Subgroups — #18081769

Order 4864.

486 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564559	C₂	1	2	yes	maximal
2	C₈×C₄⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926297	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926290	C₂	1	2	yes	maximal
5	C₂×C₁₉×#500	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×#10597292	—	2	4	no	—
9	C₂×C₈×Dic₁₉	—	2	1	no	—
10	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
11	#11926298	—	2	1	no	—
12	#11926297	—	2	2	no	—
13	#11926290	—	2	4	no	—
14	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₁₉×#500	—	2	2	no	—
16	C₁₉×#500	C₄	1	4	yes	complemented
17	C₂×C₁₉×#107	—	2	1	no	—
18	C₂×C₁₉×#107	—	2	1	no	—
19	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₉	—	2	2	no	—
21	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
23	C₁₉×C₄⋊Q₈	—	2	2	no	—
24	C₁₉×C₄⋊Q₈	—	2	2	no	—
25	C₁₉×C₄⋊Q₈	—	2	2	no	—
26	C₁₉×C₄⋊Q₈	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
29	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₉	D₄	1	2	yes	—
31	C₂×C₄×Dic₁₉	—	2	1	no	—
32	C₂²×C₁₉⋊C₈	—	2	1	no	—
33	C₂²×C₁₉⋊C₈	—	2	2	no	—
34	C₁₉×#107	—	2	4	no	—
35	C₁₉×#107	—	2	4	no	—
36	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₁₉	D₄	1	2	yes	—
38	#10597292	—	4	8	no	—
39	C₁₉⋊#95	—	2	1	no	—
40	C₈×Dic₁₉	—	4	2	no	—
41	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
42	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
43	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
44	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
45	C₂×Q₈×C₁₉	—	2	4	no	—
46	C₂×Q₈×C₁₉	—	4	2	no	—
47	C₂×Q₈×C₁₉	—	4	4	no	—
48	C₂×Q₈×C₁₉	—	4	4	no	—
49	C₂²×C₄×C₁₉	—	4	2	no	—
50	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
53	C₂²×Dic₁₉	—	2	2	no	—
54	C₂×C₈×C₁₉	—	2	1	no	—
55	C₂×C₈×C₁₉	—	2	1	no	—
56	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
59	C₂×C₈×C₁₉	Q₁₆	1	2	yes	—
60	C₄⋊C₄×C₁₉	—	2	4	no	—
61	C₄⋊C₄×C₁₉	—	2	4	no	—
62	C₄⋊C₄×C₁₉	—	4	2	no	—
63	C₄⋊C₄×C₁₉	—	4	2	no	—
64	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₁₉	#41	1	1	yes	characteristic
68	C₄⋊Dic₁₉	—	2	2	no	—
69	C₄⋊Dic₁₉	D₈	1	4	yes	—
70	C₄×Dic₁₉	—	4	1	no	—
71	C₄×Dic₁₉	—	4	1	no	—
72	C₂×C₁₉⋊C₈	—	2	2	no	—
73	C₂×C₁₉⋊C₈	—	4	1	no	—
74	C₂×C₁₉⋊C₈	—	4	1	no	—
75	C₂×C₁₉⋊C₈	—	4	2	no	—
76	C₂×C₁₉⋊C₈	—	4	4	no	—
77	#9731	—	19	1	no	maximal, Sylow
78	C₂³×C₁₉	#121	1	1	yes	characteristic
79	Q₈×C₁₉	—	4	8	no	—
80	Q₈×C₁₉	—	8	2	no	—
81	Q₈×C₁₉	—	8	2	no	—
82	C₂×C₄×C₁₉	—	2	2	no	—
83	C₂×C₄×C₁₉	—	4	4	no	—
84	C₂×C₄×C₁₉	—	8	4	no	—
85	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
89	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
90	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
91	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₉	#135	1	2	yes	—
95	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
96	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
97	C₂×Dic₁₉	—	2	4	no	—
98	C₂×Dic₁₉	—	4	2	no	—
99	C₂×Dic₁₉	—	4	2	no	—
100	C₈×C₁₉	—	2	2	no	—
101	C₈×C₁₉	#103	1	4	yes	complemented
102	C₁₉⋊C₈	—	4	2	no	—
103	C₁₉⋊C₈	—	8	2	no	—
104	C₂×#500	D₁₉	1	1	yes	characteristic
105	#1981	—	19	2	no	—
106	C₈×C₄⋊C₄	—	19	1	no	—
107	C₂×#335	—	19	1	no	—
108	C₂×#332	—	19	2	no	—
109	C₂²×C₁₉	#390	1	1	yes	characteristic
110	C₂²×C₁₉	#461	1	1	yes	characteristic
111	C₂²×C₁₉	#462	1	1	yes	characteristic
112	C₂²×C₁₉	#466	1	2	yes	—
113	C₂²×C₁₉	#473	1	2	yes	—
114	C₄×C₁₉	—	2	1	no	—
115	C₄×C₁₉	—	2	1	no	—
116	C₄×C₁₉	—	8	4	no	—
117	C₄×C₁₉	C₂×#103	1	1	yes	characteristic
118	C₄×C₁₉	#419	1	1	yes	characteristic
119	C₄×C₁₉	C₄×D₈	1	2	yes	—
120	C₄×C₁₉	#461	1	1	yes	characteristic
121	C₄×C₁₉	#462	1	1	yes	characteristic
122	C₄×C₁₉	#493	1	1	yes	characteristic
123	C₄×C₁₉	#495	1	1	yes	characteristic
124	Dic₁₉	—	4	4	no	—
125	C₂×C₄⋊Q₈	C₂×D₁₉	1	2	yes	—
126	#500	—	2	2	no	—
127	#500	Dic₁₉	1	4	yes	complemented
128	C₂×#107	—	2	1	no	—
129	C₂×#107	—	2	1	no	—
130	C₂×C₄⋊C₈	—	19	1	no	—
131	C₂×C₄⋊C₈	—	19	2	no	—
132	C₂×#103	—	38	4	no	—
133	C₂×C₄×C₈	—	38	1	no	—
134	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
135	C₄×C₄⋊C₄	—	19	1	no	—
136	#336	—	38	1	no	—
137	#335	—	38	2	no	—
138	#332	—	38	4	no	—
139	C₂×C₁₉	#1673	1	2	yes	—
140	C₂×C₁₉	#1674	1	2	yes	—
141	C₂×C₁₉	#1970	1	1	yes	characteristic
142	C₂×C₁₉	#1979	1	1	yes	characteristic
143	C₂×C₁₉	#1981	1	1	yes	characteristic
144	C₂²×Q₈	—	2	2	no	—
145	C₂²×C₈	—	38	1	no	—
146	C₂²×C₈	—	38	2	no	—
147	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
148	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	C₂×Dic₁₉	1	4	yes	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	19	2	no	—
156	C₂×C₄²	—	38	1	no	—
157	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
158	#107	—	2	4	no	—
159	#107	—	2	4	no	—
160	C₄⋊C₈	—	19	2	no	—
161	C₄⋊C₈	—	38	2	no	—
162	C₄⋊C₈	—	38	1	no	—
163	C₄⋊C₈	—	38	2	no	—
164	#103	—	76	8	no	—
165	C₄×C₈	—	76	2	no	—
166	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
167	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
168	#95	—	38	1	no	—
169	C₁₉	#9731	1	1	yes	characteristic, complemented, Sylow
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	—	4	4	no	—
173	C₂×Q₈	—	4	4	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	38	2	no	—
176	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
177	C₂²×C₄	#64543	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	38	2	no	—
182	C₂×C₈	—	76	1	no	—
183	C₂×C₈	—	76	1	no	—
184	C₂×C₈	—	76	2	no	—
185	C₂×C₈	—	76	4	no	—
186	C₂×C₈	#64527	1	2	yes	—
187	C₂×C₈	#64528	1	2	yes	—
188	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
189	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	19	4	no	—
195	C₄⋊C₄	—	38	2	no	—
196	C₄²	—	76	1	no	—
197	C₄²	—	76	1	no	—
198	C₄²	D₄×D₁₉	1	1	yes	characteristic
199	C₄²	#64543	1	1	yes	characteristic
200	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
201	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
202	C₂³	#10597384	1	1	yes	characteristic
203	Q₈	—	4	8	no	—
204	Q₈	—	8	2	no	—
205	Q₈	—	8	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	8	4	no	—
209	C₂×C₄	—	38	4	no	—
210	C₂×C₄	—	76	2	no	—
211	C₂×C₄	—	76	2	no	—
212	C₂×C₄	D₈×D₁₉	1	2	yes	—
213	C₂×C₄	#10597358	1	2	yes	—
214	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
215	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
216	C₂×C₄	#10597382	1	1	yes	characteristic
217	C₂×C₄	#10597384	1	1	yes	characteristic
218	C₂×C₄	#10597386	1	1	yes	characteristic
219	C₂×C₄	C₂×#64527	1	1	yes	characteristic
220	C₂×C₄	#10597388	1	1	yes	characteristic
221	C₂×C₄	#10597388	1	1	yes	characteristic
222	C₂×C₄	C₂×#64528	1	1	yes	characteristic
223	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	76	2	no	—
226	C₈	—	152	2	no	—
227	C₈	#10597292	1	4	yes	complemented
228	C₂²	#11926447	1	1	yes	characteristic
229	C₂²	#11926746	1	1	yes	characteristic
230	C₂²	#11926751	1	1	yes	characteristic
231	C₂²	#11926777	1	2	yes	—
232	C₂²	#11926816	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	8	4	no	—
236	C₄	—	76	4	no	—
237	C₄	C₂×#10597292	1	1	yes	characteristic
238	C₄	#11926553	1	1	yes	characteristic
239	C₄	D₈×Dic₁₉	1	2	yes	—
240	C₄	#11926746	1	1	yes	characteristic
241	C₄	#11926751	1	1	yes	characteristic
242	C₄	#11926919	1	1	yes	characteristic
243	C₄	#11926927	1	1	yes	characteristic
244	C₂	#14563021	1	2	yes	—
245	C₂	#14563026	1	2	yes	—
246	C₂	#14564502	1	1	yes	characteristic
247	C₂	#14564551	1	1	yes	characteristic
248	C₂	#14564559	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597384
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#500	C₂
socle, prime radical	C₂³×C₁₉	#121
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#500	D₁₉
hypocenter	C₁₉	#9731
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
16. C₁₉×#500	C₄	semidirect
101. C₈×C₁₉	#103	semidirect
127. #500	Dic₁₉	semidirect
169. C₁₉	#9731	semidirect
227. C₈	#10597292	semidirect

Maximal subgroups

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

subgroup	index
#14564559	2
C₈×C₄⋊Dic₁₉	2
C₂×#11926297	2
C₂×#11926290	2
C₂×C₁₉×#500	2
#9731	19