Subgroups — #27290004

Order 13568.

766 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133152	C₂	1	2	yes	maximal
2	C₂²×#13133128	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810387	C₂	1	2	yes	maximal
4	#20274409	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂³×#88322	C₂²	1	1	yes	characteristic
8	C₂³×#88322	C₂²	1	1	yes	characteristic
9	C₂×#13133152	—	2	4	no	—
10	C₂×#13133152	—	2	4	no	—
11	C₂×#13133152	C₄	1	8	yes	complemented
12	C₂×#13133128	—	2	4	no	—
13	C₂×#13133128	C₄	1	4	yes	—
14	#16810387	—	2	4	no	—
15	#16810378	—	2	2	no	—
16	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#336	—	2	2	no	—
19	C₂×#11719917	—	2	2	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	C₂×C₄	1	4	yes	—
25	C₂²×#88322	—	2	2	no	—
26	C₂²×#88322	—	2	2	no	—
27	C₂²×#88322	—	2	2	no	—
28	C₂²×#88322	C₂×C₄	1	2	yes	—
29	C₂²×#88322	C₂×C₄	1	2	yes	—
30	C₂²×#88322	C₂×C₄	1	4	yes	—
31	C₂²×#88322	D₄	1	2	yes	—
32	C₂²×#88322	D₄	1	4	yes	—
33	C₂²×#88322	D₄	1	2	yes	—
34	C₂²×#88322	Q₈	1	2	yes	—
35	C₂²×#88322	Q₈	1	2	yes	—
36	#13133152	—	2	16	no	—
37	#13133128	—	2	8	no	—
38	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
39	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₅₃	—	2	2	no	—
41	C₂×M₁₆×C₅₃	—	2	2	no	—
42	C₂×M₁₆×C₅₃	—	2	2	no	—
43	C₂×M₁₆×C₅₃	—	2	2	no	—
44	C₂×M₁₆×C₅₃	—	2	2	no	—
45	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₅₃	—	2	2	no	—
47	C₂×C₄²×C₅₃	—	2	2	no	—
48	C₂×C₄²×C₅₃	—	2	2	no	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅₃	D₄	1	2	yes	—
51	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	—	2	4	no	—
53	C₂²×C₄×C₅₃	—	4	4	no	—
54	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	#41	1	2	yes	—
64	C₂²×C₄×C₅₃	#41	1	2	yes	—
65	M₁₆×C₅₃	—	2	4	no	—
66	M₁₆×C₅₃	—	2	4	no	—
67	M₁₆×C₅₃	—	4	2	no	—
68	M₁₆×C₅₃	—	4	2	no	—
69	C₂×C₈×C₅₃	—	2	4	no	—
70	C₂×C₈×C₅₃	—	4	2	no	—
71	C₂×C₈×C₅₃	—	4	2	no	—
72	C₄²×C₅₃	—	2	2	no	—
73	C₄²×C₅₃	—	2	2	no	—
74	C₄²×C₅₃	—	2	4	no	—
75	#11719917	—	2	4	no	—
76	#11719917	—	2	4	no	—
77	#11719917	—	4	2	no	—
78	#11719917	—	4	2	no	—
79	C₂×#88322	—	2	2	no	—
80	C₂×#88322	—	2	2	no	—
81	C₂×#88322	—	2	4	no	—
82	C₂×#88322	—	2	4	no	—
83	C₂×#88322	—	2	4	no	—
84	C₂×#88322	—	4	2	no	—
85	C₂×#88322	—	4	2	no	—
86	C₂×#88322	C₄²	1	4	yes	—
87	C₂×#88322	C₂²⋊C₄	1	4	yes	—
88	C₂×#88322	C₂²⋊C₄	1	8	yes	—
89	C₂×#88322	C₄⋊C₄	1	4	yes	—
90	C₂×#88322	C₄⋊C₄	1	4	yes	—
91	C₂×#88322	C₄⋊C₄	1	8	yes	—
92	C₂³×C₅₃	—	2	4	no	—
93	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
95	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
96	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
97	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	—	2	4	no	—
99	C₂×C₄×C₅₃	—	2	8	no	—
100	C₂×C₄×C₅₃	—	4	8	no	—
101	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₅₃	#121	1	2	yes	—
113	C₂×C₄×C₅₃	#121	1	2	yes	—
114	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₅₃	#124	1	2	yes	—
116	C₂×C₄×C₅₃	#125	1	2	yes	—
117	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
119	C₈×C₅₃	—	4	4	no	—
120	#88322	—	2	4	no	—
121	#88322	—	4	4	no	—
122	#88322	#95	1	8	yes	—
123	#10865	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	—	2	4	no	—
125	C₂²×C₅₃	—	2	2	no	—
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	—	2	4	no	—
128	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₅₃	#380	1	1	yes	characteristic
130	C₂²×C₅₃	#384	1	1	yes	characteristic
131	C₂²×C₅₃	#389	1	2	yes	—
132	C₂²×C₅₃	#391	1	2	yes	—
133	C₂²×C₅₃	#430	1	4	yes	—
134	C₄×C₅₃	—	4	4	no	—
135	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
136	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₅₃	#380	1	1	yes	characteristic
138	C₄×C₅₃	#382	1	1	yes	characteristic
139	C₄×C₅₃	#389	1	2	yes	—
140	C₄×C₅₃	#390	1	2	yes	—
141	C₂²×#108	—	53	2	no	—
142	C₂²×#97	—	53	1	no	—
143	C₂×#341	—	53	2	no	—
144	#1728	—	53	1	no	—
145	#1728	D₅₃	1	1	yes	characteristic
146	C₂×C₅₃	—	2	2	no	—
147	C₂×C₅₃	—	2	2	no	—
148	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
149	C₂×C₅₃	#1778	1	2	yes	—
150	C₂×C₅₃	#1811	1	2	yes	—
151	C₂×C₅₃	#1849	1	2	yes	—
152	C₂²×M₁₆	—	53	2	no	—
153	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
154	C₂³×C₈	—	53	1	no	—
155	C₂³×C₈	—	53	1	no	—
156	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
157	C₂×#108	—	53	8	no	—
158	C₂×#108	—	106	4	no	—
159	C₂×#108	—	106	4	no	—
160	C₂×#97	—	53	4	no	—
161	C₂×#97	—	106	4	no	—
162	#341	—	106	4	no	—
163	#336	—	2	2	no	—
164	#336	—	106	2	no	—
165	C₅₃	#10865	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₄	Dic₁₀₆	1	2	yes	—
167	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	53	4	no	—
174	C₂×M₁₆	—	106	2	no	—
175	C₂×M₁₆	—	106	2	no	—
176	C₂×M₁₆	—	106	2	no	—
177	C₂×M₁₆	—	106	2	no	—
178	C₂×M₁₆	—	106	2	no	—
179	C₂×M₁₆	C₄×D₅₃	1	4	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	53	2	no	—
182	C₂²×C₈	—	53	4	no	—
183	C₂²×C₈	—	53	4	no	—
184	C₂²×C₈	—	53	2	no	—
185	C₂²×C₈	—	53	2	no	—
186	C₂²×C₈	—	53	2	no	—
187	C₂²×C₈	—	53	2	no	—
188	C₂²×C₈	—	53	2	no	—
189	C₂²×C₈	—	106	2	no	—
190	C₂²×C₈	—	106	2	no	—
191	C₂²×C₈	—	106	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	C₄×D₅₃	1	2	yes	—
195	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
196	#108	—	106	16	no	—
197	#97	—	106	8	no	—
198	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	#11719919	1	4	yes	—
202	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
204	C₂²×C₄	#11719945	1	2	yes	—
205	C₂²×C₄	D₄×D₅₃	1	2	yes	—
206	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
207	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
208	C₂²×C₄	#11719947	1	2	yes	—
209	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
210	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	106	4	no	—
217	M₁₆	—	106	4	no	—
218	M₁₆	—	212	2	no	—
219	M₁₆	—	212	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	53	4	no	—
224	C₂×C₈	—	53	4	no	—
225	C₂×C₈	—	53	8	no	—
226	C₂×C₈	—	53	8	no	—
227	C₂×C₈	—	53	4	no	—
228	C₂×C₈	—	53	4	no	—
229	C₂×C₈	—	106	2	no	—
230	C₂×C₈	—	106	2	no	—
231	C₂×C₈	—	106	4	no	—
232	C₂×C₈	—	106	4	no	—
233	C₂×C₈	—	106	4	no	—
234	C₂×C₈	—	212	2	no	—
235	C₂×C₈	—	212	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	4	no	—
239	C₂³	—	2	4	no	—
240	C₂³	C₂×#11719919	1	1	yes	characteristic
241	C₂³	C₄×Dic₁₀₆	1	2	yes	—
242	C₂³	#13133194	1	2	yes	—
243	C₂³	#13133216	1	1	yes	characteristic
244	C₂³	#13133216	1	1	yes	characteristic
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	8	no	—
248	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
249	C₂×C₄	C₂²⋊C₄×D₅₃	1	2	yes	—
250	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
251	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
252	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
253	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
254	C₂×C₄	#13133173	1	1	yes	characteristic
255	C₂×C₄	#13133174	1	1	yes	characteristic
256	C₂×C₄	#13133177	1	2	yes	—
257	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
258	C₂×C₄	#13133182	1	2	yes	—
259	C₂×C₄	#13133189	1	2	yes	—
260	C₂×C₄	#13133190	1	2	yes	—
261	C₂×C₄	#13133195	1	2	yes	—
262	C₂×C₄	#13133203	1	2	yes	—
263	C₂×C₄	#13133207	1	2	yes	—
264	C₂×C₄	#13133213	1	1	yes	characteristic
265	C₂×C₄	#13133215	1	1	yes	characteristic
266	C₈	—	4	4	no	—
267	C₈	—	53	8	no	—
268	C₈	—	106	4	no	—
269	C₈	—	212	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	C₄×#11719919	1	1	yes	characteristic
275	C₂²	#16810481	1	1	yes	characteristic
276	C₂²	#16810492	1	1	yes	characteristic
277	C₂²	#16810520	1	2	yes	—
278	C₂²	#16810527	1	2	yes	—
279	C₂²	#16810671	1	4	yes	—
280	C₄	—	4	4	no	—
281	C₄	#95×D₅₃	1	1	yes	characteristic
282	C₄	#16810474	1	1	yes	characteristic
283	C₄	#16810482	1	1	yes	characteristic
284	C₄	#16810487	1	1	yes	characteristic
285	C₄	#16810518	1	2	yes	—
286	C₄	#16810523	1	2	yes	—
287	C₂	—	2	2	no	—
288	C₂	—	2	2	no	—
289	C₂	#20272943	1	1	yes	characteristic
290	C₂	#20274622	1	2	yes	—
291	C₂	#20274755	1	2	yes	—
292	C₂	#20274889	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1728	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1728	D₅₃
hypocenter	C₅₃	#10865
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
11. C₂×#13133152	C₄	semidirect
165. C₅₃	#10865	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133152	2
C₂²×#13133128	2
C₂×#16810387	2
#20274409	2
C₅₃×#1728	2
#10865	53