Subgroups — #27271235

Order 13568.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810386	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1808	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₅₃	—	2	1	no	—
5	C₂²×#11719908	—	2	1	no	—
6	C₂²⋊#11719908	—	2	1	no	—
7	C₂²⋊#11719908	C₄	1	2	yes	complemented
8	#16810386	—	2	1	no	—
9	#16810386	C₄	1	2	yes	complemented
10	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#107×C₅₃	C₄	1	2	yes	complemented
12	C₅₃×#380	—	2	1	no	—
13	C₅₃×#340	—	2	1	no	—
14	C₂³×Dic₅₃	—	4	1	no	—
15	C₂×C₄⋊Dic₅₃	—	2	4	no	—
16	C₂×C₄⋊Dic₅₃	—	2	2	no	—
17	C₂×C₄⋊Dic₅₃	—	4	2	no	—
18	C₂×C₄⋊Dic₅₃	—	4	1	no	—
19	C₂×#11719908	—	2	2	no	—
20	C₂×#11719908	—	4	2	no	—
21	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	—	4	2	no	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
31	#107×C₅₃	—	4	2	no	—
32	#95×C₅₃	—	4	1	no	—
33	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	—	2	2	no	—
36	C₂²×C₄×C₅₃	—	4	2	no	—
37	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
43	C₂²×Dic₅₃	—	4	4	no	—
44	C₂²×Dic₅₃	—	4	2	no	—
45	C₂²×Dic₅₃	—	8	2	no	—
46	C₂²×Dic₅₃	—	8	1	no	—
47	C₂²×Dic₅₃	—	8	1	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	—	4	1	no	—
51	C₂×C₈×C₅₃	—	4	1	no	—
52	C₂×C₈×C₅₃	—	4	1	no	—
53	C₂×C₈×C₅₃	—	4	1	no	—
54	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
58	C₄⋊C₄×C₅₃	—	4	2	no	—
59	C₄⋊C₄×C₅₃	—	4	2	no	—
60	C₂²⋊C₄×C₅₃	—	8	1	no	—
61	C₂²⋊C₄×C₅₃	—	8	1	no	—
62	C₄⋊Dic₅₃	—	2	4	no	—
63	C₄⋊Dic₅₃	—	4	4	no	—
64	C₄⋊Dic₅₃	—	4	2	no	—
65	#11719908	—	4	2	no	—
66	C₂³×C₅₃	—	2	2	no	—
67	C₂³×C₅₃	—	2	2	no	—
68	C₂³×C₅₃	#102	1	1	yes	characteristic
69	C₂³×C₅₃	#102	1	1	yes	characteristic
70	C₂³×C₅₃	#107	1	1	yes	characteristic
71	C₂³×C₅₃	D₁₆	1	1	yes	characteristic
72	C₂³×C₅₃	SD₃₂	1	1	yes	characteristic
73	C₂³×C₅₃	SD₃₂	1	1	yes	characteristic
74	C₂³×C₅₃	Q₃₂	1	1	yes	characteristic
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	8	2	no	—
83	C₂×C₄×C₅₃	—	8	2	no	—
84	C₂×C₄×C₅₃	—	8	2	no	—
85	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#99	1	2	yes	—
87	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#104	1	2	yes	—
90	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
91	C₂×Dic₅₃	—	4	4	no	—
92	C₂×Dic₅₃	—	8	4	no	—
93	C₂×Dic₅₃	—	8	4	no	—
94	C₂×Dic₅₃	—	8	2	no	—
95	C₂×Dic₅₃	—	8	2	no	—
96	C₈×C₅₃	—	2	1	no	—
97	C₈×C₅₃	—	2	1	no	—
98	C₈×C₅₃	—	2	1	no	—
99	C₈×C₅₃	—	2	1	no	—
100	#7258	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	—	2	1	no	—
102	C₂²×C₅₃	—	2	1	no	—
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	—	2	2	no	—
105	C₂²×C₅₃	—	2	2	no	—
106	C₂²×C₅₃	—	2	2	no	—
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	#327	1	2	yes	—
110	C₂²×C₅₃	#340	1	1	yes	characteristic
111	C₂²×C₅₃	#357	1	1	yes	characteristic
112	C₂²×C₅₃	#357	1	1	yes	characteristic
113	C₂²×C₅₃	#358	1	1	yes	characteristic
114	C₂²×C₅₃	#358	1	1	yes	characteristic
115	C₂²×C₅₃	#360	1	2	yes	—
116	C₂²×C₅₃	#366	1	1	yes	characteristic
117	C₂²×C₅₃	#367	1	1	yes	characteristic
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	16	2	no	—
121	C₄×C₅₃	#328	1	2	yes	—
122	C₄×C₅₃	#339	1	1	yes	characteristic
123	C₄×C₅₃	#342	1	1	yes	characteristic
124	Dic₅₃	—	8	4	no	—
125	C₂×C₂²⋊C₁₆	—	53	1	no	—
126	#1808	D₅₃	1	1	yes	characteristic
127	C₂×#340	—	53	1	no	—
128	C₂×C₅₃	—	2	1	no	—
129	C₂×C₅₃	—	2	1	no	—
130	C₂×C₅₃	—	2	1	no	—
131	C₂×C₅₃	—	2	1	no	—
132	C₂×C₅₃	#1294	1	1	yes	characteristic
133	C₂×C₅₃	#1347	1	2	yes	—
134	C₂×C₅₃	#1348	1	2	yes	—
135	C₂×C₅₃	#1380	1	1	yes	characteristic
136	C₂×C₅₃	#1386	1	1	yes	characteristic
137	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
138	C₂²×C₄⋊C₄	—	106	1	no	—
139	C₂²×C₁₆	—	106	1	no	—
140	C₂×#107	Dic₅₃	1	2	yes	complemented
141	#380	—	2	1	no	—
142	C₂²⋊C₁₆	—	53	2	no	—
143	C₂²⋊C₁₆	—	106	1	no	—
144	#340	—	2	1	no	—
145	#340	—	53	2	no	—
146	#340	—	106	1	no	—
147	C₅₃	#7258	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₄	—	212	1	no	—
149	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
150	C₂²×C₈	—	4	2	no	—
151	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
152	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
153	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
154	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
155	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
156	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	106	4	no	—
159	C₂×C₄⋊C₄	—	106	2	no	—
160	C₂×C₄⋊C₄	—	212	2	no	—
161	C₂×C₄⋊C₄	—	212	1	no	—
162	C₂×C₂²⋊C₄	—	4	1	no	—
163	C₂×C₁₆	—	106	2	no	—
164	C₂×C₁₆	—	212	2	no	—
165	#107	—	4	2	no	—
166	#95	—	4	1	no	—
167	C₂⁴	#11719922	1	1	yes	characteristic
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	—	424	2	no	—
172	C₂²×C₄	—	424	1	no	—
173	C₂²×C₄	—	424	1	no	—
174	C₂²×C₄	—	212	4	no	—
175	C₂²×C₄	—	212	2	no	—
176	C₂²×C₄	#11719919	1	1	yes	characteristic
177	C₂²×C₄	#11719921	1	1	yes	characteristic
178	C₂²×C₄	#11719923	1	1	yes	characteristic
179	C₂²×C₄	#11719924	1	1	yes	characteristic
180	C₂²×C₄	#11719925	1	1	yes	characteristic
181	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	4	1	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	—	4	1	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
189	C₂×C₈	#11719919	1	1	yes	characteristic
190	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
191	C₂×C₈	#11719921	1	1	yes	characteristic
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	106	4	no	—
195	C₄⋊C₄	—	212	4	no	—
196	C₄⋊C₄	—	212	2	no	—
197	C₂²⋊C₄	—	8	1	no	—
198	C₂²⋊C₄	—	8	1	no	—
199	C₁₆	—	212	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	#13133137	1	1	yes	characteristic
203	C₂³	#13133138	1	1	yes	characteristic
204	C₂³	#13133150	1	1	yes	characteristic
205	C₂³	#13133158	1	1	yes	characteristic
206	C₂³	#13133160	1	1	yes	characteristic
207	C₂³	#13133161	1	1	yes	characteristic
208	C₂³	#13133163	1	1	yes	characteristic
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	—	424	4	no	—
220	C₂×C₄	—	424	4	no	—
221	C₂×C₄	—	424	2	no	—
222	C₂×C₄	—	424	2	no	—
223	C₂×C₄	—	212	4	no	—
224	C₂×C₄	#13133124	1	1	yes	characteristic
225	C₂×C₄	#13133132	1	2	yes	—
226	C₂×C₄	#13133140	1	1	yes	characteristic
227	C₂×C₄	#13133141	1	1	yes	characteristic
228	C₂×C₄	#13133144	1	2	yes	—
229	C₂×C₄	#13133148	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	#16810357	1	2	yes	—
243	C₂²	#16810385	1	1	yes	characteristic
244	C₂²	#16810423	1	1	yes	characteristic
245	C₂²	#16810424	1	1	yes	characteristic
246	C₂²	#16810426	1	1	yes	characteristic
247	C₂²	#16810427	1	1	yes	characteristic
248	C₂²	#16810433	1	2	yes	—
249	C₂²	#16810448	1	1	yes	characteristic
250	C₂²	#16810450	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	16	2	no	—
254	C₄	—	424	4	no	—
255	C₄	#16810360	1	2	yes	—
256	C₄	#16810382	1	1	yes	characteristic
257	C₄	#16810390	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	#20272593	1	1	yes	characteristic
263	C₂	#20272724	1	2	yes	—
264	C₂	#20272727	1	2	yes	—
265	C₂	#20272806	1	1	yes	characteristic
266	C₂	#20272820	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133138
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₂³×C₈	C₂×D₅₃
Fitting	C₅₃×#1808	C₂
socle	C₂³×C₅₃	#102
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂³×C₈×C₅₃	C₂²
hypercenter	#1808	D₅₃
hypocenter	C₅₃	#7258
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1808

Complements

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

N	H ≅ G/N	type
7. C₂²⋊#11719908	C₄	semidirect
9. #16810386	C₄	semidirect
11. C₂×#107×C₅₃	C₄	semidirect
140. C₂×#107	Dic₅₃	semidirect
147. C₅₃	#7258	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#11719908	2
C₂×#16810386	2
C₅₃×#1808	2
#7258	53