Subgroups — C₁₉×#12907 #18018866

Order 4864.

342 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#367	C₂	1	2	yes	maximal
2	C₂×C₁₉×#366	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1846	C₂	1	2	yes	maximal
5	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
6	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
7	C₂×C₁₉×#107	C₂²	1	4	yes	—
8	C₂×C₁₉×#106	—	2	2	no	—
9	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
10	C₁₉×#382	—	2	2	no	—
11	C₁₉×#367	—	2	4	no	—
12	C₁₉×#366	—	2	4	no	—
13	C₄×C₁₆×C₁₉	C₄	1	4	yes	complemented
14	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
17	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₁₉	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₁₉	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₁₉	D₄	1	2	yes	—
21	C₂×C₁₆×C₁₉	D₄	1	2	yes	—
22	C₂×C₁₆×C₁₉	Q₈	1	2	yes	—
23	C₂×C₁₆×C₁₉	Q₈	1	2	yes	—
24	C₁₉×#107	—	2	4	no	—
25	C₁₉×#107	—	2	4	no	—
26	C₁₉×#106	—	4	4	no	—
27	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₉	Q₈	1	2	yes	—
29	C₁₉×#95	—	4	4	no	—
30	C₂²×C₄×C₁₉	—	4	4	no	—
31	C₂²×C₄×C₁₉	D₈	1	2	yes	—
32	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
34	C₂×C₈×C₁₉	D₈	1	2	yes	—
35	C₂×C₈×C₁₉	Q₁₆	1	2	yes	—
36	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
42	C₄⋊C₄×C₁₉	—	4	4	no	—
43	C₄⋊C₄×C₁₉	—	4	4	no	—
44	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
45	C₄²×C₁₉	Q₁₆	1	2	yes	—
46	C₁₆×C₁₉	C₄⋊C₄	1	4	yes	—
47	C₁₆×C₁₉	C₄⋊C₄	1	4	yes	—
48	#12907	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₁₉	—	2	4	no	—
51	C₂×C₄×C₁₉	—	8	4	no	—
52	C₂×C₄×C₁₉	—	8	8	no	—
53	C₂×C₄×C₁₉	#107	1	4	yes	—
54	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
56	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
60	C₈×C₁₉	#107	1	4	yes	—
61	C₈×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₈×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₈×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₈×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×#367	C₂×C₁₉	1	2	yes	—
66	C₂×#366	C₂×C₁₉	1	2	yes	—
67	C₂×C₄×C₁₆	C₂×C₁₉	1	1	yes	characteristic
68	#1846	C₂×C₁₉	1	2	yes	—
69	C₂²×C₁₉	C₂×#107	1	1	yes	characteristic
70	C₂²×C₁₉	#493	1	1	yes	characteristic
71	C₂²×C₁₉	#500	1	1	yes	characteristic
72	C₂²×C₁₉	#508	1	4	yes	—
73	C₄×C₁₉	—	2	4	no	—
74	C₄×C₁₉	—	16	4	no	—
75	C₄×C₁₉	#384	1	1	yes	characteristic
76	C₄×C₁₉	C₂×#107	1	1	yes	characteristic
77	C₄×C₁₉	#494	1	1	yes	characteristic
78	C₄×C₁₉	#500	1	1	yes	characteristic
79	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
80	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
81	C₂×#107	C₂²×C₁₉	1	4	yes	—
82	C₂×#106	—	2	2	no	—
83	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
84	#382	—	2	2	no	—
85	#367	—	2	4	no	—
86	#366	—	2	4	no	—
87	C₄×C₁₆	C₄×C₁₉	1	4	yes	complemented
88	C₂×C₁₉	#1845	1	1	yes	characteristic
89	C₂×C₁₉	#2158	1	2	yes	—
90	C₂×C₁₉	#2249	1	2	yes	—
91	C₂×C₁₉	#2253	1	2	yes	—
92	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
93	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
94	C₂×C₄⋊C₄	—	2	4	no	—
95	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
96	C₂×C₁₆	C₂×C₄×C₁₉	1	2	yes	—
97	C₂×C₁₆	C₂×C₄×C₁₉	1	2	yes	—
98	C₂×C₁₆	D₄×C₁₉	1	2	yes	—
99	C₂×C₁₆	D₄×C₁₉	1	2	yes	—
100	C₂×C₁₆	Q₈×C₁₉	1	2	yes	—
101	C₂×C₁₆	Q₈×C₁₉	1	2	yes	—
102	#107	—	2	4	no	—
103	#107	—	2	4	no	—
104	#106	—	4	4	no	—
105	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
106	C₄×C₈	Q₈×C₁₉	1	2	yes	—
107	#95	—	4	4	no	—
108	C₁₉	#12907	1	1	yes	characteristic, direct product factor, Sylow
109	C₂²×C₄	—	4	4	no	—
110	C₂²×C₄	D₈×C₁₉	1	2	yes	—
111	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
112	C₂×C₈	C₄⋊C₄×C₁₉	1	2	yes	—
113	C₂×C₈	D₈×C₁₉	1	2	yes	—
114	C₂×C₈	Q₁₆×C₁₉	1	2	yes	—
115	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
116	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
117	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
118	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
119	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
120	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
121	C₄⋊C₄	—	4	4	no	—
122	C₄⋊C₄	—	4	4	no	—
123	C₄²	C₄⋊C₄×C₁₉	1	2	yes	—
124	C₄²	Q₁₆×C₁₉	1	2	yes	—
125	C₁₆	C₄⋊C₄×C₁₉	1	4	yes	—
126	C₁₆	C₄⋊C₄×C₁₉	1	4	yes	—
127	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	8	4	no	—
130	C₂×C₄	—	8	8	no	—
131	C₂×C₄	C₁₉×#107	1	4	yes	—
132	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
133	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
134	C₂×C₄	C₁₉×C₄⋊Q₈	1	1	yes	characteristic
135	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
136	C₂×C₄	C₂×Q₁₆×C₁₉	1	1	yes	characteristic
137	C₂×C₄	C₂×Q₁₆×C₁₉	1	1	yes	characteristic
138	C₈	C₁₉×#107	1	4	yes	—
139	C₈	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
140	C₈	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
141	C₈	C₁₉×C₄⋊Q₈	1	1	yes	characteristic
142	C₈	C₁₉×C₄⋊Q₈	1	1	yes	characteristic
143	C₂²	C₂×C₁₉×#107	1	1	yes	characteristic
144	C₂²	C₁₉×#493	1	1	yes	characteristic
145	C₂²	C₁₉×#500	1	1	yes	characteristic
146	C₂²	C₁₉×#508	1	4	yes	—
147	C₄	—	2	4	no	—
148	C₄	—	16	4	no	—
149	C₄	C₁₉×#384	1	1	yes	characteristic
150	C₄	C₂×C₁₉×#107	1	1	yes	characteristic
151	C₄	C₁₉×#494	1	1	yes	characteristic
152	C₄	C₁₉×#500	1	1	yes	characteristic
153	C₂	C₁₉×#1845	1	1	yes	characteristic
154	C₂	C₁₉×#2158	1	2	yes	—
155	C₂	C₁₉×#2249	1	2	yes	—
156	C₂	C₁₉×#2253	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄×C₁₆×C₁₉	C₄	semidirect
48. #12907	C₁₉	direct (N × H)
87. C₄×C₁₆	C₄×C₁₉	semidirect
108. C₁₉	#12907	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#367	2
C₂×C₁₉×#366	2
C₂×C₄×C₁₆×C₁₉	2
C₁₉×#1846	2
#12907	19

Subgroups — C19×#12907 #18018866

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12907 #18018866