Subgroups — #27277136

Order 13568.

542 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274872	C₂	1	1	yes	maximal, characteristic, complemented
2	#20274715	C₂	1	2	yes	maximal
3	C₂×#16810374	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272599	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133149	—	2	1	no	—
7	C₂×#13133149	—	2	1	no	—
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	C₂×#13133138	—	2	2	no	—
10	#16810493	C₂²	1	2	yes	—
11	#16810385	—	2	2	no	—
12	#16810374	—	2	2	no	—
13	C₂³×C₈×C₅₃	—	2	1	no	—
14	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₅₃	—	2	2	no	—
16	C₈×D₄×C₅₃	C₄	1	4	yes	complemented
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	—	2	2	no	—
22	C₂×C₄⋊Dic₅₃	—	2	2	no	—
23	C₂×C₄⋊Dic₅₃	—	2	2	no	—
24	#13133149	—	4	2	no	—
25	#13133149	—	4	2	no	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	—	2	2	no	—
28	#13133138	—	4	4	no	—
29	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₅₃	—	2	1	no	—
31	C₂²×C₈×C₅₃	—	2	4	no	—
32	C₂²×C₈×C₅₃	—	2	1	no	—
33	C₂²×C₈×C₅₃	—	4	2	no	—
34	C₂²×C₈×C₅₃	—	4	2	no	—
35	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
37	C₄×D₄×C₅₃	—	2	1	no	—
38	C₄×D₄×C₅₃	—	2	1	no	—
39	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
40	C₄×D₄×C₅₃	Q₈	1	2	yes	—
41	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₅₃	—	2	1	no	—
45	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₂²⋊C₈×C₅₃	—	2	4	no	—
47	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₅₃	D₄	1	2	yes	—
49	C₂⁴×C₅₃	—	2	1	no	—
50	C₂×D₄×C₅₃	—	2	1	no	—
51	C₂×D₄×C₅₃	—	2	2	no	—
52	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂×D₄×C₅₃	SD₁₆	1	4	yes	—
54	C₂²×C₄×C₅₃	—	2	1	no	—
55	C₂²×C₄×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	—	4	1	no	—
58	C₂²×C₄×C₅₃	—	4	1	no	—
59	C₂²×C₄×C₅₃	—	4	1	no	—
60	C₂²×C₄×C₅₃	—	4	1	no	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
64	C₂²×Dic₅₃	—	4	2	no	—
65	C₂×C₈×C₅₃	—	2	1	no	—
66	C₂×C₈×C₅₃	—	2	1	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	—	4	1	no	—
69	C₂×C₈×C₅₃	—	4	1	no	—
70	C₂×C₈×C₅₃	—	4	8	no	—
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₅₃	D₈	1	2	yes	—
74	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
75	C₄⋊C₄×C₅₃	—	2	1	no	—
76	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
77	C₂²⋊C₄×C₅₃	—	2	2	no	—
78	C₂²⋊C₄×C₅₃	—	2	2	no	—
79	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
82	C₄²×C₅₃	#41	1	1	yes	characteristic
83	C₄⋊Dic₅₃	—	4	2	no	—
84	C₄⋊Dic₅₃	—	4	2	no	—
85	C₄⋊Dic₅₃	—	4	2	no	—
86	C₄⋊Dic₅₃	—	4	2	no	—
87	C₄⋊Dic₅₃	—	8	2	no	—
88	C₂×#88322	—	4	2	no	—
89	C₂×#88322	—	4	2	no	—
90	C₂×#88322	—	4	2	no	—
91	C₂³×C₅₃	—	2	2	no	—
92	C₂³×C₅₃	—	2	4	no	—
93	C₂³×C₅₃	—	4	1	no	—
94	C₂³×C₅₃	—	4	1	no	—
95	C₂³×C₅₃	—	4	2	no	—
96	C₂³×C₅₃	#121	1	1	yes	characteristic
97	D₄×C₅₃	—	2	2	no	—
98	D₄×C₅₃	—	2	4	no	—
99	D₄×C₅₃	#106	1	4	yes	complemented
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	4	1	no	—
103	C₂×C₄×C₅₃	—	4	1	no	—
104	C₂×C₄×C₅₃	—	4	2	no	—
105	C₂×C₄×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
117	C₂×C₄×C₅₃	#135	1	2	yes	—
118	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
120	C₂×Dic₅₃	—	8	2	no	—
121	C₂×Dic₅₃	—	8	2	no	—
122	C₂×Dic₅₃	—	8	2	no	—
123	C₈×C₅₃	—	2	2	no	—
124	C₈×C₅₃	—	4	2	no	—
125	C₈×C₅₃	#102	1	4	yes	complemented
126	#88322	—	8	2	no	—
127	#8642	—	53	1	no	maximal, Sylow
128	C₂²×C₅₃	—	2	4	no	—
129	C₂²×C₅₃	—	4	2	no	—
130	C₂²×C₅₃	—	4	2	no	—
131	C₂²×C₅₃	—	4	4	no	—
132	C₂²×C₅₃	—	4	4	no	—
133	C₂²×C₅₃	#380	1	1	yes	characteristic
134	C₂²×C₅₃	#459	1	1	yes	characteristic
135	C₂²×C₅₃	#461	1	1	yes	characteristic
136	C₂²×C₅₃	#465	1	2	yes	—
137	C₂²×C₅₃	#471	1	2	yes	—
138	C₄×C₅₃	—	2	1	no	—
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	—	4	2	no	—
141	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
142	C₄×C₅₃	#417	1	1	yes	characteristic
143	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
144	C₄×C₅₃	#427	1	1	yes	characteristic
145	C₄×C₅₃	#465	1	2	yes	—
146	C₄×C₅₃	#474	1	1	yes	characteristic
147	C₄×C₅₃	#476	1	1	yes	characteristic
148	Dic₅₃	—	16	2	no	—
149	C₂×C₈×D₄	D₅₃	1	1	yes	characteristic
150	#1844	—	53	1	no	—
151	#1802	—	53	2	no	—
152	C₂×#334	—	53	1	no	—
153	#1296	—	53	2	no	—
154	C₂×C₅₃	—	4	4	no	—
155	C₂×C₅₃	#1665	1	2	yes	—
156	C₂×C₅₃	#1671	1	2	yes	—
157	C₂×C₅₃	#1802	1	1	yes	characteristic
158	C₂×C₅₃	#1807	1	1	yes	characteristic
159	C₂×C₅₃	#1815	1	1	yes	characteristic
160	C₂³×C₈	—	2	1	no	—
161	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
162	C₈×D₄	—	2	2	no	—
163	C₈×D₄	Dic₅₃	1	4	yes	complemented
164	C₂×#106	—	106	1	no	—
165	C₂×#106	—	106	1	no	—
166	C₂×C₄⋊C₈	—	53	2	no	—
167	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
168	C₂×#102	—	106	2	no	—
169	C₂×C₂²⋊C₈	—	2	1	no	—
170	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
171	#384	—	53	2	no	—
172	#340	—	106	2	no	—
173	#334	—	106	2	no	—
174	C₅₃	#8642	1	1	yes	characteristic, complemented, Sylow
175	C₂²×D₄	D₂₁₂	1	1	yes	characteristic
176	C₂³×C₄	—	2	1	no	—
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	2	4	no	—
179	C₂²×C₈	—	4	2	no	—
180	C₂²×C₈	—	4	2	no	—
181	C₂²×C₈	—	106	2	no	—
182	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
183	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
184	C₄×D₄	—	2	1	no	—
185	C₄×D₄	—	2	1	no	—
186	C₄×D₄	Dic₁₀₆	1	2	yes	—
187	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
188	C₂×C₄⋊C₄	—	106	2	no	—
189	C₂×C₄⋊C₄	—	106	2	no	—
190	C₂×C₄⋊C₄	—	106	2	no	—
191	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
192	C₂×C₂²⋊C₄	—	2	1	no	—
193	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
194	#106	—	212	2	no	—
195	#106	—	212	2	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	106	2	no	—
198	C₄⋊C₈	—	106	2	no	—
199	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
200	#102	—	212	4	no	—
201	C₂²⋊C₈	—	2	4	no	—
202	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
203	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
204	C₂⁴	—	2	1	no	—
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	#11719913	1	4	yes	—
208	C₂×D₄	C₄⋊Dic₅₃	1	2	yes	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	212	2	no	—
217	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
218	C₂²×C₄	#11719945	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	8	no	—
226	C₂×C₈	—	212	2	no	—
227	C₂×C₈	—	212	2	no	—
228	C₂×C₈	—	212	2	no	—
229	C₂×C₈	#11719922	1	2	yes	—
230	C₂×C₈	#11719923	1	2	yes	—
231	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
232	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	424	2	no	—
235	C₄⋊C₄	—	212	2	no	—
236	C₄⋊C₄	—	212	2	no	—
237	C₄⋊C₄	—	212	2	no	—
238	C₄⋊C₄	—	212	2	no	—
239	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
243	C₄²	#11719945	1	1	yes	characteristic
244	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
245	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
246	C₂³	—	2	4	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	4	1	no	—
249	C₂³	—	4	1	no	—
250	C₂³	—	4	2	no	—
251	C₂³	#13133191	1	1	yes	characteristic
252	D₄	—	2	4	no	—
253	D₄	—	2	2	no	—
254	D₄	#13133149	1	4	yes	complemented
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	424	2	no	—
264	C₂×C₄	—	424	2	no	—
265	C₂×C₄	—	424	2	no	—
266	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
268	C₂×C₄	#13133176	1	1	yes	characteristic
269	C₂×C₄	#13133191	1	1	yes	characteristic
270	C₂×C₄	#13133196	1	1	yes	characteristic
271	C₂×C₄	#13133196	1	1	yes	characteristic
272	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
273	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
274	C₂×C₄	C₂×#11719913	1	2	yes	—
275	C₂×C₄	#13133239	1	2	yes	—
276	C₂×C₄	#13133242	1	1	yes	characteristic
277	C₂×C₄	#13133247	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	4	2	no	—
280	C₈	—	424	2	no	—
281	C₈	#13133138	1	4	yes	complemented
282	C₂²	—	2	4	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	4	4	no	—
287	C₂²	#16810483	1	1	yes	characteristic
288	C₂²	#16810818	1	1	yes	characteristic
289	C₂²	#16810828	1	1	yes	characteristic
290	C₂²	#16810852	1	2	yes	—
291	C₂²	#16810886	1	2	yes	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	2	no	—
295	C₄	—	848	2	no	—
296	C₄	C₂×#13133138	1	1	yes	characteristic
297	C₄	#16810624	1	1	yes	characteristic
298	C₄	C₂×#13133149	1	1	yes	characteristic
299	C₄	#16810661	1	1	yes	characteristic
300	C₄	#16810852	1	2	yes	—
301	C₄	#16810903	1	1	yes	characteristic
302	C₄	#16810913	1	1	yes	characteristic
303	C₂	—	4	4	no	—
304	C₂	#20274056	1	2	yes	—
305	C₂	#20274090	1	2	yes	—
306	C₂	#20274715	1	1	yes	characteristic
307	C₂	#20274740	1	1	yes	characteristic
308	C₂	#20274774	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133191
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₈×D₄×C₅₃	C₂
socle	C₂³×C₅₃	#121
prime radical	C₂²×D₄×C₅₃	D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈×D₄	D₅₃
hypocenter	C₅₃	#8642
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #20274872	C₂	semidirect
3. C₂×#16810374	C₂	semidirect
4. #20272599	C₂	semidirect
16. C₈×D₄×C₅₃	C₄	semidirect
99. D₄×C₅₃	#106	semidirect
125. C₈×C₅₃	#102	semidirect
163. C₈×D₄	Dic₅₃	semidirect
174. C₅₃	#8642	semidirect
254. D₄	#13133149	semidirect
281. C₈	#13133138	semidirect

Maximal subgroups

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

subgroup	index
#20274872	2
#20274715	2
C₂×#16810374	2
#20272599	2
C₂×C₈×D₄×C₅₃	2
#8642	53