Subgroups — #18071996

Order 4864.

718 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926596	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₁₉	C₂	1	1	yes	maximal, characteristic
3	#14561470	C₂	1	1	yes	maximal, characteristic, complemented
4	#14561469	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1277	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	#11926596	—	2	4	no	—
9	#11926596	C₄	1	8	yes	—
10	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
11	C₂×(C₂×D₁₉)⋊C₈	—	2	2	no	—
12	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₁₉	—	2	2	no	—
14	C₂×C₈×Dic₁₉	C₄	1	2	yes	—
15	C₄×C₈×D₁₉	—	2	4	no	—
16	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×D₁₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×D₁₉	Q₈	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	—	2	2	no	—
22	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
24	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₁₉	D₄	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	2	yes	—
29	C₂×C₈×D₁₉	—	2	4	no	—
30	C₂×C₈×D₁₉	—	2	4	no	—
31	C₂×C₈×D₁₉	C₈	1	8	yes	complemented
32	C₄²×D₁₉	C₂×C₄	1	4	yes	—
33	C₄²×D₁₉	D₄	1	4	yes	—
34	C₁₉×C₄⋊C₈	—	2	2	no	—
35	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
36	C₄×C₈×C₁₉	—	2	2	no	—
37	(C₂×D₁₉)⋊C₈	—	2	8	no	—
38	C₈×Dic₁₉	—	2	8	no	—
39	C₈×Dic₁₉	—	2	4	no	—
40	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
41	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
42	C₄×C₁₉⋊C₈	—	2	2	no	—
43	C₂³×D₁₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
47	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₉	—	2	1	no	—
51	C₂×C₄×D₁₉	—	2	2	no	—
52	C₂×C₄×D₁₉	—	2	1	no	—
53	C₂×C₄×D₁₉	—	2	4	no	—
54	C₂×C₄×D₁₉	C₄²	1	2	yes	—
55	C₂×C₄×D₁₉	C₂²⋊C₄	1	4	yes	—
56	C₂×C₄×D₁₉	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×D₁₉	C₄⋊C₄	1	4	yes	—
58	C₂×C₄×D₁₉	C₂×C₈	1	4	yes	—
59	C₂×C₄×D₁₉	M₁₆	1	4	yes	—
60	C₂×C₈×C₁₉	—	2	2	no	—
61	C₂×C₈×C₁₉	—	2	2	no	—
62	C₂×C₈×C₁₉	—	2	4	no	—
63	C₂×C₈×C₁₉	—	4	2	no	—
64	C₂×C₈×C₁₉	—	4	2	no	—
65	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
66	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
67	C₄²×C₁₉	C₂×D₄	1	2	yes	—
68	C₄×Dic₁₉	C₄²	1	2	yes	—
69	C₄×Dic₁₉	C₂²⋊C₄	1	4	yes	—
70	C₄×Dic₁₉	C₂²⋊C₄	1	2	yes	—
71	C₄×Dic₁₉	C₄⋊C₄	1	4	yes	—
72	C₂×C₁₉⋊C₈	—	2	2	no	—
73	C₂×C₁₉⋊C₈	—	2	2	no	—
74	C₂×C₁₉⋊C₈	—	2	4	no	—
75	C₂×C₁₉⋊C₈	—	4	2	no	—
76	C₂×C₁₉⋊C₈	—	4	2	no	—
77	C₂×C₁₉⋊C₈	C₂×C₈	1	4	yes	—
78	C₈×D₁₉	—	2	16	no	—
79	#7872	—	19	1	no	maximal, Sylow
80	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×D₁₉	—	2	1	no	—
82	C₂²×D₁₉	—	2	1	no	—
83	C₂²×D₁₉	—	2	2	no	—
84	C₂²×D₁₉	#95	1	2	yes	—
85	C₂²×D₁₉	C₄⋊C₈	1	4	yes	—
86	C₂×C₄×C₁₉	—	2	2	no	—
87	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
90	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
92	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
94	C₂×C₄×C₁₉	#135	1	2	yes	—
95	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
96	C₂×Dic₁₉	—	2	1	no	—
97	C₂×Dic₁₉	—	2	1	no	—
98	C₂×Dic₁₉	#95	1	2	yes	—
99	C₂×Dic₁₉	#95	1	2	yes	—
100	C₂×Dic₁₉	#95	1	2	yes	—
101	C₂×Dic₁₉	C₄⋊C₈	1	4	yes	—
102	C₂×Dic₁₉	#108	1	4	yes	—
103	C₄×D₁₉	—	2	4	no	—
104	C₄×D₁₉	—	2	4	no	—
105	C₄×D₁₉	—	2	8	no	—
106	C₄×D₁₉	C₄×C₈	1	4	yes	—
107	C₄×D₁₉	#97	1	4	yes	—
108	C₄×D₁₉	C₂²⋊C₈	1	8	yes	—
109	C₈×C₁₉	—	2	4	no	—
110	C₈×C₁₉	—	4	4	no	—
111	C₁₉⋊C₈	—	2	4	no	—
112	C₁₉⋊C₈	—	4	4	no	—
113	C₂×#431	—	19	2	no	—
114	C₂²×C₄×C₈	—	19	1	no	—
115	#1277	—	19	2	no	—
116	#1277	—	19	1	no	—
117	#1277	D₁₉	1	1	yes	characteristic, complemented
118	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
119	C₂²×C₁₉	C₂×C₄⋊C₈	1	2	yes	—
120	C₂²×C₁₉	#427	1	1	yes	characteristic
121	C₂²×C₁₉	#427	1	1	yes	characteristic
122	C₂²×C₁₉	C₂×#108	1	2	yes	—
123	C₂×D₁₉	—	2	4	no	—
124	C₂×D₁₉	—	2	4	no	—
125	C₂×D₁₉	—	2	2	no	—
126	C₂×D₁₉	—	2	2	no	—
127	C₂×D₁₉	#336	1	4	yes	—
128	C₄×C₁₉	—	2	2	no	—
129	C₄×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
130	C₄×C₁₉	C₂×#97	1	1	yes	characteristic
131	C₄×C₁₉	C₂×C₂²⋊C₈	1	2	yes	—
132	C₄×C₁₉	#416	1	2	yes	—
133	C₄×C₁₉	#416	1	2	yes	—
134	Dic₁₉	—	2	2	no	—
135	Dic₁₉	#336	1	4	yes	—
136	Dic₁₉	#341	1	4	yes	—
137	C₂³×C₈	—	19	2	no	—
138	C₂²×C₄²	—	19	1	no	—
139	#431	—	19	8	no	—
140	#431	—	38	4	no	—
141	C₂×C₄⋊C₈	—	19	2	no	—
142	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
143	C₂×C₂²⋊C₈	—	38	2	no	—
144	C₂×C₄×C₈	—	19	2	no	—
145	C₂×C₄×C₈	—	19	1	no	—
146	C₂×C₄×C₈	—	38	4	no	—
147	C₂×C₄×C₈	—	38	2	no	—
148	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
149	C₂×C₁₉	#1564	1	2	yes	—
150	C₂×C₁₉	#1564	1	2	yes	—
151	C₂×C₁₉	C₂×#336	1	1	yes	characteristic
152	C₂×C₁₉	#1735	1	1	yes	characteristic
153	C₂×C₁₉	C₂×#341	1	1	yes	characteristic
154	D₁₉	—	2	4	no	—
155	C₂³×C₄	—	19	1	no	—
156	C₂³×C₄	—	19	1	no	—
157	C₂³×C₄	—	19	1	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	19	2	no	—
160	C₂²×C₈	—	19	8	no	—
161	C₂²×C₈	—	38	4	no	—
162	C₂²×C₈	—	38	4	no	—
163	C₂²×C₈	—	38	2	no	—
164	C₂²×C₈	C₄×D₁₉	1	2	yes	—
165	C₂×C₄²	—	19	1	no	—
166	C₂×C₄²	—	19	1	no	—
167	C₂×C₄²	—	19	1	no	—
168	C₂×C₄²	—	19	4	no	—
169	C₂×C₄²	—	19	4	no	—
170	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	19	4	no	—
173	C₄⋊C₈	—	38	2	no	—
174	C₄⋊C₈	C₄×D₁₉	1	4	yes	—
175	C₂²⋊C₈	—	38	8	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	38	4	no	—
178	C₄×C₈	—	38	8	no	—
179	C₄×C₈	—	38	2	no	—
180	C₁₉	#7872	1	1	yes	characteristic, complemented, Sylow
181	C₂⁴	—	19	1	no	—
182	C₂²×C₄	—	19	1	no	—
183	C₂²×C₄	—	19	1	no	—
184	C₂²×C₄	—	19	2	no	—
185	C₂²×C₄	—	19	2	no	—
186	C₂²×C₄	—	19	4	no	—
187	C₂²×C₄	—	19	4	no	—
188	C₂²×C₄	—	19	1	no	—
189	C₂²×C₄	—	19	4	no	—
190	C₂²×C₄	—	19	4	no	—
191	C₂²×C₄	—	38	4	no	—
192	C₂²×C₄	—	38	1	no	—
193	C₂²×C₄	—	38	2	no	—
194	C₂²×C₄	—	38	1	no	—
195	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
196	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
197	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	19	4	no	—
204	C₂×C₈	—	38	16	no	—
205	C₂×C₈	—	38	2	no	—
206	C₂×C₈	—	38	2	no	—
207	C₂×C₈	—	38	4	no	—
208	C₂×C₈	—	76	2	no	—
209	C₂×C₈	—	76	2	no	—
210	C₂×C₈	C₈×D₁₉	1	4	yes	—
211	C₄²	—	19	2	no	—
212	C₄²	—	19	4	no	—
213	C₄²	—	19	2	no	—
214	C₄²	—	19	4	no	—
215	C₄²	C₂×C₄×D₁₉	1	2	yes	—
216	C₄²	D₄×D₁₉	1	2	yes	—
217	C₂³	—	19	4	no	—
218	C₂³	—	19	2	no	—
219	C₂³	—	38	1	no	—
220	C₂³	—	38	1	no	—
221	C₂³	—	38	2	no	—
222	C₂³	C₄⋊C₄×D₁₉	1	1	yes	characteristic
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	19	4	no	—
225	C₂×C₄	—	19	4	no	—
226	C₂×C₄	—	19	2	no	—
227	C₂×C₄	—	19	2	no	—
228	C₂×C₄	—	19	4	no	—
229	C₂×C₄	—	19	4	no	—
230	C₂×C₄	—	19	8	no	—
231	C₂×C₄	—	19	2	no	—
232	C₂×C₄	—	38	8	no	—
233	C₂×C₄	—	38	1	no	—
234	C₂×C₄	—	38	1	no	—
235	C₂×C₄	—	38	4	no	—
236	C₂×C₄	—	38	4	no	—
237	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
238	C₂×C₄	C₂²⋊C₄×D₁₉	1	2	yes	—
239	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
240	C₂×C₄	C₄⋊C₄×D₁₉	1	2	yes	—
241	C₂×C₄	C₂×C₈×D₁₉	1	2	yes	—
242	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
243	C₂×C₄	#10597358	1	1	yes	characteristic
244	C₂×C₄	#10597362	1	2	yes	—
245	C₂×C₄	#10597365	1	1	yes	characteristic
246	C₈	—	2	4	no	—
247	C₈	—	4	4	no	—
248	C₈	—	38	4	no	—
249	C₈	—	76	4	no	—
250	C₂²	—	19	4	no	—
251	C₂²	—	38	2	no	—
252	C₂²	—	38	2	no	—
253	C₂²	—	38	4	no	—
254	C₂²	—	38	4	no	—
255	C₂²	#95×D₁₉	1	1	yes	characteristic
256	C₂²	C₄⋊C₈×D₁₉	1	2	yes	—
257	C₂²	#11926579	1	1	yes	characteristic
258	C₂²	#11926581	1	1	yes	characteristic
259	C₂²	#108×D₁₉	1	2	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	19	4	no	—
262	C₄	—	19	4	no	—
263	C₄	—	38	2	no	—
264	C₄	C₄×C₈×D₁₉	1	1	yes	characteristic
265	C₄	#97×D₁₉	1	1	yes	characteristic
266	C₄	C₂²⋊C₈×D₁₉	1	2	yes	—
267	C₄	#11926537	1	2	yes	—
268	C₄	#11926539	1	2	yes	—
269	C₂	—	38	4	no	—
270	C₂	#14562380	1	2	yes	—
271	C₂	#14562382	1	2	yes	—
272	C₂	D₁₉×#336	1	1	yes	characteristic
273	C₂	#14563350	1	1	yes	characteristic
274	C₂	D₁₉×#341	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₄×C₁₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1277	C₂
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂³×D₁₉	C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1277	D₁₉
hypocenter	C₁₉	#7872
LCS	G ▷ 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
3. #14561470	C₂	semidirect
4. #14561469	C₂	semidirect
5. C₁₉×#1277	C₂	semidirect
31. C₂×C₈×D₁₉	C₈	semidirect
117. #1277	D₁₉	semidirect
180. C₁₉	#7872	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926596	2
C₂×C₄×C₈×D₁₉	2
#14561470	2
#14561469	2
C₁₉×#1277	2
#7872	19