Subgroups — #20288479

Order 6784.

1660 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	C₂×#13133262	C₂	1	2	yes	maximal, complemented
2	#16811277	C₂	1	4	yes	maximal, complemented
3	#16811268	C₂	1	8	yes	maximal, complemented
4	D₄×D₂₁₂	C₂	1	4	yes	maximal, complemented
5	#16811204	C₂	1	2	yes	maximal, complemented
6	#16811185	C₂	1	4	yes	maximal, complemented
7	#16811173	C₂	1	2	yes	maximal, complemented
8	C₂×#13133191	C₂	1	4	yes	maximal, complemented
9	C₂×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#13133262	—	2	16	no	—
11	C₂×#11719945	C₂²	1	4	yes	complemented
12	C₂×#11719947	C₂²	1	8	yes	complemented
13	C₂²×D₂₁₂	C₂²	1	4	yes	complemented
14	C₂²×C₂²⋊D₅₃	C₂²	1	8	yes	complemented
15	C₂×D₄×D₅₃	C₂²	1	8	yes	complemented
16	#13133217	C₂²	1	2	yes	complemented
17	#13133214	C₂²	1	2	yes	complemented
18	#13133205	C₂²	1	4	yes	complemented
19	#13133201	C₂²	1	16	yes	complemented
20	#13133197	C₂²	1	8	yes	complemented
21	#13133192	C₂²	1	8	yes	complemented
22	#13133191	C₂²	1	8	yes	complemented
23	#13133191	C₂²	1	8	yes	complemented
24	#13133191	C₂²	1	4	yes	complemented
25	#13133191	C₂²	1	8	yes	complemented
26	#13133191	C₂²	1	4	yes	complemented
27	#13133187	C₂²	1	16	yes	complemented
28	C₄×D₂₁₂	C₂²	1	4	yes	complemented
29	C₄×D₂₁₂	C₂²	1	4	yes	complemented
30	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	complemented
31	C₂×#11719921	C₂²	1	8	yes	complemented
32	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
33	C₂³×C₄×C₅₃	C₂²	1	2	yes	complemented
34	C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
35	C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
36	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
37	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
38	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
39	C₂⁴×C₅₃	C₂³	1	2	yes	—
40	C₂³×D₅₃	C₂³	1	8	yes	—
41	C₂×D₄×C₅₃	D₄	1	8	yes	complemented
42	C₂×D₄×C₅₃	C₂³	1	4	yes	—
43	C₂²×C₄×C₅₃	—	4	4	no	—
44	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
46	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
49	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₅₃	—	2	32	no	—
51	C₂×C₂²⋊D₅₃	C₂³	1	16	yes	—
52	C₂×C₂²⋊D₅₃	C₂³	1	8	yes	—
53	C₂²×Dic₅₃	C₂³	1	8	yes	—
54	#11719947	—	2	32	no	—
55	D₄×D₅₃	—	2	32	no	—
56	#11719945	—	2	16	no	—
57	C₂×D₂₁₂	—	4	8	no	—
58	C₂×D₂₁₂	C₂³	1	4	yes	—
59	C₂×D₂₁₂	C₂³	1	4	yes	—
60	C₂×D₂₁₂	C₂³	1	4	yes	—
61	C₂×D₂₁₂	C₂³	1	8	yes	—
62	C₂×C₄×D₅₃	C₂³	1	8	yes	—
63	C₂×Dic₁₀₆	C₂³	1	4	yes	—
64	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
65	C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
66	C₄²×C₅₃	C₂³	1	2	yes	—
67	C₄²×C₅₃	C₂³	1	2	yes	—
68	#11719921	C₂³	1	8	yes	—
69	#11719921	C₂³	1	16	yes	—
70	#11719921	C₂³	1	8	yes	—
71	C₄⋊Dic₅₃	C₂³	1	4	yes	—
72	C₄⋊Dic₅₃	C₂³	1	8	yes	—
73	C₄⋊Dic₅₃	C₂³	1	4	yes	—
74	C₂³×C₅₃	—	4	2	no	—
75	C₂³×C₅₃	—	4	2	no	—
76	C₂³×C₅₃	C₂×D₄	1	8	yes	complemented
77	C₂³×C₅₃	C₂⁴	1	4	yes	—
78	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
79	C₂²×D₅₃	—	2	16	no	—
80	C₂²×D₅₃	—	4	8	no	—
81	C₂²×D₅₃	—	4	8	no	—
82	C₂²×D₅₃	C₂⁴	1	8	yes	—
83	D₄×C₅₃	C₂×D₄	1	16	yes	complemented
84	C₂×C₄×C₅₃	—	2	8	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	4	4	no	—
88	C₂×C₄×C₅₃	—	4	1	no	—
89	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	complemented
90	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
91	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
92	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
94	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
95	C₂²⋊D₅₃	—	2	16	no	—
96	C₂²⋊D₅₃	—	2	32	no	—
97	C₂×Dic₅₃	—	2	16	no	—
98	C₂×Dic₅₃	C₂⁴	1	8	yes	—
99	D₂₁₂	—	2	8	no	—
100	D₂₁₂	—	2	8	no	—
101	D₂₁₂	—	4	8	no	—
102	D₂₁₂	—	4	4	no	—
103	C₄×D₅₃	—	2	16	no	—
104	Dic₁₀₆	—	2	8	no	—
105	C₂²×C₅₃	—	2	1	no	—
106	C₂²×C₅₃	—	2	4	no	—
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	—	2	4	no	—
109	C₂²×C₅₃	—	4	8	no	—
110	C₂²×C₅₃	C₂²×D₄	1	8	yes	—
111	C₂²×C₅₃	C₂²×D₄	1	2	yes	—
112	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
113	C₂×D₅₃	—	2	8	no	—
114	C₂×D₅₃	—	2	8	no	—
115	C₂×D₅₃	—	2	8	no	—
116	C₂×D₅₃	—	8	16	no	—
117	C₄×C₅₃	—	2	4	no	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	2	no	—
120	C₄×C₅₃	C₂²×D₄	1	4	yes	—
121	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	complemented
122	Dic₅₃	—	2	8	no	—
123	#3496	—	53	1	no	maximal, Sylow
124	C₂×C₅₃	—	2	8	no	—
125	C₂×C₅₃	—	2	2	no	—
126	C₂×C₅₃	C₂³×D₄	1	1	yes	characteristic
127	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
128	C₂×C₅₃	#585	1	1	yes	characteristic
129	D₅₃	—	4	8	no	—
130	C₂×H(5,𝔽₂)	—	53	2	no	—
131	#547	—	53	4	no	—
132	#546	—	53	8	no	—
133	D₄²	—	53	4	no	—
134	#537	—	53	2	no	—
135	#534	—	53	4	no	—
136	#532	—	53	2	no	—
137	C₂×#121	—	53	4	no	—
138	C₂×C₄×D₄	D₅₃	1	1	yes	characteristic, complemented
139	C₅₃	#3496	1	1	yes	characteristic, complemented, Sylow
140	H(5,𝔽₂)	—	106	16	no	—
141	C₂×#41	—	53	8	no	—
142	C₂×#41	—	53	4	no	—
143	C₂²×D₄	—	53	8	no	—
144	C₂²×D₄	—	53	8	no	—
145	C₂²×D₄	—	53	4	no	—
146	C₂²×D₄	C₂×D₅₃	1	1	yes	characteristic
147	C₂³×C₄	C₂×D₅₃	1	2	yes	complemented
148	C₄⋊Q₈	—	53	2	no	—
149	#127	—	53	2	no	—
150	#124	—	53	4	no	—
151	#123	—	53	16	no	—
152	C₂²⋊Q₈	—	53	8	no	—
153	#121	—	53	4	no	—
154	#121	—	53	8	no	—
155	#121	—	53	8	no	—
156	#121	—	53	8	no	—
157	#121	—	53	4	no	—
158	#121	—	53	8	no	—
159	#120	—	53	16	no	—
160	C₄×D₄	—	53	4	no	—
161	C₄×D₄	—	53	4	no	—
162	C₄×D₄	C₂×D₅₃	1	4	yes	complemented
163	C₄×D₄	C₂×D₅₃	1	4	yes	complemented
164	C₂×C₄⋊C₄	—	53	4	no	—
165	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
166	C₂×C₂²⋊C₄	—	53	8	no	—
167	C₂×C₂²⋊C₄	C₂×D₅₃	1	2	yes	complemented
168	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
169	C₂⁴	—	53	8	no	—
170	C₂⁴	C₂²×D₅₃	1	2	yes	—
171	#41	—	106	32	no	—
172	#41	—	106	16	no	—
173	C₂×Q₈	—	53	4	no	—
174	C₂×D₄	—	53	16	no	—
175	C₂×D₄	—	53	8	no	—
176	C₂×D₄	—	53	4	no	—
177	C₂×D₄	—	53	4	no	—
178	C₂×D₄	—	53	4	no	—
179	C₂×D₄	—	53	8	no	—
180	C₂×D₄	—	106	32	no	—
181	C₂×D₄	—	106	32	no	—
182	C₂×D₄	—	212	8	no	—
183	C₂×D₄	D₂₁₂	1	8	yes	complemented
184	C₂×D₄	C₂²×D₅₃	1	4	yes	—
185	C₂²×C₄	—	4	4	no	—
186	C₂²×C₄	—	53	8	no	—
187	C₂²×C₄	—	53	8	no	—
188	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
190	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
192	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
193	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
194	C₄⋊C₄	—	53	4	no	—
195	C₄⋊C₄	—	53	8	no	—
196	C₄⋊C₄	—	53	4	no	—
197	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
198	C₂²⋊C₄	—	53	8	no	—
199	C₂²⋊C₄	—	53	16	no	—
200	C₂²⋊C₄	—	53	8	no	—
201	C₂²⋊C₄	C₂²×D₅₃	1	8	yes	—
202	C₄²	C₂²×D₅₃	1	2	yes	—
203	C₄²	C₂²×D₅₃	1	2	yes	—
204	C₂³	—	4	2	no	—
205	C₂³	—	4	2	no	—
206	C₂³	—	53	8	no	—
207	C₂³	—	106	16	no	—
208	C₂³	—	212	8	no	—
209	C₂³	—	212	8	no	—
210	C₂³	C₂×D₂₁₂	1	8	yes	complemented
211	C₂³	C₂³×D₅₃	1	4	yes	—
212	C₂³	C₂³×D₅₃	1	1	yes	characteristic
213	Q₈	—	106	8	no	—
214	D₄	—	106	32	no	—
215	D₄	—	106	16	no	—
216	D₄	—	106	8	no	—
217	D₄	—	106	8	no	—
218	D₄	—	212	4	no	—
219	D₄	—	212	8	no	—
220	D₄	C₂×D₂₁₂	1	16	yes	complemented
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	53	8	no	—
227	C₂×C₄	—	106	16	no	—
228	C₂×C₄	—	106	16	no	—
229	C₂×C₄	C₂×D₂₁₂	1	4	yes	complemented
230	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
231	C₂×C₄	C₂³×D₅₃	1	2	yes	—
232	C₂×C₄	C₂³×D₅₃	1	4	yes	—
233	C₂×C₄	C₂³×D₅₃	1	2	yes	—
234	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
235	C₂²	—	2	1	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	4	8	no	—
240	C₂²	—	106	8	no	—
241	C₂²	—	106	8	no	—
242	C₂²	—	106	8	no	—
243	C₂²	—	424	16	no	—
244	C₂²	C₂²×D₂₁₂	1	8	yes	—
245	C₂²	C₂²×D₂₁₂	1	2	yes	—
246	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	4	no	—
249	C₄	—	2	2	no	—
250	C₄	—	106	8	no	—
251	C₄	C₂²×D₂₁₂	1	4	yes	—
252	C₄	#13133262	1	2	yes	complemented
253	C₂	—	2	2	no	—
254	C₂	—	2	8	no	—
255	C₂	—	212	8	no	—
256	C₂	C₂³×D₂₁₂	1	1	yes	characteristic
257	C₂	C₂×#13133262	1	1	yes	characteristic
258	C₂	#16811459	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₂×C₄×D₄×C₅₃	C₂
hypercenter	C₂×C₄×D₄	D₅₃
hypocenter	C₅₃	#3496
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#13133262	C₂	semidirect
2. #16811277	C₂	semidirect
3. #16811268	C₂	semidirect
4. D₄×D₂₁₂	C₂	semidirect
5. #16811204	C₂	semidirect
6. #16811185	C₂	semidirect
7. #16811173	C₂	semidirect
8. C₂×#13133191	C₂	semidirect
9. C₂×C₄×D₄×C₅₃	C₂	semidirect
11. C₂×#11719945	C₂²	semidirect
12. C₂×#11719947	C₂²	semidirect
13. C₂²×D₂₁₂	C₂²	semidirect
14. C₂²×C₂²⋊D₅₃	C₂²	semidirect
15. C₂×D₄×D₅₃	C₂²	semidirect
16. #13133217	C₂²	semidirect
17. #13133214	C₂²	semidirect
18. #13133205	C₂²	semidirect
19. #13133201	C₂²	semidirect
20. #13133197	C₂²	semidirect
21. #13133192	C₂²	semidirect
22. #13133191	C₂²	semidirect
23. #13133191	C₂²	semidirect
24. #13133191	C₂²	semidirect
25. #13133191	C₂²	semidirect
26. #13133191	C₂²	semidirect
27. #13133187	C₂²	semidirect
28. C₄×D₂₁₂	C₂²	semidirect
29. C₄×D₂₁₂	C₂²	semidirect
30. C₂×C₄⋊Dic₅₃	C₂²	semidirect
31. C₂×#11719921	C₂²	semidirect
33. C₂³×C₄×C₅₃	C₂²	semidirect
34. C₄×D₄×C₅₃	C₂²	semidirect
35. C₄×D₄×C₅₃	C₂²	semidirect
36. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
37. C₂×C₂²⋊C₄×C₅₃	C₂²	semidirect
38. C₂×C₄²×C₅₃	C₂²	semidirect
41. C₂×D₄×C₅₃	D₄	semidirect
76. C₂³×C₅₃	C₂×D₄	semidirect
83. D₄×C₅₃	C₂×D₄	semidirect
89. C₂×C₄×C₅₃	C₂×D₄	semidirect
121. C₄×C₅₃	H(5,𝔽₂)	semidirect
138. C₂×C₄×D₄	D₅₃	semidirect
139. C₅₃	#3496	semidirect
147. C₂³×C₄	C₂×D₅₃	semidirect
162. C₄×D₄	C₂×D₅₃	semidirect
163. C₄×D₄	C₂×D₅₃	semidirect
165. C₂×C₄⋊C₄	C₂×D₅₃	semidirect
167. C₂×C₂²⋊C₄	C₂×D₅₃	semidirect
168. C₂×C₄²	C₂×D₅₃	semidirect
183. C₂×D₄	D₂₁₂	semidirect
210. C₂³	C₂×D₂₁₂	semidirect
220. D₄	C₂×D₂₁₂	semidirect
229. C₂×C₄	C₂×D₂₁₂	semidirect
252. C₄	#13133262	semidirect