Subgroups — C₂×#20272813 #27298992

Order 13568.

614 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133153	C₂	1	1	yes	maximal, characteristic
3	#20272813	C₂	1	4	yes	maximal, direct product factor
4	C₂²×#106×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	—	2	1	no	—
6	C₂×#13133157	—	2	2	no	—
7	C₂×#13133157	—	2	2	no	—
8	C₂×#13133157	C₄	1	2	yes	complemented
9	C₂×#13133157	C₂²	1	2	yes	—
10	C₂²×#11719908	C₄	1	2	yes	complemented
11	C₂×#13133153	—	2	2	no	—
12	C₂×#13133153	—	2	1	no	—
13	C₂×#13133153	C₄	1	4	yes	complemented
14	C₂×#13133153	C₂²	1	2	yes	—
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
17	C₂×#106×C₅₃	—	2	1	no	—
18	C₂×#106×C₅₃	—	2	2	no	—
19	C₂×#106×C₅₃	C₄	1	4	yes	—
20	C₂×#106×C₅₃	C₂²	1	2	yes	—
21	C₂×#11719917	—	2	4	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	4	2	no	—
24	C₂×#11719917	—	4	1	no	—
25	C₂²×#88322	—	4	1	no	—
26	#13133157	—	2	4	no	—
27	#13133157	—	2	2	no	—
28	#13133157	—	2	2	no	—
29	C₂×#11719908	—	2	4	no	—
30	C₂×#11719908	C₂×C₄	1	4	yes	complemented
31	#13133153	—	2	4	no	—
32	#13133153	—	2	2	no	—
33	#13133153	C₂×C₄	1	4	yes	complemented
34	C₂³×C₄×C₅₃	—	4	1	no	—
35	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	D₄	1	2	yes	—
41	C₂²×C₈×C₅₃	D₄	1	2	yes	—
42	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
47	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
48	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
49	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
50	#106×C₅₃	—	2	4	no	—
51	#106×C₅₃	—	2	2	no	—
52	#106×C₅₃	C₂×C₄	1	4	yes	—
53	C₂⁴×C₅₃	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	—	4	4	no	—
55	C₂²×C₄×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	—	8	4	no	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	D₈	1	2	yes	—
62	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₄²	1	2	yes	—
68	C₂×C₈×C₅₃	C₄²	1	2	yes	—
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
78	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
85	C₄⋊C₄×C₅₃	—	2	4	no	—
86	C₄⋊C₄×C₅₃	—	4	4	no	—
87	C₄⋊C₄×C₅₃	—	4	2	no	—
88	#11719917	—	2	4	no	—
89	#11719917	—	4	4	no	—
90	#11719917	—	4	2	no	—
91	C₂×#88322	—	4	4	no	—
92	C₂×#88322	—	4	2	no	—
93	#11719908	—	2	4	no	—
94	C₂³×C₅₃	—	2	4	no	—
95	C₂³×C₅₃	#102	1	2	yes	—
96	C₂³×C₅₃	#103	1	2	yes	—
97	C₂³×C₅₃	#106	1	1	yes	characteristic
98	C₂³×C₅₃	#106	1	1	yes	characteristic
99	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅₃	—	4	4	no	—
101	C₂×C₄×C₅₃	—	8	8	no	—
102	C₂×C₄×C₅₃	—	8	4	no	—
103	C₂×C₄×C₅₃	#95	1	2	yes	—
104	C₂×C₄×C₅₃	#95	1	2	yes	—
105	C₂×C₄×C₅₃	#102	1	2	yes	—
106	C₂×C₄×C₅₃	#102	1	2	yes	—
107	C₂×C₄×C₅₃	#102	1	2	yes	—
108	C₂×C₄×C₅₃	#103	1	2	yes	—
109	C₂×C₄×C₅₃	#103	1	2	yes	—
110	C₂×C₄×C₅₃	#103	1	2	yes	—
111	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#107	1	2	yes	—
114	C₂×C₄×C₅₃	#107	1	2	yes	—
115	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
121	C₈×C₅₃	#95	1	2	yes	—
122	C₈×C₅₃	#95	1	2	yes	—
123	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
124	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	#88322	—	4	4	no	—
128	C₂×#1383	—	53	1	no	maximal, Sylow
129	C₂²×C₅₃	—	2	4	no	—
130	C₂²×C₅₃	—	2	4	no	—
131	C₂²×C₅₃	—	2	2	no	—
132	C₂²×C₅₃	—	2	2	no	—
133	C₂²×C₅₃	#340	1	2	yes	—
134	C₂²×C₅₃	#340	1	2	yes	—
135	C₂²×C₅₃	#365	1	4	yes	—
136	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
137	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
138	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
139	C₄×C₅₃	—	8	4	no	—
140	C₄×C₅₃	#340	1	2	yes	—
141	C₄×C₅₃	#340	1	2	yes	—
142	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
143	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
144	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
145	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
146	C₂²×M₃₂	—	53	1	no	—
147	C₂²×#108	—	53	1	no	—
148	C₂²×#106	D₅₃	1	1	yes	characteristic
149	#1383	—	53	4	no	—
150	C₂×C₅₃	—	2	1	no	—
151	C₂×C₅₃	—	2	2	no	—
152	C₂×C₅₃	—	2	1	no	—
153	C₂×C₅₃	#1383	1	4	yes	complemented
154	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
155	C₂×C₅₃	C₂×#365	1	2	yes	—
156	C₂²×M₁₆	—	106	1	no	—
157	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
158	C₂²×C₄⋊C₄	—	2	1	no	—
159	C₂×M₃₂	—	53	2	no	—
160	C₂×M₃₂	—	53	2	no	—
161	C₂×M₃₂	—	106	2	no	—
162	C₂×M₃₂	—	106	2	no	—
163	C₂²×C₁₆	—	53	2	no	—
164	C₂×#108	—	53	4	no	—
165	C₂×#108	—	53	2	no	—
166	C₂×#108	—	106	2	no	—
167	C₂×#108	—	106	1	no	—
168	C₂×#106	—	2	2	no	—
169	C₂×#106	—	2	1	no	—
170	C₂×#106	Dic₅₃	1	4	yes	—
171	C₂×#106	C₂×D₅₃	1	2	yes	—
172	C₅₃	C₂×#1383	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	—	4	1	no	—
174	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
175	C₂×M₁₆	—	106	2	no	—
176	C₂×M₁₆	—	106	4	no	—
177	C₂×M₁₆	—	212	2	no	—
178	C₂×M₁₆	—	212	1	no	—
179	C₂²×C₈	—	212	1	no	—
180	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
181	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
182	C₂²×C₈	C₄×D₅₃	1	2	yes	—
183	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
184	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
185	C₂²×C₈	D₂₁₂	1	2	yes	—
186	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
187	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
188	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	4	no	—
192	C₂×C₄⋊C₄	—	4	2	no	—
193	C₂×C₄⋊C₄	—	4	1	no	—
194	M₃₂	—	106	2	no	—
195	M₃₂	—	106	2	no	—
196	M₃₂	—	106	4	no	—
197	C₂×C₁₆	—	53	4	no	—
198	C₂×C₁₆	—	106	4	no	—
199	#108	—	53	4	no	—
200	#108	—	106	2	no	—
201	#108	—	106	4	no	—
202	#106	—	2	2	no	—
203	#106	—	2	4	no	—
204	#106	C₂×Dic₅₃	1	4	yes	—
205	C₂⁴	#11719923	1	1	yes	characteristic
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	8	4	no	—
209	C₂²×C₄	#11719919	1	1	yes	characteristic
210	C₂²×C₄	#11719919	1	1	yes	characteristic
211	C₂²×C₄	#11719921	1	2	yes	—
212	C₂²×C₄	#11719922	1	2	yes	—
213	C₂²×C₄	#11719923	1	1	yes	characteristic
214	C₂²×C₄	#11719924	1	1	yes	characteristic
215	C₂²×C₄	#11719924	1	1	yes	characteristic
216	C₂²×C₄	#11719925	1	2	yes	—
217	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
218	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
219	M₁₆	—	106	4	no	—
220	M₁₆	—	212	4	no	—
221	M₁₆	—	212	2	no	—
222	C₂×C₈	—	212	4	no	—
223	C₂×C₈	—	212	2	no	—
224	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
225	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
226	C₂×C₈	#11719919	1	1	yes	characteristic
227	C₂×C₈	#11719919	1	1	yes	characteristic
228	C₂×C₈	#11719919	1	2	yes	—
229	C₂×C₈	#11719919	1	2	yes	—
230	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
231	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
232	C₂×C₈	#11719921	1	2	yes	—
233	C₂×C₈	#11719921	1	2	yes	—
234	C₂×C₈	#11719921	1	2	yes	—
235	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
236	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
239	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
240	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
241	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₁₆	—	106	4	no	—
246	C₂³	—	2	4	no	—
247	C₂³	#13133138	1	2	yes	—
248	C₂³	#13133140	1	2	yes	—
249	C₂³	#13133148	1	1	yes	characteristic
250	C₂³	#13133148	1	1	yes	characteristic
251	C₂³	C₂×#11719923	1	1	yes	characteristic
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	8	4	no	—
254	C₂×C₄	—	8	8	no	—
255	C₂×C₄	#13133124	1	2	yes	—
256	C₂×C₄	#13133124	1	2	yes	—
257	C₂×C₄	#13133137	1	2	yes	—
258	C₂×C₄	#13133137	1	2	yes	—
259	C₂×C₄	#13133138	1	2	yes	—
260	C₂×C₄	#13133140	1	2	yes	—
261	C₂×C₄	#13133141	1	2	yes	—
262	C₂×C₄	#13133141	1	2	yes	—
263	C₂×C₄	#13133148	1	1	yes	characteristic
264	C₂×C₄	#13133148	1	1	yes	characteristic
265	C₂×C₄	#13133150	1	2	yes	—
266	C₂×C₄	#13133150	1	2	yes	—
267	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
268	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
269	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
270	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
271	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
272	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
273	C₈	—	212	4	no	—
274	C₈	#13133124	1	2	yes	—
275	C₈	#13133124	1	2	yes	—
276	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
277	C₈	C₂×#11719921	1	1	yes	characteristic
278	C₈	C₂×#11719919	1	1	yes	characteristic
279	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	#16810385	1	2	yes	—
285	C₂²	#16810385	1	2	yes	—
286	C₂²	#16810445	1	4	yes	—
287	C₂²	C₂×#13133138	1	1	yes	characteristic
288	C₂²	C₂×#13133140	1	1	yes	characteristic
289	C₂²	C₂×#13133148	1	1	yes	characteristic
290	C₄	—	8	4	no	—
291	C₄	#16810385	1	2	yes	—
292	C₄	#16810385	1	2	yes	—
293	C₄	C₂×#13133124	1	1	yes	characteristic
294	C₄	C₂×#13133137	1	1	yes	characteristic
295	C₄	C₂×#13133141	1	1	yes	characteristic
296	C₄	C₂×#13133150	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	#20272813	1	4	yes	direct product factor
301	C₂	C₂×#16810385	1	1	yes	characteristic
302	C₂	C₂×#16810445	1	2	yes	—

	subgroup	G/H
center	C₂³	#13133148
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×#106×C₅₃	C₂
socle	C₂³×C₅₃	#106
prime radical	C₂⁴×C₅₃	SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×#106	D₅₃
hypocenter	C₅₃	C₂×#1383
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
3. #20272813	C₂	direct (N × H)
8. C₂×#13133157	C₄	semidirect
10. C₂²×#11719908	C₄	semidirect
13. C₂×#13133153	C₄	semidirect
30. C₂×#11719908	C₂×C₄	semidirect
33. #13133153	C₂×C₄	semidirect
153. C₂×C₅₃	#1383	semidirect
172. C₅₃	C₂×#1383	semidirect
300. C₂	#20272813	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133157	2
C₂²×#13133153	2
#20272813	2
C₂²×#106×C₅₃	2
C₂×#1383	53

Subgroups — C2×#20272813 #27298992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272813 #27298992