Subgroups — C₂×#20276539 #27537247

Order 13568.

754 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133156	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
4	#20276539	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810402	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810402	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16810399	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₂₁₂	C₄	1	1	yes	characteristic
12	C₂×#13133156	—	2	4	no	—
13	C₂×#13133156	C₂²	1	4	yes	complemented
14	C₂²×#11719908	C₂²	1	1	yes	characteristic
15	C₂²×#11719908	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₅₃	—	2	4	no	—
17	C₂×C₁₆×D₅₃	C₂²	1	4	yes	complemented
18	C₈×D₂₁₂	C₄	1	4	yes	—
19	C₈×D₂₁₂	C₂²	1	4	yes	—
20	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
21	C₂×#13133129	C₄	1	1	yes	characteristic
22	C₂×#13133129	C₄	1	1	yes	characteristic
23	#16810402	C₂²	1	4	yes	complemented
24	#16810402	C₂²	1	4	yes	complemented
25	#16810399	C₂²	1	4	yes	complemented
26	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
30	C₂²×D₂₁₂	C₈	1	1	yes	characteristic
31	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
34	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₅₃	—	2	2	no	—
36	C₂×C₈×D₅₃	—	2	2	no	—
37	C₂×C₈×D₅₃	—	2	2	no	—
38	C₂×C₈×D₅₃	—	2	2	no	—
39	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₅₃	C₂³	1	2	yes	—
42	C₂×C₈×D₅₃	C₂³	1	2	yes	—
43	C₄×D₂₁₂	C₈	1	4	yes	—
44	C₄×D₂₁₂	C₂×C₄	1	4	yes	—
45	C₂×C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
46	C₂×#11719921	C₈	1	1	yes	characteristic
47	C₂×#11719921	C₈	1	1	yes	characteristic
48	#13133156	—	2	8	no	—
49	C₂×#11719908	—	2	2	no	—
50	C₂×#11719908	D₄	1	4	yes	complemented
51	C₂×#11719908	C₂³	1	2	yes	—
52	C₂×#11719908	C₂³	1	2	yes	—
53	C₁₆×D₅₃	—	2	8	no	—
54	C₄⋊#88322	C₂×C₄	1	4	yes	—
55	#13133129	C₂×C₄	1	4	yes	—
56	#13133129	C₂×C₄	1	4	yes	—
57	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₁₆×C₅₃	—	2	2	no	—
61	C₂×C₁₆×C₅₃	—	2	2	no	—
62	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
63	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
64	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₅₃	C₂³	1	2	yes	—
66	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
67	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
73	C₂×D₂₁₂	—	2	4	no	—
74	C₂×D₂₁₂	C₂×C₈	1	4	yes	—
75	C₂×C₄×D₅₃	—	2	2	no	—
76	C₂×C₄×D₅₃	—	2	2	no	—
77	C₂×C₄×D₅₃	—	2	2	no	—
78	C₂×C₄×D₅₃	—	2	2	no	—
79	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
80	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
81	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
82	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₅₃	#41	1	2	yes	—
88	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
89	C₄²×C₅₃	C₂×C₈	1	2	yes	—
90	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
91	#11719921	C₂×C₈	1	4	yes	—
92	#11719921	C₂×C₈	1	4	yes	—
93	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
94	C₂×#88322	—	2	2	no	—
95	C₂×#88322	—	2	2	no	—
96	C₂×#88322	C₂²×C₄	1	2	yes	—
97	C₂×#88322	C₂²×C₄	1	2	yes	—
98	C₈×D₅₃	—	2	4	no	—
99	C₈×D₅₃	—	2	4	no	—
100	C₈×D₅₃	—	2	2	no	—
101	C₈×D₅₃	—	2	2	no	—
102	C₈×D₅₃	—	2	2	no	—
103	C₈×D₅₃	—	2	2	no	—
104	C₁₆×C₅₃	—	2	2	no	—
105	C₁₆×C₅₃	—	2	2	no	—
106	#11719908	—	2	2	no	—
107	#11719908	C₂×D₄	1	4	yes	complemented
108	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
109	C₂²×D₅₃	—	2	2	no	—
110	C₂²×D₅₃	—	2	2	no	—
111	C₂²×D₅₃	—	2	2	no	—
112	C₂²×D₅₃	—	2	2	no	—
113	C₂²×D₅₃	—	2	2	no	—
114	C₂²×D₅₃	M₃₂	1	4	yes	complemented
115	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
116	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
121	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	#131	1	2	yes	—
124	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×Dic₅₃	—	2	2	no	—
126	C₂×Dic₅₃	—	2	2	no	—
127	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
128	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
129	D₂₁₂	—	2	8	no	—
130	C₄×D₅₃	—	2	2	no	—
131	C₄×D₅₃	—	2	2	no	—
132	C₄×D₅₃	—	2	2	no	—
133	C₄×D₅₃	—	2	2	no	—
134	C₄×D₅₃	—	2	4	no	—
135	C₄×D₅₃	—	2	4	no	—
136	C₈×C₅₃	—	2	2	no	—
137	C₈×C₅₃	C₄×D₄	1	2	yes	—
138	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
140	#88322	—	2	2	no	—
141	#88322	—	2	2	no	—
142	C₂×#2168	—	53	1	no	maximal, Sylow
143	C₂²×C₅₃	C₈×D₄	1	2	yes	—
144	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
145	C₂²×C₅₃	#504	1	2	yes	—
146	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
147	C₂×D₅₃	—	2	2	no	—
148	C₂×D₅₃	—	2	2	no	—
149	C₂×D₅₃	—	2	8	no	—
150	C₂×D₅₃	—	2	2	no	—
151	C₂×D₅₃	—	2	2	no	—
152	C₂×D₅₃	—	2	2	no	—
153	C₂×D₅₃	—	4	4	no	—
154	C₂×D₅₃	C₂×M₃₂	1	4	yes	complemented
155	C₄×C₅₃	—	2	2	no	—
156	C₄×C₅₃	—	2	2	no	—
157	C₄×C₅₃	C₈×D₄	1	2	yes	—
158	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
159	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
160	Dic₅₃	—	2	2	no	—
161	Dic₅₃	—	2	2	no	—
162	C₂²×M₃₂	—	53	1	no	—
163	C₂³×C₁₆	—	53	1	no	—
164	C₂×C₈×D₄	—	53	1	no	—
165	#2168	—	53	8	no	—
166	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
167	C₂×C₂²⋊C₁₆	—	53	1	no	—
168	C₂×C₂²⋊C₁₆	—	53	1	no	—
169	C₂×#346	—	53	1	no	—
170	C₂×C₅₃	#2168	1	4	yes	complemented
171	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
172	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
173	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
174	D₅₃	—	2	4	no	—
175	D₅₃	—	4	2	no	—
176	C₂³×C₈	—	53	1	no	—
177	C₂³×C₈	—	53	1	no	—
178	C₂×C₄×D₄	—	53	1	no	—
179	C₂×M₃₂	—	53	4	no	—
180	C₂×M₃₂	—	106	4	no	—
181	C₂²×C₁₆	—	53	1	no	—
182	C₂²×C₁₆	—	53	1	no	—
183	C₂²×C₁₆	—	53	4	no	—
184	C₂²×C₁₆	—	106	4	no	—
185	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
186	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
187	C₈×D₄	—	53	4	no	—
188	C₈×D₄	—	53	4	no	—
189	C₂×C₄⋊C₈	—	53	1	no	—
190	C₂×C₂²⋊C₈	—	53	1	no	—
191	C₂×C₂²⋊C₈	—	53	1	no	—
192	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
193	C₄⋊C₁₆	C₂×D₅₃	1	4	yes	complemented
194	C₂²⋊C₁₆	—	53	4	no	—
195	C₂²⋊C₁₆	—	53	4	no	—
196	#346	—	53	4	no	—
197	C₅₃	C₂×#2168	1	1	yes	characteristic, complemented, Sylow
198	C₂²×D₄	—	53	1	no	—
199	C₂³×C₄	—	53	1	no	—
200	C₂³×C₄	—	53	1	no	—
201	C₂²×C₈	—	53	1	no	—
202	C₂²×C₈	—	53	1	no	—
203	C₂²×C₈	—	53	2	no	—
204	C₂²×C₈	—	53	2	no	—
205	C₂²×C₈	—	53	2	no	—
206	C₂²×C₈	—	53	2	no	—
207	C₂²×C₈	—	106	2	no	—
208	C₂²×C₈	—	106	2	no	—
209	C₂²×C₈	—	106	2	no	—
210	C₂²×C₈	—	106	2	no	—
211	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
213	C₄×D₄	—	53	4	no	—
214	C₄×D₄	—	53	4	no	—
215	C₂×C₄⋊C₄	—	53	1	no	—
216	C₂×C₂²⋊C₄	—	53	1	no	—
217	C₂×C₂²⋊C₄	—	53	1	no	—
218	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
219	M₃₂	—	106	8	no	—
220	C₂×C₁₆	—	2	2	no	—
221	C₂×C₁₆	—	2	2	no	—
222	C₂×C₁₆	—	53	4	no	—
223	C₂×C₁₆	—	53	2	no	—
224	C₂×C₁₆	—	53	2	no	—
225	C₂×C₁₆	—	106	8	no	—
226	C₂×C₁₆	—	106	2	no	—
227	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
228	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
229	C₄⋊C₈	—	53	4	no	—
230	C₂²⋊C₈	—	53	4	no	—
231	C₂²⋊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	C₂⁴	—	53	1	no	—
236	C₂×D₄	—	53	4	no	—
237	C₂×D₄	—	106	4	no	—
238	C₂²×C₄	—	53	1	no	—
239	C₂²×C₄	—	53	1	no	—
240	C₂²×C₄	—	53	2	no	—
241	C₂²×C₄	—	53	2	no	—
242	C₂²×C₄	—	53	2	no	—
243	C₂²×C₄	—	53	2	no	—
244	C₂²×C₄	—	106	2	no	—
245	C₂²×C₄	—	106	2	no	—
246	C₂²×C₄	—	106	2	no	—
247	C₂²×C₄	—	106	2	no	—
248	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
249	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	53	2	no	—
253	C₂×C₈	—	53	2	no	—
254	C₂×C₈	—	106	2	no	—
255	C₂×C₈	—	106	2	no	—
256	C₂×C₈	—	106	4	no	—
257	C₂×C₈	—	106	4	no	—
258	C₂×C₈	—	106	2	no	—
259	C₂×C₈	—	106	2	no	—
260	C₂×C₈	—	106	2	no	—
261	C₂×C₈	—	106	2	no	—
262	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
263	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
264	C₂×C₈	D₄×D₅₃	1	2	yes	—
265	C₂×C₈	#11719949	1	2	yes	—
266	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
267	C₄⋊C₄	—	53	4	no	—
268	C₂²⋊C₄	—	53	4	no	—
269	C₂²⋊C₄	—	53	4	no	—
270	C₄²	C₈×D₅₃	1	2	yes	—
271	C₄²	C₂×C₄×D₅₃	1	2	yes	—
272	C₁₆	—	2	2	no	—
273	C₁₆	—	2	2	no	—
274	C₁₆	—	53	4	no	—
275	C₁₆	—	106	2	no	—
276	C₂³	—	53	4	no	—
277	C₂³	—	53	2	no	—
278	C₂³	—	53	2	no	—
279	C₂³	—	106	2	no	—
280	C₂³	—	106	2	no	—
281	C₂³	—	106	2	no	—
282	C₂³	—	106	2	no	—
283	C₂³	—	106	2	no	—
284	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
285	D₄	—	106	8	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	53	2	no	—
289	C₂×C₄	—	53	2	no	—
290	C₂×C₄	—	106	2	no	—
291	C₂×C₄	—	106	2	no	—
292	C₂×C₄	—	106	2	no	—
293	C₂×C₄	—	106	2	no	—
294	C₂×C₄	—	106	2	no	—
295	C₂×C₄	—	106	2	no	—
296	C₂×C₄	—	106	4	no	—
297	C₂×C₄	—	106	4	no	—
298	C₂×C₄	#13133179	1	2	yes	—
299	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
300	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
301	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
302	C₂×C₄	#13133225	1	2	yes	—
303	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	106	2	no	—
306	C₈	—	106	2	no	—
307	C₈	#13133179	1	2	yes	—
308	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
309	C₈	C₂×#11719949	1	1	yes	characteristic
310	C₂²	—	53	4	no	—
311	C₂²	—	106	8	no	—
312	C₂²	—	106	2	no	—
313	C₂²	—	106	2	no	—
314	C₂²	—	106	2	no	—
315	C₂²	—	106	2	no	—
316	C₂²	—	106	2	no	—
317	C₂²	—	212	4	no	—
318	C₂²	#16810688	1	2	yes	—
319	C₂²	M₃₂×D₅₃	1	2	yes	—
320	C₂²	#16811039	1	2	yes	—
321	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	106	2	no	—
325	C₄	—	106	2	no	—
326	C₄	#16810688	1	2	yes	—
327	C₄	C₂×#13133179	1	1	yes	characteristic
328	C₄	C₂×#13133225	1	1	yes	characteristic
329	C₂	—	106	4	no	—
330	C₂	—	212	2	no	—
331	C₂	#20276539	1	4	yes	direct product factor
332	C₂	C₂×#16810688	1	1	yes	characteristic
333	C₂	C₂×M₃₂×D₅₃	1	1	yes	characteristic
334	C₂	C₂×#16811039	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂²×D₂₁₂	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	C₂×#2168
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133156	C₂	semidirect
2. C₂²×C₁₆×D₅₃	C₂	semidirect
4. #20276539	C₂	direct (N × H)
5. C₂×#16810402	C₂	semidirect
6. C₂×#16810402	C₂	semidirect
7. C₂×#16810399	C₂	semidirect
8. C₂×C₅₃×C₄⋊C₁₆	C₂	semidirect
13. C₂×#13133156	C₂²	semidirect
17. C₂×C₁₆×D₅₃	C₂²	semidirect
23. #16810402	C₂²	semidirect
24. #16810402	C₂²	semidirect
25. #16810399	C₂²	semidirect
29. C₅₃×C₄⋊C₁₆	C₂²	semidirect
50. C₂×#11719908	D₄	semidirect
107. #11719908	C₂×D₄	semidirect
114. C₂²×D₅₃	M₃₂	semidirect
154. C₂×D₅₃	C₂×M₃₂	semidirect
166. C₂×C₄⋊C₁₆	D₅₃	semidirect
170. C₂×C₅₃	#2168	semidirect
193. C₄⋊C₁₆	C₂×D₅₃	semidirect
197. C₅₃	C₂×#2168	semidirect
331. C₂	#20276539	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×#13133156	2
C₂²×C₁₆×D₅₃	2
C₂×C₈×D₂₁₂	2
#20276539	2
C₂×#16810402	2
C₂×#16810399	2
C₂×C₅₃×C₄⋊C₁₆	2
C₂×#2168	53

Subgroups — C2×#20276539 #27537247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276539 #27537247