Subgroups — C₂×#20276788 #27538469

Order 13568.

790 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811059	C₂	1	2	yes	maximal, complemented
2	C₂×#16810741	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276788	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810444	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810436	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13133239	C₂²	1	1	yes	characteristic
8	C₂×#13133223	—	2	1	no	—
9	#16811059	—	2	4	no	—
10	#16811059	C₂²	1	8	yes	complemented
11	C₂×#13133161	—	2	2	no	—
12	C₂×#13133158	—	2	2	no	—
13	C₂×#13133157	C₂²	1	2	yes	—
14	#16810741	—	2	2	no	—
15	#16810741	C₂²	1	4	yes	complemented
16	C₂×#13133153	C₂²	1	1	yes	characteristic
17	C₂×#13133145	—	2	1	no	—
18	#16810444	C₂²	1	4	yes	complemented
19	#16810436	C₂²	1	8	yes	complemented
20	C₂²×D₈×C₅₃	—	2	1	no	—
21	C₂²×D₈×C₅₃	C₂²	1	2	yes	—
22	C₂×#127×C₅₃	—	2	1	no	—
23	C₅₃×#493	—	2	2	no	—
24	C₅₃×#493	C₂²	1	4	yes	complemented
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#11719945	—	2	1	no	—
27	#13133239	—	2	1	no	—
28	#13133239	—	2	1	no	—
29	#13133239	D₄	1	2	yes	complemented
30	#13133239	C₂³	1	2	yes	—
31	C₂×Dic₂₁₂	D₄	1	1	yes	characteristic, complemented
32	C₂×#11719913	—	2	1	no	—
33	C₂×D₄₂₄	D₄	1	1	yes	characteristic, complemented
34	#13133223	—	2	4	no	—
35	#13133223	—	4	2	no	—
36	C₂×#11719917	—	2	1	no	—
37	C₂×#11719912	—	4	1	no	—
38	C₂×C₈×D₅₃	—	4	1	no	—
39	#13133161	—	2	8	no	—
40	#13133158	—	2	8	no	—
41	#13133157	—	2	2	no	—
42	#13133157	C₂³	1	4	yes	—
43	C₂×#11719908	—	2	2	no	—
44	#13133153	—	2	1	no	—
45	#13133153	C₂³	1	2	yes	—
46	#13133145	—	2	4	no	—
47	C₂²×D₄×C₅₃	—	2	2	no	—
48	C₂²×D₄×C₅₃	—	4	2	no	—
49	C₂×D₈×C₅₃	—	2	2	no	—
50	C₂×D₈×C₅₃	—	2	4	no	—
51	C₂×D₈×C₅₃	—	2	2	no	—
52	C₂×D₈×C₅₃	—	2	2	no	—
53	C₂×D₈×C₅₃	—	2	2	no	—
54	C₂×D₈×C₅₃	—	2	2	no	—
55	C₂×D₈×C₅₃	—	4	1	no	—
56	C₂×D₈×C₅₃	—	4	1	no	—
57	C₂×D₈×C₅₃	—	4	2	no	—
58	C₂×D₈×C₅₃	C₂³	1	4	yes	—
59	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	#127×C₅₃	—	2	4	no	—
62	#127×C₅₃	—	4	2	no	—
63	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
64	C₄×C₈×C₅₃	D₄	1	2	yes	—
65	C₄×C₈×C₅₃	C₂³	1	2	yes	—
66	C₂⁴×C₅₃	—	4	2	no	—
67	C₂×D₄×C₅₃	—	2	4	no	—
68	C₂×D₄×C₅₃	—	4	8	no	—
69	C₂×D₄×C₅₃	—	4	2	no	—
70	C₂×D₄×C₅₃	—	4	2	no	—
71	C₂×D₄×C₅₃	—	4	2	no	—
72	C₂×D₄×C₅₃	—	4	2	no	—
73	C₂×D₄×C₅₃	—	4	2	no	—
74	C₂×D₄×C₅₃	—	8	8	no	—
75	C₂²×C₄×C₅₃	—	2	1	no	—
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₂²⋊D₅₃	—	4	1	no	—
78	#11719945	—	2	2	no	—
79	#11719945	—	2	2	no	—
80	#11719945	—	4	1	no	—
81	#11719945	—	4	1	no	—
82	C₂×D₂₁₂	—	2	1	no	—
83	C₂×C₄×D₅₃	—	4	1	no	—
84	C₂×Dic₁₀₆	—	2	1	no	—
85	D₈×C₅₃	—	2	4	no	—
86	D₈×C₅₃	—	2	4	no	—
87	D₈×C₅₃	—	4	2	no	—
88	D₈×C₅₃	—	4	2	no	—
89	D₈×C₅₃	—	4	4	no	—
90	D₈×C₅₃	—	4	4	no	—
91	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
92	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
93	C₂×C₈×C₅₃	D₈	1	2	yes	—
94	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
99	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
100	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
101	C₄²×C₅₃	—	2	1	no	—
102	C₄²×C₅₃	C₂×D₄	1	2	yes	—
103	#11719917	—	2	2	no	—
104	#11719917	—	4	1	no	—
105	C₂×#88322	—	4	1	no	—
106	Dic₂₁₂	—	2	1	no	—
107	Dic₂₁₂	C₂×D₄	1	2	yes	complemented
108	D₄₂₄	—	2	1	no	—
109	D₄₂₄	C₂×D₄	1	2	yes	complemented
110	#11719913	—	2	2	no	—
111	#11719913	—	2	2	no	—
112	#11719912	—	4	4	no	—
113	C₈×D₅₃	—	4	4	no	—
114	#11719908	—	2	4	no	—
115	C₂³×C₅₃	—	4	4	no	—
116	C₂³×C₅₃	—	8	8	no	—
117	C₂³×C₅₃	—	8	2	no	—
118	C₂³×C₅₃	—	8	2	no	—
119	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂²×D₅₃	—	4	1	no	—
121	D₄×C₅₃	—	4	4	no	—
122	D₄×C₅₃	—	4	4	no	—
123	D₄×C₅₃	—	8	16	no	—
124	D₄×C₅₃	—	8	2	no	—
125	D₄×C₅₃	—	8	2	no	—
126	C₂×C₄×C₅₃	—	2	4	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
134	C₂²⋊D₅₃	—	4	2	no	—
135	C₂²⋊D₅₃	—	4	2	no	—
136	C₂×Dic₅₃	—	4	1	no	—
137	D₂₁₂	—	2	2	no	—
138	D₂₁₂	—	4	1	no	—
139	C₄×D₅₃	—	4	2	no	—
140	C₄×D₅₃	—	4	2	no	—
141	Dic₁₀₆	—	2	2	no	—
142	Dic₁₀₆	—	4	1	no	—
143	C₈×C₅₃	#121	1	1	yes	characteristic
144	C₈×C₅₃	#121	1	1	yes	characteristic
145	C₈×C₅₃	C₂×D₈	1	2	yes	—
146	C₈×C₅₃	C₂×D₈	1	2	yes	—
147	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
149	#88322	—	4	2	no	—
150	C₂×#2213	—	53	1	no	maximal, Sylow
151	C₂²×C₅₃	—	2	2	no	—
152	C₂²×C₅₃	—	8	8	no	—
153	C₂²×C₅₃	—	8	4	no	—
154	C₂²×C₅₃	—	16	4	no	—
155	C₂²×C₅₃	—	16	4	no	—
156	C₂²×C₅₃	#459	1	1	yes	characteristic
157	C₂²×C₅₃	#459	1	1	yes	characteristic
158	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
159	C₂×D₅₃	—	4	2	no	—
160	C₂×D₅₃	—	8	2	no	—
161	C₄×C₅₃	—	2	4	no	—
162	C₄×C₅₃	#459	1	1	yes	characteristic
163	C₄×C₅₃	#459	1	1	yes	characteristic
164	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
165	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
166	Dic₅₃	—	4	2	no	—
167	C₂×#509	—	53	2	no	—
168	C₂×#493	D₅₃	1	1	yes	characteristic, complemented
169	C₂×#443	—	53	1	no	—
170	#2213	—	53	8	no	—
171	C₂×#364	—	53	1	no	—
172	C₂×#361	—	53	2	no	—
173	C₂×C₅₃	—	2	1	no	—
174	C₂×C₅₃	—	2	1	no	—
175	C₂×C₅₃	—	16	4	no	—
176	C₂×C₅₃	#2213	1	2	yes	complemented
177	C₂×C₅₃	C₂×#459	1	1	yes	characteristic
178	D₅₃	—	8	2	no	—
179	C₂×#135	—	53	1	no	—
180	C₂²×D₈	—	2	1	no	—
181	C₂²×D₈	C₂×D₅₃	1	2	yes	—
182	C₂×#131	—	106	1	no	—
183	C₂×#127	—	2	1	no	—
184	#509	—	53	8	no	—
185	#509	—	106	4	no	—
186	C₂×SD₃₂	—	106	2	no	—
187	C₂×D₁₆	—	106	2	no	—
188	C₂×M₃₂	—	53	2	no	—
189	#493	—	2	2	no	—
190	#493	C₂×D₅₃	1	4	yes	complemented
191	#443	—	53	4	no	—
192	#443	—	106	2	no	—
193	C₂×#108	—	53	1	no	—
194	C₂×#104	—	106	1	no	—
195	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
196	#364	—	53	4	no	—
197	#361	—	53	8	no	—
198	C₅₃	C₂×#2213	1	1	yes	characteristic, complemented, Sylow
199	C₂×#41	—	106	1	no	—
200	C₂²×D₄	—	2	2	no	—
201	C₂²×D₄	—	4	2	no	—
202	#135	—	53	2	no	—
203	#135	—	53	2	no	—
204	#135	—	106	1	no	—
205	#135	—	106	1	no	—
206	C₂×Q₁₆	—	53	1	no	—
207	C₂×SD₁₆	—	106	1	no	—
208	C₂×D₈	—	2	2	no	—
209	C₂×D₈	—	2	2	no	—
210	C₂×D₈	—	2	2	no	—
211	C₂×D₈	—	2	2	no	—
212	C₂×D₈	—	2	4	no	—
213	C₂×D₈	—	2	2	no	—
214	C₂×D₈	—	4	1	no	—
215	C₂×D₈	—	4	1	no	—
216	C₂×D₈	—	4	2	no	—
217	C₂×D₈	—	53	1	no	—
218	C₂×D₈	C₂²×D₅₃	1	4	yes	—
219	#131	—	106	4	no	—
220	#131	—	212	2	no	—
221	C₂×M₁₆	—	106	1	no	—
222	C₂×M₁₆	—	212	1	no	—
223	C₂²×C₈	—	212	1	no	—
224	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
225	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
226	#127	—	2	4	no	—
227	#127	—	4	2	no	—
228	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
229	SD₃₂	—	106	8	no	—
230	D₁₆	—	106	8	no	—
231	M₃₂	—	53	4	no	—
232	M₃₂	—	106	2	no	—
233	C₂×C₁₆	—	106	2	no	—
234	#108	—	53	2	no	—
235	#108	—	106	1	no	—
236	#104	—	106	4	no	—
237	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
238	C₄×C₈	C₂²×D₅₃	1	2	yes	—
239	C₂⁴	—	4	2	no	—
240	#41	—	106	2	no	—
241	#41	—	106	2	no	—
242	#41	—	212	1	no	—
243	#41	—	212	1	no	—
244	C₂×Q₈	—	106	1	no	—
245	C₂×D₄	—	2	4	no	—
246	C₂×D₄	—	4	8	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	8	8	no	—
253	C₂×D₄	—	106	1	no	—
254	C₂×D₄	—	212	1	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	212	1	no	—
257	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
258	Q₁₆	—	53	2	no	—
259	Q₁₆	—	106	1	no	—
260	SD₁₆	—	106	2	no	—
261	SD₁₆	—	106	2	no	—
262	D₈	—	2	4	no	—
263	D₈	—	2	4	no	—
264	D₈	—	4	2	no	—
265	D₈	—	4	2	no	—
266	D₈	—	4	4	no	—
267	D₈	—	4	4	no	—
268	D₈	—	53	2	no	—
269	D₈	—	106	1	no	—
270	M₁₆	—	106	2	no	—
271	M₁₆	—	212	1	no	—
272	M₁₆	—	212	4	no	—
273	C₂×C₈	—	212	1	no	—
274	C₂×C₈	—	212	4	no	—
275	C₂×C₈	#11719922	1	1	yes	characteristic
276	C₂×C₈	#11719922	1	1	yes	characteristic
277	C₂×C₈	#11719922	1	2	yes	—
278	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
279	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
280	C₂×C₈	#11719947	1	1	yes	characteristic
281	C₂×C₈	#11719947	1	1	yes	characteristic
282	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
283	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
284	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
285	C₄²	—	2	1	no	—
286	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
287	C₁₆	—	106	4	no	—
288	C₂³	—	4	4	no	—
289	C₂³	—	8	8	no	—
290	C₂³	—	8	2	no	—
291	C₂³	—	8	2	no	—
292	C₂³	—	212	1	no	—
293	C₂³	#13133243	1	1	yes	characteristic
294	Q₈	—	106	2	no	—
295	Q₈	—	212	1	no	—
296	D₄	—	4	4	no	—
297	D₄	—	4	4	no	—
298	D₄	—	8	16	no	—
299	D₄	—	8	2	no	—
300	D₄	—	8	2	no	—
301	D₄	—	106	2	no	—
302	D₄	—	212	2	no	—
303	D₄	—	212	2	no	—
304	D₄	—	212	1	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	212	1	no	—
308	C₂×C₄	—	212	2	no	—
309	C₂×C₄	—	212	2	no	—
310	C₂×C₄	#13133188	1	1	yes	characteristic
311	C₂×C₄	#13133188	1	1	yes	characteristic
312	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
313	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
314	C₂×C₄	#13133243	1	1	yes	characteristic
315	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
316	C₈	—	212	2	no	—
317	C₈	#13133188	1	1	yes	characteristic
318	C₈	#13133188	1	1	yes	characteristic
319	C₈	C₂×#11719922	1	2	yes	—
320	C₈	C₂×#11719922	1	2	yes	—
321	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
322	C₈	C₂×#11719947	1	1	yes	characteristic
323	C₂²	—	2	2	no	—
324	C₂²	—	8	8	no	—
325	C₂²	—	8	4	no	—
326	C₂²	—	16	4	no	—
327	C₂²	—	16	4	no	—
328	C₂²	—	424	2	no	—
329	C₂²	—	212	2	no	—
330	C₂²	#16810815	1	1	yes	characteristic
331	C₂²	#16810815	1	1	yes	characteristic
332	C₂²	C₂×#13133243	1	1	yes	characteristic
333	C₄	—	2	4	no	—
334	C₄	—	212	2	no	—
335	C₄	#16810815	1	1	yes	characteristic
336	C₄	#16810815	1	1	yes	characteristic
337	C₄	C₂×#13133188	1	1	yes	characteristic
338	C₄	C₂²×#11719922	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	16	4	no	—
342	C₂	—	424	2	no	—
343	C₂	#20276788	1	2	yes	direct product factor
344	C₂	C₂×#16810815	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810815
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#493	C₂
socle	C₂²×C₅₃	#459
hypercenter	C₂×#493	D₅₃
hypocenter	C₅₃	C₂×#2213
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#16811059	C₂	semidirect
2. C₂×#16810741	C₂	semidirect
3. #20276788	C₂	direct (N × H)
4. C₂×#16810444	C₂	semidirect
5. C₂×#16810436	C₂	semidirect
6. C₂×C₅₃×#493	C₂	semidirect
10. #16811059	C₂²	semidirect
15. #16810741	C₂²	semidirect
18. #16810444	C₂²	semidirect
19. #16810436	C₂²	semidirect
24. C₅₃×#493	C₂²	semidirect
29. #13133239	D₄	semidirect
31. C₂×Dic₂₁₂	D₄	semidirect
33. C₂×D₄₂₄	D₄	semidirect
107. Dic₂₁₂	C₂×D₄	semidirect
109. D₄₂₄	C₂×D₄	semidirect
168. C₂×#493	D₅₃	semidirect
176. C₂×C₅₃	#2213	semidirect
190. #493	C₂×D₅₃	semidirect
198. C₅₃	C₂×#2213	semidirect
343. C₂	#20276788	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×#16811059	2
C₂×#16810741	2
#20276788	2
C₂×#16810444	2
C₂×#16810436	2
C₂×C₅₃×#493	2
C₂×#2213	53

Subgroups — C2×#20276788 #27538469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276788 #27538469