Subgroups — #18073003

Order 4864.

726 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₁₉	C₂	1	1	yes	maximal, characteristic
2	C₄×(C₂×D₁₉)⋊C₈	C₂	1	2	yes	maximal
3	C₂×C₈×C₁₉⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₁₉⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1273	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₁₉	C₄	1	2	yes	—
7	C₂×C₄²×D₁₉	C₂²	1	1	yes	characteristic
8	C₂×(C₂×D₁₉)⋊C₈	C₄	1	4	yes	—
9	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₉	C₄	1	2	yes	—
12	C₄×C₈×D₁₉	—	2	4	no	—
13	#11926298	C₄	1	4	yes	—
14	C₈×C₁₉⋊C₈	—	2	4	no	—
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
17	C₂²×C₄×D₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×D₁₉	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
22	C₂×C₄×Dic₁₉	C₂×C₄	1	2	yes	—
23	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×C₈×D₁₉	—	2	4	no	—
27	C₂×C₈×D₁₉	—	2	4	no	—
28	C₂×C₈×D₁₉	C₈	1	8	yes	complemented
29	C₄²×D₁₉	—	2	2	no	—
30	C₄²×D₁₉	—	2	1	no	—
31	C₄²×D₁₉	—	2	1	no	—
32	C₄×C₈×C₁₉	—	2	2	no	—
33	C₄×C₈×C₁₉	—	2	2	no	—
34	C₄×C₈×C₁₉	—	2	2	no	—
35	(C₂×D₁₉)⋊C₈	C₈	1	16	yes	complemented
36	C₈×Dic₁₉	—	2	4	no	—
37	C₄×C₁₉⋊C₈	—	2	2	no	—
38	C₄×C₁₉⋊C₈	—	2	2	no	—
39	C₄×C₁₉⋊C₈	D₄	1	4	yes	—
40	C₂³×D₁₉	C₄²	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
43	C₂²×Dic₁₉	C₄²	1	1	yes	characteristic
44	C₂²×Dic₁₉	C₄²	1	2	yes	—
45	C₂×C₄×D₁₉	—	2	1	no	—
46	C₂×C₄×D₁₉	—	2	1	no	—
47	C₂×C₄×D₁₉	—	2	2	no	—
48	C₂×C₄×D₁₉	—	2	2	no	—
49	C₂×C₄×D₁₉	—	2	2	no	—
50	C₂×C₄×D₁₉	—	2	4	no	—
51	C₂×C₄×D₁₉	C₂×C₈	1	8	yes	—
52	C₂×C₄×D₁₉	C₂×C₈	1	4	yes	—
53	C₂×C₈×C₁₉	—	2	2	no	—
54	C₂×C₈×C₁₉	—	2	4	no	—
55	C₂×C₈×C₁₉	—	2	2	no	—
56	C₂×C₈×C₁₉	—	2	4	no	—
57	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
58	C₂×C₈×C₁₉	C₂×C₈	1	8	yes	complemented
59	C₄²×C₁₉	C₂×D₄	1	2	yes	—
60	C₄²×C₁₉	#41	1	1	yes	characteristic
61	C₄²×C₁₉	#41	1	1	yes	characteristic
62	C₄×Dic₁₉	—	2	1	no	—
63	C₄×Dic₁₉	—	2	1	no	—
64	C₄×Dic₁₉	—	2	2	no	—
65	C₄×Dic₁₉	—	2	2	no	—
66	C₂×C₁₉⋊C₈	—	2	4	no	—
67	C₂×C₁₉⋊C₈	—	2	4	no	—
68	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	4	yes	—
69	C₂×C₁₉⋊C₈	C₂×C₈	1	4	yes	—
70	C₂×C₁₉⋊C₈	C₂×C₈	1	8	yes	complemented
71	C₂×C₁₉⋊C₈	M₁₆	1	4	yes	—
72	C₈×D₁₉	—	2	16	no	—
73	C₈×C₂²⋊C₈	—	19	1	no	maximal, Sylow
74	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
75	C₂²×D₁₉	—	2	1	no	—
76	C₂²×D₁₉	—	2	1	no	—
77	C₂²×D₁₉	—	2	2	no	—
78	C₂²×D₁₉	C₄×C₈	1	4	yes	—
79	C₂²×D₁₉	C₄×C₈	1	2	yes	—
80	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
82	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₁₉	C₂²×C₈	1	4	yes	—
86	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
87	C₂×C₄×C₁₉	#131	1	4	yes	—
88	C₂×Dic₁₉	—	2	1	no	—
89	C₂×Dic₁₉	—	2	2	no	—
90	C₂×Dic₁₉	—	2	1	no	—
91	C₂×Dic₁₉	—	2	2	no	—
92	C₂×Dic₁₉	C₄×C₈	1	2	yes	—
93	C₂×Dic₁₉	C₄×C₈	1	4	yes	—
94	C₄×D₁₉	—	2	4	no	—
95	C₄×D₁₉	—	2	4	no	—
96	C₄×D₁₉	—	2	8	no	—
97	C₄×D₁₉	—	2	8	no	—
98	C₈×C₁₉	—	2	4	no	—
99	C₈×C₁₉	—	2	8	no	—
100	C₁₉⋊C₈	—	2	8	no	—
101	C₁₉⋊C₈	C₂²⋊C₈	1	8	yes	complemented
102	C₂²×C₄×C₈	—	19	1	no	—
103	C₄×C₂²⋊C₈	—	19	2	no	—
104	C₂×C₈²	—	19	2	no	—
105	#1273	—	19	1	no	—
106	#1273	D₁₉	1	1	yes	characteristic, complemented
107	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
108	C₂²×C₁₉	C₂×C₄×C₈	1	2	yes	—
109	C₂²×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
110	C₂²×C₁₉	C₄×M₁₆	1	1	yes	characteristic
111	C₂²×C₁₉	#405	1	2	yes	—
112	C₂×D₁₉	—	2	4	no	—
113	C₂×D₁₉	—	2	2	no	—
114	C₂×D₁₉	—	2	2	no	—
115	C₂×D₁₉	—	2	4	no	—
116	C₂×D₁₉	C₈²	1	4	yes	complemented
117	C₄×C₁₉	C₂×C₂²⋊C₈	1	2	yes	—
118	C₄×C₁₉	#431	1	2	yes	—
119	C₄×C₁₉	C₈×D₄	1	4	yes	—
120	C₄×C₁₉	C₈×D₄	1	4	yes	—
121	Dic₁₉	—	2	2	no	—
122	Dic₁₉	—	2	4	no	—
123	C₂³×C₈	—	19	2	no	—
124	C₂²×C₄²	—	19	1	no	—
125	C₂×C₂²⋊C₈	—	19	4	no	—
126	C₂×C₄×C₈	—	19	1	no	—
127	C₂×C₄×C₈	—	19	2	no	—
128	C₂×C₄×C₈	—	19	2	no	—
129	C₂×C₄×C₈	—	38	4	no	—
130	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂×D₁₉	1	2	yes	—
132	#336	—	19	4	no	—
133	C₈²	—	38	4	no	—
134	C₂×C₁₉	C₂×C₈²	1	1	yes	characteristic
135	C₂×C₁₉	C₈×M₁₆	1	2	yes	—
136	C₂×C₁₉	#1450	1	1	yes	characteristic
137	C₂×C₁₉	C₄×C₂²⋊C₈	1	1	yes	characteristic
138	C₂×C₁₉	C₈×C₂²⋊C₄	1	2	yes	—
139	D₁₉	—	2	4	no	—
140	C₂³×C₄	—	19	1	no	—
141	C₂³×C₄	—	19	2	no	—
142	C₂²×C₈	—	19	2	no	—
143	C₂²×C₈	—	19	4	no	—
144	C₂²×C₈	—	19	8	no	—
145	C₂²×C₈	—	38	4	no	—
146	C₂²×C₈	—	38	4	no	—
147	C₂²×C₈	C₄×D₁₉	1	2	yes	—
148	C₂²×C₈	C₄×D₁₉	1	4	yes	—
149	C₂×C₄²	—	19	2	no	—
150	C₂×C₄²	—	19	1	no	—
151	C₂×C₄²	—	38	2	no	—
152	C₂×C₄²	—	38	1	no	—
153	C₂×C₄²	—	38	1	no	—
154	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
155	C₂²⋊C₈	—	19	16	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	19	4	no	—
160	C₄×C₈	—	38	4	no	—
161	C₄×C₈	—	38	2	no	—
162	C₄×C₈	—	38	2	no	—
163	C₁₉	C₈×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
164	C₂⁴	—	19	1	no	—
165	C₂²×C₄	—	19	1	no	—
166	C₂²×C₄	—	19	2	no	—
167	C₂²×C₄	—	19	8	no	—
168	C₂²×C₄	—	19	4	no	—
169	C₂²×C₄	—	38	1	no	—
170	C₂²×C₄	—	38	2	no	—
171	C₂²×C₄	—	38	1	no	—
172	C₂²×C₄	—	38	2	no	—
173	C₂²×C₄	—	38	2	no	—
174	C₂²×C₄	—	38	4	no	—
175	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	19	4	no	—
182	C₂×C₈	—	19	4	no	—
183	C₂×C₈	—	19	4	no	—
184	C₂×C₈	—	19	8	no	—
185	C₂×C₈	—	38	16	no	—
186	C₂×C₈	—	38	4	no	—
187	C₂×C₈	—	38	4	no	—
188	C₂×C₈	C₈×D₁₉	1	4	yes	—
189	C₂×C₈	C₈×D₁₉	1	8	yes	complemented
190	C₄²	—	38	1	no	—
191	C₄²	—	38	2	no	—
192	C₄²	—	38	1	no	—
193	C₄²	—	38	2	no	—
194	C₄²	D₄×D₁₉	1	2	yes	—
195	C₄²	#64543	1	1	yes	characteristic
196	C₄²	#64545	1	1	yes	characteristic
197	C₂³	—	19	2	no	—
198	C₂³	—	19	4	no	—
199	C₂³	—	38	1	no	—
200	C₂³	—	38	1	no	—
201	C₂³	—	38	2	no	—
202	C₂³	C₄²×D₁₉	1	1	yes	characteristic
203	C₂×C₄	—	19	2	no	—
204	C₂×C₄	—	19	4	no	—
205	C₂×C₄	—	38	1	no	—
206	C₂×C₄	—	38	2	no	—
207	C₂×C₄	—	38	1	no	—
208	C₂×C₄	—	38	2	no	—
209	C₂×C₄	—	38	4	no	—
210	C₂×C₄	—	38	8	no	—
211	C₂×C₄	—	38	8	no	—
212	C₂×C₄	—	38	4	no	—
213	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
214	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
215	C₂×C₄	#10597338	1	1	yes	characteristic
216	C₂×C₄	#10597339	1	2	yes	—
217	C₂×C₄	C₂×C₈×D₁₉	1	2	yes	—
218	C₂×C₄	C₂×C₈×D₁₉	1	4	yes	—
219	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
220	C₂×C₄	#10597353	1	4	yes	—
221	C₈	—	2	8	no	—
222	C₈	—	2	4	no	—
223	C₈	—	19	8	no	—
224	C₈	—	38	8	no	—
225	C₂²	—	19	4	no	—
226	C₂²	—	38	2	no	—
227	C₂²	—	38	4	no	—
228	C₂²	—	38	4	no	—
229	C₂²	—	38	2	no	—
230	C₂²	#11926391	1	1	yes	characteristic
231	C₂²	C₄×C₈×D₁₉	1	1	yes	characteristic
232	C₂²	C₄×C₈×D₁₉	1	2	yes	—
233	C₂²	#11926496	1	1	yes	characteristic
234	C₂²	#11926501	1	2	yes	—
235	C₄	—	38	2	no	—
236	C₄	—	38	4	no	—
237	C₄	C₂²⋊C₈×D₁₉	1	2	yes	—
238	C₄	#11926596	1	2	yes	—
239	C₄	(C₂²⋊D₁₉)⋊C₈	1	4	yes	—
240	C₄	#11926609	1	4	yes	—
241	C₂	—	38	4	no	—
242	C₂	C₈²×D₁₉	1	1	yes	characteristic
243	C₂	#14561914	1	2	yes	—
244	C₂	#14561920	1	1	yes	characteristic
245	C₂	#14563392	1	1	yes	characteristic
246	C₂	#14563405	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₁₉
commutator	C₂×C₁₉	C₂×C₈²
Fitting	C₁₉×#1273	C₂
socle	C₂³×C₁₉	C₂×C₄²
prime radical	C₂³×D₁₉	C₄²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1273	D₁₉
hypocenter	C₁₉	C₈×C₂²⋊C₈
LCS	G ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄² ◁ #1273

Complements

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

N	H ≅ G/N	type
3. C₂×C₈×C₁₉⋊C₈	C₂	semidirect
4. C₁₉⋊#1273	C₂	semidirect
5. C₁₉×#1273	C₂	semidirect
28. C₂×C₈×D₁₉	C₈	semidirect
35. (C₂×D₁₉)⋊C₈	C₈	semidirect
58. C₂×C₈×C₁₉	C₂×C₈	semidirect
70. C₂×C₁₉⋊C₈	C₂×C₈	semidirect
101. C₁₉⋊C₈	C₂²⋊C₈	semidirect
106. #1273	D₁₉	semidirect
116. C₂×D₁₉	C₈²	semidirect
163. C₁₉	C₈×C₂²⋊C₈	semidirect
189. C₂×C₈	C₈×D₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₁₉	2
C₄×(C₂×D₁₉)⋊C₈	2
C₂×C₈×C₁₉⋊C₈	2
C₁₉⋊#1273	2
C₁₉×#1273	2
C₈×C₂²⋊C₈	19