Subgroups — #27403908

Order 13568.

754 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20277680	C₂	1	2	yes	maximal
2	#20275587	C₂	1	2	yes	maximal
3	#20275358	C₂	1	4	yes	maximal
4	#20274879	C₂	1	1	yes	maximal, characteristic
5	#20274730	C₂	1	2	yes	maximal
6	#20274614	C₂	1	2	yes	maximal
7	#20274582	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133151	C₂²	1	2	yes	—
10	C₂×#13133149	C₂²	1	2	yes	—
11	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
12	C₂×#13133141	C₂²	1	4	yes	—
13	C₂×#13133127	C₂²	1	2	yes	—
14	#16810530	C₂²	1	4	yes	—
15	#16810509	—	2	4	no	—
16	#16810491	—	2	2	no	—
17	#16810489	C₂²	1	2	yes	—
18	C₄⋊C₄×Dic₅₃	C₂²	1	4	yes	—
19	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
20	#16810469	—	2	2	no	—
21	#16810385	C₂²	1	4	yes	—
22	#16810377	C₂²	1	2	yes	—
23	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₅₃	C₄	1	8	yes	complemented
26	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#103×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂²×#88322	C₂³	1	2	yes	—
30	C₂×C₄⋊Dic₅₃	D₄	1	2	yes	—
31	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
32	C₂×C₄×Dic₅₃	—	2	4	no	—
33	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
34	#13133151	—	4	2	no	—
35	#13133149	—	2	4	no	—
36	C₄⋊#88322	—	2	2	no	—
37	#13133141	—	4	4	no	—
38	#13133127	—	2	4	no	—
39	#13133124	—	2	2	no	—
40	#13133124	—	2	2	no	—
41	#13133124	—	2	4	no	—
42	#13133124	—	2	4	no	—
43	#13133124	—	2	4	no	—
44	#13133124	D₄	1	2	yes	—
45	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
46	C₂×Q₁₆×C₅₃	—	2	2	no	—
47	C₂×Q₁₆×C₅₃	—	2	2	no	—
48	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₅₃	—	2	4	no	—
52	C₄×Q₈×C₅₃	C₂×C₄	1	8	yes	—
53	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₅₃	—	2	2	no	—
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	#107×C₅₃	C₂×C₄	1	4	yes	—
58	#103×C₅₃	C₂×C₄	1	8	yes	—
59	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
60	C₂×Q₈×C₅₃	—	2	2	no	—
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	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	—	2	2	no	—
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₅₃	—	2	4	no	—
72	C₂²×Dic₅₃	C₂×D₄	1	4	yes	—
73	Q₁₆×C₅₃	—	2	8	no	—
74	C₂×C₈×C₅₃	—	2	1	no	—
75	C₂×C₈×C₅₃	—	2	1	no	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₅₃	#41	1	2	yes	—
79	C₂×C₈×C₅₃	#41	1	2	yes	—
80	C₄⋊C₄×C₅₃	—	2	8	no	—
81	C₄⋊C₄×C₅₃	—	2	2	no	—
82	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
83	C₄²×C₅₃	—	2	8	no	—
84	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
85	C₄²×C₅₃	#41	1	2	yes	—
86	C₄⋊Dic₅₃	—	2	4	no	—
87	C₄⋊Dic₅₃	—	2	4	no	—
88	C₄⋊Dic₅₃	—	4	2	no	—
89	C₄×Dic₅₃	—	2	2	no	—
90	C₄×Dic₅₃	—	2	2	no	—
91	C₄×Dic₅₃	—	4	8	no	—
92	C₂×#88322	—	2	2	no	—
93	C₂×#88322	—	2	2	no	—
94	C₂×#88322	—	2	2	no	—
95	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₅₃	—	2	4	no	—
97	Q₈×C₅₃	—	2	4	no	—
98	Q₈×C₅₃	—	4	2	no	—
99	Q₈×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	—	4	2	no	—
105	C₂×C₄×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	—	4	4	no	—
107	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
115	C₂×Dic₅₃	—	2	4	no	—
116	C₂×Dic₅₃	—	2	4	no	—
117	C₂×Dic₅₃	—	2	4	no	—
118	C₂×Dic₅₃	—	4	4	no	—
119	C₂×Dic₅₃	—	4	4	no	—
120	C₂×Dic₅₃	—	4	4	no	—
121	C₈×C₅₃	—	2	2	no	—
122	C₈×C₅₃	#117	1	4	yes	—
123	#88322	—	4	2	no	—
124	#22815	—	53	1	no	maximal, Sylow
125	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
126	C₂²×C₅₃	#547	1	1	yes	characteristic
127	C₂²×C₅₃	#550	1	1	yes	characteristic
128	C₂²×C₅₃	#575	1	2	yes	—
129	C₂²×C₅₃	#578	1	2	yes	—
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	4	4	no	—
133	C₄×C₅₃	—	4	4	no	—
134	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
135	C₄×C₅₃	#520	1	1	yes	characteristic
136	C₄×C₅₃	#555	1	1	yes	characteristic
137	C₄×C₅₃	#556	1	1	yes	characteristic
138	Dic₅₃	—	4	4	no	—
139	Dic₅₃	—	8	4	no	—
140	C₂×C₄×Q₁₆	D₅₃	1	1	yes	characteristic
141	#2356	—	53	2	no	—
142	#1971	—	53	2	no	—
143	#1928	—	53	4	no	—
144	#1846	—	53	1	no	—
145	#1805	—	53	2	no	—
146	#1776	—	53	2	no	—
147	#1770	—	53	1	no	—
148	C₂×C₅₃	#2354	1	1	yes	characteristic
149	C₂×C₅₃	#2943	1	1	yes	characteristic
150	C₂×C₅₃	#2950	1	1	yes	characteristic
151	C₂×C₅₃	#3352	1	2	yes	—
152	C₂×C₅₃	#3402	1	2	yes	—
153	C₂²×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
154	C₂×C₄×Q₈	C₂×D₅₃	1	2	yes	—
155	C₄×Q₁₆	Dic₅₃	1	8	yes	complemented
156	C₂×#107	—	53	2	no	—
157	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
158	C₂×#106	—	53	2	no	—
159	C₂×C₄⋊C₈	—	53	1	no	—
160	C₂×#103	—	53	4	no	—
161	C₂×#103	C₂×D₅₃	1	2	yes	—
162	C₂×#97	—	53	2	no	—
163	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
164	#391	—	53	4	no	—
165	#387	—	106	4	no	—
166	#383	—	106	2	no	—
167	#382	—	53	2	no	—
168	C₄×C₄⋊C₄	—	53	1	no	—
169	C₄×C₄⋊C₄	—	53	4	no	—
170	#376	—	106	2	no	—
171	#340	—	53	4	no	—
172	#336	—	53	2	no	—
173	C₅₃	#22815	1	1	yes	characteristic, complemented, Sylow
174	C₂²×Q₈	C₂²×D₅₃	1	2	yes	—
175	C₂×Q₁₆	—	2	2	no	—
176	C₂×Q₁₆	—	2	2	no	—
177	C₂×Q₁₆	C₂×Dic₅₃	1	4	yes	—
178	C₂²×C₈	—	53	2	no	—
179	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
181	C₄×Q₈	—	2	4	no	—
182	C₄×Q₈	C₂×Dic₅₃	1	8	yes	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	53	4	no	—
185	C₂×C₄⋊C₄	—	53	2	no	—
186	C₂×C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	53	2	no	—
189	C₂×C₄²	—	106	4	no	—
190	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
191	#107	—	212	2	no	—
192	#107	C₂×Dic₅₃	1	4	yes	—
193	#106	—	106	4	no	—
194	C₄⋊C₈	—	106	2	no	—
195	#103	—	212	4	no	—
196	#103	C₂×Dic₅₃	1	8	yes	—
197	#97	—	106	4	no	—
198	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
199	#95	—	53	2	no	—
200	#95	—	106	4	no	—
201	#95	—	106	4	no	—
202	#95	—	106	4	no	—
203	#95	—	106	2	no	—
204	#95	—	106	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	C₂²×Dic₅₃	1	4	yes	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	53	4	no	—
214	C₂²×C₄	—	106	4	no	—
215	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
218	Q₁₆	—	2	8	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	106	2	no	—
222	C₂×C₈	—	106	2	no	—
223	C₂×C₈	—	106	2	no	—
224	C₂×C₈	#11719945	1	2	yes	—
225	C₂×C₈	#11719945	1	2	yes	—
226	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
227	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
228	C₄⋊C₄	—	2	8	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	106	4	no	—
231	C₄⋊C₄	—	106	4	no	—
232	C₄⋊C₄	—	212	2	no	—
233	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
234	C₄²	—	2	8	no	—
235	C₄²	—	106	2	no	—
236	C₄²	—	106	2	no	—
237	C₄²	—	212	8	no	—
238	C₄²	#11719947	1	2	yes	—
239	C₄²	C₂²×Dic₅₃	1	2	yes	—
240	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	2	4	no	—
243	Q₈	—	4	2	no	—
244	Q₈	—	4	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	106	4	no	—
253	C₂×C₄	—	106	4	no	—
254	C₂×C₄	—	106	4	no	—
255	C₂×C₄	—	212	4	no	—
256	C₂×C₄	—	212	4	no	—
257	C₂×C₄	—	212	4	no	—
258	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
259	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
260	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
261	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
262	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
263	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
264	C₂×C₄	#13133263	1	1	yes	characteristic
265	C₂×C₄	#13133264	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	212	2	no	—
268	C₈	#13133176	1	4	yes	—
269	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
270	C₂²	#16811278	1	1	yes	characteristic
271	C₂²	#16811294	1	1	yes	characteristic
272	C₂²	#16811420	1	2	yes	—
273	C₂²	#16811436	1	2	yes	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	—	4	4	no	—
278	C₄	—	424	4	no	—
279	C₄	—	212	4	no	—
280	C₄	C₂×#13133176	1	1	yes	characteristic
281	C₄	#16811106	1	1	yes	characteristic
282	C₄	#16811338	1	1	yes	characteristic
283	C₄	#16811343	1	1	yes	characteristic
284	C₂	#20277664	1	1	yes	characteristic
285	C₂	#20283516	1	1	yes	characteristic
286	C₂	#20283583	1	1	yes	characteristic
287	C₂	#20287291	1	2	yes	—
288	C₂	#20287792	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₄×Q₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₅₃
hypocenter	C₅₃	#22815
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
25. C₄×Q₁₆×C₅₃	C₄	semidirect
155. C₄×Q₁₆	Dic₅₃	semidirect
173. C₅₃	#22815	semidirect

Maximal subgroups

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

subgroup	index
#20277680	2
#20275587	2
#20275358	2
#20274879	2
#20274730	2
#20274614	2
#20274582	2
C₂×C₄×Q₁₆×C₅₃	2
#22815	53