Subgroups — #18072982

Order 4864.

354 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563321	C₂	1	1	yes	maximal, characteristic
2	#14561918	C₂	1	1	yes	maximal, characteristic
3	#14561465	C₂	1	2	yes	maximal
4	#14561463	C₂	1	2	yes	maximal
5	C₁₉×#1450	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	2	yes	—
7	C₂×#10597260	C₂²	1	1	yes	characteristic
8	C₂×#10597272	C₄	1	2	yes	—
9	C₂×C₈×Dic₁₉	C₄	1	2	yes	—
10	#11926390	C₂²	1	1	yes	characteristic
11	#11926265	C₄	1	2	yes	complemented
12	C₈×C₁₉⋊C₈	C₄	1	2	yes	complemented
13	C₁₉×#431	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#322	C₄	1	2	yes	complemented
16	C₈²×C₁₉	C₄	1	2	yes	complemented
17	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
20	C₂×C₄×Dic₁₉	C₂×C₄	1	2	yes	—
21	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
22	C₁₉×C₄⋊C₈	—	2	2	no	—
23	C₁₉×C₂²⋊C₈	—	2	2	no	—
24	C₄×C₈×C₁₉	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₁₉⋊#95	—	2	2	no	—
30	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
31	#10597272	C₈	1	8	yes	complemented
32	C₈×Dic₁₉	—	2	4	no	—
33	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
34	#10597260	—	2	2	no	—
35	C₄×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
36	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	#41	1	2	yes	—
38	C₂²×Dic₁₉	—	2	2	no	—
39	C₂×C₈×C₁₉	—	2	4	no	—
40	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₁₉	M₁₆	1	4	yes	—
46	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₁₉	#41	1	2	yes	—
49	C₄×Dic₁₉	—	2	2	no	—
50	C₄×Dic₁₉	C₂×C₈	1	4	yes	—
51	C₂×C₁₉⋊C₈	—	2	2	no	—
52	C₂×C₁₉⋊C₈	—	2	2	no	—
53	C₂×C₁₉⋊C₈	—	2	4	no	—
54	C₂×C₁₉⋊C₈	C₂×C₈	1	4	yes	—
55	#8138	—	19	1	no	maximal, Sylow
56	C₂³×C₁₉	#117	1	1	yes	characteristic
57	C₂×C₄×C₁₉	—	4	2	no	—
58	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
59	C₂×C₄×C₁₉	#117	1	2	yes	—
60	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
61	C₂×C₄×C₁₉	#117	1	2	yes	—
62	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
63	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
64	C₂×Dic₁₉	—	2	4	no	—
65	C₂×Dic₁₉	—	4	4	no	—
66	C₈×C₁₉	—	4	4	no	—
67	C₈×C₁₉	C₄×C₈	1	2	yes	—
68	C₈×C₁₉	C₄×C₈	1	2	yes	—
69	C₈×C₁₉	#97	1	2	yes	—
70	C₈×C₁₉	#97	1	2	yes	—
71	C₁₉⋊C₈	—	2	4	no	—
72	C₁₉⋊C₈	—	4	4	no	—
73	#1726	—	19	1	no	—
74	#1450	—	19	1	no	—
75	#1450	D₁₉	1	1	yes	characteristic
76	#1275	—	19	2	no	—
77	#1274	—	19	2	no	—
78	C₂²×C₁₉	—	2	2	no	—
79	C₂²×C₁₉	#376	1	1	yes	characteristic
80	C₂²×C₁₉	#431	1	2	yes	—
81	C₄×C₁₉	—	2	2	no	—
82	C₄×C₁₉	—	2	2	no	—
83	C₄×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
84	C₄×C₁₉	C₂×#97	1	1	yes	characteristic
85	C₄×C₁₉	#431	1	2	yes	—
86	Dic₁₉	—	4	4	no	—
87	#431	—	19	2	no	—
88	#431	C₂×D₁₉	1	2	yes	—
89	C₂×#97	—	19	2	no	—
90	C₂×#97	—	19	1	no	—
91	C₂×C₄×C₈	—	19	2	no	—
92	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
93	#376	—	19	1	no	—
94	#322	—	19	2	no	—
95	#322	Dic₁₉	1	2	yes	complemented
96	C₈²	—	19	2	no	—
97	C₈²	Dic₁₉	1	2	yes	complemented
98	C₂×C₁₉	—	2	2	no	—
99	C₂×C₁₉	#1744	1	1	yes	characteristic
100	C₂×C₁₉	#1747	1	2	yes	—
101	C₂²×C₈	—	19	2	no	—
102	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
103	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
104	C₂×C₄²	—	19	2	no	—
105	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	38	2	no	—
108	C₂²⋊C₈	—	2	2	no	—
109	C₂²⋊C₈	—	38	2	no	—
110	#97	—	19	8	no	—
111	#97	—	38	2	no	—
112	C₄×C₈	—	19	4	no	—
113	C₄×C₈	—	38	4	no	—
114	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
115	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
116	C₄×C₈	C₂×Dic₁₉	1	4	yes	—
117	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
118	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
119	#95	—	38	2	no	—
120	C₁₉	#8138	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₄	—	38	2	no	—
122	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
123	C₂²×C₄	#64541	1	2	yes	—
124	C₂×C₈	—	2	4	no	—
125	C₂×C₈	—	19	4	no	—
126	C₂×C₈	—	38	2	no	—
127	C₂×C₈	—	38	2	no	—
128	C₂×C₈	—	38	4	no	—
129	C₂×C₈	C₈×D₁₉	1	4	yes	—
130	C₂×C₈	#64515	1	4	yes	—
131	C₂×C₈	C₄×Dic₁₉	1	1	yes	characteristic
132	C₂×C₈	C₄×Dic₁₉	1	1	yes	characteristic
133	C₂×C₈	C₄×Dic₁₉	1	1	yes	characteristic
134	C₂×C₈	C₄×Dic₁₉	1	1	yes	characteristic
135	C₄²	—	19	4	no	—
136	C₄²	—	38	2	no	—
137	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
138	C₄²	#64541	1	2	yes	—
139	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
140	C₂³	#10597318	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	38	4	no	—
143	C₂×C₄	—	76	4	no	—
144	C₂×C₄	#10597318	1	2	yes	—
145	C₂×C₄	#10597318	1	1	yes	characteristic
146	C₂×C₄	C₂×C₈×D₁₉	1	2	yes	—
147	C₂×C₄	C₂×#64515	1	2	yes	—
148	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
149	C₂×C₄	#10597372	1	2	yes	—
150	C₈	—	4	4	no	—
151	C₈	—	38	4	no	—
152	C₈	—	76	4	no	—
153	C₈	C₈×Dic₁₉	1	2	yes	—
154	C₈	C₈×Dic₁₉	1	2	yes	—
155	C₈	#10597272	1	2	yes	—
156	C₈	#10597272	1	2	yes	—
157	C₂²	—	2	2	no	—
158	C₂²	#11926390	1	1	yes	characteristic
159	C₂²	#11926595	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	76	4	no	—
163	C₄	C₂×C₈×Dic₁₉	1	1	yes	characteristic
164	C₄	C₂×#10597272	1	1	yes	characteristic
165	C₄	#11926595	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	#14563380	1	1	yes	characteristic
168	C₂	#14563390	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₁₉
commutator	C₄×C₁₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1450	C₂
socle	C₂²×C₁₉	#376
prime radical	C₂³×C₁₉	#117
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1450	D₁₉
hypocenter	C₁₉	#8138
LCS	G ▷ C₄×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
11. #11926265	C₄	semidirect
12. C₈×C₁₉⋊C₈	C₄	semidirect
15. C₁₉×#322	C₄	semidirect
16. C₈²×C₁₉	C₄	semidirect
31. #10597272	C₈	semidirect
95. #322	Dic₁₉	semidirect
97. C₈²	Dic₁₉	semidirect
120. C₁₉	#8138	semidirect

Maximal subgroups

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

subgroup	index
#14563321	2
#14561918	2
#14561465	2
#14561463	2
C₁₉×#1450	2
#8138	19