Subgroups — C₁₉×#13291 #18019250

Order 4864.

306 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3163	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2110	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1833	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1389	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1386	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1384	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1383	C₂	1	1	yes	maximal, characteristic
8	C₁₉×#544	—	2	1	no	—
9	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
11	C₁₉×#501	—	2	2	no	—
12	C₁₉×#499	C₄	1	2	yes	complemented
13	C₁₉×#498	C₄	1	2	yes	complemented
14	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
15	C₁₉×#428	—	2	1	no	—
16	C₁₉×#427	C₂²	1	1	yes	characteristic
17	C₁₉×#427	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×#106	C₂²	1	1	yes	characteristic
19	C₁₉×#405	C₂²	1	1	yes	characteristic
20	C₁₉×#347	—	2	2	no	—
21	C₁₉×#344	—	2	1	no	—
22	C₁₉×H(3,ℤ/4)	—	2	1	no	—
23	C₂×M₁₆×C₁₉	—	2	1	no	—
24	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
26	C₁₉×C₄⋊Q₈	—	2	2	no	—
27	C₁₉×C₄⋊Q₈	—	4	1	no	—
28	C₁₉×#125	—	2	2	no	—
29	C₁₉×#125	—	4	1	no	—
30	C₁₉×C₂²⋊Q₈	—	4	2	no	—
31	C₁₉×#117	—	2	1	no	—
32	C₁₉×#117	—	2	1	no	—
33	C₁₉×#117	D₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
35	C₁₉×M₃₂	—	2	1	no	—
36	C₁₉×M₃₂	—	2	1	no	—
37	C₁₉×M₃₂	—	2	1	no	—
38	C₁₉×M₃₂	Q₈	1	2	yes	complemented
39	C₂×C₁₆×C₁₉	—	2	1	no	—
40	C₂×C₁₆×C₁₉	Q₈	1	2	yes	complemented
41	C₁₉×#108	—	2	1	no	—
42	C₁₉×#108	—	2	1	no	—
43	C₁₉×#107	—	2	2	no	—
44	C₁₉×#107	—	2	1	no	—
45	C₁₉×#107	—	2	1	no	—
46	C₁₉×#106	—	2	2	no	—
47	C₁₉×#106	—	2	1	no	—
48	C₁₉×#106	—	2	1	no	—
49	C₁₉×#106	C₂×C₄	1	2	yes	—
50	C₁₉×#106	C₂×C₄	1	1	yes	characteristic
51	C₁₉×#106	C₂×C₄	1	1	yes	characteristic
52	C₁₉×#97	D₄	1	2	yes	—
53	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
54	C₂×Q₈×C₁₉	—	4	2	no	—
55	C₂²×C₄×C₁₉	—	4	1	no	—
56	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
57	M₁₆×C₁₉	—	2	1	no	—
58	M₁₆×C₁₉	—	2	1	no	—
59	M₁₆×C₁₉	—	4	1	no	—
60	M₁₆×C₁₉	—	4	1	no	—
61	C₂×C₈×C₁₉	—	4	1	no	—
62	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
68	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
69	C₄⋊C₄×C₁₉	—	2	1	no	—
70	C₄⋊C₄×C₁₉	—	2	2	no	—
71	C₄⋊C₄×C₁₉	—	2	1	no	—
72	C₄⋊C₄×C₁₉	—	2	1	no	—
73	C₄⋊C₄×C₁₉	—	4	2	no	—
74	C₄⋊C₄×C₁₉	—	4	2	no	—
75	C₄⋊C₄×C₁₉	—	4	2	no	—
76	C₄⋊C₄×C₁₉	—	4	1	no	—
77	C₄⋊C₄×C₁₉	—	4	1	no	—
78	C₂²⋊C₄×C₁₉	—	2	1	no	—
79	C₂²⋊C₄×C₁₉	—	4	1	no	—
80	C₂²⋊C₄×C₁₉	—	4	1	no	—
81	C₄²×C₁₉	—	4	1	no	—
82	C₄²×C₁₉	—	4	1	no	—
83	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
84	C₁₆×C₁₉	—	2	2	no	—
85	C₁₆×C₁₉	—	2	2	no	—
86	#13291	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
88	Q₈×C₁₉	—	16	2	no	—
89	C₂×C₄×C₁₉	—	2	2	no	—
90	C₂×C₄×C₁₉	—	4	1	no	—
91	C₂×C₄×C₁₉	—	4	1	no	—
92	C₂×C₄×C₁₉	—	4	2	no	—
93	C₂×C₄×C₁₉	—	8	1	no	—
94	C₂×C₄×C₁₉	—	16	1	no	—
95	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
97	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
100	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
101	C₈×C₁₉	—	2	2	no	—
102	C₈×C₁₉	—	8	1	no	—
103	C₈×C₁₉	C₄×Q₈	1	1	yes	characteristic
104	C₈×C₁₉	C₄×Q₈	1	1	yes	characteristic
105	C₈×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
106	C₈×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
107	#3163	C₂×C₁₉	1	1	yes	characteristic
108	#2110	C₂×C₁₉	1	1	yes	characteristic
109	#1833	C₂×C₁₉	1	1	yes	characteristic
110	#1389	C₂×C₁₉	1	1	yes	characteristic
111	#1386	C₂×C₁₉	1	1	yes	characteristic
112	#1384	C₂×C₁₉	1	1	yes	characteristic
113	#1383	C₂×C₁₉	1	1	yes	characteristic
114	C₂²×C₁₉	—	4	1	no	—
115	C₂²×C₁₉	#418	1	1	yes	characteristic
116	C₂²×C₁₉	#489	1	1	yes	characteristic
117	C₂²×C₁₉	#501	1	1	yes	characteristic
118	C₄×C₁₉	—	4	2	no	—
119	C₄×C₁₉	—	8	1	no	—
120	C₄×C₁₉	—	8	2	no	—
121	C₄×C₁₉	—	8	1	no	—
122	C₄×C₁₉	—	16	1	no	—
123	C₄×C₁₉	#391	1	1	yes	characteristic
124	C₄×C₁₉	#419	1	1	yes	characteristic
125	C₄×C₁₉	#489	1	1	yes	characteristic
126	C₄×C₁₉	#501	1	1	yes	characteristic
127	#544	—	2	1	no	—
128	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
129	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
130	#501	—	2	2	no	—
131	#499	C₄×C₁₉	1	2	yes	complemented
132	#498	C₄×C₁₉	1	2	yes	complemented
133	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
134	#428	—	2	1	no	—
135	#427	C₂²×C₁₉	1	1	yes	characteristic
136	#427	C₂²×C₁₉	1	1	yes	characteristic
137	C₂×#106	C₂²×C₁₉	1	1	yes	characteristic
138	#405	C₂²×C₁₉	1	1	yes	characteristic
139	#347	—	2	2	no	—
140	#344	—	2	1	no	—
141	H(3,ℤ/4)	—	2	1	no	—
142	C₂×C₁₉	—	2	1	no	—
143	C₂×C₁₉	—	2	1	no	—
144	C₂×C₁₉	—	2	1	no	—
145	C₂×C₁₉	#1993	1	1	yes	characteristic
146	C₂×M₁₆	—	2	1	no	—
147	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
148	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	—	4	1	no	—
151	#125	—	2	2	no	—
152	#125	—	4	1	no	—
153	C₂²⋊Q₈	—	4	2	no	—
154	#117	—	2	1	no	—
155	#117	—	2	1	no	—
156	#117	D₄×C₁₉	1	1	yes	characteristic
157	C₂×C₄⋊C₄	—	2	1	no	—
158	M₃₂	—	2	1	no	—
159	M₃₂	—	2	1	no	—
160	M₃₂	—	2	1	no	—
161	M₃₂	Q₈×C₁₉	1	2	yes	complemented
162	C₂×C₁₆	—	2	1	no	—
163	C₂×C₁₆	Q₈×C₁₉	1	2	yes	complemented
164	#108	—	2	1	no	—
165	#108	—	2	1	no	—
166	#107	—	2	2	no	—
167	#107	—	2	1	no	—
168	#107	—	2	1	no	—
169	#106	—	2	2	no	—
170	#106	—	2	1	no	—
171	#106	—	2	1	no	—
172	#106	C₂×C₄×C₁₉	1	2	yes	—
173	#106	C₂×C₄×C₁₉	1	1	yes	characteristic
174	#106	C₂×C₄×C₁₉	1	1	yes	characteristic
175	#97	D₄×C₁₉	1	2	yes	—
176	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
177	C₁₉	#13291	1	1	yes	characteristic, direct product factor, Sylow
178	C₂×Q₈	—	4	2	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
181	M₁₆	—	2	1	no	—
182	M₁₆	—	2	1	no	—
183	M₁₆	—	4	1	no	—
184	M₁₆	—	4	1	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	C₂²⋊C₄×C₁₉	1	2	yes	—
187	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
188	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
189	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
190	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
191	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
192	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	1	no	—
201	C₄⋊C₄	—	4	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	4	1	no	—
204	C₂²⋊C₄	—	4	1	no	—
205	C₄²	—	4	1	no	—
206	C₄²	—	4	1	no	—
207	C₄²	C₂²⋊C₄×C₁₉	1	2	yes	—
208	C₁₆	—	2	2	no	—
209	C₁₆	—	2	2	no	—
210	C₂³	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
211	Q₈	—	16	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	8	1	no	—
217	C₂×C₄	—	16	1	no	—
218	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
219	C₂×C₄	C₁₉×#124	1	1	yes	characteristic
220	C₂×C₄	C₁₉×C₄⋊Q₈	1	1	yes	characteristic
221	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
223	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	8	1	no	—
226	C₈	C₄×Q₈×C₁₉	1	1	yes	characteristic
227	C₈	C₄×Q₈×C₁₉	1	1	yes	characteristic
228	C₈	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
229	C₈	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
230	C₂²	—	4	1	no	—
231	C₂²	C₁₉×#418	1	1	yes	characteristic
232	C₂²	C₁₉×#489	1	1	yes	characteristic
233	C₂²	C₁₉×#501	1	1	yes	characteristic
234	C₄	—	4	2	no	—
235	C₄	—	8	2	no	—
236	C₄	—	8	1	no	—
237	C₄	—	8	1	no	—
238	C₄	—	16	1	no	—
239	C₄	C₁₉×#391	1	1	yes	characteristic
240	C₄	C₁₉×#419	1	1	yes	characteristic
241	C₄	C₁₉×#489	1	1	yes	characteristic
242	C₄	C₁₉×#501	1	1	yes	characteristic
243	C₂	—	2	1	no	—
244	C₂	—	2	1	no	—
245	C₂	—	2	1	no	—
246	C₂	C₁₉×#1993	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#1993
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂³×C₁₉	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#405 ◁ G

Complements

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

N	H ≅ G/N	type
12. C₁₉×#499	C₄	semidirect
13. C₁₉×#498	C₄	semidirect
38. C₁₉×M₃₂	Q₈	semidirect
40. C₂×C₁₆×C₁₉	Q₈	semidirect
86. #13291	C₁₉	direct (N × H)
131. #499	C₄×C₁₉	semidirect
132. #498	C₄×C₁₉	semidirect
161. M₃₂	Q₈×C₁₉	semidirect
163. C₂×C₁₆	Q₈×C₁₉	semidirect
177. C₁₉	#13291	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉×#3163	2
C₁₉×#2110	2
C₁₉×#1833	2
C₁₉×#1389	2
C₁₉×#1386	2
C₁₉×#1384	2
C₁₉×#1383	2
#13291	19

Subgroups — C19×#13291 #18019250

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13291 #18019250