Subgroups — #18081597

Order 4864.

498 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	#14563787	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926293	C₂	1	2	yes	maximal
4	C₂×#11926290	C₂	1	2	yes	maximal
5	C₁₉×#2862	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
10	#11926299	—	2	1	no	—
11	#11926293	—	2	4	no	—
12	#11926290	—	2	4	no	—
13	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₉×#391	—	2	1	no	—
16	C₁₉×#391	—	2	1	no	—
17	C₁₉×#391	C₄	1	4	yes	—
18	C₁₉×#382	—	2	2	no	—
19	C₄³×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₉	—	2	1	no	—
21	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
22	C₁₉×C₄⋊Q₈	—	2	2	no	—
23	C₁₉×C₄⋊Q₈	—	2	2	no	—
24	C₁₉×C₄⋊Q₈	—	2	2	no	—
25	C₁₉×C₄⋊Q₈	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	D₄	1	2	yes	—
34	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
35	C₂²×C₁₉⋊C₈	—	2	2	no	—
36	C₂²×C₁₉⋊C₈	—	2	2	no	—
37	C₁₉×#95	—	2	4	no	—
38	C₁₉×#95	—	2	4	no	—
39	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
40	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
41	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
42	C₄⋊(C₁₉⋊C₈)	—	4	4	no	—
43	C₄⋊(C₁₉⋊C₈)	D₄	1	4	yes	—
44	C₄×C₁₉⋊C₈	—	4	2	no	—
45	C₂×Q₈×C₁₉	—	2	4	no	—
46	C₂×Q₈×C₁₉	—	4	4	no	—
47	C₂×Q₈×C₁₉	—	4	2	no	—
48	C₂×Q₈×C₁₉	—	4	2	no	—
49	C₂×Q₈×C₁₉	—	4	1	no	—
50	C₂×Q₈×C₁₉	—	4	1	no	—
51	C₂²×C₄×C₁₉	—	2	2	no	—
52	C₂²×C₄×C₁₉	—	2	2	no	—
53	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₄⋊C₄×C₁₉	—	2	4	no	—
59	C₄⋊C₄×C₁₉	—	4	2	no	—
60	C₄⋊C₄×C₁₉	—	4	1	no	—
61	C₄⋊C₄×C₁₉	—	4	1	no	—
62	C₄⋊C₄×C₁₉	—	4	1	no	—
63	C₄⋊C₄×C₁₉	—	4	1	no	—
64	C₄²×C₁₉	—	2	2	no	—
65	C₄²×C₁₉	—	2	1	no	—
66	C₄²×C₁₉	—	2	1	no	—
67	C₄²×C₁₉	—	4	2	no	—
68	C₄²×C₁₉	SD₁₆	1	4	yes	—
69	C₄²×C₁₉	Q₁₆	1	4	yes	—
70	C₄²×C₁₉	C₂×D₄	1	2	yes	—
71	C₄²×C₁₉	#41	1	2	yes	—
72	C₂×C₁₉⋊C₈	—	2	4	no	—
73	C₂×C₁₉⋊C₈	—	4	2	no	—
74	C₂×C₁₉⋊C₈	—	4	2	no	—
75	C₂×C₁₉⋊C₈	—	4	2	no	—
76	C₂×C₁₉⋊C₈	—	4	4	no	—
77	#9697	—	19	1	no	maximal, Sylow
78	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	Q₈×C₁₉	—	8	4	no	—
80	Q₈×C₁₉	—	8	2	no	—
81	Q₈×C₁₉	—	8	2	no	—
82	C₂×C₄×C₁₉	—	2	2	no	—
83	C₂×C₄×C₁₉	—	2	2	no	—
84	C₂×C₄×C₁₉	—	2	2	no	—
85	C₂×C₄×C₁₉	—	2	2	no	—
86	C₂×C₄×C₁₉	—	2	2	no	—
87	C₂×C₄×C₁₉	—	4	2	no	—
88	C₂×C₄×C₁₉	—	4	2	no	—
89	C₂×C₄×C₁₉	—	4	4	no	—
90	C₂×C₄×C₁₉	—	4	2	no	—
91	C₂×C₄×C₁₉	—	4	2	no	—
92	C₂×C₄×C₁₉	#103	1	4	yes	—
93	C₂×C₄×C₁₉	#103	1	4	yes	—
94	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₁₉	#121	1	2	yes	—
96	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
97	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
98	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
99	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
100	C₂×C₄×C₁₉	#135	1	2	yes	—
101	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
103	C₁₉⋊C₈	—	4	4	no	—
104	C₁₉⋊C₈	—	8	2	no	—
105	#2862	D₁₉	1	1	yes	characteristic
106	#1842	—	19	1	no	—
107	C₄×C₄⋊C₈	—	19	1	no	—
108	C₂×#333	—	19	2	no	—
109	C₂×#332	—	19	2	no	—
110	C₂²×C₁₉	#390	1	1	yes	characteristic
111	C₂²×C₁₉	#413	1	2	yes	—
112	C₂²×C₁₉	C₂×#103	1	2	yes	—
113	C₂²×C₁₉	#416	1	1	yes	characteristic
114	C₂²×C₁₉	#417	1	1	yes	characteristic
115	C₄×C₁₉	—	2	4	no	—
116	C₄×C₁₉	—	2	4	no	—
117	C₄×C₁₉	—	2	2	no	—
118	C₄×C₁₉	—	8	2	no	—
119	C₄×C₁₉	—	8	2	no	—
120	C₄×C₁₉	#463	1	2	yes	—
121	C₄×C₁₉	#464	1	2	yes	—
122	C₄×C₁₉	#487	1	2	yes	—
123	C₄×C₁₉	#492	1	1	yes	characteristic
124	C₄×C₁₉	#494	1	1	yes	characteristic
125	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
126	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
127	C₂×C₄⋊C₈	—	19	2	no	—
128	C₂×C₄⋊C₈	—	19	2	no	—
129	C₂×C₄⋊C₈	—	38	2	no	—
130	C₂×C₄×C₈	—	38	1	no	—
131	#391	—	2	1	no	—
132	#391	—	2	1	no	—
133	#391	Dic₁₉	1	4	yes	—
134	#382	—	2	2	no	—
135	C₄³	C₂×D₁₉	1	1	yes	characteristic
136	#336	—	38	1	no	—
137	#333	—	38	4	no	—
138	#332	—	38	4	no	—
139	C₂×C₁₉	#1542	1	2	yes	—
140	C₂×C₁₉	#1550	1	2	yes	—
141	C₂×C₁₉	#1956	1	1	yes	characteristic
142	C₂×C₁₉	#1967	1	1	yes	characteristic
143	C₂×C₁₉	#1983	1	1	yes	characteristic
144	C₂²×Q₈	—	2	1	no	—
145	C₂²×Q₈	C₂×Dic₁₉	1	2	yes	—
146	C₂²×C₈	—	38	2	no	—
147	C₂²×C₈	—	38	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
156	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
157	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
158	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
159	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
160	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
161	C₄⋊C₈	—	19	4	no	—
162	C₄⋊C₈	—	38	2	no	—
163	C₄⋊C₈	—	38	2	no	—
164	C₄⋊C₈	—	38	2	no	—
165	C₄⋊C₈	—	76	4	no	—
166	C₄×C₈	—	76	2	no	—
167	#95	—	2	4	no	—
168	#95	—	2	4	no	—
169	C₁₉	#9697	1	1	yes	characteristic, complemented, Sylow
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	4	4	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	4	1	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂×Q₈	—	4	1	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
179	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
180	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
183	C₂×C₈	—	38	4	no	—
184	C₂×C₈	—	76	2	no	—
185	C₂×C₈	—	76	2	no	—
186	C₂×C₈	—	76	2	no	—
187	C₂×C₈	—	76	4	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	1	no	—
190	C₄⋊C₄	—	4	1	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	1	no	—
193	C₄⋊C₄	—	4	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	4	2	no	—
198	C₄²	#64527	1	4	yes	—
199	C₄²	#64528	1	4	yes	—
200	C₄²	D₄×D₁₉	1	2	yes	—
201	C₄²	#64543	1	2	yes	—
202	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
203	Q₈	—	8	4	no	—
204	Q₈	—	8	2	no	—
205	Q₈	—	8	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	#10597292	1	4	yes	—
217	C₂×C₄	#10597292	1	4	yes	—
218	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
219	C₂×C₄	#10597382	1	1	yes	characteristic
220	C₂×C₄	#10597384	1	2	yes	—
221	C₂×C₄	#10597386	1	1	yes	characteristic
222	C₂×C₄	C₂×#64527	1	2	yes	—
223	C₂×C₄	C₂×#64528	1	2	yes	—
224	C₂×C₄	#10597395	1	1	yes	characteristic
225	C₂×C₄	#10597396	1	2	yes	—
226	C₂×C₄	#10597397	1	1	yes	characteristic
227	C₈	—	76	4	no	—
228	C₈	—	152	2	no	—
229	C₂²	#11926447	1	1	yes	characteristic
230	C₂²	#11926528	1	2	yes	—
231	C₂²	C₂×#10597292	1	2	yes	—
232	C₂²	#11926540	1	1	yes	characteristic
233	C₂²	#11926545	1	1	yes	characteristic
234	C₄	—	2	4	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	4	no	—
237	C₄	—	8	2	no	—
238	C₄	—	8	2	no	—
239	C₄	#11926756	1	2	yes	—
240	C₄	#11926763	1	2	yes	—
241	C₄	#11926894	1	2	yes	—
242	C₄	#11926918	1	1	yes	characteristic
243	C₄	#11926923	1	1	yes	characteristic
244	C₂	#14562292	1	2	yes	—
245	C₂	#14562325	1	2	yes	—
246	C₂	#14564438	1	1	yes	characteristic
247	C₂	#14564485	1	1	yes	characteristic
248	C₂	#14564568	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2862	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2862	D₁₉
hypocenter	C₁₉	#9697
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

Complements

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

N	H ≅ G/N	type
169. C₁₉	#9697	semidirect

Maximal subgroups

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

subgroup	index
#14563787	2
C₄×C₄⋊(C₁₉⋊C₈)	2
C₂×#11926293	2
C₂×#11926290	2
C₁₉×#2862	2
#9697	19