Subgroups — C₁₉×#33909 #18039868

Order 4864.

468 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3342	C₂	1	2	yes	maximal
2	C₁₉×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#2237	C₂	1	4	yes	maximal, complemented
4	C₁₉×#2225	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2216	C₂	1	2	yes	maximal, complemented
6	C₄×C₁₉×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×C₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
8	C₁₉×#2158	C₂	1	2	yes	maximal, complemented
9	C₁₉×#556	—	2	2	no	—
10	C₂×C₁₉×Q₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₁₉	C₂²	1	2	yes	—
12	C₁₉×#500	C₂²	1	2	yes	—
13	C₁₉×#485	—	2	4	no	—
14	C₁₉×#475	—	2	2	no	—
15	C₁₉×#467	C₂²	1	4	yes	—
16	C₄×Q₁₆×C₁₉	C₂²	1	2	yes	—
17	C₈×D₄×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#427	C₂²	1	4	yes	—
19	C₁₉×#367	C₂²	1	4	yes	—
20	C₁₉×#366	C₂²	1	1	yes	characteristic
21	C₁₉×#366	C₂²	1	2	yes	—
22	C₁₉×#366	C₂²	1	2	yes	—
23	C₁₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₁₉×#358	C₂²	1	2	yes	—
25	C₁₉×#358	C₂²	1	4	yes	—
26	C₁₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
27	C₄×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
28	C₂×Q₁₆×C₁₉	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
30	C₁₉×C₄⋊Q₈	—	2	2	no	—
31	C₁₉×#126	—	4	4	no	—
32	C₁₉×#124	—	4	2	no	—
33	C₁₉×C₂²⋊Q₈	—	2	4	no	—
34	C₄×Q₈×C₁₉	—	2	2	no	—
35	C₄×Q₈×C₁₉	—	4	2	no	—
36	C₄×D₄×C₁₉	D₄	1	1	yes	characteristic
37	C₁₉×#117	—	2	4	no	—
38	C₁₉×Q₃₂	—	2	2	no	—
39	C₂×C₁₆×C₁₉	—	2	4	no	—
40	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
41	C₂×C₁₆×C₁₉	C₂³	1	2	yes	—
42	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
43	C₁₉×#107	—	2	2	no	—
44	C₁₉×#107	C₂³	1	4	yes	—
45	C₁₉×#107	C₂³	1	2	yes	—
46	C₁₉×#106	—	2	4	no	—
47	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
48	C₁₉×#103	—	2	2	no	—
49	C₁₉×#103	—	2	4	no	—
50	C₁₉×C₂²⋊C₈	D₄	1	2	yes	—
51	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
52	C₂×Q₈×C₁₉	—	2	2	no	—
53	C₂×Q₈×C₁₉	—	4	2	no	—
54	C₂×D₄×C₁₉	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
56	Q₁₆×C₁₉	—	2	2	no	—
57	Q₁₆×C₁₉	—	2	2	no	—
58	C₂×C₈×C₁₉	—	2	2	no	—
59	C₂×C₈×C₁₉	—	2	2	no	—
60	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄×C₁₉	—	2	2	no	—
64	C₄⋊C₄×C₁₉	—	2	4	no	—
65	C₄⋊C₄×C₁₉	—	4	2	no	—
66	C₄⋊C₄×C₁₉	—	4	4	no	—
67	C₄⋊C₄×C₁₉	—	4	2	no	—
68	C₄⋊C₄×C₁₉	D₈	1	1	yes	characteristic
69	C₂²⋊C₄×C₁₉	—	4	4	no	—
70	C₂²⋊C₄×C₁₉	—	4	4	no	—
71	C₂²⋊C₄×C₁₉	D₈	1	2	yes	—
72	C₄²×C₁₉	—	4	4	no	—
73	C₄²×C₁₉	—	4	2	no	—
74	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₁₆×C₁₉	—	2	2	no	—
76	C₁₆×C₁₉	—	2	1	no	—
77	C₁₆×C₁₉	#41	1	2	yes	—
78	#33909	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₁₉	C₂×D₈	1	2	yes	—
80	Q₈×C₁₉	—	4	2	no	—
81	Q₈×C₁₉	—	4	2	no	—
82	Q₈×C₁₉	—	8	4	no	—
83	D₄×C₁₉	—	2	2	no	—
84	C₂×C₄×C₁₉	—	2	2	no	—
85	C₂×C₄×C₁₉	—	2	2	no	—
86	C₂×C₄×C₁₉	—	4	2	no	—
87	C₂×C₄×C₁₉	—	4	4	no	—
88	C₂×C₄×C₁₉	—	4	2	no	—
89	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
93	C₈×C₁₉	—	2	2	no	—
94	C₈×C₁₉	—	2	1	no	—
95	C₈×C₁₉	C₂×#41	1	1	yes	characteristic
96	C₈×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
97	#3342	C₂×C₁₉	1	2	yes	—
98	#2252	C₂×C₁₉	1	1	yes	characteristic, complemented
99	#2237	C₂×C₁₉	1	4	yes	complemented
100	#2225	C₂×C₁₉	1	2	yes	complemented
101	#2216	C₂×C₁₉	1	2	yes	complemented
102	C₄×Q₃₂	C₂×C₁₉	1	1	yes	characteristic, complemented
103	D₄×C₁₆	C₂×C₁₉	1	1	yes	characteristic
104	#2158	C₂×C₁₉	1	2	yes	complemented
105	C₂²×C₁₉	—	2	2	no	—
106	C₂²×C₁₉	—	2	2	no	—
107	C₂²×C₁₉	—	2	2	no	—
108	C₂²×C₁₉	C₂²×D₈	1	1	yes	characteristic
109	C₄×C₁₉	—	2	2	no	—
110	C₄×C₁₉	—	2	1	no	—
111	C₄×C₁₉	—	8	2	no	—
112	C₄×C₁₉	—	8	4	no	—
113	C₄×C₁₉	—	8	2	no	—
114	C₄×C₁₉	#508	1	2	yes	—
115	C₄×C₁₉	#547	1	1	yes	characteristic
116	C₄×C₁₉	#578	1	1	yes	characteristic
117	#556	—	2	2	no	—
118	C₂×Q₃₂	C₂²×C₁₉	1	1	yes	characteristic
119	C₂²×C₁₆	C₂²×C₁₉	1	2	yes	—
120	#500	C₂²×C₁₉	1	2	yes	—
121	#485	—	2	4	no	—
122	#475	—	2	2	no	—
123	#467	C₂²×C₁₉	1	4	yes	—
124	C₄×Q₁₆	C₂²×C₁₉	1	2	yes	—
125	C₈×D₄	C₂²×C₁₉	1	1	yes	characteristic
126	#427	C₂²×C₁₉	1	4	yes	—
127	#367	C₂²×C₁₉	1	4	yes	—
128	#366	C₂²×C₁₉	1	1	yes	characteristic
129	#366	C₂²×C₁₉	1	2	yes	—
130	#366	C₂²×C₁₉	1	2	yes	—
131	C₄⋊C₁₆	C₂²×C₁₉	1	1	yes	characteristic
132	#358	C₂²×C₁₉	1	2	yes	—
133	#358	C₂²×C₁₉	1	4	yes	—
134	C₂²⋊C₁₆	C₂²×C₁₉	1	2	yes	—
135	C₄×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
136	C₂×C₁₉	—	4	2	no	—
137	C₂×C₁₉	#3334	1	1	yes	characteristic
138	C₂×C₁₉	C₂×#508	1	1	yes	characteristic
139	C₂×C₁₉	#3417	1	1	yes	characteristic
140	C₂×Q₁₆	C₂³×C₁₉	1	2	yes	—
141	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
142	C₄⋊Q₈	—	2	2	no	—
143	#126	—	4	4	no	—
144	#124	—	4	2	no	—
145	C₂²⋊Q₈	—	2	4	no	—
146	C₄×Q₈	—	2	2	no	—
147	C₄×Q₈	—	4	2	no	—
148	C₄×D₄	D₄×C₁₉	1	1	yes	characteristic
149	#117	—	2	4	no	—
150	Q₃₂	—	2	2	no	—
151	C₂×C₁₆	—	2	4	no	—
152	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
153	C₂×C₁₆	C₂³×C₁₉	1	2	yes	—
154	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
155	#107	—	2	2	no	—
156	#107	C₂³×C₁₉	1	4	yes	—
157	#107	C₂³×C₁₉	1	2	yes	—
158	#106	—	2	4	no	—
159	C₄⋊C₈	D₄×C₁₉	1	1	yes	characteristic
160	#103	—	2	2	no	—
161	#103	—	2	4	no	—
162	C₂²⋊C₈	D₄×C₁₉	1	2	yes	—
163	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
164	C₁₉	#33909	1	1	yes	characteristic, direct product factor, Sylow
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×D₄	D₈×C₁₉	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
169	Q₁₆	—	2	2	no	—
170	Q₁₆	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
174	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
175	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	4	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	D₈×C₁₉	1	1	yes	characteristic
182	C₂²⋊C₄	—	4	4	no	—
183	C₂²⋊C₄	—	4	4	no	—
184	C₂²⋊C₄	D₈×C₁₉	1	2	yes	—
185	C₄²	—	4	4	no	—
186	C₄²	—	4	2	no	—
187	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
188	C₁₆	—	2	2	no	—
189	C₁₆	—	2	1	no	—
190	C₁₆	#41×C₁₉	1	2	yes	—
191	C₂³	C₂×D₈×C₁₉	1	2	yes	—
192	Q₈	—	4	2	no	—
193	Q₈	—	4	2	no	—
194	Q₈	—	8	4	no	—
195	D₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	C₂×D₈×C₁₉	1	2	yes	—
202	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
203	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
204	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	2	1	no	—
207	C₈	C₂×#41×C₁₉	1	1	yes	characteristic
208	C₈	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	C₂²×D₈×C₁₉	1	1	yes	characteristic
213	C₄	—	2	2	no	—
214	C₄	—	2	1	no	—
215	C₄	—	8	2	no	—
216	C₄	—	8	4	no	—
217	C₄	—	8	2	no	—
218	C₄	C₁₉×#508	1	2	yes	—
219	C₄	C₁₉×#547	1	1	yes	characteristic
220	C₄	C₁₉×#578	1	1	yes	characteristic
221	C₂	—	4	2	no	—
222	C₂	C₁₉×#3334	1	1	yes	characteristic
223	C₂	C₂×C₁₉×#508	1	1	yes	characteristic
224	C₂	C₁₉×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₉
prime radical	C₂×D₄×C₁₉	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄×D₄×C₁₉ ◁ C₈×D₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2252	C₂	semidirect
3. C₁₉×#2237	C₂	semidirect
4. C₁₉×#2225	C₂	semidirect
5. C₁₉×#2216	C₂	semidirect
6. C₄×C₁₉×Q₃₂	C₂	semidirect
8. C₁₉×#2158	C₂	semidirect
78. #33909	C₁₉	direct (N × H)
98. #2252	C₂×C₁₉	semidirect
99. #2237	C₂×C₁₉	semidirect
100. #2225	C₂×C₁₉	semidirect
101. #2216	C₂×C₁₉	semidirect
102. C₄×Q₃₂	C₂×C₁₉	semidirect
104. #2158	C₂×C₁₉	semidirect
164. C₁₉	#33909	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×#3342	2
C₁₉×#2252	2
C₁₉×#2237	2
C₁₉×#2225	2
C₁₉×#2216	2
C₄×C₁₉×Q₃₂	2
D₄×C₁₆×C₁₉	2
C₁₉×#2158	2
#33909	19

Subgroups — C19×#33909 #18039868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#33909 #18039868