Subgroups — #27276942

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	#20274695	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810375	C₂	1	2	yes	maximal, complemented
4	#20272549	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1797	C₂	1	1	yes	maximal, characteristic
6	#16810676	—	2	4	no	—
7	C₂×C₄⋊#88322	C₄	1	2	yes	—
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	C₂×C₂²⋊#88322	—	2	1	no	—
10	C₂×C₂²⋊#88322	C₄	1	2	yes	—
11	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	#16810378	—	2	1	no	—
14	#16810375	—	2	4	no	—
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
18	C₅₃×#336	—	2	1	no	—
19	C₂²×#88322	—	2	2	no	—
20	C₂²×#88322	C₂×C₄	1	2	yes	—
21	C₂²×#88322	C₂×C₄	1	2	yes	—
22	C₄⋊#88322	—	2	4	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₂²⋊#88322	—	2	4	no	—
26	C₂²⋊#88322	—	4	2	no	—
27	C₄×#88322	—	2	2	no	—
28	C₄×#88322	D₄	1	4	yes	—
29	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	—	2	2	no	—
34	C₂×C₄²×C₅₃	—	2	2	no	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	—	2	2	no	—
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₂²⋊C₈×C₅₃	—	4	2	no	—
42	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	—	2	2	no	—
45	C₂²×C₄×C₅₃	—	2	1	no	—
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	—	2	2	no	—
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	—	4	1	no	—
51	C₂²×C₄×C₅₃	—	4	1	no	—
52	C₂²×C₄×C₅₃	C₄⋊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	D₈	1	2	yes	—
56	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₅₃	—	4	2	no	—
61	C₂×C₈×C₅₃	—	4	2	no	—
62	C₂×C₈×C₅₃	—	4	2	no	—
63	C₄²×C₅₃	—	2	1	no	—
64	C₄²×C₅₃	—	2	1	no	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	C₂×D₄	1	2	yes	—
68	C₄²×C₅₃	#41	1	2	yes	—
69	C₂×#88322	—	2	2	no	—
70	C₂×#88322	—	2	2	no	—
71	C₂×#88322	—	2	2	no	—
72	C₂×#88322	—	2	2	no	—
73	C₂×#88322	—	2	2	no	—
74	C₂×#88322	—	4	2	no	—
75	C₂×#88322	—	4	2	no	—
76	C₂×#88322	—	4	2	no	—
77	C₂×#88322	C₂²⋊C₄	1	4	yes	—
78	C₂³×C₅₃	—	2	1	no	—
79	C₂³×C₅₃	—	2	1	no	—
80	C₂³×C₅₃	—	2	1	no	—
81	C₂³×C₅₃	—	2	1	no	—
82	C₂³×C₅₃	—	2	1	no	—
83	C₂³×C₅₃	—	2	1	no	—
84	C₂³×C₅₃	#107	1	2	yes	—
85	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
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	4	no	—
91	C₂×C₄×C₅₃	—	2	4	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	4	4	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	#107	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#121	1	2	yes	—
105	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
106	C₂×C₄×C₅₃	#131	1	2	yes	—
107	C₂×C₄×C₅₃	#131	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#135	1	2	yes	—
111	C₈×C₅₃	—	8	2	no	—
112	#88322	—	2	4	no	—
113	#88322	—	4	2	no	—
114	#88322	—	8	2	no	—
115	#8599	—	53	1	no	maximal, Sylow
116	C₂²×C₅₃	—	2	2	no	—
117	C₂²×C₅₃	—	2	2	no	—
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	—	4	1	no	—
120	C₂²×C₅₃	—	4	1	no	—
121	C₂²×C₅₃	—	4	1	no	—
122	C₂²×C₅₃	—	4	1	no	—
123	C₂²×C₅₃	#380	1	1	yes	characteristic
124	C₂²×C₅₃	#424	1	1	yes	characteristic
125	C₂²×C₅₃	#424	1	1	yes	characteristic
126	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
127	C₂²×C₅₃	#427	1	1	yes	characteristic
128	C₂²×C₅₃	#430	1	1	yes	characteristic
129	C₂²×C₅₃	#430	1	1	yes	characteristic
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	4	1	no	—
133	C₄×C₅₃	—	4	1	no	—
134	C₄×C₅₃	#408	1	2	yes	—
135	C₄×C₅₃	#433	1	2	yes	—
136	C₄×C₅₃	#466	1	2	yes	—
137	C₄×C₅₃	#467	1	2	yes	—
138	C₂×#431	—	53	1	no	—
139	#1797	—	53	1	no	—
140	#1797	D₅₃	1	1	yes	characteristic
141	C₂×#335	—	53	2	no	—
142	#1277	—	53	2	no	—
143	C₂×C₅₃	—	4	2	no	—
144	C₂×C₅₃	#1571	1	1	yes	characteristic
145	C₂×C₅₃	#1571	1	1	yes	characteristic
146	C₂×C₅₃	#1572	1	1	yes	characteristic
147	C₂×C₅₃	#1572	1	1	yes	characteristic
148	C₂×C₅₃	#1798	1	1	yes	characteristic
149	C₂×C₅₃	#1808	1	1	yes	characteristic
150	C₂×C₅₃	#1811	1	1	yes	characteristic
151	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
152	#431	—	106	4	no	—
153	C₂×C₄⋊C₈	—	53	2	no	—
154	C₂×C₄⋊C₈	—	53	2	no	—
155	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
156	C₂×C₂²⋊C₈	—	2	1	no	—
157	C₂×C₂²⋊C₈	—	53	2	no	—
158	C₂×C₂²⋊C₈	—	106	1	no	—
159	C₂×C₄×C₈	—	53	1	no	—
160	C₂×C₄×C₈	—	53	1	no	—
161	#336	—	2	1	no	—
162	#336	—	106	1	no	—
163	#335	—	106	4	no	—
164	C₅₃	#8599	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
166	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
167	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	53	2	no	—
170	C₂²×C₈	—	53	2	no	—
171	C₂²×C₈	—	106	2	no	—
172	C₂×C₄²	—	2	2	no	—
173	C₂×C₄²	—	2	2	no	—
174	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
175	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
176	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
177	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	106	4	no	—
181	C₄⋊C₈	—	106	2	no	—
182	C₄⋊C₈	—	106	2	no	—
183	C₂²⋊C₈	—	4	2	no	—
184	C₂²⋊C₈	—	106	4	no	—
185	C₂²⋊C₈	—	212	2	no	—
186	C₄×C₈	—	53	4	no	—
187	C₄×C₈	—	106	2	no	—
188	C₂⁴	#11719919	1	1	yes	characteristic
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	#11719919	1	1	yes	characteristic
199	C₂²×C₄	#11719919	1	1	yes	characteristic
200	C₂²×C₄	#11719919	1	1	yes	characteristic
201	C₂²×C₄	#11719922	1	2	yes	—
202	C₂²×C₄	#11719925	1	2	yes	—
203	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	53	4	no	—
210	C₂×C₈	—	106	2	no	—
211	C₂×C₈	—	106	2	no	—
212	C₂×C₈	—	106	2	no	—
213	C₂×C₈	—	106	2	no	—
214	C₂×C₈	—	106	2	no	—
215	C₂×C₈	—	212	2	no	—
216	C₂×C₈	—	212	2	no	—
217	C₂×C₈	—	212	2	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	D₄×D₅₃	1	2	yes	—
223	C₄²	#11719947	1	2	yes	—
224	C₂³	—	2	1	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	1	no	—
230	C₂³	#13133150	1	2	yes	—
231	C₂³	C₂×#11719919	1	1	yes	characteristic
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	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	—	4	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	#13133150	1	2	yes	—
248	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
249	C₂×C₄	#13133173	1	1	yes	characteristic
250	C₂×C₄	#13133189	1	2	yes	—
251	C₂×C₄	#13133194	1	2	yes	—
252	C₂×C₄	#13133223	1	2	yes	—
253	C₂×C₄	#13133225	1	2	yes	—
254	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
255	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
256	C₂×C₄	#13133237	1	2	yes	—
257	C₈	—	8	2	no	—
258	C₈	—	106	4	no	—
259	C₈	—	424	2	no	—
260	C₈	—	212	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	#16810481	1	1	yes	characteristic
269	C₂²	#16810649	1	1	yes	characteristic
270	C₂²	#16810651	1	1	yes	characteristic
271	C₂²	C₂×#13133150	1	1	yes	characteristic
272	C₂²	#16810659	1	1	yes	characteristic
273	C₂²	#16810671	1	1	yes	characteristic
274	C₂²	#16810671	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	#16810588	1	2	yes	—
280	C₄	#16810681	1	2	yes	—
281	C₄	#16810857	1	2	yes	—
282	C₄	#16810862	1	2	yes	—
283	C₂	—	4	2	no	—
284	C₂	#20273489	1	1	yes	characteristic
285	C₂	#20273491	1	1	yes	characteristic
286	C₂	#20273493	1	1	yes	characteristic
287	C₂	#20273496	1	1	yes	characteristic
288	C₂	#20274697	1	1	yes	characteristic
289	C₂	#20274742	1	1	yes	characteristic
290	C₂	#20274755	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1797	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1797	D₅₃
hypocenter	C₅₃	#8599
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
3. C₂×#16810375	C₂	semidirect
4. #20272549	C₂	semidirect
164. C₅₃	#8599	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
#20274695	2
C₂×#16810375	2
#20272549	2
C₅₃×#1797	2
#8599	53