Subgroups — C₂×#20276486 #27537102

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810658	C₂	1	1	yes	maximal, characteristic
4	#20276486	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810447	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810446	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16810445	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×M₁₆×D₅₃	—	2	1	no	—
10	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
11	C₂×#13133175	—	2	1	no	—
12	C₂×#13133157	C₂²	1	1	yes	characteristic
13	C₂×#13133156	C₄	1	2	yes	complemented
14	M₃₂×D₅₃	C₄	1	4	yes	complemented
15	M₃₂×D₅₃	C₂²	1	4	yes	—
16	C₂×C₁₆×D₅₃	C₄	1	2	yes	complemented
17	C₂×#13133153	C₂²	1	1	yes	characteristic
18	C₂×#13133152	—	2	1	no	—
19	#108×D₅₃	—	2	2	no	—
20	#108×D₅₃	C₂²	1	4	yes	—
21	#16810658	—	2	2	no	—
22	#16810658	C₂²	1	4	yes	—
23	C₂×#13133150	—	2	1	no	—
24	C₂×#13133149	C₂²	1	1	yes	characteristic
25	#16810447	C₂²	1	4	yes	complemented
26	#16810446	C₂²	1	4	yes	complemented
27	#16810445	C₂²	1	4	yes	complemented
28	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#365	C₂²	1	4	yes	complemented
32	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
33	C₂×#11719917	—	2	1	no	—
34	C₂×#11719912	—	4	1	no	—
35	M₁₆×D₅₃	—	2	4	no	—
36	M₁₆×D₅₃	—	4	2	no	—
37	C₂²×#88322	D₄	1	1	yes	characteristic
38	C₂×C₈×D₅₃	—	4	1	no	—
39	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₅₃	D₄	1	2	yes	—
43	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
44	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
45	C₂×C₈×D₅₃	Q₈	1	2	yes	—
46	C₂×C₈×D₅₃	C₂³	1	2	yes	—
47	#13133175	—	2	4	no	—
48	#13133175	—	4	2	no	—
49	C₂×C₄⋊Dic₅₃	—	2	1	no	—
50	C₂×#11719921	—	4	1	no	—
51	C₂×C₄×Dic₅₃	—	4	1	no	—
52	#13133157	C₂×C₄	1	2	yes	—
53	#13133157	C₂³	1	2	yes	—
54	#13133156	C₂×C₄	1	8	yes	complemented
55	C₂×#11719908	C₂×C₄	1	2	yes	—
56	C₁₆×D₅₃	C₂×C₄	1	8	yes	complemented
57	#13133153	—	2	1	no	—
58	#13133153	C₂³	1	2	yes	—
59	#13133152	—	2	4	no	—
60	#13133150	—	2	4	no	—
61	#13133149	—	2	1	no	—
62	#13133149	C₂³	1	2	yes	—
63	C₂×M₁₆×C₅₃	—	2	1	no	—
64	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
66	M₃₂×C₅₃	C₂×C₄	1	2	yes	—
67	M₃₂×C₅₃	C₂³	1	2	yes	—
68	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
69	#108×C₅₃	—	2	1	no	—
70	#108×C₅₃	C₂³	1	2	yes	—
71	#106×C₅₃	—	2	1	no	—
72	#106×C₅₃	C₂³	1	2	yes	—
73	C₂³×D₅₃	SD₁₆	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	—	4	1	no	—
75	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₅₃	—	4	1	no	—
77	C₂²×Dic₅₃	SD₁₆	1	1	yes	characteristic
78	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
81	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
82	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
83	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
84	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
85	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
86	M₁₆×C₅₃	—	2	2	no	—
87	M₁₆×C₅₃	—	4	1	no	—
88	C₂×C₈×C₅₃	—	4	1	no	—
89	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₅₃	—	2	2	no	—
96	C₄⋊C₄×C₅₃	—	4	1	no	—
97	#11719921	—	4	4	no	—
98	C₄⋊Dic₅₃	—	2	2	no	—
99	C₄⋊Dic₅₃	—	4	1	no	—
100	C₄×Dic₅₃	—	4	4	no	—
101	#11719917	—	2	2	no	—
102	#11719917	—	4	1	no	—
103	C₂×#88322	—	4	1	no	—
104	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
105	C₂×#88322	C₄⋊C₄	1	2	yes	—
106	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
107	C₂×#88322	C₂×D₄	1	2	yes	—
108	#11719912	—	4	4	no	—
109	C₈×D₅₃	—	4	4	no	—
110	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
111	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
112	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
113	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
114	C₈×D₅₃	C₂²×C₄	1	2	yes	—
115	C₈×D₅₃	C₂²×C₄	1	2	yes	—
116	C₈×D₅₃	C₂×Q₈	1	2	yes	—
117	C₈×D₅₃	C₂×Q₈	1	2	yes	—
118	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
119	#11719908	C₂²×C₄	1	4	yes	—
120	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
121	C₂²×D₅₃	—	4	2	no	—
122	C₂²×D₅₃	#106	1	1	yes	characteristic
123	C₂²×D₅₃	#106	1	1	yes	characteristic
124	C₂²×D₅₃	#106	1	2	yes	—
125	C₂²×D₅₃	C₂×SD₁₆	1	2	yes	—
126	C₂×C₄×C₅₃	—	4	2	no	—
127	C₂×C₄×C₅₃	—	8	2	no	—
128	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×Dic₅₃	—	4	2	no	—
135	C₂×Dic₅₃	—	8	2	no	—
136	C₂×Dic₅₃	#106	1	2	yes	—
137	C₂×Dic₅₃	#106	1	1	yes	characteristic
138	C₂×Dic₅₃	#106	1	1	yes	characteristic
139	C₂×Dic₅₃	C₂×SD₁₆	1	2	yes	—
140	C₄×D₅₃	#106	1	2	yes	—
141	C₄×D₅₃	#106	1	2	yes	—
142	C₄×D₅₃	#106	1	2	yes	—
143	C₄×D₅₃	#106	1	2	yes	—
144	C₄×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
145	C₄×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
146	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
147	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
148	C₈×C₅₃	—	4	2	no	—
149	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
152	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
153	#88322	—	4	2	no	—
154	#88322	C₂×C₄⋊C₄	1	2	yes	—
155	#88322	C₂×C₄⋊C₄	1	2	yes	—
156	C₂×#2155	—	53	1	no	maximal, Sylow
157	C₂²×C₅₃	—	2	2	no	—
158	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
159	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
160	C₂²×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
161	C₂×D₅₃	—	2	2	no	—
162	C₂×D₅₃	—	2	2	no	—
163	C₂×D₅₃	—	4	2	no	—
164	C₂×D₅₃	—	4	2	no	—
165	C₂×D₅₃	C₂×#106	1	2	yes	—
166	C₂×D₅₃	C₂×#106	1	2	yes	—
167	C₄×C₅₃	—	8	2	no	—
168	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
169	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
170	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
171	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
172	Dic₅₃	—	8	2	no	—
173	Dic₅₃	C₂×#106	1	2	yes	—
174	Dic₅₃	C₂×#106	1	2	yes	—
175	C₂²×M₃₂	—	53	1	no	—
176	C₂²×#108	—	53	1	no	—
177	C₂×#427	—	53	1	no	—
178	#2155	—	53	8	no	—
179	C₂×#365	—	53	1	no	—
180	C₂×#365	—	53	1	no	—
181	C₂×#365	—	53	1	no	—
182	C₂×#365	D₅₃	1	1	yes	characteristic, complemented
183	C₂×C₅₃	—	2	1	no	—
184	C₂×C₅₃	—	2	1	no	—
185	C₂×C₅₃	#2155	1	2	yes	complemented
186	C₂×C₅₃	C₂²×#106	1	1	yes	characteristic
187	D₅₃	—	2	2	no	—
188	D₅₃	—	2	2	no	—
189	C₂²×M₁₆	—	106	1	no	—
190	C₂³×C₈	—	53	1	no	—
191	C₂×#117	—	106	1	no	—
192	C₂×M₃₂	—	53	1	no	—
193	C₂×M₃₂	—	53	4	no	—
194	C₂×M₃₂	—	53	4	no	—
195	C₂×M₃₂	—	53	2	no	—
196	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
197	C₂²×C₁₆	—	53	2	no	—
198	C₂×#108	—	53	1	no	—
199	C₂×#108	—	53	4	no	—
200	C₂×#108	—	106	2	no	—
201	C₂×#108	—	106	1	no	—
202	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
203	#427	—	53	4	no	—
204	#427	—	106	2	no	—
205	C₂×#107	—	106	1	no	—
206	C₂×#106	—	53	1	no	—
207	C₂×#106	C₂×D₅₃	1	1	yes	characteristic
208	#365	—	53	4	no	—
209	#365	—	53	4	no	—
210	#365	—	53	4	no	—
211	#365	C₂×D₅₃	1	4	yes	complemented
212	C₅₃	C₂×#2155	1	1	yes	characteristic, complemented, Sylow
213	C₂³×C₄	—	53	1	no	—
214	C₂×M₁₆	—	2	1	no	—
215	C₂×M₁₆	—	106	4	no	—
216	C₂×M₁₆	—	106	1	no	—
217	C₂×M₁₆	—	212	1	no	—
218	C₂×M₁₆	—	212	2	no	—
219	C₂²×C₈	—	53	1	no	—
220	C₂²×C₈	—	53	1	no	—
221	C₂²×C₈	—	53	2	no	—
222	C₂²×C₈	—	53	1	no	—
223	C₂²×C₈	—	53	2	no	—
224	C₂²×C₈	—	53	2	no	—
225	C₂²×C₈	—	53	1	no	—
226	C₂²×C₈	—	53	1	no	—
227	C₂²×C₈	—	53	2	no	—
228	C₂²×C₈	—	212	1	no	—
229	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
230	#117	—	106	4	no	—
231	#117	—	212	2	no	—
232	C₂×C₄⋊C₄	—	2	1	no	—
233	C₂×C₄⋊C₄	—	106	1	no	—
234	C₂×C₂²⋊C₄	—	212	1	no	—
235	C₂×C₄²	—	212	1	no	—
236	M₃₂	—	53	2	no	—
237	M₃₂	—	53	2	no	—
238	M₃₂	—	53	8	no	—
239	M₃₂	C₄×D₅₃	1	2	yes	—
240	M₃₂	C₂²×D₅₃	1	2	yes	—
241	C₂×C₁₆	—	53	2	no	—
242	C₂×C₁₆	—	53	8	no	—
243	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
244	#108	—	2	1	no	—
245	#108	—	53	2	no	—
246	#108	—	106	4	no	—
247	#108	—	106	1	no	—
248	#108	C₂²×D₅₃	1	2	yes	—
249	#107	—	106	4	no	—
250	#106	—	2	1	no	—
251	#106	—	53	2	no	—
252	#106	—	106	1	no	—
253	#106	C₂²×D₅₃	1	2	yes	—
254	C₂⁴	—	53	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	53	1	no	—
257	C₂²×C₄	—	53	2	no	—
258	C₂²×C₄	—	53	1	no	—
259	C₂²×C₄	—	53	1	no	—
260	C₂²×C₄	—	53	2	no	—
261	C₂²×C₄	—	53	1	no	—
262	C₂²×C₄	—	53	2	no	—
263	C₂²×C₄	—	53	1	no	—
264	C₂²×C₄	—	53	2	no	—
265	C₂²×C₄	—	212	1	no	—
266	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
267	M₁₆	—	2	2	no	—
268	M₁₆	—	4	1	no	—
269	M₁₆	—	106	2	no	—
270	M₁₆	—	212	4	no	—
271	M₁₆	—	212	1	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	53	2	no	—
274	C₂×C₈	—	53	1	no	—
275	C₂×C₈	—	53	1	no	—
276	C₂×C₈	—	53	2	no	—
277	C₂×C₈	—	53	2	no	—
278	C₂×C₈	—	53	2	no	—
279	C₂×C₈	—	53	2	no	—
280	C₂×C₈	—	53	2	no	—
281	C₂×C₈	—	53	2	no	—
282	C₂×C₈	—	53	2	no	—
283	C₂×C₈	—	53	2	no	—
284	C₂×C₈	—	53	2	no	—
285	C₂×C₈	—	212	4	no	—
286	C₂×C₈	—	212	1	no	—
287	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
288	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
289	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
290	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
291	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
292	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	106	2	no	—
296	C₄⋊C₄	—	212	1	no	—
297	C₂²⋊C₄	—	212	4	no	—
298	C₄²	—	212	4	no	—
299	C₁₆	—	53	4	no	—
300	C₁₆	C₂×C₄×D₅₃	1	4	yes	—
301	C₂³	—	53	1	no	—
302	C₂³	—	53	1	no	—
303	C₂³	—	53	2	no	—
304	C₂³	—	53	2	no	—
305	C₂³	—	212	2	no	—
306	C₂³	SD₁₆×D₅₃	1	1	yes	characteristic
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	8	2	no	—
309	C₂×C₄	—	53	2	no	—
310	C₂×C₄	—	53	2	no	—
311	C₂×C₄	—	53	1	no	—
312	C₂×C₄	—	53	1	no	—
313	C₂×C₄	—	53	2	no	—
314	C₂×C₄	—	53	2	no	—
315	C₂×C₄	—	53	2	no	—
316	C₂×C₄	—	53	2	no	—
317	C₂×C₄	—	53	2	no	—
318	C₂×C₄	—	53	2	no	—
319	C₂×C₄	—	53	2	no	—
320	C₂×C₄	—	53	2	no	—
321	C₂×C₄	—	424	2	no	—
322	C₂×C₄	—	212	2	no	—
323	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
324	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
325	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
326	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
327	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
329	C₈	—	4	2	no	—
330	C₈	—	53	2	no	—
331	C₈	—	53	2	no	—
332	C₈	—	212	2	no	—
333	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
334	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
335	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
336	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
337	C₂²	—	2	2	no	—
338	C₂²	—	53	2	no	—
339	C₂²	—	53	2	no	—
340	C₂²	—	106	2	no	—
341	C₂²	—	106	2	no	—
342	C₂²	—	212	2	no	—
343	C₂²	—	212	2	no	—
344	C₂²	#106×D₅₃	1	1	yes	characteristic
345	C₂²	#106×D₅₃	1	1	yes	characteristic
346	C₂²	C₂×SD₁₆×D₅₃	1	1	yes	characteristic
347	C₄	—	8	2	no	—
348	C₄	—	53	2	no	—
349	C₄	—	53	2	no	—
350	C₄	—	424	2	no	—
351	C₄	#106×D₅₃	1	1	yes	characteristic
352	C₄	#106×D₅₃	1	1	yes	characteristic
353	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
354	C₄	C₂×SD₁₆×D₅₃	1	1	yes	characteristic
355	C₂	—	2	1	no	—
356	C₂	—	2	1	no	—
357	C₂	—	106	2	no	—
358	C₂	—	106	2	no	—
359	C₂	#20276486	1	2	yes	direct product factor
360	C₂	C₂×#106×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#106×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#365	C₂
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂³×D₅₃	SD₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#365	D₅₃
hypocenter	C₅₃	C₂×#2155
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

Complements

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

N	H ≅ G/N	type
4. #20276486	C₂	direct (N × H)
5. C₂×#16810447	C₂	semidirect
6. C₂×#16810446	C₂	semidirect
7. C₂×#16810445	C₂	semidirect
8. C₂×C₅₃×#365	C₂	semidirect
13. C₂×#13133156	C₄	semidirect
14. M₃₂×D₅₃	C₄	semidirect
16. C₂×C₁₆×D₅₃	C₄	semidirect
25. #16810447	C₂²	semidirect
26. #16810446	C₂²	semidirect
27. #16810445	C₂²	semidirect
31. C₅₃×#365	C₂²	semidirect
54. #13133156	C₂×C₄	semidirect
56. C₁₆×D₅₃	C₂×C₄	semidirect
182. C₂×#365	D₅₃	semidirect
185. C₂×C₅₃	#2155	semidirect
211. #365	C₂×D₅₃	semidirect
212. C₅₃	C₂×#2155	semidirect
359. C₂	#20276486	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×M₃₂×D₅₃	2
C₂×#108×D₅₃	2
C₂×#16810658	2
#20276486	2
C₂×#16810447	2
C₂×#16810446	2
C₂×#16810445	2
C₂×C₅₃×#365	2
C₂×#2155	53

Subgroups — C2×#20276486 #27537102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276486 #27537102