Subgroups — #27272153

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133217	C₂	1	1	yes	maximal, characteristic
2	#20274461	C₂	1	1	yes	maximal, characteristic
3	#20272539	C₂	1	4	yes	maximal
4	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133217	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₀₆	—	2	1	no	—
7	C₂×C₄×#88322	—	2	1	no	—
8	#16810529	—	2	1	no	—
9	#16810521	—	2	1	no	—
10	#16810521	C₂²	1	2	yes	—
11	#16810493	C₄	1	1	yes	characteristic
12	C₄×C₄⋊Dic₅₃	—	2	1	no	—
13	C₄×C₄⋊Dic₅₃	C₄	1	2	yes	—
14	#16810378	—	2	1	no	—
15	#16810378	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
17	C₄³×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#336	—	2	1	no	—
19	C₅₃×#336	C₂²	1	2	yes	—
20	C₂²×Dic₁₀₆	—	2	1	no	—
21	C₂²×#88322	—	2	2	no	—
22	#13133217	—	2	2	no	—
23	#13133217	C₈	1	4	yes	complemented
24	C₄×Dic₁₀₆	—	4	2	no	—
25	C₄×Dic₁₀₆	—	4	2	no	—
26	C₂×C₄⋊Dic₅₃	—	2	1	no	—
27	C₂×C₄⋊Dic₅₃	—	2	1	no	—
28	C₂×C₄⋊Dic₅₃	—	2	1	no	—
29	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
31	C₂×#11719919	—	2	2	no	—
32	C₂×C₄×Dic₅₃	—	2	2	no	—
33	C₄×#88322	—	2	4	no	—
34	#13133124	—	2	2	no	—
35	#13133124	—	2	2	no	—
36	C₂²×C₈×C₅₃	—	2	2	no	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₅₃	—	4	2	no	—
45	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₅₃	—	2	2	no	—
53	C₂²×Dic₅₃	—	2	2	no	—
54	C₂×Dic₁₀₆	—	2	4	no	—
55	C₂×Dic₁₀₆	—	4	2	no	—
56	C₂×Dic₁₀₆	—	4	2	no	—
57	C₂×C₈×C₅₃	—	4	2	no	—
58	C₂×C₈×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	—	4	2	no	—
60	C₄²×C₅₃	—	2	2	no	—
61	C₄²×C₅₃	—	2	2	no	—
62	C₄²×C₅₃	—	2	2	no	—
63	C₄²×C₅₃	—	4	1	no	—
64	C₄²×C₅₃	—	4	1	no	—
65	C₄²×C₅₃	C₂×C₈	1	2	yes	—
66	C₄²×C₅₃	M₁₆	1	2	yes	—
67	C₄²×C₅₃	D₈	1	2	yes	—
68	C₄²×C₅₃	SD₁₆	1	2	yes	—
69	C₄⋊Dic₅₃	—	2	4	no	—
70	C₄⋊Dic₅₃	—	2	2	no	—
71	C₄⋊Dic₅₃	—	4	2	no	—
72	C₄⋊Dic₅₃	—	4	2	no	—
73	C₄⋊Dic₅₃	—	4	2	no	—
74	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
75	#11719919	—	4	2	no	—
76	#11719919	—	4	2	no	—
77	C₄×Dic₅₃	—	4	4	no	—
78	C₂×#88322	—	2	4	no	—
79	C₂×#88322	—	4	2	no	—
80	C₂×#88322	—	4	2	no	—
81	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	C₂²⋊C₈	1	2	yes	—
95	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
96	C₂×C₄×C₅₃	#101	1	2	yes	—
97	C₂×C₄×C₅₃	#102	1	2	yes	—
98	C₂×C₄×C₅₃	#102	1	2	yes	—
99	C₂×C₄×C₅₃	#104	1	2	yes	—
100	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#137	1	2	yes	—
105	C₂×Dic₅₃	—	2	4	no	—
106	C₂×Dic₅₃	—	4	2	no	—
107	C₂×Dic₅₃	—	4	2	no	—
108	C₂×Dic₅₃	—	4	2	no	—
109	C₂×Dic₅₃	—	4	2	no	—
110	C₂×Dic₅₃	—	4	2	no	—
111	Dic₁₀₆	—	4	8	no	—
112	C₈×C₅₃	—	8	2	no	—
113	#88322	—	4	4	no	—
114	#7588	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
116	C₂²×C₅₃	#410	1	1	yes	characteristic
117	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
118	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
119	C₂²×C₅₃	#419	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
121	C₂²×C₅₃	#421	1	1	yes	characteristic
122	C₄×C₅₃	—	2	2	no	—
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	—	2	1	no	—
125	C₄×C₅₃	—	2	2	no	—
126	C₄×C₅₃	—	2	2	no	—
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	2	2	no	—
130	C₄×C₅₃	D₄⋊C₈	1	4	yes	—
131	Dic₅₃	—	4	4	no	—
132	Dic₅₃	—	8	2	no	—
133	C₄×C₄⋊Q₈	—	53	1	no	—
134	#1831	D₅₃	1	1	yes	characteristic
135	#1744	—	53	1	no	—
136	#1271	—	53	4	no	—
137	C₂×C₅₃	#1461	1	1	yes	characteristic
138	C₂×C₅₃	C₂×D₄⋊C₈	1	1	yes	characteristic
139	C₂×C₅₃	#1479	1	1	yes	characteristic
140	C₂×C₅₃	#1506	1	1	yes	characteristic
141	C₂×C₅₃	#1517	1	1	yes	characteristic
142	C₂×C₅₃	#1531	1	1	yes	characteristic
143	C₂×C₅₃	#1546	1	1	yes	characteristic
144	C₂×C₄⋊Q₈	—	53	1	no	—
145	C₂×C₄×Q₈	—	106	1	no	—
146	C₂×C₄⋊C₈	—	2	1	no	—
147	C₂×C₄×C₈	—	106	1	no	—
148	#391	—	106	1	no	—
149	#389	—	53	2	no	—
150	#389	—	106	1	no	—
151	#384	—	53	1	no	—
152	C₄×C₄⋊C₄	—	53	2	no	—
153	C₄×C₄⋊C₄	—	106	1	no	—
154	C₄³	C₂×D₅₃	1	1	yes	characteristic
155	#336	—	2	1	no	—
156	#336	—	53	2	no	—
157	#336	—	106	1	no	—
158	#336	C₂×D₅₃	1	2	yes	—
159	C₅₃	#7588	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₈	—	106	1	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	106	2	no	—
163	C₄⋊Q₈	—	53	4	no	—
164	C₄⋊Q₈	—	106	2	no	—
165	C₄×Q₈	—	212	2	no	—
166	C₄×Q₈	—	212	2	no	—
167	C₂×C₄⋊C₄	—	53	2	no	—
168	C₂×C₄⋊C₄	—	53	2	no	—
169	C₂×C₄⋊C₄	—	106	1	no	—
170	C₂×C₄⋊C₄	—	106	2	no	—
171	C₂×C₄⋊C₄	—	106	1	no	—
172	C₂×C₄⋊C₄	—	106	1	no	—
173	C₂×C₄²	—	106	2	no	—
174	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
175	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
176	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
177	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
178	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
180	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
181	C₄⋊C₈	—	4	2	no	—
182	C₄×C₈	—	106	4	no	—
183	#95	—	106	2	no	—
184	#95	—	106	2	no	—
185	C₂×Q₈	—	106	4	no	—
186	C₂×Q₈	—	212	2	no	—
187	C₂×Q₈	—	212	2	no	—
188	C₂²×C₄	—	106	2	no	—
189	C₂²×C₄	—	106	2	no	—
190	C₂²×C₄	#11719921	1	1	yes	characteristic
191	C₂²×C₄	#11719921	1	1	yes	characteristic
192	C₂²×C₄	#11719921	1	1	yes	characteristic
193	C₂²×C₄	#11719921	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	106	4	no	—
201	C₂×C₈	—	212	2	no	—
202	C₂×C₈	—	212	2	no	—
203	C₄⋊C₄	—	53	4	no	—
204	C₄⋊C₄	—	106	4	no	—
205	C₄⋊C₄	—	106	2	no	—
206	C₄⋊C₄	—	212	2	no	—
207	C₄⋊C₄	—	212	2	no	—
208	C₄⋊C₄	—	212	2	no	—
209	C₄⋊C₄	—	212	2	no	—
210	C₄⋊C₄	—	212	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	4	1	no	—
215	C₄²	—	4	1	no	—
216	C₄²	—	212	4	no	—
217	C₄²	C₈×D₅₃	1	2	yes	—
218	C₄²	#11719912	1	2	yes	—
219	C₄²	#11719922	1	2	yes	—
220	C₄²	#11719924	1	2	yes	—
221	C₂³	C₂×#11719921	1	1	yes	characteristic
222	Q₈	—	212	8	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	106	4	no	—
236	C₂×C₄	—	212	2	no	—
237	C₂×C₄	—	212	2	no	—
238	C₂×C₄	—	212	2	no	—
239	C₂×C₄	—	212	2	no	—
240	C₂×C₄	—	212	2	no	—
241	C₂×C₄	#13133129	1	2	yes	—
242	C₂×C₄	#13133129	1	2	yes	—
243	C₂×C₄	#13133135	1	2	yes	—
244	C₂×C₄	#13133137	1	2	yes	—
245	C₂×C₄	#13133137	1	2	yes	—
246	C₂×C₄	#13133143	1	2	yes	—
247	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
248	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
249	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
250	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
251	C₂×C₄	#13133250	1	2	yes	—
252	C₈	—	8	2	no	—
253	C₈	—	212	4	no	—
254	C₂²	C₂×#13133129	1	1	yes	characteristic
255	C₂²	#16810596	1	1	yes	characteristic
256	C₂²	C₂×#13133135	1	1	yes	characteristic
257	C₂²	C₂×#13133137	1	1	yes	characteristic
258	C₂²	#16810633	1	1	yes	characteristic
259	C₂²	C₂×#13133143	1	1	yes	characteristic
260	C₂²	#16810641	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	424	2	no	—
270	C₄	—	212	4	no	—
271	C₄	#16810350	1	4	yes	—
272	C₂	#20273038	1	1	yes	characteristic
273	C₂	C₂×#16810350	1	1	yes	characteristic
274	C₂	#20273111	1	1	yes	characteristic
275	C₂	#20273228	1	1	yes	characteristic
276	C₂	#20273274	1	1	yes	characteristic
277	C₂	#20273331	1	1	yes	characteristic
278	C₂	#20273388	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719921
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1831	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1831	D₅₃
hypocenter	C₅₃	#7588
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
23. #13133217	C₈	semidirect
159. C₅₃	#7588	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133217	2
#20274461	2
#20272539	2
C₅₃×#1831	2
#7588	53