Subgroups — #27282966

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275510	C₂	1	2	yes	maximal, complemented
2	C₂×#16810374	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊#88322	C₂	1	2	yes	maximal, complemented
4	C₂×#16810344	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₂²⋊#88322	—	2	4	no	—
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	#16810374	—	2	2	no	—
11	D₄⋊#88322	—	2	4	no	—
12	#16810344	—	2	2	no	—
13	C₂²×D₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
15	C₄×D₈×C₅₃	—	2	2	no	—
16	C₄×D₈×C₅₃	—	2	2	no	—
17	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
18	C₂×#102×C₅₃	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	2	no	—
21	C₂²×#88322	—	2	1	no	—
22	C₂²×#88322	D₄	1	2	yes	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₂²⋊#88322	—	4	8	no	—
26	C₄×#88322	—	2	1	no	—
27	C₄×#88322	—	2	1	no	—
28	C₄×#88322	—	2	1	no	—
29	C₄×#88322	D₄	1	2	yes	—
30	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₅₃	—	2	2	no	—
32	C₂×D₈×C₅₃	—	2	2	no	—
33	C₂×D₈×C₅₃	—	2	2	no	—
34	C₂×D₈×C₅₃	C₈	1	4	yes	complemented
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×D₄×C₅₃	—	2	2	no	—
38	C₄×D₄×C₅₃	—	2	2	no	—
39	C₄×D₄×C₅₃	—	2	2	no	—
40	C₄×D₄×C₅₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	#107×C₅₃	—	2	2	no	—
45	#102×C₅₃	—	2	4	no	—
46	C₄×C₈×C₅₃	D₄	1	2	yes	—
47	C₄×C₈×C₅₃	D₄	1	2	yes	—
48	C₂⁴×C₅₃	—	2	2	no	—
49	C₂×D₄×C₅₃	—	2	2	no	—
50	C₂×D₄×C₅₃	—	2	2	no	—
51	C₂×D₄×C₅₃	—	2	2	no	—
52	C₂×D₄×C₅₃	—	2	2	no	—
53	C₂×D₄×C₅₃	—	2	2	no	—
54	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	—	4	2	no	—
57	C₂²×C₄×C₅₃	—	4	2	no	—
58	C₂²×C₄×C₅₃	—	4	4	no	—
59	C₂²×C₄×C₅₃	—	4	4	no	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
63	D₈×C₅₃	—	2	8	no	—
64	C₂×C₈×C₅₃	—	2	1	no	—
65	C₂×C₈×C₅₃	—	2	1	no	—
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
70	C₄⋊C₄×C₅₃	—	2	2	no	—
71	C₄⋊C₄×C₅₃	—	2	2	no	—
72	C₂²⋊C₄×C₅₃	—	4	2	no	—
73	C₂²⋊C₄×C₅₃	—	4	2	no	—
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅₃	#41	1	1	yes	characteristic
78	C₂×#88322	—	2	2	no	—
79	C₂×#88322	—	2	2	no	—
80	C₂×#88322	—	2	4	no	—
81	C₂×#88322	—	4	2	no	—
82	C₂×#88322	—	4	2	no	—
83	C₂×#88322	—	4	4	no	—
84	C₂³×C₅₃	—	2	4	no	—
85	C₂³×C₅₃	—	4	2	no	—
86	C₂³×C₅₃	—	4	2	no	—
87	C₂³×C₅₃	—	4	4	no	—
88	C₂³×C₅₃	—	4	4	no	—
89	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
90	D₄×C₅₃	—	2	4	no	—
91	D₄×C₅₃	—	2	4	no	—
92	D₄×C₅₃	—	4	2	no	—
93	D₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	4	4	no	—
97	C₂×C₄×C₅₃	—	8	4	no	—
98	C₂×C₄×C₅₃	—	8	2	no	—
99	C₂×C₄×C₅₃	—	8	2	no	—
100	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
110	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
111	C₈×C₅₃	—	2	2	no	—
112	C₈×C₅₃	C₂²⋊C₈	1	4	yes	complemented
113	#88322	—	4	2	no	—
114	#88322	—	4	2	no	—
115	#88322	—	8	2	no	—
116	#9659	—	53	1	no	maximal, Sylow
117	C₂²×C₅₃	—	4	8	no	—
118	C₂²×C₅₃	—	4	4	no	—
119	C₂²×C₅₃	—	8	4	no	—
120	C₂²×C₅₃	—	8	2	no	—
121	C₂²×C₅₃	—	8	2	no	—
122	C₂²×C₅₃	#390	1	1	yes	characteristic
123	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
124	C₂²×C₅₃	#436	1	1	yes	characteristic
125	C₂²×C₅₃	#442	1	2	yes	—
126	C₂²×C₅₃	#444	1	2	yes	—
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	8	2	no	—
130	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₄×C₅₃	#408	1	1	yes	characteristic
132	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
133	C₄×C₅₃	#436	1	1	yes	characteristic
134	C₄×C₅₃	#469	1	2	yes	—
135	C₄×C₅₃	#496	1	2	yes	—
136	C₂×C₄×D₈	D₅₃	1	1	yes	characteristic
137	#1954	—	53	2	no	—
138	C₂×#334	—	53	1	no	—
139	C₂×D₄⋊C₈	—	53	2	no	—
140	C₂×#322	—	53	1	no	—
141	C₂×C₅₃	—	8	4	no	—
142	C₂×C₅₃	#1580	1	2	yes	—
143	C₂×C₅₃	#1594	1	2	yes	—
144	C₂×C₅₃	#1954	1	1	yes	characteristic
145	C₂×C₅₃	#1972	1	1	yes	characteristic
146	C₂×C₅₃	#1975	1	1	yes	characteristic
147	C₂²×D₈	Dic₅₃	1	1	yes	characteristic
148	C₂×C₄×D₄	C₂×D₅₃	1	2	yes	—
149	C₄×D₈	—	2	2	no	—
150	C₄×D₈	—	2	2	no	—
151	C₂×#107	Dic₅₃	1	1	yes	characteristic
152	C₂×C₄⋊C₈	—	53	2	no	—
153	C₂×#102	Dic₅₃	1	2	yes	—
154	C₂×C₂²⋊C₈	—	106	4	no	—
155	C₂×C₄×C₈	—	53	1	no	—
156	C₂×C₄×C₈	—	53	1	no	—
157	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
158	#334	—	106	2	no	—
159	D₄⋊C₈	—	106	4	no	—
160	#322	—	106	2	no	—
161	C₅₃	#9659	1	1	yes	characteristic, complemented, Sylow
162	C₂²×D₄	C₂×Dic₅₃	1	2	yes	—
163	C₂³×C₄	—	2	2	no	—
164	C₂×D₈	—	2	2	no	—
165	C₂×D₈	—	2	2	no	—
166	C₂×D₈	#88322	1	4	yes	complemented
167	C₂²×C₈	—	53	2	no	—
168	C₂²×C₈	—	106	1	no	—
169	C₂²×C₈	—	106	2	no	—
170	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
171	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
172	C₄×D₄	—	2	2	no	—
173	C₄×D₄	—	2	2	no	—
174	C₄×D₄	—	2	2	no	—
175	C₄×D₄	—	2	2	no	—
176	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
177	C₂×C₂²⋊C₄	—	2	2	no	—
178	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
179	#107	—	2	2	no	—
180	C₄⋊C₈	—	106	2	no	—
181	C₄⋊C₈	—	106	2	no	—
182	#102	—	2	4	no	—
183	C₂²⋊C₈	—	212	8	no	—
184	C₄×C₈	—	53	2	no	—
185	C₄×C₈	—	106	1	no	—
186	C₄×C₈	—	106	1	no	—
187	C₄×C₈	—	106	1	no	—
188	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
189	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
190	C₂⁴	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	C₂×#88322	1	4	yes	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
203	C₂²×C₄	#11719947	1	1	yes	characteristic
204	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
205	D₈	—	2	8	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	106	2	no	—
209	C₂×C₈	—	106	2	no	—
210	C₂×C₈	—	106	4	no	—
211	C₂×C₈	—	212	2	no	—
212	C₂×C₈	—	212	4	no	—
213	C₂×C₈	—	212	2	no	—
214	C₂×C₈	C₂×#88322	1	2	yes	—
215	C₂×C₈	#11719917	1	2	yes	—
216	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
217	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₂²⋊C₄	—	4	2	no	—
222	C₄²	D₄×D₅₃	1	1	yes	characteristic
223	C₄²	#11719947	1	1	yes	characteristic
224	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
226	C₂³	—	2	4	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	4	no	—
230	C₂³	—	4	4	no	—
231	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
232	D₄	—	2	4	no	—
233	D₄	—	2	4	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	4	no	—
242	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
243	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
244	C₂×C₄	#13133188	1	1	yes	characteristic
245	C₂×C₄	#13133188	1	1	yes	characteristic
246	C₂×C₄	#13133204	1	1	yes	characteristic
247	C₂×C₄	#13133213	1	1	yes	characteristic
248	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
249	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
250	C₂×C₄	#13133224	1	1	yes	characteristic
251	C₂×C₄	#13133224	1	1	yes	characteristic
252	C₂×C₄	#13133241	1	4	yes	—
253	C₈	—	2	2	no	—
254	C₈	—	424	2	no	—
255	C₈	—	212	2	no	—
256	C₈	—	212	2	no	—
257	C₈	C₂²⋊#88322	1	4	yes	complemented
258	C₂²	—	4	4	no	—
259	C₂²	—	4	8	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	—	8	4	no	—
263	C₂²	#16810526	1	1	yes	characteristic
264	C₂²	D₄×#88322	1	1	yes	characteristic
265	C₂²	#16810701	1	1	yes	characteristic
266	C₂²	#16810735	1	2	yes	—
267	C₂²	#16810745	1	2	yes	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	8	2	no	—
271	C₄	C₂×C₂²⋊#88322	1	1	yes	characteristic
272	C₄	#16810590	1	1	yes	characteristic
273	C₄	D₄×#88322	1	1	yes	characteristic
274	C₄	#16810701	1	1	yes	characteristic
275	C₄	#16810873	1	2	yes	—
276	C₄	#16811011	1	2	yes	—
277	C₂	—	8	4	no	—
278	C₂	#20273539	1	2	yes	—
279	C₂	#20273629	1	2	yes	—
280	C₂	#20275510	1	1	yes	characteristic
281	C₂	#20275592	1	1	yes	characteristic
282	C₂	#20275609	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₄×D₈×C₅₃	C₂
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂²×D₈×C₅₃	C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₄×D₈	D₅₃
hypocenter	C₅₃	#9659
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #20275510	C₂	semidirect
2. C₂×#16810374	C₂	semidirect
3. C₂×D₄⋊#88322	C₂	semidirect
4. C₂×#16810344	C₂	semidirect
34. C₂×D₈×C₅₃	C₈	semidirect
112. C₈×C₅₃	C₂²⋊C₈	semidirect
161. C₅₃	#9659	semidirect
166. C₂×D₈	#88322	semidirect
257. C₈	C₂²⋊#88322	semidirect

Maximal subgroups

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

subgroup	index
#20275510	2
C₂×#16810374	2
C₂×D₄⋊#88322	2
C₂×#16810344	2
C₂×C₄×D₈×C₅₃	2
#9659	53