Subgroups — #27277009

Order 13568.

486 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×C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
2	#20274704	C₂	1	2	yes	maximal
3	#20273471	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272589	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₅₃	C₄	1	2	yes	—
7	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
8	C₈×D₂₁₂	—	2	1	no	—
9	C₈×D₂₁₂	—	2	1	no	—
10	C₈×D₂₁₂	—	2	1	no	—
11	C₈×D₂₁₂	—	2	1	no	—
12	#16810647	C₂²	1	2	yes	—
13	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
14	C₂×#13133133	—	2	2	no	—
15	C₂×#13133129	C₄	1	2	yes	—
16	#16810391	—	2	2	no	—
17	#16810374	—	2	1	no	—
18	#16810374	—	2	1	no	—
19	C₅₃×#423	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#334	—	2	1	no	—
22	C₅₃×#334	—	2	1	no	—
23	C₂²×D₂₁₂	D₄	1	1	yes	characteristic
24	C₂²×C₄×D₅₃	C₂×C₄	1	2	yes	—
25	C₂×#11719917	—	2	2	no	—
26	C₂²×#88322	C₂×C₄	1	2	yes	—
27	C₂×C₈×D₅₃	—	2	2	no	—
28	C₂×C₈×D₅₃	—	2	2	no	—
29	C₂×C₈×D₅₃	—	4	4	no	—
30	C₄×D₂₁₂	—	2	1	no	—
31	C₄×D₂₁₂	—	2	1	no	—
32	C₄×D₂₁₂	—	2	1	no	—
33	C₄×D₂₁₂	—	2	1	no	—
34	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
35	C₂×#11719921	Q₈	1	2	yes	—
36	C₄⋊#88322	—	2	1	no	—
37	C₄⋊#88322	—	2	1	no	—
38	C₄⋊#88322	—	2	2	no	—
39	C₄⋊#88322	—	2	2	no	—
40	#13133133	—	4	4	no	—
41	#13133129	—	2	2	no	—
42	#13133129	—	2	2	no	—
43	C₂×M₁₆×C₅₃	—	2	2	no	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₅₃	—	2	2	no	—
48	C₄⋊C₈×C₅₃	—	2	2	no	—
49	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₂³×D₅₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
58	C₂×D₂₁₂	—	2	1	no	—
59	C₂×D₂₁₂	—	2	1	no	—
60	C₂×D₂₁₂	—	2	2	no	—
61	C₂×D₂₁₂	SD₁₆	1	4	yes	—
62	C₂×C₄×D₅₃	—	2	2	no	—
63	C₂×C₄×D₅₃	—	2	2	no	—
64	C₂×C₄×D₅₃	—	4	2	no	—
65	C₂×C₄×D₅₃	—	4	2	no	—
66	M₁₆×C₅₃	—	4	2	no	—
67	M₁₆×C₅₃	—	4	2	no	—
68	C₂×C₈×C₅₃	—	2	2	no	—
69	C₂×C₈×C₅₃	—	4	1	no	—
70	C₂×C₈×C₅₃	—	4	2	no	—
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅₃	#41	1	1	yes	characteristic
78	C₄²×C₅₃	#41	1	1	yes	characteristic
79	#11719921	—	2	2	no	—
80	#11719921	—	2	2	no	—
81	C₄⋊Dic₅₃	—	2	1	no	—
82	C₄⋊Dic₅₃	—	2	1	no	—
83	#11719917	—	4	2	no	—
84	#11719917	—	4	2	no	—
85	C₂×#88322	—	2	2	no	—
86	C₂×#88322	—	4	2	no	—
87	C₂×#88322	—	4	2	no	—
88	C₈×D₅₃	—	4	4	no	—
89	C₈×D₅₃	—	4	4	no	—
90	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂²×D₅₃	—	2	2	no	—
92	C₂²×D₅₃	—	2	2	no	—
93	C₂²×D₅₃	—	4	2	no	—
94	C₂²×D₅₃	—	4	2	no	—
95	C₂²×D₅₃	#106	1	4	yes	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	4	1	no	—
99	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
106	C₂×C₄×C₅₃	#135	1	2	yes	—
107	C₂×Dic₅₃	—	2	2	no	—
108	C₂×Dic₅₃	—	4	2	no	—
109	D₂₁₂	—	2	4	no	—
110	D₂₁₂	—	2	4	no	—
111	C₄×D₅₃	—	4	4	no	—
112	C₄×D₅₃	—	4	2	no	—
113	C₄×D₅₃	—	4	2	no	—
114	C₈×C₅₃	—	2	1	no	—
115	C₈×C₅₃	—	2	1	no	—
116	C₈×C₅₃	—	2	1	no	—
117	C₈×C₅₃	—	8	2	no	—
118	C₈×C₅₃	#100	1	2	yes	complemented
119	#88322	—	4	2	no	—
120	#88322	—	8	2	no	—
121	#8614	—	53	1	no	maximal, Sylow
122	C₂²×C₅₃	—	2	1	no	—
123	C₂²×C₅₃	—	2	1	no	—
124	C₂²×C₅₃	#380	1	1	yes	characteristic
125	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
126	C₂²×C₅₃	#427	1	1	yes	characteristic
127	C₂×D₅₃	—	2	4	no	—
128	C₂×D₅₃	—	4	4	no	—
129	C₂×D₅₃	—	4	2	no	—
130	C₂×D₅₃	—	4	2	no	—
131	C₂×D₅₃	—	4	4	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	2	1	no	—
135	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
136	C₄×C₅₃	#413	1	1	yes	characteristic
137	C₄×C₅₃	#465	1	2	yes	—
138	C₄×C₅₃	#465	1	2	yes	—
139	Dic₅₃	—	4	2	no	—
140	C₂×C₈×D₄	—	53	1	no	—
141	#1800	—	53	2	no	—
142	#1566	—	53	1	no	—
143	#1566	D₅₃	1	1	yes	characteristic, complemented
144	#1292	—	53	2	no	—
145	C₂×C₅₃	—	2	1	no	—
146	C₂×C₅₃	—	2	1	no	—
147	C₂×C₅₃	#1800	1	1	yes	characteristic
148	C₂×C₅₃	#1807	1	1	yes	characteristic
149	C₂×C₅₃	#1813	1	1	yes	characteristic
150	D₅₃	—	4	4	no	—
151	C₂³×C₈	—	53	2	no	—
152	C₂×C₄×D₄	—	53	1	no	—
153	C₈×D₄	—	106	1	no	—
154	C₈×D₄	—	106	1	no	—
155	C₈×D₄	—	106	1	no	—
156	C₈×D₄	—	106	1	no	—
157	#423	—	53	2	no	—
158	#423	C₂×D₅₃	1	2	yes	—
159	C₂×C₄⋊C₈	—	53	1	no	—
160	C₂×#100	—	106	2	no	—
161	C₂×C₂²⋊C₈	—	53	2	no	—
162	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
163	#343	—	106	2	no	—
164	#334	—	2	1	no	—
165	#334	—	2	1	no	—
166	#334	—	106	1	no	—
167	#334	—	106	1	no	—
168	C₅₃	#8614	1	1	yes	characteristic, complemented, Sylow
169	C₂²×D₄	—	53	1	no	—
170	C₂³×C₄	—	53	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	106	2	no	—
173	C₂²×C₈	—	53	2	no	—
174	C₂²×C₈	—	106	2	no	—
175	C₂²×C₈	—	106	2	no	—
176	C₂²×C₈	—	212	4	no	—
177	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
178	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
179	C₄×D₄	—	106	1	no	—
180	C₄×D₄	—	106	1	no	—
181	C₄×D₄	—	106	1	no	—
182	C₄×D₄	—	106	1	no	—
183	C₂×C₄⋊C₄	—	53	1	no	—
184	C₂×C₂²⋊C₄	—	53	2	no	—
185	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	—	106	1	no	—
189	C₄⋊C₈	—	106	2	no	—
190	C₄⋊C₈	—	106	2	no	—
191	C₄⋊C₈	—	106	1	no	—
192	#100	—	212	4	no	—
193	C₂²⋊C₈	—	106	2	no	—
194	C₂²⋊C₈	—	106	2	no	—
195	C₄×C₈	D₂₁₂	1	1	yes	characteristic
196	C₄×C₈	D₂₁₂	1	1	yes	characteristic
197	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
198	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
199	C₂⁴	—	53	2	no	—
200	C₂×D₄	—	53	4	no	—
201	C₂×D₄	—	106	1	no	—
202	C₂×D₄	—	106	1	no	—
203	C₂×D₄	—	106	2	no	—
204	C₂²×C₄	—	53	2	no	—
205	C₂²×C₄	—	106	2	no	—
206	C₂²×C₄	—	106	2	no	—
207	C₂²×C₄	—	212	2	no	—
208	C₂²×C₄	—	212	2	no	—
209	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
210	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
211	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
212	M₁₆	—	4	2	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	212	2	no	—
215	M₁₆	—	212	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	106	2	no	—
220	C₂×C₈	—	212	4	no	—
221	C₂×C₈	—	212	2	no	—
222	C₂×C₈	—	212	4	no	—
223	C₂×C₈	—	212	2	no	—
224	C₂×C₈	#11719921	1	1	yes	characteristic
225	C₂×C₈	#11719921	1	1	yes	characteristic
226	C₂×C₈	#11719921	1	1	yes	characteristic
227	C₂×C₈	#11719921	1	1	yes	characteristic
228	C₄⋊C₄	—	106	1	no	—
229	C₄⋊C₄	—	106	1	no	—
230	C₂²⋊C₄	—	106	2	no	—
231	C₂²⋊C₄	—	106	2	no	—
232	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
233	C₄²	#11719947	1	1	yes	characteristic
234	C₄²	#11719949	1	1	yes	characteristic
235	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
236	C₂³	—	53	4	no	—
237	C₂³	—	106	2	no	—
238	C₂³	—	106	2	no	—
239	C₂³	—	212	2	no	—
240	C₂³	—	212	2	no	—
241	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
242	D₄	—	106	4	no	—
243	D₄	—	106	4	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	106	2	no	—
248	C₂×C₄	—	212	4	no	—
249	C₂×C₄	—	212	2	no	—
250	C₂×C₄	—	212	2	no	—
251	C₂×C₄	—	212	2	no	—
252	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
253	C₂×C₄	#13133175	1	1	yes	characteristic
254	C₂×C₄	#13133188	1	1	yes	characteristic
255	C₂×C₄	#13133192	1	1	yes	characteristic
256	C₂×C₄	#13133193	1	1	yes	characteristic
257	C₂×C₄	#13133197	1	1	yes	characteristic
258	C₂×C₄	SD₁₆×D₅₃	1	2	yes	—
259	C₂×C₄	#13133238	1	2	yes	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	8	2	no	—
264	C₈	—	424	2	no	—
265	C₈	—	212	2	no	—
266	C₈	#13133133	1	2	yes	complemented
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	106	4	no	—
270	C₂²	—	212	4	no	—
271	C₂²	—	212	4	no	—
272	C₂²	—	212	2	no	—
273	C₂²	—	212	2	no	—
274	C₂²	#16810480	1	1	yes	characteristic
275	C₂²	#106×D₅₃	1	1	yes	characteristic
276	C₂²	#16810658	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	212	2	no	—
281	C₄	C₂×#13133133	1	1	yes	characteristic
282	C₄	#16810605	1	1	yes	characteristic
283	C₄	#16810849	1	2	yes	—
284	C₄	#16810853	1	2	yes	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	212	4	no	—
288	C₂	#20274704	1	1	yes	characteristic
289	C₂	#20274737	1	1	yes	characteristic
290	C₂	#20274762	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810480
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1566	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂²×D₂₁₂	D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1566	D₅₃
hypocenter	C₅₃	#8614
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
3. #20273471	C₂	semidirect
4. #20272589	C₂	semidirect
5. C₅₃×#1566	C₂	semidirect
118. C₈×C₅₃	#100	semidirect
143. #1566	D₅₃	semidirect
168. C₅₃	#8614	semidirect
266. C₈	#13133133	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₂₁₂	2
#20274704	2
#20273471	2
#20272589	2
C₅₃×#1566	2
#8614	53