Subgroups — C₂×#20272798 #27298950

Order 13568.

630 conjugacy classes of subgroups.

The subgroup lattice has 328 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×#13133127	C₂	1	1	yes	maximal, characteristic
3	#20272798	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
7	C₂×#13133157	—	2	1	no	—
8	C₂×#13133157	—	2	2	no	—
9	C₂×#13133157	C₄	1	4	yes	complemented
10	C₂×#13133157	C₂²	1	2	yes	—
11	C₂²×#11719908	—	2	1	no	—
12	C₂×#13133127	—	2	1	no	—
13	C₂×#13133127	—	2	1	no	—
14	C₂×#13133127	—	2	1	no	—
15	C₂×#13133127	—	2	1	no	—
16	C₂×#13133127	C₄	1	2	yes	—
17	C₂×#13133127	C₂²	1	2	yes	—
18	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×M₃₂×C₅₃	—	2	2	no	—
20	C₂×M₃₂×C₅₃	—	2	1	no	—
21	C₂×M₃₂×C₅₃	C₄	1	4	yes	complemented
22	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
23	C₂²×C₁₆×C₅₃	—	2	1	no	—
24	C₂³×Dic₅₃	C₈	1	2	yes	—
25	C₂²×#88322	—	2	1	no	—
26	C₂²×#88322	—	2	1	no	—
27	C₂²×#88322	—	2	2	no	—
28	C₂²×#88322	C₈	1	4	yes	—
29	C₂²×#88322	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₅₃	—	2	1	no	—
31	C₂×C₄×Dic₅₃	—	2	1	no	—
32	C₂×C₄×Dic₅₃	—	2	1	no	—
33	C₂×C₄×Dic₅₃	—	2	1	no	—
34	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
35	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
36	#13133157	—	2	4	no	—
37	#13133157	—	2	2	no	—
38	#13133157	C₂×C₄	1	4	yes	complemented
39	C₂×#11719908	—	2	4	no	—
40	C₂×#11719908	—	2	2	no	—
41	#13133127	—	2	2	no	—
42	#13133127	—	2	2	no	—
43	#13133127	—	2	2	no	—
44	#13133127	—	2	2	no	—
45	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	D₄	1	2	yes	—
54	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	M₃₂×C₅₃	—	2	4	no	—
57	M₃₂×C₅₃	—	2	2	no	—
58	M₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
59	C₂×C₁₆×C₅₃	—	2	4	no	—
60	C₂×C₁₆×C₅₃	—	2	2	no	—
61	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
69	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×Dic₅₃	—	2	2	no	—
73	C₂²×Dic₅₃	—	2	2	no	—
74	C₂²×Dic₅₃	—	4	4	no	—
75	C₂²×Dic₅₃	C₂×C₈	1	4	yes	—
76	C₂×C₈×C₅₃	C₄²	1	2	yes	—
77	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
85	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
86	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
87	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
88	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
94	C₄×Dic₅₃	—	2	2	no	—
95	C₄×Dic₅₃	—	2	2	no	—
96	C₄×Dic₅₃	—	2	2	no	—
97	C₄×Dic₅₃	—	2	2	no	—
98	C₂×#88322	—	2	4	no	—
99	C₂×#88322	—	2	4	no	—
100	C₂×#88322	—	2	2	no	—
101	C₂×#88322	—	2	2	no	—
102	C₂×#88322	C₂×C₈	1	4	yes	—
103	C₁₆×C₅₃	—	2	4	no	—
104	#11719908	—	2	4	no	—
105	C₂³×C₅₃	—	2	4	no	—
106	C₂³×C₅₃	C₄×C₈	1	2	yes	—
107	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
108	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
109	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
110	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#95	1	2	yes	—
112	C₂×C₄×C₅₃	#95	1	2	yes	—
113	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
114	C₂×C₄×C₅₃	#97	1	2	yes	—
115	C₂×C₄×C₅₃	#97	1	2	yes	—
116	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
119	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
120	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
121	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
122	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
123	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
129	C₂×Dic₅₃	—	2	4	no	—
130	C₂×Dic₅₃	—	4	4	no	—
131	C₂×Dic₅₃	—	4	4	no	—
132	C₂×Dic₅₃	—	4	4	no	—
133	C₈×C₅₃	#95	1	2	yes	—
134	C₈×C₅₃	#95	1	2	yes	—
135	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	#88322	—	2	4	no	—
140	C₂×#1377	—	53	1	no	maximal, Sylow
141	C₂²×C₅₃	—	2	2	no	—
142	C₂²×C₅₃	—	2	2	no	—
143	C₂²×C₅₃	—	2	4	no	—
144	C₂²×C₅₃	—	2	4	no	—
145	C₂²×C₅₃	#336	1	2	yes	—
146	C₂²×C₅₃	#336	1	2	yes	—
147	C₂²×C₅₃	#349	1	4	yes	—
148	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
149	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
150	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
151	C₄×C₅₃	#336	1	2	yes	—
152	C₄×C₅₃	#336	1	2	yes	—
153	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
154	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
155	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
156	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
157	Dic₅₃	—	4	4	no	—
158	C₂²×M₃₂	—	53	1	no	—
159	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
160	C₂²×#97	—	53	1	no	—
161	#1377	—	53	4	no	—
162	C₂×C₅₃	—	2	2	no	—
163	C₂×C₅₃	—	2	1	no	—
164	C₂×C₅₃	—	2	1	no	—
165	C₂×C₅₃	#1377	1	4	yes	complemented
166	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
167	C₂×C₅₃	C₂×#349	1	2	yes	—
168	C₂³×C₈	—	53	1	no	—
169	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
170	C₂²×C₄²	—	53	1	no	—
171	C₂×M₃₂	—	2	1	no	—
172	C₂×M₃₂	—	2	2	no	—
173	C₂×M₃₂	—	53	4	no	—
174	C₂×M₃₂	—	53	2	no	—
175	C₂×M₃₂	—	106	1	no	—
176	C₂×M₃₂	—	106	2	no	—
177	C₂×M₃₂	Dic₅₃	1	4	yes	complemented
178	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
179	C₂²×C₁₆	—	2	1	no	—
180	C₂²×C₁₆	—	106	1	no	—
181	C₂×#97	—	53	2	no	—
182	C₂×#97	—	53	2	no	—
183	C₂×#97	—	106	1	no	—
184	C₂×#97	—	106	1	no	—
185	C₂×#97	—	106	1	no	—
186	C₂×#97	—	106	1	no	—
187	C₅₃	C₂×#1377	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₄	—	53	2	no	—
189	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
190	C₂²×C₈	—	53	2	no	—
191	C₂²×C₈	—	53	4	no	—
192	C₂²×C₈	—	106	2	no	—
193	C₂²×C₈	—	106	1	no	—
194	C₂²×C₈	—	106	1	no	—
195	C₂²×C₈	Dic₁₀₆	1	2	yes	—
196	C₂²×C₈	C₄×D₅₃	1	2	yes	—
197	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
198	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
199	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
200	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
201	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
202	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
203	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
204	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
205	C₂×C₄²	—	53	2	no	—
206	C₂×C₄²	—	53	2	no	—
207	C₂×C₄²	—	106	1	no	—
208	C₂×C₄²	—	106	1	no	—
209	C₂×C₄²	—	106	1	no	—
210	C₂×C₄²	—	106	1	no	—
211	M₃₂	—	2	2	no	—
212	M₃₂	—	2	4	no	—
213	M₃₂	—	53	4	no	—
214	M₃₂	—	106	2	no	—
215	M₃₂	—	106	4	no	—
216	M₃₂	C₂×Dic₅₃	1	4	yes	complemented
217	C₂×C₁₆	—	2	2	no	—
218	C₂×C₁₆	—	2	4	no	—
219	C₂×C₁₆	—	106	2	no	—
220	C₂×C₁₆	—	106	4	no	—
221	#97	—	106	2	no	—
222	#97	—	106	2	no	—
223	#97	—	106	2	no	—
224	#97	—	106	2	no	—
225	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
226	C₂²×C₄	—	53	4	no	—
227	C₂²×C₄	—	106	2	no	—
228	C₂²×C₄	—	106	2	no	—
229	C₂²×C₄	—	212	4	no	—
230	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
231	C₂²×C₄	C₈×D₅₃	1	2	yes	—
232	C₂²×C₄	#11719912	1	2	yes	—
233	C₂²×C₄	#11719912	1	1	yes	characteristic
234	C₂²×C₄	#11719912	1	1	yes	characteristic
235	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
236	C₂²×C₄	#11719919	1	2	yes	—
237	C₂²×C₄	#11719921	1	1	yes	characteristic
238	C₂²×C₄	#11719921	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
240	C₂×C₈	—	53	4	no	—
241	C₂×C₈	—	106	4	no	—
242	C₂×C₈	—	106	4	no	—
243	C₂×C₈	—	106	2	no	—
244	C₂×C₈	—	106	2	no	—
245	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
246	C₂×C₈	#11719919	1	2	yes	—
247	C₂×C₈	#11719919	1	2	yes	—
248	C₂×C₈	#11719919	1	2	yes	—
249	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
250	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
251	C₂×C₈	#11719921	1	1	yes	characteristic
252	C₂×C₈	#11719921	1	1	yes	characteristic
253	C₂×C₈	#11719921	1	2	yes	—
254	C₂×C₈	#11719921	1	2	yes	—
255	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
256	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
257	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
258	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
259	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
260	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
263	C₄²	—	106	2	no	—
264	C₄²	—	106	2	no	—
265	C₄²	—	106	2	no	—
266	C₄²	—	106	2	no	—
267	C₁₆	—	2	4	no	—
268	C₁₆	—	106	4	no	—
269	C₂³	—	2	4	no	—
270	C₂³	C₈×Dic₅₃	1	2	yes	—
271	C₂³	#13133129	1	1	yes	characteristic
272	C₂³	#13133129	1	1	yes	characteristic
273	C₂³	#13133146	1	2	yes	—
274	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
275	C₂×C₄	—	106	4	no	—
276	C₂×C₄	—	212	4	no	—
277	C₂×C₄	—	212	4	no	—
278	C₂×C₄	—	212	4	no	—
279	C₂×C₄	#13133124	1	2	yes	—
280	C₂×C₄	#13133124	1	2	yes	—
281	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
282	C₂×C₄	#13133127	1	2	yes	—
283	C₂×C₄	#13133127	1	2	yes	—
284	C₂×C₄	#13133129	1	2	yes	—
285	C₂×C₄	#13133129	1	2	yes	—
286	C₂×C₄	#13133129	1	1	yes	characteristic
287	C₂×C₄	#13133129	1	1	yes	characteristic
288	C₂×C₄	#13133146	1	2	yes	—
289	C₂×C₄	#13133146	1	2	yes	—
290	C₂×C₄	#13133146	1	2	yes	—
291	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
293	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
294	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
295	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
296	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
297	C₈	—	106	4	no	—
298	C₈	#13133124	1	2	yes	—
299	C₈	#13133124	1	2	yes	—
300	C₈	C₂×#11719921	1	1	yes	characteristic
301	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
302	C₈	C₂×#11719919	1	1	yes	characteristic
303	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	4	no	—
307	C₂²	—	2	4	no	—
308	C₂²	#16810377	1	2	yes	—
309	C₂²	#16810377	1	2	yes	—
310	C₂²	#16810404	1	4	yes	—
311	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
312	C₂²	C₂×#13133129	1	1	yes	characteristic
313	C₂²	C₂×#13133146	1	1	yes	characteristic
314	C₄	—	212	4	no	—
315	C₄	#16810377	1	2	yes	—
316	C₄	#16810377	1	2	yes	—
317	C₄	C₂×#13133124	1	1	yes	characteristic
318	C₄	C₂×#13133127	1	1	yes	characteristic
319	C₄	C₂×#13133129	1	1	yes	characteristic
320	C₄	C₂×#13133146	1	1	yes	characteristic
321	C₂	—	2	1	no	—
322	C₂	—	2	2	no	—
323	C₂	—	2	1	no	—
324	C₂	#20272798	1	4	yes	direct product factor
325	C₂	C₂×#16810377	1	1	yes	characteristic
326	C₂	C₂×#16810404	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#11719921
commutator	C₂²×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂³×C₅₃	C₂²⋊C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	C₂×#1377
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20272798	C₂	direct (N × H)
9. C₂×#13133157	C₄	semidirect
21. C₂×M₃₂×C₅₃	C₄	semidirect
38. #13133157	C₂×C₄	semidirect
58. M₃₂×C₅₃	C₂×C₄	semidirect
165. C₂×C₅₃	#1377	semidirect
177. C₂×M₃₂	Dic₅₃	semidirect
187. C₅₃	C₂×#1377	semidirect
216. M₃₂	C₂×Dic₅₃	semidirect
324. C₂	#20272798	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×#13133127	2
#20272798	2
C₂²×M₃₂×C₅₃	2
C₂×#1377	53

Subgroups — C2×#20272798 #27298950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272798 #27298950