Subgroups — #18069724

Order 4864.

262 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₁₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926297	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1845	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₉⋊C₁₆	—	2	1	no	—
5	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
6	C₄⋊(C₁₉⋊C₁₆)	—	2	1	no	—
7	C₄⋊(C₁₉⋊C₁₆)	C₄	1	2	yes	complemented
8	#11926297	—	2	1	no	—
9	#11926297	C₄	1	2	yes	complemented
10	C₂×C₁₉×#107	—	2	1	no	—
11	C₂×C₁₉×#107	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₁₉×#384	—	2	1	no	—
14	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
17	C₂×C₄⋊C₄×C₁₉	—	4	1	no	—
18	C₂×C₄⋊C₄×C₁₉	—	4	1	no	—
19	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
20	C₂²×C₁₉⋊C₈	—	4	1	no	—
21	C₁₉×#107	—	4	1	no	—
22	C₁₉×#107	—	4	1	no	—
23	C₁₉×#107	—	4	2	no	—
24	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
27	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
28	C₂×C₁₉⋊C₁₆	—	2	2	no	—
29	C₂×C₁₉⋊C₁₆	—	4	1	no	—
30	C₂×C₁₉⋊C₁₆	—	4	1	no	—
31	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
32	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
33	C₂²×C₄×C₁₉	—	4	2	no	—
34	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₉	—	2	1	no	—
38	C₂×C₈×C₁₉	—	2	1	no	—
39	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
44	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₁₉	Q₁₆	1	1	yes	characteristic
47	C₄⋊C₄×C₁₉	—	4	2	no	—
48	C₄⋊C₄×C₁₉	—	4	2	no	—
49	C₄⋊C₄×C₁₉	—	8	2	no	—
50	C₄⋊C₄×C₁₉	—	8	1	no	—
51	C₄⋊C₄×C₁₉	—	8	1	no	—
52	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
54	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
55	C₄²×C₁₉	Q₁₆	1	1	yes	characteristic
56	C₂×C₁₉⋊C₈	—	4	2	no	—
57	C₂×C₁₉⋊C₈	—	8	1	no	—
58	C₂×C₁₉⋊C₈	—	8	1	no	—
59	C₁₉⋊C₁₆	—	4	2	no	—
60	#7274	—	19	1	no	maximal, Sylow
61	C₂³×C₁₉	#102	1	1	yes	characteristic
62	C₂×C₄×C₁₉	—	2	1	no	—
63	C₂×C₄×C₁₉	—	2	1	no	—
64	C₂×C₄×C₁₉	—	8	2	no	—
65	C₂×C₄×C₁₉	—	8	2	no	—
66	C₂×C₄×C₁₉	—	8	2	no	—
67	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
68	C₂×C₄×C₁₉	#100	1	1	yes	characteristic
69	C₂×C₄×C₁₉	#101	1	1	yes	characteristic
70	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
71	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
72	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
73	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
74	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
75	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
76	C₂×C₄×C₁₉	#107	1	1	yes	characteristic
77	C₂×C₄×C₁₉	D₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₉	SD₃₂	1	1	yes	characteristic
79	C₂×C₄×C₁₉	SD₃₂	1	1	yes	characteristic
80	C₂×C₄×C₁₉	Q₃₂	1	1	yes	characteristic
81	C₈×C₁₉	—	2	1	no	—
82	C₈×C₁₉	—	2	1	no	—
83	C₈×C₁₉	#102	1	1	yes	characteristic
84	C₈×C₁₉	#103	1	1	yes	characteristic
85	C₈×C₁₉	#106	1	1	yes	characteristic
86	C₈×C₁₉	#107	1	1	yes	characteristic
87	C₁₉⋊C₈	—	8	2	no	—
88	C₂×C₄⋊C₁₆	—	19	1	no	—
89	#1845	D₁₉	1	1	yes	characteristic
90	C₂×#335	—	19	1	no	—
91	C₂²×C₁₉	#331	1	1	yes	characteristic
92	C₂²×C₁₉	#332	1	1	yes	characteristic
93	C₂²×C₁₉	#340	1	1	yes	characteristic
94	C₂²×C₁₉	#357	1	1	yes	characteristic
95	C₂²×C₁₉	#358	1	1	yes	characteristic
96	C₂²×C₁₉	#361	1	1	yes	characteristic
97	C₂²×C₁₉	#362	1	1	yes	characteristic
98	C₄×C₁₉	—	2	1	no	—
99	C₄×C₁₉	—	2	1	no	—
100	C₄×C₁₉	—	16	2	no	—
101	C₄×C₁₉	#332	1	1	yes	characteristic
102	C₄×C₁₉	#333	1	1	yes	characteristic
103	C₄×C₁₉	#340	1	1	yes	characteristic
104	C₄×C₁₉	#343	1	1	yes	characteristic
105	C₄×C₁₉	#357	1	1	yes	characteristic
106	C₄×C₁₉	#358	1	1	yes	characteristic
107	C₄×C₁₉	#366	1	1	yes	characteristic
108	C₄×C₁₉	#367	1	1	yes	characteristic
109	C₂²×C₁₆	—	38	1	no	—
110	C₂×#107	—	2	1	no	—
111	C₂×#107	Dic₁₉	1	2	yes	—
112	C₂×C₄⋊C₈	—	38	1	no	—
113	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
114	#384	—	2	1	no	—
115	C₄⋊C₁₆	—	19	2	no	—
116	C₄⋊C₁₆	—	38	1	no	—
117	#335	—	19	2	no	—
118	#335	—	38	1	no	—
119	C₂×C₁₉	#1296	1	1	yes	characteristic
120	C₂×C₁₉	#1361	1	1	yes	characteristic
121	C₂×C₁₉	#1362	1	1	yes	characteristic
122	C₂×C₁₉	#1364	1	1	yes	characteristic
123	C₂×C₁₉	#1365	1	1	yes	characteristic
124	C₂×C₁₉	#1380	1	1	yes	characteristic
125	C₂×C₁₉	#1387	1	1	yes	characteristic
126	C₂²×C₈	—	76	1	no	—
127	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
128	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₄⋊C₄	—	4	1	no	—
132	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
133	C₂×C₁₆	—	38	2	no	—
134	C₂×C₁₆	—	76	1	no	—
135	C₂×C₁₆	—	76	1	no	—
136	#107	—	4	1	no	—
137	#107	—	4	1	no	—
138	#107	—	4	2	no	—
139	C₄⋊C₈	—	38	2	no	—
140	C₄⋊C₈	—	76	1	no	—
141	C₄×C₈	Dic₃₈	1	1	yes	characteristic
142	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
143	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
144	C₄×C₈	D₇₆	1	1	yes	characteristic
145	C₁₉	#7274	1	1	yes	characteristic, complemented, Sylow
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	#64525	1	1	yes	characteristic
148	C₂²×C₄	#64526	1	1	yes	characteristic
149	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	76	2	no	—
153	C₂×C₈	—	152	1	no	—
154	C₂×C₈	—	152	1	no	—
155	C₂×C₈	C₄×Dic₁₉	1	1	yes	characteristic
156	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
157	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
158	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
159	C₂×C₈	#64525	1	1	yes	characteristic
160	C₂×C₈	#64526	1	1	yes	characteristic
161	C₂×C₈	#64527	1	1	yes	characteristic
162	C₂×C₈	#64528	1	1	yes	characteristic
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	8	2	no	—
166	C₄⋊C₄	—	8	1	no	—
167	C₄⋊C₄	—	8	1	no	—
168	C₄²	Dic₁₉⋊C₄	1	1	yes	characteristic
169	C₄²	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
170	C₄²	#64527	1	1	yes	characteristic
171	C₄²	#64528	1	1	yes	characteristic
172	C₁₆	—	76	2	no	—
173	C₂³	#10597289	1	1	yes	characteristic
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	#10597264	1	1	yes	characteristic
180	C₂×C₄	#10597265	1	1	yes	characteristic
181	C₂×C₄	#10597266	1	1	yes	characteristic
182	C₂×C₄	#10597267	1	1	yes	characteristic
183	C₂×C₄	#10597283	1	1	yes	characteristic
184	C₂×C₄	#10597284	1	1	yes	characteristic
185	C₂×C₄	#10597285	1	1	yes	characteristic
186	C₂×C₄	#10597286	1	1	yes	characteristic
187	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
188	C₂×C₄	#10597289	1	1	yes	characteristic
189	C₂×C₄	#10597290	1	1	yes	characteristic
190	C₂×C₄	#10597292	1	1	yes	characteristic
191	C₂×C₄	#10597292	1	1	yes	characteristic
192	C₂×C₄	#10597293	1	1	yes	characteristic
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	152	2	no	—
196	C₈	#10597264	1	1	yes	characteristic
197	C₈	#10597265	1	1	yes	characteristic
198	C₈	#10597266	1	1	yes	characteristic
199	C₈	#10597267	1	1	yes	characteristic
200	C₂²	#11926288	1	1	yes	characteristic
201	C₂²	#11926290	1	1	yes	characteristic
202	C₂²	#11926306	1	1	yes	characteristic
203	C₂²	#11926345	1	1	yes	characteristic
204	C₂²	#11926348	1	1	yes	characteristic
205	C₂²	#11926357	1	1	yes	characteristic
206	C₂²	#11926360	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	16	2	no	—
210	C₄	#11926290	1	1	yes	characteristic
211	C₄	#11926293	1	1	yes	characteristic
212	C₄	#11926306	1	1	yes	characteristic
213	C₄	#11926313	1	1	yes	characteristic
214	C₄	#11926344	1	1	yes	characteristic
215	C₄	#11926347	1	1	yes	characteristic
216	C₄	#11926369	1	1	yes	characteristic
217	C₄	#11926371	1	1	yes	characteristic
218	C₂	#14561518	1	1	yes	characteristic
219	C₂	#14561681	1	1	yes	characteristic
220	C₂	#14561684	1	1	yes	characteristic
221	C₂	#14561690	1	1	yes	characteristic
222	C₂	#14561693	1	1	yes	characteristic
223	C₂	#14561726	1	1	yes	characteristic
224	C₂	#14561743	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597289
commutator	C₂×C₈×C₁₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₁₉
Fitting	C₁₉×#1845	C₂
socle, prime radical	C₂³×C₁₉	#102
cosocle	C₂×C₄×C₈×C₁₉	C₂²
hypercenter	#1845	D₁₉
hypocenter	C₁₉	#7274
LCS	G ▷ C₂×C₈×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
7. C₄⋊(C₁₉⋊C₁₆)	C₄	semidirect
9. #11926297	C₄	semidirect
145. C₁₉	#7274	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₁₉⋊C₁₆)	2
C₂×#11926297	2
C₁₉×#1845	2
#7274	19