Subgroups — #18082299

Order 4864.

526 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564597	C₂	1	2	yes	maximal
2	#14564438	C₂	1	2	yes	maximal
3	#14563795	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₉×#500	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597275	—	2	2	no	—
7	C₂×#10597292	—	2	4	no	—
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
10	#11926414	C₂²	1	2	yes	—
11	#11926306	—	2	4	no	—
12	C₈×C₁₉⋊C₈	—	2	2	no	—
13	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₁₉×#500	—	2	2	no	—
15	C₁₉×#500	C₄	1	4	yes	complemented
16	C₂×C₁₉×#107	—	2	2	no	—
17	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₂²×Q₈×C₁₉	—	2	2	no	—
19	C₂²×C₈×C₁₉	D₄	1	2	yes	—
20	C₁₉×C₄⋊Q₈	—	2	2	no	—
21	C₁₉×C₄⋊Q₈	—	2	2	no	—
22	C₁₉×C₄⋊Q₈	—	2	2	no	—
23	C₁₉×C₄⋊Q₈	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
26	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₉	—	2	2	no	—
28	C₂×C₄⋊Dic₁₉	—	2	2	no	—
29	C₂×C₄⋊Dic₁₉	—	2	2	no	—
30	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
31	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
32	C₁₉×#107	—	2	8	no	—
33	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₁₉	D₄	1	2	yes	—
35	#10597292	—	4	8	no	—
36	#10597275	—	4	4	no	—
37	C₄×C₁₉⋊C₈	—	2	1	no	—
38	C₄×C₁₉⋊C₈	—	2	1	no	—
39	C₄×C₁₉⋊C₈	—	2	1	no	—
40	C₄×C₁₉⋊C₈	—	2	1	no	—
41	C₂×Q₈×C₁₉	—	2	4	no	—
42	C₂×Q₈×C₁₉	—	4	2	no	—
43	C₂×Q₈×C₁₉	—	4	4	no	—
44	C₂×Q₈×C₁₉	—	4	4	no	—
45	C₂²×C₄×C₁₉	—	4	2	no	—
46	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₁₉	—	4	2	no	—
50	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
51	C₂×C₈×C₁₉	SD₁₆	1	4	yes	—
52	C₂×C₈×C₁₉	Q₁₆	1	4	yes	—
53	C₄⋊C₄×C₁₉	—	2	4	no	—
54	C₄⋊C₄×C₁₉	—	2	4	no	—
55	C₄⋊C₄×C₁₉	—	4	2	no	—
56	C₄⋊C₄×C₁₉	—	4	2	no	—
57	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₁₉	#41	1	1	yes	characteristic
60	C₄²×C₁₉	#41	1	1	yes	characteristic
61	C₄⋊Dic₁₉	—	4	4	no	—
62	C₄⋊Dic₁₉	—	4	4	no	—
63	C₄⋊Dic₁₉	—	4	4	no	—
64	C₂×C₁₉⋊C₈	—	2	2	no	—
65	C₂×C₁₉⋊C₈	—	2	2	no	—
66	C₂×C₁₉⋊C₈	—	2	2	no	—
67	C₂×C₁₉⋊C₈	—	2	2	no	—
68	C₂×C₁₉⋊C₈	D₈	1	4	yes	—
69	C₂×C₁₉⋊C₈	Q₁₆	1	4	yes	—
70	#9823	—	19	1	no	maximal, Sylow
71	C₂³×C₁₉	#127	1	1	yes	characteristic
72	Q₈×C₁₉	—	4	8	no	—
73	Q₈×C₁₉	—	8	4	no	—
74	C₂×C₄×C₁₉	—	4	4	no	—
75	C₂×C₄×C₁₉	—	8	2	no	—
76	C₂×C₄×C₁₉	—	8	2	no	—
77	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
81	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
82	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
83	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
85	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
86	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
87	C₂×C₄×C₁₉	#135	1	2	yes	—
88	C₂×C₄×C₁₉	#135	1	2	yes	—
89	C₂×Dic₁₉	—	8	2	no	—
90	C₂×Dic₁₉	—	8	2	no	—
91	C₂×Dic₁₉	—	8	2	no	—
92	C₈×C₁₉	#103	1	8	yes	complemented
93	C₁₉⋊C₈	—	2	4	no	—
94	C₁₉⋊C₈	—	2	4	no	—
95	C₂×#500	D₁₉	1	1	yes	characteristic
96	#1990	—	19	2	no	—
97	#1956	—	19	2	no	—
98	#1845	—	19	1	no	—
99	C₂×C₈²	—	19	1	no	—
100	C₂²×C₁₉	#390	1	1	yes	characteristic
101	C₂²×C₁₉	#492	1	1	yes	characteristic
102	C₂²×C₁₉	#493	1	1	yes	characteristic
103	C₂²×C₁₉	#494	1	1	yes	characteristic
104	C₂²×C₁₉	#494	1	1	yes	characteristic
105	C₂²×C₁₉	#495	1	1	yes	characteristic
106	C₂²×C₁₉	#495	1	1	yes	characteristic
107	C₄×C₁₉	—	8	4	no	—
108	C₄×C₁₉	C₂×#103	1	2	yes	—
109	C₄×C₁₉	C₄×D₈	1	2	yes	—
110	C₄×C₁₉	C₄×Q₁₆	1	2	yes	—
111	C₄×C₁₉	#466	1	2	yes	—
112	C₄×C₁₉	#467	1	2	yes	—
113	C₄×C₁₉	#487	1	2	yes	—
114	Dic₁₉	—	16	2	no	—
115	C₂×C₄⋊Q₈	C₂×D₁₉	1	2	yes	—
116	#500	—	2	2	no	—
117	#500	Dic₁₉	1	4	yes	complemented
118	C₂×#107	—	2	2	no	—
119	C₂×#107	—	38	2	no	—
120	C₂×#103	—	38	4	no	—
121	C₂×C₄×C₈	—	19	1	no	—
122	C₂×C₄×C₈	—	19	1	no	—
123	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
124	#384	—	19	2	no	—
125	#340	—	38	4	no	—
126	C₈²	—	38	2	no	—
127	C₂×C₁₉	#1706	1	1	yes	characteristic
128	C₂×C₁₉	#1707	1	1	yes	characteristic
129	C₂×C₁₉	#1710	1	1	yes	characteristic
130	C₂×C₁₉	#1713	1	1	yes	characteristic
131	C₂×C₁₉	#1956	1	1	yes	characteristic
132	C₂×C₁₉	#1969	1	1	yes	characteristic
133	C₂×C₁₉	#1970	1	1	yes	characteristic
134	C₂²×Q₈	—	2	2	no	—
135	C₂²×C₈	—	19	2	no	—
136	C₂²×C₈	—	19	2	no	—
137	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
138	C₄⋊Q₈	—	2	2	no	—
139	C₄⋊Q₈	—	2	2	no	—
140	C₄⋊Q₈	—	2	2	no	—
141	C₄⋊Q₈	C₂×Dic₁₉	1	4	yes	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₄⋊C₄	—	38	2	no	—
145	C₂×C₄⋊C₄	—	38	2	no	—
146	C₂×C₄⋊C₄	—	38	2	no	—
147	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
148	#107	—	2	8	no	—
149	#107	—	76	4	no	—
150	#103	—	76	8	no	—
151	C₄×C₈	—	38	1	no	—
152	C₄×C₈	—	38	1	no	—
153	C₄×C₈	—	38	1	no	—
154	C₄×C₈	—	38	1	no	—
155	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
156	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
157	C₁₉	#9823	1	1	yes	characteristic, complemented, Sylow
158	C₂×Q₈	—	2	4	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	4	no	—
161	C₂×Q₈	—	4	4	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	—	76	2	no	—
164	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
165	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
167	C₂×C₈	—	19	4	no	—
168	C₂×C₈	—	19	4	no	—
169	C₂×C₈	—	38	2	no	—
170	C₂×C₈	—	38	2	no	—
171	C₂×C₈	—	38	2	no	—
172	C₂×C₈	—	38	2	no	—
173	C₂×C₈	#64527	1	4	yes	—
174	C₂×C₈	#64528	1	4	yes	—
175	C₂×C₈	C₂²⋊Dic₁₉	1	4	yes	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	76	4	no	—
181	C₄⋊C₄	—	76	4	no	—
182	C₄⋊C₄	—	76	4	no	—
183	C₄²	#64543	1	1	yes	characteristic
184	C₄²	#64543	1	1	yes	characteristic
185	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
186	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
187	C₂³	#10597386	1	1	yes	characteristic
188	Q₈	—	4	8	no	—
189	Q₈	—	8	4	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	152	2	no	—
194	C₂×C₄	—	152	2	no	—
195	C₂×C₄	—	152	2	no	—
196	C₂×C₄	D₈×D₁₉	1	2	yes	—
197	C₂×C₄	#10597358	1	2	yes	—
198	C₂×C₄	Q₁₆×D₁₉	1	2	yes	—
199	C₂×C₄	#10597365	1	2	yes	—
200	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
201	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
202	C₂×C₄	#10597382	1	1	yes	characteristic
203	C₂×C₄	#10597384	1	1	yes	characteristic
204	C₂×C₄	#10597384	1	1	yes	characteristic
205	C₂×C₄	C₂×#64527	1	2	yes	—
206	C₂×C₄	C₂×#64528	1	2	yes	—
207	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
208	C₈	—	38	4	no	—
209	C₈	—	38	4	no	—
210	C₈	#10597292	1	8	yes	complemented
211	C₂²	#11926447	1	1	yes	characteristic
212	C₂²	#11926918	1	1	yes	characteristic
213	C₂²	#11926919	1	1	yes	characteristic
214	C₂²	#11926922	1	1	yes	characteristic
215	C₂²	#11926923	1	1	yes	characteristic
216	C₂²	#11926927	1	1	yes	characteristic
217	C₂²	#11926929	1	1	yes	characteristic
218	C₄	—	8	4	no	—
219	C₄	—	304	2	no	—
220	C₄	C₂×#10597292	1	2	yes	—
221	C₄	D₈×Dic₁₉	1	2	yes	—
222	C₄	Q₁₆×Dic₁₉	1	2	yes	—
223	C₄	#11926777	1	2	yes	—
224	C₄	#11926782	1	2	yes	—
225	C₄	#11926894	1	2	yes	—
226	C₂	#14563228	1	1	yes	characteristic
227	C₂	#14563233	1	1	yes	characteristic
228	C₂	#14563244	1	1	yes	characteristic
229	C₂	#14563252	1	1	yes	characteristic
230	C₂	#14564438	1	1	yes	characteristic
231	C₂	#14564497	1	1	yes	characteristic
232	C₂	#14564502	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597386
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#500	C₂
socle, prime radical	C₂³×C₁₉	#127
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#500	D₁₉
hypocenter	C₁₉	#9823
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
15. C₁₉×#500	C₄	semidirect
92. C₈×C₁₉	#103	semidirect
117. #500	Dic₁₉	semidirect
157. C₁₉	#9823	semidirect
210. C₈	#10597292	semidirect

Maximal subgroups

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

subgroup	index
#14564597	2
#14564438	2
#14563795	2
C₂×C₈×C₁₉⋊C₈	2
C₂×C₁₉×#500	2
#9823	19