Subgroups — #18076209

Order 4864.

362 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563787	C₂	1	1	yes	maximal, characteristic
2	#14563381	C₂	1	1	yes	maximal, characteristic
3	#14561469	C₂	1	4	yes	maximal
4	C₁₉×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
6	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
9	#11926299	C₄	1	2	yes	—
10	#11926299	C₄	1	2	yes	—
11	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
12	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
13	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
14	C₄³×C₁₉	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₉	—	2	2	no	—
16	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
23	C₂²×C₁₉⋊C₈	—	2	2	no	—
24	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
25	C₁₉×C₄⋊C₈	—	2	4	no	—
26	C₁₉×C₄⋊C₈	—	4	2	no	—
27	C₁₉×C₄⋊C₈	—	4	2	no	—
28	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
29	C₄⋊(C₁₉⋊C₈)	—	4	2	no	—
30	C₄⋊(C₁₉⋊C₈)	—	4	2	no	—
31	C₄×C₁₉⋊C₈	—	2	2	no	—
32	C₄×C₁₉⋊C₈	—	2	2	no	—
33	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₁₉	—	4	2	no	—
41	C₂×C₈×C₁₉	—	4	2	no	—
42	C₂×C₈×C₁₉	—	4	2	no	—
43	C₄²×C₁₉	—	2	2	no	—
44	C₄²×C₁₉	—	2	2	no	—
45	C₄²×C₁₉	—	4	1	no	—
46	C₄²×C₁₉	—	4	1	no	—
47	C₄²×C₁₉	D₈	1	2	yes	—
48	C₄²×C₁₉	SD₁₆	1	2	yes	—
49	C₄²×C₁₉	SD₁₆	1	2	yes	—
50	C₄²×C₁₉	Q₁₆	1	2	yes	—
51	C₄²×C₁₉	#41	1	2	yes	—
52	C₄²×C₁₉	#41	1	2	yes	—
53	C₂×C₁₉⋊C₈	—	2	4	no	—
54	C₂×C₁₉⋊C₈	—	2	4	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	—	4	2	no	—
59	#8787	—	19	1	no	maximal, Sylow
60	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₉	—	2	1	no	—
62	C₂×C₄×C₁₉	—	2	1	no	—
63	C₂×C₄×C₁₉	—	2	1	no	—
64	C₂×C₄×C₁₉	—	2	1	no	—
65	C₂×C₄×C₁₉	—	2	1	no	—
66	C₂×C₄×C₁₉	—	2	1	no	—
67	C₂×C₄×C₁₉	—	2	1	no	—
68	C₂×C₄×C₁₉	—	2	1	no	—
69	C₂×C₄×C₁₉	—	2	1	no	—
70	C₂×C₄×C₁₉	—	2	1	no	—
71	C₂×C₄×C₁₉	#106	1	2	yes	—
72	C₂×C₄×C₁₉	#106	1	2	yes	—
73	C₂×C₄×C₁₉	#107	1	2	yes	—
74	C₂×C₄×C₁₉	#107	1	2	yes	—
75	C₂×C₄×C₁₉	#117	1	2	yes	—
76	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
77	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
78	C₂×C₄×C₁₉	#125	1	2	yes	—
79	C₂×C₄×C₁₉	#131	1	2	yes	—
80	C₂×C₄×C₁₉	#131	1	2	yes	—
81	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
85	C₈×C₁₉	—	8	2	no	—
86	C₁₉⋊C₈	—	4	4	no	—
87	C₁₉⋊C₈	—	8	2	no	—
88	#1842	—	19	1	no	—
89	#1842	D₁₉	1	1	yes	characteristic
90	#1744	—	19	1	no	—
91	#1277	—	19	4	no	—
92	C₂²×C₁₉	#382	1	1	yes	characteristic
93	C₂²×C₁₉	#424	1	1	yes	characteristic
94	C₂²×C₁₉	#424	1	1	yes	characteristic
95	C₂²×C₁₉	C₂×#106	1	1	yes	characteristic
96	C₂²×C₁₉	C₂×#107	1	1	yes	characteristic
97	C₂²×C₁₉	#430	1	1	yes	characteristic
98	C₂²×C₁₉	#430	1	1	yes	characteristic
99	C₄×C₁₉	—	2	2	no	—
100	C₄×C₁₉	—	2	2	no	—
101	C₄×C₁₉	—	2	2	no	—
102	C₄×C₁₉	—	2	2	no	—
103	C₄×C₁₉	—	2	2	no	—
104	C₄×C₁₉	#433	1	4	yes	—
105	C₄×C₁₉	#486	1	1	yes	characteristic
106	C₄×C₁₉	#486	1	1	yes	characteristic
107	C₄×C₁₉	#487	1	1	yes	characteristic
108	C₄×C₁₉	#487	1	1	yes	characteristic
109	C₂×C₄⋊C₈	—	2	1	no	—
110	C₂×C₄⋊C₈	—	2	1	no	—
111	C₂×C₄⋊C₈	—	19	2	no	—
112	C₂×C₄⋊C₈	—	38	1	no	—
113	C₂×C₄⋊C₈	—	38	1	no	—
114	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
115	C₂×C₄×C₈	—	19	2	no	—
116	C₄³	C₂×D₁₉	1	1	yes	characteristic
117	#336	—	19	2	no	—
118	#336	—	19	2	no	—
119	C₂×C₁₉	#1570	1	1	yes	characteristic
120	C₂×C₁₉	#1570	1	1	yes	characteristic
121	C₂×C₁₉	#1571	1	1	yes	characteristic
122	C₂×C₁₉	#1571	1	1	yes	characteristic
123	C₂×C₁₉	#1832	1	1	yes	characteristic
124	C₂×C₁₉	#1834	1	1	yes	characteristic
125	C₂×C₁₉	#1838	1	1	yes	characteristic
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	19	4	no	—
128	C₂²×C₈	—	38	2	no	—
129	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
130	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
131	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
132	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
133	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
134	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
135	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
136	C₄⋊C₈	—	2	4	no	—
137	C₄⋊C₈	—	4	2	no	—
138	C₄⋊C₈	—	4	2	no	—
139	C₄⋊C₈	—	38	4	no	—
140	C₄⋊C₈	—	76	2	no	—
141	C₄⋊C₈	—	76	2	no	—
142	C₄×C₈	—	38	2	no	—
143	C₄×C₈	—	38	2	no	—
144	C₁₉	#8787	1	1	yes	characteristic, complemented, Sylow
145	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
146	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
147	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
148	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
149	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	38	4	no	—
156	C₂×C₈	—	38	4	no	—
157	C₂×C₈	—	38	4	no	—
158	C₂×C₈	—	76	2	no	—
159	C₂×C₈	—	76	2	no	—
160	C₂×C₈	—	76	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	4	1	no	—
164	C₄²	—	4	1	no	—
165	C₄²	#64525	1	2	yes	—
166	C₄²	#64526	1	2	yes	—
167	C₄²	#64527	1	2	yes	—
168	C₄²	#64528	1	2	yes	—
169	C₄²	#64543	1	2	yes	—
170	C₄²	#64545	1	2	yes	—
171	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	#10597264	1	2	yes	—
183	C₂×C₄	#10597264	1	2	yes	—
184	C₂×C₄	#10597265	1	2	yes	—
185	C₂×C₄	#10597265	1	2	yes	—
186	C₂×C₄	#10597336	1	2	yes	—
187	C₂×C₄	#10597338	1	2	yes	—
188	C₂×C₄	#10597347	1	2	yes	—
189	C₂×C₄	#10597353	1	2	yes	—
190	C₂×C₄	C₂×#64525	1	1	yes	characteristic
191	C₂×C₄	C₂×#64526	1	1	yes	characteristic
192	C₂×C₄	#10597382	1	1	yes	characteristic
193	C₂×C₄	C₂×#64527	1	1	yes	characteristic
194	C₂×C₄	C₂×#64528	1	1	yes	characteristic
195	C₂×C₄	#10597393	1	1	yes	characteristic
196	C₈	—	8	2	no	—
197	C₈	—	76	4	no	—
198	C₈	—	152	2	no	—
199	C₂²	#11926409	1	1	yes	characteristic
200	C₂²	#11926570	1	1	yes	characteristic
201	C₂²	#11926572	1	1	yes	characteristic
202	C₂²	C₂×#10597265	1	1	yes	characteristic
203	C₂²	C₂×#10597264	1	1	yes	characteristic
204	C₂²	#11926592	1	1	yes	characteristic
205	C₂²	#11926592	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	2	2	no	—
211	C₄	#11926604	1	4	yes	—
212	C₄	#11926890	1	1	yes	characteristic
213	C₄	#11926892	1	1	yes	characteristic
214	C₄	#11926894	1	1	yes	characteristic
215	C₄	#11926896	1	1	yes	characteristic
216	C₂	#14562405	1	1	yes	characteristic
217	C₂	#14562407	1	1	yes	characteristic
218	C₂	#14562409	1	1	yes	characteristic
219	C₂	#14562411	1	1	yes	characteristic
220	C₂	#14563754	1	1	yes	characteristic
221	C₂	#14563762	1	1	yes	characteristic
222	C₂	#14563775	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1842	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1842	D₁₉
hypocenter	C₁₉	#8787
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
144. C₁₉	#8787	semidirect

Maximal subgroups

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

subgroup	index
#14563787	2
#14563381	2
#14561469	2
C₁₉×#1842	2
#8787	19