Subgroups — C₂×C₁₉×#1747 #18026328

Order 4864.

930 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#405	C₂	1	3	yes	maximal
2	C₁₉×#1747	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₁₉×#338	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉×H(3,ℤ/4)	C₂	1	3	yes	maximal
5	C₂²×M₁₆×C₁₉	C₂²	1	3	yes	—
6	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
7	C₂×C₁₉×#117	C₂²	1	3	yes	—
8	C₁₉×#405	C₄	1	12	yes	complemented
9	C₁₉×#405	C₂²	1	12	yes	—
10	C₂×C₁₉×#97	C₄	1	6	yes	complemented
11	C₂×C₄×C₈×C₁₉	C₄	1	6	yes	complemented
12	C₁₉×#338	C₂²	1	4	yes	—
13	C₁₉×H(3,ℤ/4)	C₂²	1	12	yes	—
14	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
15	C₂×M₁₆×C₁₉	—	2	6	no	—
16	C₂×M₁₆×C₁₉	—	2	6	no	—
17	C₂×M₁₆×C₁₉	C₂×C₄	1	6	yes	—
18	C₂×M₁₆×C₁₉	C₂³	1	6	yes	—
19	C₂²×C₈×C₁₉	C₂×C₄	1	6	yes	—
20	C₂²×C₈×C₁₉	C₂×C₄	1	3	yes	—
21	C₂²×C₈×C₁₉	C₂×C₄	1	6	yes	—
22	C₂²×C₈×C₁₉	C₂×C₄	1	3	yes	—
23	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
24	C₁₉×#117	—	2	3	no	—
25	C₁₉×#117	—	2	3	no	—
26	C₁₉×#117	C₂×C₄	1	6	yes	—
27	C₁₉×#117	C₂³	1	6	yes	—
28	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
29	C₂×C₂²⋊C₄×C₁₉	—	2	3	no	—
30	C₂×C₄²×C₁₉	C₂×C₄	1	6	yes	—
31	C₁₉×#97	C₂×C₄	1	24	yes	complemented
32	C₄×C₈×C₁₉	C₂×C₄	1	24	yes	complemented
33	C₂⁴×C₁₉	#41	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	—	2	6	no	—
35	C₂²×C₄×C₁₉	C₂²×C₄	1	3	yes	—
36	C₂²×C₄×C₁₉	C₂²×C₄	1	3	yes	—
37	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
38	C₂²×C₄×C₁₉	#41	1	3	yes	—
39	C₂²×C₄×C₁₉	#41	1	3	yes	—
40	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
41	M₁₆×C₁₉	—	2	12	no	—
42	M₁₆×C₁₉	—	2	12	no	—
43	C₂×C₈×C₁₉	—	2	6	no	—
44	C₂×C₈×C₁₉	—	2	6	no	—
45	C₂×C₈×C₁₉	C₄²	1	6	yes	—
46	C₂×C₈×C₁₉	C₄²	1	3	yes	—
47	C₂×C₈×C₁₉	C₄²	1	6	yes	—
48	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₁₉	C₂²×C₄	1	12	yes	—
50	C₂×C₈×C₁₉	C₂²×C₄	1	6	yes	—
51	C₂×C₈×C₁₉	C₂²×C₄	1	6	yes	—
52	C₄⋊C₄×C₁₉	—	2	6	no	—
53	C₄⋊C₄×C₁₉	—	2	6	no	—
54	C₂²⋊C₄×C₁₉	—	2	6	no	—
55	C₂²⋊C₄×C₁₉	—	2	6	no	—
56	C₄²×C₁₉	—	2	6	no	—
57	C₄²×C₁₉	C₂²×C₄	1	12	yes	—
58	C₂×#1747	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₁₉	—	4	2	no	—
60	C₂³×C₁₉	#117	1	3	yes	—
61	C₂³×C₁₉	#117	1	3	yes	—
62	C₂³×C₁₉	C₂×#41	1	1	yes	characteristic
63	C₂×C₄×C₁₉	—	2	12	no	—
64	C₂×C₄×C₁₉	—	4	12	no	—
65	C₂×C₄×C₁₉	C₂×C₄²	1	3	yes	—
66	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₁₉	#117	1	3	yes	—
68	C₂×C₄×C₁₉	#117	1	3	yes	—
69	C₂×C₄×C₁₉	#117	1	6	yes	—
70	C₂×C₄×C₁₉	#117	1	3	yes	—
71	C₂×C₄×C₁₉	#117	1	3	yes	—
72	C₂×C₄×C₁₉	C₂³×C₄	1	3	yes	—
73	C₂×C₄×C₁₉	C₂×#41	1	3	yes	—
74	C₈×C₁₉	—	2	12	no	—
75	C₈×C₁₉	C₂×C₄²	1	6	yes	—
76	C₈×C₁₉	C₂×C₄²	1	2	yes	—
77	C₂×#405	C₂×C₁₉	1	3	yes	—
78	#1747	C₂×C₁₉	1	8	yes	direct product factor
79	C₂×#338	C₂×C₁₉	1	1	yes	characteristic
80	C₂×H(3,ℤ/4)	C₂×C₁₉	1	3	yes	—
81	C₂²×C₁₉	—	2	6	no	—
82	C₂²×C₁₉	—	4	3	no	—
83	C₂²×C₁₉	—	4	1	no	—
84	C₂²×C₁₉	#376	1	3	yes	—
85	C₂²×C₁₉	#376	1	1	yes	characteristic
86	C₂²×C₁₉	C₂×#117	1	3	yes	—
87	C₄×C₁₉	—	4	12	no	—
88	C₄×C₁₉	#376	1	3	yes	—
89	C₄×C₁₉	#376	1	1	yes	characteristic
90	C₄×C₁₉	C₂²×C₄²	1	1	yes	characteristic
91	C₄×C₁₉	C₂×#117	1	3	yes	—
92	C₂²×M₁₆	C₂²×C₁₉	1	3	yes	—
93	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
94	C₂×#117	C₂²×C₁₉	1	3	yes	—
95	#405	C₄×C₁₉	1	12	yes	complemented
96	#405	C₂²×C₁₉	1	12	yes	—
97	C₂×#97	C₄×C₁₉	1	6	yes	complemented
98	C₂×C₄×C₈	C₄×C₁₉	1	6	yes	complemented
99	#338	C₂²×C₁₉	1	4	yes	—
100	H(3,ℤ/4)	C₂²×C₁₉	1	12	yes	—
101	C₂×C₁₉	—	2	3	no	—
102	C₂×C₁₉	—	2	3	no	—
103	C₂×C₁₉	#1747	1	2	yes	direct product factor
104	C₂×C₁₉	C₂×#376	1	1	yes	characteristic
105	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
106	C₂×M₁₆	—	2	6	no	—
107	C₂×M₁₆	—	2	6	no	—
108	C₂×M₁₆	C₂×C₄×C₁₉	1	6	yes	—
109	C₂×M₁₆	C₂³×C₁₉	1	6	yes	—
110	C₂²×C₈	C₂×C₄×C₁₉	1	6	yes	—
111	C₂²×C₈	C₂×C₄×C₁₉	1	3	yes	—
112	C₂²×C₈	C₂×C₄×C₁₉	1	3	yes	—
113	C₂²×C₈	C₂×C₄×C₁₉	1	6	yes	—
114	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
115	#117	—	2	3	no	—
116	#117	—	2	3	no	—
117	#117	C₂×C₄×C₁₉	1	6	yes	—
118	#117	C₂³×C₁₉	1	6	yes	—
119	C₂×C₄⋊C₄	—	2	3	no	—
120	C₂×C₂²⋊C₄	—	2	3	no	—
121	C₂×C₄²	C₂×C₄×C₁₉	1	6	yes	—
122	#97	C₂×C₄×C₁₉	1	24	yes	complemented
123	C₄×C₈	C₂×C₄×C₁₉	1	24	yes	complemented
124	C₁₉	C₂×#1747	1	1	yes	characteristic, direct product factor, Sylow
125	C₂⁴	#41×C₁₉	1	1	yes	characteristic
126	C₂²×C₄	—	2	6	no	—
127	C₂²×C₄	C₂²×C₄×C₁₉	1	3	yes	—
128	C₂²×C₄	C₂²×C₄×C₁₉	1	3	yes	—
129	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
130	C₂²×C₄	#41×C₁₉	1	3	yes	—
131	C₂²×C₄	#41×C₁₉	1	3	yes	—
132	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
133	M₁₆	—	2	12	no	—
134	M₁₆	—	2	12	no	—
135	C₂×C₈	—	2	6	no	—
136	C₂×C₈	—	2	6	no	—
137	C₂×C₈	C₄²×C₁₉	1	3	yes	—
138	C₂×C₈	C₄²×C₁₉	1	6	yes	—
139	C₂×C₈	C₄²×C₁₉	1	6	yes	—
140	C₂×C₈	C₄²×C₁₉	1	1	yes	characteristic
141	C₂×C₈	C₂²×C₄×C₁₉	1	12	yes	—
142	C₂×C₈	C₂²×C₄×C₁₉	1	6	yes	—
143	C₂×C₈	C₂²×C₄×C₁₉	1	6	yes	—
144	C₄⋊C₄	—	2	6	no	—
145	C₄⋊C₄	—	2	6	no	—
146	C₂²⋊C₄	—	2	6	no	—
147	C₂²⋊C₄	—	2	6	no	—
148	C₄²	—	2	6	no	—
149	C₄²	C₂²×C₄×C₁₉	1	12	yes	—
150	C₂³	—	4	2	no	—
151	C₂³	C₁₉×#117	1	3	yes	—
152	C₂³	C₁₉×#117	1	3	yes	—
153	C₂³	C₂×#41×C₁₉	1	1	yes	characteristic
154	C₂×C₄	—	2	12	no	—
155	C₂×C₄	—	4	12	no	—
156	C₂×C₄	C₂×C₄²×C₁₉	1	3	yes	—
157	C₂×C₄	C₂×C₄²×C₁₉	1	1	yes	characteristic
158	C₂×C₄	C₁₉×#117	1	3	yes	—
159	C₂×C₄	C₁₉×#117	1	3	yes	—
160	C₂×C₄	C₁₉×#117	1	3	yes	—
161	C₂×C₄	C₁₉×#117	1	3	yes	—
162	C₂×C₄	C₁₉×#117	1	6	yes	—
163	C₂×C₄	C₂³×C₄×C₁₉	1	3	yes	—
164	C₂×C₄	C₂×#41×C₁₉	1	3	yes	—
165	C₈	—	2	12	no	—
166	C₈	C₂×C₄²×C₁₉	1	6	yes	—
167	C₈	C₂×C₄²×C₁₉	1	2	yes	—
168	C₂²	—	2	6	no	—
169	C₂²	—	4	3	no	—
170	C₂²	—	4	1	no	—
171	C₂²	C₁₉×#376	1	3	yes	—
172	C₂²	C₁₉×#376	1	1	yes	characteristic
173	C₂²	C₂×C₁₉×#117	1	3	yes	—
174	C₄	—	4	12	no	—
175	C₄	C₁₉×#376	1	1	yes	characteristic
176	C₄	C₁₉×#376	1	3	yes	—
177	C₄	C₂²×C₄²×C₁₉	1	1	yes	characteristic
178	C₄	C₂×C₁₉×#117	1	3	yes	—
179	C₂	—	2	3	no	—
180	C₂	—	2	3	no	—
181	C₂	C₁₉×#1747	1	2	yes	direct product factor
182	C₂	C₂×C₁₉×#376	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₉	C₄²
commutator	C₄	C₂²×C₄²×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle	C₂²×C₁₉	#376
prime radical	C₂⁴×C₁₉	#41
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#1747	C₂	direct (N × H)
8. C₁₉×#405	C₄	semidirect
10. C₂×C₁₉×#97	C₄	semidirect
11. C₂×C₄×C₈×C₁₉	C₄	semidirect
31. C₁₉×#97	C₂×C₄	semidirect
32. C₄×C₈×C₁₉	C₂×C₄	semidirect
58. C₂×#1747	C₁₉	direct (N × H)
78. #1747	C₂×C₁₉	direct (N × H)
95. #405	C₄×C₁₉	semidirect
97. C₂×#97	C₄×C₁₉	semidirect
98. C₂×C₄×C₈	C₄×C₁₉	semidirect
103. C₂×C₁₉	#1747	direct (N × H)
122. #97	C₂×C₄×C₁₉	semidirect
123. C₄×C₈	C₂×C₄×C₁₉	semidirect
124. C₁₉	C₂×#1747	direct (N × H)
181. C₂	C₁₉×#1747	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×#405	2
C₁₉×#1747	2
C₂×C₁₉×#338	2
C₂×C₁₉×H(3,ℤ/4)	2
C₂×#1747	19

Subgroups — C2×C19×#1747 #18026328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#1747 #18026328