Subgroups — C₂×#20272825 #27299016

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×#13133151	C₂	1	1	yes	maximal, characteristic
3	#20272825	C₂	1	4	yes	maximal, direct product factor
4	C₂²×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₅₃	—	2	1	no	—
6	C₂×#13133157	—	2	1	no	—
7	C₂×#13133157	—	2	2	no	—
8	C₂×#13133157	C₄	1	4	yes	complemented
9	C₂×#13133157	C₂²	1	2	yes	—
10	C₂²×#11719908	—	2	1	no	—
11	C₂×#13133151	—	2	1	no	—
12	C₂×#13133151	—	2	1	no	—
13	C₂×#13133151	—	2	1	no	—
14	C₂×#13133151	C₄	1	2	yes	—
15	C₂×#13133151	C₄	1	2	yes	—
16	C₂×#13133151	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₅₃	—	2	1	no	—
18	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#108×C₅₃	—	2	1	no	—
20	C₂×#108×C₅₃	—	2	2	no	—
21	C₂×#108×C₅₃	C₄	1	4	yes	complemented
22	C₂×#108×C₅₃	C₂²	1	2	yes	—
23	C₂³×Dic₅₃	—	4	1	no	—
24	C₂×C₄⋊Dic₅₃	—	2	2	no	—
25	C₂×C₄⋊Dic₅₃	—	2	2	no	—
26	C₂×C₄⋊Dic₅₃	—	2	2	no	—
27	C₂×C₄⋊Dic₅₃	—	4	1	no	—
28	C₂×C₄⋊Dic₅₃	—	4	1	no	—
29	C₂×C₄⋊Dic₅₃	—	4	1	no	—
30	#13133157	—	2	4	no	—
31	#13133157	—	2	2	no	—
32	#13133157	C₂×C₄	1	4	yes	complemented
33	C₂×#11719908	—	2	4	no	—
34	C₂×#11719908	—	2	2	no	—
35	#13133151	—	2	2	no	—
36	#13133151	—	2	2	no	—
37	#13133151	—	2	2	no	—
38	#13133151	C₂×C₄	1	4	yes	—
39	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
40	C₂×M₁₆×C₅₃	—	2	4	no	—
41	C₂×M₁₆×C₅₃	—	2	2	no	—
42	C₂×M₁₆×C₅₃	—	4	1	no	—
43	C₂×M₁₆×C₅₃	—	4	2	no	—
44	C₂²×C₈×C₅₃	—	4	1	no	—
45	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	D₄	1	2	yes	—
52	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
54	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	#108×C₅₃	—	2	4	no	—
56	#108×C₅₃	—	2	2	no	—
57	#108×C₅₃	C₂×C₄	1	4	yes	complemented
58	C₂⁴×C₅₃	SD₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₅₃	—	4	2	no	—
70	C₂²×Dic₅₃	—	4	2	no	—
71	C₂²×Dic₅₃	—	4	2	no	—
72	C₂²×Dic₅₃	—	8	4	no	—
73	M₁₆×C₅₃	—	2	4	no	—
74	M₁₆×C₅₃	—	4	4	no	—
75	M₁₆×C₅₃	—	4	2	no	—
76	C₂×C₈×C₅₃	—	4	4	no	—
77	C₂×C₈×C₅₃	—	4	2	no	—
78	C₂×C₈×C₅₃	C₄²	1	2	yes	—
79	C₂×C₈×C₅₃	C₄²	1	2	yes	—
80	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₅₃	C₂²⋊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	C₄⋊C₄	1	2	yes	—
89	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
90	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
96	C₄⋊Dic₅₃	—	2	4	no	—
97	C₄⋊Dic₅₃	—	4	2	no	—
98	C₄⋊Dic₅₃	—	4	2	no	—
99	C₄⋊Dic₅₃	—	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	#95	1	2	yes	—
109	C₂×C₄×C₅₃	#95	1	2	yes	—
110	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#102	1	2	yes	—
113	C₂×C₄×C₅₃	#102	1	2	yes	—
114	C₂×C₄×C₅₃	#103	1	2	yes	—
115	C₂×C₄×C₅₃	#103	1	2	yes	—
116	C₂×C₄×C₅₃	#103	1	2	yes	—
117	C₂×C₄×C₅₃	#106	1	2	yes	—
118	C₂×C₄×C₅₃	#107	1	2	yes	—
119	C₂×C₄×C₅₃	#107	1	2	yes	—
120	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×Dic₅₃	—	4	4	no	—
127	C₂×Dic₅₃	—	8	4	no	—
128	C₂×Dic₅₃	—	8	4	no	—
129	C₂×Dic₅₃	—	8	4	no	—
130	C₈×C₅₃	—	4	4	no	—
131	C₈×C₅₃	#95	1	2	yes	—
132	C₈×C₅₃	#95	1	2	yes	—
133	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
134	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
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	#340	1	2	yes	—
152	C₄×C₅₃	#340	1	2	yes	—
153	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
154	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
155	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
156	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
157	Dic₅₃	—	8	4	no	—
158	C₂²×M₃₂	—	53	1	no	—
159	C₂²×#108	D₅₃	1	1	yes	characteristic
160	C₂²×#107	—	53	1	no	—
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	—	2	1	no	—
170	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
171	C₂²×C₄⋊C₄	—	106	1	no	—
172	C₂×M₃₂	—	53	4	no	—
173	C₂×M₃₂	—	53	2	no	—
174	C₂×M₃₂	—	106	1	no	—
175	C₂×M₃₂	—	106	2	no	—
176	C₂²×C₁₆	—	106	1	no	—
177	C₂×#108	—	2	2	no	—
178	C₂×#108	—	2	1	no	—
179	C₂×#108	Dic₅₃	1	4	yes	complemented
180	C₂×#108	C₂×D₅₃	1	2	yes	—
181	C₂×#107	—	53	2	no	—
182	C₂×#107	—	53	2	no	—
183	C₂×#107	—	53	2	no	—
184	C₂×#107	—	106	1	no	—
185	C₂×#107	—	106	1	no	—
186	C₂×#107	—	106	1	no	—
187	C₅₃	C₂×#1387	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₄	—	212	1	no	—
189	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	4	no	—
192	C₂×M₁₆	—	4	1	no	—
193	C₂×M₁₆	—	4	2	no	—
194	C₂²×C₈	—	4	1	no	—
195	C₂²×C₈	Dic₁₀₆	1	2	yes	—
196	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
197	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
198	C₂²×C₈	C₄×D₅₃	1	2	yes	—
199	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
200	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
201	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
202	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
203	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
205	C₂×C₄⋊C₄	—	106	2	no	—
206	C₂×C₄⋊C₄	—	106	2	no	—
207	C₂×C₄⋊C₄	—	106	2	no	—
208	C₂×C₄⋊C₄	—	212	1	no	—
209	C₂×C₄⋊C₄	—	212	1	no	—
210	C₂×C₄⋊C₄	—	212	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	—	2	2	no	—
217	#108	—	2	4	no	—
218	#108	C₂×Dic₅₃	1	4	yes	complemented
219	#107	—	53	4	no	—
220	#107	—	106	2	no	—
221	#107	—	106	2	no	—
222	#107	—	106	2	no	—
223	C₂⁴	#11719924	1	1	yes	characteristic
224	C₂²×C₄	—	424	4	no	—
225	C₂²×C₄	—	212	2	no	—
226	C₂²×C₄	—	212	2	no	—
227	C₂²×C₄	—	212	2	no	—
228	C₂²×C₄	#11719919	1	2	yes	—
229	C₂²×C₄	#11719921	1	1	yes	characteristic
230	C₂²×C₄	#11719921	1	1	yes	characteristic
231	C₂²×C₄	#11719922	1	1	yes	characteristic
232	C₂²×C₄	#11719922	1	1	yes	characteristic
233	C₂²×C₄	#11719923	1	2	yes	—
234	C₂²×C₄	#11719924	1	1	yes	characteristic
235	C₂²×C₄	#11719925	1	2	yes	—
236	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
237	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
238	M₁₆	—	2	4	no	—
239	M₁₆	—	4	4	no	—
240	M₁₆	—	4	2	no	—
241	C₂×C₈	—	4	4	no	—
242	C₂×C₈	—	4	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	1	yes	characteristic
251	C₂×C₈	#11719921	1	1	yes	characteristic
252	C₂×C₈	#11719921	1	2	yes	—
253	C₂×C₈	#11719921	1	2	yes	—
254	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
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	—	106	4	no	—
262	C₄⋊C₄	—	212	2	no	—
263	C₄⋊C₄	—	212	2	no	—
264	C₄⋊C₄	—	212	2	no	—
265	C₁₆	—	106	4	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	#13133137	1	1	yes	characteristic
269	C₂³	#13133137	1	1	yes	characteristic
270	C₂³	#13133141	1	2	yes	—
271	C₂³	#13133148	1	2	yes	—
272	C₂³	C₂×#11719924	1	1	yes	characteristic
273	C₂×C₄	—	424	4	no	—
274	C₂×C₄	—	424	4	no	—
275	C₂×C₄	—	424	4	no	—
276	C₂×C₄	—	212	4	no	—
277	C₂×C₄	#13133124	1	2	yes	—
278	C₂×C₄	#13133124	1	2	yes	—
279	C₂×C₄	#13133137	1	1	yes	characteristic
280	C₂×C₄	#13133137	1	1	yes	characteristic
281	C₂×C₄	#13133138	1	2	yes	—
282	C₂×C₄	#13133138	1	2	yes	—
283	C₂×C₄	#13133140	1	2	yes	—
284	C₂×C₄	#13133140	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×#11719923	1	1	yes	characteristic
294	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
295	C₈	—	4	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	#16810435	1	2	yes	—
311	C₂²	#16810438	1	2	yes	—
312	C₂²	C₂×#13133137	1	1	yes	characteristic
313	C₂²	C₂×#13133141	1	1	yes	characteristic
314	C₂²	C₂×#13133148	1	1	yes	characteristic
315	C₄	—	424	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×#13133138	1	1	yes	characteristic
320	C₄	C₂×#13133140	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	#20272825	1	4	yes	direct product factor
326	C₂	C₂×#16810385	1	1	yes	characteristic
327	C₂	C₂×#16810435	1	1	yes	characteristic
328	C₂	C₂×#16810438	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133137
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×#108×C₅₃	C₂
socle	C₂³×C₅₃	#102
prime radical	C₂⁴×C₅₃	SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×#108	D₅₃
hypocenter	C₅₃	C₂×#1387
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
3. #20272825	C₂	direct (N × H)
8. C₂×#13133157	C₄	semidirect
21. C₂×#108×C₅₃	C₄	semidirect
32. #13133157	C₂×C₄	semidirect
57. #108×C₅₃	C₂×C₄	semidirect
165. C₂×C₅₃	#1387	semidirect
179. C₂×#108	Dic₅₃	semidirect
187. C₅₃	C₂×#1387	semidirect
218. #108	C₂×Dic₅₃	semidirect
325. C₂	#20272825	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×#13133151	2
#20272825	2
C₂²×#108×C₅₃	2
C₂×#1387	53

Subgroups — C2×#20272825 #27299016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272825 #27299016