Subgroups — C₁₉×#62817 #18068776

Order 4864.

9494 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3532	C₂	1	24	yes	maximal, complemented
2	C₁₉×#3490	C₂	1	18	yes	maximal, complemented
3	C₁₉×#3484	C₂	1	18	yes	maximal, complemented
4	C₂²×C₁₉×#120	C₂	1	3	yes	maximal, complemented
5	C₂⁶×C₁₉	C₂²	1	1	yes	characteristic, complemented
6	C₂×C₁₉×H(5,𝔽₂)	C₂²	1	6	yes	complemented
7	C₂³×D₄×C₁₉	C₂²	1	9	yes	complemented
8	C₁₉×H(3,𝔽₄)	C₂²	1	8	yes	complemented
9	C₁₉×#560	C₂²	1	24	yes	complemented
10	C₁₉×#551	C₂²	1	72	yes	complemented
11	C₁₉×#546	C₂²	1	144	yes	complemented
12	D₄²×C₁₉	C₂²	1	72	yes	complemented
13	C₁₉×#543	C₂²	1	6	yes	complemented
14	C₁₉×#538	C₂²	1	72	yes	complemented
15	C₁₉×#535	C₂²	1	36	yes	complemented
16	C₁₉×#534	C₂²	1	18	yes	complemented
17	C₁₉×#534	C₂²	1	36	yes	complemented
18	C₁₉×#525	C₂²	1	36	yes	complemented
19	C₂×C₁₉×#120	C₂²	1	36	yes	complemented
20	C₂×C₁₉×#120	C₂²	1	36	yes	complemented
21	C₂×C₁₉×#120	C₂²	1	12	yes	complemented
22	C₁₉×#518	C₂²	1	18	yes	complemented
23	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	9	yes	complemented
24	C₂⁵×C₁₉	—	4	12	no	—
25	C₂⁵×C₁₉	C₂³	1	9	yes	—
26	C₂⁵×C₁₉	C₂³	1	6	yes	—
27	C₂⁵×C₁₉	C₂³	1	3	yes	complemented
28	C₁₉×H(5,𝔽₂)	—	4	24	no	—
29	C₂×#41×C₁₉	C₂³	1	36	yes	complemented
30	C₂²×D₄×C₁₉	—	4	36	no	—
31	C₂²×D₄×C₁₉	C₂³	1	18	yes	—
32	C₂²×D₄×C₁₉	C₂³	1	18	yes	complemented
33	C₂²×D₄×C₁₉	C₂³	1	72	yes	complemented
34	C₂²×D₄×C₁₉	C₂³	1	36	yes	complemented
35	C₂³×C₄×C₁₉	C₂³	1	9	yes	complemented
36	C₁₉×#127	C₂³	1	24	yes	complemented
37	C₁₉×#126	C₂³	1	48	yes	complemented
38	C₁₉×#124	C₂³	1	72	yes	complemented
39	C₁₉×#123	C₂³	1	72	yes	complemented
40	C₁₉×#123	C₂³	1	72	yes	complemented
41	C₁₉×C₂²⋊Q₈	C₂³	1	72	yes	complemented
42	C₁₉×#121	C₂³	1	144	yes	complemented
43	C₁₉×#121	C₂³	1	72	yes	complemented
44	C₁₉×#120	C₂³	1	72	yes	—
45	C₁₉×#120	C₂³	1	24	yes	complemented
46	C₁₉×#120	C₂³	1	72	yes	complemented
47	C₁₉×#120	C₂³	1	24	yes	complemented
48	C₁₉×#120	C₂³	1	72	yes	complemented
49	C₁₉×#120	C₂³	1	24	yes	complemented
50	C₄×D₄×C₁₉	C₂³	1	144	yes	complemented
51	C₁₉×#117	C₂³	1	36	yes	complemented
52	C₂×C₂²⋊C₄×C₁₉	C₂³	1	18	yes	—
53	C₂×C₂²⋊C₄×C₁₉	C₂³	1	18	yes	complemented
54	C₂×C₂²⋊C₄×C₁₉	C₂³	1	72	yes	complemented
55	C₂×C₂²⋊C₄×C₁₉	C₂³	1	36	yes	complemented
56	C₂⁴×C₁₉	—	2	36	no	—
57	C₂⁴×C₁₉	—	4	36	no	—
58	C₂⁴×C₁₉	—	4	36	no	—
59	C₂⁴×C₁₉	—	4	18	no	—
60	C₂⁴×C₁₉	—	4	64	no	—
61	C₂⁴×C₁₉	C₂⁴	1	3	yes	—
62	C₂⁴×C₁₉	C₂⁴	1	18	yes	—
63	C₂⁴×C₁₉	C₂⁴	1	9	yes	—
64	C₂⁴×C₁₉	C₂⁴	1	18	yes	—
65	C₂⁴×C₁₉	C₂⁴	1	2	yes	—
66	C₂⁴×C₁₉	C₂⁴	1	3	yes	—
67	C₂⁴×C₁₉	C₂⁴	1	4	yes	complemented
68	#41×C₁₉	—	4	72	no	—
69	C₂×Q₈×C₁₉	C₂⁴	1	12	yes	complemented
70	C₂×D₄×C₁₉	—	2	72	no	—
71	C₂×D₄×C₁₉	—	4	36	no	—
72	C₂×D₄×C₁₉	—	4	144	no	—
73	C₂×D₄×C₁₉	—	4	72	no	—
74	C₂×D₄×C₁₉	C₂⁴	1	72	yes	—
75	C₂×D₄×C₁₉	C₂⁴	1	72	yes	—
76	C₂×D₄×C₁₉	C₂⁴	1	36	yes	complemented
77	C₂²×C₄×C₁₉	—	4	18	no	—
78	C₂²×C₄×C₁₉	C₂⁴	1	18	yes	—
79	C₂²×C₄×C₁₉	C₂⁴	1	36	yes	—
80	C₂²×C₄×C₁₉	C₂⁴	1	36	yes	complemented
81	C₄⋊C₄×C₁₉	C₂⁴	1	72	yes	complemented
82	C₂²⋊C₄×C₁₉	C₂⁴	1	72	yes	—
83	C₂²⋊C₄×C₁₉	C₂⁴	1	72	yes	—
84	C₂²⋊C₄×C₁₉	C₂⁴	1	72	yes	complemented
85	C₄²×C₁₉	C₂⁴	1	24	yes	complemented
86	#62817	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₁₉	—	2	18	no	—
88	C₂³×C₁₉	—	2	72	no	—
89	C₂³×C₁₉	—	2	18	no	—
90	C₂³×C₁₉	—	2	36	no	—
91	C₂³×C₁₉	—	4	6	no	—
92	C₂³×C₁₉	—	4	18	no	—
93	C₂³×C₁₉	—	4	18	no	—
94	C₂³×C₁₉	—	4	6	no	—
95	C₂³×C₁₉	—	4	36	no	—
96	C₂³×C₁₉	—	4	12	no	—
97	C₂³×C₁₉	—	4	144	no	—
98	C₂³×C₁₉	—	4	96	no	—
99	C₂³×C₁₉	—	8	24	no	—
100	C₂³×C₁₉	H(5,𝔽₂)	1	12	yes	complemented
101	C₂³×C₁₉	C₂⁵	1	6	yes	—
102	C₂³×C₁₉	C₂⁵	1	12	yes	—
103	C₂³×C₁₉	C₂⁵	1	9	yes	—
104	Q₈×C₁₉	—	4	12	no	—
105	D₄×C₁₉	—	2	144	no	—
106	D₄×C₁₉	—	4	72	no	—
107	D₄×C₁₉	—	4	36	no	—
108	C₂×C₄×C₁₉	—	2	36	no	—
109	C₂×C₄×C₁₉	—	4	36	no	—
110	C₂×C₄×C₁₉	—	4	36	no	—
111	C₂×C₄×C₁₉	C₂⁵	1	36	yes	—
112	#3532	C₂×C₁₉	1	24	yes	complemented
113	#3490	C₂×C₁₉	1	18	yes	complemented
114	#3484	C₂×C₁₉	1	18	yes	complemented
115	C₂²×#120	C₂×C₁₉	1	3	yes	complemented
116	C₂²×C₁₉	—	2	24	no	—
117	C₂²×C₁₉	—	2	18	no	—
118	C₂²×C₁₉	—	2	36	no	—
119	C₂²×C₁₉	—	2	18	no	—
120	C₂²×C₁₉	—	4	12	no	—
121	C₂²×C₁₉	—	4	72	no	—
122	C₂²×C₁₉	—	4	36	no	—
123	C₂²×C₁₉	—	4	8	no	—
124	C₂²×C₁₉	—	8	36	no	—
125	C₂²×C₁₉	—	16	4	no	—
126	C₂²×C₁₉	C₂×H(5,𝔽₂)	1	18	yes	—
127	C₂²×C₁₉	C₂⁶	1	1	yes	characteristic
128	C₄×C₁₉	—	2	36	no	—
129	C₂⁶	C₂²×C₁₉	1	1	yes	characteristic, complemented
130	C₂×H(5,𝔽₂)	C₂²×C₁₉	1	6	yes	complemented
131	C₂³×D₄	C₂²×C₁₉	1	9	yes	complemented
132	H(3,𝔽₄)	C₂²×C₁₉	1	8	yes	complemented
133	#560	C₂²×C₁₉	1	24	yes	complemented
134	#551	C₂²×C₁₉	1	72	yes	complemented
135	#546	C₂²×C₁₉	1	144	yes	complemented
136	D₄²	C₂²×C₁₉	1	72	yes	complemented
137	#543	C₂²×C₁₉	1	6	yes	complemented
138	#538	C₂²×C₁₉	1	72	yes	complemented
139	#535	C₂²×C₁₉	1	36	yes	complemented
140	#534	C₂²×C₁₉	1	18	yes	complemented
141	#534	C₂²×C₁₉	1	36	yes	complemented
142	#525	C₂²×C₁₉	1	36	yes	complemented
143	C₂×#120	C₂²×C₁₉	1	36	yes	complemented
144	C₂×#120	C₂²×C₁₉	1	36	yes	complemented
145	C₂×#120	C₂²×C₁₉	1	12	yes	complemented
146	#518	C₂²×C₁₉	1	18	yes	complemented
147	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	9	yes	complemented
148	C₂×C₁₉	—	2	18	no	—
149	C₂×C₁₉	—	4	6	no	—
150	C₂×C₁₉	—	4	12	no	—
151	C₂×C₁₉	C₂²×H(5,𝔽₂)	1	3	yes	—
152	C₂⁵	—	4	12	no	—
153	C₂⁵	C₂³×C₁₉	1	9	yes	—
154	C₂⁵	C₂³×C₁₉	1	6	yes	—
155	C₂⁵	C₂³×C₁₉	1	3	yes	complemented
156	H(5,𝔽₂)	—	4	24	no	—
157	C₂×#41	C₂³×C₁₉	1	36	yes	complemented
158	C₂²×D₄	—	4	36	no	—
159	C₂²×D₄	C₂³×C₁₉	1	18	yes	—
160	C₂²×D₄	C₂³×C₁₉	1	18	yes	complemented
161	C₂²×D₄	C₂³×C₁₉	1	72	yes	complemented
162	C₂²×D₄	C₂³×C₁₉	1	36	yes	complemented
163	C₂³×C₄	C₂³×C₁₉	1	9	yes	complemented
164	#127	C₂³×C₁₉	1	24	yes	complemented
165	#126	C₂³×C₁₉	1	48	yes	complemented
166	#124	C₂³×C₁₉	1	72	yes	complemented
167	#123	C₂³×C₁₉	1	72	yes	complemented
168	#123	C₂³×C₁₉	1	72	yes	complemented
169	C₂²⋊Q₈	C₂³×C₁₉	1	72	yes	complemented
170	#121	C₂³×C₁₉	1	144	yes	complemented
171	#121	C₂³×C₁₉	1	72	yes	complemented
172	#120	C₂³×C₁₉	1	72	yes	—
173	#120	C₂³×C₁₉	1	24	yes	complemented
174	#120	C₂³×C₁₉	1	72	yes	complemented
175	#120	C₂³×C₁₉	1	24	yes	complemented
176	#120	C₂³×C₁₉	1	72	yes	complemented
177	#120	C₂³×C₁₉	1	24	yes	complemented
178	C₄×D₄	C₂³×C₁₉	1	144	yes	complemented
179	#117	C₂³×C₁₉	1	36	yes	complemented
180	C₂×C₂²⋊C₄	C₂³×C₁₉	1	18	yes	—
181	C₂×C₂²⋊C₄	C₂³×C₁₉	1	18	yes	complemented
182	C₂×C₂²⋊C₄	C₂³×C₁₉	1	72	yes	complemented
183	C₂×C₂²⋊C₄	C₂³×C₁₉	1	36	yes	complemented
184	C₁₉	#62817	1	1	yes	characteristic, direct product factor, Sylow
185	C₂⁴	—	2	36	no	—
186	C₂⁴	—	4	36	no	—
187	C₂⁴	—	4	36	no	—
188	C₂⁴	—	4	18	no	—
189	C₂⁴	—	4	64	no	—
190	C₂⁴	C₂⁴×C₁₉	1	3	yes	—
191	C₂⁴	C₂⁴×C₁₉	1	18	yes	—
192	C₂⁴	C₂⁴×C₁₉	1	9	yes	—
193	C₂⁴	C₂⁴×C₁₉	1	18	yes	—
194	C₂⁴	C₂⁴×C₁₉	1	2	yes	—
195	C₂⁴	C₂⁴×C₁₉	1	3	yes	—
196	C₂⁴	C₂⁴×C₁₉	1	4	yes	complemented
197	#41	—	4	72	no	—
198	C₂×Q₈	C₂⁴×C₁₉	1	12	yes	complemented
199	C₂×D₄	—	2	72	no	—
200	C₂×D₄	—	4	36	no	—
201	C₂×D₄	—	4	144	no	—
202	C₂×D₄	—	4	72	no	—
203	C₂×D₄	C₂⁴×C₁₉	1	72	yes	—
204	C₂×D₄	C₂⁴×C₁₉	1	72	yes	—
205	C₂×D₄	C₂⁴×C₁₉	1	36	yes	complemented
206	C₂²×C₄	—	4	18	no	—
207	C₂²×C₄	C₂⁴×C₁₉	1	18	yes	—
208	C₂²×C₄	C₂⁴×C₁₉	1	36	yes	—
209	C₂²×C₄	C₂⁴×C₁₉	1	36	yes	complemented
210	C₄⋊C₄	C₂⁴×C₁₉	1	72	yes	complemented
211	C₂²⋊C₄	C₂⁴×C₁₉	1	72	yes	—
212	C₂²⋊C₄	C₂⁴×C₁₉	1	72	yes	—
213	C₂²⋊C₄	C₂⁴×C₁₉	1	72	yes	complemented
214	C₄²	C₂⁴×C₁₉	1	24	yes	complemented
215	C₂³	—	2	18	no	—
216	C₂³	—	2	72	no	—
217	C₂³	—	2	18	no	—
218	C₂³	—	2	36	no	—
219	C₂³	—	4	6	no	—
220	C₂³	—	4	18	no	—
221	C₂³	—	4	18	no	—
222	C₂³	—	4	6	no	—
223	C₂³	—	4	36	no	—
224	C₂³	—	4	12	no	—
225	C₂³	—	4	144	no	—
226	C₂³	—	4	96	no	—
227	C₂³	—	8	24	no	—
228	C₂³	C₁₉×H(5,𝔽₂)	1	12	yes	complemented
229	C₂³	C₂⁵×C₁₉	1	6	yes	—
230	C₂³	C₂⁵×C₁₉	1	12	yes	—
231	C₂³	C₂⁵×C₁₉	1	9	yes	—
232	Q₈	—	4	12	no	—
233	D₄	—	2	144	no	—
234	D₄	—	4	72	no	—
235	D₄	—	4	36	no	—
236	C₂×C₄	—	2	36	no	—
237	C₂×C₄	—	4	36	no	—
238	C₂×C₄	—	4	36	no	—
239	C₂×C₄	C₂⁵×C₁₉	1	36	yes	—
240	C₂²	—	2	24	no	—
241	C₂²	—	2	18	no	—
242	C₂²	—	2	36	no	—
243	C₂²	—	2	18	no	—
244	C₂²	—	4	12	no	—
245	C₂²	—	4	72	no	—
246	C₂²	—	4	36	no	—
247	C₂²	—	4	8	no	—
248	C₂²	—	8	36	no	—
249	C₂²	—	16	4	no	—
250	C₂²	C₂×C₁₉×H(5,𝔽₂)	1	18	yes	—
251	C₂²	C₂⁶×C₁₉	1	1	yes	characteristic
252	C₄	—	2	36	no	—
253	C₂	—	2	18	no	—
254	C₂	—	4	6	no	—
255	C₂	—	4	12	no	—
256	C₂	C₂²×C₁₉×H(5,𝔽₂)	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂⁶
commutator, Frattini, cosocle	C₂²	C₂⁶×C₁₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#3532	C₂	semidirect
2. C₁₉×#3490	C₂	semidirect
3. C₁₉×#3484	C₂	semidirect
4. C₂²×C₁₉×#120	C₂	semidirect
5. C₂⁶×C₁₉	C₂²	semidirect
6. C₂×C₁₉×H(5,𝔽₂)	C₂²	semidirect
7. C₂³×D₄×C₁₉	C₂²	semidirect
8. C₁₉×H(3,𝔽₄)	C₂²	semidirect
9. C₁₉×#560	C₂²	semidirect
10. C₁₉×#551	C₂²	semidirect
11. C₁₉×#546	C₂²	semidirect
12. D₄²×C₁₉	C₂²	semidirect
13. C₁₉×#543	C₂²	semidirect
14. C₁₉×#538	C₂²	semidirect
15. C₁₉×#535	C₂²	semidirect
16. C₁₉×#534	C₂²	semidirect
17. C₁₉×#534	C₂²	semidirect
18. C₁₉×#525	C₂²	semidirect
19. C₂×C₁₉×#120	C₂²	semidirect
20. C₂×C₁₉×#120	C₂²	semidirect
21. C₂×C₁₉×#120	C₂²	semidirect
22. C₁₉×#518	C₂²	semidirect
23. C₂²×C₂²⋊C₄×C₁₉	C₂²	semidirect
27. C₂⁵×C₁₉	C₂³	semidirect
29. C₂×#41×C₁₉	C₂³	semidirect
32. C₂²×D₄×C₁₉	C₂³	semidirect
33. C₂²×D₄×C₁₉	C₂³	semidirect
34. C₂²×D₄×C₁₉	C₂³	semidirect
35. C₂³×C₄×C₁₉	C₂³	semidirect
36. C₁₉×#127	C₂³	semidirect
37. C₁₉×#126	C₂³	semidirect
38. C₁₉×#124	C₂³	semidirect
39. C₁₉×#123	C₂³	semidirect
40. C₁₉×#123	C₂³	semidirect
41. C₁₉×C₂²⋊Q₈	C₂³	semidirect
42. C₁₉×#121	C₂³	semidirect
43. C₁₉×#121	C₂³	semidirect
45. C₁₉×#120	C₂³	semidirect
46. C₁₉×#120	C₂³	semidirect
47. C₁₉×#120	C₂³	semidirect
48. C₁₉×#120	C₂³	semidirect
49. C₁₉×#120	C₂³	semidirect
50. C₄×D₄×C₁₉	C₂³	semidirect
51. C₁₉×#117	C₂³	semidirect
53. C₂×C₂²⋊C₄×C₁₉	C₂³	semidirect
54. C₂×C₂²⋊C₄×C₁₉	C₂³	semidirect
55. C₂×C₂²⋊C₄×C₁₉	C₂³	semidirect
67. C₂⁴×C₁₉	C₂⁴	semidirect
69. C₂×Q₈×C₁₉	C₂⁴	semidirect
76. C₂×D₄×C₁₉	C₂⁴	semidirect
80. C₂²×C₄×C₁₉	C₂⁴	semidirect
81. C₄⋊C₄×C₁₉	C₂⁴	semidirect
84. C₂²⋊C₄×C₁₉	C₂⁴	semidirect
85. C₄²×C₁₉	C₂⁴	semidirect
86. #62817	C₁₉	direct (N × H)
100. C₂³×C₁₉	H(5,𝔽₂)	semidirect
112. #3532	C₂×C₁₉	semidirect
113. #3490	C₂×C₁₉	semidirect
114. #3484	C₂×C₁₉	semidirect
115. C₂²×#120	C₂×C₁₉	semidirect
129. C₂⁶	C₂²×C₁₉	semidirect
130. C₂×H(5,𝔽₂)	C₂²×C₁₉	semidirect
131. C₂³×D₄	C₂²×C₁₉	semidirect
132. H(3,𝔽₄)	C₂²×C₁₉	semidirect
133. #560	C₂²×C₁₉	semidirect
134. #551	C₂²×C₁₉	semidirect
135. #546	C₂²×C₁₉	semidirect
136. D₄²	C₂²×C₁₉	semidirect
137. #543	C₂²×C₁₉	semidirect
138. #538	C₂²×C₁₉	semidirect
139. #535	C₂²×C₁₉	semidirect
140. #534	C₂²×C₁₉	semidirect
141. #534	C₂²×C₁₉	semidirect
142. #525	C₂²×C₁₉	semidirect
143. C₂×#120	C₂²×C₁₉	semidirect
144. C₂×#120	C₂²×C₁₉	semidirect
145. C₂×#120	C₂²×C₁₉	semidirect
146. #518	C₂²×C₁₉	semidirect
147. C₂²×C₂²⋊C₄	C₂²×C₁₉	semidirect
155. C₂⁵	C₂³×C₁₉	semidirect
157. C₂×#41	C₂³×C₁₉	semidirect
160. C₂²×D₄	C₂³×C₁₉	semidirect
161. C₂²×D₄	C₂³×C₁₉	semidirect
162. C₂²×D₄	C₂³×C₁₉	semidirect
163. C₂³×C₄	C₂³×C₁₉	semidirect
164. #127	C₂³×C₁₉	semidirect
165. #126	C₂³×C₁₉	semidirect
166. #124	C₂³×C₁₉	semidirect
167. #123	C₂³×C₁₉	semidirect
168. #123	C₂³×C₁₉	semidirect
169. C₂²⋊Q₈	C₂³×C₁₉	semidirect
170. #121	C₂³×C₁₉	semidirect
171. #121	C₂³×C₁₉	semidirect
173. #120	C₂³×C₁₉	semidirect
174. #120	C₂³×C₁₉	semidirect
175. #120	C₂³×C₁₉	semidirect
176. #120	C₂³×C₁₉	semidirect
177. #120	C₂³×C₁₉	semidirect
178. C₄×D₄	C₂³×C₁₉	semidirect
179. #117	C₂³×C₁₉	semidirect
181. C₂×C₂²⋊C₄	C₂³×C₁₉	semidirect
182. C₂×C₂²⋊C₄	C₂³×C₁₉	semidirect
183. C₂×C₂²⋊C₄	C₂³×C₁₉	semidirect
184. C₁₉	#62817	direct (N × H)
196. C₂⁴	C₂⁴×C₁₉	semidirect
198. C₂×Q₈	C₂⁴×C₁₉	semidirect
205. C₂×D₄	C₂⁴×C₁₉	semidirect
209. C₂²×C₄	C₂⁴×C₁₉	semidirect
210. C₄⋊C₄	C₂⁴×C₁₉	semidirect
213. C₂²⋊C₄	C₂⁴×C₁₉	semidirect
214. C₄²	C₂⁴×C₁₉	semidirect
228. C₂³	C₁₉×H(5,𝔽₂)	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₉×#3532	2
C₁₉×#3490	2
C₁₉×#3484	2
C₂²×C₁₉×#120	2
#62817	19

Subgroups — C19×#62817 #18068776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#62817 #18068776