Subgroups — C₁₉×#12004 #18017963

Order 4864.

270 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2150	C₂	1	2	yes	maximal
2	C₂×C₈²×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1370	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₉×M₃₂	—	2	2	no	—
5	C₂×C₄×C₈×C₁₉	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₉	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
8	C₁₉×C₄⋊C₁₆	—	2	2	no	—
9	C₁₉×C₄⋊C₁₆	C₂²	1	4	yes	—
10	C₈²×C₁₉	C₄	1	2	yes	—
11	C₈²×C₁₉	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₉	—	2	1	no	—
13	C₂²×C₈×C₁₉	C₈	1	2	yes	—
14	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
16	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₁₉×M₃₂	—	4	4	no	—
18	C₂×C₁₆×C₁₉	—	2	4	no	—
19	C₄×C₈×C₁₉	—	2	1	no	—
20	C₄×C₈×C₁₉	C₈	1	1	yes	characteristic
21	C₄×C₈×C₁₉	C₈	1	1	yes	characteristic
22	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
26	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₁₉	M₁₆	1	1	yes	characteristic
31	C₂×C₈×C₁₉	—	2	4	no	—
32	C₂×C₈×C₁₉	—	2	2	no	—
33	C₂×C₈×C₁₉	—	2	2	no	—
34	C₂×C₈×C₁₉	—	2	4	no	—
35	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₁₉	D₈	1	2	yes	—
40	C₂×C₈×C₁₉	Q₁₆	1	2	yes	—
41	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₁₉	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₁₉	M₁₆	1	1	yes	characteristic
46	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
47	C₁₆×C₁₉	—	4	4	no	—
48	#12004	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂³×C₁₉	C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₁₉	—	2	2	no	—
51	C₂×C₄×C₁₉	—	2	2	no	—
52	C₂×C₄×C₁₉	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₉	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₁₉	C₄⋊C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₉	#107	1	2	yes	—
56	C₂×C₄×C₁₉	#108	1	2	yes	—
57	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
60	C₈×C₁₉	—	2	4	no	—
61	C₈×C₁₉	—	2	1	no	—
62	C₈×C₁₉	—	2	1	no	—
63	C₈×C₁₉	—	2	2	no	—
64	C₈×C₁₉	#107	1	2	yes	—
65	C₈×C₁₉	M₃₂	1	4	yes	complemented
66	C₈×C₁₉	C₂×D₈	1	1	yes	characteristic
67	C₈×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
68	#2150	C₂×C₁₉	1	2	yes	—
69	C₂×C₈²	C₂×C₁₉	1	1	yes	characteristic
70	#1370	C₂×C₁₉	1	4	yes	complemented
71	C₂²×C₁₉	—	2	1	no	—
72	C₂²×C₁₉	—	2	1	no	—
73	C₂²×C₁₉	#335	1	2	yes	—
74	C₂²×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
75	C₄×C₁₉	—	2	1	no	—
76	C₄×C₁₉	—	2	1	no	—
77	C₄×C₁₉	#335	1	2	yes	—
78	C₄×C₁₉	#364	1	2	yes	—
79	C₄×C₁₉	C₂×#107	1	1	yes	characteristic
80	C₄×C₁₉	C₂×#108	1	1	yes	characteristic
81	C₄×C₁₉	C₂×M₃₂	1	2	yes	—
82	C₂×M₃₂	—	2	2	no	—
83	C₂×C₄×C₈	C₄×C₁₉	1	1	yes	characteristic
84	C₂×C₄×C₈	C₄×C₁₉	1	1	yes	characteristic
85	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
86	C₄⋊C₁₆	—	2	2	no	—
87	C₄⋊C₁₆	C₂²×C₁₉	1	4	yes	—
88	C₈²	C₄×C₁₉	1	2	yes	—
89	C₈²	C₂²×C₁₉	1	2	yes	—
90	C₂×C₁₉	—	2	2	no	—
91	C₂×C₁₉	C₂×#335	1	1	yes	characteristic
92	C₂×C₁₉	#2150	1	1	yes	characteristic
93	C₂×C₁₉	C₂×#364	1	1	yes	characteristic
94	C₂²×C₈	—	2	1	no	—
95	C₂²×C₈	C₈×C₁₉	1	2	yes	—
96	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
97	C₂²×C₈	Q₈×C₁₉	1	1	yes	characteristic
98	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
99	M₃₂	—	4	4	no	—
100	C₂×C₁₆	—	2	4	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	C₈×C₁₉	1	1	yes	characteristic
103	C₄×C₈	C₈×C₁₉	1	1	yes	characteristic
104	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
105	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
106	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
107	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
108	C₄×C₈	Q₈×C₁₉	1	1	yes	characteristic
109	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
110	C₁₉	#12004	1	1	yes	characteristic, direct product factor, Sylow
111	C₂²×C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₈×C₁₉	1	1	yes	characteristic
113	C₂²×C₄	M₁₆×C₁₉	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
119	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
120	C₂×C₈	C₂×C₈×C₁₉	1	2	yes	—
121	C₂×C₈	C₂×C₈×C₁₉	1	2	yes	—
122	C₂×C₈	D₈×C₁₉	1	2	yes	—
123	C₂×C₈	Q₁₆×C₁₉	1	2	yes	—
124	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
125	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
126	C₄²	C₄⋊C₄×C₁₉	1	1	yes	characteristic
127	C₄²	C₂×C₈×C₁₉	1	1	yes	characteristic
128	C₄²	M₁₆×C₁₉	1	1	yes	characteristic
129	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
130	C₁₆	—	4	4	no	—
131	C₂³	C₁₉×C₄⋊C₈	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	C₁₉×C₄⋊C₈	1	1	yes	characteristic
135	C₂×C₄	C₁₉×C₄⋊C₈	1	1	yes	characteristic
136	C₂×C₄	C₁₉×C₄⋊C₈	1	1	yes	characteristic
137	C₂×C₄	C₁₉×#107	1	2	yes	—
138	C₂×C₄	C₁₉×#108	1	2	yes	—
139	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
140	C₂×C₄	C₂²×C₈×C₁₉	1	1	yes	characteristic
141	C₂×C₄	C₂×M₁₆×C₁₉	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	2	no	—
146	C₈	C₁₉×#107	1	2	yes	—
147	C₈	C₁₉×M₃₂	1	4	yes	complemented
148	C₈	C₂×D₈×C₁₉	1	1	yes	characteristic
149	C₈	C₂×Q₁₆×C₁₉	1	1	yes	characteristic
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	C₁₉×#335	1	2	yes	—
153	C₂²	C₂×C₁₉×C₄⋊C₈	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	C₁₉×#335	1	2	yes	—
157	C₄	C₁₉×#364	1	2	yes	—
158	C₄	C₂×C₁₉×#107	1	1	yes	characteristic
159	C₄	C₂×C₁₉×#108	1	1	yes	characteristic
160	C₄	C₂×C₁₉×M₃₂	1	2	yes	—
161	C₂	—	2	2	no	—
162	C₂	C₂×C₁₉×#335	1	1	yes	characteristic
163	C₂	C₁₉×#2150	1	1	yes	characteristic
164	C₂	C₂×C₁₉×#364	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
3. C₁₉×#1370	C₂	semidirect
48. #12004	C₁₉	direct (N × H)
65. C₈×C₁₉	M₃₂	semidirect
70. #1370	C₂×C₁₉	semidirect
110. C₁₉	#12004	direct (N × H)
147. C₈	C₁₉×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#2150	2
C₂×C₈²×C₁₉	2
C₁₉×#1370	2
#12004	19

Subgroups — C19×#12004 #18017963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12004 #18017963