Subgroups — C₂×#20272806 #27298968

Order 13568.

694 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	C₂³×#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133151	C₂	1	1	yes	maximal, characteristic
3	#20272806	C₂	1	4	yes	maximal, direct product factor
4	C₂²×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₅₃	—	2	1	no	—
6	C₂²×#11719908	—	2	2	no	—
7	C₂²×#11719908	—	2	2	no	—
8	C₂²×#11719908	C₄	1	4	yes	complemented
9	C₂²×#11719908	C₂²	1	2	yes	—
10	C₂×#13133151	—	2	2	no	—
11	C₂×#13133151	—	2	1	no	—
12	C₂×#13133151	C₄	1	4	yes	complemented
13	C₂×#13133151	C₂²	1	2	yes	—
14	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
16	C₂×#107×C₅₃	—	2	1	no	—
17	C₂×#107×C₅₃	—	2	2	no	—
18	C₂×#107×C₅₃	C₄	1	4	yes	complemented
19	C₂×#107×C₅₃	C₂²	1	2	yes	—
20	C₂³×Dic₅₃	—	4	1	no	—
21	C₂×C₄⋊Dic₅₃	—	2	4	no	—
22	C₂×C₄⋊Dic₅₃	—	2	2	no	—
23	C₂×C₄⋊Dic₅₃	—	4	1	no	—
24	C₂×C₄⋊Dic₅₃	—	4	2	no	—
25	C₂×#11719908	—	2	8	no	—
26	C₂×#11719908	—	2	2	no	—
27	C₂×#11719908	—	2	2	no	—
28	C₂×#11719908	C₂×C₄	1	4	yes	complemented
29	#13133151	—	2	4	no	—
30	#13133151	—	2	2	no	—
31	#13133151	C₂×C₄	1	4	yes	complemented
32	C₂³×C₄×C₅₃	—	4	1	no	—
33	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₅₃	D₄	1	2	yes	—
38	C₂²×C₈×C₅₃	D₄	1	2	yes	—
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
41	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
45	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
46	#107×C₅₃	—	2	4	no	—
47	#107×C₅₃	—	2	2	no	—
48	#107×C₅₃	C₂×C₄	1	4	yes	complemented
49	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	—	4	4	no	—
51	C₂²×C₄×C₅₃	—	4	2	no	—
52	C₂²×C₄×C₅₃	—	8	2	no	—
53	C₂²×C₄×C₅₃	—	8	2	no	—
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
59	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
60	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₅₃	—	4	4	no	—
63	C₂²×Dic₅₃	—	4	2	no	—
64	C₂²×Dic₅₃	—	8	2	no	—
65	C₂²×Dic₅₃	—	8	2	no	—
66	C₂×C₈×C₅₃	C₄²	1	4	yes	—
67	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
70	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
72	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
79	C₄⋊C₄×C₅₃	—	2	4	no	—
80	C₄⋊C₄×C₅₃	—	4	4	no	—
81	C₄⋊C₄×C₅₃	—	4	2	no	—
82	C₄⋊Dic₅₃	—	2	4	no	—
83	C₄⋊Dic₅₃	—	4	4	no	—
84	C₄⋊Dic₅₃	—	4	2	no	—
85	#11719908	—	2	4	no	—
86	C₂³×C₅₃	#102	1	2	yes	—
87	C₂³×C₅₃	#102	1	2	yes	—
88	C₂³×C₅₃	#107	1	2	yes	—
89	C₂³×C₅₃	D₁₆	1	2	yes	—
90	C₂³×C₅₃	SD₃₂	1	2	yes	—
91	C₂³×C₅₃	SD₃₂	1	2	yes	—
92	C₂³×C₅₃	Q₃₂	1	2	yes	—
93	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	—	4	4	no	—
95	C₂×C₄×C₅₃	—	8	8	no	—
96	C₂×C₄×C₅₃	—	8	2	no	—
97	C₂×C₄×C₅₃	—	8	2	no	—
98	C₂×C₄×C₅₃	#95	1	4	yes	—
99	C₂×C₄×C₅₃	#102	1	2	yes	—
100	C₂×C₄×C₅₃	#102	1	2	yes	—
101	C₂×C₄×C₅₃	#103	1	4	yes	—
102	C₂×C₄×C₅₃	#103	1	4	yes	—
103	C₂×C₄×C₅₃	#106	1	4	yes	—
104	C₂×C₄×C₅₃	#107	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
111	C₂×Dic₅₃	—	4	4	no	—
112	C₂×Dic₅₃	—	8	8	no	—
113	C₂×Dic₅₃	—	8	2	no	—
114	C₂×Dic₅₃	—	8	2	no	—
115	C₈×C₅₃	#95	1	4	yes	—
116	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
117	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×#1380	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	#340	1	4	yes	—
122	C₂²×C₅₃	#357	1	4	yes	—
123	C₂²×C₅₃	#357	1	4	yes	—
124	C₂²×C₅₃	#358	1	4	yes	—
125	C₂²×C₅₃	#358	1	4	yes	—
126	C₂²×C₅₃	#366	1	4	yes	—
127	C₂²×C₅₃	#367	1	4	yes	—
128	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
129	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
130	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
131	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
132	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
133	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
134	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
135	C₄×C₅₃	—	8	4	no	—
136	C₄×C₅₃	#340	1	4	yes	—
137	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
138	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
139	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
140	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
141	Dic₅₃	—	8	4	no	—
142	C₂³×C₁₆	—	53	1	no	—
143	C₂²×#107	—	53	1	no	—
144	C₂²×#107	D₅₃	1	1	yes	characteristic
145	#1380	—	53	4	no	—
146	C₂×C₅₃	#1380	1	8	yes	complemented
147	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
148	C₂×C₅₃	C₂×#357	1	1	yes	characteristic
149	C₂×C₅₃	C₂×#357	1	1	yes	characteristic
150	C₂×C₅₃	C₂×#358	1	1	yes	characteristic
151	C₂×C₅₃	C₂×#358	1	1	yes	characteristic
152	C₂×C₅₃	C₂×#366	1	1	yes	characteristic
153	C₂×C₅₃	C₂×#367	1	1	yes	characteristic
154	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
155	C₂²×C₄⋊C₄	—	2	1	no	—
156	C₂²×C₄⋊C₄	—	106	1	no	—
157	C₂²×C₁₆	—	53	4	no	—
158	C₂²×C₁₆	—	53	2	no	—
159	C₂²×C₁₆	—	106	2	no	—
160	C₂²×C₁₆	—	106	2	no	—
161	C₂×#107	—	2	2	no	—
162	C₂×#107	—	2	1	no	—
163	C₂×#107	—	53	4	no	—
164	C₂×#107	—	53	2	no	—
165	C₂×#107	—	106	2	no	—
166	C₂×#107	—	106	1	no	—
167	C₂×#107	Dic₅₃	1	4	yes	complemented
168	C₂×#107	C₂×D₅₃	1	2	yes	—
169	C₅₃	C₂×#1380	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	—	4	1	no	—
171	C₂³×C₄	—	212	1	no	—
172	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
173	C₂²×C₈	Dic₁₀₆	1	2	yes	—
174	C₂²×C₈	C₄×D₅₃	1	2	yes	—
175	C₂²×C₈	C₄×D₅₃	1	2	yes	—
176	C₂²×C₈	D₂₁₂	1	2	yes	—
177	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
178	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
179	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	4	no	—
183	C₂×C₄⋊C₄	—	4	1	no	—
184	C₂×C₄⋊C₄	—	4	2	no	—
185	C₂×C₄⋊C₄	—	106	2	no	—
186	C₂×C₄⋊C₄	—	106	4	no	—
187	C₂×C₄⋊C₄	—	212	1	no	—
188	C₂×C₄⋊C₄	—	212	2	no	—
189	C₂×C₁₆	—	53	4	no	—
190	C₂×C₁₆	—	106	2	no	—
191	C₂×C₁₆	—	106	2	no	—
192	C₂×C₁₆	—	106	8	no	—
193	#107	—	2	2	no	—
194	#107	—	2	4	no	—
195	#107	—	53	4	no	—
196	#107	—	106	2	no	—
197	#107	—	106	4	no	—
198	#107	C₂×Dic₅₃	1	4	yes	complemented
199	C₂⁴	#11719922	1	1	yes	characteristic
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	8	2	no	—
203	C₂²×C₄	—	8	2	no	—
204	C₂²×C₄	—	424	2	no	—
205	C₂²×C₄	—	424	2	no	—
206	C₂²×C₄	—	212	4	no	—
207	C₂²×C₄	—	212	2	no	—
208	C₂²×C₄	#11719919	1	2	yes	—
209	C₂²×C₄	#11719921	1	2	yes	—
210	C₂²×C₄	#11719922	1	1	yes	characteristic
211	C₂²×C₄	#11719923	1	2	yes	—
212	C₂²×C₄	#11719924	1	2	yes	—
213	C₂²×C₄	#11719925	1	2	yes	—
214	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
216	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
217	C₂×C₈	#11719919	1	2	yes	—
218	C₂×C₈	#11719919	1	4	yes	—
219	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
220	C₂×C₈	#11719921	1	2	yes	—
221	C₂×C₈	#11719921	1	4	yes	—
222	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
223	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
226	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
227	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
228	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	4	4	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	106	4	no	—
233	C₄⋊C₄	—	212	4	no	—
234	C₄⋊C₄	—	212	2	no	—
235	C₁₆	—	106	4	no	—
236	C₂³	#13133137	1	2	yes	—
237	C₂³	#13133138	1	2	yes	—
238	C₂³	#13133150	1	2	yes	—
239	C₂³	#13133158	1	2	yes	—
240	C₂³	#13133160	1	2	yes	—
241	C₂³	#13133161	1	2	yes	—
242	C₂³	#13133163	1	2	yes	—
243	C₂³	C₂×#11719922	1	1	yes	characteristic
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	8	2	no	—
246	C₂×C₄	—	8	2	no	—
247	C₂×C₄	—	8	8	no	—
248	C₂×C₄	—	424	2	no	—
249	C₂×C₄	—	424	2	no	—
250	C₂×C₄	—	424	8	no	—
251	C₂×C₄	—	212	4	no	—
252	C₂×C₄	#13133124	1	4	yes	—
253	C₂×C₄	#13133137	1	2	yes	—
254	C₂×C₄	#13133138	1	2	yes	—
255	C₂×C₄	#13133140	1	4	yes	—
256	C₂×C₄	#13133141	1	4	yes	—
257	C₂×C₄	#13133148	1	4	yes	—
258	C₂×C₄	#13133150	1	2	yes	—
259	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
260	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
261	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
262	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
263	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
264	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
265	C₈	#13133124	1	4	yes	—
266	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
267	C₈	C₂×#11719921	1	1	yes	characteristic
268	C₈	C₂×#11719919	1	1	yes	characteristic
269	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
270	C₂²	#16810385	1	4	yes	—
271	C₂²	#16810423	1	4	yes	—
272	C₂²	#16810424	1	4	yes	—
273	C₂²	#16810426	1	4	yes	—
274	C₂²	#16810427	1	4	yes	—
275	C₂²	#16810448	1	4	yes	—
276	C₂²	#16810450	1	4	yes	—
277	C₂²	C₂×#13133137	1	1	yes	characteristic
278	C₂²	C₂×#13133138	1	1	yes	characteristic
279	C₂²	C₂×#13133150	1	1	yes	characteristic
280	C₂²	C₂×#13133158	1	1	yes	characteristic
281	C₂²	C₂×#13133160	1	1	yes	characteristic
282	C₂²	C₂×#13133161	1	1	yes	characteristic
283	C₂²	C₂×#13133163	1	1	yes	characteristic
284	C₄	—	8	4	no	—
285	C₄	—	424	4	no	—
286	C₄	#16810385	1	4	yes	—
287	C₄	C₂×#13133124	1	1	yes	characteristic
288	C₄	C₂×#13133140	1	1	yes	characteristic
289	C₄	C₂×#13133141	1	1	yes	characteristic
290	C₄	C₂×#13133148	1	1	yes	characteristic
291	C₂	#20272806	1	8	yes	direct product factor
292	C₂	C₂×#16810385	1	1	yes	characteristic
293	C₂	C₂×#16810423	1	1	yes	characteristic
294	C₂	C₂×#16810424	1	1	yes	characteristic
295	C₂	C₂×#16810426	1	1	yes	characteristic
296	C₂	C₂×#16810427	1	1	yes	characteristic
297	C₂	C₂×#16810448	1	1	yes	characteristic
298	C₂	C₂×#16810450	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#11719922
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×#107×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×#107	D₅₃
hypocenter	C₅₃	C₂×#1380
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
3. #20272806	C₂	direct (N × H)
8. C₂²×#11719908	C₄	semidirect
12. C₂×#13133151	C₄	semidirect
18. C₂×#107×C₅₃	C₄	semidirect
28. C₂×#11719908	C₂×C₄	semidirect
31. #13133151	C₂×C₄	semidirect
48. #107×C₅₃	C₂×C₄	semidirect
146. C₂×C₅₃	#1380	semidirect
167. C₂×#107	Dic₅₃	semidirect
169. C₅₃	C₂×#1380	semidirect
198. #107	C₂×Dic₅₃	semidirect
291. C₂	#20272806	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11719908	2
C₂²×#13133151	2
#20272806	2
C₂²×#107×C₅₃	2
C₂×#1380	53

Subgroups — C2×#20272806 #27298968

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272806 #27298968