Subgroups — #27270642

Order 13568.

468 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274695	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1434	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88322	—	2	1	no	—
5	C₂×C₂²⋊#88322	—	2	1	no	—
6	C₂×C₂²⋊#88322	—	2	1	no	—
7	C₂×C₄×#88322	C₄	1	2	yes	complemented
8	#16810378	—	2	1	no	—
9	#16810378	C₄	1	2	yes	complemented
10	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#95×C₅₃	—	2	1	no	—
12	C₂×#95×C₅₃	—	2	1	no	—
13	C₂×#95×C₅₃	C₄	1	2	yes	—
14	C₂²×#88322	—	2	2	no	—
15	C₂²×#88322	—	2	2	no	—
16	C₄⋊#88322	—	4	2	no	—
17	C₂²⋊#88322	—	2	4	no	—
18	C₂²⋊#88322	—	4	1	no	—
19	C₂²⋊#88322	—	4	1	no	—
20	C₄×#88322	—	4	2	no	—
21	C₂³×C₄×C₅₃	—	2	2	no	—
22	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	—	4	2	no	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
30	#95×C₅₃	—	2	2	no	—
31	#95×C₅₃	—	4	2	no	—
32	#95×C₅₃	—	4	1	no	—
33	#95×C₅₃	—	4	1	no	—
34	#95×C₅₃	C₈	1	4	yes	complemented
35	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	—	2	4	no	—
39	C₂²×C₄×C₅₃	—	4	1	no	—
40	C₂²×C₄×C₅₃	—	4	1	no	—
41	C₂²×C₄×C₅₃	—	4	1	no	—
42	C₂²×C₄×C₅₃	—	4	1	no	—
43	C₂²×C₄×C₅₃	—	4	1	no	—
44	C₂²×C₄×C₅₃	—	4	2	no	—
45	C₂²×C₄×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	—	4	2	no	—
47	C₂²×C₄×C₅₃	—	4	2	no	—
48	C₂²×C₄×C₅₃	—	4	2	no	—
49	C₂²×C₄×C₅₃	—	4	2	no	—
50	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
58	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
62	C₄²×C₅₃	—	2	2	no	—
63	C₄²×C₅₃	—	2	2	no	—
64	C₄²×C₅₃	—	4	1	no	—
65	C₄²×C₅₃	—	4	1	no	—
66	C₂×#88322	—	2	4	no	—
67	C₂×#88322	—	4	2	no	—
68	C₂×#88322	—	4	2	no	—
69	C₂×#88322	—	4	2	no	—
70	C₂×#88322	—	4	2	no	—
71	C₂×#88322	—	4	2	no	—
72	C₂³×C₅₃	—	2	1	no	—
73	C₂³×C₅₃	—	2	1	no	—
74	C₂³×C₅₃	—	2	1	no	—
75	C₂³×C₅₃	—	2	1	no	—
76	C₂³×C₅₃	#95	1	1	yes	characteristic
77	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
78	C₂³×C₅₃	#102	1	1	yes	characteristic
79	C₂³×C₅₃	#103	1	1	yes	characteristic
80	C₂³×C₅₃	#104	1	2	yes	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	4	1	no	—
97	C₂×C₄×C₅₃	—	4	1	no	—
98	C₂×C₄×C₅₃	—	4	1	no	—
99	C₂×C₄×C₅₃	—	4	1	no	—
100	C₂×C₄×C₅₃	—	4	4	no	—
101	C₂×C₄×C₅₃	—	4	4	no	—
102	C₂×C₄×C₅₃	—	4	4	no	—
103	C₂×C₄×C₅₃	—	4	4	no	—
104	C₂×C₄×C₅₃	—	8	2	no	—
105	C₂×C₄×C₅₃	—	8	2	no	—
106	C₂×C₄×C₅₃	—	8	2	no	—
107	C₂×C₄×C₅₃	—	8	2	no	—
108	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#99	1	2	yes	—
111	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
114	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
116	#88322	—	4	4	no	—
117	#88322	—	8	2	no	—
118	#7024	—	53	1	no	maximal, Sylow
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	—	2	2	no	—
121	C₂²×C₅₃	—	2	1	no	—
122	C₂²×C₅₃	—	2	1	no	—
123	C₂²×C₅₃	—	2	2	no	—
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	2	1	no	—
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	D₄⋊C₈	1	2	yes	—
128	C₂²×C₅₃	Q₈⋊C₈	1	2	yes	—
129	C₂²×C₅₃	#336	1	1	yes	characteristic
130	C₂²×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
131	C₂²×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
132	C₂²×C₅₃	#339	1	1	yes	characteristic
133	C₂²×C₅₃	#340	1	1	yes	characteristic
134	C₂²×C₅₃	#342	1	1	yes	characteristic
135	C₂²×C₅₃	#344	1	1	yes	characteristic
136	C₄×C₅₃	—	2	1	no	—
137	C₄×C₅₃	—	2	1	no	—
138	C₄×C₅₃	—	2	2	no	—
139	C₄×C₅₃	—	2	2	no	—
140	C₄×C₅₃	—	2	1	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	2	1	no	—
144	C₄×C₅₃	—	4	1	no	—
145	C₄×C₅₃	—	8	4	no	—
146	#1797	—	53	1	no	—
147	C₄×C₂²⋊C₈	—	53	1	no	—
148	#1434	D₅₃	1	1	yes	characteristic
149	C₂×C₅₃	—	2	2	no	—
150	C₂×C₅₃	—	2	2	no	—
151	C₂×C₅₃	#1274	1	1	yes	characteristic
152	C₂×C₅₃	#1279	1	1	yes	characteristic
153	C₂×C₅₃	#1282	1	1	yes	characteristic
154	C₂×C₅₃	#1284	1	1	yes	characteristic
155	C₂×C₅₃	#1286	1	1	yes	characteristic
156	C₂×C₅₃	#1288	1	1	yes	characteristic
157	C₂×C₅₃	#1289	1	1	yes	characteristic
158	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
159	C₂×C₄⋊C₈	—	106	1	no	—
160	C₂×C₂²⋊C₈	—	106	1	no	—
161	C₂×C₂²⋊C₈	—	106	1	no	—
162	C₂×C₄×C₈	—	53	2	no	—
163	C₂×#95	—	2	1	no	—
164	C₂×#95	—	2	1	no	—
165	C₂×#95	Dic₅₃	1	2	yes	—
166	#336	—	53	2	no	—
167	#336	—	106	1	no	—
168	C₅₃	#7024	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₄	—	2	2	no	—
170	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
171	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
172	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
173	C₂²×C₈	—	106	2	no	—
174	C₂²×C₈	—	106	2	no	—
175	C₂×C₄²	—	4	2	no	—
176	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
177	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
178	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
179	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
180	C₄⋊C₈	—	212	2	no	—
181	C₂²⋊C₈	—	106	4	no	—
182	C₂²⋊C₈	—	212	1	no	—
183	C₂²⋊C₈	—	212	1	no	—
184	C₄×C₈	—	212	2	no	—
185	#95	—	2	2	no	—
186	#95	—	4	2	no	—
187	#95	—	4	1	no	—
188	#95	—	4	1	no	—
189	#95	#88322	1	4	yes	complemented
190	C₂⁴	C₂²⋊Dic₅₃	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	C₂×#88322	1	2	yes	—
206	C₂²×C₄	#11719917	1	2	yes	—
207	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
208	C₂²×C₄	#11719919	1	1	yes	characteristic
209	C₂²×C₄	#11719919	1	1	yes	characteristic
210	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
211	C₂²×C₄	#11719921	1	1	yes	characteristic
212	C₂²×C₄	#11719921	1	1	yes	characteristic
213	C₂²×C₄	#11719922	1	1	yes	characteristic
214	C₂²×C₄	#11719923	1	1	yes	characteristic
215	C₂²×C₄	#11719924	1	1	yes	characteristic
216	C₂²×C₄	#11719925	1	1	yes	characteristic
217	C₂×C₈	—	106	4	no	—
218	C₂×C₈	—	212	2	no	—
219	C₂×C₈	—	212	2	no	—
220	C₂×C₈	—	212	2	no	—
221	C₂×C₈	—	212	2	no	—
222	C₂×C₈	—	212	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	4	1	no	—
226	C₄²	—	4	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	1	no	—
231	C₂³	#13133124	1	1	yes	characteristic
232	C₂³	C₂²⋊#88322	1	2	yes	—
233	C₂³	#13133138	1	1	yes	characteristic
234	C₂³	#13133141	1	1	yes	characteristic
235	C₂³	#13133144	1	2	yes	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	8	2	no	—
260	C₂×C₄	—	8	2	no	—
261	C₂×C₄	—	8	2	no	—
262	C₂×C₄	—	8	2	no	—
263	C₂×C₄	C₄×#88322	1	1	yes	characteristic
264	C₂×C₄	#13133128	1	1	yes	characteristic
265	C₂×C₄	#13133131	1	2	yes	—
266	C₂×C₄	#13133137	1	1	yes	characteristic
267	C₂×C₄	#13133140	1	1	yes	characteristic
268	C₂×C₄	C₄⋊#88322	1	2	yes	—
269	C₂×C₄	#13133148	1	1	yes	characteristic
270	C₂×C₄	#13133150	1	1	yes	characteristic
271	C₈	—	424	2	no	—
272	C₈	—	212	4	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	D₄⋊#88322	1	2	yes	—
282	C₂²	Q₈⋊#88322	1	2	yes	—
283	C₂²	#16810378	1	1	yes	characteristic
284	C₂²	#16810379	1	1	yes	characteristic
285	C₂²	#16810379	1	1	yes	characteristic
286	C₂²	#16810382	1	1	yes	characteristic
287	C₂²	#16810385	1	1	yes	characteristic
288	C₂²	#16810389	1	1	yes	characteristic
289	C₂²	#16810393	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	1	no	—
299	C₄	—	8	4	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	2	no	—
302	C₂	#20272544	1	1	yes	characteristic
303	C₂	#20272556	1	1	yes	characteristic
304	C₂	#20272563	1	1	yes	characteristic
305	C₂	#20272567	1	1	yes	characteristic
306	C₂	#20272573	1	1	yes	characteristic
307	C₂	#20272579	1	1	yes	characteristic
308	C₂	#20272582	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133124
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₅₃
Fitting	C₅₃×#1434	C₂
socle	C₂³×C₅₃	#95
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₂²×C₄²×C₅₃	C₂²
hypercenter	#1434	D₅₃
hypocenter	C₅₃	#7024
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1434

Complements

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

N	H ≅ G/N	type
7. C₂×C₄×#88322	C₄	semidirect
9. #16810378	C₄	semidirect
34. #95×C₅₃	C₈	semidirect
168. C₅₃	#7024	semidirect
189. #95	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20274695	2
C₄×C₂²⋊#88322	2
C₅₃×#1434	2
#7024	53