Subgroups — C₂×#20272823 #27299014

Order 13568.

614 conjugacy classes of subgroups.

The subgroup lattice has 330 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	#20272823	C₂	1	4	yes	maximal, direct product factor
4	C₂²×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	—	2	1	no	—
6	C₂×#13133157	—	2	2	no	—
7	C₂×#13133157	—	2	1	no	—
8	C₂×#13133157	C₄	1	4	yes	complemented
9	C₂×#13133157	C₂²	1	2	yes	—
10	C₂²×#11719908	—	2	1	no	—
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×#107×C₅₃	—	2	1	no	—
18	C₂×#107×C₅₃	—	2	1	no	—
19	C₂×#107×C₅₃	—	2	1	no	—
20	C₂×#107×C₅₃	C₄	1	2	yes	—
21	C₂×#107×C₅₃	C₄	1	2	yes	—
22	C₂×#107×C₅₃	C₂²	1	2	yes	—
23	C₂×#11719917	—	2	4	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	—	4	1	no	—
26	C₂×#11719917	—	4	2	no	—
27	C₂²×#88322	—	4	1	no	—
28	#13133157	—	2	4	no	—
29	#13133157	—	2	2	no	—
30	#13133157	C₂×C₄	1	4	yes	complemented
31	C₂×#11719908	—	2	4	no	—
32	C₂×#11719908	—	2	2	no	—
33	#13133153	—	2	4	no	—
34	#13133153	—	2	2	no	—
35	#13133153	C₂×C₄	1	4	yes	complemented
36	C₂³×C₄×C₅₃	—	4	1	no	—
37	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₅₃	D₄	1	2	yes	—
43	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
50	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
51	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
52	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
53	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
54	#107×C₅₃	—	2	2	no	—
55	#107×C₅₃	—	2	2	no	—
56	#107×C₅₃	—	2	2	no	—
57	#107×C₅₃	C₂×C₄	1	4	yes	—
58	C₂⁴×C₅₃	SD₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	—	4	2	no	—
60	C₂²×C₄×C₅₃	—	4	2	no	—
61	C₂²×C₄×C₅₃	—	4	2	no	—
62	C₂²×C₄×C₅₃	—	8	4	no	—
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
71	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₄²	1	2	yes	—
74	C₂×C₈×C₅₃	C₄²	1	2	yes	—
75	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₅₃	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
91	C₄⋊C₄×C₅₃	—	2	4	no	—
92	C₄⋊C₄×C₅₃	—	4	2	no	—
93	C₄⋊C₄×C₅₃	—	4	2	no	—
94	C₄⋊C₄×C₅₃	—	4	2	no	—
95	#11719917	—	2	4	no	—
96	#11719917	—	4	4	no	—
97	#11719917	—	4	2	no	—
98	C₂×#88322	—	4	4	no	—
99	C₂×#88322	—	4	2	no	—
100	#11719908	—	2	4	no	—
101	C₂³×C₅₃	—	2	2	no	—
102	C₂³×C₅₃	—	2	2	no	—
103	C₂³×C₅₃	#102	1	1	yes	characteristic
104	C₂³×C₅₃	#102	1	1	yes	characteristic
105	C₂³×C₅₃	#103	1	2	yes	—
106	C₂³×C₅₃	#106	1	2	yes	—
107	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₅₃	—	4	4	no	—
109	C₂×C₄×C₅₃	—	8	4	no	—
110	C₂×C₄×C₅₃	—	8	4	no	—
111	C₂×C₄×C₅₃	—	8	4	no	—
112	C₂×C₄×C₅₃	#95	1	2	yes	—
113	C₂×C₄×C₅₃	#95	1	2	yes	—
114	C₂×C₄×C₅₃	#102	1	2	yes	—
115	C₂×C₄×C₅₃	#102	1	2	yes	—
116	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
117	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#103	1	2	yes	—
119	C₂×C₄×C₅₃	#103	1	2	yes	—
120	C₂×C₄×C₅₃	#103	1	2	yes	—
121	C₂×C₄×C₅₃	#106	1	2	yes	—
122	C₂×C₄×C₅₃	#107	1	2	yes	—
123	C₂×C₄×C₅₃	#107	1	2	yes	—
124	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
130	C₈×C₅₃	#95	1	2	yes	—
131	C₈×C₅₃	#95	1	2	yes	—
132	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
133	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	#88322	—	4	4	no	—
137	C₂×#1387	—	53	1	no	maximal, Sylow
138	C₂²×C₅₃	—	2	1	no	—
139	C₂²×C₅₃	—	2	1	no	—
140	C₂²×C₅₃	—	2	1	no	—
141	C₂²×C₅₃	—	2	1	no	—
142	C₂²×C₅₃	—	2	4	no	—
143	C₂²×C₅₃	—	2	4	no	—
144	C₂²×C₅₃	#340	1	2	yes	—
145	C₂²×C₅₃	#340	1	2	yes	—
146	C₂²×C₅₃	#361	1	2	yes	—
147	C₂²×C₅₃	#362	1	2	yes	—
148	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
149	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
150	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
151	C₄×C₅₃	—	8	4	no	—
152	C₄×C₅₃	#340	1	2	yes	—
153	C₄×C₅₃	#340	1	2	yes	—
154	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
155	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
156	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
157	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
158	C₂²×M₃₂	—	53	1	no	—
159	C₂²×#108	—	53	1	no	—
160	C₂²×#107	D₅₃	1	1	yes	characteristic
161	#1387	—	53	4	no	—
162	C₂×C₅₃	—	2	2	no	—
163	C₂×C₅₃	—	2	1	no	—
164	C₂×C₅₃	—	2	1	no	—
165	C₂×C₅₃	#1387	1	4	yes	complemented
166	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
167	C₂×C₅₃	C₂×#361	1	1	yes	characteristic
168	C₂×C₅₃	C₂×#362	1	1	yes	characteristic
169	C₂²×M₁₆	—	106	1	no	—
170	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
171	C₂²×C₄⋊C₄	—	2	1	no	—
172	C₂×M₃₂	—	53	4	no	—
173	C₂×M₃₂	—	53	2	no	—
174	C₂×M₃₂	—	106	2	no	—
175	C₂×M₃₂	—	106	1	no	—
176	C₂²×C₁₆	—	106	1	no	—
177	C₂×#108	—	53	4	no	—
178	C₂×#108	—	53	2	no	—
179	C₂×#108	—	106	2	no	—
180	C₂×#108	—	106	1	no	—
181	C₂×#107	—	2	1	no	—
182	C₂×#107	—	2	1	no	—
183	C₂×#107	—	2	1	no	—
184	C₂×#107	Dic₅₃	1	2	yes	—
185	C₂×#107	Dic₅₃	1	2	yes	—
186	C₂×#107	C₂×D₅₃	1	2	yes	—
187	C₅₃	C₂×#1387	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₄	—	4	1	no	—
189	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
190	C₂×M₁₆	—	106	2	no	—
191	C₂×M₁₆	—	106	4	no	—
192	C₂×M₁₆	—	212	1	no	—
193	C₂×M₁₆	—	212	2	no	—
194	C₂²×C₈	—	212	1	no	—
195	C₂²×C₈	Dic₁₀₆	1	2	yes	—
196	C₂²×C₈	C₄×D₅₃	1	2	yes	—
197	C₂²×C₈	C₄×D₅₃	1	2	yes	—
198	C₂²×C₈	D₂₁₂	1	2	yes	—
199	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
200	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
201	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
202	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
203	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	4	1	no	—
209	C₂×C₄⋊C₄	—	4	1	no	—
210	C₂×C₄⋊C₄	—	4	1	no	—
211	M₃₂	—	53	4	no	—
212	M₃₂	—	106	2	no	—
213	M₃₂	—	106	4	no	—
214	C₂×C₁₆	—	106	2	no	—
215	C₂×C₁₆	—	106	4	no	—
216	#108	—	53	4	no	—
217	#108	—	106	2	no	—
218	#108	—	106	4	no	—
219	#107	—	2	2	no	—
220	#107	—	2	2	no	—
221	#107	—	2	2	no	—
222	#107	C₂×Dic₅₃	1	4	yes	—
223	C₂⁴	#11719923	1	1	yes	characteristic
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	—	8	4	no	—
228	C₂²×C₄	#11719919	1	2	yes	—
229	C₂²×C₄	#11719921	1	2	yes	—
230	C₂²×C₄	#11719922	1	1	yes	characteristic
231	C₂²×C₄	#11719922	1	1	yes	characteristic
232	C₂²×C₄	#11719923	1	1	yes	characteristic
233	C₂²×C₄	#11719924	1	2	yes	—
234	C₂²×C₄	#11719925	1	2	yes	—
235	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
236	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
238	M₁₆	—	106	4	no	—
239	M₁₆	—	212	4	no	—
240	M₁₆	—	212	2	no	—
241	C₂×C₈	—	212	4	no	—
242	C₂×C₈	—	212	2	no	—
243	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
244	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
245	C₂×C₈	#11719919	1	2	yes	—
246	C₂×C₈	#11719919	1	2	yes	—
247	C₂×C₈	#11719919	1	2	yes	—
248	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
249	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
250	C₂×C₈	#11719921	1	2	yes	—
251	C₂×C₈	#11719921	1	2	yes	—
252	C₂×C₈	#11719921	1	2	yes	—
253	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
254	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
255	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
256	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
257	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
258	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
259	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₁₆	—	106	4	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	#13133138	1	1	yes	characteristic
269	C₂³	#13133138	1	1	yes	characteristic
270	C₂³	#13133140	1	2	yes	—
271	C₂³	#13133148	1	2	yes	—
272	C₂³	C₂×#11719923	1	1	yes	characteristic
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	8	4	no	—
275	C₂×C₄	—	8	4	no	—
276	C₂×C₄	—	8	4	no	—
277	C₂×C₄	#13133124	1	2	yes	—
278	C₂×C₄	#13133124	1	2	yes	—
279	C₂×C₄	#13133137	1	2	yes	—
280	C₂×C₄	#13133137	1	2	yes	—
281	C₂×C₄	#13133138	1	1	yes	characteristic
282	C₂×C₄	#13133138	1	1	yes	characteristic
283	C₂×C₄	#13133140	1	2	yes	—
284	C₂×C₄	#13133141	1	2	yes	—
285	C₂×C₄	#13133141	1	2	yes	—
286	C₂×C₄	#13133148	1	2	yes	—
287	C₂×C₄	#13133150	1	2	yes	—
288	C₂×C₄	#13133150	1	2	yes	—
289	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
290	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
291	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
292	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
293	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
294	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
295	C₈	—	212	4	no	—
296	C₈	#13133124	1	2	yes	—
297	C₈	#13133124	1	2	yes	—
298	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
299	C₈	C₂×#11719921	1	1	yes	characteristic
300	C₈	C₂×#11719919	1	1	yes	characteristic
301	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
302	C₂²	—	2	4	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	#16810385	1	2	yes	—
309	C₂²	#16810385	1	2	yes	—
310	C₂²	#16810436	1	2	yes	—
311	C₂²	#16810439	1	2	yes	—
312	C₂²	C₂×#13133138	1	1	yes	characteristic
313	C₂²	C₂×#13133140	1	1	yes	characteristic
314	C₂²	C₂×#13133148	1	1	yes	characteristic
315	C₄	—	8	4	no	—
316	C₄	#16810385	1	2	yes	—
317	C₄	#16810385	1	2	yes	—
318	C₄	C₂×#13133124	1	1	yes	characteristic
319	C₄	C₂×#13133137	1	1	yes	characteristic
320	C₄	C₂×#13133141	1	1	yes	characteristic
321	C₄	C₂×#13133150	1	1	yes	characteristic
322	C₂	—	2	1	no	—
323	C₂	—	2	2	no	—
324	C₂	—	2	1	no	—
325	C₂	#20272823	1	4	yes	direct product factor
326	C₂	C₂×#16810385	1	1	yes	characteristic
327	C₂	C₂×#16810436	1	1	yes	characteristic
328	C₂	C₂×#16810439	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133138
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×#107×C₅₃	C₂
socle	C₂³×C₅₃	#102
prime radical	C₂⁴×C₅₃	SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×#107	D₅₃
hypocenter	C₅₃	C₂×#1387
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
3. #20272823	C₂	direct (N × H)
8. C₂×#13133157	C₄	semidirect
13. C₂×#13133153	C₄	semidirect
30. #13133157	C₂×C₄	semidirect
35. #13133153	C₂×C₄	semidirect
165. C₂×C₅₃	#1387	semidirect
187. C₅₃	C₂×#1387	semidirect
325. C₂	#20272823	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
#20272823	2
C₂²×#107×C₅₃	2
C₂×#1387	53

Subgroups — C2×#20272823 #27299014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272823 #27299014