Subgroups — C₁₉×#33404 #18039363

Order 4864.

636 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3407	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2916	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2912	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#2155	C₂	1	6	yes	maximal, complemented
5	C₂×C₁₉×#365	C₂	1	6	yes	maximal, complemented
6	C₁₉×#568	—	2	1	no	—
7	C₂×C₁₉×#131	C₂²	1	1	yes	characteristic
8	C₁₉×#520	—	2	1	no	—
9	C₁₉×#504	C₄	1	8	yes	complemented
10	C₂×C₁₉×M₃₂	C₂²	1	6	yes	—
11	C₁₉×#430	—	2	3	no	—
12	C₂×C₁₉×#108	C₂²	1	6	yes	—
13	C₁₉×#428	—	2	3	no	—
14	C₁₉×#427	C₂²	1	3	yes	—
15	C₂×C₁₉×#106	C₂²	1	3	yes	—
16	C₁₉×#365	C₂²	1	12	yes	—
17	C₁₉×#365	C₂²	1	4	yes	—
18	C₂×#41×C₁₉	D₄	1	1	yes	characteristic
19	C₁₉×#131	—	4	4	no	—
20	C₁₉×#131	C₂×C₄	1	4	yes	—
21	C₁₉×#131	Q₈	1	4	yes	—
22	C₂×M₁₆×C₁₉	—	2	6	no	—
23	C₂×M₁₆×C₁₉	D₄	1	3	yes	—
24	C₂²×C₈×C₁₉	C₂³	1	3	yes	—
25	C₄×Q₈×C₁₉	—	4	1	no	—
26	C₄×D₄×C₁₉	—	4	3	no	—
27	C₁₉×#117	—	2	3	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
29	C₁₉×M₃₂	C₂×C₄	1	12	yes	complemented
30	C₁₉×M₃₂	C₂³	1	4	yes	—
31	C₂×C₁₆×C₁₉	C₂×C₄	1	12	yes	complemented
32	C₁₉×#108	—	2	6	no	—
33	C₁₉×#108	C₂³	1	4	yes	—
34	C₁₉×#107	—	2	3	no	—
35	C₁₉×#106	—	2	3	no	—
36	C₁₉×#106	C₂³	1	3	yes	—
37	C₁₉×#106	C₂³	1	1	yes	characteristic
38	#41×C₁₉	C₄⋊C₄	1	4	yes	—
39	#41×C₁₉	SD₁₆	1	4	yes	—
40	C₂×Q₈×C₁₉	SD₁₆	1	1	yes	characteristic
41	C₂×D₄×C₁₉	SD₁₆	1	3	yes	—
42	C₂²×C₄×C₁₉	—	4	3	no	—
43	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
44	M₁₆×C₁₉	—	2	4	no	—
45	M₁₆×C₁₉	—	4	6	no	—
46	M₁₆×C₁₉	C₄⋊C₄	1	12	yes	—
47	C₂×C₈×C₁₉	—	4	6	no	—
48	C₂×C₈×C₁₉	C₂²×C₄	1	6	yes	—
49	C₂×C₈×C₁₉	C₂×D₄	1	3	yes	—
50	C₂×C₈×C₁₉	C₂×Q₈	1	6	yes	—
51	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
52	C₄⋊C₄×C₁₉	—	2	3	no	—
53	C₄⋊C₄×C₁₉	—	2	1	no	—
54	C₄⋊C₄×C₁₉	—	4	3	no	—
55	C₂²⋊C₄×C₁₉	—	4	3	no	—
56	C₄²×C₁₉	—	4	3	no	—
57	C₁₆×C₁₉	C₂²×C₄	1	8	yes	—
58	#33404	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₁₉	C₂×SD₁₆	1	3	yes	—
60	Q₈×C₁₉	#106	1	4	yes	—
61	D₄×C₁₉	#106	1	12	yes	—
62	C₂×C₄×C₁₉	—	4	1	no	—
63	C₂×C₄×C₁₉	—	4	3	no	—
64	C₂×C₄×C₁₉	—	16	3	no	—
65	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	6	yes	—
66	C₂×C₄×C₁₉	C₂×SD₁₆	1	6	yes	—
67	C₂×C₄×C₁₉	C₂×SD₁₆	1	3	yes	—
68	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
69	C₈×C₁₉	—	4	4	no	—
70	C₈×C₁₉	C₂×C₄⋊C₄	1	6	yes	—
71	C₈×C₁₉	C₂³×C₄	1	1	yes	characteristic
72	C₈×C₁₉	C₂²×Q₈	1	1	yes	characteristic
73	#3407	C₂×C₁₉	1	1	yes	characteristic
74	#2916	C₂×C₁₉	1	1	yes	characteristic
75	#2912	C₂×C₁₉	1	1	yes	characteristic
76	#2155	C₂×C₁₉	1	6	yes	complemented
77	C₂×#365	C₂×C₁₉	1	6	yes	complemented
78	C₂²×C₁₉	—	4	3	no	—
79	C₂²×C₁₉	C₂×#106	1	6	yes	—
80	C₂²×C₁₉	C₂²×SD₁₆	1	1	yes	characteristic
81	C₄×C₁₉	—	8	1	no	—
82	C₄×C₁₉	—	8	3	no	—
83	C₄×C₁₉	C₂×#106	1	6	yes	—
84	C₄×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
85	C₄×C₁₉	C₂²×SD₁₆	1	1	yes	characteristic
86	#568	—	2	1	no	—
87	C₂×#131	C₂²×C₁₉	1	1	yes	characteristic
88	#520	—	2	1	no	—
89	#504	C₄×C₁₉	1	8	yes	complemented
90	C₂×M₃₂	C₂²×C₁₉	1	6	yes	—
91	#430	—	2	3	no	—
92	C₂×#108	C₂²×C₁₉	1	6	yes	—
93	#428	—	2	3	no	—
94	#427	C₂²×C₁₉	1	3	yes	—
95	C₂×#106	C₂²×C₁₉	1	3	yes	—
96	#365	C₂²×C₁₉	1	12	yes	—
97	#365	C₂²×C₁₉	1	4	yes	—
98	C₂×C₁₉	—	2	1	no	—
99	C₂×C₁₉	—	2	6	no	—
100	C₂×C₁₉	C₂²×#106	1	1	yes	characteristic
101	C₂×#41	D₄×C₁₉	1	1	yes	characteristic
102	#131	—	4	4	no	—
103	#131	C₂×C₄×C₁₉	1	4	yes	—
104	#131	Q₈×C₁₉	1	4	yes	—
105	C₂×M₁₆	—	2	6	no	—
106	C₂×M₁₆	D₄×C₁₉	1	3	yes	—
107	C₂²×C₈	C₂³×C₁₉	1	3	yes	—
108	C₄×Q₈	—	4	1	no	—
109	C₄×D₄	—	4	3	no	—
110	#117	—	2	3	no	—
111	C₂×C₄⋊C₄	—	2	3	no	—
112	M₃₂	C₂×C₄×C₁₉	1	12	yes	complemented
113	M₃₂	C₂³×C₁₉	1	4	yes	—
114	C₂×C₁₆	C₂×C₄×C₁₉	1	12	yes	complemented
115	#108	—	2	6	no	—
116	#108	C₂³×C₁₉	1	4	yes	—
117	#107	—	2	3	no	—
118	#106	—	2	3	no	—
119	#106	C₂³×C₁₉	1	3	yes	—
120	#106	C₂³×C₁₉	1	1	yes	characteristic
121	C₁₉	#33404	1	1	yes	characteristic, direct product factor, Sylow
122	#41	C₄⋊C₄×C₁₉	1	4	yes	—
123	#41	SD₁₆×C₁₉	1	4	yes	—
124	C₂×Q₈	SD₁₆×C₁₉	1	1	yes	characteristic
125	C₂×D₄	SD₁₆×C₁₉	1	3	yes	—
126	C₂²×C₄	—	4	3	no	—
127	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
128	M₁₆	—	2	4	no	—
129	M₁₆	—	4	6	no	—
130	M₁₆	C₄⋊C₄×C₁₉	1	12	yes	—
131	C₂×C₈	—	4	6	no	—
132	C₂×C₈	C₂²×C₄×C₁₉	1	6	yes	—
133	C₂×C₈	C₂×D₄×C₁₉	1	3	yes	—
134	C₂×C₈	C₂×Q₈×C₁₉	1	6	yes	—
135	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
136	C₄⋊C₄	—	2	3	no	—
137	C₄⋊C₄	—	2	1	no	—
138	C₄⋊C₄	—	4	3	no	—
139	C₂²⋊C₄	—	4	3	no	—
140	C₄²	—	4	3	no	—
141	C₁₆	C₂²×C₄×C₁₉	1	8	yes	—
142	C₂³	C₂×SD₁₆×C₁₉	1	3	yes	—
143	Q₈	C₁₉×#106	1	4	yes	—
144	D₄	C₁₉×#106	1	12	yes	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	3	no	—
147	C₂×C₄	—	16	3	no	—
148	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	6	yes	—
149	C₂×C₄	C₂×SD₁₆×C₁₉	1	3	yes	—
150	C₂×C₄	C₂×SD₁₆×C₁₉	1	6	yes	—
151	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
152	C₈	—	4	4	no	—
153	C₈	C₂×C₄⋊C₄×C₁₉	1	6	yes	—
154	C₈	C₂³×C₄×C₁₉	1	1	yes	characteristic
155	C₈	C₂²×Q₈×C₁₉	1	1	yes	characteristic
156	C₂²	—	4	3	no	—
157	C₂²	C₂×C₁₉×#106	1	6	yes	—
158	C₂²	C₂²×SD₁₆×C₁₉	1	1	yes	characteristic
159	C₄	—	8	3	no	—
160	C₄	—	8	1	no	—
161	C₄	C₂×C₁₉×#106	1	6	yes	—
162	C₄	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
163	C₄	C₂²×SD₁₆×C₁₉	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	2	6	no	—
166	C₂	C₂²×C₁₉×#106	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	C₂²×#106
commutator	C₈	C₂³×C₄×C₁₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₉
prime radical	C₂×#41×C₁₉	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×#41×C₁₉ ◁ C₂×C₁₉×#131 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₉×#2155	C₂	semidirect
5. C₂×C₁₉×#365	C₂	semidirect
9. C₁₉×#504	C₄	semidirect
29. C₁₉×M₃₂	C₂×C₄	semidirect
31. C₂×C₁₆×C₁₉	C₂×C₄	semidirect
58. #33404	C₁₉	direct (N × H)
76. #2155	C₂×C₁₉	semidirect
77. C₂×#365	C₂×C₁₉	semidirect
89. #504	C₄×C₁₉	semidirect
112. M₃₂	C₂×C₄×C₁₉	semidirect
114. C₂×C₁₆	C₂×C₄×C₁₉	semidirect
121. C₁₉	#33404	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×#3407	2
C₁₉×#2916	2
C₁₉×#2912	2
C₁₉×#2155	2
C₂×C₁₉×#365	2
#33404	19

Subgroups — C19×#33404 #18039363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#33404 #18039363