Subgroups — C₂×#20276931 #27538978

Order 13568.

604 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810877	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810670	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276931	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810445	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810437	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810435	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16810404	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic
10	C₂²×#11719912	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×D₅₃	—	2	1	no	—
12	C₂×#13133192	—	2	1	no	—
13	C₂×#13133157	C₂²	1	1	yes	characteristic
14	#16810877	—	2	2	no	—
15	#16810877	C₂²	1	4	yes	—
16	#16810670	—	2	2	no	—
17	#16810670	C₂²	1	4	yes	complemented
18	C₂×#13133153	C₂²	1	1	yes	characteristic
19	C₂×#13133152	—	2	1	no	—
20	C₂×#13133137	—	2	1	no	—
21	#16810445	C₂²	1	4	yes	complemented
22	#16810437	C₂²	1	4	yes	complemented
23	#16810435	C₂²	1	4	yes	complemented
24	#16810404	C₂²	1	4	yes	complemented
25	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#365	C₂²	1	4	yes	complemented
29	C₂²×D₂₁₂	—	2	1	no	—
30	C₂²×D₂₁₂	—	4	1	no	—
31	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
32	C₂×D₄₂₄	—	2	1	no	—
33	C₂×D₄₂₄	—	2	1	no	—
34	C₂×D₄₂₄	—	2	1	no	—
35	C₂×D₄₂₄	—	2	1	no	—
36	C₂×D₄₂₄	—	2	1	no	—
37	C₂×D₄₂₄	C₂³	1	2	yes	—
38	C₂×#11719917	—	2	1	no	—
39	C₂×#11719912	—	2	1	no	—
40	C₂×#11719912	—	2	1	no	—
41	C₂×#11719912	—	2	1	no	—
42	C₂×#11719912	—	2	1	no	—
43	C₂×#11719912	—	4	1	no	—
44	C₂×#11719912	D₄	1	2	yes	—
45	C₂×#11719912	C₂³	1	2	yes	—
46	M₁₆×D₅₃	—	2	4	no	—
47	M₁₆×D₅₃	—	4	2	no	—
48	C₂²×#88322	D₄	1	1	yes	characteristic
49	C₂×C₈×D₅₃	—	4	1	no	—
50	#13133192	—	2	4	no	—
51	#13133192	—	4	2	no	—
52	C₂×#11719921	—	4	1	no	—
53	#13133157	—	2	1	no	—
54	#13133157	C₂³	1	2	yes	—
55	C₂×#11719908	—	2	1	no	—
56	#13133153	—	2	1	no	—
57	#13133153	C₂³	1	2	yes	—
58	#13133152	—	2	4	no	—
59	#13133137	—	2	4	no	—
60	C₂×M₁₆×C₅₃	—	2	1	no	—
61	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
63	M₃₂×C₅₃	—	2	1	no	—
64	M₃₂×C₅₃	C₂³	1	2	yes	—
65	C₂×C₁₆×C₅₃	—	2	1	no	—
66	#108×C₅₃	—	2	1	no	—
67	#108×C₅₃	C₂³	1	2	yes	—
68	#106×C₅₃	—	2	1	no	—
69	#106×C₅₃	C₂³	1	2	yes	—
70	C₂³×D₅₃	—	4	1	no	—
71	C₂³×D₅₃	SD₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	—	4	1	no	—
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×Dic₅₃	SD₁₆	1	1	yes	characteristic
75	C₂×D₂₁₂	—	2	2	no	—
76	C₂×D₂₁₂	—	4	1	no	—
77	C₂×D₂₁₂	—	4	1	no	—
78	C₂×D₂₁₂	—	4	1	no	—
79	C₂×D₂₁₂	—	4	1	no	—
80	C₂×D₂₁₂	—	4	4	no	—
81	C₂×D₂₁₂	—	4	1	no	—
82	C₂×D₂₁₂	—	8	4	no	—
83	C₂×C₄×D₅₃	—	2	1	no	—
84	C₂×C₄×D₅₃	—	2	1	no	—
85	C₂×C₄×D₅₃	—	2	1	no	—
86	C₂×C₄×D₅₃	—	2	1	no	—
87	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
88	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
89	M₁₆×C₅₃	—	2	2	no	—
90	M₁₆×C₅₃	—	4	1	no	—
91	C₂×C₈×C₅₃	—	4	1	no	—
92	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
94	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
95	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
96	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₅₃	—	2	2	no	—
99	C₄⋊C₄×C₅₃	—	4	1	no	—
100	#11719921	—	4	4	no	—
101	#11719917	—	2	2	no	—
102	#11719917	—	4	1	no	—
103	C₂×#88322	—	2	1	no	—
104	C₂×#88322	—	2	1	no	—
105	C₂×#88322	—	4	1	no	—
106	C₂×#88322	C₂×D₄	1	2	yes	—
107	D₄₂₄	—	2	2	no	—
108	D₄₂₄	—	2	2	no	—
109	D₄₂₄	—	4	1	no	—
110	D₄₂₄	—	4	1	no	—
111	#11719912	—	2	2	no	—
112	#11719912	—	2	2	no	—
113	#11719912	—	2	2	no	—
114	#11719912	—	2	2	no	—
115	#11719912	—	4	4	no	—
116	C₈×D₅₃	—	4	4	no	—
117	C₁₆×C₅₃	—	2	2	no	—
118	#11719908	—	2	2	no	—
119	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
120	C₂²×D₅₃	—	2	1	no	—
121	C₂²×D₅₃	—	2	1	no	—
122	C₂²×D₅₃	—	4	2	no	—
123	C₂²×D₅₃	—	4	2	no	—
124	C₂²×D₅₃	—	8	2	no	—
125	C₂²×D₅₃	—	8	2	no	—
126	C₂²×D₅₃	—	8	1	no	—
127	C₂²×D₅₃	—	8	1	no	—
128	C₂²×D₅₃	C₂×SD₁₆	1	2	yes	—
129	C₂×C₄×C₅₃	—	4	2	no	—
130	C₂×C₄×C₅₃	—	8	2	no	—
131	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
132	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₅₃	—	2	1	no	—
138	C₂×Dic₅₃	—	2	1	no	—
139	C₂×Dic₅₃	C₂×SD₁₆	1	2	yes	—
140	D₂₁₂	—	4	2	no	—
141	D₂₁₂	—	4	2	no	—
142	D₂₁₂	—	8	8	no	—
143	D₂₁₂	—	8	1	no	—
144	D₂₁₂	—	8	1	no	—
145	C₄×D₅₃	—	2	2	no	—
146	C₄×D₅₃	—	2	2	no	—
147	C₄×D₅₃	—	2	2	no	—
148	C₄×D₅₃	—	2	2	no	—
149	C₈×C₅₃	—	4	2	no	—
150	C₈×C₅₃	#123	1	1	yes	characteristic
151	C₈×C₅₃	#123	1	1	yes	characteristic
152	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
153	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
154	#88322	—	2	2	no	—
155	#88322	—	4	2	no	—
156	C₂×#2238	—	53	1	no	maximal, Sylow
157	C₂²×C₅₃	—	2	2	no	—
158	C₂²×C₅₃	#481	1	1	yes	characteristic
159	C₂²×C₅₃	#481	1	1	yes	characteristic
160	C₂²×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
161	C₂×D₅₃	—	2	2	no	—
162	C₂×D₅₃	—	4	2	no	—
163	C₂×D₅₃	—	4	2	no	—
164	C₂×D₅₃	—	8	1	no	—
165	C₂×D₅₃	—	8	2	no	—
166	C₂×D₅₃	—	8	2	no	—
167	C₂×D₅₃	—	8	2	no	—
168	C₂×D₅₃	—	16	2	no	—
169	C₂×D₅₃	—	16	2	no	—
170	C₄×C₅₃	—	8	2	no	—
171	C₄×C₅₃	#481	1	1	yes	characteristic
172	C₄×C₅₃	#481	1	1	yes	characteristic
173	C₄×C₅₃	C₂×#123	1	1	yes	characteristic
174	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	Dic₅₃	—	2	2	no	—
176	C₂×#469	—	53	1	no	—
177	C₂×#430	—	53	1	no	—
178	#2238	—	53	8	no	—
179	C₂×#365	—	53	1	no	—
180	C₂×#365	D₅₃	1	1	yes	characteristic, complemented
181	C₂×#361	—	53	1	no	—
182	C₂×#361	—	53	1	no	—
183	C₂×#349	—	53	1	no	—
184	C₂×C₅₃	—	2	1	no	—
185	C₂×C₅₃	—	2	1	no	—
186	C₂×C₅₃	#2238	1	2	yes	complemented
187	C₂×C₅₃	C₂×#481	1	1	yes	characteristic
188	D₅₃	—	4	2	no	—
189	D₅₃	—	16	2	no	—
190	C₂²×D₈	—	53	1	no	—
191	C₂²×M₁₆	—	53	1	no	—
192	C₂²×M₁₆	—	106	1	no	—
193	C₂×#121	—	106	1	no	—
194	C₂×M₃₂	—	53	1	no	—
195	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
196	#469	—	53	4	no	—
197	#469	—	106	2	no	—
198	#430	—	53	4	no	—
199	#430	—	106	2	no	—
200	C₂×#108	—	53	1	no	—
201	C₂×#108	—	106	1	no	—
202	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
203	C₂×#106	C₂×D₅₃	1	1	yes	characteristic
204	C₂×#102	—	106	1	no	—
205	#365	—	53	4	no	—
206	#365	C₂×D₅₃	1	4	yes	complemented
207	#361	—	53	4	no	—
208	#361	—	53	4	no	—
209	#349	—	53	4	no	—
210	C₅₃	C₂×#2238	1	1	yes	characteristic, complemented, Sylow
211	C₂²×D₄	—	106	1	no	—
212	C₂²×D₄	—	212	1	no	—
213	C₂³×C₄	—	53	1	no	—
214	C₂×D₈	—	53	2	no	—
215	C₂×D₈	—	106	1	no	—
216	C₂×D₈	—	106	1	no	—
217	C₂×D₈	—	106	1	no	—
218	C₂×D₈	—	106	1	no	—
219	C₂×D₈	—	106	1	no	—
220	C₂×M₁₆	—	2	1	no	—
221	C₂×M₁₆	—	53	2	no	—
222	C₂×M₁₆	—	53	2	no	—
223	C₂×M₁₆	—	106	1	no	—
224	C₂×M₁₆	—	106	1	no	—
225	C₂×M₁₆	—	106	1	no	—
226	C₂×M₁₆	—	106	1	no	—
227	C₂×M₁₆	—	106	4	no	—
228	C₂×M₁₆	—	106	1	no	—
229	C₂×M₁₆	—	212	1	no	—
230	C₂×M₁₆	—	212	2	no	—
231	C₂²×C₈	—	53	1	no	—
232	C₂²×C₈	—	212	1	no	—
233	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
234	#121	—	106	4	no	—
235	#121	—	212	2	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₂²⋊C₄	—	212	1	no	—
238	M₃₂	—	2	1	no	—
239	M₃₂	—	53	2	no	—
240	M₃₂	—	106	1	no	—
241	M₃₂	C₂²×D₅₃	1	2	yes	—
242	C₂×C₁₆	—	2	1	no	—
243	C₂×C₁₆	—	106	1	no	—
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	#106	—	2	1	no	—
250	#106	C₂²×D₅₃	1	2	yes	—
251	#102	—	106	4	no	—
252	C₂⁴	—	53	1	no	—
253	C₂⁴	—	212	1	no	—
254	C₂×D₄	—	106	2	no	—
255	C₂×D₄	—	424	4	no	—
256	C₂×D₄	—	212	4	no	—
257	C₂×D₄	—	212	1	no	—
258	C₂×D₄	—	212	1	no	—
259	C₂×D₄	—	212	1	no	—
260	C₂×D₄	—	212	1	no	—
261	C₂×D₄	—	212	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	53	1	no	—
264	C₂²×C₄	—	53	2	no	—
265	C₂²×C₄	—	53	2	no	—
266	C₂²×C₄	—	106	1	no	—
267	C₂²×C₄	—	106	1	no	—
268	C₂²×C₄	—	106	1	no	—
269	C₂²×C₄	—	106	1	no	—
270	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
271	D₈	—	106	2	no	—
272	D₈	—	106	2	no	—
273	D₈	—	212	1	no	—
274	D₈	—	212	1	no	—
275	M₁₆	—	2	2	no	—
276	M₁₆	—	4	1	no	—
277	M₁₆	—	106	2	no	—
278	M₁₆	—	106	2	no	—
279	M₁₆	—	106	2	no	—
280	M₁₆	—	106	2	no	—
281	M₁₆	—	106	2	no	—
282	M₁₆	—	212	4	no	—
283	M₁₆	—	212	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	53	2	no	—
286	C₂×C₈	—	106	1	no	—
287	C₂×C₈	—	106	1	no	—
288	C₂×C₈	—	212	4	no	—
289	C₂×C₈	—	212	1	no	—
290	C₂×C₈	#11719945	1	1	yes	characteristic
291	C₂×C₈	#11719945	1	1	yes	characteristic
292	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
293	C₂×C₈	#11719949	1	1	yes	characteristic
294	C₂×C₈	#11719949	1	1	yes	characteristic
295	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₂²⋊C₄	—	212	4	no	—
299	C₁₆	—	2	2	no	—
300	C₁₆	—	106	2	no	—
301	C₂³	—	53	2	no	—
302	C₂³	—	106	1	no	—
303	C₂³	—	106	1	no	—
304	C₂³	—	424	2	no	—
305	C₂³	—	424	2	no	—
306	C₂³	—	424	1	no	—
307	C₂³	—	424	1	no	—
308	C₂³	—	212	2	no	—
309	C₂³	—	212	2	no	—
310	C₂³	SD₁₆×D₅₃	1	1	yes	characteristic
311	D₄	—	424	8	no	—
312	D₄	—	424	1	no	—
313	D₄	—	424	1	no	—
314	D₄	—	212	2	no	—
315	D₄	—	212	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	8	2	no	—
318	C₂×C₄	—	53	2	no	—
319	C₂×C₄	—	106	1	no	—
320	C₂×C₄	—	106	1	no	—
321	C₂×C₄	—	106	2	no	—
322	C₂×C₄	—	106	2	no	—
323	C₂×C₄	—	106	2	no	—
324	C₂×C₄	—	106	2	no	—
325	C₂×C₄	#13133202	1	1	yes	characteristic
326	C₂×C₄	#13133202	1	1	yes	characteristic
327	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
328	C₂×C₄	#13133244	1	1	yes	characteristic
329	C₂×C₄	#13133244	1	1	yes	characteristic
330	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
331	C₈	—	4	2	no	—
332	C₈	—	106	2	no	—
333	C₈	—	212	2	no	—
334	C₈	#13133202	1	1	yes	characteristic
335	C₈	#13133202	1	1	yes	characteristic
336	C₈	C₂×#11719945	1	1	yes	characteristic
337	C₈	C₂×#11719949	1	1	yes	characteristic
338	C₂²	—	2	2	no	—
339	C₂²	—	848	2	no	—
340	C₂²	—	848	2	no	—
341	C₂²	—	106	2	no	—
342	C₂²	—	424	2	no	—
343	C₂²	—	424	2	no	—
344	C₂²	—	424	1	no	—
345	C₂²	—	424	2	no	—
346	C₂²	—	212	2	no	—
347	C₂²	—	212	2	no	—
348	C₂²	#16810943	1	1	yes	characteristic
349	C₂²	#16810943	1	1	yes	characteristic
350	C₂²	C₂×SD₁₆×D₅₃	1	1	yes	characteristic
351	C₄	—	8	2	no	—
352	C₄	—	106	2	no	—
353	C₄	#16810943	1	1	yes	characteristic
354	C₄	#16810943	1	1	yes	characteristic
355	C₄	C₂×#13133202	1	1	yes	characteristic
356	C₄	C₂×#13133244	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	848	2	no	—
360	C₂	—	212	2	no	—
361	C₂	#20276931	1	2	yes	direct product factor
362	C₂	C₂×#16810943	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810943
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#365	C₂
socle	C₂²×C₅₃	#481
prime radical	C₂×#16810877	C₂
hypercenter	C₂×#365	D₅₃
hypocenter	C₅₃	C₂×#2238
LCS	G ▷ C₂×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×#16810670	C₂	semidirect
3. #20276931	C₂	direct (N × H)
4. C₂×#16810445	C₂	semidirect
5. C₂×#16810437	C₂	semidirect
6. C₂×#16810435	C₂	semidirect
7. C₂×#16810404	C₂	semidirect
8. C₂×C₅₃×#365	C₂	semidirect
17. #16810670	C₂²	semidirect
21. #16810445	C₂²	semidirect
22. #16810437	C₂²	semidirect
23. #16810435	C₂²	semidirect
24. #16810404	C₂²	semidirect
28. C₅₃×#365	C₂²	semidirect
180. C₂×#365	D₅₃	semidirect
186. C₂×C₅₃	#2238	semidirect
206. #365	C₂×D₅₃	semidirect
210. C₅₃	C₂×#2238	semidirect
361. C₂	#20276931	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×#16810877	2
C₂×#16810670	2
#20276931	2
C₂×#16810445	2
C₂×#16810437	2
C₂×#16810435	2
C₂×#16810404	2
C₂×C₅₃×#365	2
C₂×#2238	53

Subgroups — C2×#20276931 #27538978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276931 #27538978