Subgroups — C₁₉×#20762 #18026721

Order 4864.

910 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	C₂²×C₁₉×#97	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2285	C₂	1	2	yes	maximal
3	C₁₉×#1836	C₂	1	8	yes	maximal, complemented
4	C₁₉×#1735	C₂	1	4	yes	maximal
5	C₂³×C₈×C₁₉	C₂²	1	2	yes	—
6	C₂×C₁₉×#117	C₂²	1	4	yes	—
7	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
8	C₂×C₁₉×#97	C₄	1	8	yes	complemented
9	C₂×C₁₉×#97	C₂²	1	4	yes	—
10	C₁₉×#384	C₂²	1	4	yes	—
11	C₁₉×#382	C₂²	1	4	yes	—
12	C₁₉×#380	—	2	4	no	—
13	C₄×C₄⋊C₄×C₁₉	—	2	4	no	—
14	C₁₉×#340	C₂²	1	16	yes	—
15	C₂³×C₄×C₁₉	D₄	1	2	yes	—
16	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
17	C₂²×C₈×C₁₉	—	2	8	no	—
18	C₂²×C₈×C₁₉	C₂×C₄	1	8	yes	—
19	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
20	C₁₉×#117	—	2	16	no	—
21	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
22	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
23	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
24	C₂×C₄²×C₁₉	—	2	4	no	—
25	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₁₉	D₄	1	2	yes	—
27	C₂×C₄²×C₁₉	Q₈	1	4	yes	—
28	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
29	C₁₉×#97	C₂×C₄	1	16	yes	—
30	C₁₉×#95	—	2	8	no	—
31	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	—	2	8	no	—
33	C₂²×C₄×C₁₉	—	4	4	no	—
34	C₂²×C₄×C₁₉	C₄⋊C₄	1	8	yes	—
35	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
36	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
37	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
39	C₂²×C₄×C₁₉	#41	1	8	yes	—
40	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
41	C₂×C₈×C₁₉	—	2	16	no	—
42	C₂×C₈×C₁₉	—	2	8	no	—
43	C₂×C₈×C₁₉	C₂²×C₄	1	8	yes	—
44	C₄⋊C₄×C₁₉	—	2	16	no	—
45	C₄⋊C₄×C₁₉	—	4	16	no	—
46	C₂²⋊C₄×C₁₉	—	4	8	no	—
47	C₄²×C₁₉	—	4	8	no	—
48	C₄²×C₁₉	C₄⋊C₄	1	8	yes	—
49	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
50	C₄²×C₁₉	C₂×Q₈	1	4	yes	—
51	#20762	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂³×C₁₉	—	2	4	no	—
53	C₂³×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
54	C₂³×C₁₉	#124	1	4	yes	—
55	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₉	—	2	8	no	—
57	C₂×C₄×C₁₉	—	4	8	no	—
58	C₂×C₄×C₁₉	—	4	8	no	—
59	C₂×C₄×C₁₉	—	4	16	no	—
60	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	8	yes	—
61	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
62	C₂×C₄×C₁₉	#117	1	8	yes	—
63	C₂×C₄×C₁₉	#124	1	4	yes	—
64	C₂×C₄×C₁₉	#125	1	8	yes	—
65	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₁₉	C₂×#41	1	4	yes	—
68	C₈×C₁₉	—	2	8	no	—
69	C₂²×#97	C₂×C₁₉	1	1	yes	characteristic
70	#2285	C₂×C₁₉	1	2	yes	—
71	#1836	C₂×C₁₉	1	8	yes	complemented
72	#1735	C₂×C₁₉	1	4	yes	—
73	C₂²×C₁₉	—	2	8	no	—
74	C₂²×C₁₉	—	2	4	no	—
75	C₂²×C₁₉	#382	1	4	yes	—
76	C₂²×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₁₉	C₂×#124	1	2	yes	—
78	C₂²×C₁₉	#575	1	4	yes	—
79	C₄×C₁₉	—	2	8	no	—
80	C₄×C₁₉	—	8	8	no	—
81	C₄×C₁₉	#382	1	4	yes	—
82	C₄×C₁₉	C₂×#117	1	2	yes	—
83	C₄×C₁₉	C₂×#125	1	2	yes	—
84	C₂³×C₈	C₂²×C₁₉	1	2	yes	—
85	C₂×#117	C₂²×C₁₉	1	4	yes	—
86	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
87	C₂×#97	C₄×C₁₉	1	8	yes	complemented
88	C₂×#97	C₂²×C₁₉	1	4	yes	—
89	#384	C₂²×C₁₉	1	4	yes	—
90	#382	C₂²×C₁₉	1	4	yes	—
91	#380	—	2	4	no	—
92	C₄×C₄⋊C₄	—	2	4	no	—
93	#340	C₂²×C₁₉	1	16	yes	—
94	C₂×C₁₉	—	2	4	no	—
95	C₂×C₁₉	C₂×#382	1	1	yes	characteristic
96	C₂×C₁₉	#2911	1	2	yes	—
97	C₂×C₁₉	#3135	1	4	yes	—
98	C₂³×C₄	D₄×C₁₉	1	2	yes	—
99	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
100	C₂²×C₈	—	2	8	no	—
101	C₂²×C₈	C₂×C₄×C₁₉	1	8	yes	—
102	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
103	#117	—	2	16	no	—
104	C₂×C₄⋊C₄	—	2	8	no	—
105	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
106	C₂×C₂²⋊C₄	—	2	4	no	—
107	C₂×C₄²	—	2	4	no	—
108	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
109	C₂×C₄²	D₄×C₁₉	1	2	yes	—
110	C₂×C₄²	Q₈×C₁₉	1	4	yes	—
111	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
112	#97	C₂×C₄×C₁₉	1	16	yes	—
113	#95	—	2	8	no	—
114	C₁₉	#20762	1	1	yes	characteristic, direct product factor, Sylow
115	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
116	C₂²×C₄	—	2	8	no	—
117	C₂²×C₄	—	4	4	no	—
118	C₂²×C₄	C₄⋊C₄×C₁₉	1	8	yes	—
119	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
120	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
121	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
122	C₂²×C₄	C₂×Q₈×C₁₉	1	2	yes	—
123	C₂²×C₄	#41×C₁₉	1	8	yes	—
124	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
125	C₂×C₈	—	2	16	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	C₂²×C₄×C₁₉	1	8	yes	—
128	C₄⋊C₄	—	2	16	no	—
129	C₄⋊C₄	—	4	16	no	—
130	C₂²⋊C₄	—	4	8	no	—
131	C₄²	—	4	8	no	—
132	C₄²	C₄⋊C₄×C₁₉	1	8	yes	—
133	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
134	C₄²	C₂×Q₈×C₁₉	1	4	yes	—
135	C₂³	—	2	4	no	—
136	C₂³	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
137	C₂³	C₁₉×#124	1	4	yes	—
138	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	4	8	no	—
141	C₂×C₄	—	4	8	no	—
142	C₂×C₄	—	4	16	no	—
143	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	8	yes	—
144	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
145	C₂×C₄	C₁₉×#117	1	8	yes	—
146	C₂×C₄	C₁₉×#124	1	4	yes	—
147	C₂×C₄	C₁₉×#125	1	8	yes	—
148	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
149	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
150	C₂×C₄	C₂×#41×C₁₉	1	4	yes	—
151	C₈	—	2	8	no	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	8	no	—
154	C₂²	C₁₉×#382	1	4	yes	—
155	C₂²	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
156	C₂²	C₂×C₁₉×#124	1	2	yes	—
157	C₂²	C₁₉×#575	1	4	yes	—
158	C₄	—	2	8	no	—
159	C₄	—	8	8	no	—
160	C₄	C₁₉×#382	1	4	yes	—
161	C₄	C₂×C₁₉×#117	1	2	yes	—
162	C₄	C₂×C₁₉×#125	1	2	yes	—
163	C₂	—	2	4	no	—
164	C₂	C₂×C₁₉×#382	1	1	yes	characteristic
165	C₂	C₁₉×#2911	1	2	yes	—
166	C₂	C₁₉×#3135	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle	C₂³×C₁₉	C₂²×D₄
prime radical	C₂⁴×C₁₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₉ ◁ C₂²×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#1836	C₂	semidirect
8. C₂×C₁₉×#97	C₄	semidirect
51. #20762	C₁₉	direct (N × H)
71. #1836	C₂×C₁₉	semidirect
87. C₂×#97	C₄×C₁₉	semidirect
114. C₁₉	#20762	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×#97	2
C₁₉×#2285	2
C₁₉×#1836	2
C₁₉×#1735	2
#20762	19

Subgroups — C19×#20762 #18026721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#20762 #18026721