Subgroups — #18081824

Order 4864.

474 conjugacy classes of subgroups.

The subgroup lattice has 244 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	#14563502	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926295	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×#10597272	—	2	1	no	—
10	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
11	#11926298	—	2	1	no	—
12	#11926295	—	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	#10597272	—	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	4	no	—
69	C₄×Dic₁₉	—	4	1	no	—
70	C₄×Dic₁₉	—	4	1	no	—
71	C₂×C₁₉⋊C₈	—	2	2	no	—
72	C₂×C₁₉⋊C₈	—	4	1	no	—
73	C₂×C₁₉⋊C₈	—	4	1	no	—
74	C₂×C₁₉⋊C₈	—	4	2	no	—
75	C₂×C₁₉⋊C₈	—	4	4	no	—
76	#9740	—	19	1	no	maximal, Sylow
77	C₂³×C₁₉	#121	1	1	yes	characteristic
78	Q₈×C₁₉	—	4	8	no	—
79	Q₈×C₁₉	—	8	2	no	—
80	Q₈×C₁₉	—	8	2	no	—
81	C₂×C₄×C₁₉	—	2	2	no	—
82	C₂×C₄×C₁₉	—	4	4	no	—
83	C₂×C₄×C₁₉	—	8	4	no	—
84	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
88	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
89	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
90	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	4	yes	—
93	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
94	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
95	C₂×Dic₁₉	—	4	2	no	—
96	C₂×Dic₁₉	—	4	2	no	—
97	C₂×Dic₁₉	—	4	2	no	—
98	C₈×C₁₉	—	2	2	no	—
99	C₈×C₁₉	#103	1	4	yes	complemented
100	C₁₉⋊C₈	—	4	2	no	—
101	C₁₉⋊C₈	—	8	2	no	—
102	C₂×#500	D₁₉	1	1	yes	characteristic
103	#1981	—	19	2	no	—
104	#1770	—	19	1	no	—
105	C₂×#334	—	19	1	no	—
106	C₂×#332	—	19	2	no	—
107	C₂²×C₁₉	#390	1	1	yes	characteristic
108	C₂²×C₁₉	#461	1	1	yes	characteristic
109	C₂²×C₁₉	#462	1	1	yes	characteristic
110	C₂²×C₁₉	#469	1	2	yes	—
111	C₂²×C₁₉	#473	1	2	yes	—
112	C₄×C₁₉	—	2	1	no	—
113	C₄×C₁₉	—	2	1	no	—
114	C₄×C₁₉	—	8	4	no	—
115	C₄×C₁₉	C₂×#103	1	1	yes	characteristic
116	C₄×C₁₉	#419	1	1	yes	characteristic
117	C₄×C₁₉	#442	1	2	yes	—
118	C₄×C₁₉	#461	1	1	yes	characteristic
119	C₄×C₁₉	#462	1	1	yes	characteristic
120	C₄×C₁₉	#496	1	2	yes	—
121	Dic₁₉	—	8	2	no	—
122	C₂×C₄⋊Q₈	C₂×D₁₉	1	2	yes	—
123	#500	—	2	2	no	—
124	#500	Dic₁₉	1	4	yes	complemented
125	C₂×#107	—	2	1	no	—
126	C₂×#107	—	2	1	no	—
127	C₂×C₄⋊C₈	—	19	1	no	—
128	C₂×C₄⋊C₈	—	19	2	no	—
129	C₂×#103	—	38	4	no	—
130	C₂×#97	—	38	1	no	—
131	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
132	C₄×C₄⋊C₄	—	19	1	no	—
133	#336	—	38	1	no	—
134	#334	—	38	2	no	—
135	#332	—	38	4	no	—
136	C₂×C₁₉	#1693	1	2	yes	—
137	C₂×C₁₉	#1694	1	2	yes	—
138	C₂×C₁₉	#1972	1	1	yes	characteristic
139	C₂×C₁₉	#1979	1	1	yes	characteristic
140	C₂×C₁₉	#1981	1	1	yes	characteristic
141	C₂²×Q₈	—	2	2	no	—
142	C₂²×C₈	—	38	1	no	—
143	C₂²×C₈	—	38	2	no	—
144	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
145	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
146	C₄⋊Q₈	—	2	2	no	—
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	C₂×Dic₁₉	1	4	yes	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	19	2	no	—
153	C₂×C₄²	—	38	1	no	—
154	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
155	#107	—	2	4	no	—
156	#107	—	2	4	no	—
157	C₄⋊C₈	—	19	2	no	—
158	C₄⋊C₈	—	38	2	no	—
159	C₄⋊C₈	—	38	1	no	—
160	C₄⋊C₈	—	38	2	no	—
161	#103	—	76	8	no	—
162	#97	—	76	2	no	—
163	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
164	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
165	#95	—	38	1	no	—
166	C₁₉	#9740	1	1	yes	characteristic, complemented, Sylow
167	C₂×Q₈	—	2	4	no	—
168	C₂×Q₈	—	4	2	no	—
169	C₂×Q₈	—	4	4	no	—
170	C₂×Q₈	—	4	4	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	—	38	2	no	—
173	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
174	C₂²×C₄	#64543	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	38	2	no	—
179	C₂×C₈	—	76	1	no	—
180	C₂×C₈	—	76	1	no	—
181	C₂×C₈	—	76	2	no	—
182	C₂×C₈	—	76	4	no	—
183	C₂×C₈	#64527	1	2	yes	—
184	C₂×C₈	#64528	1	2	yes	—
185	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
186	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	38	4	no	—
192	C₄²	—	76	1	no	—
193	C₄²	—	76	1	no	—
194	C₄²	D₄×D₁₉	1	1	yes	characteristic
195	C₄²	#64543	1	1	yes	characteristic
196	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
197	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
198	C₂³	#10597384	1	1	yes	characteristic
199	Q₈	—	4	8	no	—
200	Q₈	—	8	2	no	—
201	Q₈	—	8	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	—	76	2	no	—
206	C₂×C₄	—	76	2	no	—
207	C₂×C₄	—	76	2	no	—
208	C₂×C₄	#10597357	1	4	yes	—
209	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
210	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
211	C₂×C₄	#10597382	1	1	yes	characteristic
212	C₂×C₄	#10597384	1	1	yes	characteristic
213	C₂×C₄	#10597386	1	1	yes	characteristic
214	C₂×C₄	C₂×#64527	1	1	yes	characteristic
215	C₂×C₄	#10597388	1	1	yes	characteristic
216	C₂×C₄	#10597388	1	1	yes	characteristic
217	C₂×C₄	C₂×#64528	1	1	yes	characteristic
218	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
219	C₈	—	2	2	no	—
220	C₈	—	76	2	no	—
221	C₈	—	152	2	no	—
222	C₈	#10597292	1	4	yes	complemented
223	C₂²	#11926447	1	1	yes	characteristic
224	C₂²	#11926746	1	1	yes	characteristic
225	C₂²	#11926751	1	1	yes	characteristic
226	C₂²	#11926794	1	2	yes	—
227	C₂²	#11926816	1	2	yes	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	8	4	no	—
231	C₄	—	152	2	no	—
232	C₄	C₂×#10597292	1	1	yes	characteristic
233	C₄	#11926553	1	1	yes	characteristic
234	C₄	#11926656	1	2	yes	—
235	C₄	#11926746	1	1	yes	characteristic
236	C₄	#11926751	1	1	yes	characteristic
237	C₄	#11926932	1	2	yes	—
238	C₂	#14563145	1	2	yes	—
239	C₂	#14563150	1	2	yes	—
240	C₂	#14564512	1	1	yes	characteristic
241	C₂	#14564551	1	1	yes	characteristic
242	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	#9740
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
99. C₈×C₁₉	#103	semidirect
124. #500	Dic₁₉	semidirect
166. C₁₉	#9740	semidirect
222. 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
#14563502	2
C₂×#11926295	2
C₂×#11926290	2
C₂×C₁₉×#500	2
#9740	19