Subgroups — C₂×#364×D₅₃ #27537086

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 344 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	2	yes	maximal
2	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810444	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16810443	C₂	1	2	yes	maximal, complemented
5	#364×D₅₃	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₅₃	C₄	1	1	yes	characteristic
10	C₂×#13133157	C₂²	1	2	yes	—
11	C₂×#13133156	—	2	2	no	—
12	M₃₂×D₅₃	—	2	4	no	—
13	M₃₂×D₅₃	C₂²	1	8	yes	—
14	C₂×C₁₆×D₅₃	—	2	2	no	—
15	C₂×C₄×#88322	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₄×C₈×D₅₃	C₄	1	4	yes	—
19	C₄×C₈×D₅₃	C₂²	1	4	yes	—
20	#16810444	C₂²	1	4	yes	complemented
21	#16810443	C₂²	1	8	yes	complemented
22	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#364	C₂²	1	4	yes	complemented
25	C₂²×C₄×D₅₃	—	2	1	no	—
26	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
28	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×D₅₃	C₈	1	2	yes	—
30	C₂×C₈×D₅₃	C₈	1	4	yes	—
31	C₂×C₈×D₅₃	C₈	1	2	yes	—
32	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
34	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
35	C₂×C₈×D₅₃	D₄	1	2	yes	—
36	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
37	C₂×C₈×D₅₃	Q₈	1	2	yes	—
38	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
39	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
40	C₂×C₈×D₅₃	C₂³	1	2	yes	—
41	C₂×C₄×Dic₅₃	—	2	1	no	—
42	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄²×D₅₃	—	2	2	no	—
44	C₄²×D₅₃	C₂×C₄	1	4	yes	—
45	#13133157	—	2	2	no	—
46	#13133157	C₂³	1	4	yes	—
47	#13133156	—	2	8	no	—
48	C₂×#11719908	—	2	2	no	—
49	C₁₆×D₅₃	—	2	8	no	—
50	C₄×#88322	C₂×C₄	1	4	yes	—
51	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
52	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
53	C₈×Dic₅₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
57	M₃₂×C₅₃	—	2	2	no	—
58	M₃₂×C₅₃	C₂³	1	4	yes	—
59	C₂×C₁₆×C₅₃	—	2	2	no	—
60	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₅₃	C₂³	1	2	yes	—
62	C₂³×D₅₃	M₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	—	2	1	no	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×Dic₅₃	—	2	1	no	—
66	C₂²×Dic₅₃	M₁₆	1	1	yes	characteristic
67	C₂×C₄×D₅₃	—	2	8	no	—
68	C₂×C₄×D₅₃	—	2	4	no	—
69	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
70	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
73	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
74	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
75	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
76	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
86	C₄²×C₅₃	—	2	1	no	—
87	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
88	C₄×Dic₅₃	—	2	1	no	—
89	C₄×Dic₅₃	—	2	4	no	—
90	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
91	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
92	C₂×#88322	C₄⋊C₄	1	2	yes	—
93	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
94	C₂×#88322	C₂×C₈	1	2	yes	—
95	C₂×#88322	C₂×C₈	1	2	yes	—
96	C₂×#88322	C₂²×C₄	1	2	yes	—
97	C₂×#88322	C₂²×C₄	1	2	yes	—
98	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
99	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
100	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
101	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
102	C₈×D₅₃	C₂×C₈	1	8	yes	—
103	C₈×D₅₃	C₂×C₈	1	8	yes	—
104	C₈×D₅₃	C₂×D₄	1	2	yes	—
105	C₈×D₅₃	C₂×D₄	1	2	yes	—
106	C₈×D₅₃	C₂×Q₈	1	2	yes	—
107	C₈×D₅₃	C₂×Q₈	1	2	yes	—
108	C₁₆×C₅₃	—	2	4	no	—
109	#11719908	—	2	4	no	—
110	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
111	C₂²×D₅₃	—	2	4	no	—
112	C₂²×D₅₃	C₄⋊C₈	1	1	yes	characteristic
113	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
114	C₂²×D₅₃	C₄⋊C₈	1	1	yes	characteristic
115	C₂²×D₅₃	C₂×M₁₆	1	2	yes	—
116	C₂×C₄×C₅₃	—	2	4	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×Dic₅₃	—	2	4	no	—
125	C₂×Dic₅₃	—	2	2	no	—
126	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
127	C₂×Dic₅₃	C₄⋊C₈	1	1	yes	characteristic
128	C₂×Dic₅₃	C₄⋊C₈	1	1	yes	characteristic
129	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
130	C₄×D₅₃	—	2	16	no	—
131	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
132	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
133	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
134	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
135	C₄×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
136	C₄×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
137	C₄×D₅₃	C₂²×C₈	1	2	yes	—
138	C₄×D₅₃	C₂²×C₈	1	2	yes	—
139	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₈×C₅₃	C₂²×C₈	1	4	yes	—
142	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
143	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
144	#88322	C₂×C₄⋊C₄	1	2	yes	—
145	#88322	C₂×C₄⋊C₄	1	2	yes	—
146	#88322	C₂²×C₈	1	4	yes	—
147	C₂²×#364	—	53	1	no	maximal, Sylow
148	C₂²×C₅₃	—	2	2	no	—
149	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
150	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
151	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
152	C₂×D₅₃	—	2	2	no	—
153	C₂×D₅₃	—	2	4	no	—
154	C₂×D₅₃	—	2	4	no	—
155	C₂×D₅₃	#364	1	4	yes	direct product factor
156	C₂×D₅₃	C₂×C₄⋊C₈	1	2	yes	—
157	C₂×D₅₃	C₂×C₄⋊C₈	1	2	yes	—
158	C₄×C₅₃	—	2	4	no	—
159	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
160	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
161	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
162	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
163	Dic₅₃	—	2	4	no	—
164	Dic₅₃	C₂×C₄⋊C₈	1	2	yes	—
165	Dic₅₃	C₂×C₄⋊C₈	1	2	yes	—
166	C₂²×M₃₂	—	53	2	no	—
167	C₂²×C₄×C₈	—	53	1	no	—
168	C₂×#364	—	53	2	no	—
169	C₂×#364	—	53	1	no	—
170	C₂×#364	—	53	8	no	—
171	C₂×#364	D₅₃	1	1	yes	characteristic, direct product factor
172	C₂×C₅₃	—	2	1	no	—
173	C₂×C₅₃	—	2	1	no	—
174	C₂×C₅₃	C₂×#364	1	2	yes	complemented
175	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
176	D₅₃	—	2	2	no	—
177	D₅₃	C₂×#364	1	4	yes	direct product factor
178	C₂³×C₈	—	53	1	no	—
179	C₂³×C₈	—	53	1	no	—
180	C₂²×C₄²	—	53	1	no	—
181	C₂×M₃₂	—	53	2	no	—
182	C₂×M₃₂	—	53	8	no	—
183	C₂×M₃₂	—	106	2	no	—
184	C₂×M₃₂	—	106	4	no	—
185	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
186	C₂²×C₁₆	—	106	2	no	—
187	C₂×C₄×C₈	—	53	1	no	—
188	C₂×C₄×C₈	—	53	1	no	—
189	C₂×C₄×C₈	—	53	1	no	—
190	C₂×C₄×C₈	—	53	4	no	—
191	C₂×C₄×C₈	—	53	4	no	—
192	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
193	#364	—	53	8	no	—
194	#364	—	53	4	no	—
195	#364	C₂×D₅₃	1	4	yes	direct product factor
196	C₅₃	C₂²×#364	1	1	yes	characteristic, complemented, Sylow
197	C₂³×C₄	—	53	1	no	—
198	C₂³×C₄	—	106	1	no	—
199	C₂²×C₈	—	53	1	no	—
200	C₂²×C₈	—	53	1	no	—
201	C₂²×C₈	—	53	2	no	—
202	C₂²×C₈	—	53	4	no	—
203	C₂²×C₈	—	53	2	no	—
204	C₂²×C₈	—	53	1	no	—
205	C₂²×C₈	—	53	1	no	—
206	C₂²×C₈	—	53	2	no	—
207	C₂²×C₈	—	53	1	no	—
208	C₂²×C₈	—	53	1	no	—
209	C₂²×C₈	—	53	2	no	—
210	C₂²×C₈	—	53	2	no	—
211	C₂²×C₈	—	53	4	no	—
212	C₂²×C₈	—	53	2	no	—
213	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
215	C₂×C₄²	—	53	1	no	—
216	C₂×C₄²	—	53	4	no	—
217	C₂×C₄²	—	106	2	no	—
218	C₂×C₄²	—	106	1	no	—
219	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
220	M₃₂	—	2	2	no	—
221	M₃₂	—	53	4	no	—
222	M₃₂	—	106	8	no	—
223	M₃₂	—	106	2	no	—
224	M₃₂	C₂²×D₅₃	1	4	yes	—
225	C₂×C₁₆	—	2	2	no	—
226	C₂×C₁₆	—	106	8	no	—
227	C₂×C₁₆	—	106	2	no	—
228	C₄×C₈	—	53	2	no	—
229	C₄×C₈	—	53	4	no	—
230	C₄×C₈	—	53	2	no	—
231	C₄×C₈	—	53	4	no	—
232	C₄×C₈	C₄×D₅₃	1	2	yes	—
233	C₄×C₈	C₂²×D₅₃	1	2	yes	—
234	C₂⁴	—	53	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	53	1	no	—
237	C₂²×C₄	—	53	2	no	—
238	C₂²×C₄	—	53	1	no	—
239	C₂²×C₄	—	53	1	no	—
240	C₂²×C₄	—	53	1	no	—
241	C₂²×C₄	—	53	2	no	—
242	C₂²×C₄	—	53	1	no	—
243	C₂²×C₄	—	53	2	no	—
244	C₂²×C₄	—	53	2	no	—
245	C₂²×C₄	—	106	4	no	—
246	C₂²×C₄	—	106	8	no	—
247	C₂²×C₄	—	106	1	no	—
248	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
249	C₂×C₈	—	53	2	no	—
250	C₂×C₈	—	53	1	no	—
251	C₂×C₈	—	53	1	no	—
252	C₂×C₈	—	53	2	no	—
253	C₂×C₈	—	53	2	no	—
254	C₂×C₈	—	53	2	no	—
255	C₂×C₈	—	53	2	no	—
256	C₂×C₈	—	53	2	no	—
257	C₂×C₈	—	53	2	no	—
258	C₂×C₈	—	53	2	no	—
259	C₂×C₈	—	53	2	no	—
260	C₂×C₈	—	53	8	no	—
261	C₂×C₈	—	53	8	no	—
262	C₂×C₈	—	53	2	no	—
263	C₂×C₈	—	53	2	no	—
264	C₂×C₈	—	53	2	no	—
265	C₂×C₈	—	53	2	no	—
266	C₂×C₈	C₈×D₅₃	1	2	yes	—
267	C₂×C₈	C₈×D₅₃	1	2	yes	—
268	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
270	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
271	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
272	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
273	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
275	C₄²	—	2	1	no	—
276	C₄²	—	53	2	no	—
277	C₄²	—	106	4	no	—
278	C₄²	—	106	1	no	—
279	C₄²	C₂×C₄×D₅₃	1	2	yes	—
280	C₁₆	—	2	4	no	—
281	C₁₆	—	106	4	no	—
282	C₂³	—	53	1	no	—
283	C₂³	—	53	2	no	—
284	C₂³	—	53	2	no	—
285	C₂³	—	53	1	no	—
286	C₂³	—	106	4	no	—
287	C₂³	M₁₆×D₅₃	1	1	yes	characteristic
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	53	2	no	—
291	C₂×C₄	—	53	2	no	—
292	C₂×C₄	—	53	1	no	—
293	C₂×C₄	—	53	1	no	—
294	C₂×C₄	—	53	2	no	—
295	C₂×C₄	—	53	2	no	—
296	C₂×C₄	—	53	2	no	—
297	C₂×C₄	—	53	2	no	—
298	C₂×C₄	—	53	2	no	—
299	C₂×C₄	—	53	2	no	—
300	C₂×C₄	—	53	2	no	—
301	C₂×C₄	—	53	2	no	—
302	C₂×C₄	—	106	16	no	—
303	C₂×C₄	—	106	4	no	—
304	C₂×C₄	—	106	2	no	—
305	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
306	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
307	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
309	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
310	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
311	C₈	—	53	2	no	—
312	C₈	—	53	2	no	—
313	C₈	—	53	4	no	—
314	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
315	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
316	C₈	C₂×C₈×D₅₃	1	4	yes	—
317	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
318	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
319	C₂²	—	2	2	no	—
320	C₂²	—	53	2	no	—
321	C₂²	—	53	4	no	—
322	C₂²	—	53	2	no	—
323	C₂²	—	106	4	no	—
324	C₂²	—	106	4	no	—
325	C₂²	—	106	2	no	—
326	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
327	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
328	C₂²	C₂×M₁₆×D₅₃	1	1	yes	characteristic
329	C₄	—	2	4	no	—
330	C₄	—	53	2	no	—
331	C₄	—	53	2	no	—
332	C₄	—	106	4	no	—
333	C₄	C₄⋊C₈×D₅₃	1	1	yes	characteristic
334	C₄	C₄⋊C₈×D₅₃	1	1	yes	characteristic
335	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
336	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
337	C₂	—	2	1	no	—
338	C₂	—	2	1	no	—
339	C₂	—	53	4	no	—
340	C₂	—	106	2	no	—
341	C₂	#364×D₅₃	1	2	yes	direct product factor
342	C₂	C₂×C₄⋊C₈×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	C₂²×#364
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
3. C₂×#16810444	C₂	semidirect
4. C₂×#16810443	C₂	semidirect
5. #364×D₅₃	C₂	direct (N × H)
6. C₂×C₅₃×#364	C₂	semidirect
20. #16810444	C₂²	semidirect
21. #16810443	C₂²	semidirect
24. C₅₃×#364	C₂²	semidirect
155. C₂×D₅₃	#364	direct (N × H)
171. C₂×#364	D₅₃	direct (N × H)
174. C₂×C₅₃	C₂×#364	semidirect
177. D₅₃	C₂×#364	direct (N × H)
195. #364	C₂×D₅₃	direct (N × H)
196. C₅₃	C₂²×#364	semidirect
341. C₂	#364×D₅₃	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×C₄×C₈×D₅₃	2
C₂×#16810444	2
C₂×#16810443	2
#364×D₅₃	2
C₂×C₅₃×#364	2
C₂²×#364	53

Subgroups — C2×#364×D53 #27537086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#364×D₅₃ #27537086