Subgroups — #18073719

Order 4864.

470 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₈×Dic₁₉	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₁₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926294	C₂	1	2	yes	maximal
4	#14561470	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₁₉	—	2	2	no	—
9	C₂×C₈×Dic₁₉	C₄	1	2	yes	—
10	C₂×C₈×Dic₁₉	C₄	1	2	yes	—
11	C₄²×Dic₁₉	C₂²	1	1	yes	characteristic
12	#11926298	—	2	2	no	—
13	#11926294	C₄	1	8	yes	complemented
14	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₉	—	2	2	no	—
17	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
19	C₂×C₄×Dic₁₉	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₉	Q₈	1	2	yes	—
23	C₂²×C₁₉⋊C₈	—	2	2	no	—
24	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₁₉×C₄⋊C₈	—	2	2	no	—
27	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₁₉	—	2	2	no	—
29	C₈×Dic₁₉	—	2	4	no	—
30	C₈×Dic₁₉	—	4	4	no	—
31	C₈×Dic₁₉	C₈	1	8	yes	complemented
32	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
33	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
34	C₄×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
38	C₂²×Dic₁₉	C₄⋊C₄	1	2	yes	—
39	C₂²×Dic₁₉	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₁₉	—	2	2	no	—
41	C₂×C₈×C₁₉	—	2	2	no	—
42	C₂×C₈×C₁₉	—	2	4	no	—
43	C₂×C₈×C₁₉	—	4	2	no	—
44	C₂×C₈×C₁₉	—	4	2	no	—
45	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
46	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
47	C₄²×C₁₉	#41	1	2	yes	—
48	C₄×Dic₁₉	—	2	1	no	—
49	C₄×Dic₁₉	—	2	1	no	—
50	C₄×Dic₁₉	—	2	4	no	—
51	C₄×Dic₁₉	C₂×C₈	1	4	yes	—
52	C₄×Dic₁₉	M₁₆	1	4	yes	—
53	C₄×Dic₁₉	SD₁₆	1	4	yes	—
54	C₂×C₁₉⋊C₈	—	2	2	no	—
55	C₂×C₁₉⋊C₈	—	2	4	no	—
56	C₂×C₁₉⋊C₈	—	4	2	no	—
57	C₂×C₁₉⋊C₈	—	4	2	no	—
58	C₂×C₁₉⋊C₈	C₄²	1	2	yes	—
59	C₂×C₁₉⋊C₈	C₄²	1	2	yes	—
60	C₂×C₁₉⋊C₈	C₂×C₈	1	4	yes	—
61	C₄×#334	—	19	1	no	maximal, Sylow
62	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₉	—	2	2	no	—
64	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
66	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
68	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
69	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
70	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
71	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
72	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
73	C₂×Dic₁₉	—	2	2	no	—
74	C₂×Dic₁₉	—	2	2	no	—
75	C₂×Dic₁₉	C₄⋊C₈	1	4	yes	—
76	C₂×Dic₁₉	C₄⋊C₈	1	4	yes	—
77	C₂×Dic₁₉	#106	1	4	yes	—
78	C₂×Dic₁₉	#108	1	4	yes	—
79	C₈×C₁₉	—	2	4	no	—
80	C₈×C₁₉	—	4	4	no	—
81	C₁₉⋊C₈	—	2	2	no	—
82	C₁₉⋊C₈	—	4	4	no	—
83	C₁₉⋊C₈	C₄×C₈	1	4	yes	complemented
84	C₄×C₄⋊C₈	—	19	2	no	—
85	C₄²×C₈	—	19	1	no	—
86	C₂×#334	—	19	2	no	—
87	#1277	—	19	1	no	—
88	#1277	D₁₉	1	1	yes	characteristic
89	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₁₉	C₂×C₄⋊C₈	1	2	yes	—
91	C₂²×C₁₉	C₂×#106	1	1	yes	characteristic
92	C₂²×C₁₉	#427	1	1	yes	characteristic
93	C₂²×C₁₉	C₂×#108	1	2	yes	—
94	C₄×C₁₉	—	2	2	no	—
95	C₄×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
96	C₄×C₁₉	C₄×M₁₆	1	1	yes	characteristic
97	C₄×C₁₉	#431	1	2	yes	—
98	C₄×C₁₉	C₄×SD₁₆	1	2	yes	—
99	C₄×C₁₉	C₄×SD₁₆	1	2	yes	—
100	Dic₁₉	—	2	4	no	—
101	Dic₁₉	#334	1	8	yes	complemented
102	C₂×C₄⋊C₈	—	19	2	no	—
103	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
104	C₂×C₄×C₈	—	19	2	no	—
105	C₂×C₄×C₈	—	19	1	no	—
106	C₂×C₄×C₈	—	19	2	no	—
107	C₂×C₄×C₈	—	38	2	no	—
108	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
109	C₄³	—	19	1	no	—
110	#336	—	38	2	no	—
111	#334	—	19	8	no	—
112	C₂×C₁₉	C₂×#334	1	2	yes	—
113	C₂×C₁₉	#1564	1	2	yes	—
114	C₂×C₁₉	C₄×C₄⋊C₈	1	1	yes	characteristic
115	C₂×C₁₉	C₄×#106	1	1	yes	characteristic
116	C₂×C₁₉	C₄×#108	1	1	yes	characteristic
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	19	1	no	—
119	C₂²×C₈	—	19	1	no	—
120	C₂²×C₈	—	38	2	no	—
121	C₂²×C₈	C₄×D₁₉	1	2	yes	—
122	C₂×C₄²	—	19	2	no	—
123	C₂×C₄²	—	19	1	no	—
124	C₂×C₄²	—	19	1	no	—
125	C₂×C₄²	—	19	2	no	—
126	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	19	4	no	—
129	C₄⋊C₈	—	38	2	no	—
130	C₄⋊C₈	C₄×D₁₉	1	4	yes	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	19	8	no	—
133	C₄×C₈	—	19	4	no	—
134	C₄×C₈	—	38	4	no	—
135	C₄×C₈	—	76	4	no	—
136	C₁₉	C₄×#334	1	1	yes	characteristic, complemented, Sylow
137	C₂²×C₄	—	19	2	no	—
138	C₂²×C₄	—	19	2	no	—
139	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
140	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
141	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	4	2	no	—
147	C₂×C₈	—	19	2	no	—
148	C₂×C₈	—	19	4	no	—
149	C₂×C₈	—	19	2	no	—
150	C₂×C₈	—	38	2	no	—
151	C₂×C₈	—	38	4	no	—
152	C₂×C₈	—	76	2	no	—
153	C₂×C₈	—	76	2	no	—
154	C₂×C₈	C₈×D₁₉	1	4	yes	—
155	C₄²	—	19	4	no	—
156	C₄²	—	19	4	no	—
157	C₄²	—	19	4	no	—
158	C₄²	—	38	4	no	—
159	C₄²	—	38	1	no	—
160	C₄²	—	38	1	no	—
161	C₄²	C₂×C₄×D₁₉	1	2	yes	—
162	C₄²	#64543	1	2	yes	—
163	C₂³	C₄⋊C₄×D₁₉	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	19	4	no	—
166	C₂×C₄	—	19	4	no	—
167	C₂×C₄	—	19	4	no	—
168	C₂×C₄	—	19	4	no	—
169	C₂×C₄	—	38	2	no	—
170	C₂×C₄	—	38	2	no	—
171	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
172	C₂×C₄	#10597323	1	1	yes	characteristic
173	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
174	C₂×C₄	#10597333	1	2	yes	—
175	C₂×C₄	C₂×C₈×D₁₉	1	2	yes	—
176	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
177	C₂×C₄	#10597358	1	1	yes	characteristic
178	C₂×C₄	SD₁₆×D₁₉	1	2	yes	—
179	C₂×C₄	#10597365	1	1	yes	characteristic
180	C₈	—	2	4	no	—
181	C₈	—	4	4	no	—
182	C₈	—	19	4	no	—
183	C₈	—	38	2	no	—
184	C₈	—	76	4	no	—
185	C₂²	#11926395	1	1	yes	characteristic
186	C₂²	C₄⋊C₈×D₁₉	1	2	yes	—
187	C₂²	#106×D₁₉	1	1	yes	characteristic
188	C₂²	#11926581	1	1	yes	characteristic
189	C₂²	#108×D₁₉	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	19	8	no	—
192	C₄	—	38	4	no	—
193	C₄	C₄×C₈×D₁₉	1	1	yes	characteristic
194	C₄	#11926496	1	1	yes	characteristic
195	C₄	#11926594	1	2	yes	—
196	C₄	#11926629	1	2	yes	—
197	C₄	#11926635	1	2	yes	—
198	C₂	D₁₉×#334	1	2	yes	—
199	C₂	#14562382	1	2	yes	—
200	C₂	#14563483	1	1	yes	characteristic
201	C₂	#14563523	1	1	yes	characteristic
202	C₂	#14563536	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₄×C₁₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1277	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1277	D₁₉
hypocenter	C₁₉	C₄×#334
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
13. #11926294	C₄	semidirect
31. C₈×Dic₁₉	C₈	semidirect
83. C₁₉⋊C₈	C₄×C₈	semidirect
101. Dic₁₉	#334	semidirect
136. C₁₉	C₄×#334	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₈×Dic₁₉	2
C₄×C₈×Dic₁₉	2
C₂×#11926294	2
#14561470	2
C₁₉×#1277	2
C₄×#334	19