2	4	no	—
53	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
54	#95×C₅₃	—	2	2	no	—
55	#95×C₅₃	—	2	4	no	—
56	#95×C₅₃	—	2	4	no	—
57	#95×C₅₃	—	2	2	no	—
58	#95×C₅₃	—	2	4	no	—
59	#95×C₅₃	D₄	1	1	yes	characteristic
60	#95×C₅₃	D₄	1	1	yes	characteristic
61	C₂×Q₈×C₅₃	—	2	2	no	—
62	C₂×Q₈×C₅₃	—	2	2	no	—
63	C₂×Q₈×C₅₃	—	2	2	no	—
64	C₂×Q₈×C₅₃	—	2	2	no	—
65	C₂×Q₈×C₅₃	—	2	2	no	—
66	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₅₃	—	2	2	no	—
68	C₂²×C₄×C₅₃	—	2	2	no	—
69	C₂²×C₄×C₅₃	—	2	4	no	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
75	Q₁₆×C₅₃	—	2	8	no	—
76	C₂×C₈×C₅₃	—	2	1	no	—
77	C₂×C₈×C₅₃	—	2	1	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	—	2	2	no	—
81	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₅₃	#41	1	2	yes	—
84	C₂×C₈×C₅₃	#41	1	2	yes	—
85	C₄⋊C₄×C₅₃	—	2	8	no	—
86	C₄⋊C₄×C₅₃	—	2	4	no	—
87	C₄⋊C₄×C₅₃	—	2	2	no	—
88	C₄⋊C₄×C₅₃	—	2	4	no	—
89	C₄⋊C₄×C₅₃	—	2	4	no	—
90	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
91	C₄²×C₅₃	—	2	8	no	—
92	C₄²×C₅₃	—	2	2	no	—
93	C₄²×C₅₃	—	2	2	no	—
94	C₄²×C₅₃	—	4	8	no	—
95	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₅₃	#41	1	2	yes	—
97	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₅₃	—	2	4	no	—
99	Q₈×C₅₃	—	2	4	no	—
100	Q₈×C₅₃	—	4	2	no	—
101	Q₈×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	4	no	—
107	C₂×C₄×C₅₃	—	2	4	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	4	2	no	—
112	C₂×C₄×C₅₃	—	4	2	no	—
113	C₂×C₄×C₅₃	—	4	4	no	—
114	C₂×C₄×C₅₃	—	4	4	no	—
115	C₂×C₄×C₅₃	—	4	4	no	—
116	C₂×C₄×C₅₃	—	4	4	no	—
117	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₅₃	#135	1	4	yes	—
120	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
126	C₈×C₅₃	—	2	2	no	—
127	C₈×C₅₃	—	4	2	no	—
128	C₈×C₅₃	#117	1	4	yes	—
129	#22809	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₅₃	#547	1	1	yes	characteristic
132	C₂²×C₅₃	#550	1	1	yes	characteristic
133	C₂²×C₅₃	C₂×#135	1	2	yes	—
134	C₂²×C₅₃	#578	1	2	yes	—
135	C₄×C₅₃	—	2	4	no	—
136	C₄×C₅₃	—	2	2	no	—
137	C₄×C₅₃	—	2	2	no	—
138	C₄×C₅₃	—	4	2	no	—
139	C₄×C₅₃	—	4	4	no	—
140	C₄×C₅₃	—	4	4	no	—
141	C₄×C₅₃	—	8	4	no	—
142	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
143	C₄×C₅₃	#520	1	1	yes	characteristic
144	C₄×C₅₃	#555	1	1	yes	characteristic
145	C₄×C₅₃	#556	1	1	yes	characteristic
146	C₂×C₄×Q₁₆	C₂×C₅₃	1	1	yes	characteristic
147	#2356	C₂×C₅₃	1	2	yes	—
148	#1969	C₂×C₅₃	1	2	yes	—
149	#1927	C₂×C₅₃	1	4	yes	—
150	#1846	C₂×C₅₃	1	1	yes	characteristic
151	#1805	C₂×C₅₃	1	2	yes	—
152	C₄×#107	C₂×C₅₃	1	2	yes	—
153	C₈×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
154	C₂×C₅₃	#2354	1	1	yes	characteristic
155	C₂×C₅₃	C₄×#135	1	1	yes	characteristic
156	C₂×C₅₃	#2950	1	1	yes	characteristic
157	C₂×C₅₃	#3336	1	2	yes	—
158	C₂×C₅₃	#3393	1	2	yes	—
159	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
160	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
161	C₄×Q₁₆	C₄×C₅₃	1	8	yes	complemented
162	C₂×#107	C₂²×C₅₃	1	2	yes	—
163	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
164	C₂×#106	C₂²×C₅₃	1	2	yes	—
165	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
166	C₂×#103	C₂²×C₅₃	1	4	yes	—
167	C₂×#103	C₂²×C₅₃	1	2	yes	—
168	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
169	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
170	#391	C₂²×C₅₃	1	4	yes	—
171	#387	—	2	4	no	—
172	#383	—	2	2	no	—
173	#382	C₂²×C₅₃	1	2	yes	—
174	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
175	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
176	#376	—	2	2	no	—
177	#340	C₂²×C₅₃	1	4	yes	—
178	#336	C₂²×C₅₃	1	2	yes	—
179	C₅₃	#22809	1	1	yes	characteristic, direct product factor, Sylow
180	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	C₂×C₄×C₅₃	1	4	yes	—
184	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
185	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
187	C₄×Q₈	—	2	4	no	—
188	C₄×Q₈	C₂×C₄×C₅₃	1	8	yes	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
191	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
192	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	4	no	—
195	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
196	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
197	#107	—	2	4	no	—
198	#107	C₂×C₄×C₅₃	1	4	yes	—
199	#106	—	4	2	no	—
200	C₄⋊C₈	—	2	2	no	—
201	#103	—	4	4	no	—
202	#103	C₂×C₄×C₅₃	1	8	yes	—
203	C₄×C₈	—	2	4	no	—
204	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
205	#95	—	2	2	no	—
206	#95	—	2	4	no	—
207	#95	—	2	4	no	—
208	#95	—	2	2	no	—
209	#95	—	2	4	no	—
210	#95	D₄×C₅₃	1	1	yes	characteristic
211	#95	D₄×C₅₃	1	1	yes	characteristic
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
225	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
226	Q₁₆	—	2	8	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
233	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
234	C₂×C₈	#41×C₅₃	1	2	yes	—
235	C₂×C₈	#41×C₅₃	1	2	yes	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
242	C₄²	—	2	8	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	4	8	no	—
246	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
247	C₄²	#41×C₅₃	1	2	yes	—
248	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
249	Q₈	—	2	4	no	—
250	Q₈	—	2	4	no	—
251	Q₈	—	4	2	no	—
252	Q₈	—	4	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
269	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
270	C₂×C₄	#135×C₅₃	1	4	yes	—
271	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
272	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
273	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
274	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
275	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
276	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	4	2	no	—
279	C₈	#117×C₅₃	1	4	yes	—
280	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
281	C₂²	C₅₃×#547	1	1	yes	characteristic
282	C₂²	C₅₃×#550	1	1	yes	characteristic
283	C₂²	C₂×#135×C₅₃	1	2	yes	—
284	C₂²	C₅₃×#578	1	2	yes	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	4	no	—
288	C₄	—	4	4	no	—
289	C₄	—	4	2	no	—
290	C₄	—	4	4	no	—
291	C₄	—	8	4	no	—
292	C₄	C₂×#117×C₅₃	1	1	yes	characteristic
293	C₄	C₅₃×#520	1	1	yes	characteristic
294	C₄	C₅₃×#555	1	1	yes	characteristic
295	C₄	C₅₃×#556	1	1	yes	characteristic
296	C₂	C₅₃×#2354	1	1	yes	characteristic
297	C₂	C₄×#135×C₅₃	1	1	yes	characteristic
298	C₂	C₅₃×#2950	1	1	yes	characteristic
299	C₂	C₅₃×#3336	1	2	yes	—
300	C₂	C₅₃×#3393	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. C₄×Q₁₆×C₅₃	C₄	semidirect
129. #22809	C₅₃	direct (N × H)
161. C₄×Q₁₆	C₄×C₅₃	semidirect
179. C₅₃	#22809	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×C₄×Q₁₆×C₅₃	2
C₅₃×#2356	2
C₅₃×#1969	2
C₅₃×#1927	2
C₅₃×#1846	2
C₅₃×#1805	2
C₄×#107×C₅₃	2
C₈×C₄⋊C₄×C₅₃	2
#22809	53

Subgroups — C53×#22809 #27230321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22809 #27230321