Subgroups — C₂×#16811389 #20289424

Order 6784.

1460 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11719945	C₂	1	1	yes	maximal, characteristic
2	#16811389	C₂	1	16	yes	maximal, direct product factor
3	C₂×#13133225	C₂	1	4	yes	maximal, complemented
4	C₂×#13133223	C₂	1	4	yes	maximal, complemented
5	C₂²×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×M₁₆×D₅₃	C₂	1	4	yes	maximal, complemented
7	C₂²×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719945	C₄	1	2	yes	—
9	C₂×#11719945	C₄	1	2	yes	—
10	C₂×#11719945	C₄	1	8	yes	—
11	C₂×#11719945	C₂²	1	2	yes	—
12	C₂×#11719945	C₂²	1	2	yes	—
13	C₂×#11719945	C₂²	1	8	yes	—
14	C₂²×Dic₁₀₆	C₄	1	1	yes	characteristic
15	C₂²×D₂₁₂	C₄	1	1	yes	characteristic
16	C₂²×C₂²⋊D₅₃	C₄	1	2	yes	—
17	C₂²×C₄×D₅₃	C₂²	1	2	yes	—
18	#13133225	C₂²	1	32	yes	complemented
19	#13133223	C₂²	1	32	yes	complemented
20	C₂×#11719917	C₂²	1	2	yes	—
21	C₂×#11719917	C₂²	1	2	yes	complemented
22	C₂×#11719917	C₂²	1	8	yes	complemented
23	C₂×#11719912	C₂²	1	8	yes	complemented
24	M₁₆×D₅₃	C₂²	1	32	yes	complemented
25	C₂²×#88322	C₂²	1	2	yes	complemented
26	C₂×C₈×D₅₃	C₂²	1	8	yes	complemented
27	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
29	C₂×M₁₆×C₅₃	C₂²	1	2	yes	complemented
30	C₂×M₁₆×C₅₃	C₂²	1	8	yes	complemented
31	C₂²×C₈×C₅₃	C₂²	1	2	yes	complemented
32	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×D₅₃	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
38	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
42	C₂×C₂²⋊D₅₃	C₂×C₄	1	4	yes	—
43	C₂×C₂²⋊D₅₃	C₂×C₄	1	4	yes	—
44	C₂×C₂²⋊D₅₃	C₂×C₄	1	16	yes	—
45	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
46	#11719945	C₂×C₄	1	8	yes	—
47	#11719945	C₂×C₄	1	8	yes	—
48	#11719945	C₂×C₄	1	16	yes	—
49	#11719945	C₂×C₄	1	16	yes	—
50	#11719945	C₂³	1	8	yes	—
51	#11719945	C₂³	1	8	yes	—
52	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
53	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
54	C₂×D₂₁₂	C₂×C₄	1	8	yes	—
55	C₂×C₄×D₅₃	C₂×C₄	1	4	yes	—
56	C₂×C₄×D₅₃	C₂×C₄	1	8	yes	—
57	C₂×C₄×D₅₃	C₂³	1	4	yes	—
58	C₂×C₄×D₅₃	C₂³	1	8	yes	—
59	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
60	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
61	C₂×Dic₁₀₆	C₂×C₄	1	8	yes	—
62	M₁₆×C₅₃	C₂³	1	8	yes	—
63	M₁₆×C₅₃	C₂³	1	8	yes	complemented
64	C₂×C₈×C₅₃	C₂³	1	4	yes	—
65	C₂×C₈×C₅₃	C₂³	1	8	yes	complemented
66	#11719917	C₂³	1	8	yes	—
67	#11719917	C₂³	1	8	yes	complemented
68	C₂×#88322	C₂³	1	4	yes	—
69	C₂×#88322	C₂³	1	8	yes	complemented
70	#11719912	C₂³	1	32	yes	complemented
71	C₈×D₅₃	C₂³	1	32	yes	complemented
72	C₂³×C₅₃	—	4	2	no	—
73	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂³×C₅₃	C₂²×C₄	1	4	yes	—
77	C₂²×D₅₃	—	4	4	no	—
78	C₂²×D₅₃	C₂²×C₄	1	4	yes	—
79	C₂²×D₅₃	C₂²×C₄	1	8	yes	—
80	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
81	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
88	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
89	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
90	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
91	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
92	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
93	C₂²⋊D₅₃	C₂²×C₄	1	16	yes	—
94	C₂²⋊D₅₃	C₂²×C₄	1	16	yes	—
95	C₂×Dic₅₃	C₂²×C₄	1	8	yes	—
96	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
97	D₂₁₂	C₂²×C₄	1	8	yes	—
98	D₂₁₂	C₂²×C₄	1	8	yes	—
99	C₄×D₅₃	C₂²×C₄	1	8	yes	—
100	C₄×D₅₃	C₂²×C₄	1	16	yes	—
101	C₄×D₅₃	C₂⁴	1	8	yes	—
102	Dic₁₀₆	C₂²×C₄	1	8	yes	—
103	Dic₁₀₆	C₂²×C₄	1	8	yes	—
104	C₈×C₅₃	C₂⁴	1	8	yes	—
105	#88322	C₂⁴	1	8	yes	—
106	C₂²×C₅₃	—	2	2	no	—
107	C₂²×C₅₃	—	2	2	no	—
108	C₂²×C₅₃	—	2	2	no	—
109	C₂²×C₅₃	—	4	4	no	—
110	C₂²×C₅₃	C₂³×C₄	1	2	yes	—
111	C₂²×C₅₃	C₂³×C₄	1	2	yes	—
112	C₂²×C₅₃	C₂³×C₄	1	2	yes	—
113	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×D₅₃	—	2	8	no	—
115	C₂×D₅₃	—	4	8	no	—
116	C₂×D₅₃	C₂³×C₄	1	8	yes	—
117	C₄×C₅₃	C₂³×C₄	1	2	yes	—
118	C₄×C₅₃	C₂³×C₄	1	2	yes	—
119	C₄×C₅₃	C₂³×C₄	1	2	yes	—
120	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
121	C₄×C₅₃	C₂⁵	1	1	yes	characteristic
122	Dic₅₃	C₂³×C₄	1	8	yes	—
123	C₂×#568	—	53	1	no	maximal, Sylow
124	C₂×C₅₃	—	2	2	no	—
125	C₂×C₅₃	—	2	2	no	—
126	C₂×C₅₃	—	2	2	no	—
127	C₂×C₅₃	#568	1	2	yes	complemented
128	C₂×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
129	D₅₃	—	2	8	no	—
130	C₂²×#41	—	53	1	no	—
131	#568	—	53	16	no	—
132	C₂×#131	—	53	4	no	—
133	C₂×#131	—	53	4	no	—
134	C₂²×M₁₆	—	53	1	no	—
135	C₂²×M₁₆	—	53	4	no	—
136	C₂²×M₁₆	D₅₃	1	1	yes	characteristic, complemented
137	C₅₃	C₂×#568	1	1	yes	characteristic, complemented, Sylow
138	C₂×#41	—	53	2	no	—
139	C₂×#41	—	53	2	no	—
140	C₂×#41	—	53	8	no	—
141	C₂×#41	—	53	2	no	—
142	C₂×#41	—	53	2	no	—
143	C₂×#41	—	53	8	no	—
144	C₂²×Q₈	—	53	1	no	—
145	C₂²×D₄	—	53	2	no	—
146	C₂²×D₄	—	53	1	no	—
147	C₂³×C₄	—	53	2	no	—
148	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
149	#131	—	53	32	no	—
150	#131	—	53	32	no	—
151	C₂×M₁₆	—	53	2	no	—
152	C₂×M₁₆	—	53	2	no	—
153	C₂×M₁₆	—	53	8	no	—
154	C₂×M₁₆	—	53	32	no	—
155	C₂×M₁₆	—	53	8	no	—
156	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
157	C₂×M₁₆	C₂×D₅₃	1	2	yes	complemented
158	C₂×M₁₆	C₂×D₅₃	1	8	yes	complemented
159	C₂²×C₈	—	53	2	no	—
160	C₂²×C₈	—	53	8	no	—
161	C₂²×C₈	C₂×D₅₃	1	2	yes	complemented
162	C₂⁴	—	53	2	no	—
163	C₂⁴	C₄×D₅₃	1	1	yes	characteristic
164	#41	—	53	8	no	—
165	#41	—	53	8	no	—
166	#41	—	53	16	no	—
167	#41	—	53	16	no	—
168	#41	—	53	8	no	—
169	#41	—	53	8	no	—
170	C₂×Q₈	—	53	2	no	—
171	C₂×Q₈	—	53	2	no	—
172	C₂×Q₈	—	53	8	no	—
173	C₂×D₄	—	53	4	no	—
174	C₂×D₄	—	53	4	no	—
175	C₂×D₄	—	53	16	no	—
176	C₂×D₄	—	53	2	no	—
177	C₂×D₄	—	53	2	no	—
178	C₂×D₄	—	53	8	no	—
179	C₂²×C₄	—	53	2	no	—
180	C₂²×C₄	—	53	4	no	—
181	C₂²×C₄	—	53	8	no	—
182	C₂²×C₄	—	53	8	no	—
183	C₂²×C₄	—	53	4	no	—
184	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
186	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
187	C₂²×C₄	C₄×D₅₃	1	4	yes	—
188	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
189	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
192	M₁₆	—	53	8	no	—
193	M₁₆	—	53	8	no	—
194	M₁₆	—	53	32	no	—
195	M₁₆	C₂²×D₅₃	1	8	yes	—
196	M₁₆	C₂²×D₅₃	1	8	yes	complemented
197	C₂×C₈	—	53	8	no	—
198	C₂×C₈	—	53	4	no	—
199	C₂×C₈	—	53	32	no	—
200	C₂×C₈	C₂²×D₅₃	1	4	yes	—
201	C₂×C₈	C₂²×D₅₃	1	8	yes	complemented
202	C₂³	—	4	2	no	—
203	C₂³	—	53	4	no	—
204	C₂³	—	53	8	no	—
205	C₂³	—	212	4	no	—
206	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
207	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
208	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
209	C₂³	C₂×C₄×D₅₃	1	4	yes	—
210	Q₈	—	53	8	no	—
211	Q₈	—	53	8	no	—
212	D₄	—	53	16	no	—
213	D₄	—	53	16	no	—
214	D₄	—	53	8	no	—
215	D₄	—	53	8	no	—
216	C₂×C₄	—	53	8	no	—
217	C₂×C₄	—	53	4	no	—
218	C₂×C₄	—	53	16	no	—
219	C₂×C₄	—	53	8	no	—
220	C₂×C₄	—	53	8	no	—
221	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
222	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
223	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
224	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
227	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
228	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
229	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
230	C₂×C₄	C₂³×D₅₃	1	2	yes	—
231	C₂×C₄	C₂³×D₅₃	1	2	yes	—
232	C₂×C₄	C₂³×D₅₃	1	2	yes	—
233	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
234	C₈	—	53	8	no	—
235	C₈	C₂³×D₅₃	1	8	yes	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	4	4	no	—
240	C₂²	—	53	8	no	—
241	C₂²	—	106	8	no	—
242	C₂²	—	212	8	no	—
243	C₂²	C₂²×C₄×D₅₃	1	2	yes	—
244	C₂²	C₂²×C₄×D₅₃	1	2	yes	—
245	C₂²	C₂²×C₄×D₅₃	1	2	yes	—
246	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
247	C₄	—	53	8	no	—
248	C₄	C₂²×C₄×D₅₃	1	2	yes	—
249	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
250	C₄	C₂²×C₄×D₅₃	1	2	yes	—
251	C₄	C₂²×C₄×D₅₃	1	2	yes	—
252	C₄	C₂⁴×D₅₃	1	1	yes	characteristic
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	—	2	2	no	—
256	C₂	—	106	8	no	—
257	C₂	#16811389	1	2	yes	direct product factor
258	C₂	C₂³×C₄×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₅₃
commutator	C₂×C₅₃	C₂⁴×C₄
Frattini	C₄	C₂⁴×D₅₃
Fitting	C₂²×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂³×C₄
hypercenter	C₂²×M₁₆	D₅₃
hypocenter	C₅₃	C₂×#568
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. #16811389	C₂	direct (N × H)
3. C₂×#13133225	C₂	semidirect
4. C₂×#13133223	C₂	semidirect
5. C₂²×#11719917	C₂	semidirect
6. C₂×M₁₆×D₅₃	C₂	semidirect
7. C₂²×M₁₆×C₅₃	C₂	semidirect
18. #13133225	C₂²	semidirect
19. #13133223	C₂²	semidirect
21. C₂×#11719917	C₂²	semidirect
22. C₂×#11719917	C₂²	semidirect
23. C₂×#11719912	C₂²	semidirect
24. M₁₆×D₅₃	C₂²	semidirect
25. C₂²×#88322	C₂²	semidirect
26. C₂×C₈×D₅₃	C₂²	semidirect
29. C₂×M₁₆×C₅₃	C₂²	semidirect
30. C₂×M₁₆×C₅₃	C₂²	semidirect
31. C₂²×C₈×C₅₃	C₂²	semidirect
63. M₁₆×C₅₃	C₂³	semidirect
65. C₂×C₈×C₅₃	C₂³	semidirect
67. #11719917	C₂³	semidirect
69. C₂×#88322	C₂³	semidirect
70. #11719912	C₂³	semidirect
71. C₈×D₅₃	C₂³	semidirect
127. C₂×C₅₃	#568	semidirect
136. C₂²×M₁₆	D₅₃	semidirect
137. C₅₃	C₂×#568	semidirect
157. C₂×M₁₆	C₂×D₅₃	semidirect
158. C₂×M₁₆	C₂×D₅₃	semidirect
161. C₂²×C₈	C₂×D₅₃	semidirect
196. M₁₆	C₂²×D₅₃	semidirect
201. C₂×C₈	C₂²×D₅₃	semidirect
257. C₂	#16811389	direct (N × H)

Subgroups — C2×#16811389 #20289424

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811389 #20289424