Subgroups — C₁₉×#3209 #14561066

Order 2432.

384 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#494	C₂	1	2	yes	maximal, complemented
4	C₁₉×#487	C₂	1	2	yes	maximal, complemented
5	C₁₉×#462	C₂	1	4	yes	maximal, complemented
6	C₁₉×#451	C₂	1	4	yes	maximal
7	C₁₉×#431	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₁₉	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₁₉	C₂²	1	8	yes	—
10	C₁₉×C₄⋊Q₈	—	2	1	no	—
11	C₁₉×C₄⋊Q₈	—	2	1	no	—
12	C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₉×#125	—	2	1	no	—
17	C₁₉×C₂²⋊Q₈	—	2	1	no	—
18	C₁₉×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₄×Q₈×C₁₉	—	2	1	no	—
20	C₄×Q₈×C₁₉	C₂²	1	2	yes	—
21	C₁₉×#117	—	2	1	no	—
22	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
23	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
24	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
25	C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
26	C₁₉×#103	C₂²	1	8	yes	—
27	C₁₉×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₄×C₈×C₁₉	C₂²	1	2	yes	—
29	C₂×Q₈×C₁₉	—	2	1	no	—
30	C₂×Q₈×C₁₉	—	2	2	no	—
31	C₂×Q₈×C₁₉	—	4	4	no	—
32	C₂×Q₈×C₁₉	C₂³	1	2	yes	—
33	C₂×Q₈×C₁₉	C₂³	1	4	yes	—
34	C₂²×C₄×C₁₉	—	2	2	no	—
35	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
38	Q₁₆×C₁₉	—	4	8	no	—
39	C₂×C₈×C₁₉	C₂³	1	4	yes	—
40	C₄⋊C₄×C₁₉	—	2	2	no	—
41	C₄⋊C₄×C₁₉	—	2	2	no	—
42	C₄⋊C₄×C₁₉	—	2	2	no	—
43	C₄⋊C₄×C₁₉	—	2	2	no	—
44	C₄⋊C₄×C₁₉	—	2	1	no	—
45	C₄⋊C₄×C₁₉	—	2	1	no	—
46	C₄⋊C₄×C₁₉	—	2	1	no	—
47	C₄⋊C₄×C₁₉	—	2	2	no	—
48	C₄⋊C₄×C₁₉	—	2	2	no	—
49	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
50	C₂²⋊C₄×C₁₉	—	2	2	no	—
51	C₄²×C₁₉	—	2	2	no	—
52	C₄²×C₁₉	D₄	1	1	yes	characteristic
53	C₄²×C₁₉	D₄	1	1	yes	characteristic
54	C₄²×C₁₉	C₂³	1	1	yes	characteristic
55	C₄²×C₁₉	C₂³	1	1	yes	characteristic
56	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
57	Q₈×C₁₉	—	2	4	no	—
58	Q₈×C₁₉	—	2	8	no	—
59	Q₈×C₁₉	—	4	4	no	—
60	Q₈×C₁₉	—	4	2	no	—
61	C₂×C₄×C₁₉	—	2	2	no	—
62	C₂×C₄×C₁₉	—	2	2	no	—
63	C₂×C₄×C₁₉	—	2	2	no	—
64	C₂×C₄×C₁₉	—	2	2	no	—
65	C₂×C₄×C₁₉	—	2	4	no	—
66	C₂×C₄×C₁₉	—	4	4	no	—
67	C₂×C₄×C₁₉	Q₁₆	1	4	yes	complemented
68	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
74	C₈×C₁₉	—	2	4	no	—
75	#3209	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂²×C₁₉	—	2	1	no	—
77	C₂²×C₁₉	—	2	1	no	—
78	C₂²×C₁₉	C₂×Q₁₆	1	2	yes	—
79	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
80	C₄×C₁₉	—	2	1	no	—
81	C₄×C₁₉	—	2	1	no	—
82	C₄×C₁₉	—	4	4	no	—
83	C₄×C₁₉	—	4	2	no	—
84	C₄×C₁₉	—	4	2	no	—
85	C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
86	C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
87	C₄×C₁₉	#137	1	2	yes	—
88	C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
89	#533	C₂×C₁₉	1	1	yes	characteristic
90	C₂×C₄⋊Q₈	C₂×C₁₉	1	1	yes	characteristic
91	#494	C₂×C₁₉	1	2	yes	complemented
92	#487	C₂×C₁₉	1	2	yes	complemented
93	#462	C₂×C₁₉	1	4	yes	complemented
94	#451	C₂×C₁₉	1	4	yes	—
95	#431	C₂×C₁₉	1	1	yes	characteristic
96	C₂×C₁₉	—	2	2	no	—
97	C₂×C₁₉	#535	1	1	yes	characteristic
98	C₂×C₁₉	C₂²×Q₁₆	1	1	yes	characteristic
99	C₂×C₁₉	C₂×#137	1	1	yes	characteristic
100	C₂²×Q₈	C₂²×C₁₉	1	2	yes	—
101	C₂×Q₁₆	C₂²×C₁₉	1	8	yes	—
102	C₄⋊Q₈	—	2	1	no	—
103	C₄⋊Q₈	—	2	1	no	—
104	C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
105	C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
106	C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
107	C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
108	#125	—	2	1	no	—
109	C₂²⋊Q₈	—	2	1	no	—
110	C₂²⋊Q₈	C₂²×C₁₉	1	2	yes	—
111	C₄×Q₈	—	2	1	no	—
112	C₄×Q₈	C₂²×C₁₉	1	2	yes	—
113	#117	—	2	1	no	—
114	C₂×C₄⋊C₄	—	2	1	no	—
115	C₂×C₄⋊C₄	—	2	1	no	—
116	C₂×C₄²	C₂²×C₁₉	1	1	yes	characteristic
117	C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
118	#103	C₂²×C₁₉	1	8	yes	—
119	C₂²⋊C₈	C₂²×C₁₉	1	2	yes	—
120	C₄×C₈	C₂²×C₁₉	1	2	yes	—
121	C₁₉	#3209	1	1	yes	characteristic, direct product factor, Sylow
122	C₂×Q₈	—	2	1	no	—
123	C₂×Q₈	—	2	2	no	—
124	C₂×Q₈	—	4	4	no	—
125	C₂×Q₈	C₂³×C₁₉	1	4	yes	—
126	C₂×Q₈	C₂³×C₁₉	1	2	yes	—
127	C₂²×C₄	—	2	2	no	—
128	C₂²×C₄	D₄×C₁₉	1	1	yes	characteristic
129	C₂²×C₄	D₄×C₁₉	1	1	yes	characteristic
130	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
131	Q₁₆	—	4	8	no	—
132	C₂×C₈	C₂³×C₁₉	1	4	yes	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	—	2	2	no	—
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	—	2	1	no	—
138	C₄⋊C₄	—	2	1	no	—
139	C₄⋊C₄	—	2	1	no	—
140	C₄⋊C₄	—	2	2	no	—
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
143	C₂²⋊C₄	—	2	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	D₄×C₁₉	1	1	yes	characteristic
146	C₄²	D₄×C₁₉	1	1	yes	characteristic
147	C₄²	C₂³×C₁₉	1	1	yes	characteristic
148	C₄²	C₂³×C₁₉	1	1	yes	characteristic
149	C₂³	C₂×D₄×C₁₉	1	1	yes	characteristic
150	Q₈	—	2	4	no	—
151	Q₈	—	2	8	no	—
152	Q₈	—	4	4	no	—
153	Q₈	—	4	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	Q₁₆×C₁₉	1	4	yes	complemented
161	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
162	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
163	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
164	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
165	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
166	C₂×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
167	C₈	—	2	4	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	C₂×Q₁₆×C₁₉	1	2	yes	—
171	C₂²	C₂²×D₄×C₁₉	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	4	4	no	—
175	C₄	—	4	2	no	—
176	C₄	—	4	2	no	—
177	C₄	C₂×Q₁₆×C₁₉	1	2	yes	—
178	C₄	C₂×Q₁₆×C₁₉	1	2	yes	—
179	C₄	C₁₉×#137	1	2	yes	—
180	C₄	C₁₉×H(5,𝔽₂)	1	2	yes	—
181	C₂	—	2	2	no	—
182	C₂	C₁₉×#535	1	1	yes	characteristic
183	C₂	C₂²×Q₁₆×C₁₉	1	1	yes	characteristic
184	C₂	C₂×C₁₉×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₉
prime radical	C₂³×C₁₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#494	C₂	semidirect
4. C₁₉×#487	C₂	semidirect
5. C₁₉×#462	C₂	semidirect
67. C₂×C₄×C₁₉	Q₁₆	semidirect
75. #3209	C₁₉	direct (N × H)
91. #494	C₂×C₁₉	semidirect
92. #487	C₂×C₁₉	semidirect
93. #462	C₂×C₁₉	semidirect
121. C₁₉	#3209	direct (N × H)
160. C₂×C₄	Q₁₆×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#533	2
C₂×C₁₉×C₄⋊Q₈	2
C₁₉×#494	2
C₁₉×#487	2
C₁₉×#462	2
C₁₉×#451	2
C₁₉×#431	2
#3209	19

Subgroups — C19×#3209 #14561066

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#3209 #14561066