Subgroups — C₂×#20276884 #27538820

Order 13568.

558 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811020	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810740	C₂	1	1	yes	maximal, characteristic
3	#20276884	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810446	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810445	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810443	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16810433	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅₃×#360	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133224	—	2	1	no	—
10	C₂×#13133216	—	2	1	no	—
11	C₂×#13133157	C₂²	1	1	yes	characteristic
12	#16811020	—	2	2	no	—
13	#16811020	C₂²	1	4	yes	complemented
14	#16810740	—	2	2	no	—
15	#16810740	C₂²	1	4	yes	—
16	C₂×#13133153	C₂²	1	1	yes	characteristic
17	C₂×#13133149	C₂²	1	1	yes	characteristic
18	C₂×#13133148	—	2	1	no	—
19	C₂×#13133144	—	2	1	no	—
20	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
21	#16810446	C₂²	1	4	yes	complemented
22	#16810445	C₂²	1	4	yes	complemented
23	#16810443	C₂²	1	4	yes	complemented
24	#16810433	C₂²	1	4	yes	—
25	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#360	C₂²	1	4	yes	—
29	C₂²×Dic₁₀₆	—	4	1	no	—
30	#13133224	—	2	4	no	—
31	#13133224	—	4	2	no	—
32	C₂×#11719917	—	2	1	no	—
33	C₂×#11719917	—	4	1	no	—
34	C₂²×#88322	—	4	1	no	—
35	C₂²×#88322	D₄	1	1	yes	characteristic
36	#13133216	—	2	4	no	—
37	#13133216	—	4	2	no	—
38	C₂×C₄⋊Dic₅₃	—	2	1	no	—
39	C₂×#11719919	—	4	1	no	—
40	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
41	#13133157	—	2	1	no	—
42	#13133157	C₂³	1	2	yes	—
43	C₂×#11719908	—	2	1	no	—
44	#13133153	—	2	1	no	—
45	#13133153	C₂³	1	2	yes	—
46	#13133149	—	2	1	no	—
47	#13133149	C₂³	1	2	yes	—
48	#13133148	—	2	4	no	—
49	#13133144	—	2	4	no	—
50	C₈×Dic₅₃	D₄	1	2	yes	—
51	C₈×Dic₅₃	C₂³	1	2	yes	—
52	C₂×#41×C₅₃	—	2	1	no	—
53	#135×C₅₃	—	2	1	no	—
54	#135×C₅₃	—	2	1	no	—
55	#135×C₅₃	Q₈	1	2	yes	complemented
56	#135×C₅₃	C₂³	1	2	yes	—
57	C₂×Q₁₆×C₅₃	Q₈	1	1	yes	characteristic, complemented
58	C₂×SD₁₆×C₅₃	—	2	1	no	—
59	C₂×D₈×C₅₃	Q₈	1	1	yes	characteristic, complemented
60	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
62	M₃₂×C₅₃	—	2	1	no	—
63	M₃₂×C₅₃	C₂³	1	2	yes	—
64	C₂×C₁₆×C₅₃	—	2	1	no	—
65	#106×C₅₃	—	2	1	no	—
66	#106×C₅₃	C₂³	1	2	yes	—
67	#41×C₅₃	—	2	2	no	—
68	#41×C₅₃	—	2	2	no	—
69	#41×C₅₃	—	4	1	no	—
70	#41×C₅₃	—	4	1	no	—
71	C₂×Q₈×C₅₃	—	2	1	no	—
72	C₂×D₄×C₅₃	—	2	1	no	—
73	C₂×D₄×C₅₃	—	4	1	no	—
74	C₂²×C₄×C₅₃	—	4	1	no	—
75	C₂²×C₄×C₅₃	—	4	1	no	—
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₅₃	—	2	1	no	—
78	C₂²×Dic₅₃	—	4	1	no	—
79	C₂×Dic₁₀₆	—	4	4	no	—
80	C₂×Dic₁₀₆	—	8	4	no	—
81	Q₁₆×C₅₃	—	2	1	no	—
82	Q₁₆×C₅₃	C₂×Q₈	1	2	yes	complemented
83	SD₁₆×C₅₃	—	2	2	no	—
84	SD₁₆×C₅₃	—	2	2	no	—
85	D₈×C₅₃	—	2	1	no	—
86	D₈×C₅₃	C₂×Q₈	1	2	yes	complemented
87	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
91	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₅₃	—	2	2	no	—
94	C₄⋊C₄×C₅₃	—	4	1	no	—
95	C₄⋊Dic₅₃	—	2	2	no	—
96	C₄⋊Dic₅₃	—	4	1	no	—
97	#11719919	—	4	4	no	—
98	C₄×Dic₅₃	—	2	1	no	—
99	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
100	#11719917	—	2	2	no	—
101	#11719917	—	4	1	no	—
102	#11719917	—	4	4	no	—
103	C₂×#88322	—	4	1	no	—
104	C₂×#88322	—	4	4	no	—
105	C₂×#88322	SD₁₆	1	1	yes	characteristic
106	C₂×#88322	SD₁₆	1	1	yes	characteristic
107	C₂×#88322	SD₁₆	1	2	yes	—
108	C₂×#88322	C₂×D₄	1	2	yes	—
109	C₁₆×C₅₃	—	2	2	no	—
110	#11719908	—	2	2	no	—
111	C₂³×C₅₃	—	4	1	no	—
112	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
113	Q₈×C₅₃	—	2	2	no	—
114	Q₈×C₅₃	—	4	1	no	—
115	D₄×C₅₃	—	2	2	no	—
116	D₄×C₅₃	—	4	2	no	—
117	D₄×C₅₃	—	4	2	no	—
118	D₄×C₅₃	—	4	1	no	—
119	C₂×C₄×C₅₃	—	4	1	no	—
120	C₂×C₄×C₅₃	—	4	2	no	—
121	C₂×C₄×C₅₃	—	4	2	no	—
122	C₂×C₄×C₅₃	—	4	2	no	—
123	C₂×C₄×C₅₃	—	8	2	no	—
124	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₅₃	—	2	4	no	—
131	C₂×Dic₅₃	—	2	2	no	—
132	C₂×Dic₅₃	—	4	2	no	—
133	C₂×Dic₅₃	—	8	2	no	—
134	Dic₁₀₆	—	8	8	no	—
135	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
136	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
137	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
138	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
139	#88322	—	4	2	no	—
140	#88322	C₂×SD₁₆	1	2	yes	—
141	#88322	C₂×SD₁₆	1	2	yes	—
142	C₂×#2230	—	53	1	no	maximal, Sylow
143	C₂²×C₅₃	—	2	2	no	—
144	C₂²×C₅₃	—	4	2	no	—
145	C₂²×C₅₃	—	8	2	no	—
146	C₂²×C₅₃	#476	1	1	yes	characteristic
147	C₂²×C₅₃	#476	1	1	yes	characteristic
148	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₄×C₅₃	—	4	2	no	—
150	C₄×C₅₃	—	8	2	no	—
151	C₄×C₅₃	#476	1	1	yes	characteristic
152	C₄×C₅₃	#476	1	1	yes	characteristic
153	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
154	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
155	Dic₅₃	—	2	4	no	—
156	Dic₅₃	—	8	2	no	—
157	C₂×#498	—	53	1	no	—
158	C₂×#443	—	53	1	no	—
159	#2230	—	53	8	no	—
160	C₂×#365	—	53	1	no	—
161	C₂×#365	—	53	1	no	—
162	C₂×#364	—	53	1	no	—
163	C₂×#360	—	53	1	no	—
164	C₂×#360	D₅₃	1	1	yes	characteristic
165	C₂×C₅₃	—	2	1	no	—
166	C₂×C₅₃	—	2	1	no	—
167	C₂×C₅₃	—	8	2	no	—
168	C₂×C₅₃	#2230	1	2	yes	complemented
169	C₂×C₅₃	C₂×#476	1	1	yes	characteristic
170	C₂×#135	C₂×D₅₃	1	1	yes	characteristic
171	C₂×#131	—	106	1	no	—
172	C₂×C₄⋊Q₈	—	106	1	no	—
173	C₂×M₃₂	—	53	1	no	—
174	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
175	#498	—	53	4	no	—
176	#498	—	106	2	no	—
177	#443	—	53	4	no	—
178	#443	—	106	2	no	—
179	C₂×#108	—	53	1	no	—
180	C₂×#106	—	53	1	no	—
181	C₂×#106	—	106	1	no	—
182	C₂×#106	C₂×D₅₃	1	1	yes	characteristic
183	C₂×#104	—	106	1	no	—
184	C₂×C₄×C₈	—	53	1	no	—
185	#365	—	53	4	no	—
186	#365	—	53	4	no	—
187	#364	—	53	4	no	—
188	#360	—	53	4	no	—
189	#360	C₂×D₅₃	1	4	yes	—
190	C₅₃	C₂×#2230	1	1	yes	characteristic, complemented, Sylow
191	C₂×#41	—	2	1	no	—
192	C₂²×Q₈	—	212	1	no	—
193	#135	—	2	1	no	—
194	#135	—	2	1	no	—
195	#135	Dic₁₀₆	1	2	yes	complemented
196	#135	C₂²×D₅₃	1	2	yes	—
197	C₂×Q₁₆	Dic₁₀₆	1	1	yes	characteristic, complemented
198	C₂×SD₁₆	—	2	1	no	—
199	C₂×D₈	Dic₁₀₆	1	1	yes	characteristic, complemented
200	#131	—	106	4	no	—
201	#131	—	212	2	no	—
202	C₂×M₁₆	—	106	1	no	—
203	C₂×M₁₆	—	212	1	no	—
204	C₂²×C₈	—	53	1	no	—
205	C₂²×C₈	—	212	1	no	—
206	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
207	C₄⋊Q₈	—	106	4	no	—
208	C₄⋊Q₈	—	212	2	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	106	1	no	—
211	C₂×C₄⋊C₄	—	212	1	no	—
212	C₂×C₄²	—	53	1	no	—
213	M₃₂	—	2	1	no	—
214	M₃₂	—	53	2	no	—
215	M₃₂	—	106	1	no	—
216	M₃₂	C₂²×D₅₃	1	2	yes	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	106	1	no	—
219	#108	—	53	2	no	—
220	#108	—	106	1	no	—
221	#106	—	2	1	no	—
222	#106	—	53	2	no	—
223	#106	—	106	4	no	—
224	#106	—	106	1	no	—
225	#106	C₂²×D₅₃	1	2	yes	—
226	#104	—	106	4	no	—
227	C₄×C₈	—	53	2	no	—
228	C₄×C₈	—	53	2	no	—
229	#41	—	2	2	no	—
230	#41	—	2	2	no	—
231	#41	—	4	1	no	—
232	#41	—	4	1	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	424	4	no	—
235	C₂×Q₈	—	212	4	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	106	1	no	—
241	C₂²×C₄	—	212	1	no	—
242	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
243	Q₁₆	—	2	1	no	—
244	Q₁₆	C₂×Dic₁₀₆	1	2	yes	complemented
245	SD₁₆	—	2	2	no	—
246	SD₁₆	—	2	2	no	—
247	D₈	—	2	1	no	—
248	D₈	C₂×Dic₁₀₆	1	2	yes	complemented
249	M₁₆	—	106	2	no	—
250	M₁₆	—	212	4	no	—
251	M₁₆	—	212	1	no	—
252	C₂×C₈	—	53	1	no	—
253	C₂×C₈	—	53	1	no	—
254	C₂×C₈	—	53	2	no	—
255	C₂×C₈	—	53	2	no	—
256	C₂×C₈	—	212	4	no	—
257	C₂×C₈	—	212	1	no	—
258	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
259	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
260	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
261	C₂×C₈	#11719947	1	1	yes	characteristic
262	C₂×C₈	#11719947	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	106	2	no	—
267	C₄⋊C₄	—	212	4	no	—
268	C₄⋊C₄	—	212	1	no	—
269	C₄²	—	53	2	no	—
270	C₄²	—	106	1	no	—
271	C₁₆	—	2	2	no	—
272	C₁₆	—	106	2	no	—
273	C₂³	—	4	1	no	—
274	C₂³	#13133241	1	1	yes	characteristic
275	Q₈	—	2	2	no	—
276	Q₈	—	4	1	no	—
277	Q₈	—	424	8	no	—
278	D₄	—	2	2	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	2	no	—
281	D₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	106	4	no	—
288	C₂×C₄	—	106	2	no	—
289	C₂×C₄	—	424	2	no	—
290	C₂×C₄	—	212	2	no	—
291	C₂×C₄	#13133194	1	1	yes	characteristic
292	C₂×C₄	#13133194	1	1	yes	characteristic
293	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
294	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
295	C₂×C₄	#13133241	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
297	C₈	—	53	2	no	—
298	C₈	—	53	2	no	—
299	C₈	—	212	2	no	—
300	C₈	#13133194	1	1	yes	characteristic
301	C₈	#13133194	1	1	yes	characteristic
302	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
303	C₈	C₂×#11719947	1	1	yes	characteristic
304	C₂²	—	2	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	#16810911	1	1	yes	characteristic
308	C₂²	#16810911	1	1	yes	characteristic
309	C₂²	C₂×#13133241	1	1	yes	characteristic
310	C₄	—	4	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	106	4	no	—
313	C₄	—	424	2	no	—
314	C₄	#16810911	1	1	yes	characteristic
315	C₄	#16810911	1	1	yes	characteristic
316	C₄	C₂×#13133194	1	1	yes	characteristic
317	C₄	C₂×SD₁₆×D₅₃	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	8	2	no	—
321	C₂	#20276884	1	2	yes	direct product factor
322	C₂	C₂×#16810911	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810911
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#360	C₂
socle	C₂²×C₅₃	#476
prime radical	C₂×#135×C₅₃	C₂²
hypercenter	C₂×#360	D₅₃
hypocenter	C₅₃	C₂×#2230
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#360

Complements

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

N	H ≅ G/N	type
1. C₂×#16811020	C₂	semidirect
3. #20276884	C₂	direct (N × H)
4. C₂×#16810446	C₂	semidirect
5. C₂×#16810445	C₂	semidirect
6. C₂×#16810443	C₂	semidirect
13. #16811020	C₂²	semidirect
21. #16810446	C₂²	semidirect
22. #16810445	C₂²	semidirect
23. #16810443	C₂²	semidirect
55. #135×C₅₃	Q₈	semidirect
57. C₂×Q₁₆×C₅₃	Q₈	semidirect
59. C₂×D₈×C₅₃	Q₈	semidirect
82. Q₁₆×C₅₃	C₂×Q₈	semidirect
86. D₈×C₅₃	C₂×Q₈	semidirect
168. C₂×C₅₃	#2230	semidirect
190. C₅₃	C₂×#2230	semidirect
195. #135	Dic₁₀₆	semidirect
197. C₂×Q₁₆	Dic₁₀₆	semidirect
199. C₂×D₈	Dic₁₀₆	semidirect
244. Q₁₆	C₂×Dic₁₀₆	semidirect
248. D₈	C₂×Dic₁₀₆	semidirect
321. C₂	#20276884	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×#16811020	2
C₂×#16810740	2
#20276884	2
C₂×#16810446	2
C₂×#16810445	2
C₂×#16810443	2
C₂×#16810433	2
C₂×C₅₃×#360	2
C₂×#2230	53

Subgroups — C2×#20276884 #27538820

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276884 #27538820