Subgroups — C₂×#20275466 #27387505

Order 13568.

934 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810661	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810651	C₂	1	2	yes	maximal
3	C₂²×#13133127	C₂	1	1	yes	maximal, characteristic
4	#20275466	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810384	C₂	1	2	yes	maximal
6	C₂²×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#88322	C₂²	1	1	yes	characteristic
8	C₂×#13133176	C₂²	1	2	yes	—
9	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
10	#16810661	C₄	1	4	yes	complemented
11	#16810661	C₂²	1	4	yes	—
12	C₂×#13133151	C₄	1	2	yes	complemented
13	C₂×#13133149	C₄	1	2	yes	complemented
14	#16810651	—	2	4	no	—
15	#16810651	C₂²	1	8	yes	—
16	C₂×#13133146	—	2	4	no	—
17	C₂×#13133127	—	2	4	no	—
18	C₂×#13133127	C₂²	1	4	yes	—
19	#16810384	C₂²	1	8	yes	—
20	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
21	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#108×C₅₃	—	2	4	no	—
23	C₂×#108×C₅₃	C₂²	1	4	yes	—
24	C₂³×Dic₅₃	—	2	1	no	—
25	C₂²×#88322	—	2	2	no	—
26	C₂²×#88322	—	2	2	no	—
27	C₂²×#88322	D₄	1	2	yes	—
28	C₂²×#88322	D₄	1	2	yes	—
29	C₂²×#88322	C₂³	1	2	yes	—
30	#13133176	—	2	2	no	—
31	#13133176	—	2	2	no	—
32	#13133176	C₂×C₄	1	4	yes	—
33	#13133176	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
36	C₂×C₄×Dic₅₃	—	2	2	no	—
37	C₂×C₄×Dic₅₃	—	2	2	no	—
38	C₂×C₄×Dic₅₃	—	2	1	no	—
39	C₂×C₄×Dic₅₃	Q₈	1	4	yes	—
40	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
41	#13133151	C₂×C₄	1	8	yes	complemented
42	#13133149	C₂×C₄	1	8	yes	complemented
43	#13133146	—	2	16	no	—
44	#13133127	—	2	8	no	—
45	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×M₁₆×C₅₃	—	2	2	no	—
47	C₂×M₁₆×C₅₃	—	2	4	no	—
48	C₂×M₁₆×C₅₃	—	2	4	no	—
49	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
50	C₂²×C₈×C₅₃	—	2	2	no	—
51	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
54	C₂²×C₈×C₅₃	D₄	1	4	yes	—
55	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
56	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
57	#108×C₅₃	—	2	8	no	—
58	C₂⁴×C₅₃	#41	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	#41	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₅₃	—	2	4	no	—
70	C₂²×Dic₅₃	—	2	4	no	—
71	C₂²×Dic₅₃	—	2	2	no	—
72	C₂²×Dic₅₃	—	4	4	no	—
73	M₁₆×C₅₃	—	2	4	no	—
74	M₁₆×C₅₃	—	2	4	no	—
75	M₁₆×C₅₃	—	4	4	no	—
76	C₂×C₈×C₅₃	—	2	4	no	—
77	C₂×C₈×C₅₃	—	4	4	no	—
78	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
79	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
80	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
81	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
84	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
85	C₂²⋊Dic₅₃	—	2	4	no	—
86	C₂²⋊Dic₅₃	—	2	4	no	—
87	C₄⋊Dic₅₃	—	2	4	no	—
88	C₄⋊Dic₅₃	C₂²×C₄	1	8	yes	—
89	C₄×Dic₅₃	—	2	4	no	—
90	C₄×Dic₅₃	—	2	4	no	—
91	C₄×Dic₅₃	—	2	2	no	—
92	C₄×Dic₅₃	—	2	4	no	—
93	C₄×Dic₅₃	C₂×Q₈	1	4	yes	—
94	C₂×#88322	—	2	4	no	—
95	C₂×#88322	—	2	4	no	—
96	C₂×#88322	—	2	2	no	—
97	C₂×#88322	—	2	2	no	—
98	C₂×#88322	C₂×D₄	1	4	yes	—
99	C₂³×C₅₃	—	2	4	no	—
100	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
101	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
102	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
103	C₂³×C₅₃	#125	1	2	yes	—
104	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
110	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₅₃	#121	1	2	yes	—
112	C₂×C₄×C₅₃	#121	1	2	yes	—
113	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₅₃	#125	1	2	yes	—
115	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
117	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
123	C₂×Dic₅₃	—	2	8	no	—
124	C₂×Dic₅₃	—	2	4	no	—
125	C₂×Dic₅₃	—	4	4	no	—
126	C₂×Dic₅₃	—	4	8	no	—
127	C₂×Dic₅₃	—	4	8	no	—
128	C₈×C₅₃	—	4	4	no	—
129	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
130	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
131	#88322	—	2	4	no	—
132	C₂×#1946	—	53	1	no	maximal, Sylow
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	—	2	4	no	—
136	C₂²×C₅₃	—	2	4	no	—
137	C₂²×C₅₃	#389	1	2	yes	—
138	C₂²×C₅₃	#389	1	2	yes	—
139	C₂²×C₅₃	#444	1	4	yes	—
140	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
142	C₂²×C₅₃	C₂×#125	1	1	yes	characteristic
143	C₄×C₅₃	#389	1	2	yes	—
144	C₄×C₅₃	#389	1	2	yes	—
145	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
147	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
148	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
149	Dic₅₃	—	4	4	no	—
150	Dic₅₃	—	4	8	no	—
151	C₂²×#108	D₅₃	1	1	yes	characteristic
152	C₂×#427	—	53	1	no	—
153	C₂×#424	—	53	2	no	—
154	C₂²×#97	—	53	1	no	—
155	#1946	—	53	8	no	—
156	C₂×#339	—	53	2	no	—
157	C₂×C₅₃	—	2	1	no	—
158	C₂×C₅₃	—	2	1	no	—
159	C₂×C₅₃	—	2	2	no	—
160	C₂×C₅₃	#1946	1	4	yes	complemented
161	C₂×C₅₃	C₂×#389	1	1	yes	characteristic
162	C₂×C₅₃	C₂×#444	1	2	yes	—
163	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
164	C₂³×C₈	—	53	1	no	—
165	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
166	C₂×#117	—	53	2	no	—
167	C₂²×C₄²	—	53	1	no	—
168	C₂×#108	—	2	4	no	—
169	C₂×#108	C₂×D₅₃	1	4	yes	—
170	#427	—	53	4	no	—
171	#427	—	53	4	no	—
172	C₂×#107	—	53	2	no	—
173	C₂×#106	—	53	2	no	—
174	#424	—	53	8	no	—
175	#424	—	106	4	no	—
176	C₂×C₄⋊C₈	—	106	4	no	—
177	C₂×#97	—	53	4	no	—
178	C₂×#97	—	106	4	no	—
179	#339	—	53	8	no	—
180	C₅₃	C₂×#1946	1	1	yes	characteristic, complemented, Sylow
181	C₂³×C₄	—	106	1	no	—
182	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	4	no	—
185	C₂×M₁₆	—	2	4	no	—
186	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	53	2	no	—
189	C₂²×C₈	—	53	2	no	—
190	C₂²×C₈	—	53	2	no	—
191	C₂²×C₈	—	106	2	no	—
192	C₂²×C₈	—	106	2	no	—
193	C₂²×C₈	Dic₁₀₆	1	4	yes	—
194	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
195	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
196	C₂²×C₈	C₄×D₅₃	1	2	yes	—
197	C₂²×C₈	C₂²⋊D₅₃	1	4	yes	—
198	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
199	#117	—	53	4	no	—
200	#117	—	53	4	no	—
201	#117	—	106	2	no	—
202	#117	—	106	2	no	—
203	C₂×C₄⋊C₄	—	53	4	no	—
204	C₂×C₂²⋊C₄	—	106	2	no	—
205	C₂×C₄²	—	53	4	no	—
206	C₂×C₄²	—	53	2	no	—
207	C₂×C₄²	—	106	2	no	—
208	C₂×C₄²	—	106	1	no	—
209	C₂×C₄²	—	106	2	no	—
210	#108	—	2	8	no	—
211	#107	—	53	8	no	—
212	#106	—	53	8	no	—
213	C₄⋊C₈	—	106	16	no	—
214	#97	—	106	8	no	—
215	C₂⁴	#11719949	1	1	yes	characteristic
216	C₂²×C₄	—	106	4	no	—
217	C₂²×C₄	—	106	4	no	—
218	C₂²×C₄	—	106	2	no	—
219	C₂²×C₄	—	212	4	no	—
220	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
221	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
223	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
224	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
225	C₂²×C₄	#11719947	1	2	yes	—
226	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
227	C₂²×C₄	#11719949	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
229	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
230	M₁₆	—	2	4	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	4	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	4	4	no	—
235	C₂×C₈	—	53	4	no	—
236	C₂×C₈	—	106	4	no	—
237	C₂×C₈	—	106	4	no	—
238	C₂×C₈	—	106	2	no	—
239	C₂×C₈	—	106	2	no	—
240	C₂×C₈	#11719919	1	4	yes	—
241	C₂×C₈	#11719919	1	4	yes	—
242	C₂×C₈	#11719919	1	8	yes	—
243	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
244	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
245	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
246	C₂×C₈	C₂×C₂²⋊D₅₃	1	4	yes	—
247	C₄⋊C₄	—	53	8	no	—
248	C₄⋊C₄	—	106	4	no	—
249	C₂²⋊C₄	—	106	4	no	—
250	C₂²⋊C₄	—	106	4	no	—
251	C₄²	—	53	4	no	—
252	C₄²	—	106	4	no	—
253	C₄²	—	106	4	no	—
254	C₄²	—	106	4	no	—
255	C₄²	—	106	2	no	—
256	C₂³	—	2	4	no	—
257	C₂³	#13133179	1	1	yes	characteristic
258	C₂³	#13133179	1	1	yes	characteristic
259	C₂³	#13133193	1	2	yes	—
260	C₂³	#13133208	1	2	yes	—
261	C₂³	C₂×#11719949	1	1	yes	characteristic
262	C₂×C₄	—	106	8	no	—
263	C₂×C₄	—	106	4	no	—
264	C₂×C₄	—	212	8	no	—
265	C₂×C₄	—	212	4	no	—
266	C₂×C₄	—	212	8	no	—
267	C₂×C₄	C₂×#11719919	1	2	yes	—
268	C₂×C₄	C₂×#11719919	1	2	yes	—
269	C₂×C₄	#13133179	1	1	yes	characteristic
270	C₂×C₄	#13133179	1	1	yes	characteristic
271	C₂×C₄	#13133182	1	2	yes	—
272	C₂×C₄	#13133182	1	2	yes	—
273	C₂×C₄	#13133188	1	2	yes	—
274	C₂×C₄	#13133188	1	2	yes	—
275	C₂×C₄	#13133193	1	2	yes	—
276	C₂×C₄	#13133208	1	2	yes	—
277	C₂×C₄	#13133216	1	2	yes	—
278	C₂×C₄	#13133216	1	2	yes	—
279	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
280	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
282	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
283	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
284	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
285	C₈	—	4	4	no	—
286	C₈	—	106	4	no	—
287	C₈	C₂×#11719919	1	4	yes	—
288	C₈	C₂×#11719919	1	4	yes	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	#16810519	1	2	yes	—
294	C₂²	#16810519	1	2	yes	—
295	C₂²	#16810744	1	4	yes	—
296	C₂²	C₂×#13133179	1	1	yes	characteristic
297	C₂²	C₂×#13133193	1	1	yes	characteristic
298	C₂²	C₂×#13133208	1	1	yes	characteristic
299	C₄	—	212	4	no	—
300	C₄	—	212	8	no	—
301	C₄	#16810519	1	2	yes	—
302	C₄	#16810519	1	2	yes	—
303	C₄	C₂²×#11719919	1	1	yes	characteristic
304	C₄	C₂×#13133182	1	1	yes	characteristic
305	C₄	C₂×#13133188	1	1	yes	characteristic
306	C₄	C₂×#13133216	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	1	no	—
310	C₂	#20275466	1	4	yes	direct product factor
311	C₂	C₂×#16810519	1	1	yes	characteristic
312	C₂	C₂×#16810744	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×#108×C₅₃	C₂
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂²×#108	D₅₃
hypocenter	C₅₃	C₂×#1946
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
4. #20275466	C₂	direct (N × H)
10. #16810661	C₄	semidirect
12. C₂×#13133151	C₄	semidirect
13. C₂×#13133149	C₄	semidirect
41. #13133151	C₂×C₄	semidirect
42. #13133149	C₂×C₄	semidirect
160. C₂×C₅₃	#1946	semidirect
180. C₅₃	C₂×#1946	semidirect
310. C₂	#20275466	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16810661	2
C₂×#16810651	2
C₂²×#13133127	2
#20275466	2
C₂×#16810384	2
C₂²×#108×C₅₃	2
C₂×#1946	53

Subgroups — C2×#20275466 #27387505

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20275466 #27387505