Subgroups — #27275240

Order 13568.

462 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133146	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133146	C₂	1	1	yes	maximal, characteristic
3	#20274460	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
5	#20272542	C₂	1	1	yes	maximal, characteristic
6	#20272542	C₂	1	1	yes	maximal, characteristic
7	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133146	C₄	1	2	yes	—
9	C₂×#13133146	C₄	1	2	yes	—
10	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
11	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
14	#16810377	C₄	1	2	yes	—
15	#16810377	C₄	1	2	yes	—
16	#16810377	C₄	1	2	yes	—
17	#16810377	C₄	1	2	yes	—
18	C₈×#88322	—	2	2	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#322	—	2	2	no	—
23	C₂²×#88322	C₂×C₄	1	2	yes	—
24	C₂²×#88322	C₂×C₄	1	2	yes	—
25	C₂²×#88322	C₂×C₄	1	2	yes	—
26	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	#13133146	—	2	4	no	—
33	#13133146	C₈	1	8	yes	complemented
34	C₄×#88322	—	2	1	no	—
35	C₄×#88322	—	2	1	no	—
36	C₄×#88322	—	2	1	no	—
37	C₄×#88322	—	2	1	no	—
38	C₄×#88322	—	2	1	no	—
39	C₄×#88322	—	2	1	no	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₅₃	—	2	1	no	—
45	C₄×C₈×C₅₃	—	2	1	no	—
46	C₄×C₈×C₅₃	—	2	1	no	—
47	C₄×C₈×C₅₃	—	2	1	no	—
48	C₄×C₈×C₅₃	D₄	1	2	yes	—
49	C₄×C₈×C₅₃	Q₈	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	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×Dic₅₃	C₄²	1	1	yes	characteristic
54	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
55	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
56	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
65	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
66	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₅₃	#41	1	1	yes	characteristic
69	C₄²×C₅₃	#41	1	1	yes	characteristic
70	C₄×Dic₅₃	—	2	2	no	—
71	C₄×Dic₅₃	—	2	2	no	—
72	C₄×Dic₅₃	—	2	2	no	—
73	C₄×Dic₅₃	—	2	2	no	—
74	C₄×Dic₅₃	—	2	2	no	—
75	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
76	C₂×#88322	—	2	2	no	—
77	C₂×#88322	—	2	2	no	—
78	C₂×#88322	—	2	2	no	—
79	C₂×#88322	—	2	2	no	—
80	C₂×#88322	—	2	2	no	—
81	C₂×#88322	—	2	2	no	—
82	C₂×#88322	—	2	2	no	—
83	C₂×#88322	C₂×C₈	1	4	yes	—
84	C₂×#88322	M₁₆	1	4	yes	—
85	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
93	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
94	C₂×C₄×C₅₃	#131	1	2	yes	—
95	C₂×C₄×C₅₃	#131	1	2	yes	—
96	C₂×C₄×C₅₃	#131	1	2	yes	—
97	C₂×C₄×C₅₃	#131	1	2	yes	—
98	C₂×Dic₅₃	—	2	1	no	—
99	C₂×Dic₅₃	—	2	1	no	—
100	C₂×Dic₅₃	—	2	1	no	—
101	C₂×Dic₅₃	—	2	1	no	—
102	C₂×Dic₅₃	—	2	1	no	—
103	C₂×Dic₅₃	—	2	1	no	—
104	C₂×Dic₅₃	—	2	1	no	—
105	C₂×Dic₅₃	—	2	1	no	—
106	C₂×Dic₅₃	—	2	1	no	—
107	C₂×Dic₅₃	—	2	1	no	—
108	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
109	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
110	C₈×C₅₃	—	2	4	no	—
111	C₈×C₅₃	—	2	4	no	—
112	C₈×C₅₃	—	4	2	no	—
113	#88322	—	2	4	no	—
114	#88322	—	2	4	no	—
115	#88322	—	4	2	no	—
116	#8264	—	53	1	no	maximal, Sylow
117	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
119	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
120	C₂²×C₅₃	#405	1	1	yes	characteristic
121	C₂²×C₅₃	#405	1	1	yes	characteristic
122	C₂²×C₅₃	#405	1	1	yes	characteristic
123	C₂²×C₅₃	#405	1	1	yes	characteristic
124	C₄×C₅₃	#424	1	2	yes	—
125	C₄×C₅₃	#433	1	2	yes	—
126	C₄×C₅₃	C₈×D₄	1	2	yes	—
127	C₄×C₅₃	#435	1	2	yes	—
128	C₄×C₅₃	C₈×Q₈	1	2	yes	—
129	C₄×C₅₃	#446	1	2	yes	—
130	Dic₅₃	—	2	2	no	—
131	Dic₅₃	—	2	2	no	—
132	Dic₅₃	—	4	1	no	—
133	Dic₅₃	—	4	1	no	—
134	C₄×C₄⋊C₈	—	53	1	no	—
135	C₄×C₄⋊C₈	—	53	1	no	—
136	#1744	—	53	1	no	—
137	C₂×#322	D₅₃	1	1	yes	characteristic
138	C₂×C₈²	—	53	1	no	—
139	#1273	—	53	1	no	—
140	#1273	—	53	1	no	—
141	C₂×C₅₃	C₈×M₁₆	1	1	yes	characteristic
142	C₂×C₅₃	#1450	1	1	yes	characteristic
143	C₂×C₅₃	#1453	1	1	yes	characteristic
144	C₂×C₅₃	#1454	1	1	yes	characteristic
145	C₂×C₅₃	#1768	1	1	yes	characteristic
146	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₅₃	#1771	1	1	yes	characteristic
148	C₂×C₄⋊C₈	—	53	2	no	—
149	C₂×C₄⋊C₈	—	53	2	no	—
150	C₂×C₄×C₈	—	53	1	no	—
151	C₂×C₄×C₈	—	53	1	no	—
152	C₂×C₄×C₈	—	53	1	no	—
153	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
156	C₄³	—	53	1	no	—
157	#336	—	53	2	no	—
158	#336	—	53	2	no	—
159	#336	—	53	2	no	—
160	#336	—	53	2	no	—
161	#322	—	2	2	no	—
162	C₈²	—	106	2	no	—
163	C₅₃	#8264	1	1	yes	characteristic, complemented, Sylow
164	C₂²×C₈	—	53	2	no	—
165	C₂²×C₈	—	53	2	no	—
166	C₂²×C₈	—	53	2	no	—
167	C₂²×C₈	C₄×D₅₃	1	2	yes	—
168	C₂²×C₈	C₄×D₅₃	1	2	yes	—
169	C₂²×C₈	C₄×D₅₃	1	2	yes	—
170	C₂×C₄²	—	53	1	no	—
171	C₂×C₄²	—	53	1	no	—
172	C₂×C₄²	—	53	1	no	—
173	C₂×C₄²	—	53	1	no	—
174	C₂×C₄²	—	53	1	no	—
175	C₂×C₄²	—	53	1	no	—
176	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
177	C₄⋊C₈	—	53	8	no	—
178	C₄⋊C₈	—	106	4	no	—
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	106	1	no	—
184	C₄×C₈	—	106	1	no	—
185	C₄×C₈	—	106	1	no	—
186	C₄×C₈	—	106	1	no	—
187	C₄×C₈	—	106	1	no	—
188	C₄×C₈	—	106	1	no	—
189	C₄×C₈	Dic₁₀₆	1	2	yes	—
190	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
191	C₂²×C₄	—	53	1	no	—
192	C₂²×C₄	—	53	1	no	—
193	C₂²×C₄	—	53	1	no	—
194	C₂²×C₄	—	53	1	no	—
195	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	53	4	no	—
206	C₂×C₈	—	53	4	no	—
207	C₂×C₈	—	106	2	no	—
208	C₂×C₈	—	106	2	no	—
209	C₂×C₈	—	106	2	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	C₈×D₅₃	1	4	yes	—
215	C₂×C₈	#11719919	1	4	yes	—
216	C₄²	—	53	4	no	—
217	C₄²	—	106	2	no	—
218	C₄²	—	106	2	no	—
219	C₄²	—	106	2	no	—
220	C₄²	—	106	2	no	—
221	C₄²	—	106	2	no	—
222	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
223	C₄²	#11719945	1	1	yes	characteristic
224	C₄²	#11719945	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
226	C₂³	C₄²×D₅₃	1	1	yes	characteristic
227	C₂×C₄	—	53	2	no	—
228	C₂×C₄	—	53	2	no	—
229	C₂×C₄	—	106	1	no	—
230	C₂×C₄	—	106	1	no	—
231	C₂×C₄	—	106	1	no	—
232	C₂×C₄	—	106	1	no	—
233	C₂×C₄	—	106	1	no	—
234	C₂×C₄	—	106	1	no	—
235	C₂×C₄	—	106	1	no	—
236	C₂×C₄	—	106	1	no	—
237	C₂×C₄	—	106	1	no	—
238	C₂×C₄	—	106	1	no	—
239	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
240	C₂×C₄	#13133174	1	1	yes	characteristic
241	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
242	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
243	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
244	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
245	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
246	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
247	C₂×C₄	#13133223	1	2	yes	—
248	C₂×C₄	#13133225	1	2	yes	—
249	C₂×C₄	#13133225	1	2	yes	—
250	C₂×C₄	#13133225	1	2	yes	—
251	C₈	—	2	4	no	—
252	C₈	—	2	4	no	—
253	C₈	—	4	2	no	—
254	C₈	—	106	4	no	—
255	C₈	—	106	4	no	—
256	C₈	—	212	2	no	—
257	C₂²	C₄×#11719919	1	1	yes	characteristic
258	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
259	C₂²	#16810575	1	1	yes	characteristic
260	C₂²	#16810577	1	1	yes	characteristic
261	C₂²	#16810580	1	1	yes	characteristic
262	C₂²	#16810580	1	1	yes	characteristic
263	C₂²	#16810580	1	1	yes	characteristic
264	C₄	—	106	2	no	—
265	C₄	—	106	2	no	—
266	C₄	—	212	1	no	—
267	C₄	—	212	1	no	—
268	C₄	#16810651	1	2	yes	—
269	C₄	#16810685	1	2	yes	—
270	C₄	C₈×C₂²⋊D₅₃	1	2	yes	—
271	C₄	#16810692	1	2	yes	—
272	C₄	C₈×Dic₁₀₆	1	2	yes	—
273	C₄	#16810751	1	2	yes	—
274	C₂	#20272994	1	1	yes	characteristic
275	C₂	#20273000	1	1	yes	characteristic
276	C₂	#20273011	1	1	yes	characteristic
277	C₂	#20273016	1	1	yes	characteristic
278	C₂	#20274575	1	1	yes	characteristic
279	C₂	C₈×#11719919	1	1	yes	characteristic
280	C₂	#20274591	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×C₅₃×#322	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#8264
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
33. #13133146	C₈	semidirect
163. C₅₃	#8264	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133146	2
#20274460	2
C₂×C₈×#88322	2
#20272542	2
C₂×C₅₃×#322	2
#8264	53