Subgroups — C₁₉×#60665 #18066624

Order 4864.

1834 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3503	C₂	1	2	yes	maximal
2	C₁₉×#3165	C₂	1	6	yes	maximal, complemented
3	C₁₉×#3163	C₂	1	12	yes	maximal, complemented
4	C₁₉×#3160	C₂	1	3	yes	maximal, complemented
5	C₂×C₁₉×#498	C₂	1	3	yes	maximal, complemented
6	C₁₉×#2912	C₂	1	4	yes	maximal
7	C₄×C₁₉×#131	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₉×#131	C₂²	1	2	yes	—
9	Q₈²×C₁₉	—	2	2	no	—
10	C₁₉×#557	—	2	2	no	—
11	C₁₉×#554	—	2	6	no	—
12	D₄×Q₈×C₁₉	—	2	6	no	—
13	C₁₉×#544	C₂²	1	12	yes	—
14	C₁₉×#533	C₂²	1	6	yes	—
15	C₂×C₁₉×C₄⋊Q₈	C₂²	1	6	yes	—
16	C₁₉×#520	C₂²	1	4	yes	—
17	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#501	C₂²	1	24	yes	—
19	C₁₉×#500	C₂²	1	6	yes	—
20	C₁₉×#499	C₂²	1	12	yes	—
21	C₁₉×#498	C₂²	1	3	yes	—
22	C₁₉×#498	C₂²	1	12	yes	—
23	C₁₉×#498	C₂²	1	6	yes	—
24	C₁₉×#498	C₂²	1	1	yes	characteristic
25	C₁₉×#428	C₂²	1	24	yes	—
26	C₁₉×#427	C₂²	1	12	yes	—
27	C₂×C₁₉×#106	C₂²	1	12	yes	—
28	C₁₉×#405	C₂²	1	6	yes	—
29	C₄×M₁₆×C₁₉	C₂²	1	3	yes	—
30	C₂×C₄×C₈×C₁₉	C₂²	1	3	yes	—
31	C₂×#41×C₁₉	C₂³	1	1	yes	characteristic
32	C₂²×Q₈×C₁₉	—	2	6	no	—
33	C₁₉×#131	Q₈	1	16	yes	complemented
34	C₂×M₁₆×C₁₉	C₂³	1	6	yes	—
35	C₂²×C₈×C₁₉	C₂³	1	6	yes	—
36	C₁₉×C₄⋊Q₈	—	2	12	no	—
37	C₁₉×C₄⋊Q₈	—	2	6	no	—
38	C₁₉×C₄⋊Q₈	—	2	6	no	—
39	C₁₉×C₄⋊Q₈	C₂³	1	6	yes	—
40	C₁₉×C₄⋊Q₈	C₂³	1	12	yes	—
41	C₁₉×C₄⋊Q₈	C₂³	1	2	yes	—
42	C₁₉×#125	—	2	12	no	—
43	C₁₉×#125	C₂³	1	12	yes	—
44	C₁₉×C₂²⋊Q₈	—	2	6	no	—
45	C₁₉×C₂²⋊Q₈	—	2	6	no	—
46	C₁₉×C₂²⋊Q₈	—	2	24	no	—
47	C₄×Q₈×C₁₉	—	2	4	no	—
48	C₄×Q₈×C₁₉	—	2	6	no	—
49	C₄×Q₈×C₁₉	—	2	6	no	—
50	C₄×Q₈×C₁₉	D₄	1	2	yes	—
51	C₄×D₄×C₁₉	—	2	12	no	—
52	C₄×D₄×C₁₉	D₄	1	6	yes	—
53	C₁₉×#117	C₂³	1	12	yes	—
54	C₁₉×#117	C₂³	1	3	yes	—
55	C₂×C₄⋊C₄×C₁₉	—	2	6	no	—
56	C₂×C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
57	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
58	C₁₉×#107	C₂³	1	24	yes	—
59	C₁₉×#106	C₂³	1	12	yes	—
60	C₁₉×#106	C₂³	1	24	yes	—
61	C₁₉×#106	C₂³	1	4	yes	—
62	C₁₉×#97	C₂³	1	6	yes	—
63	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₁₉	C₂³	1	3	yes	—
65	C₄×C₈×C₁₉	C₂³	1	6	yes	—
66	#41×C₁₉	C₂×Q₈	1	8	yes	—
67	C₂×Q₈×C₁₉	—	2	2	no	—
68	C₂×Q₈×C₁₉	—	2	6	no	—
69	C₂×Q₈×C₁₉	—	2	12	no	—
70	C₂×Q₈×C₁₉	—	8	12	no	—
71	C₂×Q₈×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₂×D₄×C₁₉	C₂×D₄	1	3	yes	—
73	C₂²×C₄×C₁₉	—	2	12	no	—
74	C₂²×C₄×C₁₉	C₂×D₄	1	6	yes	—
75	C₂²×C₄×C₁₉	C₂⁴	1	3	yes	—
76	M₁₆×C₁₉	C₂×Q₈	1	24	yes	complemented
77	C₂×C₈×C₁₉	C₂×Q₈	1	24	yes	complemented
78	C₂×C₈×C₁₉	C₂⁴	1	2	yes	—
79	C₂×C₈×C₁₉	C₂⁴	1	6	yes	—
80	C₄⋊C₄×C₁₉	—	2	6	no	—
81	C₄⋊C₄×C₁₉	—	2	12	no	—
82	C₄⋊C₄×C₁₉	—	2	12	no	—
83	C₄⋊C₄×C₁₉	—	2	2	no	—
84	C₄⋊C₄×C₁₉	—	2	12	no	—
85	C₄⋊C₄×C₁₉	—	2	12	no	—
86	C₄⋊C₄×C₁₉	C₂×D₄	1	6	yes	—
87	C₄⋊C₄×C₁₉	C₂⁴	1	12	yes	—
88	C₄⋊C₄×C₁₉	C₂⁴	1	4	yes	—
89	C₂²⋊C₄×C₁₉	—	2	12	no	—
90	C₂²⋊C₄×C₁₉	C₂×D₄	1	6	yes	—
91	C₄²×C₁₉	—	2	12	no	—
92	C₄²×C₁₉	C₂×D₄	1	6	yes	—
93	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
94	C₄²×C₁₉	C₂⁴	1	3	yes	—
95	#60665	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₁₉	C₂²×D₄	1	3	yes	—
97	Q₈×C₁₉	—	4	4	no	—
98	Q₈×C₁₉	—	4	12	no	—
99	Q₈×C₁₉	—	8	12	no	—
100	Q₈×C₁₉	C₄⋊Q₈	1	4	yes	—
101	D₄×C₁₉	C₄⋊Q₈	1	12	yes	—
102	C₂×C₄×C₁₉	—	2	12	no	—
103	C₂×C₄×C₁₉	—	2	4	no	—
104	C₂×C₄×C₁₉	—	2	12	no	—
105	C₂×C₄×C₁₉	—	8	12	no	—
106	C₂×C₄×C₁₉	C₂²×D₄	1	6	yes	—
107	C₂×C₄×C₁₉	C₂²×D₄	1	3	yes	—
108	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
109	C₂×C₄×C₁₉	C₂²×Q₈	1	12	yes	—
110	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
111	C₈×C₁₉	C₂²×Q₈	1	4	yes	—
112	C₈×C₁₉	C₂²×Q₈	1	12	yes	—
113	#3503	C₂×C₁₉	1	2	yes	—
114	#3165	C₂×C₁₉	1	6	yes	complemented
115	#3163	C₂×C₁₉	1	12	yes	complemented
116	#3160	C₂×C₁₉	1	3	yes	complemented
117	C₂×#498	C₂×C₁₉	1	3	yes	complemented
118	#2912	C₂×C₁₉	1	4	yes	—
119	C₄×#131	C₂×C₁₉	1	1	yes	characteristic
120	C₂²×C₁₉	—	2	6	no	—
121	C₂²×C₁₉	C₂×C₄⋊Q₈	1	6	yes	—
122	C₂²×C₁₉	C₂³×D₄	1	1	yes	characteristic
123	C₄×C₁₉	—	2	6	no	—
124	C₄×C₁₉	—	4	12	no	—
125	C₄×C₁₉	—	4	4	no	—
126	C₄×C₁₉	C₂×C₄⋊Q₈	1	6	yes	—
127	C₄×C₁₉	#577	1	4	yes	—
128	C₄×C₁₉	C₂³×Q₈	1	2	yes	—
129	C₂×#131	C₂²×C₁₉	1	2	yes	—
130	Q₈²	—	2	2	no	—
131	#557	—	2	2	no	—
132	#554	—	2	6	no	—
133	D₄×Q₈	—	2	6	no	—
134	#544	C₂²×C₁₉	1	12	yes	—
135	#533	C₂²×C₁₉	1	6	yes	—
136	C₂×C₄⋊Q₈	C₂²×C₁₉	1	6	yes	—
137	#520	C₂²×C₁₉	1	4	yes	—
138	C₄×#41	C₂²×C₁₉	1	1	yes	characteristic
139	#501	C₂²×C₁₉	1	24	yes	—
140	#500	C₂²×C₁₉	1	6	yes	—
141	#499	C₂²×C₁₉	1	12	yes	—
142	#498	C₂²×C₁₉	1	3	yes	—
143	#498	C₂²×C₁₉	1	12	yes	—
144	#498	C₂²×C₁₉	1	6	yes	—
145	#498	C₂²×C₁₉	1	1	yes	characteristic
146	#428	C₂²×C₁₉	1	24	yes	—
147	#427	C₂²×C₁₉	1	12	yes	—
148	C₂×#106	C₂²×C₁₉	1	12	yes	—
149	#405	C₂²×C₁₉	1	6	yes	—
150	C₄×M₁₆	C₂²×C₁₉	1	3	yes	—
151	C₂×C₄×C₈	C₂²×C₁₉	1	3	yes	—
152	C₂×C₁₉	—	2	6	no	—
153	C₂×C₁₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
154	C₂×C₁₉	C₂×#577	1	2	yes	—
155	C₂×#41	C₂³×C₁₉	1	1	yes	characteristic
156	C₂²×Q₈	—	2	6	no	—
157	#131	Q₈×C₁₉	1	16	yes	complemented
158	C₂×M₁₆	C₂³×C₁₉	1	6	yes	—
159	C₂²×C₈	C₂³×C₁₉	1	6	yes	—
160	C₄⋊Q₈	—	2	12	no	—
161	C₄⋊Q₈	—	2	6	no	—
162	C₄⋊Q₈	—	2	6	no	—
163	C₄⋊Q₈	C₂³×C₁₉	1	6	yes	—
164	C₄⋊Q₈	C₂³×C₁₉	1	12	yes	—
165	C₄⋊Q₈	C₂³×C₁₉	1	2	yes	—
166	#125	—	2	12	no	—
167	#125	C₂³×C₁₉	1	12	yes	—
168	C₂²⋊Q₈	—	2	6	no	—
169	C₂²⋊Q₈	—	2	6	no	—
170	C₂²⋊Q₈	—	2	24	no	—
171	C₄×Q₈	—	2	4	no	—
172	C₄×Q₈	—	2	6	no	—
173	C₄×Q₈	—	2	6	no	—
174	C₄×Q₈	D₄×C₁₉	1	2	yes	—
175	C₄×D₄	—	2	12	no	—
176	C₄×D₄	D₄×C₁₉	1	6	yes	—
177	#117	C₂³×C₁₉	1	12	yes	—
178	#117	C₂³×C₁₉	1	3	yes	—
179	C₂×C₄⋊C₄	—	2	6	no	—
180	C₂×C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
181	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
182	#107	C₂³×C₁₉	1	24	yes	—
183	#106	C₂³×C₁₉	1	12	yes	—
184	#106	C₂³×C₁₉	1	24	yes	—
185	#106	C₂³×C₁₉	1	4	yes	—
186	#97	C₂³×C₁₉	1	6	yes	—
187	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
188	C₄×C₈	C₂³×C₁₉	1	3	yes	—
189	C₄×C₈	C₂³×C₁₉	1	6	yes	—
190	C₁₉	#60665	1	1	yes	characteristic, direct product factor, Sylow
191	#41	C₂×Q₈×C₁₉	1	8	yes	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	—	2	6	no	—
194	C₂×Q₈	—	2	12	no	—
195	C₂×Q₈	—	8	12	no	—
196	C₂×Q₈	C₂×D₄×C₁₉	1	1	yes	characteristic
197	C₂×D₄	C₂×D₄×C₁₉	1	3	yes	—
198	C₂²×C₄	—	2	12	no	—
199	C₂²×C₄	C₂×D₄×C₁₉	1	6	yes	—
200	C₂²×C₄	C₂⁴×C₁₉	1	3	yes	—
201	M₁₆	C₂×Q₈×C₁₉	1	24	yes	complemented
202	C₂×C₈	C₂×Q₈×C₁₉	1	24	yes	complemented
203	C₂×C₈	C₂⁴×C₁₉	1	2	yes	—
204	C₂×C₈	C₂⁴×C₁₉	1	6	yes	—
205	C₄⋊C₄	—	2	6	no	—
206	C₄⋊C₄	—	2	12	no	—
207	C₄⋊C₄	—	2	12	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	12	no	—
210	C₄⋊C₄	—	2	12	no	—
211	C₄⋊C₄	C₂×D₄×C₁₉	1	6	yes	—
212	C₄⋊C₄	C₂⁴×C₁₉	1	12	yes	—
213	C₄⋊C₄	C₂⁴×C₁₉	1	4	yes	—
214	C₂²⋊C₄	—	2	12	no	—
215	C₂²⋊C₄	C₂×D₄×C₁₉	1	6	yes	—
216	C₄²	—	2	12	no	—
217	C₄²	C₂×D₄×C₁₉	1	6	yes	—
218	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
219	C₄²	C₂⁴×C₁₉	1	3	yes	—
220	C₂³	C₂²×D₄×C₁₉	1	3	yes	—
221	Q₈	—	4	12	no	—
222	Q₈	—	4	4	no	—
223	Q₈	—	8	12	no	—
224	Q₈	C₁₉×C₄⋊Q₈	1	4	yes	—
225	D₄	C₁₉×C₄⋊Q₈	1	12	yes	—
226	C₂×C₄	—	2	12	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	12	no	—
229	C₂×C₄	—	8	12	no	—
230	C₂×C₄	C₂²×D₄×C₁₉	1	6	yes	—
231	C₂×C₄	C₂²×D₄×C₁₉	1	3	yes	—
232	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
233	C₂×C₄	C₂²×Q₈×C₁₉	1	12	yes	—
234	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
235	C₈	C₂²×Q₈×C₁₉	1	4	yes	—
236	C₈	C₂²×Q₈×C₁₉	1	12	yes	—
237	C₂²	—	2	6	no	—
238	C₂²	C₂×C₁₉×C₄⋊Q₈	1	6	yes	—
239	C₂²	C₂³×D₄×C₁₉	1	1	yes	characteristic
240	C₄	—	2	6	no	—
241	C₄	—	4	12	no	—
242	C₄	—	4	4	no	—
243	C₄	C₂×C₁₉×C₄⋊Q₈	1	6	yes	—
244	C₄	C₁₉×#577	1	4	yes	—
245	C₄	C₂³×Q₈×C₁₉	1	2	yes	—
246	C₂	—	2	6	no	—
247	C₂	C₂²×C₁₉×C₄⋊Q₈	1	1	yes	characteristic
248	C₂	C₂×C₁₉×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₉
prime radical	C₂×#41×C₁₉	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄×#41×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#3165	C₂	semidirect
3. C₁₉×#3163	C₂	semidirect
4. C₁₉×#3160	C₂	semidirect
5. C₂×C₁₉×#498	C₂	semidirect
33. C₁₉×#131	Q₈	semidirect
76. M₁₆×C₁₉	C₂×Q₈	semidirect
77. C₂×C₈×C₁₉	C₂×Q₈	semidirect
95. #60665	C₁₉	direct (N × H)
114. #3165	C₂×C₁₉	semidirect
115. #3163	C₂×C₁₉	semidirect
116. #3160	C₂×C₁₉	semidirect
117. C₂×#498	C₂×C₁₉	semidirect
157. #131	Q₈×C₁₉	semidirect
190. C₁₉	#60665	direct (N × H)
201. M₁₆	C₂×Q₈×C₁₉	semidirect
202. C₂×C₈	C₂×Q₈×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#3503	2
C₁₉×#3165	2
C₁₉×#3163	2
C₁₉×#3160	2
C₂×C₁₉×#498	2
C₁₉×#2912	2
C₄×C₁₉×#131	2
#60665	19

Subgroups — C19×#60665 #18066624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#60665 #18066624