Subgroups — #27289734

Order 13568.

650 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×#16810579	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810395	C₂	1	2	yes	maximal
3	#20274448	C₂	1	1	yes	maximal, characteristic
4	#20274413	C₂	1	2	yes	maximal
5	C₅₃×#1742	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	1	yes	characteristic
7	C₂³×#88322	C₂²	1	1	yes	characteristic
8	C₂×#13133176	C₂²	1	2	yes	—
9	#16810579	—	2	4	no	—
10	C₂×#13133128	C₄	1	2	yes	complemented
11	C₂×C₄×#88322	C₄	1	2	yes	complemented
12	#16810395	—	2	4	no	—
13	#16810390	C₄	1	4	yes	—
14	#16810377	—	2	4	no	—
15	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#343	C₄	1	4	yes	complemented
18	C₂×#11719917	—	2	2	no	—
19	C₂×#11719917	—	2	2	no	—
20	C₂×#11719917	D₄	1	2	yes	—
21	C₂×#11719917	Q₈	1	2	yes	—
22	C₂²×#88322	—	2	2	no	—
23	C₂²×#88322	—	2	1	no	—
24	C₂²×#88322	—	2	1	no	—
25	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
26	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂²×#88322	C₂×C₄	1	2	yes	—
28	C₂²×#88322	D₄	1	4	yes	—
29	#13133176	—	2	2	no	—
30	#13133176	—	2	2	no	—
31	#13133176	—	2	2	no	—
32	#13133176	—	2	2	no	—
33	C₂×C₄⋊Dic₅₃	—	2	2	no	—
34	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₄×Dic₅₃	—	2	2	no	—
36	#13133128	—	2	4	no	—
37	C₄×#88322	—	2	4	no	—
38	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₅₃	—	2	2	no	—
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	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₅₃	—	2	2	no	—
46	#117×C₅₃	—	2	2	no	—
47	#117×C₅₃	—	2	2	no	—
48	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
51	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	#41	1	2	yes	—
59	C₂²×C₄×C₅₃	#41	1	2	yes	—
60	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	#41	1	2	yes	—
62	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
63	C₂²×Dic₅₃	—	2	4	no	—
64	M₁₆×C₅₃	—	2	4	no	—
65	M₁₆×C₅₃	—	2	4	no	—
66	M₁₆×C₅₃	—	4	2	no	—
67	M₁₆×C₅₃	—	4	2	no	—
68	C₂×C₈×C₅₃	—	2	4	no	—
69	C₂×C₈×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	—	4	2	no	—
71	C₄⋊C₄×C₅₃	—	2	4	no	—
72	C₂²⋊C₄×C₅₃	—	2	4	no	—
73	C₄²×C₅₃	—	2	2	no	—
74	C₄²×C₅₃	—	2	2	no	—
75	C₂²⋊Dic₅₃	—	2	4	no	—
76	C₂²⋊Dic₅₃	—	2	4	no	—
77	C₄⋊Dic₅₃	—	4	2	no	—
78	C₄⋊Dic₅₃	—	4	2	no	—
79	C₄×Dic₅₃	—	4	2	no	—
80	C₄×Dic₅₃	—	4	2	no	—
81	#11719917	—	2	4	no	—
82	#11719917	—	2	4	no	—
83	C₂×#88322	—	2	2	no	—
84	C₂×#88322	—	2	1	no	—
85	C₂×#88322	—	2	1	no	—
86	C₂×#88322	—	2	1	no	—
87	C₂×#88322	—	2	1	no	—
88	C₂×#88322	—	2	2	no	—
89	C₂×#88322	—	2	4	no	—
90	C₂×#88322	—	2	4	no	—
91	C₂×#88322	C₂²⋊C₄	1	2	yes	—
92	C₂×#88322	C₂²⋊C₄	1	2	yes	—
93	C₂×#88322	C₄⋊C₄	1	4	yes	—
94	C₂³×C₅₃	—	4	2	no	—
95	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
96	C₂³×C₅₃	#117	1	2	yes	—
97	C₂³×C₅₃	C₄×D₄	1	2	yes	—
98	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
99	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	—	4	4	no	—
102	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#117	1	2	yes	—
109	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#121	1	2	yes	—
113	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₅₃	#123	1	2	yes	—
115	C₂×C₄×C₅₃	#123	1	2	yes	—
116	C₂×C₄×C₅₃	#124	1	2	yes	—
117	C₂×C₄×C₅₃	#125	1	2	yes	—
118	C₂×C₄×C₅₃	#126	1	2	yes	—
119	C₂×C₄×C₅₃	#126	1	2	yes	—
120	C₂×Dic₅₃	—	4	8	no	—
121	C₂×Dic₅₃	—	4	4	no	—
122	C₈×C₅₃	—	4	4	no	—
123	#88322	—	2	4	no	—
124	#88322	—	2	2	no	—
125	#88322	—	2	2	no	—
126	#10803	—	53	1	no	maximal, Sylow
127	C₂²×C₅₃	—	2	4	no	—
128	C₂²×C₅₃	—	2	2	no	—
129	C₂²×C₅₃	—	4	2	no	—
130	C₂²×C₅₃	—	4	1	no	—
131	C₂²×C₅₃	—	4	1	no	—
132	C₂²×C₅₃	#376	1	1	yes	characteristic
133	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₅₃	#387	1	2	yes	—
136	C₂²×C₅₃	#388	1	2	yes	—
137	C₄×C₅₃	—	4	4	no	—
138	C₄×C₅₃	#380	1	1	yes	characteristic
139	C₄×C₅₃	#381	1	1	yes	characteristic
140	C₄×C₅₃	#382	1	1	yes	characteristic
141	C₄×C₅₃	#383	1	1	yes	characteristic
142	C₄×C₅₃	#387	1	2	yes	—
143	C₄×C₅₃	#388	1	2	yes	—
144	Dic₅₃	—	8	4	no	—
145	C₂×#405	—	53	1	no	—
146	C₂×#344	—	53	2	no	—
147	#1742	D₅₃	1	1	yes	characteristic
148	#1740	—	53	1	no	—
149	#1729	—	53	2	no	—
150	C₂×C₅₃	—	2	2	no	—
151	C₂×C₅₃	—	2	1	no	—
152	C₂×C₅₃	—	2	1	no	—
153	C₂×C₅₃	—	2	2	no	—
154	C₂×C₅₃	#1434	1	1	yes	characteristic
155	C₂×C₅₃	#1757	1	2	yes	—
156	C₂²×M₁₆	—	53	1	no	—
157	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
158	C₂³×C₈	—	53	1	no	—
159	C₂×#117	—	53	2	no	—
160	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
161	#405	—	106	4	no	—
162	C₂×#97	—	53	2	no	—
163	C₂×C₄×C₈	—	53	2	no	—
164	#344	—	106	4	no	—
165	#343	Dic₅₃	1	4	yes	complemented
166	#342	—	53	4	no	—
167	#336	—	106	4	no	—
168	C₅₃	#10803	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	53	2	no	—
176	C₂×M₁₆	—	53	2	no	—
177	C₂×M₁₆	—	106	2	no	—
178	C₂×M₁₆	—	106	2	no	—
179	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	53	1	no	—
182	C₂²×C₈	—	53	1	no	—
183	C₂²×C₈	—	53	2	no	—
184	C₂²×C₈	—	53	4	no	—
185	C₂²×C₈	—	106	2	no	—
186	C₂²×C₈	—	106	1	no	—
187	C₂²×C₈	—	106	1	no	—
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	#117	—	106	2	no	—
191	#117	—	106	2	no	—
192	#117	—	106	2	no	—
193	#117	—	106	2	no	—
194	C₂×C₄⋊C₄	—	106	2	no	—
195	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
196	C₂×C₂²⋊C₄	—	53	4	no	—
197	C₂×C₂²⋊C₄	C₄×D₅₃	1	2	yes	—
198	C₂×C₄²	—	106	2	no	—
199	C₂×C₄²	C₄×D₅₃	1	2	yes	—
200	#97	—	106	4	no	—
201	C₄×C₈	—	106	4	no	—
202	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
203	C₂²×C₄	—	106	4	no	—
204	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
205	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
206	C₂²×C₄	#11719945	1	2	yes	—
207	C₂²×C₄	#11719945	1	2	yes	—
208	C₂²×C₄	D₄×D₅₃	1	2	yes	—
209	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
210	C₂²×C₄	#11719947	1	2	yes	—
211	C₂²×C₄	#11719947	1	1	yes	characteristic
212	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
213	C₂²×C₄	#11719949	1	1	yes	characteristic
214	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	106	4	no	—
220	M₁₆	—	106	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	53	2	no	—
225	C₂×C₈	—	53	2	no	—
226	C₂×C₈	—	53	4	no	—
227	C₂×C₈	—	106	2	no	—
228	C₂×C₈	—	106	1	no	—
229	C₂×C₈	—	106	1	no	—
230	C₂×C₈	—	106	2	no	—
231	C₂×C₈	—	106	1	no	—
232	C₂×C₈	—	106	1	no	—
233	C₂×C₈	—	106	4	no	—
234	C₂×C₈	—	106	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	212	2	no	—
237	C₄⋊C₄	—	212	2	no	—
238	C₂²⋊C₄	—	2	4	no	—
239	C₂²⋊C₄	—	106	4	no	—
240	C₂²⋊C₄	—	106	4	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	212	2	no	—
244	C₄²	—	212	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
247	C₂³	#13133174	1	2	yes	—
248	C₂³	#13133177	1	2	yes	—
249	C₂³	#13133179	1	1	yes	characteristic
250	C₂³	#13133182	1	1	yes	characteristic
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	212	8	no	—
254	C₂×C₄	—	212	4	no	—
255	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
256	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
257	C₂×C₄	#13133173	1	1	yes	characteristic
258	C₂×C₄	#13133174	1	1	yes	characteristic
259	C₂×C₄	#13133174	1	1	yes	characteristic
260	C₂×C₄	#13133175	1	1	yes	characteristic
261	C₂×C₄	#13133176	1	2	yes	—
262	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
263	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
264	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
265	C₂×C₄	#13133190	1	2	yes	—
266	C₂×C₄	#13133195	1	2	yes	—
267	C₂×C₄	#13133199	1	2	yes	—
268	C₂×C₄	#13133202	1	2	yes	—
269	C₂×C₄	#13133203	1	2	yes	—
270	C₂×C₄	#13133207	1	2	yes	—
271	C₂×C₄	#13133210	1	2	yes	—
272	C₂×C₄	#13133212	1	2	yes	—
273	C₈	—	4	4	no	—
274	C₈	—	106	4	no	—
275	C₈	—	106	2	no	—
276	C₈	—	106	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	#16810469	1	1	yes	characteristic
283	C₂²	C₂²⋊C₄×Dic₅₃	1	1	yes	characteristic
284	C₂²	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
285	C₂²	#16810505	1	2	yes	—
286	C₂²	#16810512	1	2	yes	—
287	C₄	—	4	4	no	—
288	C₄	—	424	4	no	—
289	C₄	#16810482	1	1	yes	characteristic
290	C₄	#16810485	1	1	yes	characteristic
291	C₄	#16810487	1	1	yes	characteristic
292	C₄	#16810490	1	1	yes	characteristic
293	C₄	#16810509	1	2	yes	—
294	C₄	#16810516	1	2	yes	—
295	C₂	—	2	2	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	#20272940	1	1	yes	characteristic
300	C₂	#20274518	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1742	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1742	D₅₃
hypocenter	C₅₃	#10803
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1742

Complements

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

N	H ≅ G/N	type
10. C₂×#13133128	C₄	semidirect
11. C₂×C₄×#88322	C₄	semidirect
17. C₅₃×#343	C₄	semidirect
165. #343	Dic₅₃	semidirect
168. C₅₃	#10803	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810579	2
C₂×#16810395	2
#20274448	2
#20274413	2
C₅₃×#1742	2
#10803	53