Subgroups — #27403900

Order 13568.

770 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20277672	C₂	1	2	yes	maximal
2	#20275587	C₂	1	2	yes	maximal
3	#20275351	C₂	1	4	yes	maximal
4	#20274872	C₂	1	1	yes	maximal, characteristic
5	#20274720	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×#13133149	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	#16810493	C₂²	1	2	yes	—
17	#16810489	—	2	2	no	—
18	C₄⋊C₄×Dic₅₃	C₂²	1	4	yes	—
19	C₄×C₄⋊Dic₅₃	—	2	2	no	—
20	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
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	—	2	2	no	—
31	C₂×C₄⋊Dic₅₃	—	2	2	no	—
32	C₂×C₄⋊Dic₅₃	D₄	1	2	yes	—
33	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
34	C₂×C₄×Dic₅₃	—	2	4	no	—
35	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
36	#13133149	—	2	4	no	—
37	#13133149	—	4	2	no	—
38	C₄⋊#88322	—	2	2	no	—
39	#13133141	—	4	4	no	—
40	#13133127	—	2	4	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	—	2	4	no	—
89	C₄⋊Dic₅₃	—	4	4	no	—
90	C₄⋊Dic₅₃	—	8	2	no	—
91	C₄×Dic₅₃	—	2	2	no	—
92	C₄×Dic₅₃	—	2	2	no	—
93	C₄×Dic₅₃	—	8	4	no	—
94	C₂×#88322	—	2	2	no	—
95	C₂×#88322	—	2	2	no	—
96	C₂×#88322	—	2	2	no	—
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	4	no	—
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₅₃	#137	1	4	yes	—
112	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
118	C₂×Dic₅₃	—	2	4	no	—
119	C₂×Dic₅₃	—	2	4	no	—
120	C₂×Dic₅₃	—	2	4	no	—
121	C₂×Dic₅₃	—	4	4	no	—
122	C₂×Dic₅₃	—	4	4	no	—
123	C₂×Dic₅₃	—	4	4	no	—
124	C₈×C₅₃	—	2	2	no	—
125	C₈×C₅₃	#117	1	4	yes	—
126	#88322	—	4	2	no	—
127	#22814	—	53	1	no	maximal, Sylow
128	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂²×C₅₃	#547	1	1	yes	characteristic
130	C₂²×C₅₃	#550	1	1	yes	characteristic
131	C₂²×C₅₃	C₂×#137	1	2	yes	—
132	C₂²×C₅₃	#577	1	2	yes	—
133	C₄×C₅₃	—	2	2	no	—
134	C₄×C₅₃	—	4	4	no	—
135	C₄×C₅₃	—	4	4	no	—
136	C₄×C₅₃	#441	1	4	yes	—
137	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
138	C₄×C₅₃	#520	1	1	yes	characteristic
139	C₄×C₅₃	#555	1	1	yes	characteristic
140	C₄×C₅₃	#556	1	1	yes	characteristic
141	Dic₅₃	—	4	4	no	—
142	Dic₅₃	—	8	4	no	—
143	C₂×C₄×Q₁₆	D₅₃	1	1	yes	characteristic
144	#2355	—	53	2	no	—
145	#1971	—	53	2	no	—
146	#1927	—	53	4	no	—
147	#1844	—	53	1	no	—
148	#1803	—	53	2	no	—
149	#1776	—	53	2	no	—
150	#1770	—	53	1	no	—
151	C₂×C₅₃	#2354	1	1	yes	characteristic
152	C₂×C₅₃	C₂×#441	1	1	yes	characteristic
153	C₂×C₅₃	#2952	1	1	yes	characteristic
154	C₂×C₅₃	#3356	1	2	yes	—
155	C₂×C₅₃	#3400	1	2	yes	—
156	C₂²×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
157	C₂×C₄×Q₈	C₂×D₅₃	1	2	yes	—
158	C₄×Q₁₆	Dic₅₃	1	8	yes	complemented
159	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
160	C₂×#106	—	53	2	no	—
161	C₂×#106	—	53	2	no	—
162	C₂×C₄⋊C₈	—	53	1	no	—
163	C₂×#103	—	53	4	no	—
164	C₂×#103	C₂×D₅₃	1	2	yes	—
165	C₂×#97	—	53	2	no	—
166	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
167	#391	—	53	4	no	—
168	#387	—	106	4	no	—
169	#384	—	53	2	no	—
170	#382	—	106	2	no	—
171	C₄×C₄⋊C₄	—	53	1	no	—
172	C₄×C₄⋊C₄	—	53	4	no	—
173	C₄×C₄⋊C₄	—	106	2	no	—
174	#340	—	53	4	no	—
175	#336	—	53	2	no	—
176	C₅₃	#22814	1	1	yes	characteristic, complemented, Sylow
177	C₂²×Q₈	C₂²×D₅₃	1	2	yes	—
178	C₂×Q₁₆	—	2	2	no	—
179	C₂×Q₁₆	—	2	2	no	—
180	C₂×Q₁₆	C₂×Dic₅₃	1	4	yes	—
181	C₂²×C₈	—	53	2	no	—
182	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
184	C₄×Q₈	—	2	4	no	—
185	C₄×Q₈	C₂×Dic₅₃	1	8	yes	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	53	2	no	—
188	C₂×C₄⋊C₄	—	53	4	no	—
189	C₂×C₄⋊C₄	—	106	2	no	—
190	C₂×C₄⋊C₄	—	106	2	no	—
191	C₂×C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	53	2	no	—
194	C₂×C₄²	—	106	4	no	—
195	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
196	#107	C₂×Dic₅₃	1	4	yes	—
197	#106	—	106	4	no	—
198	#106	—	212	2	no	—
199	C₄⋊C₈	—	106	2	no	—
200	#103	—	212	4	no	—
201	#103	C₂×Dic₅₃	1	8	yes	—
202	#97	—	106	4	no	—
203	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
204	#95	—	53	2	no	—
205	#95	—	106	4	no	—
206	#95	—	106	4	no	—
207	#95	—	106	4	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	C₂²×Dic₅₃	1	4	yes	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	53	4	no	—
217	C₂²×C₄	—	106	4	no	—
218	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
219	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
220	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
221	Q₁₆	—	2	8	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	106	2	no	—
225	C₂×C₈	—	106	2	no	—
226	C₂×C₈	—	106	2	no	—
227	C₂×C₈	#11719945	1	2	yes	—
228	C₂×C₈	#11719945	1	2	yes	—
229	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
230	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
231	C₄⋊C₄	—	2	8	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	106	4	no	—
234	C₄⋊C₄	—	106	4	no	—
235	C₄⋊C₄	—	106	4	no	—
236	C₄⋊C₄	—	424	2	no	—
237	C₄⋊C₄	—	212	4	no	—
238	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
239	C₄²	—	2	8	no	—
240	C₄²	—	106	2	no	—
241	C₄²	—	106	2	no	—
242	C₄²	—	424	4	no	—
243	C₄²	#11719947	1	2	yes	—
244	C₄²	C₂²×Dic₅₃	1	2	yes	—
245	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
246	Q₈	—	2	4	no	—
247	Q₈	—	2	4	no	—
248	Q₈	—	4	2	no	—
249	Q₈	—	4	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	106	4	no	—
258	C₂×C₄	—	106	4	no	—
259	C₂×C₄	—	106	4	no	—
260	C₂×C₄	—	212	4	no	—
261	C₂×C₄	—	212	4	no	—
262	C₂×C₄	—	212	4	no	—
263	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
264	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
265	C₂×C₄	#13133249	1	4	yes	—
266	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
267	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
268	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
269	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
270	C₂×C₄	#13133263	1	1	yes	characteristic
271	C₂×C₄	#13133264	1	1	yes	characteristic
272	C₈	—	2	2	no	—
273	C₈	—	212	2	no	—
274	C₈	#13133176	1	4	yes	—
275	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
276	C₂²	#16811278	1	1	yes	characteristic
277	C₂²	#16811294	1	1	yes	characteristic
278	C₂²	C₂×#13133249	1	2	yes	—
279	C₂²	#16811430	1	2	yes	—
280	C₄	—	2	2	no	—
281	C₄	—	4	4	no	—
282	C₄	—	4	4	no	—
283	C₄	—	424	4	no	—
284	C₄	—	212	4	no	—
285	C₄	#16810730	1	4	yes	—
286	C₄	C₂×#13133176	1	1	yes	characteristic
287	C₄	#16811106	1	1	yes	characteristic
288	C₄	#16811338	1	1	yes	characteristic
289	C₄	#16811343	1	1	yes	characteristic
290	C₂	#20277664	1	1	yes	characteristic
291	C₂	C₂×#16810730	1	1	yes	characteristic
292	C₂	#20283599	1	1	yes	characteristic
293	C₂	#20287332	1	2	yes	—
294	C₂	#20287766	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	#22814
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
158. C₄×Q₁₆	Dic₅₃	semidirect
176. C₅₃	#22814	semidirect

Maximal subgroups

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

subgroup	index
#20277672	2
#20275587	2
#20275351	2
#20274872	2
#20274720	2
#20274614	2
#20274582	2
C₂×C₄×Q₁₆×C₅₃	2
#22814	53