Subgroups — #28303023

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290570	C₂	1	2	yes	maximal
2	C₂×#16811484	C₂	1	2	yes	maximal, complemented
3	#20289545	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88324	C₂²	1	4	yes	—
6	C₂×#13133274	—	2	2	no	—
7	#16811489	—	2	2	no	—
8	#16811484	—	2	4	no	—
9	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
10	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×#88322	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×D₅₃	—	2	4	no	—
16	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
17	C₂²×#88324	—	2	4	no	—
18	C₄⋊#88324	—	2	4	no	—
19	C₄⋊#88324	—	2	4	no	—
20	#13133274	—	4	4	no	—
21	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
24	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
25	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×D₅₃	—	2	2	no	—
27	C₂×C₈×D₅₃	—	2	2	no	—
28	C₂×C₈×D₅₃	—	2	2	no	—
29	C₂×C₈×D₅₃	—	2	2	no	—
30	C₂×C₈×D₅₃	—	2	2	no	—
31	C₂×C₈×D₅₃	C₈	1	4	yes	complemented
32	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
33	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
34	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
35	C₄²×D₅₃	—	2	2	no	—
36	C₄²×D₅₃	—	2	2	no	—
37	C₄×#88322	—	2	2	no	—
38	C₈×Dic₅₃	—	2	2	no	—
39	C₈×Dic₅₃	D₄	1	4	yes	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	—	2	2	no	—
44	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×D₅₃	—	2	2	no	—
52	C₂×C₄×D₅₃	—	2	2	no	—
53	C₂×C₄×D₅₃	—	2	1	no	—
54	C₂×C₄×D₅₃	—	2	1	no	—
55	C₂×C₄×D₅₃	—	2	1	no	—
56	C₂×C₄×D₅₃	—	2	1	no	—
57	C₂×C₄×D₅₃	—	2	2	no	—
58	C₂×C₄×D₅₃	—	2	2	no	—
59	C₂×C₄×D₅₃	—	2	2	no	—
60	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
61	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
62	C₂×C₄×D₅₃	SD₁₆	1	4	yes	—
63	C₂×#88324	—	4	4	no	—
64	C₂×#88324	—	4	4	no	—
65	C₂×#88324	—	4	4	no	—
66	C₂×C₈×C₅₃	—	2	1	no	—
67	C₂×C₈×C₅₃	—	2	1	no	—
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
70	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
71	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
72	C₄²×C₅₃	—	2	1	no	—
73	C₄²×C₅₃	—	2	1	no	—
74	C₄×Dic₅₃	—	2	2	no	—
75	C₄×Dic₅₃	—	2	2	no	—
76	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
77	C₄×Dic₅₃	#41	1	2	yes	—
78	C₂×#88322	—	2	1	no	—
79	C₂×#88322	—	2	1	no	—
80	C₂×#88322	—	2	1	no	—
81	C₂×#88322	—	2	1	no	—
82	C₂×#88322	C₂²⋊C₄	1	2	yes	—
83	C₂×#88322	C₂×C₈	1	2	yes	—
84	C₈×D₅₃	—	2	8	no	—
85	C₈×D₅₃	—	2	8	no	—
86	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×D₅₃	—	2	1	no	—
88	C₂²×D₅₃	—	2	1	no	—
89	C₂²×D₅₃	—	2	1	no	—
90	C₂²×D₅₃	—	2	1	no	—
91	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
92	C₂²×D₅₃	#106	1	2	yes	—
93	C₂²×D₅₃	#108	1	2	yes	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	#106	1	2	yes	—
99	C₂×C₄×C₅₃	#108	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂×Dic₅₃	—	2	1	no	—
107	C₂×Dic₅₃	—	2	1	no	—
108	C₂×Dic₅₃	—	2	1	no	—
109	C₂×Dic₅₃	—	2	1	no	—
110	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
111	C₂×Dic₅₃	#121	1	2	yes	—
112	C₂×Dic₅₃	C₂²⋊Q₈	1	2	yes	—
113	C₂×Dic₅₃	C₂×SD₁₆	1	2	yes	—
114	C₂×Dic₅₃	#135	1	2	yes	—
115	C₄×D₅₃	—	2	8	no	—
116	C₄×D₅₃	—	2	2	no	—
117	C₄×D₅₃	—	2	2	no	—
118	C₄×D₅₃	—	2	2	no	—
119	C₄×D₅₃	—	2	2	no	—
120	C₄×D₅₃	—	2	8	no	—
121	#88324	—	8	4	no	—
122	C₈×C₅₃	—	2	2	no	—
123	C₈×C₅₃	C₂²⋊C₈	1	4	yes	complemented
124	#88322	—	2	2	no	—
125	#88322	—	2	2	no	—
126	#8595	—	53	1	no	maximal, Sylow
127	C₂²×C₅₃	#380	1	1	yes	characteristic
128	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
129	C₂²×C₅₃	#423	1	1	yes	characteristic
130	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
131	C₂²×C₅₃	#427	1	1	yes	characteristic
132	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
133	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
134	C₂×D₅₃	—	2	2	no	—
135	C₂×D₅₃	—	2	2	no	—
136	C₂×D₅₃	—	2	2	no	—
137	C₂×D₅₃	—	2	2	no	—
138	C₂×D₅₃	—	2	2	no	—
139	C₂×D₅₃	—	2	2	no	—
140	C₂×D₅₃	#334	1	4	yes	complemented
141	C₄×C₅₃	—	2	2	no	—
142	C₄×C₅₃	—	2	2	no	—
143	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
144	C₄×C₅₃	#407	1	1	yes	characteristic
145	C₄×C₅₃	#431	1	1	yes	characteristic
146	C₄×C₅₃	#432	1	1	yes	characteristic
147	Dic₅₃	—	2	2	no	—
148	Dic₅₃	—	2	2	no	—
149	Dic₅₃	#465	1	4	yes	—
150	C₂²×C₄×C₈	—	53	1	no	—
151	#1797	—	53	2	no	—
152	C₂×#334	—	53	2	no	—
153	#1277	—	53	2	no	—
154	C₂×C₅₃	C₂×#334	1	1	yes	characteristic
155	C₂×C₅₃	#1564	1	1	yes	characteristic
156	C₂×C₅₃	#1566	1	1	yes	characteristic
157	C₂×C₅₃	#1568	1	1	yes	characteristic
158	C₂×C₅₃	#1797	1	1	yes	characteristic
159	C₂×C₅₃	#1807	1	1	yes	characteristic
160	C₂×C₅₃	#1810	1	1	yes	characteristic
161	D₅₃	—	2	4	no	—
162	C₂³×C₈	—	53	1	no	—
163	C₂³×C₈	—	53	1	no	—
164	C₂²×C₄²	—	53	1	no	—
165	C₂×C₄⋊C₈	—	53	4	no	—
166	C₂×C₂²⋊C₈	—	106	2	no	—
167	C₂×C₄×C₈	—	53	1	no	—
168	C₂×C₄×C₈	—	53	1	no	—
169	C₂×C₄×C₈	—	53	1	no	—
170	C₂×C₄×C₈	—	106	4	no	—
171	C₂×C₄×C₈	#6157	1	1	yes	characteristic
172	#336	—	106	2	no	—
173	#334	—	106	4	no	—
174	C₅₃	#8595	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₄	—	53	1	no	—
176	C₂³×C₄	—	53	1	no	—
177	C₂³×C₄	—	53	1	no	—
178	C₂²×C₈	—	53	1	no	—
179	C₂²×C₈	—	53	1	no	—
180	C₂²×C₈	—	53	4	no	—
181	C₂²×C₈	—	106	2	no	—
182	C₂²×C₈	—	106	2	no	—
183	C₂²×C₈	—	106	2	no	—
184	C₂²×C₈	—	106	2	no	—
185	C₂²×C₈	—	106	2	no	—
186	C₂²×C₈	—	106	4	no	—
187	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
188	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
189	C₂×C₄²	—	53	1	no	—
190	C₂×C₄²	—	53	1	no	—
191	C₂×C₄²	—	53	1	no	—
192	C₂×C₄²	—	106	2	no	—
193	C₂×C₄²	—	106	2	no	—
194	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
195	C₄⋊C₈	—	106	4	no	—
196	C₄⋊C₈	—	106	4	no	—
197	C₂²⋊C₈	—	212	4	no	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	—	53	4	no	—
200	C₄×C₈	—	106	2	no	—
201	C₄×C₈	—	106	2	no	—
202	C₂⁴	—	53	1	no	—
203	C₂²×C₄	—	53	1	no	—
204	C₂²×C₄	—	53	1	no	—
205	C₂²×C₄	—	53	1	no	—
206	C₂²×C₄	—	53	4	no	—
207	C₂²×C₄	—	53	2	no	—
208	C₂²×C₄	—	53	2	no	—
209	C₂²×C₄	—	106	2	no	—
210	C₂²×C₄	—	106	2	no	—
211	C₂²×C₄	—	106	2	no	—
212	C₂²×C₄	—	106	2	no	—
213	C₂²×C₄	—	106	2	no	—
214	C₂²×C₄	—	106	1	no	—
215	C₂²×C₄	—	106	1	no	—
216	C₂²×C₄	—	106	1	no	—
217	C₂²×C₄	—	106	1	no	—
218	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
219	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
220	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	53	2	no	—
224	C₂×C₈	—	53	2	no	—
225	C₂×C₈	—	106	8	no	—
226	C₂×C₈	—	106	1	no	—
227	C₂×C₈	—	106	1	no	—
228	C₂×C₈	—	106	1	no	—
229	C₂×C₈	—	106	1	no	—
230	C₂×C₈	—	106	8	no	—
231	C₂×C₈	—	212	4	no	—
232	C₂×C₈	—	212	4	no	—
233	C₂×C₈	—	212	4	no	—
234	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
235	C₂×C₈	#11719936	1	2	yes	—
236	C₂×C₈	#11719936	1	2	yes	—
237	C₂×C₈	C₂²⋊#6157	1	2	yes	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	53	2	no	—
241	C₄²	—	53	2	no	—
242	C₄²	—	106	2	no	—
243	C₄²	—	106	2	no	—
244	C₂³	—	53	2	no	—
245	C₂³	—	53	2	no	—
246	C₂³	—	53	2	no	—
247	C₂³	—	106	1	no	—
248	C₂³	—	106	1	no	—
249	C₂³	—	106	1	no	—
250	C₂³	—	106	1	no	—
251	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	53	2	no	—
257	C₂×C₄	—	53	2	no	—
258	C₂×C₄	—	53	2	no	—
259	C₂×C₄	—	53	2	no	—
260	C₂×C₄	—	53	2	no	—
261	C₂×C₄	—	106	8	no	—
262	C₂×C₄	—	106	8	no	—
263	C₂×C₄	—	106	1	no	—
264	C₂×C₄	—	106	1	no	—
265	C₂×C₄	—	106	1	no	—
266	C₂×C₄	—	106	1	no	—
267	C₂×C₄	—	106	2	no	—
268	C₂×C₄	—	106	2	no	—
269	C₂×C₄	—	106	2	no	—
270	C₂×C₄	—	106	2	no	—
271	C₂×C₄	#13133285	1	2	yes	—
272	C₂×C₄	#13133287	1	2	yes	—
273	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
274	C₂×C₄	#13133296	1	1	yes	characteristic
275	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
276	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
277	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
278	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	106	2	no	—
281	C₈	—	106	2	no	—
282	C₈	—	424	4	no	—
283	C₈	#13133274	1	4	yes	complemented
284	C₂²	—	53	4	no	—
285	C₂²	—	106	2	no	—
286	C₂²	—	106	2	no	—
287	C₂²	—	106	2	no	—
288	C₂²	—	106	2	no	—
289	C₂²	—	106	2	no	—
290	C₂²	—	106	2	no	—
291	C₂²	#16811565	1	1	yes	characteristic
292	C₂²	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
293	C₂²	#16811627	1	1	yes	characteristic
294	C₂²	C₂×#13133285	1	1	yes	characteristic
295	C₂²	#16811633	1	1	yes	characteristic
296	C₂²	C₂×#13133287	1	1	yes	characteristic
297	C₂²	D₅₃⋊#108	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	53	4	no	—
301	C₄	—	106	2	no	—
302	C₄	—	106	2	no	—
303	C₄	C₂×#13133274	1	1	yes	characteristic
304	C₄	#16811594	1	1	yes	characteristic
305	C₄	#16811642	1	1	yes	characteristic
306	C₄	#16811645	1	1	yes	characteristic
307	C₂	—	106	4	no	—
308	C₂	D₅₃⋊#334	1	1	yes	characteristic
309	C₂	#20290263	1	1	yes	characteristic
310	C₂	#20290270	1	1	yes	characteristic
311	C₂	#20290276	1	1	yes	characteristic
312	C₂	#20290570	1	1	yes	characteristic
313	C₂	#20290591	1	1	yes	characteristic
314	C₂	#20290594	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂³×D₅₃	C₄⋊C₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#8595
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16811484	C₂	semidirect
3. #20289545	C₂	semidirect
31. C₂×C₈×D₅₃	C₈	semidirect
123. C₈×C₅₃	C₂²⋊C₈	semidirect
140. C₂×D₅₃	#334	semidirect
174. C₅₃	#8595	semidirect
283. C₈	#13133274	semidirect

Maximal subgroups

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

subgroup	index
#20290570	2
C₂×#16811484	2
#20289545	2
C₂×C₄×C₈×D₅₃	2
#8595	53