Subgroups — #27288803

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	#20275651	C₂	1	1	yes	maximal, characteristic
2	#20274717	C₂	1	2	yes	maximal
3	#20272598	C₂	1	2	yes	maximal
4	#20272550	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
9	C₂×#13133140	—	2	2	no	—
10	#16810385	—	2	2	no	—
11	#16810377	—	2	1	no	—
12	#16810377	—	2	1	no	—
13	C₂×#107×C₅₃	—	2	1	no	—
14	C₂×#107×C₅₃	—	2	1	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#384	C₂²	1	2	yes	—
17	C₂²×Dic₁₀₆	D₄	1	1	yes	characteristic
18	C₂²×#88322	—	2	2	no	—
19	C₂²×#88322	—	2	1	no	—
20	C₄×Dic₁₀₆	—	2	2	no	—
21	C₄×Dic₁₀₆	—	2	2	no	—
22	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
23	C₂×#11719919	—	2	1	no	—
24	C₂×C₄×Dic₅₃	—	2	1	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	—	2	2	no	—
28	#13133140	—	4	4	no	—
29	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
34	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	#107×C₅₃	—	4	2	no	—
36	#107×C₅₃	—	4	2	no	—
37	C₄×C₈×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	—	4	2	no	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
42	C₂²×Dic₅₃	—	2	1	no	—
43	C₂²×Dic₅₃	—	2	1	no	—
44	C₂×Dic₁₀₆	—	2	2	no	—
45	C₂×Dic₁₀₆	—	2	2	no	—
46	C₂×Dic₁₀₆	—	2	1	no	—
47	C₂×Dic₁₀₆	D₈	1	2	yes	—
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	SD₁₆	1	2	yes	—
53	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
54	C₄⋊C₄×C₅₃	—	4	2	no	—
55	C₄⋊C₄×C₅₃	—	4	2	no	—
56	C₄⋊C₄×C₅₃	—	4	2	no	—
57	C₄⋊C₄×C₅₃	—	4	2	no	—
58	C₄⋊C₄×C₅₃	—	8	2	no	—
59	C₄²×C₅₃	#41	1	2	yes	—
60	C₄²×C₅₃	#41	1	2	yes	—
61	C₄⋊Dic₅₃	—	2	2	no	—
62	#11719919	—	4	2	no	—
63	C₄×Dic₅₃	—	4	1	no	—
64	C₄×Dic₅₃	—	4	1	no	—
65	C₂×#88322	—	4	2	no	—
66	C₂×#88322	—	4	1	no	—
67	C₂×#88322	—	4	2	no	—
68	C₂×#88322	—	4	1	no	—
69	C₂×#88322	—	4	2	no	—
70	C₂×#88322	—	4	1	no	—
71	C₂³×C₅₃	#121	1	1	yes	characteristic
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	8	2	no	—
75	C₂×C₄×C₅₃	—	8	2	no	—
76	C₂×C₄×C₅₃	—	8	2	no	—
77	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#123	1	2	yes	—
79	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#126	1	2	yes	—
81	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
85	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#137	1	2	yes	—
88	C₂×Dic₅₃	—	2	2	no	—
89	C₂×Dic₅₃	—	4	1	no	—
90	C₂×Dic₅₃	—	4	1	no	—
91	C₂×Dic₅₃	—	4	1	no	—
92	C₂×Dic₅₃	—	4	1	no	—
93	C₂×Dic₅₃	—	4	1	no	—
94	Dic₁₀₆	—	2	4	no	—
95	Dic₁₀₆	—	4	2	no	—
96	C₈×C₅₃	—	2	2	no	—
97	C₈×C₅₃	—	2	2	no	—
98	#88322	—	8	2	no	—
99	#88322	—	8	1	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	—	16	2	no	—
110	C₄×C₅₃	#480	1	2	yes	—
111	C₄×C₅₃	#483	1	1	yes	characteristic
112	C₄×C₅₃	#484	1	1	yes	characteristic
113	C₄×C₅₃	#485	1	2	yes	—
114	C₄×C₅₃	#487	1	1	yes	characteristic
115	C₄×C₅₃	#489	1	1	yes	characteristic
116	Dic₅₃	—	4	2	no	—
117	Dic₅₃	—	8	1	no	—
118	#1984	—	53	1	no	—
119	#1845	D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
131	C₂×#107	—	2	1	no	—
132	C₂×#107	—	2	1	no	—
133	C₂×C₄⋊C₈	—	53	2	no	—
134	C₂×C₄⋊C₈	—	53	1	no	—
135	C₂×#103	—	106	2	no	—
136	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
137	#384	C₂×D₅₃	1	2	yes	—
138	#340	—	106	2	no	—
139	#336	—	106	1	no	—
140	#336	—	106	1	no	—
141	C₅₃	#10617	1	1	yes	characteristic, complemented, Sylow
142	C₂²×Q₈	—	53	1	no	—
143	C₂²×C₈	—	106	2	no	—
144	C₂²×C₈	—	106	1	no	—
145	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
147	C₄×Q₈	—	106	2	no	—
148	C₄×Q₈	—	106	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	53	1	no	—
153	C₂×C₄⋊C₄	—	106	1	no	—
154	C₂×C₄²	—	106	1	no	—
155	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
156	#107	—	4	2	no	—
157	#107	—	4	2	no	—
158	C₄⋊C₈	—	106	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	—	53	2	no	—
164	C₂×Q₈	—	106	1	no	—
165	C₂×Q₈	—	106	2	no	—
166	C₂×Q₈	—	106	2	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	—	106	1	no	—
169	C₂²×C₄	—	106	1	no	—
170	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
171	C₂²×C₄	#11719947	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	—	212	2	no	—
178	C₂×C₈	—	212	1	no	—
179	C₂×C₈	—	212	2	no	—
180	C₂×C₈	—	212	1	no	—
181	C₂×C₈	—	212	1	no	—
182	C₂×C₈	—	212	2	no	—
183	C₂×C₈	#11719923	1	2	yes	—
184	C₂×C₈	#11719925	1	2	yes	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	8	2	no	—
190	C₄⋊C₄	—	106	2	no	—
191	C₄⋊C₄	—	212	2	no	—
192	C₄²	—	212	1	no	—
193	C₄²	—	212	1	no	—
194	C₄²	#11719945	1	2	yes	—
195	C₄²	#11719949	1	2	yes	—
196	C₂³	#13133188	1	1	yes	characteristic
197	Q₈	—	106	4	no	—
198	Q₈	—	212	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	—	8	2	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	—	106	2	no	—
205	C₂×C₄	—	212	1	no	—
206	C₂×C₄	—	212	1	no	—
207	C₂×C₄	—	212	1	no	—
208	C₂×C₄	—	212	1	no	—
209	C₂×C₄	—	212	1	no	—
210	C₂×C₄	#13133200	1	1	yes	characteristic
211	C₂×C₄	#13133202	1	2	yes	—
212	C₂×C₄	#13133206	1	1	yes	characteristic
213	C₂×C₄	#13133212	1	2	yes	—
214	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
215	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
216	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
217	C₂×C₄	#13133236	1	1	yes	characteristic
218	C₂×C₄	#13133243	1	1	yes	characteristic
219	C₂×C₄	#13133248	1	1	yes	characteristic
220	C₂×C₄	#13133249	1	2	yes	—
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	424	2	no	—
224	C₈	—	424	1	no	—
225	C₂²	#16810549	1	1	yes	characteristic
226	C₂²	#16810820	1	1	yes	characteristic
227	C₂²	#16810830	1	1	yes	characteristic
228	C₂²	#16810856	1	1	yes	characteristic
229	C₂²	#16810878	1	1	yes	characteristic
230	C₂²	#16810890	1	1	yes	characteristic
231	C₂²	#16810895	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	16	2	no	—
234	C₄	—	424	1	no	—
235	C₄	—	212	2	no	—
236	C₄	#16810938	1	2	yes	—
237	C₄	#16810953	1	1	yes	characteristic
238	C₄	#16810958	1	1	yes	characteristic
239	C₄	#16810966	1	2	yes	—
240	C₄	#16810975	1	1	yes	characteristic
241	C₄	#16810985	1	1	yes	characteristic
242	C₂	#20274072	1	1	yes	characteristic
243	C₂	#20274106	1	1	yes	characteristic
244	C₂	#20274206	1	1	yes	characteristic
245	C₂	#20274240	1	1	yes	characteristic
246	C₂	#20276124	1	1	yes	characteristic
247	C₂	#20276139	1	1	yes	characteristic
248	C₂	#20276156	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133188
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1845	C₂
socle, prime radical	C₂³×C₅₃	#121
hypercenter	#1845	D₅₃
hypocenter	C₅₃	#10617
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

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
#20275651	2
#20274717	2
#20272598	2
#20272550	2
C₅₃×#1845	2
#10617	53