Subgroups — #27288802

Order 13568.

366 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274875	C₂	1	1	yes	maximal, characteristic
2	#20274720	C₂	1	2	yes	maximal
3	#20272599	C₂	1	2	yes	maximal
4	#20272550	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1984	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133151	—	2	1	no	—
7	C₂×#13133151	—	2	1	no	—
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	C₂×#13133141	—	2	2	no	—
10	#16810493	C₂²	1	2	yes	—
11	#16810385	—	2	2	no	—
12	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#336	—	2	1	no	—
16	C₅₃×#336	—	2	1	no	—
17	C₂²×#88322	—	2	2	no	—
18	C₂×C₄⋊Dic₅₃	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	#13133151	—	4	2	no	—
22	#13133151	—	4	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	#13133141	—	4	4	no	—
26	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	—	2	1	no	—
28	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₄×Q₈×C₅₃	—	2	2	no	—
31	C₄×Q₈×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	—	2	1	no	—
35	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	—	2	2	no	—
37	C₄×C₈×C₅₃	—	2	2	no	—
38	C₂×Q₈×C₅₃	—	2	1	no	—
39	C₂×Q₈×C₅₃	—	2	2	no	—
40	C₂×Q₈×C₅₃	—	2	2	no	—
41	C₂×Q₈×C₅₃	D₈	1	2	yes	—
42	C₂²×C₄×C₅₃	—	2	1	no	—
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
47	C₂²×Dic₅₃	—	4	2	no	—
48	C₂×C₈×C₅₃	—	2	1	no	—
49	C₂×C₈×C₅₃	—	2	1	no	—
50	C₂×C₈×C₅₃	—	2	1	no	—
51	C₂×C₈×C₅₃	—	2	1	no	—
52	C₂×C₈×C₅₃	—	4	1	no	—
53	C₂×C₈×C₅₃	—	4	1	no	—
54	C₂×C₈×C₅₃	—	4	1	no	—
55	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
56	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
57	C₄⋊C₄×C₅₃	—	2	2	no	—
58	C₄⋊C₄×C₅₃	—	4	2	no	—
59	C₄²×C₅₃	—	4	1	no	—
60	C₄²×C₅₃	—	4	1	no	—
61	C₄²×C₅₃	#41	1	2	yes	—
62	C₄²×C₅₃	#41	1	2	yes	—
63	C₄⋊Dic₅₃	—	4	2	no	—
64	C₄⋊Dic₅₃	—	4	2	no	—
65	C₄⋊Dic₅₃	—	4	2	no	—
66	C₄⋊Dic₅₃	—	4	2	no	—
67	C₄⋊Dic₅₃	—	8	2	no	—
68	C₂×#88322	—	4	2	no	—
69	C₂×#88322	—	4	2	no	—
70	C₂×#88322	—	4	2	no	—
71	C₂³×C₅₃	#121	1	1	yes	characteristic
72	Q₈×C₅₃	—	2	4	no	—
73	Q₈×C₅₃	—	4	2	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	4	1	no	—
78	C₂×C₄×C₅₃	—	4	1	no	—
79	C₂×C₄×C₅₃	—	4	1	no	—
80	C₂×C₄×C₅₃	—	4	1	no	—
81	C₂×C₄×C₅₃	—	4	1	no	—
82	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#123	1	2	yes	—
84	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#126	1	2	yes	—
86	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
90	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#137	1	2	yes	—
93	C₂×Dic₅₃	—	8	2	no	—
94	C₂×Dic₅₃	—	8	2	no	—
95	C₂×Dic₅₃	—	8	2	no	—
96	C₈×C₅₃	—	2	2	no	—
97	C₈×C₅₃	—	2	2	no	—
98	C₈×C₅₃	—	8	1	no	—
99	#88322	—	8	2	no	—
100	#10617	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	#397	1	1	yes	characteristic
102	C₂²×C₅₃	#460	1	1	yes	characteristic
103	C₂²×C₅₃	#462	1	1	yes	characteristic
104	C₂²×C₅₃	#466	1	1	yes	characteristic
105	C₂²×C₅₃	#470	1	1	yes	characteristic
106	C₂²×C₅₃	#472	1	1	yes	characteristic
107	C₂²×C₅₃	#473	1	1	yes	characteristic
108	C₄×C₅₃	—	2	2	no	—
109	C₄×C₅₃	—	4	2	no	—
110	C₄×C₅₃	—	8	1	no	—
111	C₄×C₅₃	#480	1	2	yes	—
112	C₄×C₅₃	#483	1	1	yes	characteristic
113	C₄×C₅₃	#484	1	1	yes	characteristic
114	C₄×C₅₃	#485	1	2	yes	—
115	C₄×C₅₃	#487	1	1	yes	characteristic
116	C₄×C₅₃	#489	1	1	yes	characteristic
117	Dic₅₃	—	16	2	no	—
118	#1984	D₅₃	1	1	yes	characteristic
119	#1845	—	53	1	no	—
120	#1803	—	53	2	no	—
121	#1296	—	53	2	no	—
122	#1277	—	53	1	no	—
123	C₂×C₅₃	#1668	1	1	yes	characteristic
124	C₂×C₅₃	#1674	1	1	yes	characteristic
125	C₂×C₅₃	#1690	1	1	yes	characteristic
126	C₂×C₅₃	#1696	1	1	yes	characteristic
127	C₂×C₅₃	#2072	1	1	yes	characteristic
128	C₂×C₅₃	#2075	1	1	yes	characteristic
129	C₂×C₅₃	#2078	1	1	yes	characteristic
130	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
131	C₂×#107	—	106	1	no	—
132	C₂×#107	—	106	1	no	—
133	C₂×C₄⋊C₈	—	53	2	no	—
134	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
135	C₂×#103	—	106	2	no	—
136	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
137	#384	—	53	2	no	—
138	#340	—	106	2	no	—
139	#336	—	2	1	no	—
140	#336	—	2	1	no	—
141	C₅₃	#10617	1	1	yes	characteristic, complemented, Sylow
142	C₂²×Q₈	D₂₁₂	1	1	yes	characteristic
143	C₂²×C₈	—	2	1	no	—
144	C₂²×C₈	—	106	2	no	—
145	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
147	C₄×Q₈	—	2	2	no	—
148	C₄×Q₈	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	—	106	2	no	—
151	C₂×C₄⋊C₄	—	106	2	no	—
152	C₂×C₄⋊C₄	—	106	2	no	—
153	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
156	#107	—	212	2	no	—
157	#107	—	212	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	106	2	no	—
160	C₄⋊C₈	—	106	2	no	—
161	#103	—	212	4	no	—
162	C₄×C₈	—	2	2	no	—
163	C₂×Q₈	—	2	1	no	—
164	C₂×Q₈	—	2	2	no	—
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	D₄₂₄	1	2	yes	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	212	2	no	—
170	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
171	C₂²×C₄	#11719945	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	212	2	no	—
181	C₂×C₈	—	212	2	no	—
182	C₂×C₈	—	212	2	no	—
183	C₂×C₈	#11719924	1	2	yes	—
184	C₂×C₈	#11719925	1	2	yes	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	424	2	no	—
188	C₄⋊C₄	—	212	2	no	—
189	C₄⋊C₄	—	212	2	no	—
190	C₄⋊C₄	—	212	2	no	—
191	C₄⋊C₄	—	212	2	no	—
192	C₄²	—	4	1	no	—
193	C₄²	—	4	1	no	—
194	C₄²	#11719947	1	2	yes	—
195	C₄²	#11719947	1	2	yes	—
196	C₂³	#13133191	1	1	yes	characteristic
197	Q₈	—	2	4	no	—
198	Q₈	—	4	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	424	2	no	—
208	C₂×C₄	—	424	2	no	—
209	C₂×C₄	—	424	2	no	—
210	C₂×C₄	#13133198	1	1	yes	characteristic
211	C₂×C₄	#13133201	1	2	yes	—
212	C₂×C₄	#13133204	1	1	yes	characteristic
213	C₂×C₄	#13133210	1	2	yes	—
214	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
215	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
216	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
217	C₂×C₄	#13133239	1	1	yes	characteristic
218	C₂×C₄	#13133242	1	1	yes	characteristic
219	C₂×C₄	#13133247	1	1	yes	characteristic
220	C₂×C₄	#13133250	1	2	yes	—
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	8	1	no	—
224	C₈	—	424	2	no	—
225	C₂²	#16810548	1	1	yes	characteristic
226	C₂²	#16810823	1	1	yes	characteristic
227	C₂²	#16810833	1	1	yes	characteristic
228	C₂²	#16810859	1	1	yes	characteristic
229	C₂²	#16810881	1	1	yes	characteristic
230	C₂²	#16810893	1	1	yes	characteristic
231	C₂²	#16810898	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	8	1	no	—
235	C₄	—	848	2	no	—
236	C₄	#16810937	1	2	yes	—
237	C₄	#16810951	1	1	yes	characteristic
238	C₄	#16810956	1	1	yes	characteristic
239	C₄	#16810965	1	2	yes	—
240	C₄	#16810973	1	1	yes	characteristic
241	C₄	#16810982	1	1	yes	characteristic
242	C₂	#20274075	1	1	yes	characteristic
243	C₂	#20274109	1	1	yes	characteristic
244	C₂	#20274209	1	1	yes	characteristic
245	C₂	#20274243	1	1	yes	characteristic
246	C₂	#20276123	1	1	yes	characteristic
247	C₂	#20276138	1	1	yes	characteristic
248	C₂	#20276155	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133191
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1984	C₂
socle, prime radical	C₂³×C₅₃	#121
hypercenter	#1984	D₅₃
hypocenter	C₅₃	#10617
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

Complements

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

N	H ≅ G/N	type
141. C₅₃	#10617	semidirect

Maximal subgroups

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

subgroup	index
#20274875	2
#20274720	2
#20272599	2
#20272550	2
C₅₃×#1984	2
#10617	53