Subgroups — C₂×#20273016 #27349717

Order 13568.

802 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810674	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810674	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810674	C₂	1	1	yes	maximal, characteristic
4	#20273016	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810345	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810344	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
10	#16810674	C₄	1	4	yes	—
11	#16810674	C₄	1	4	yes	—
12	#16810674	C₄	1	4	yes	—
13	#16810674	C₂²	1	4	yes	—
14	#16810674	C₂²	1	4	yes	—
15	#16810674	C₂²	1	4	yes	—
16	C₂×#13133146	C₄	1	2	yes	—
17	C₂×#13133146	C₄	1	2	yes	—
18	C₂×#13133146	C₄	1	2	yes	—
19	C₂×#13133129	C₄	1	2	yes	—
20	C₂×#13133129	C₄	1	2	yes	—
21	C₂×#13133129	C₄	1	2	yes	—
22	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
23	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
24	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
25	#16810345	C₂²	1	4	yes	complemented
26	#16810344	C₂²	1	4	yes	complemented
27	C₈×#88322	C₂²	1	4	yes	complemented
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#322	C₂²	1	4	yes	complemented
32	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×#88322	C₂×C₄	1	2	yes	—
36	C₂²×#88322	C₂×C₄	1	2	yes	—
37	C₂²×#88322	C₂×C₄	1	2	yes	—
38	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄²×D₅₃	C₂×C₄	1	2	yes	—
42	C₄²×D₅₃	C₂×C₄	1	2	yes	—
43	C₄²×D₅₃	C₂×C₄	1	2	yes	—
44	C₄²×D₅₃	C₂³	1	2	yes	—
45	#13133146	C₂×C₄	1	8	yes	—
46	#13133146	C₂×C₄	1	8	yes	—
47	#13133146	C₂×C₄	1	8	yes	—
48	#13133129	C₂×C₄	1	8	yes	—
49	#13133129	C₂×C₄	1	8	yes	—
50	#13133129	C₂×C₄	1	8	yes	—
51	C₄×#88322	C₂×C₄	1	2	yes	—
52	C₄×#88322	C₂×C₄	1	2	yes	—
53	C₄×#88322	C₂×C₄	1	2	yes	—
54	C₄×#88322	C₂³	1	2	yes	—
55	C₄×#88322	C₂³	1	2	yes	—
56	C₄×#88322	C₂³	1	2	yes	—
57	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
58	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₅₃	C₂³	1	2	yes	—
65	C₄×C₈×C₅₃	C₂³	1	2	yes	—
66	C₄×C₈×C₅₃	C₂³	1	2	yes	—
67	C₂³×D₅₃	C₄²	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
72	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
73	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
74	C₂×C₄×D₅₃	—	2	4	no	—
75	C₂×C₄×D₅₃	—	2	4	no	—
76	C₂×C₄×D₅₃	—	2	4	no	—
77	C₂×C₄×D₅₃	C₄²	1	2	yes	—
78	C₂×C₄×D₅₃	C₄²	1	2	yes	—
79	C₂×C₄×D₅₃	C₄²	1	2	yes	—
80	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
81	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
82	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₅₃	—	2	4	no	—
84	C₂×C₈×C₅₃	—	2	4	no	—
85	C₂×C₈×C₅₃	—	2	4	no	—
86	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
89	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
93	C₄×Dic₅₃	C₄²	1	2	yes	—
94	C₄×Dic₅₃	C₄²	1	2	yes	—
95	C₄×Dic₅₃	C₄²	1	2	yes	—
96	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
97	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
98	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
99	C₂×#88322	—	2	4	no	—
100	C₂×#88322	—	2	4	no	—
101	C₂×#88322	—	2	4	no	—
102	C₂×#88322	C₂²×C₄	1	4	yes	—
103	C₂×#88322	C₂²×C₄	1	4	yes	—
104	C₂×#88322	C₂²×C₄	1	4	yes	—
105	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
106	C₂²×D₅₃	—	2	2	no	—
107	C₂²×D₅₃	—	2	2	no	—
108	C₂²×D₅₃	—	2	2	no	—
109	C₂²×D₅₃	C₂×C₄²	1	2	yes	—
110	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#131	1	4	yes	—
114	C₂×C₄×C₅₃	#131	1	4	yes	—
115	C₂×C₄×C₅₃	#131	1	4	yes	—
116	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×Dic₅₃	—	2	2	no	—
120	C₂×Dic₅₃	—	2	2	no	—
121	C₂×Dic₅₃	—	2	2	no	—
122	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
123	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
124	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
125	C₄×D₅₃	—	2	4	no	—
126	C₄×D₅₃	—	2	4	no	—
127	C₄×D₅₃	—	2	4	no	—
128	C₄×D₅₃	—	2	4	no	—
129	C₄×D₅₃	—	2	4	no	—
130	C₄×D₅₃	—	2	4	no	—
131	C₈×C₅₃	—	2	4	no	—
132	C₈×C₅₃	—	2	4	no	—
133	C₈×C₅₃	—	2	4	no	—
134	#88322	—	2	4	no	—
135	#88322	—	2	4	no	—
136	#88322	—	2	4	no	—
137	C₂×#1454	—	53	1	no	maximal, Sylow
138	C₂²×C₅₃	#405	1	2	yes	—
139	C₂²×C₅₃	#405	1	2	yes	—
140	C₂²×C₅₃	#405	1	2	yes	—
141	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
142	C₂×D₅₃	—	2	2	no	—
143	C₂×D₅₃	—	2	2	no	—
144	C₂×D₅₃	—	2	2	no	—
145	C₂×D₅₃	—	4	4	no	—
146	C₄×C₅₃	#405	1	2	yes	—
147	C₄×C₅₃	#405	1	2	yes	—
148	C₄×C₅₃	#405	1	2	yes	—
149	C₄×C₅₃	C₂×#131	1	2	yes	—
150	C₄×C₅₃	C₂×#131	1	2	yes	—
151	C₄×C₅₃	C₂×#131	1	2	yes	—
152	Dic₅₃	—	2	2	no	—
153	Dic₅₃	—	2	2	no	—
154	Dic₅₃	—	2	2	no	—
155	C₂×#431	—	53	1	no	—
156	C₂×#431	—	53	1	no	—
157	C₂×#431	—	53	1	no	—
158	#1454	—	53	8	no	—
159	C₂×#322	—	53	1	no	—
160	C₂×#322	—	53	1	no	—
161	C₂×#322	D₅₃	1	1	yes	characteristic, complemented
162	C₂×C₈²	—	53	1	no	—
163	C₂×C₅₃	#1454	1	4	yes	complemented
164	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
165	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
166	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
167	D₅₃	—	4	2	no	—
168	C₂²×C₄²	—	53	1	no	—
169	#431	—	53	4	no	—
170	#431	—	53	4	no	—
171	#431	—	53	4	no	—
172	#431	—	53	4	no	—
173	#431	—	53	4	no	—
174	#431	—	53	4	no	—
175	C₂×C₄⋊C₈	—	53	2	no	—
176	C₂×C₄⋊C₈	—	53	2	no	—
177	C₂×C₄⋊C₈	—	53	2	no	—
178	C₂×C₂²⋊C₈	—	53	2	no	—
179	C₂×C₂²⋊C₈	—	53	2	no	—
180	C₂×C₂²⋊C₈	—	53	2	no	—
181	C₂×C₄×C₈	—	53	1	no	—
182	C₂×C₄×C₈	—	53	1	no	—
183	C₂×C₄×C₈	—	53	1	no	—
184	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
185	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
186	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
187	#322	—	53	4	no	—
188	#322	—	53	4	no	—
189	#322	C₂×D₅₃	1	4	yes	complemented
190	C₈²	—	53	4	no	—
191	C₅₃	C₂×#1454	1	1	yes	characteristic, complemented, Sylow
192	C₂³×C₄	—	53	1	no	—
193	C₂³×C₄	—	53	1	no	—
194	C₂³×C₄	—	53	1	no	—
195	C₂²×C₈	—	53	2	no	—
196	C₂²×C₈	—	53	2	no	—
197	C₂²×C₈	—	53	2	no	—
198	C₂²×C₈	C₄×D₅₃	1	2	yes	—
199	C₂²×C₈	C₄×D₅₃	1	2	yes	—
200	C₂²×C₈	C₄×D₅₃	1	2	yes	—
201	C₂×C₄²	—	53	1	no	—
202	C₂×C₄²	—	53	1	no	—
203	C₂×C₄²	—	53	1	no	—
204	C₂×C₄²	—	53	2	no	—
205	C₂×C₄²	—	53	2	no	—
206	C₂×C₄²	—	53	2	no	—
207	C₂×C₄²	—	53	2	no	—
208	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
209	C₄⋊C₈	—	53	8	no	—
210	C₄⋊C₈	—	53	8	no	—
211	C₄⋊C₈	—	53	8	no	—
212	C₂²⋊C₈	—	53	8	no	—
213	C₂²⋊C₈	—	53	8	no	—
214	C₂²⋊C₈	—	53	8	no	—
215	C₄×C₈	—	53	2	no	—
216	C₄×C₈	—	53	2	no	—
217	C₄×C₈	—	53	2	no	—
218	C₄×C₈	—	53	2	no	—
219	C₄×C₈	—	53	2	no	—
220	C₄×C₈	—	53	2	no	—
221	C₄×C₈	C₄×D₅₃	1	2	yes	—
222	C₄×C₈	C₄×D₅₃	1	2	yes	—
223	C₄×C₈	C₄×D₅₃	1	2	yes	—
224	C₄×C₈	C₂²×D₅₃	1	2	yes	—
225	C₄×C₈	C₂²×D₅₃	1	2	yes	—
226	C₄×C₈	C₂²×D₅₃	1	2	yes	—
227	C₂⁴	—	53	1	no	—
228	C₂²×C₄	—	53	1	no	—
229	C₂²×C₄	—	53	1	no	—
230	C₂²×C₄	—	53	1	no	—
231	C₂²×C₄	—	53	2	no	—
232	C₂²×C₄	—	53	2	no	—
233	C₂²×C₄	—	53	2	no	—
234	C₂²×C₄	—	53	2	no	—
235	C₂²×C₄	—	53	2	no	—
236	C₂²×C₄	—	53	2	no	—
237	C₂²×C₄	—	106	4	no	—
238	C₂²×C₄	—	106	4	no	—
239	C₂²×C₄	—	106	4	no	—
240	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	53	4	no	—
247	C₂×C₈	—	53	4	no	—
248	C₂×C₈	—	53	4	no	—
249	C₂×C₈	—	106	4	no	—
250	C₂×C₈	—	106	4	no	—
251	C₂×C₈	—	106	4	no	—
252	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
253	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
254	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
255	C₄²	—	53	2	no	—
256	C₄²	—	53	2	no	—
257	C₄²	—	53	2	no	—
258	C₄²	—	53	2	no	—
259	C₄²	—	53	2	no	—
260	C₄²	—	53	2	no	—
261	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
262	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
263	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
264	C₄²	C₂³×D₅₃	1	1	yes	characteristic
265	C₂³	—	53	2	no	—
266	C₂³	—	106	2	no	—
267	C₂³	—	106	2	no	—
268	C₂³	—	106	2	no	—
269	C₂³	C₄²×D₅₃	1	1	yes	characteristic
270	C₂×C₄	—	53	2	no	—
271	C₂×C₄	—	53	2	no	—
272	C₂×C₄	—	53	2	no	—
273	C₂×C₄	—	106	2	no	—
274	C₂×C₄	—	106	2	no	—
275	C₂×C₄	—	106	2	no	—
276	C₂×C₄	—	106	4	no	—
277	C₂×C₄	—	106	4	no	—
278	C₂×C₄	—	106	4	no	—
279	C₂×C₄	—	106	4	no	—
280	C₂×C₄	—	106	4	no	—
281	C₂×C₄	—	106	4	no	—
282	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
283	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
284	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
285	C₂×C₄	#13133223	1	4	yes	—
286	C₂×C₄	#13133225	1	4	yes	—
287	C₂×C₄	#13133225	1	4	yes	—
288	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
291	C₈	—	2	4	no	—
292	C₈	—	2	4	no	—
293	C₈	—	2	4	no	—
294	C₈	—	106	4	no	—
295	C₈	—	106	4	no	—
296	C₈	—	106	4	no	—
297	C₂²	—	106	2	no	—
298	C₂²	—	106	2	no	—
299	C₂²	—	106	2	no	—
300	C₂²	—	212	4	no	—
301	C₂²	#16810577	1	2	yes	—
302	C₂²	#16810580	1	2	yes	—
303	C₂²	#16810580	1	2	yes	—
304	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
305	C₄	—	106	2	no	—
306	C₄	—	106	2	no	—
307	C₄	—	106	2	no	—
308	C₄	#16810577	1	2	yes	—
309	C₄	#16810580	1	2	yes	—
310	C₄	#16810580	1	2	yes	—
311	C₄	C₂×#13133223	1	2	yes	—
312	C₄	C₂×#13133225	1	2	yes	—
313	C₄	C₂×#13133225	1	2	yes	—
314	C₂	—	212	2	no	—
315	C₂	#20273016	1	4	yes	direct product factor
316	C₂	C₂×#16810577	1	1	yes	characteristic
317	C₂	C₂×#16810580	1	1	yes	characteristic
318	C₂	C₂×#16810580	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₄²
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#322	C₂
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂³×D₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	C₂×#1454
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
4. #20273016	C₂	direct (N × H)
5. C₂×#16810345	C₂	semidirect
6. C₂×#16810344	C₂	semidirect
7. C₂×C₈×#88322	C₂	semidirect
8. C₂×C₅₃×#322	C₂	semidirect
25. #16810345	C₂²	semidirect
26. #16810344	C₂²	semidirect
27. C₈×#88322	C₂²	semidirect
31. C₅₃×#322	C₂²	semidirect
161. C₂×#322	D₅₃	semidirect
163. C₂×C₅₃	#1454	semidirect
189. #322	C₂×D₅₃	semidirect
191. C₅₃	C₂×#1454	semidirect
315. C₂	#20273016	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×#16810674	2
#20273016	2
C₂×#16810345	2
C₂×#16810344	2
C₂×C₈×#88322	2
C₂×C₅₃×#322	2
C₂×#1454	53

Subgroups — C2×#20273016 #27349717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20273016 #27349717