Subgroups — C₂×#20276612 #27537532

Order 13568.

710 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811061	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810744	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810734	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276612	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810435	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810432	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16810400	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133239	C₂²	1	1	yes	characteristic
10	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic
11	C₂×#13133225	—	2	1	no	—
12	C₂×#13133179	—	2	1	no	—
13	#16811061	C₄	1	4	yes	complemented
14	#16811061	C₂²	1	4	yes	—
15	C₂×#13133162	C₄	1	2	yes	complemented
16	C₂×D₈₄₈	C₄	1	2	yes	complemented
17	C₂×#13133157	C₂²	1	1	yes	characteristic
18	#16810744	—	2	2	no	—
19	#16810744	C₂²	1	4	yes	complemented
20	#16810734	—	2	2	no	—
21	#16810734	C₂²	1	4	yes	complemented
22	C₂×#13133143	—	2	1	no	—
23	C₂×#13133137	—	2	1	no	—
24	C₂×#13133127	C₂²	1	1	yes	characteristic
25	#16810435	C₂²	1	4	yes	complemented
26	#16810432	C₂²	1	4	yes	complemented
27	#16810400	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	C₂×#11719945	—	2	1	no	—
33	C₂²×D₂₁₂	—	2	1	no	—
34	C₂²×C₄×D₅₃	—	4	1	no	—
35	#13133239	—	2	1	no	—
36	#13133239	—	2	1	no	—
37	#13133239	C₂×C₄	1	2	yes	complemented
38	#13133239	C₂³	1	2	yes	—
39	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic, complemented
40	C₂×#11719913	—	2	1	no	—
41	C₂×D₄₂₄	—	2	1	no	—
42	C₂×D₄₂₄	—	2	1	no	—
43	C₂×D₄₂₄	—	2	1	no	—
44	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
45	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
46	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic, complemented
47	C₂×D₄₂₄	C₂³	1	2	yes	—
48	#13133225	—	2	4	no	—
49	#13133225	—	4	2	no	—
50	C₂×#11719912	—	4	1	no	—
51	C₂²×#88322	D₄	1	1	yes	characteristic
52	C₂×C₈×D₅₃	—	4	1	no	—
53	#13133179	—	2	4	no	—
54	#13133179	—	4	2	no	—
55	C₂×#11719921	—	4	1	no	—
56	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
57	#13133162	C₂×C₄	1	8	yes	complemented
58	D₈₄₈	C₂×C₄	1	8	yes	complemented
59	#13133157	—	2	1	no	—
60	#13133157	C₂³	1	2	yes	—
61	C₂×#11719908	—	2	1	no	—
62	#13133143	—	2	4	no	—
63	#13133137	—	2	4	no	—
64	#13133127	D₄	1	2	yes	—
65	#13133127	C₂³	1	2	yes	—
66	C₂×M₁₆×C₅₃	—	2	1	no	—
67	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
69	M₃₂×C₅₃	C₂×C₄	1	2	yes	complemented
70	M₃₂×C₅₃	C₂³	1	2	yes	—
71	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
72	#108×C₅₃	—	2	1	no	—
73	#108×C₅₃	C₂³	1	2	yes	—
74	#106×C₅₃	—	2	1	no	—
75	#106×C₅₃	C₂³	1	2	yes	—
76	C₂³×D₅₃	—	4	1	no	—
77	C₂²×C₄×C₅₃	—	4	1	no	—
78	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₂²⋊D₅₃	—	4	1	no	—
80	C₂²×Dic₅₃	—	2	1	no	—
81	#11719945	—	2	2	no	—
82	#11719945	—	2	2	no	—
83	#11719945	—	4	1	no	—
84	#11719945	—	4	1	no	—
85	C₂×D₂₁₂	—	2	2	no	—
86	C₂×D₂₁₂	—	2	2	no	—
87	C₂×D₂₁₂	—	2	1	no	—
88	C₂×D₂₁₂	—	2	2	no	—
89	C₂×D₂₁₂	—	4	1	no	—
90	C₂×D₂₁₂	—	4	1	no	—
91	C₂×D₂₁₂	—	4	1	no	—
92	C₂×C₄×D₅₃	—	4	1	no	—
93	C₂×C₄×D₅₃	—	4	4	no	—
94	C₂×C₄×D₅₃	—	8	4	no	—
95	C₂×Dic₁₀₆	—	2	1	no	—
96	M₁₆×C₅₃	—	2	2	no	—
97	M₁₆×C₅₃	—	4	1	no	—
98	C₂×C₈×C₅₃	—	4	1	no	—
99	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
103	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
104	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
105	C₄⋊C₄×C₅₃	—	2	2	no	—
106	C₄⋊C₄×C₅₃	—	4	1	no	—
107	#11719921	—	4	4	no	—
108	C₄×Dic₅₃	—	2	1	no	—
109	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
110	C₂×#88322	—	2	1	no	—
111	C₂×#88322	—	2	1	no	—
112	C₂×#88322	C₂×D₄	1	2	yes	—
113	Dic₂₁₂	—	2	1	no	—
114	Dic₂₁₂	C₂²×C₄	1	2	yes	complemented
115	D₄₂₄	—	2	2	no	—
116	D₄₂₄	—	2	2	no	—
117	D₄₂₄	—	2	1	no	—
118	D₄₂₄	—	2	2	no	—
119	D₄₂₄	C₂²×C₄	1	4	yes	—
120	D₄₂₄	C₂²×C₄	1	2	yes	complemented
121	#11719913	—	2	2	no	—
122	#11719913	—	2	2	no	—
123	#11719912	—	4	4	no	—
124	C₈×D₅₃	—	4	4	no	—
125	C₁₆×C₅₃	C₂²×C₄	1	4	yes	complemented
126	#11719908	—	2	2	no	—
127	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂²×D₅₃	—	4	2	no	—
129	C₂²×D₅₃	—	4	1	no	—
130	C₂²×D₅₃	—	4	2	no	—
131	C₂²×D₅₃	—	4	2	no	—
132	C₂²×D₅₃	—	8	2	no	—
133	C₂²×D₅₃	—	8	2	no	—
134	C₂×C₄×C₅₃	—	4	2	no	—
135	C₂×C₄×C₅₃	—	8	2	no	—
136	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
142	C₂²⋊D₅₃	—	4	2	no	—
143	C₂²⋊D₅₃	—	4	2	no	—
144	C₂×Dic₅₃	—	2	2	no	—
145	C₂×Dic₅₃	—	4	2	no	—
146	C₂×Dic₅₃	—	4	1	no	—
147	D₂₁₂	—	2	4	no	—
148	D₂₁₂	—	2	2	no	—
149	D₂₁₂	—	4	2	no	—
150	D₂₁₂	—	4	2	no	—
151	D₂₁₂	—	4	2	no	—
152	D₂₁₂	—	4	1	no	—
153	C₄×D₅₃	—	4	2	no	—
154	C₄×D₅₃	—	4	2	no	—
155	C₄×D₅₃	—	8	8	no	—
156	Dic₁₀₆	—	2	2	no	—
157	Dic₁₀₆	—	4	1	no	—
158	C₈×C₅₃	—	4	2	no	—
159	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
160	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
161	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
162	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
163	#88322	—	2	2	no	—
164	C₂×AGL(1,ℤ/16)	—	53	1	no	maximal, Sylow
165	C₂²×C₅₃	—	2	2	no	—
166	C₂²×C₅₃	#442	1	1	yes	characteristic
167	C₂²×C₅₃	#442	1	1	yes	characteristic
168	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂×D₅₃	—	4	2	no	—
170	C₂×D₅₃	—	4	4	no	—
171	C₂×D₅₃	—	8	2	no	—
172	C₂×D₅₃	—	8	2	no	—
173	C₂×D₅₃	—	8	4	no	—
174	C₂×D₅₃	—	8	2	no	—
175	C₂×D₅₃	—	8	4	no	—
176	C₄×C₅₃	—	8	2	no	—
177	C₄×C₅₃	#442	1	1	yes	characteristic
178	C₄×C₅₃	#442	1	1	yes	characteristic
179	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
180	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
181	Dic₅₃	—	4	2	no	—
182	Dic₅₃	—	4	2	no	—
183	C₂×#509	—	53	1	no	—
184	C₂×#444	—	53	1	no	—
185	C₂×#442	—	53	1	no	—
186	AGL(1,ℤ/16)	—	53	8	no	—
187	C₂×#365	D₅₃	1	1	yes	characteristic, complemented
188	C₂×#361	—	53	1	no	—
189	C₂×#360	—	53	1	no	—
190	C₂×#347	—	53	1	no	—
191	C₂×C₅₃	—	2	1	no	—
192	C₂×C₅₃	—	2	1	no	—
193	C₂×C₅₃	AGL(1,ℤ/16)	1	2	yes	complemented
194	C₂×C₅₃	C₂×#442	1	1	yes	characteristic
195	D₅₃	—	8	2	no	—
196	D₅₃	—	8	4	no	—
197	C₂×#135	—	53	1	no	—
198	C₂²×D₈	—	53	1	no	—
199	C₂×#131	—	106	1	no	—
200	C₂×C₄×D₄	—	106	1	no	—
201	#509	—	53	4	no	—
202	#509	—	53	4	no	—
203	C₂×SD₃₂	—	53	2	no	—
204	C₂×D₁₆	—	53	2	no	—
205	C₂×M₃₂	—	53	1	no	—
206	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
207	#444	—	53	4	no	—
208	#444	—	106	2	no	—
209	#442	—	53	4	no	—
210	#442	—	106	2	no	—
211	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
212	C₂×#106	C₂×D₅₃	1	1	yes	characteristic
213	C₂×#104	—	106	1	no	—
214	C₂×#102	—	106	1	no	—
215	C₂×#97	—	53	1	no	—
216	#365	C₂×D₅₃	1	4	yes	complemented
217	#361	—	53	4	no	—
218	#360	—	53	4	no	—
219	#347	—	53	4	no	—
220	C₅₃	C₂×AGL(1,ℤ/16)	1	1	yes	characteristic, complemented, Sylow
221	C₂×#41	—	106	1	no	—
222	C₂²×D₄	—	106	1	no	—
223	C₂³×C₄	—	212	1	no	—
224	#135	—	53	2	no	—
225	#135	—	53	2	no	—
226	#135	—	106	1	no	—
227	#135	—	106	1	no	—
228	C₂×Q₁₆	—	53	1	no	—
229	C₂×SD₁₆	—	106	1	no	—
230	C₂×D₈	—	53	2	no	—
231	C₂×D₈	—	53	2	no	—
232	C₂×D₈	—	53	1	no	—
233	C₂×D₈	—	53	2	no	—
234	C₂×D₈	—	106	1	no	—
235	C₂×D₈	—	106	1	no	—
236	C₂×D₈	—	106	1	no	—
237	#131	—	106	4	no	—
238	#131	—	212	2	no	—
239	C₂×M₁₆	—	2	1	no	—
240	C₂×M₁₆	—	212	1	no	—
241	C₂²×C₈	—	53	1	no	—
242	C₂²×C₈	—	212	1	no	—
243	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
244	C₄×D₄	—	106	4	no	—
245	C₄×D₄	—	212	2	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₂²⋊C₄	—	212	1	no	—
248	C₂×C₄²	—	53	1	no	—
249	SD₃₂	—	53	8	no	—
250	D₁₆	—	53	8	no	—
251	M₃₂	—	53	2	no	—
252	M₃₂	—	106	1	no	—
253	M₃₂	C₄×D₅₃	1	2	yes	complemented
254	M₃₂	C₂²×D₅₃	1	2	yes	—
255	C₂×C₁₆	—	106	1	no	—
256	C₂×C₁₆	C₄×D₅₃	1	2	yes	complemented
257	#108	—	2	1	no	—
258	#108	C₂²×D₅₃	1	2	yes	—
259	#106	—	2	1	no	—
260	#106	C₂²×D₅₃	1	2	yes	—
261	#104	—	106	4	no	—
262	#102	—	106	4	no	—
263	#97	—	53	2	no	—
264	#97	—	53	2	no	—
265	C₂⁴	—	212	1	no	—
266	#41	—	106	2	no	—
267	#41	—	106	2	no	—
268	#41	—	212	1	no	—
269	#41	—	212	1	no	—
270	C₂×Q₈	—	106	1	no	—
271	C₂×D₄	—	106	2	no	—
272	C₂×D₄	—	106	2	no	—
273	C₂×D₄	—	106	1	no	—
274	C₂×D₄	—	106	2	no	—
275	C₂×D₄	—	212	1	no	—
276	C₂×D₄	—	212	1	no	—
277	C₂×D₄	—	212	1	no	—
278	C₂×D₄	—	212	1	no	—
279	C₂²×C₄	—	4	1	no	—
280	C₂²×C₄	—	106	1	no	—
281	C₂²×C₄	—	424	4	no	—
282	C₂²×C₄	—	212	4	no	—
283	C₂²×C₄	—	212	1	no	—
284	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
285	Q₁₆	—	53	2	no	—
286	Q₁₆	—	106	1	no	—
287	SD₁₆	—	106	2	no	—
288	SD₁₆	—	106	2	no	—
289	D₈	—	53	2	no	—
290	D₈	—	53	4	no	—
291	D₈	—	106	2	no	—
292	D₈	—	106	2	no	—
293	D₈	—	106	1	no	—
294	D₈	—	106	2	no	—
295	M₁₆	—	2	2	no	—
296	M₁₆	—	4	1	no	—
297	M₁₆	—	212	4	no	—
298	C₂×C₈	—	4	1	no	—
299	C₂×C₈	—	53	2	no	—
300	C₂×C₈	—	106	1	no	—
301	C₂×C₈	—	106	1	no	—
302	C₂×C₈	—	212	4	no	—
303	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
304	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
305	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
306	C₂×C₈	#11719949	1	1	yes	characteristic
307	C₂×C₈	#11719949	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₂²⋊C₄	—	212	4	no	—
312	C₄²	—	53	2	no	—
313	C₄²	—	106	1	no	—
314	C₁₆	—	106	2	no	—
315	C₁₆	C₂×C₄×D₅₃	1	4	yes	complemented
316	C₂³	—	424	2	no	—
317	C₂³	—	424	2	no	—
318	C₂³	—	212	2	no	—
319	C₂³	—	212	2	no	—
320	C₂³	—	212	1	no	—
321	C₂³	—	212	2	no	—
322	C₂³	#13133244	1	1	yes	characteristic
323	Q₈	—	106	2	no	—
324	Q₈	—	212	1	no	—
325	D₄	—	106	4	no	—
326	D₄	—	106	2	no	—
327	D₄	—	212	2	no	—
328	D₄	—	212	2	no	—
329	D₄	—	212	2	no	—
330	D₄	—	212	2	no	—
331	D₄	—	212	2	no	—
332	D₄	—	212	1	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	8	2	no	—
335	C₂×C₄	—	106	2	no	—
336	C₂×C₄	—	424	8	no	—
337	C₂×C₄	—	212	2	no	—
338	C₂×C₄	—	212	1	no	—
339	C₂×C₄	—	212	2	no	—
340	C₂×C₄	—	212	2	no	—
341	C₂×C₄	#13133179	1	1	yes	characteristic
342	C₂×C₄	#13133179	1	1	yes	characteristic
343	C₂×C₄	#13133244	1	1	yes	characteristic
344	C₂×C₄	#13133244	1	1	yes	characteristic
345	C₂×C₄	#13133244	1	1	yes	characteristic
346	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
347	C₈	—	4	2	no	—
348	C₈	—	106	2	no	—
349	C₈	#13133179	1	1	yes	characteristic
350	C₈	#13133179	1	1	yes	characteristic
351	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
352	C₈	C₂×#11719949	1	1	yes	characteristic
353	C₂²	—	2	2	no	—
354	C₂²	—	424	2	no	—
355	C₂²	—	424	2	no	—
356	C₂²	—	424	4	no	—
357	C₂²	—	424	4	no	—
358	C₂²	—	424	2	no	—
359	C₂²	—	212	2	no	—
360	C₂²	—	212	4	no	—
361	C₂²	#16810734	1	1	yes	characteristic
362	C₂²	#16810734	1	1	yes	characteristic
363	C₂²	C₂×#13133244	1	1	yes	characteristic
364	C₄	—	8	2	no	—
365	C₄	—	212	2	no	—
366	C₄	—	212	2	no	—
367	C₄	#16810734	1	1	yes	characteristic
368	C₄	#16810734	1	1	yes	characteristic
369	C₄	C₂×#13133179	1	1	yes	characteristic
370	C₄	C₂×#13133244	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	424	4	no	—
374	C₂	—	424	2	no	—
375	C₂	#20276612	1	2	yes	direct product factor
376	C₂	C₂×#16810734	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810734
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#365	C₂
socle	C₂²×C₅₃	#442
prime radical	C₂×#16811061	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#365	D₅₃
hypocenter	C₅₃	C₂×AGL(1,ℤ/16)
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
2. C₂×#16810744	C₂	semidirect
3. C₂×#16810734	C₂	semidirect
4. #20276612	C₂	direct (N × H)
5. C₂×#16810435	C₂	semidirect
6. C₂×#16810432	C₂	semidirect
7. C₂×#16810400	C₂	semidirect
8. C₂×C₅₃×#365	C₂	semidirect
13. #16811061	C₄	semidirect
15. C₂×#13133162	C₄	semidirect
16. C₂×D₈₄₈	C₄	semidirect
19. #16810744	C₂²	semidirect
21. #16810734	C₂²	semidirect
25. #16810435	C₂²	semidirect
26. #16810432	C₂²	semidirect
27. #16810400	C₂²	semidirect
31. C₅₃×#365	C₂²	semidirect
37. #13133239	C₂×C₄	semidirect
39. C₂×Dic₂₁₂	C₂×C₄	semidirect
46. C₂×D₄₂₄	C₂×C₄	semidirect
57. #13133162	C₂×C₄	semidirect
58. D₈₄₈	C₂×C₄	semidirect
69. M₃₂×C₅₃	C₂×C₄	semidirect
71. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
114. Dic₂₁₂	C₂²×C₄	semidirect
120. D₄₂₄	C₂²×C₄	semidirect
125. C₁₆×C₅₃	C₂²×C₄	semidirect
187. C₂×#365	D₅₃	semidirect
193. C₂×C₅₃	AGL(1,ℤ/16)	semidirect
216. #365	C₂×D₅₃	semidirect
220. C₅₃	C₂×AGL(1,ℤ/16)	semidirect
253. M₃₂	C₄×D₅₃	semidirect
256. C₂×C₁₆	C₄×D₅₃	semidirect
315. C₁₆	C₂×C₄×D₅₃	semidirect
375. C₂	#20276612	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×#16811061	2
C₂×#16810744	2
C₂×#16810734	2
#20276612	2
C₂×#16810435	2
C₂×#16810432	2
C₂×#16810400	2
C₂×C₅₃×#365	2
C₂×AGL(1,ℤ/16)	53

Subgroups — C2×#20276612 #27537532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276612 #27537532