Subgroups — #18076198

Order 4864.

422 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563786	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₁₉	C₂	1	1	yes	maximal, characteristic
3	#14561470	C₂	1	1	yes	maximal, characteristic
4	#14561469	C₂	1	2	yes	maximal
5	C₁₉×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×Dic₁₉⋊C₈	—	2	2	no	—
8	C₂×Dic₁₉⋊C₈	—	2	2	no	—
9	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₁₉	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₉	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₉	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₉	C₄	1	1	yes	characteristic
14	C₄²×Dic₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₉	—	2	2	no	—
18	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
21	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
25	C₂×C₄×Dic₁₉	Q₈	1	1	yes	characteristic
26	C₂×C₄×Dic₁₉	Q₈	1	1	yes	characteristic
27	C₂²×C₁₉⋊C₈	—	2	2	no	—
28	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₁₉×C₄⋊C₈	—	2	2	no	—
31	C₁₉×C₄⋊C₈	—	2	2	no	—
32	C₄×C₈×C₁₉	—	2	2	no	—
33	Dic₁₉⋊C₈	—	4	4	no	—
34	Dic₁₉⋊C₈	—	4	4	no	—
35	C₈×Dic₁₉	—	2	2	no	—
36	C₈×Dic₁₉	—	2	2	no	—
37	C₈×Dic₁₉	—	2	2	no	—
38	C₈×Dic₁₉	C₈	1	4	yes	complemented
39	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
40	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
41	C₄×C₁₉⋊C₈	—	2	2	no	—
42	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
45	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₉	—	2	1	no	—
50	C₂×C₈×C₁₉	—	2	1	no	—
51	C₂×C₈×C₁₉	—	2	1	no	—
52	C₂×C₈×C₁₉	—	2	1	no	—
53	C₂×C₈×C₁₉	—	4	2	no	—
54	C₂×C₈×C₁₉	—	4	2	no	—
55	C₂×C₈×C₁₉	—	4	2	no	—
56	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
58	C₄²×C₁₉	#41	1	2	yes	—
59	C₄²×C₁₉	#41	1	2	yes	—
60	C₄×Dic₁₉	—	2	1	no	—
61	C₄×Dic₁₉	—	2	1	no	—
62	C₄×Dic₁₉	—	2	2	no	—
63	C₄×Dic₁₉	—	2	2	no	—
64	C₄×Dic₁₉	C₂×C₈	1	2	yes	—
65	C₄×Dic₁₉	M₁₆	1	2	yes	—
66	C₄×Dic₁₉	D₈	1	2	yes	—
67	C₄×Dic₁₉	SD₁₆	1	4	yes	—
68	C₄×Dic₁₉	Q₁₆	1	2	yes	—
69	C₂×C₁₉⋊C₈	—	2	1	no	—
70	C₂×C₁₉⋊C₈	—	2	1	no	—
71	C₂×C₁₉⋊C₈	—	2	1	no	—
72	C₂×C₁₉⋊C₈	—	2	1	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	C₂×C₈	1	2	yes	—
77	C₂×C₁₉⋊C₈	M₁₆	1	2	yes	—
78	#8784	—	19	1	no	maximal, Sylow
79	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₉	—	2	1	no	—
81	C₂×C₄×C₁₉	—	2	1	no	—
82	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
83	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
84	C₂×C₄×C₁₉	#124	1	2	yes	—
85	C₂×C₄×C₁₉	#125	1	2	yes	—
86	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
92	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
93	C₂×Dic₁₉	—	2	1	no	—
94	C₂×Dic₁₉	—	2	1	no	—
95	C₂×Dic₁₉	—	2	1	no	—
96	C₂×Dic₁₉	—	2	1	no	—
97	C₂×Dic₁₉	C₄⋊C₈	1	2	yes	—
98	C₂×Dic₁₉	C₄⋊C₈	1	2	yes	—
99	C₂×Dic₁₉	#106	1	2	yes	—
100	C₂×Dic₁₉	#106	1	2	yes	—
101	C₂×Dic₁₉	#107	1	2	yes	—
102	C₂×Dic₁₉	#107	1	2	yes	—
103	C₂×Dic₁₉	#108	1	2	yes	—
104	C₂×Dic₁₉	#108	1	2	yes	—
105	C₈×C₁₉	—	2	2	no	—
106	C₈×C₁₉	—	2	2	no	—
107	C₈×C₁₉	—	8	2	no	—
108	C₁₉⋊C₈	—	2	2	no	—
109	C₁₉⋊C₈	—	2	2	no	—
110	C₁₉⋊C₈	—	8	2	no	—
111	#1842	—	19	2	no	—
112	C₄²×C₈	—	19	1	no	—
113	#1277	—	19	2	no	—
114	#1277	—	19	1	no	—
115	#1277	D₁₉	1	1	yes	characteristic
116	C₂²×C₁₉	#382	1	1	yes	characteristic
117	C₂²×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₂²×C₁₉	#423	1	1	yes	characteristic
119	C₂²×C₁₉	C₂×#106	1	1	yes	characteristic
120	C₂²×C₁₉	C₂×#107	1	1	yes	characteristic
121	C₂²×C₁₉	C₂×#108	1	1	yes	characteristic
122	C₂²×C₁₉	C₂×#108	1	1	yes	characteristic
123	C₄×C₁₉	—	2	2	no	—
124	C₄×C₁₉	#431	1	1	yes	characteristic
125	C₄×C₁₉	#431	1	1	yes	characteristic
126	C₄×C₁₉	#432	1	1	yes	characteristic
127	C₄×C₁₉	#432	1	1	yes	characteristic
128	C₄×C₁₉	#486	1	2	yes	—
129	C₄×C₁₉	#487	1	2	yes	—
130	Dic₁₉	—	2	2	no	—
131	Dic₁₉	—	2	2	no	—
132	Dic₁₉	#334	1	4	yes	—
133	Dic₁₉	#335	1	4	yes	—
134	C₂×C₄⋊C₈	—	19	2	no	—
135	C₂×C₄⋊C₈	—	38	2	no	—
136	C₂×C₄⋊C₈	—	38	2	no	—
137	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
138	C₂×C₄×C₈	—	19	1	no	—
139	C₂×C₄×C₈	—	19	1	no	—
140	C₂×C₄×C₈	—	19	1	no	—
141	C₂×C₄×C₈	—	19	1	no	—
142	C₂×C₄×C₈	—	19	1	no	—
143	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
144	C₄³	—	19	1	no	—
145	C₂×C₁₉	C₂×#334	1	1	yes	characteristic
146	C₂×C₁₉	C₂×#335	1	1	yes	characteristic
147	C₂×C₁₉	#1566	1	1	yes	characteristic
148	C₂×C₁₉	#1567	1	1	yes	characteristic
149	C₂×C₁₉	#1831	1	1	yes	characteristic
150	C₂×C₁₉	#1834	1	1	yes	characteristic
151	C₂×C₁₉	#1837	1	1	yes	characteristic
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	19	1	no	—
154	C₂²×C₈	—	19	1	no	—
155	C₂²×C₈	—	38	2	no	—
156	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
157	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
158	C₂×C₄²	—	19	1	no	—
159	C₂×C₄²	—	19	1	no	—
160	C₂×C₄²	—	19	1	no	—
161	C₂×C₄²	—	19	1	no	—
162	C₂×C₄²	—	19	1	no	—
163	C₂×C₄²	—	19	1	no	—
164	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	38	2	no	—
168	C₄⋊C₈	—	38	2	no	—
169	C₄⋊C₈	—	76	4	no	—
170	C₄⋊C₈	—	76	4	no	—
171	C₄×C₈	—	2	2	no	—
172	C₄×C₈	—	19	4	no	—
173	C₄×C₈	—	38	2	no	—
174	C₄×C₈	—	38	2	no	—
175	C₄×C₈	—	38	2	no	—
176	C₄×C₈	—	38	2	no	—
177	C₁₉	#8784	1	1	yes	characteristic, complemented, Sylow
178	C₂²×C₄	—	19	1	no	—
179	C₂²×C₄	—	19	1	no	—
180	C₂²×C₄	—	19	1	no	—
181	C₂²×C₄	—	19	1	no	—
182	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
183	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
184	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	19	2	no	—
193	C₂×C₈	—	19	2	no	—
194	C₂×C₈	—	38	1	no	—
195	C₂×C₈	—	38	1	no	—
196	C₂×C₈	—	38	1	no	—
197	C₂×C₈	—	38	1	no	—
198	C₂×C₈	—	76	2	no	—
199	C₂×C₈	—	76	2	no	—
200	C₂×C₈	—	76	2	no	—
201	C₂×C₈	C₈×D₁₉	1	2	yes	—
202	C₂×C₈	#64515	1	2	yes	—
203	C₄²	—	19	2	no	—
204	C₄²	—	19	2	no	—
205	C₄²	—	19	2	no	—
206	C₄²	—	19	4	no	—
207	C₄²	—	19	2	no	—
208	C₄²	—	38	2	no	—
209	C₄²	—	38	2	no	—
210	C₄²	—	38	1	no	—
211	C₄²	—	38	1	no	—
212	C₄²	#64541	1	2	yes	—
213	C₄²	#64543	1	2	yes	—
214	C₂³	C₄⋊C₄×D₁₉	1	1	yes	characteristic
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	19	2	no	—
218	C₂×C₄	—	19	2	no	—
219	C₂×C₄	—	19	2	no	—
220	C₂×C₄	—	19	2	no	—
221	C₂×C₄	—	19	2	no	—
222	C₂×C₄	—	19	2	no	—
223	C₂×C₄	—	19	2	no	—
224	C₂×C₄	—	19	2	no	—
225	C₂×C₄	—	38	1	no	—
226	C₂×C₄	—	38	1	no	—
227	C₂×C₄	—	38	1	no	—
228	C₂×C₄	—	38	1	no	—
229	C₂×C₄	#10597318	1	1	yes	characteristic
230	C₂×C₄	#10597323	1	1	yes	characteristic
231	C₂×C₄	#10597331	1	2	yes	—
232	C₂×C₄	#10597335	1	2	yes	—
233	C₂×C₄	C₂×C₈×D₁₉	1	1	yes	characteristic
234	C₂×C₄	C₂×#64515	1	1	yes	characteristic
235	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
236	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
237	C₂×C₄	D₈×D₁₉	1	1	yes	characteristic
238	C₂×C₄	SD₁₆×D₁₉	1	2	yes	—
239	C₂×C₄	Q₁₆×D₁₉	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	—	8	2	no	—
243	C₈	—	38	2	no	—
244	C₈	—	38	2	no	—
245	C₈	—	152	2	no	—
246	C₂²	#11926408	1	1	yes	characteristic
247	C₂²	C₄⋊C₈×D₁₉	1	1	yes	characteristic
248	C₂²	#11926566	1	1	yes	characteristic
249	C₂²	#106×D₁₉	1	1	yes	characteristic
250	C₂²	#107×D₁₉	1	1	yes	characteristic
251	C₂²	#108×D₁₉	1	1	yes	characteristic
252	C₂²	#108×D₁₉	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	19	4	no	—
255	C₄	—	19	4	no	—
256	C₄	—	38	2	no	—
257	C₄	—	38	2	no	—
258	C₄	#11926594	1	1	yes	characteristic
259	C₄	#11926595	1	1	yes	characteristic
260	C₄	#11926598	1	1	yes	characteristic
261	C₄	#11926599	1	1	yes	characteristic
262	C₄	#11926889	1	2	yes	—
263	C₄	#11926893	1	2	yes	—
264	C₂	D₁₉×#334	1	1	yes	characteristic
265	C₂	D₁₉×#335	1	1	yes	characteristic
266	C₂	#14562389	1	1	yes	characteristic
267	C₂	#14562392	1	1	yes	characteristic
268	C₂	#14563750	1	1	yes	characteristic
269	C₂	#14563761	1	1	yes	characteristic
270	C₂	#14563771	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1277	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1277	D₁₉
hypocenter	C₁₉	#8784
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
38. C₈×Dic₁₉	C₈	semidirect
177. C₁₉	#8784	semidirect

Maximal subgroups

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

subgroup	index
#14563786	2
C₄×C₈×Dic₁₉	2
#14561470	2
#14561469	2
C₁₉×#1277	2
#8784	19