Subgroups — #27283034

Order 13568.

434 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20273463	C₂	1	1	yes	maximal, characteristic
2	#20273110	C₂	1	2	yes	maximal
3	#20273039	C₂	1	2	yes	maximal
4	#20273007	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	#16810676	C₂²	1	1	yes	characteristic
8	#16810676	C₂²	1	1	yes	characteristic
9	#16810376	—	2	1	no	—
10	#16810374	—	2	1	no	—
11	Q₈⋊#88322	—	2	4	no	—
12	#16810349	—	2	4	no	—
13	#16810345	—	2	1	no	—
14	#16810345	—	2	1	no	—
15	C₂²×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
17	C₄×Q₁₆×C₅₃	—	2	1	no	—
18	C₄×Q₁₆×C₅₃	—	2	1	no	—
19	C₄×Q₁₆×C₅₃	—	2	1	no	—
20	C₄×Q₁₆×C₅₃	—	2	1	no	—
21	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
22	C₂×#103×C₅₃	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₄⋊#88322	—	2	1	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	D₄	1	2	yes	—
27	C₂²⋊#88322	—	2	2	no	—
28	C₂²⋊#88322	—	2	1	no	—
29	C₂²⋊#88322	D₄	1	2	yes	—
30	C₄×#88322	—	2	1	no	—
31	C₄×#88322	—	2	1	no	—
32	C₄×#88322	—	2	2	no	—
33	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×Q₁₆×C₅₃	—	2	1	no	—
35	C₂×Q₁₆×C₅₃	—	2	1	no	—
36	C₂×Q₁₆×C₅₃	—	2	4	no	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₅₃	—	2	2	no	—
40	C₄×Q₈×C₅₃	—	2	2	no	—
41	C₄×Q₈×C₅₃	—	2	2	no	—
42	C₄×Q₈×C₅₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅₃	—	2	2	no	—
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	#107×C₅₃	—	2	1	no	—
48	#107×C₅₃	—	2	1	no	—
49	#103×C₅₃	—	2	2	no	—
50	#103×C₅₃	—	2	2	no	—
51	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
55	C₂×Q₈×C₅₃	—	2	2	no	—
56	C₂×Q₈×C₅₃	—	2	2	no	—
57	C₂×Q₈×C₅₃	—	2	2	no	—
58	C₂×Q₈×C₅₃	—	2	2	no	—
59	C₂×Q₈×C₅₃	—	2	2	no	—
60	C₂×Q₈×C₅₃	M₁₆	1	4	yes	—
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
66	Q₁₆×C₅₃	—	2	4	no	—
67	Q₁₆×C₅₃	—	2	4	no	—
68	C₂×C₈×C₅₃	—	2	2	no	—
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₄⋊C₄×C₅₃	—	2	2	no	—
75	C₄⋊C₄×C₅₃	—	2	2	no	—
76	C₄⋊C₄×C₅₃	—	4	2	no	—
77	C₄⋊C₄×C₅₃	—	4	2	no	—
78	C₄²×C₅₃	—	4	2	no	—
79	C₄²×C₅₃	—	4	2	no	—
80	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅₃	#41	1	1	yes	characteristic
84	C₂×#88322	—	2	2	no	—
85	C₂×#88322	—	2	2	no	—
86	C₂×#88322	—	4	2	no	—
87	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
88	Q₈×C₅₃	—	2	4	no	—
89	Q₈×C₅₃	—	2	4	no	—
90	Q₈×C₅₃	—	4	2	no	—
91	Q₈×C₅₃	—	4	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	4	no	—
94	C₂×C₄×C₅₃	—	4	1	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	8	2	no	—
99	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₅₃	#131	1	2	yes	—
107	C₈×C₅₃	—	2	1	no	—
108	C₈×C₅₃	—	2	1	no	—
109	C₈×C₅₃	—	2	1	no	—
110	C₈×C₅₃	—	2	1	no	—
111	#88322	—	4	2	no	—
112	#88322	—	4	2	no	—
113	#88322	—	8	2	no	—
114	#9671	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	—	2	1	no	—
116	C₂²×C₅₃	—	2	1	no	—
117	C₂²×C₅₃	#390	1	1	yes	characteristic
118	C₂²×C₅₃	#435	1	2	yes	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	2	1	no	—
121	C₄×C₅₃	—	2	1	no	—
122	C₄×C₅₃	—	4	4	no	—
123	C₄×C₅₃	—	8	2	no	—
124	C₄×C₅₃	#407	1	1	yes	characteristic
125	C₄×C₅₃	#408	1	1	yes	characteristic
126	C₄×C₅₃	#435	1	2	yes	—
127	C₄×C₅₃	#473	1	2	yes	—
128	C₂×C₄×Q₁₆	D₅₃	1	1	yes	characteristic
129	#1564	—	53	1	no	—
130	#1479	—	53	2	no	—
131	#1461	—	53	2	no	—
132	#1452	—	53	1	no	—
133	C₂×C₅₃	—	2	2	no	—
134	C₂×C₅₃	#1955	1	1	yes	characteristic
135	C₂×C₅₃	#1976	1	1	yes	characteristic
136	C₂×C₅₃	#1979	1	1	yes	characteristic
137	C₂²×Q₁₆	Dic₅₃	1	1	yes	characteristic
138	C₂×C₄×Q₈	C₂×D₅₃	1	2	yes	—
139	C₄×Q₁₆	—	2	1	no	—
140	C₄×Q₁₆	—	2	1	no	—
141	C₄×Q₁₆	—	2	1	no	—
142	C₄×Q₁₆	—	2	1	no	—
143	#431	—	53	1	no	—
144	#431	—	53	1	no	—
145	#431	—	53	2	no	—
146	C₂×#107	Dic₅₃	1	1	yes	characteristic
147	C₂×#103	Dic₅₃	1	2	yes	—
148	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
149	#335	—	106	1	no	—
150	#334	—	106	1	no	—
151	Q₈⋊C₈	—	106	4	no	—
152	#324	—	106	4	no	—
153	#322	—	106	1	no	—
154	#322	—	106	1	no	—
155	C₅₃	#9671	1	1	yes	characteristic, complemented, Sylow
156	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
157	C₂×Q₁₆	—	2	4	no	—
158	C₂×Q₁₆	—	2	1	no	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
161	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
162	C₄×Q₈	—	2	2	no	—
163	C₄×Q₈	—	2	2	no	—
164	C₄×Q₈	—	2	2	no	—
165	C₄×Q₈	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
168	C₂×C₄²	—	2	2	no	—
169	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
170	#107	—	2	1	no	—
171	#107	—	2	1	no	—
172	C₄⋊C₈	—	53	2	no	—
173	C₄⋊C₈	—	106	1	no	—
174	C₄⋊C₈	—	106	2	no	—
175	#103	—	2	2	no	—
176	#103	—	2	2	no	—
177	C₂²⋊C₈	—	53	2	no	—
178	C₂²⋊C₈	—	106	1	no	—
179	C₂²⋊C₈	—	106	2	no	—
180	C₄×C₈	—	106	2	no	—
181	C₄×C₈	—	106	1	no	—
182	C₄×C₈	—	106	1	no	—
183	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
184	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
185	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
186	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	#11719917	1	4	yes	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
196	C₂²×C₄	#11719947	1	1	yes	characteristic
197	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
198	Q₁₆	—	2	4	no	—
199	Q₁₆	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	106	2	no	—
203	C₂×C₈	—	106	2	no	—
204	C₂×C₈	—	212	2	no	—
205	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
207	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄²	—	4	2	no	—
214	C₄²	—	4	2	no	—
215	C₄²	D₄×D₅₃	1	1	yes	characteristic
216	C₄²	#11719947	1	1	yes	characteristic
217	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
218	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
219	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
220	Q₈	—	2	4	no	—
221	Q₈	—	2	4	no	—
222	Q₈	—	4	2	no	—
223	Q₈	—	4	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
232	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
233	C₂×C₄	#13133188	1	1	yes	characteristic
234	C₂×C₄	#13133188	1	1	yes	characteristic
235	C₂×C₄	#13133204	1	1	yes	characteristic
236	C₂×C₄	#13133213	1	1	yes	characteristic
237	C₂×C₄	C₂×#11719917	1	2	yes	—
238	C₂×C₄	#13133224	1	2	yes	—
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	424	2	no	—
244	C₈	—	212	2	no	—
245	C₈	—	212	2	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	#16810526	1	1	yes	characteristic
249	C₂²	#16810695	1	2	yes	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	4	no	—
254	C₄	—	8	2	no	—
255	C₄	#16810587	1	1	yes	characteristic
256	C₄	#16810590	1	1	yes	characteristic
257	C₄	#16810695	1	2	yes	—
258	C₄	#16810895	1	2	yes	—
259	C₂	—	2	2	no	—
260	C₂	#20275514	1	1	yes	characteristic
261	C₂	#20275614	1	1	yes	characteristic
262	C₂	#20275631	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133188
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₄×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	#390
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₄×Q₁₆	D₅₃
hypocenter	C₅₃	#9671
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
155. C₅₃	#9671	semidirect

Maximal subgroups

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

subgroup	index
#20273463	2
#20273110	2
#20273039	2
#20273007	2
C₂×C₄×Q₁₆×C₅₃	2
#9671	53