Subgroups — C₁₉×#12816 #18018775

Order 4864.

446 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₉×#135	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₉×#360	C₂	1	2	yes	maximal
4	C₁₉×#1844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1386	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₉×#135	C₂²	1	1	yes	characteristic
7	C₄×#41×C₁₉	—	2	1	no	—
8	C₂×C₁₉×M₃₂	C₂²	1	2	yes	—
9	C₄×Q₁₆×C₁₉	C₄	1	2	yes	complemented
10	C₄×SD₁₆×C₁₉	—	2	2	no	—
11	C₄×D₈×C₁₉	C₄	1	2	yes	complemented
12	C₁₉×#427	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#106	—	2	1	no	—
14	C₂×C₁₉×#106	C₂²	1	2	yes	—
15	C₁₉×#416	—	2	1	no	—
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#384	—	2	1	no	—
18	C₁₉×C₄⋊C₁₆	—	2	2	no	—
19	C₁₉×#360	—	2	4	no	—
20	C₂×#41×C₁₉	—	2	1	no	—
21	C₁₉×#135	—	2	1	no	—
22	C₁₉×#135	—	2	1	no	—
23	C₁₉×#135	D₄	1	2	yes	—
24	C₁₉×#135	Q₈	1	2	yes	—
25	C₂×Q₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂×SD₁₆×C₁₉	—	2	1	no	—
27	C₂×D₈×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
30	C₄×Q₈×C₁₉	—	2	2	no	—
31	C₄×D₄×C₁₉	—	2	2	no	—
32	C₄×D₄×C₁₉	—	4	2	no	—
33	C₁₉×#117	—	2	1	no	—
34	C₁₉×#117	—	4	1	no	—
35	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₉	—	4	2	no	—
37	C₂×C₄²×C₁₉	—	4	1	no	—
38	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
39	C₁₉×M₃₂	—	2	2	no	—
40	C₁₉×M₃₂	—	2	2	no	—
41	C₂×C₁₆×C₁₉	—	2	2	no	—
42	C₁₉×#107	C₂×C₄	1	2	yes	—
43	C₁₉×#106	—	2	1	no	—
44	C₁₉×#106	—	2	2	no	—
45	C₁₉×#106	—	2	2	no	—
46	C₁₉×#106	—	4	2	no	—
47	C₁₉×#103	—	2	2	no	—
48	C₁₉×#102	—	2	2	no	—
49	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₁₉	D₄	1	2	yes	—
51	#41×C₁₉	—	2	2	no	—
52	#41×C₁₉	—	2	2	no	—
53	#41×C₁₉	—	4	1	no	—
54	#41×C₁₉	—	4	1	no	—
55	C₂×Q₈×C₁₉	—	2	1	no	—
56	C₂×D₄×C₁₉	—	2	1	no	—
57	C₂×D₄×C₁₉	—	4	1	no	—
58	C₂²×C₄×C₁₉	—	4	1	no	—
59	C₂²×C₄×C₁₉	—	4	2	no	—
60	C₂²×C₄×C₁₉	—	4	2	no	—
61	C₂²×C₄×C₁₉	D₈	1	2	yes	—
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	Q₁₆×C₁₉	—	2	1	no	—
64	Q₁₆×C₁₉	C₄⋊C₄	1	2	yes	complemented
65	SD₁₆×C₁₉	—	2	2	no	—
66	SD₁₆×C₁₉	—	2	2	no	—
67	D₈×C₁₉	—	2	1	no	—
68	D₈×C₁₉	C₄⋊C₄	1	2	yes	complemented
69	C₂×C₈×C₁₉	—	2	1	no	—
70	C₂×C₈×C₁₉	—	2	1	no	—
71	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
77	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
78	C₄⋊C₄×C₁₉	—	2	2	no	—
79	C₄⋊C₄×C₁₉	—	4	2	no	—
80	C₄⋊C₄×C₁₉	—	4	2	no	—
81	C₄⋊C₄×C₁₉	—	4	2	no	—
82	C₄⋊C₄×C₁₉	—	8	4	no	—
83	C₂²⋊C₄×C₁₉	—	4	1	no	—
84	C₂²⋊C₄×C₁₉	—	4	2	no	—
85	C₄²×C₁₉	—	4	2	no	—
86	C₄²×C₁₉	—	4	1	no	—
87	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
89	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
90	C₁₆×C₁₉	—	4	2	no	—
91	#12816	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₁₉	—	4	1	no	—
93	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
94	Q₈×C₁₉	—	2	2	no	—
95	Q₈×C₁₉	—	4	1	no	—
96	D₄×C₁₉	—	2	2	no	—
97	D₄×C₁₉	—	4	2	no	—
98	D₄×C₁₉	—	4	2	no	—
99	D₄×C₁₉	—	4	1	no	—
100	C₂×C₄×C₁₉	—	2	4	no	—
101	C₂×C₄×C₁₉	—	4	1	no	—
102	C₂×C₄×C₁₉	—	4	2	no	—
103	C₂×C₄×C₁₉	—	4	2	no	—
104	C₂×C₄×C₁₉	—	4	2	no	—
105	C₂×C₄×C₁₉	—	8	4	no	—
106	C₂×C₄×C₁₉	—	8	4	no	—
107	C₂×C₄×C₁₉	—	8	2	no	—
108	C₂×C₄×C₁₉	#102	1	2	yes	—
109	C₂×C₄×C₁₉	#102	1	2	yes	—
110	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
116	C₈×C₁₉	—	2	2	no	—
117	C₈×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₈×C₁₉	#117	1	1	yes	characteristic
119	C₈×C₁₉	#121	1	1	yes	characteristic
120	C₈×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
121	C₄×#135	C₂×C₁₉	1	1	yes	characteristic
122	#2150	C₂×C₁₉	1	1	yes	characteristic, complemented
123	C₂×#360	C₂×C₁₉	1	2	yes	—
124	#1844	C₂×C₁₉	1	1	yes	characteristic, complemented
125	#1386	C₂×C₁₉	1	2	yes	complemented
126	C₂²×C₁₉	—	2	2	no	—
127	C₂²×C₁₉	—	4	2	no	—
128	C₂²×C₁₉	—	8	2	no	—
129	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
130	C₂²×C₁₉	#459	1	1	yes	characteristic
131	C₂²×C₁₉	#474	1	1	yes	characteristic
132	C₄×C₁₉	—	2	2	no	—
133	C₄×C₁₉	—	4	2	no	—
134	C₄×C₁₉	—	8	2	no	—
135	C₄×C₁₉	—	16	2	no	—
136	C₄×C₁₉	#360	1	4	yes	complemented
137	C₄×C₁₉	#380	1	1	yes	characteristic
138	C₄×C₁₉	#417	1	1	yes	characteristic
139	C₄×C₁₉	#461	1	1	yes	characteristic
140	C₄×C₁₉	#476	1	1	yes	characteristic
141	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
142	C₄×#41	—	2	1	no	—
143	C₂×M₃₂	C₂²×C₁₉	1	2	yes	—
144	C₄×Q₁₆	C₄×C₁₉	1	2	yes	complemented
145	C₄×SD₁₆	—	2	2	no	—
146	C₄×D₈	C₄×C₁₉	1	2	yes	complemented
147	#427	C₂²×C₁₉	1	1	yes	characteristic
148	C₂×#106	—	2	1	no	—
149	C₂×#106	C₂²×C₁₉	1	2	yes	—
150	#416	—	2	1	no	—
151	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
152	#384	—	2	1	no	—
153	C₄⋊C₁₆	—	2	2	no	—
154	#360	—	2	4	no	—
155	C₂×C₁₉	—	2	1	no	—
156	C₂×C₁₉	—	2	1	no	—
157	C₂×C₁₉	—	8	2	no	—
158	C₂×C₁₉	#1802	1	1	yes	characteristic
159	C₂×C₁₉	C₂×#360	1	2	yes	—
160	C₂×#41	—	2	1	no	—
161	#135	—	2	1	no	—
162	#135	—	2	1	no	—
163	#135	D₄×C₁₉	1	2	yes	—
164	#135	Q₈×C₁₉	1	2	yes	—
165	C₂×Q₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
166	C₂×SD₁₆	—	2	1	no	—
167	C₂×D₈	C₂×C₄×C₁₉	1	1	yes	characteristic
168	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
170	C₄×Q₈	—	2	2	no	—
171	C₄×D₄	—	2	2	no	—
172	C₄×D₄	—	4	2	no	—
173	#117	—	2	1	no	—
174	#117	—	4	1	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	4	2	no	—
177	C₂×C₄²	—	4	1	no	—
178	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
179	M₃₂	—	2	2	no	—
180	M₃₂	—	2	2	no	—
181	C₂×C₁₆	—	2	2	no	—
182	#107	C₂×C₄×C₁₉	1	2	yes	—
183	#106	—	2	1	no	—
184	#106	—	2	2	no	—
185	#106	—	2	2	no	—
186	#106	—	4	2	no	—
187	#103	—	2	2	no	—
188	#102	—	2	2	no	—
189	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
190	C₄×C₈	D₄×C₁₉	1	2	yes	—
191	C₁₉	#12816	1	1	yes	characteristic, direct product factor, Sylow
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	#41	—	4	1	no	—
195	#41	—	4	1	no	—
196	C₂×Q₈	—	2	1	no	—
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	D₈×C₁₉	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
204	Q₁₆	—	2	1	no	—
205	Q₁₆	C₄⋊C₄×C₁₉	1	2	yes	complemented
206	SD₁₆	—	2	2	no	—
207	SD₁₆	—	2	2	no	—
208	D₈	—	2	1	no	—
209	D₈	C₄⋊C₄×C₁₉	1	2	yes	complemented
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	C₂²⋊C₄×C₁₉	1	2	yes	—
213	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
214	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
215	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
216	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
217	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
218	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	8	4	no	—
224	C₂²⋊C₄	—	4	1	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₄²	—	4	1	no	—
227	C₄²	—	4	2	no	—
228	C₄²	C₂²⋊C₄×C₁₉	1	2	yes	—
229	C₄²	SD₁₆×C₁₉	1	1	yes	characteristic
230	C₄²	SD₁₆×C₁₉	1	1	yes	characteristic
231	C₁₆	—	4	2	no	—
232	C₂³	—	4	1	no	—
233	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
234	Q₈	—	2	2	no	—
235	Q₈	—	4	1	no	—
236	D₄	—	2	2	no	—
237	D₄	—	4	2	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	1	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	—	8	4	no	—
247	C₂×C₄	—	8	2	no	—
248	C₂×C₄	C₁₉×#102	1	2	yes	—
249	C₂×C₄	C₁₉×#102	1	2	yes	—
250	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
251	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
252	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
253	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
254	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
255	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
258	C₈	C₁₉×#117	1	1	yes	characteristic
259	C₈	C₁₉×#121	1	1	yes	characteristic
260	C₈	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
261	C₂²	—	2	2	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	8	2	no	—
264	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
265	C₂²	C₁₉×#459	1	1	yes	characteristic
266	C₂²	C₁₉×#474	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	8	2	no	—
270	C₄	—	16	2	no	—
271	C₄	C₁₉×#360	1	4	yes	complemented
272	C₄	C₁₉×#380	1	1	yes	characteristic
273	C₄	C₁₉×#417	1	1	yes	characteristic
274	C₄	C₁₉×#461	1	1	yes	characteristic
275	C₄	C₁₉×#476	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	8	2	no	—
279	C₂	C₁₉×#1802	1	1	yes	characteristic
280	C₂	C₂×C₁₉×#360	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂×C₁₉×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2150	C₂	semidirect
4. C₁₉×#1844	C₂	semidirect
5. C₁₉×#1386	C₂	semidirect
9. C₄×Q₁₆×C₁₉	C₄	semidirect
11. C₄×D₈×C₁₉	C₄	semidirect
64. Q₁₆×C₁₉	C₄⋊C₄	semidirect
68. D₈×C₁₉	C₄⋊C₄	semidirect
91. #12816	C₁₉	direct (N × H)
122. #2150	C₂×C₁₉	semidirect
124. #1844	C₂×C₁₉	semidirect
125. #1386	C₂×C₁₉	semidirect
136. C₄×C₁₉	#360	semidirect
144. C₄×Q₁₆	C₄×C₁₉	semidirect
146. C₄×D₈	C₄×C₁₉	semidirect
191. C₁₉	#12816	direct (N × H)
205. Q₁₆	C₄⋊C₄×C₁₉	semidirect
209. D₈	C₄⋊C₄×C₁₉	semidirect
271. C₄	C₁₉×#360	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₁₉×#135	2
C₁₉×#2150	2
C₂×C₁₉×#360	2
C₁₉×#1844	2
C₁₉×#1386	2
#12816	19

Subgroups — C19×#12816 #18018775

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12816 #18018775