Subgroups — #27283254

Order 13568.

506 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275635	C₂	1	2	yes	maximal, complemented
2	C₄×#13133151	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273210	C₂	1	2	yes	maximal, complemented
4	C₈×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	1	yes	characteristic
7	C₂×#13133151	—	2	1	no	—
8	C₂×#13133144	—	2	4	no	—
9	C₄×#11719917	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
11	#16810386	—	2	1	no	—
12	#16810357	—	2	4	no	—
13	#16810345	—	2	1	no	—
14	C₈×#88322	—	2	1	no	—
15	C₅₃×#532	C₂²	1	2	yes	—
16	C₅₃×#500	C₄	1	1	yes	characteristic, complemented
17	C₅₃×#493	C₄	1	1	yes	characteristic, complemented
18	C₅₃×#486	C₄	1	2	yes	complemented
19	C₅₃×#466	—	2	1	no	—
20	C₅₃×#466	—	2	1	no	—
21	C₄×D₈×C₅₃	—	2	1	no	—
22	C₄×D₈×C₅₃	—	2	1	no	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#11719917	D₄	1	2	yes	—
25	C₂×C₄⋊Dic₅₃	—	2	2	no	—
26	C₂×C₄×Dic₅₃	—	2	2	no	—
27	#13133151	—	2	4	no	—
28	C₄⋊#88322	—	2	1	no	—
29	#13133144	—	4	8	no	—
30	C₂²⋊#88322	D₄	1	2	yes	—
31	#13133128	—	2	1	no	—
32	C₄×#88322	—	2	1	no	—
33	C₄×#88322	—	2	1	no	—
34	#13133124	—	2	2	no	—
35	C₂×#41×C₅₃	—	2	2	no	—
36	C₂×D₈×C₅₃	—	2	1	no	—
37	C₂×D₈×C₅₃	—	2	1	no	—
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
41	#127×C₅₃	C₂×C₄	1	2	yes	—
42	#124×C₅₃	—	2	2	no	—
43	#121×C₅₃	—	2	2	no	—
44	#121×C₅₃	—	2	2	no	—
45	C₄×D₄×C₅₃	—	2	2	no	—
46	C₄×D₄×C₅₃	—	2	2	no	—
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	#107×C₅₃	—	2	1	no	—
49	#107×C₅₃	—	2	1	no	—
50	#102×C₅₃	—	2	2	no	—
51	#102×C₅₃	—	2	2	no	—
52	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
56	#41×C₅₃	—	4	4	no	—
57	#41×C₅₃	—	4	4	no	—
58	C₂×Q₈×C₅₃	—	2	2	no	—
59	C₂×D₄×C₅₃	—	2	2	no	—
60	C₂×D₄×C₅₃	—	2	2	no	—
61	C₂×D₄×C₅₃	—	2	2	no	—
62	C₂×D₄×C₅₃	—	4	2	no	—
63	C₂²×C₄×C₅₃	—	4	2	no	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₅₃	—	4	2	no	—
68	D₈×C₅₃	—	4	2	no	—
69	D₈×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	4	1	no	—
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₄⋊C₄×C₅₃	—	2	2	no	—
77	C₄⋊C₄×C₅₃	—	2	2	no	—
78	C₂²⋊C₄×C₅₃	—	4	2	no	—
79	C₂²⋊C₄×C₅₃	—	4	2	no	—
80	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₅₃	#41	1	1	yes	characteristic
83	C₄²×C₅₃	#41	1	1	yes	characteristic
84	C₄⋊Dic₅₃	—	2	4	no	—
85	C₄⋊Dic₅₃	—	4	2	no	—
86	C₄×Dic₅₃	—	4	4	no	—
87	#11719917	—	2	2	no	—
88	#11719917	D₈	1	4	yes	complemented
89	C₂×#88322	—	2	2	no	—
90	C₂×#88322	—	2	2	no	—
91	C₂³×C₅₃	—	4	2	no	—
92	C₂³×C₅₃	#127	1	1	yes	characteristic
93	Q₈×C₅₃	—	4	4	no	—
94	D₄×C₅₃	—	4	2	no	—
95	D₄×C₅₃	—	4	2	no	—
96	D₄×C₅₃	—	4	4	no	—
97	D₄×C₅₃	—	8	2	no	—
98	D₄×C₅₃	—	8	2	no	—
99	D₄×C₅₃	—	8	2	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	4	1	no	—
103	C₂×C₄×C₅₃	—	4	2	no	—
104	C₂×C₄×C₅₃	—	8	4	no	—
105	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
112	C₂×C₄×C₅₃	#135	1	2	yes	—
113	C₂×Dic₅₃	—	4	4	no	—
114	C₂×Dic₅₃	—	8	2	no	—
115	C₂×Dic₅₃	—	8	2	no	—
116	C₈×C₅₃	—	2	1	no	—
117	C₈×C₅₃	—	2	1	no	—
118	C₈×C₅₃	—	2	1	no	—
119	C₈×C₅₃	#104	1	2	yes	complemented
120	#88322	—	2	4	no	—
121	#88322	—	4	2	no	—
122	#9719	—	53	1	no	maximal, Sylow
123	C₂²×C₅₃	—	2	1	no	—
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	8	2	no	—
126	C₂²×C₅₃	—	8	2	no	—
127	C₂²×C₅₃	—	8	2	no	—
128	C₂²×C₅₃	#390	1	1	yes	characteristic
129	C₂²×C₅₃	#493	1	1	yes	characteristic
130	C₂²×C₅₃	#495	1	1	yes	characteristic
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	8	2	no	—
135	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
136	C₄×C₅₃	#421	1	1	yes	characteristic
137	C₄×C₅₃	C₄×D₈	1	2	yes	—
138	C₄×C₅₃	#466	1	2	yes	—
139	Dic₅₃	—	8	4	no	—
140	#3169	D₅₃	1	1	yes	characteristic
141	#1980	—	53	2	no	—
142	C₄×#107	—	53	1	no	—
143	#1502	—	53	2	no	—
144	C₈×M₁₆	—	53	1	no	—
145	C₂×C₅₃	—	2	1	no	—
146	C₂×C₅₃	—	2	1	no	—
147	C₂×C₅₃	—	16	2	no	—
148	C₂×C₅₃	#1970	1	1	yes	characteristic
149	C₂×C₅₃	#1976	1	1	yes	characteristic
150	C₂×C₅₃	#1980	1	1	yes	characteristic
151	#532	C₂×D₅₃	1	2	yes	—
152	#500	Dic₅₃	1	1	yes	characteristic, complemented
153	#493	Dic₅₃	1	1	yes	characteristic, complemented
154	#486	Dic₅₃	1	2	yes	complemented
155	#466	—	2	1	no	—
156	#466	—	2	1	no	—
157	C₄×D₈	—	2	1	no	—
158	C₄×D₈	—	2	1	no	—
159	#431	—	53	1	no	—
160	C₂×#107	—	106	1	no	—
161	C₂×#104	—	106	4	no	—
162	C₄×M₁₆	—	53	1	no	—
163	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
164	C₄×C₄⋊C₄	—	53	2	no	—
165	#340	—	106	1	no	—
166	#327	—	106	4	no	—
167	#322	—	106	1	no	—
168	C₈²	—	106	1	no	—
169	C₅₃	#9719	1	1	yes	characteristic, complemented, Sylow
170	C₂×#41	—	2	2	no	—
171	C₂×D₈	—	2	1	no	—
172	C₂×D₈	—	2	1	no	—
173	C₂×M₁₆	—	53	2	no	—
174	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
175	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
176	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
177	#127	C₂×Dic₅₃	1	2	yes	—
178	#124	—	2	2	no	—
179	#121	—	2	2	no	—
180	#121	—	2	2	no	—
181	C₄×D₄	—	2	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	106	2	no	—
184	C₂×C₄²	—	106	2	no	—
185	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
186	#107	—	2	1	no	—
187	#107	—	2	1	no	—
188	#107	—	106	4	no	—
189	C₄⋊C₈	—	106	1	no	—
190	#104	—	212	8	no	—
191	#102	—	2	2	no	—
192	#102	—	2	2	no	—
193	C₂²⋊C₈	—	53	2	no	—
194	#97	—	106	1	no	—
195	C₄×C₈	—	106	1	no	—
196	C₄×C₈	—	106	1	no	—
197	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
198	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
199	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
200	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
201	#95	—	106	2	no	—
202	#41	—	4	4	no	—
203	#41	—	4	4	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	212	2	no	—
211	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
212	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
214	D₈	—	4	2	no	—
215	D₈	—	4	2	no	—
216	M₁₆	—	53	4	no	—
217	M₁₆	—	106	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	106	2	no	—
221	C₂×C₈	—	106	2	no	—
222	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
223	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
224	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
225	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	106	4	no	—
229	C₄⋊C₄	—	212	2	no	—
230	C₂²⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₄²	—	212	4	no	—
233	C₄²	#11719947	1	1	yes	characteristic
234	C₄²	#11719947	1	1	yes	characteristic
235	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
236	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
237	C₂³	—	4	2	no	—
238	C₂³	#13133213	1	1	yes	characteristic
239	Q₈	—	4	4	no	—
240	D₄	—	4	2	no	—
241	D₄	—	4	2	no	—
242	D₄	—	4	4	no	—
243	D₄	—	8	2	no	—
244	D₄	—	8	2	no	—
245	D₄	—	8	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	8	4	no	—
251	C₂×C₄	—	424	2	no	—
252	C₂×C₄	—	424	2	no	—
253	C₂×C₄	—	212	4	no	—
254	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
255	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
256	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
257	C₂×C₄	#13133188	1	1	yes	characteristic
258	C₂×C₄	#13133188	1	1	yes	characteristic
259	C₂×C₄	#13133204	1	1	yes	characteristic
260	C₂×C₄	D₈×D₅₃	1	2	yes	—
261	C₂×C₄	#13133236	1	2	yes	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	106	4	no	—
266	C₈	—	212	2	no	—
267	C₈	#13133144	1	2	yes	complemented
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	#16810526	1	1	yes	characteristic
274	C₂²	#16810998	1	1	yes	characteristic
275	C₂²	#16811006	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	8	2	no	—
280	C₄	—	424	4	no	—
281	C₄	C₂×#13133144	1	1	yes	characteristic
282	C₄	#16810640	1	1	yes	characteristic
283	C₄	D₈×Dic₅₃	1	2	yes	—
284	C₄	#16810856	1	2	yes	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	16	2	no	—
288	C₂	#20275582	1	1	yes	characteristic
289	C₂	#20275614	1	1	yes	characteristic
290	C₂	#20275635	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133188
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting, prime radical	C₅₃×#3169	C₂
socle	C₂²×C₅₃	#390
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#3169	D₅₃
hypocenter	C₅₃	#9719
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #20275635	C₂	semidirect
2. C₄×#13133151	C₂	semidirect
3. #20273210	C₂	semidirect
4. C₈×#11719917	C₂	semidirect
16. C₅₃×#500	C₄	semidirect
17. C₅₃×#493	C₄	semidirect
18. C₅₃×#486	C₄	semidirect
88. #11719917	D₈	semidirect
119. C₈×C₅₃	#104	semidirect
152. #500	Dic₅₃	semidirect
153. #493	Dic₅₃	semidirect
154. #486	Dic₅₃	semidirect
169. C₅₃	#9719	semidirect
267. C₈	#13133144	semidirect

Maximal subgroups

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

subgroup	index
#20275635	2
C₄×#13133151	2
#20273210	2
C₈×#11719917	2
C₅₃×#3169	2
#9719	53