Subgroups — #27355908

Order 13568.

578 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283664	C₂	1	2	yes	maximal
2	#20273624	C₂	1	2	yes	maximal, complemented
3	#20273587	C₂	1	4	yes	maximal, complemented
4	#20273534	C₂	1	2	yes	maximal, complemented
5	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
6	#20273118	C₂	1	2	yes	maximal
7	#20273004	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#3170	C₂	1	1	yes	maximal, characteristic
9	#16810754	—	2	4	no	—
10	#16810754	C₂²	1	4	yes	—
11	Q₈×#88322	C₂²	1	4	yes	—
12	#16810676	C₂²	1	1	yes	characteristic
13	#16810650	—	2	2	no	—
14	C₂×C₄⋊#88322	C₂²	1	2	yes	—
15	#16810579	—	2	2	no	—
16	C₄×#11719917	C₂²	1	1	yes	characteristic
17	#16810374	C₂²	1	1	yes	characteristic
18	#16810374	C₂²	1	2	yes	—
19	#16810374	C₂²	1	1	yes	characteristic
20	Q₈⋊#88322	C₂²	1	4	yes	—
21	Q₈⋊#88322	C₂²	1	4	yes	—
22	#16810345	C₂²	1	2	yes	—
23	#16810344	C₂²	1	2	yes	—
24	C₅₃×#533	C₂²	1	2	yes	—
25	C₅₃×#500	C₄	1	1	yes	characteristic
26	C₅₃×#494	C₄	1	1	yes	characteristic
27	C₅₃×#487	C₄	1	2	yes	—
28	C₅₃×#467	C₄	1	2	yes	—
29	C₅₃×#467	C₄	1	2	yes	—
30	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
31	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×#11719917	D₄	1	2	yes	—
34	C₂²×#88322	—	2	2	no	—
35	C₄⋊#88322	—	2	2	no	—
36	C₄⋊#88322	—	2	1	no	—
37	C₄⋊#88322	—	2	4	no	—
38	C₄⋊#88322	—	2	4	no	—
39	C₄⋊#88322	C₂³	1	2	yes	—
40	C₄⋊#88322	C₂³	1	2	yes	—
41	C₂²⋊#88322	—	2	1	no	—
42	#13133128	—	2	4	no	—
43	#13133128	D₄	1	2	yes	—
44	C₄×#88322	—	2	4	no	—
45	C₄×#88322	C₂³	1	2	yes	—
46	C₄×#88322	C₂³	1	2	yes	—
47	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
53	#125×C₅₃	—	2	2	no	—
54	C₂²⋊Q₈×C₅₃	—	2	2	no	—
55	C₂²⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
56	C₄×Q₈×C₅₃	—	2	2	no	—
57	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
58	#117×C₅₃	—	2	2	no	—
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	#107×C₅₃	C₂×C₄	1	2	yes	—
61	#107×C₅₃	C₂×C₄	1	2	yes	—
62	#103×C₅₃	C₂×C₄	1	4	yes	—
63	#103×C₅₃	C₂×C₄	1	4	yes	—
64	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×Q₈×C₅₃	—	2	2	no	—
69	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
73	Q₁₆×C₅₃	—	2	4	no	—
74	Q₁₆×C₅₃	—	4	2	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	4	1	no	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₅₃	—	2	2	no	—
82	C₄⋊C₄×C₅₃	—	2	2	no	—
83	C₄⋊C₄×C₅₃	—	2	2	no	—
84	C₄⋊C₄×C₅₃	—	2	4	no	—
85	C₄⋊C₄×C₅₃	—	2	4	no	—
86	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
87	C₂²⋊C₄×C₅₃	—	2	4	no	—
88	C₄²×C₅₃	—	2	4	no	—
89	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅₃	#41	1	1	yes	characteristic
92	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
93	#11719917	—	2	4	no	—
94	C₂×#88322	—	2	2	no	—
95	C₂×#88322	—	2	2	no	—
96	C₂×#88322	—	2	2	no	—
97	C₂×#88322	—	4	4	no	—
98	C₂×#88322	C₂×D₄	1	4	yes	—
99	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
100	Q₈×C₅₃	—	2	4	no	—
101	Q₈×C₅₃	—	2	4	no	—
102	Q₈×C₅₃	—	8	2	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	2	4	no	—
107	C₂×C₄×C₅₃	—	4	1	no	—
108	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#137	1	4	yes	—
112	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
115	C₈×C₅₃	—	2	1	no	—
116	C₈×C₅₃	—	2	1	no	—
117	C₈×C₅₃	—	2	1	no	—
118	C₈×C₅₃	#131	1	2	yes	—
119	#88322	—	2	4	no	—
120	#88322	—	4	2	no	—
121	#88322	—	4	2	no	—
122	#88322	—	4	2	no	—
123	#18072	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	2	1	no	—
126	C₂²×C₅₃	#441	1	2	yes	—
127	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	4	4	no	—
132	C₄×C₅₃	—	4	4	no	—
133	C₄×C₅₃	#441	1	2	yes	—
134	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
135	C₄×C₅₃	#568	1	1	yes	characteristic
136	C₄×C₅₃	C₂×#137	1	2	yes	—
137	#3170	D₅₃	1	1	yes	characteristic
138	#2962	—	53	2	no	—
139	#1593	—	53	2	no	—
140	#1587	—	53	4	no	—
141	#1579	—	53	2	no	—
142	C₂×#334	—	53	1	no	—
143	#1481	—	53	2	no	—
144	#1451	—	53	1	no	—
145	C₂×C₅₃	—	2	2	no	—
146	C₂×C₅₃	#2935	1	1	yes	characteristic
147	C₂×C₅₃	C₂×#441	1	1	yes	characteristic
148	C₂×C₅₃	#2957	1	1	yes	characteristic
149	#533	C₂×D₅₃	1	2	yes	—
150	#500	Dic₅₃	1	1	yes	characteristic
151	#494	Dic₅₃	1	1	yes	characteristic
152	#487	Dic₅₃	1	2	yes	—
153	#467	Dic₅₃	1	2	yes	—
154	#467	Dic₅₃	1	2	yes	—
155	#446	—	53	4	no	—
156	#446	—	106	4	no	—
157	C₈×Q₈	—	53	4	no	—
158	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
159	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
160	#431	—	53	1	no	—
161	#424	—	106	2	no	—
162	C₂×C₄⋊C₈	—	53	2	no	—
163	#405	—	106	2	no	—
164	C₄×M₁₆	—	53	1	no	—
165	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
166	#334	—	53	1	no	—
167	#334	—	53	2	no	—
168	#334	—	53	1	no	—
169	Q₈⋊C₈	—	53	4	no	—
170	Q₈⋊C₈	—	53	4	no	—
171	#322	—	53	2	no	—
172	#322	—	53	2	no	—
173	C₅₃	#18072	1	1	yes	characteristic, complemented, Sylow
174	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
175	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
176	C₂×M₁₆	—	53	2	no	—
177	C₂²×C₈	—	106	2	no	—
178	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
179	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
180	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
181	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
182	#125	—	2	2	no	—
183	C₂²⋊Q₈	—	2	2	no	—
184	C₂²⋊Q₈	C₂×Dic₅₃	1	4	yes	—
185	C₄×Q₈	—	2	2	no	—
186	C₄×Q₈	C₂²×D₅₃	1	4	yes	—
187	#117	—	2	2	no	—
188	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
189	#107	C₂×Dic₅₃	1	2	yes	—
190	#107	C₂×Dic₅₃	1	2	yes	—
191	C₄⋊C₈	—	53	2	no	—
192	C₄⋊C₈	—	53	2	no	—
193	C₄⋊C₈	—	106	4	no	—
194	C₄⋊C₈	—	106	4	no	—
195	C₄⋊C₈	—	106	2	no	—
196	C₄⋊C₈	—	106	1	no	—
197	#103	C₂×Dic₅₃	1	4	yes	—
198	#103	C₂×Dic₅₃	1	4	yes	—
199	C₂²⋊C₈	—	106	1	no	—
200	#97	—	53	2	no	—
201	#97	—	106	4	no	—
202	C₄×C₈	—	53	2	no	—
203	C₄×C₈	—	53	2	no	—
204	C₄×C₈	—	106	4	no	—
205	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
206	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	C₂²×Dic₅₃	1	4	yes	—
211	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
212	C₂²×C₄	#11719947	1	1	yes	characteristic
213	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
214	Q₁₆	—	2	4	no	—
215	Q₁₆	—	4	2	no	—
216	M₁₆	—	106	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	53	4	no	—
220	C₂×C₈	—	106	2	no	—
221	C₂×C₈	—	106	2	no	—
222	C₂×C₈	—	106	2	no	—
223	C₂×C₈	—	212	4	no	—
224	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
225	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
226	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
227	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	D₄×D₅₃	1	1	yes	characteristic
237	C₄²	#11719947	1	1	yes	characteristic
238	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
239	C₄²	C₂³×D₅₃	1	1	yes	characteristic
240	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	2	4	no	—
243	Q₈	—	8	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
250	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
251	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
252	C₂×C₄	#13133249	1	4	yes	—
253	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
254	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
255	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	106	4	no	—
260	C₈	—	212	2	no	—
261	C₈	—	212	2	no	—
262	C₈	—	212	2	no	—
263	C₈	#13133224	1	2	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	#16810730	1	2	yes	—
267	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	4	no	—
272	C₄	—	4	4	no	—
273	C₄	#16810730	1	2	yes	—
274	C₄	C₂×#13133224	1	1	yes	characteristic
275	C₄	#16811390	1	1	yes	characteristic
276	C₄	C₂×#13133249	1	2	yes	—
277	C₂	—	2	2	no	—
278	C₂	#20283459	1	1	yes	characteristic
279	C₂	C₂×#16810730	1	1	yes	characteristic
280	C₂	#20283632	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3170	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#3170	D₅₃
hypocenter	C₅₃	#18072
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #20273624	C₂	semidirect
3. #20273587	C₂	semidirect
4. #20273534	C₂	semidirect
173. C₅₃	#18072	semidirect

Maximal subgroups

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

subgroup	index
#20283664	2
#20273624	2
#20273587	2
#20273534	2
C₂×#16810374	2
#20273118	2
#20273004	2
C₅₃×#3170	2
#18072	53