Subgroups — C₂×#20289783 #28310298

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811633	C₂	1	2	yes	maximal
2	#20289783	C₂	1	4	yes	maximal, direct product factor
3	C₂²×#13133156	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133296	—	2	2	no	—
5	#16811633	—	2	2	no	—
6	#16811633	—	2	2	no	—
7	#16811633	C₄	1	4	yes	complemented
8	#16811633	C₂²	1	4	yes	—
9	C₂×#13133286	—	2	2	no	—
10	C₂×#13133285	C₄	1	4	yes	complemented
11	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	C₂×#13133156	—	2	2	no	—
13	C₂×#13133156	—	2	2	no	—
14	C₂×#13133156	C₄	1	2	yes	complemented
15	C₂×#13133156	C₂²	1	2	yes	—
16	C₂²×#11719908	C₄	1	1	yes	characteristic, complemented
17	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
18	#13133296	—	2	4	no	—
19	#13133296	—	2	4	no	—
20	#13133296	—	4	2	no	—
21	#13133296	—	4	2	no	—
22	C₂×C₄⋊#6157	—	2	4	no	—
23	C₂×C₂²⋊#6157	—	4	2	no	—
24	C₂×C₄×#6157	—	4	2	no	—
25	#13133286	—	2	4	no	—
26	#13133286	—	2	4	no	—
27	#13133285	—	2	4	no	—
28	#13133285	C₂×C₄	1	8	yes	complemented
29	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
30	C₂²×#88322	Q₈	1	1	yes	characteristic
31	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₅₃	D₄	1	2	yes	—
35	C₂×C₈×D₅₃	D₄	1	2	yes	—
36	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
37	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
38	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
39	#13133156	—	2	4	no	—
40	#13133156	—	2	4	no	—
41	C₂×#11719908	—	2	1	no	—
42	C₂×#11719908	—	2	1	no	—
43	C₂×#11719908	C₂×C₄	1	2	yes	complemented
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₆×C₅₃	—	2	1	no	—
46	C₂×C₁₆×C₅₃	—	2	1	no	—
47	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
48	C₂³×D₅₃	D₈	1	1	yes	characteristic
49	C₂²×#6157	—	4	4	no	—
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×Dic₅₃	Q₁₆	1	1	yes	characteristic
52	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
56	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
57	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
58	C₂×C₄×D₅₃	Q₁₆	1	1	yes	characteristic
59	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²⋊#6157	—	4	4	no	—
61	C₂²⋊#6157	—	4	4	no	—
62	C₄⋊#6157	—	2	8	no	—
63	C₄⋊#6157	—	4	4	no	—
64	C₄×#6157	—	4	4	no	—
65	C₄×#6157	—	4	4	no	—
66	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
73	C₂×#88322	C₄⋊C₄	1	2	yes	—
74	C₂×#88322	C₄⋊C₄	1	2	yes	—
75	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
76	C₈×D₅₃	C₄²	1	2	yes	—
77	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
78	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
79	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
80	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
81	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
82	C₈×D₅₃	C₂²×C₄	1	2	yes	—
83	C₈×D₅₃	C₂×D₄	1	2	yes	—
84	C₁₆×C₅₃	—	2	2	no	—
85	#11719908	—	2	2	no	—
86	C₂³×C₅₃	#107	1	1	yes	characteristic
87	C₂²×D₅₃	—	4	2	no	—
88	C₂²×D₅₃	#102	1	2	yes	—
89	C₂²×D₅₃	#102	1	2	yes	—
90	C₂²×D₅₃	#107	1	1	yes	characteristic
91	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
92	C₂×#6157	—	4	8	no	—
93	C₂×#6157	—	8	8	no	—
94	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×Dic₅₃	#103	1	2	yes	—
101	C₂×Dic₅₃	#103	1	2	yes	—
102	C₂×Dic₅₃	#107	1	1	yes	characteristic
103	C₂×Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
104	C₄×D₅₃	#95	1	2	yes	—
105	C₄×D₅₃	#102	1	2	yes	—
106	C₄×D₅₃	#102	1	2	yes	—
107	C₄×D₅₃	#103	1	2	yes	—
108	C₄×D₅₃	#103	1	2	yes	—
109	C₄×D₅₃	#106	1	2	yes	—
110	C₄×D₅₃	C₂×C₂²⋊C₄	1	2	yes	—
111	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
112	C₈×C₅₃	#95	1	1	yes	characteristic
113	C₈×C₅₃	#95	1	1	yes	characteristic
114	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
115	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	#88322	#95	1	2	yes	—
117	#88322	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×#1384	—	53	1	no	maximal, Sylow
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	#340	1	1	yes	characteristic
121	C₂²×C₅₃	#340	1	1	yes	characteristic
122	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
123	C₂×D₅₃	—	2	4	no	—
124	C₂×D₅₃	—	4	2	no	—
125	C₂×D₅₃	—	4	1	no	—
126	C₂×D₅₃	—	4	1	no	—
127	C₂×D₅₃	#340	1	2	yes	—
128	C₂×D₅₃	C₂×#102	1	2	yes	—
129	#6157	—	8	8	no	—
130	C₄×C₅₃	#340	1	1	yes	characteristic
131	C₄×C₅₃	#340	1	1	yes	characteristic
132	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
133	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
134	Dic₅₃	#340	1	2	yes	—
135	Dic₅₃	C₂×#103	1	2	yes	—
136	C₂²×M₃₂	—	53	1	no	—
137	C₂×#427	—	53	2	no	—
138	#1384	—	53	4	no	—
139	C₂×C₅₃	—	2	1	no	—
140	C₂×C₅₃	—	2	1	no	—
141	C₂×C₅₃	#1384	1	2	yes	complemented
142	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
143	D₅₃	—	2	2	no	—
144	D₅₃	—	2	2	no	—
145	C₂³×C₈	—	53	1	no	—
146	C₂×#117	—	106	2	no	—
147	C₂×M₃₂	—	53	2	no	—
148	C₂×M₃₂	—	53	2	no	—
149	C₂×M₃₂	—	106	2	no	—
150	C₂×M₃₂	—	106	2	no	—
151	C₂²×C₁₆	—	53	1	no	—
152	C₂²×C₁₆	#6157	1	1	yes	characteristic, complemented
153	#427	—	53	4	no	—
154	#427	—	53	4	no	—
155	#427	—	106	2	no	—
156	#427	—	106	2	no	—
157	C₂×#107	—	106	2	no	—
158	C₂×#106	—	53	4	no	—
159	C₅₃	C₂×#1384	1	1	yes	characteristic, complemented, Sylow
160	C₂³×C₄	—	53	1	no	—
161	C₂²×C₈	—	53	1	no	—
162	C₂²×C₈	—	53	2	no	—
163	C₂²×C₈	—	53	2	no	—
164	C₂²×C₈	—	53	1	no	—
165	C₂²×C₈	—	53	1	no	—
166	C₂²×C₈	—	53	2	no	—
167	C₂²×C₈	—	53	2	no	—
168	C₂²×C₈	—	53	1	no	—
169	C₂²×C₈	—	53	1	no	—
170	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
171	#117	—	106	4	no	—
172	#117	—	106	4	no	—
173	#117	—	212	2	no	—
174	#117	—	212	2	no	—
175	C₂×C₄⋊C₄	—	106	4	no	—
176	C₂×C₂²⋊C₄	—	212	2	no	—
177	C₂×C₄²	—	212	2	no	—
178	M₃₂	—	106	4	no	—
179	M₃₂	—	106	4	no	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	53	2	no	—
183	C₂×C₁₆	—	106	1	no	—
184	C₂×C₁₆	—	106	1	no	—
185	C₂×C₁₆	C₂×#6157	1	2	yes	complemented
186	#107	—	106	4	no	—
187	#107	—	106	4	no	—
188	#106	—	53	8	no	—
189	#106	—	106	4	no	—
190	C₂⁴	—	53	1	no	—
191	C₂²×C₄	—	53	1	no	—
192	C₂²×C₄	—	53	2	no	—
193	C₂²×C₄	—	53	2	no	—
194	C₂²×C₄	—	53	2	no	—
195	C₂²×C₄	—	53	2	no	—
196	C₂²×C₄	—	53	1	no	—
197	C₂²×C₄	—	53	1	no	—
198	C₂²×C₄	—	53	1	no	—
199	C₂²×C₄	—	53	1	no	—
200	C₂²×C₄	—	212	4	no	—
201	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
202	C₂×C₈	—	53	2	no	—
203	C₂×C₈	—	53	2	no	—
204	C₂×C₈	—	53	1	no	—
205	C₂×C₈	—	53	1	no	—
206	C₂×C₈	—	53	2	no	—
207	C₂×C₈	—	53	2	no	—
208	C₂×C₈	—	53	2	no	—
209	C₂×C₈	—	53	2	no	—
210	C₂×C₈	—	53	2	no	—
211	C₂×C₈	—	53	2	no	—
212	C₂×C₈	—	53	2	no	—
213	C₂×C₈	—	53	2	no	—
214	C₂×C₈	C₄×#6157	1	1	yes	characteristic
215	C₂×C₈	C₄×#6157	1	1	yes	characteristic
216	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
217	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
219	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
220	C₄⋊C₄	—	106	8	no	—
221	C₄⋊C₄	—	212	4	no	—
222	C₂²⋊C₄	—	212	4	no	—
223	C₂²⋊C₄	—	212	4	no	—
224	C₄²	—	212	4	no	—
225	C₄²	—	212	4	no	—
226	C₁₆	—	2	2	no	—
227	C₁₆	—	106	2	no	—
228	C₂³	—	53	2	no	—
229	C₂³	—	53	1	no	—
230	C₂³	—	53	1	no	—
231	C₂³	—	53	2	no	—
232	C₂³	—	212	2	no	—
233	C₂³	#13133286	1	1	yes	characteristic
234	C₂×C₄	—	53	1	no	—
235	C₂×C₄	—	53	1	no	—
236	C₂×C₄	—	53	2	no	—
237	C₂×C₄	—	53	2	no	—
238	C₂×C₄	—	53	2	no	—
239	C₂×C₄	—	53	2	no	—
240	C₂×C₄	—	53	2	no	—
241	C₂×C₄	—	53	2	no	—
242	C₂×C₄	—	53	2	no	—
243	C₂×C₄	—	53	2	no	—
244	C₂×C₄	—	53	2	no	—
245	C₂×C₄	—	53	2	no	—
246	C₂×C₄	—	424	8	no	—
247	C₂×C₄	—	212	8	no	—
248	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
249	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
250	C₂×C₄	#13133285	1	1	yes	characteristic
251	C₂×C₄	#13133285	1	1	yes	characteristic
252	C₂×C₄	#13133286	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
254	C₈	—	53	2	no	—
255	C₈	—	53	2	no	—
256	C₈	(C₂×#6157)⋊C₄	1	1	yes	characteristic
257	C₈	(C₂×#6157)⋊C₄	1	1	yes	characteristic
258	C₈	C₂×C₄×#6157	1	1	yes	characteristic
259	C₈	C₂×C₂²⋊#6157	1	1	yes	characteristic
260	C₂²	—	2	2	no	—
261	C₂²	—	53	2	no	—
262	C₂²	—	53	2	no	—
263	C₂²	—	106	4	no	—
264	C₂²	—	212	1	no	—
265	C₂²	—	212	1	no	—
266	C₂²	—	212	2	no	—
267	C₂²	#16811497	1	1	yes	characteristic
268	C₂²	#16811497	1	1	yes	characteristic
269	C₂²	C₂×#13133286	1	1	yes	characteristic
270	C₄	—	53	2	no	—
271	C₄	—	53	2	no	—
272	C₄	—	424	8	no	—
273	C₄	#16811497	1	1	yes	characteristic
274	C₄	#16811497	1	1	yes	characteristic
275	C₄	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic
276	C₄	C₂×#13133285	1	1	yes	characteristic
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	106	2	no	—
280	C₂	—	106	2	no	—
281	C₂	#20289783	1	2	yes	direct product factor
282	C₂	C₂×#16811497	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133285
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	#340
prime radical	C₂³×D₅₃	D₈
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	C₂×#1384
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20289783	C₂	direct (N × H)
7. #16811633	C₄	semidirect
10. C₂×#13133285	C₄	semidirect
14. C₂×#13133156	C₄	semidirect
16. C₂²×#11719908	C₄	semidirect
17. C₂²×C₁₆×C₅₃	C₄	semidirect
28. #13133285	C₂×C₄	semidirect
43. C₂×#11719908	C₂×C₄	semidirect
47. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
141. C₂×C₅₃	#1384	semidirect
152. C₂²×C₁₆	#6157	semidirect
159. C₅₃	C₂×#1384	semidirect
185. C₂×C₁₆	C₂×#6157	semidirect
281. C₂	#20289783	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×#16811633	2
#20289783	2
C₂²×#13133156	2
C₂×#1384	53

Subgroups — C2×#20289783 #28310298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20289783 #28310298