Subgroups — #20289120

Order 6784.

1258 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811365	C₂	1	4	yes	maximal, complemented
2	#16811339	C₂	1	2	yes	maximal, complemented
3	#16811338	C₂	1	4	yes	maximal, complemented
4	D₄×D₂₁₂	C₂	1	4	yes	maximal, complemented
5	#16811258	C₂	1	2	yes	maximal, complemented
6	#16811189	C₂	1	8	yes	maximal, complemented
7	#16811187	C₂	1	4	yes	maximal, complemented
8	#16811173	C₂	1	2	yes	maximal, complemented
9	C₅₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11719945	C₂²	1	4	yes	complemented
11	C₂²×D₂₁₂	C₂²	1	4	yes	complemented
12	C₂×D₄×D₅₃	C₂²	1	8	yes	complemented
13	#13133217	C₂²	1	2	yes	complemented
14	#13133214	C₂²	1	2	yes	—
15	#13133213	C₂²	1	8	yes	complemented
16	#13133206	C₂²	1	8	yes	complemented
17	#13133205	C₂²	1	4	yes	complemented
18	#13133203	C₂²	1	16	yes	complemented
19	#13133192	C₂²	1	8	yes	complemented
20	#13133191	C₂²	1	8	yes	complemented
21	#13133191	C₂²	1	4	yes	complemented
22	#13133191	C₂²	1	4	yes	complemented
23	#13133188	C₂²	1	8	yes	complemented
24	#13133187	C₂²	1	16	yes	complemented
25	#13133185	C₂²	1	8	yes	complemented
26	#13133185	C₂²	1	8	yes	complemented
27	C₄×D₂₁₂	C₂²	1	4	yes	complemented
28	C₄×D₂₁₂	C₂²	1	4	yes	complemented
29	#13133176	C₂²	1	4	yes	complemented
30	#13133175	C₂²	1	8	yes	complemented
31	C₂×#41×C₅₃	C₂²	1	2	yes	complemented
32	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
33	#127×C₅₃	C₂²	1	1	yes	characteristic
34	#124×C₅₃	C₂²	1	2	yes	complemented
35	#121×C₅₃	C₂²	1	4	yes	complemented
36	C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
37	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
38	C₂³×D₅₃	C₂³	1	8	yes	—
39	#41×C₅₃	—	4	4	no	—
40	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
41	C₂×D₄×C₅₃	C₂³	1	4	yes	—
42	C₂×D₄×C₅₃	C₂³	1	4	yes	—
43	C₂×D₄×C₅₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
47	C₂×C₂²⋊D₅₃	C₂³	1	8	yes	—
48	C₂×C₂²⋊D₅₃	C₂³	1	16	yes	—
49	D₄×D₅₃	—	4	16	no	—
50	#11719945	—	4	8	no	—
51	C₂×D₂₁₂	—	4	8	no	—
52	C₂×D₂₁₂	C₂³	1	4	yes	—
53	C₂×D₂₁₂	C₂³	1	4	yes	—
54	C₂×D₂₁₂	C₂³	1	4	yes	—
55	C₂×D₂₁₂	C₂³	1	8	yes	—
56	C₂×C₄×D₅₃	C₂³	1	8	yes	—
57	C₂×Dic₁₀₆	C₂³	1	4	yes	—
58	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
59	C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
60	C₄²×C₅₃	C₂³	1	2	yes	—
61	C₄²×C₅₃	C₂³	1	1	yes	characteristic
62	C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	C₂²⋊Dic₅₃	C₂³	1	8	yes	—
64	#11719921	C₂³	1	16	yes	—
65	#11719921	C₂³	1	8	yes	—
66	#11719921	C₂³	1	8	yes	—
67	C₄⋊Dic₅₃	C₂³	1	4	yes	—
68	C₄⋊Dic₅₃	C₂³	1	4	yes	—
69	C₄×Dic₅₃	C₂³	1	8	yes	—
70	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
71	C₂³×C₅₃	C₂⁴	1	4	yes	—
72	C₂²×D₅₃	—	4	8	no	—
73	C₂²×D₅₃	—	4	16	no	—
74	C₂²×D₅₃	C₂⁴	1	8	yes	—
75	Q₈×C₅₃	—	4	2	no	—
76	D₄×C₅₃	—	2	8	no	—
77	D₄×C₅₃	—	4	4	no	—
78	D₄×C₅₃	—	4	2	no	—
79	C₂×C₄×C₅₃	—	2	8	no	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
83	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
84	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
86	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
87	C₂²⋊D₅₃	—	4	8	no	—
88	C₂²⋊D₅₃	—	4	16	no	—
89	C₂×Dic₅₃	C₂⁴	1	8	yes	—
90	D₂₁₂	—	2	8	no	—
91	D₂₁₂	—	2	8	no	—
92	D₂₁₂	—	4	8	no	—
93	D₂₁₂	—	4	4	no	—
94	C₄×D₅₃	—	2	16	no	—
95	Dic₁₀₆	—	4	4	no	—
96	C₂²×C₅₃	—	2	2	no	—
97	C₂²×C₅₃	—	2	8	no	—
98	C₂²×C₅₃	—	2	1	no	—
99	C₂²×C₅₃	—	2	4	no	—
100	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
101	C₂×D₅₃	—	2	8	no	—
102	C₂×D₅₃	—	2	8	no	—
103	C₂×D₅₃	—	2	8	no	—
104	C₂×D₅₃	—	8	16	no	—
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	2	4	no	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
109	C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
110	Dic₅₃	—	2	8	no	—
111	#3549	—	53	1	no	maximal, Sylow
112	C₂×C₅₃	—	4	4	no	—
113	C₂×C₅₃	—	4	1	no	—
114	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
115	C₂×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
116	D₅₃	—	4	8	no	—
117	H(3,𝔽₄)	—	53	4	no	—
118	#555	—	53	2	no	—
119	#555	—	53	4	no	—
120	D₄²	—	53	2	no	—
121	D₄²	—	53	4	no	—
122	#535	—	53	4	no	—
123	#535	—	53	8	no	—
124	#532	—	53	2	no	—
125	#532	D₅₃	1	1	yes	characteristic, complemented
126	C₅₃	#3549	1	1	yes	characteristic, complemented, Sylow
127	C₂×#41	—	53	4	no	—
128	C₂×#41	C₂×D₅₃	1	2	yes	complemented
129	C₂²×D₄	—	53	8	no	—
130	C₂²×D₄	—	53	4	no	—
131	C₄⋊Q₈	—	53	2	no	—
132	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
133	#127	—	53	8	no	—
134	#127	—	53	2	no	—
135	#127	C₂×D₅₃	1	1	yes	characteristic
136	#124	—	53	8	no	—
137	#124	—	53	16	no	—
138	#124	—	53	4	no	—
139	#124	C₂×D₅₃	1	2	yes	complemented
140	#121	—	53	8	no	—
141	#121	—	53	4	no	—
142	#121	—	53	4	no	—
143	#121	—	53	8	no	—
144	#121	—	53	8	no	—
145	#121	C₂×D₅₃	1	4	yes	complemented
146	#120	—	53	8	no	—
147	#120	—	53	8	no	—
148	#120	—	53	16	no	—
149	C₄×D₄	—	53	4	no	—
150	C₄×D₄	—	53	4	no	—
151	C₄×D₄	C₂×D₅₃	1	4	yes	complemented
152	#117	—	53	4	no	—
153	#117	—	53	8	no	—
154	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
155	C₂⁴	—	53	8	no	—
156	#41	—	4	4	no	—
157	#41	—	212	8	no	—
158	C₂×Q₈	—	53	4	no	—
159	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
160	C₂×D₄	—	53	16	no	—
161	C₂×D₄	—	53	8	no	—
162	C₂×D₄	—	53	4	no	—
163	C₂×D₄	—	53	4	no	—
164	C₂×D₄	—	53	4	no	—
165	C₂×D₄	—	53	8	no	—
166	C₂×D₄	—	212	8	no	—
167	C₂×D₄	—	212	16	no	—
168	C₂×D₄	C₂²×D₅₃	1	4	yes	—
169	C₂×D₄	C₂²×D₅₃	1	2	yes	—
170	C₂×D₄	C₂²×D₅₃	1	4	yes	—
171	C₂²×C₄	—	53	8	no	—
172	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
173	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
174	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
175	C₄⋊C₄	—	53	4	no	—
176	C₄⋊C₄	—	53	4	no	—
177	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
178	C₂²⋊C₄	—	53	8	no	—
179	C₂²⋊C₄	—	53	16	no	—
180	C₂²⋊C₄	—	53	8	no	—
181	C₂²⋊C₄	—	53	8	no	—
182	C₂²⋊C₄	C₂²×D₅₃	1	8	yes	—
183	C₄²	—	53	8	no	—
184	C₄²	C₂²×D₅₃	1	1	yes	characteristic
185	C₄²	C₂²×D₅₃	1	2	yes	—
186	C₄²	C₂²×D₅₃	1	1	yes	characteristic
187	C₂³	—	53	8	no	—
188	C₂³	—	212	8	no	—
189	C₂³	—	212	16	no	—
190	C₂³	C₂³×D₅₃	1	4	yes	—
191	C₂³	C₂³×D₅₃	1	1	yes	characteristic
192	Q₈	—	4	2	no	—
193	Q₈	—	212	4	no	—
194	D₄	—	2	8	no	—
195	D₄	—	4	2	no	—
196	D₄	—	4	4	no	—
197	D₄	—	106	8	no	—
198	D₄	—	106	8	no	—
199	D₄	—	212	8	no	—
200	D₄	—	212	4	no	—
201	D₄	—	212	16	no	—
202	D₄	—	212	8	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	53	8	no	—
207	C₂×C₄	—	106	16	no	—
208	C₂×C₄	C₂³×D₅₃	1	4	yes	—
209	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
210	C₂×C₄	C₂³×D₅₃	1	2	yes	—
211	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
212	C₂×C₄	C₂³×D₅₃	1	2	yes	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	8	no	—
217	C₂²	—	106	8	no	—
218	C₂²	—	106	8	no	—
219	C₂²	—	106	8	no	—
220	C₂²	—	424	16	no	—
221	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	4	no	—
224	C₄	—	2	2	no	—
225	C₄	—	106	8	no	—
226	C₄	#13133262	1	2	yes	—
227	C₄	#13133263	1	4	yes	—
228	C₂	—	4	4	no	—
229	C₂	—	4	1	no	—
230	C₂	—	212	8	no	—
231	C₂	C₂×#13133262	1	1	yes	characteristic
232	C₂	C₂×#13133263	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#532	C₂
hypercenter	#532	D₅₃
hypocenter	C₅₃	#3549
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #532

Complements

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

N	H ≅ G/N	type
1. #16811365	C₂	semidirect
2. #16811339	C₂	semidirect
3. #16811338	C₂	semidirect
4. D₄×D₂₁₂	C₂	semidirect
5. #16811258	C₂	semidirect
6. #16811189	C₂	semidirect
7. #16811187	C₂	semidirect
8. #16811173	C₂	semidirect
9. C₅₃×#532	C₂	semidirect
10. C₂×#11719945	C₂²	semidirect
11. C₂²×D₂₁₂	C₂²	semidirect
12. C₂×D₄×D₅₃	C₂²	semidirect
13. #13133217	C₂²	semidirect
15. #13133213	C₂²	semidirect
16. #13133206	C₂²	semidirect
17. #13133205	C₂²	semidirect
18. #13133203	C₂²	semidirect
19. #13133192	C₂²	semidirect
20. #13133191	C₂²	semidirect
21. #13133191	C₂²	semidirect
22. #13133191	C₂²	semidirect
23. #13133188	C₂²	semidirect
24. #13133187	C₂²	semidirect
25. #13133185	C₂²	semidirect
26. #13133185	C₂²	semidirect
27. C₄×D₂₁₂	C₂²	semidirect
28. C₄×D₂₁₂	C₂²	semidirect
29. #13133176	C₂²	semidirect
30. #13133175	C₂²	semidirect
31. C₂×#41×C₅₃	C₂²	semidirect
32. C₄⋊Q₈×C₅₃	C₂²	semidirect
34. #124×C₅₃	C₂²	semidirect
35. #121×C₅₃	C₂²	semidirect
36. C₄×D₄×C₅₃	C₂²	semidirect
37. C₂×C₄²×C₅₃	C₂²	semidirect
125. #532	D₅₃	semidirect
126. C₅₃	#3549	semidirect
128. C₂×#41	C₂×D₅₃	semidirect
132. C₄⋊Q₈	C₂×D₅₃	semidirect
139. #124	C₂×D₅₃	semidirect
145. #121	C₂×D₅₃	semidirect
151. C₄×D₄	C₂×D₅₃	semidirect
154. C₂×C₄²	C₂×D₅₃	semidirect