Subgroups — #27300018

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719912	C₂	1	1	yes	maximal, characteristic
2	#20276471	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810404	C₂	1	2	yes	maximal
4	#20272798	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11719912	C₄	1	1	yes	characteristic
7	C₂²×#11719912	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₅₃	C₄	1	1	yes	characteristic
9	C₂×#13133157	C₂²	1	2	yes	—
10	C₄×#11719912	—	2	2	no	—
11	C₄×#11719912	—	2	2	no	—
12	C₂×#13133127	C₄	1	1	yes	characteristic
13	C₂×#13133127	C₂²	1	1	yes	characteristic
14	C₂×C₄×#88322	C₄	1	1	yes	characteristic
15	C₄⋊#11719908	—	2	2	no	—
16	#16810404	—	2	4	no	—
17	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×C₄⋊C₁₆	—	2	2	no	—
20	C₂²×C₄×D₅₃	C₈	1	2	yes	—
21	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×#11719912	—	2	2	no	—
23	C₂×#11719912	—	2	1	no	—
24	C₂×#11719912	—	2	1	no	—
25	C₂×#11719912	—	2	1	no	—
26	C₂×#11719912	—	2	1	no	—
27	C₂×#11719912	—	2	2	no	—
28	C₂×#11719912	—	2	2	no	—
29	C₂×#11719912	D₄	1	2	yes	—
30	C₂×#11719912	Q₈	1	2	yes	—
31	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
32	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
34	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×D₅₃	—	2	2	no	—
36	C₄²×D₅₃	—	2	1	no	—
37	C₄²×D₅₃	—	2	1	no	—
38	#13133157	—	2	2	no	—
39	#13133157	—	2	2	no	—
40	C₂×#11719908	—	2	2	no	—
41	#13133127	—	2	2	no	—
42	#13133127	—	2	1	no	—
43	#13133127	—	2	1	no	—
44	C₄×#88322	—	2	2	no	—
45	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
48	M₃₂×C₅₃	—	2	2	no	—
49	M₃₂×C₅₃	—	2	2	no	—
50	C₂×C₁₆×C₅₃	—	2	2	no	—
51	C₄×C₈×C₅₃	D₄	1	2	yes	—
52	C₄×C₈×C₅₃	D₄	1	2	yes	—
53	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
58	C₂×C₄×D₅₃	—	2	1	no	—
59	C₂×C₄×D₅₃	—	2	1	no	—
60	C₂×C₄×D₅₃	—	2	1	no	—
61	C₂×C₄×D₅₃	—	2	1	no	—
62	C₂×C₄×D₅₃	—	2	2	no	—
63	C₂×C₄×D₅₃	—	2	2	no	—
64	C₂×C₄×D₅₃	—	4	4	no	—
65	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
66	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
67	C₂×C₈×C₅₃	—	2	1	no	—
68	C₂×C₈×C₅₃	—	2	1	no	—
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
75	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
76	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
78	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
79	C₄×Dic₅₃	—	2	2	no	—
80	C₄×Dic₅₃	—	2	2	no	—
81	C₄×Dic₅₃	—	2	1	no	—
82	C₄×Dic₅₃	—	2	1	no	—
83	C₂×#88322	—	2	1	no	—
84	C₂×#88322	—	2	1	no	—
85	C₂×#88322	—	2	1	no	—
86	C₂×#88322	—	2	1	no	—
87	C₂×#88322	C₄⋊C₄	1	2	yes	—
88	C₂×#88322	M₁₆	1	2	yes	—
89	#11719912	—	2	8	no	—
90	#11719912	—	2	2	no	—
91	#11719912	—	2	2	no	—
92	#11719912	—	2	2	no	—
93	#11719912	—	2	2	no	—
94	C₁₆×C₅₃	—	4	2	no	—
95	#11719908	—	4	2	no	—
96	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
97	C₂²×D₅₃	—	2	1	no	—
98	C₂²×D₅₃	—	2	1	no	—
99	C₂²×D₅₃	—	4	2	no	—
100	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
103	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×Dic₅₃	—	2	1	no	—
111	C₂×Dic₅₃	—	2	1	no	—
112	C₂×Dic₅₃	—	2	2	no	—
113	C₂×Dic₅₃	—	4	2	no	—
114	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
115	C₄×D₅₃	—	2	2	no	—
116	C₄×D₅₃	—	2	2	no	—
117	C₄×D₅₃	—	2	2	no	—
118	C₄×D₅₃	—	2	2	no	—
119	C₄×D₅₃	—	4	4	no	—
120	C₄×D₅₃	—	4	4	no	—
121	C₈×C₅₃	—	2	2	no	—
122	C₈×C₅₃	#121	1	1	yes	characteristic
123	C₈×C₅₃	#121	1	1	yes	characteristic
124	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
125	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
126	#88322	—	2	2	no	—
127	#88322	—	2	2	no	—
128	#12785	—	53	1	no	maximal, Sylow
129	C₂²×C₅₃	—	2	2	no	—
130	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
132	C₂²×C₅₃	#431	1	1	yes	characteristic
133	C₂×D₅₃	—	2	2	no	—
134	C₂×D₅₃	—	4	2	no	—
135	C₂×D₅₃	—	4	2	no	—
136	C₂×D₅₃	—	4	2	no	—
137	C₄×C₅₃	—	2	2	no	—
138	C₄×C₅₃	#349	1	4	yes	complemented
139	C₄×C₅₃	#380	1	1	yes	characteristic
140	C₄×C₅₃	#407	1	1	yes	characteristic
141	C₄×C₅₃	#423	1	1	yes	characteristic
142	C₄×C₅₃	#432	1	1	yes	characteristic
143	Dic₅₃	—	2	2	no	—
144	Dic₅₃	—	4	2	no	—
145	C₂×C₄×M₁₆	—	53	1	no	—
146	#2150	—	53	1	no	—
147	#2150	D₅₃	1	1	yes	characteristic, complemented
148	C₂×#349	—	53	2	no	—
149	#1377	—	53	2	no	—
150	C₂×C₅₃	—	2	1	no	—
151	C₂×C₅₃	—	2	1	no	—
152	C₂×C₅₃	#1797	1	1	yes	characteristic
153	C₂×C₅₃	C₂×#349	1	2	yes	—
154	D₅₃	—	4	2	no	—
155	C₂²×M₁₆	—	53	1	no	—
156	C₂²×M₁₆	—	53	1	no	—
157	C₂²×C₄²	—	53	1	no	—
158	C₂×M₃₂	—	53	2	no	—
159	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
160	C₄×M₁₆	—	106	2	no	—
161	C₄×M₁₆	—	106	2	no	—
162	C₂×#97	—	53	1	no	—
163	C₂×#97	—	53	1	no	—
164	C₂×C₄×C₈	—	53	1	no	—
165	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
166	C₄⋊C₁₆	—	2	2	no	—
167	C₄⋊C₁₆	—	106	2	no	—
168	#349	—	106	4	no	—
169	C₅₃	#12785	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	—	53	1	no	—
171	C₂³×C₄	—	53	2	no	—
172	C₂×M₁₆	—	53	2	no	—
173	C₂×M₁₆	—	53	2	no	—
174	C₂×M₁₆	—	106	2	no	—
175	C₂×M₁₆	—	106	2	no	—
176	C₂×M₁₆	—	106	1	no	—
177	C₂×M₁₆	—	106	1	no	—
178	C₂×M₁₆	—	106	1	no	—
179	C₂×M₁₆	—	106	1	no	—
180	C₂×M₁₆	—	106	2	no	—
181	C₂²×C₈	—	53	1	no	—
182	C₂²×C₈	—	53	1	no	—
183	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
185	C₂×C₄²	—	53	1	no	—
186	C₂×C₄²	—	53	2	no	—
187	C₂×C₄²	—	106	1	no	—
188	C₂×C₄²	—	106	1	no	—
189	C₂×C₄²	—	106	2	no	—
190	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
191	M₃₂	—	2	2	no	—
192	M₃₂	—	2	2	no	—
193	M₃₂	—	106	2	no	—
194	M₃₂	—	106	2	no	—
195	C₂×C₁₆	—	2	2	no	—
196	C₂×C₁₆	—	106	2	no	—
197	#97	—	106	1	no	—
198	#97	—	106	1	no	—
199	#97	—	106	2	no	—
200	C₄×C₈	—	106	2	no	—
201	C₄×C₈	D₂₁₂	1	2	yes	—
202	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
203	C₂⁴	—	53	1	no	—
204	C₂²×C₄	—	53	1	no	—
205	C₂²×C₄	—	53	2	no	—
206	C₂²×C₄	—	53	2	no	—
207	C₂²×C₄	—	53	2	no	—
208	C₂²×C₄	—	106	1	no	—
209	C₂²×C₄	—	106	1	no	—
210	C₂²×C₄	—	106	1	no	—
211	C₂²×C₄	—	106	1	no	—
212	C₂²×C₄	—	106	2	no	—
213	C₂²×C₄	—	106	2	no	—
214	C₂²×C₄	—	212	4	no	—
215	C₂²×C₄	C₈×D₅₃	1	2	yes	—
216	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
217	M₁₆	—	106	8	no	—
218	M₁₆	—	106	2	no	—
219	M₁₆	—	106	2	no	—
220	M₁₆	—	106	2	no	—
221	M₁₆	—	106	2	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	53	2	no	—
225	C₂×C₈	—	53	2	no	—
226	C₂×C₈	—	106	1	no	—
227	C₂×C₈	—	106	1	no	—
228	C₂×C₈	—	106	1	no	—
229	C₂×C₈	—	106	1	no	—
230	C₂×C₈	#11719921	1	2	yes	—
231	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
232	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
233	C₂×C₈	#11719947	1	1	yes	characteristic
234	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
235	C₂×C₈	#11719949	1	1	yes	characteristic
236	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
237	C₄²	—	106	2	no	—
238	C₄²	—	106	2	no	—
239	C₄²	—	106	1	no	—
240	C₄²	—	106	1	no	—
241	C₄²	#11719912	1	1	yes	characteristic
242	C₄²	#11719912	1	1	yes	characteristic
243	C₄²	#11719921	1	2	yes	—
244	C₁₆	—	4	2	no	—
245	C₁₆	—	212	2	no	—
246	C₂³	—	53	2	no	—
247	C₂³	—	106	1	no	—
248	C₂³	—	106	1	no	—
249	C₂³	—	212	2	no	—
250	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	53	2	no	—
253	C₂×C₄	—	106	1	no	—
254	C₂×C₄	—	106	1	no	—
255	C₂×C₄	—	106	2	no	—
256	C₂×C₄	—	106	2	no	—
257	C₂×C₄	—	106	2	no	—
258	C₂×C₄	—	106	2	no	—
259	C₂×C₄	—	106	2	no	—
260	C₂×C₄	—	212	4	no	—
261	C₂×C₄	—	212	4	no	—
262	C₂×C₄	—	212	2	no	—
263	C₂×C₄	#13133129	1	2	yes	—
264	C₂×C₄	#13133129	1	2	yes	—
265	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
266	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
267	C₂×C₄	#13133175	1	1	yes	characteristic
268	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
269	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
270	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	106	2	no	—
273	C₈	—	106	2	no	—
274	C₈	#13133188	1	1	yes	characteristic
275	C₈	#13133192	1	1	yes	characteristic
276	C₈	#13133193	1	1	yes	characteristic
277	C₈	#13133197	1	1	yes	characteristic
278	C₂²	—	2	2	no	—
279	C₂²	—	106	2	no	—
280	C₂²	—	212	2	no	—
281	C₂²	—	212	2	no	—
282	C₂²	—	212	2	no	—
283	C₂²	C₂×#13133129	1	1	yes	characteristic
284	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
285	C₂²	#16810675	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	106	2	no	—
288	C₄	—	212	2	no	—
289	C₄	#16810404	1	4	yes	complemented
290	C₄	#16810480	1	1	yes	characteristic
291	C₄	#16810586	1	1	yes	characteristic
292	C₄	#16810645	1	1	yes	characteristic
293	C₄	#16810679	1	1	yes	characteristic
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	—	212	2	no	—
297	C₂	#20274692	1	1	yes	characteristic
298	C₂	C₂×#16810404	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11719921
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2150	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2150	D₅₃
hypocenter	C₅₃	#12785
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #20276471	C₂	semidirect
4. #20272798	C₂	semidirect
5. C₅₃×#2150	C₂	semidirect
138. C₄×C₅₃	#349	semidirect
147. #2150	D₅₃	semidirect
169. C₅₃	#12785	semidirect
289. C₄	#16810404	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11719912	2
#20276471	2
C₂×#16810404	2
#20272798	2
C₅₃×#2150	2
#12785	53