Subgroups — C₁₉×#12014 #18017973

Order 4864.

220 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2163	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2162	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1450	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1370	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1367	C₂	1	2	yes	maximal, complemented
6	C₁₉×#431	C₄	1	1	yes	characteristic
7	C₁₉×#431	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
9	C₁₉×C₄⋊C₁₆	C₂²	1	2	yes	—
10	C₁₉×C₄⋊C₁₆	C₂²	1	2	yes	—
11	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
12	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
13	C₁₉×#346	—	2	1	no	—
14	C₄×C₁₆×C₁₉	—	2	1	no	—
15	C₁₉×#322	C₄	1	1	yes	characteristic
16	C₁₉×#322	C₄	1	1	yes	characteristic
17	C₈²×C₁₉	C₂²	1	1	yes	characteristic
18	C₈²×C₁₉	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₉	—	2	2	no	—
23	C₂×C₁₆×C₁₉	—	2	2	no	—
24	C₁₉×C₄⋊C₈	—	2	1	no	—
25	C₁₉×C₄⋊C₈	C₈	1	2	yes	—
26	C₁₉×C₂²⋊C₈	—	2	1	no	—
27	C₁₉×C₂²⋊C₈	C₈	1	2	yes	—
28	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
34	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
35	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₉	M₁₆	1	1	yes	characteristic
39	C₂×C₈×C₁₉	—	2	2	no	—
40	C₂×C₈×C₁₉	—	2	2	no	—
41	C₂×C₈×C₁₉	—	2	1	no	—
42	C₂×C₈×C₁₉	—	2	1	no	—
43	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₁₉	C₂×C₈	1	1	yes	characteristic
51	C₄²×C₁₉	M₁₆	1	1	yes	characteristic
52	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
53	C₁₆×C₁₉	—	4	2	no	—
54	C₁₆×C₁₉	—	4	2	no	—
55	#12014	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂³×C₁₉	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₉	—	2	1	no	—
58	C₂×C₄×C₁₉	—	2	1	no	—
59	C₂×C₄×C₁₉	—	2	2	no	—
60	C₂×C₄×C₁₉	—	4	1	no	—
61	C₂×C₄×C₁₉	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₉	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₉	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
67	C₈×C₁₉	—	2	1	no	—
68	C₈×C₁₉	—	2	1	no	—
69	C₈×C₁₉	—	2	2	no	—
70	C₈×C₁₉	—	2	2	no	—
71	C₈×C₁₉	—	4	1	no	—
72	C₈×C₁₉	—	4	1	no	—
73	C₈×C₁₉	#108	1	2	yes	—
74	C₈×C₁₉	#135	1	2	yes	—
75	#2163	C₂×C₁₉	1	1	yes	characteristic
76	#2162	C₂×C₁₉	1	1	yes	characteristic
77	#1450	C₂×C₁₉	1	1	yes	characteristic
78	#1370	C₂×C₁₉	1	2	yes	complemented
79	#1367	C₂×C₁₉	1	2	yes	complemented
80	C₂²×C₁₉	—	2	1	no	—
81	C₂²×C₁₉	—	2	1	no	—
82	C₂²×C₁₉	—	2	1	no	—
83	C₂²×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₄×C₁₉	—	2	1	no	—
85	C₄×C₁₉	—	2	1	no	—
86	C₄×C₁₉	—	2	1	no	—
87	C₄×C₁₉	—	2	1	no	—
88	C₄×C₁₉	#427	1	1	yes	characteristic
89	C₄×C₁₉	C₂×#108	1	1	yes	characteristic
90	C₄×C₁₉	#504	1	2	yes	—
91	#431	C₄×C₁₉	1	1	yes	characteristic
92	#431	C₄×C₁₉	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
94	C₄⋊C₁₆	C₂²×C₁₉	1	2	yes	—
95	C₄⋊C₁₆	C₂²×C₁₉	1	2	yes	—
96	C₂²⋊C₁₆	—	2	1	no	—
97	C₂²⋊C₁₆	—	2	1	no	—
98	#346	—	2	1	no	—
99	C₄×C₁₆	—	2	1	no	—
100	#322	C₄×C₁₉	1	1	yes	characteristic
101	#322	C₄×C₁₉	1	1	yes	characteristic
102	C₈²	C₂²×C₁₉	1	1	yes	characteristic
103	C₈²	C₂²×C₁₉	1	1	yes	characteristic
104	C₂×C₁₉	—	4	1	no	—
105	C₂×C₁₉	#1564	1	1	yes	characteristic
106	C₂×C₁₉	#2151	1	1	yes	characteristic
107	C₂×C₁₉	#2153	1	1	yes	characteristic
108	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
109	C₂²×C₈	Q₈×C₁₉	1	1	yes	characteristic
110	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	2	2	no	—
113	C₄⋊C₈	—	2	1	no	—
114	C₄⋊C₈	C₈×C₁₉	1	2	yes	—
115	C₂²⋊C₈	—	2	1	no	—
116	C₂²⋊C₈	C₈×C₁₉	1	2	yes	—
117	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
118	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
119	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
120	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
121	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
122	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
123	C₄×C₈	Q₈×C₁₉	1	1	yes	characteristic
124	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
125	C₁₉	#12014	1	1	yes	characteristic, direct product factor, Sylow
126	C₂²×C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
127	C₂²×C₄	C₂×C₈×C₁₉	1	1	yes	characteristic
128	C₂²×C₄	M₁₆×C₁₉	1	1	yes	characteristic
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
134	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
135	C₂×C₈	C₂×C₈×C₁₉	1	2	yes	—
136	C₂×C₈	C₂×C₈×C₁₉	1	2	yes	—
137	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
138	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
139	C₄²	C₄⋊C₄×C₁₉	1	1	yes	characteristic
140	C₄²	C₂×C₈×C₁₉	1	1	yes	characteristic
141	C₄²	M₁₆×C₁₉	1	1	yes	characteristic
142	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
143	C₁₆	—	4	2	no	—
144	C₁₆	—	4	2	no	—
145	C₂³	C₁₉×C₄⋊C₈	1	1	yes	characteristic
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	C₁₉×C₄⋊C₈	1	1	yes	characteristic
151	C₂×C₄	C₁₉×C₄⋊C₈	1	1	yes	characteristic
152	C₂×C₄	C₁₉×C₄⋊C₈	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
154	C₂×C₄	C₂²×C₈×C₁₉	1	1	yes	characteristic
155	C₂×C₄	C₂×M₁₆×C₁₉	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	2	no	—
159	C₈	—	2	2	no	—
160	C₈	—	4	1	no	—
161	C₈	—	4	1	no	—
162	C₈	C₁₉×#108	1	2	yes	—
163	C₈	C₁₉×#135	1	2	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	C₂×C₁₉×C₄⋊C₈	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	C₁₉×#427	1	1	yes	characteristic
173	C₄	C₂×C₁₉×#108	1	1	yes	characteristic
174	C₄	C₁₉×#504	1	2	yes	—
175	C₂	—	4	1	no	—
176	C₂	C₁₉×#1564	1	1	yes	characteristic
177	C₂	C₁₉×#2151	1	1	yes	characteristic
178	C₂	C₁₉×#2153	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₉
Frattini, cosocle	C₄×C₈	C₂³×C₁₉
socle	C₂²×C₁₉	C₂×C₄⋊C₈
prime radical	C₂³×C₁₉	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₉×#1370	C₂	semidirect
5. C₁₉×#1367	C₂	semidirect
55. #12014	C₁₉	direct (N × H)
78. #1370	C₂×C₁₉	semidirect
79. #1367	C₂×C₁₉	semidirect
125. C₁₉	#12014	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×#2163	2
C₁₉×#2162	2
C₁₉×#1450	2
C₁₉×#1370	2
C₁₉×#1367	2
#12014	19

Subgroups — C19×#12014 #18017973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12014 #18017973