Subgroups — C₁₉×#12812 #18018771

Order 4864.

446 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#358	C₂	1	2	yes	maximal
4	C₁₉×#1845	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1380	C₂	1	2	yes	maximal
6	C₂²×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₉	—	2	1	no	—
8	C₂²×C₁₆×C₁₉	C₂²	1	2	yes	—
9	C₄×Q₁₆×C₁₉	—	2	2	no	—
10	C₄×Q₁₆×C₁₉	C₄	1	4	yes	complemented
11	C₂×C₁₉×#107	—	2	1	no	—
12	C₂×C₁₉×#107	C₂²	1	2	yes	—
13	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×#103	—	2	1	no	—
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#384	—	2	1	no	—
17	C₁₉×C₄⋊C₁₆	—	2	2	no	—
18	C₁₉×#358	—	2	4	no	—
19	C₂²×Q₈×C₁₉	—	2	1	no	—
20	C₂×Q₁₆×C₁₉	—	2	1	no	—
21	C₂×Q₁₆×C₁₉	—	2	1	no	—
22	C₂×Q₁₆×C₁₉	—	2	1	no	—
23	C₂×Q₁₆×C₁₉	C₂×C₄	1	2	yes	—
24	C₂×Q₁₆×C₁₉	D₄	1	2	yes	—
25	C₂×Q₁₆×C₁₉	Q₈	1	2	yes	—
26	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
28	C₄×Q₈×C₁₉	—	2	4	no	—
29	C₄×Q₈×C₁₉	—	4	2	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₉	—	4	1	no	—
33	C₂×C₄⋊C₄×C₁₉	—	4	1	no	—
34	C₂×C₄⋊C₄×C₁₉	—	4	1	no	—
35	C₂×C₄²×C₁₉	—	4	1	no	—
36	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₁₆×C₁₉	—	2	2	no	—
38	C₂×C₁₆×C₁₉	—	2	2	no	—
39	C₂×C₁₆×C₁₉	—	2	2	no	—
40	C₁₉×#107	—	2	1	no	—
41	C₁₉×#107	—	2	2	no	—
42	C₁₉×#107	—	2	2	no	—
43	C₁₉×#107	—	4	2	no	—
44	C₁₉×#107	C₂×C₄	1	2	yes	—
45	C₁₉×#103	—	2	4	no	—
46	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₁₉	D₄	1	2	yes	—
48	C₂×Q₈×C₁₉	—	2	2	no	—
49	C₂×Q₈×C₁₉	—	2	2	no	—
50	C₂×Q₈×C₁₉	—	2	2	no	—
51	C₂×Q₈×C₁₉	—	4	1	no	—
52	C₂×Q₈×C₁₉	—	4	1	no	—
53	C₂×Q₈×C₁₉	—	4	1	no	—
54	C₂²×C₄×C₁₉	—	4	1	no	—
55	C₂²×C₄×C₁₉	—	4	1	no	—
56	C₂²×C₄×C₁₉	—	4	2	no	—
57	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
60	Q₁₆×C₁₉	—	2	2	no	—
61	Q₁₆×C₁₉	—	2	2	no	—
62	Q₁₆×C₁₉	—	2	2	no	—
63	Q₁₆×C₁₉	C₄⋊C₄	1	4	yes	complemented
64	C₂×C₈×C₁₉	—	2	1	no	—
65	C₂×C₈×C₁₉	—	2	1	no	—
66	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
72	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
73	C₄⋊C₄×C₁₉	—	2	2	no	—
74	C₄⋊C₄×C₁₉	—	4	4	no	—
75	C₄⋊C₄×C₁₉	—	4	1	no	—
76	C₄⋊C₄×C₁₉	—	4	2	no	—
77	C₄⋊C₄×C₁₉	—	4	2	no	—
78	C₄⋊C₄×C₁₉	—	8	2	no	—
79	C₄⋊C₄×C₁₉	—	8	2	no	—
80	C₄²×C₁₉	—	4	4	no	—
81	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₁₉	SD₁₆	1	2	yes	—
83	C₁₆×C₁₉	—	4	2	no	—
84	#12812	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
86	Q₈×C₁₉	—	2	4	no	—
87	Q₈×C₁₉	—	4	2	no	—
88	Q₈×C₁₉	—	4	2	no	—
89	Q₈×C₁₉	—	4	2	no	—
90	C₂×C₄×C₁₉	—	2	1	no	—
91	C₂×C₄×C₁₉	—	2	1	no	—
92	C₂×C₄×C₁₉	—	4	2	no	—
93	C₂×C₄×C₁₉	—	4	2	no	—
94	C₂×C₄×C₁₉	—	4	2	no	—
95	C₂×C₄×C₁₉	—	4	2	no	—
96	C₂×C₄×C₁₉	—	8	1	no	—
97	C₂×C₄×C₁₉	—	8	1	no	—
98	C₂×C₄×C₁₉	—	8	2	no	—
99	C₂×C₄×C₁₉	—	8	2	no	—
100	C₂×C₄×C₁₉	—	8	2	no	—
101	C₂×C₄×C₁₉	#102	1	2	yes	—
102	C₂×C₄×C₁₉	#102	1	2	yes	—
103	C₂×C₄×C₁₉	SD₃₂	1	2	yes	—
104	C₂×C₄×C₁₉	Q₃₂	1	2	yes	—
105	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
111	C₈×C₁₉	—	2	2	no	—
112	C₈×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₈×C₁₉	#117	1	1	yes	characteristic
114	C₈×C₁₉	#121	1	1	yes	characteristic
115	C₈×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×Q₁₆	C₂×C₁₉	1	1	yes	characteristic
117	C₂×C₄⋊C₁₆	C₂×C₁₉	1	1	yes	characteristic
118	C₂×#358	C₂×C₁₉	1	2	yes	—
119	#1845	C₂×C₁₉	1	1	yes	characteristic
120	#1380	C₂×C₁₉	1	2	yes	—
121	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
122	C₂²×C₁₉	#459	1	1	yes	characteristic
123	C₂²×C₁₉	#474	1	1	yes	characteristic
124	C₂²×C₁₉	C₂×SD₃₂	1	1	yes	characteristic
125	C₂²×C₁₉	C₂×Q₃₂	1	1	yes	characteristic
126	C₂²×C₁₉	#509	1	1	yes	characteristic
127	C₂²×C₁₉	#510	1	1	yes	characteristic
128	C₄×C₁₉	—	2	2	no	—
129	C₄×C₁₉	—	4	4	no	—
130	C₄×C₁₉	—	8	2	no	—
131	C₄×C₁₉	—	16	2	no	—
132	C₄×C₁₉	#358	1	4	yes	complemented
133	C₄×C₁₉	#380	1	1	yes	characteristic
134	C₄×C₁₉	#417	1	1	yes	characteristic
135	C₄×C₁₉	#461	1	1	yes	characteristic
136	C₄×C₁₉	#476	1	1	yes	characteristic
137	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
138	C₂×C₄×Q₈	—	2	1	no	—
139	C₂²×C₁₆	C₂²×C₁₉	1	2	yes	—
140	C₄×Q₁₆	—	2	2	no	—
141	C₄×Q₁₆	C₄×C₁₉	1	4	yes	complemented
142	C₂×#107	—	2	1	no	—
143	C₂×#107	C₂²×C₁₉	1	2	yes	—
144	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
145	C₂×#103	—	2	1	no	—
146	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
147	#384	—	2	1	no	—
148	C₄⋊C₁₆	—	2	2	no	—
149	#358	—	2	4	no	—
150	C₂×C₁₉	#1802	1	1	yes	characteristic
151	C₂×C₁₉	C₂×#358	1	1	yes	characteristic
152	C₂×C₁₉	#2139	1	1	yes	characteristic
153	C₂×C₁₉	#2207	1	1	yes	characteristic
154	C₂×C₁₉	#2209	1	1	yes	characteristic
155	C₂×C₁₉	#2225	1	1	yes	characteristic
156	C₂×C₁₉	#2227	1	1	yes	characteristic
157	C₂²×Q₈	—	2	1	no	—
158	C₂×Q₁₆	—	2	1	no	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂×Q₁₆	—	2	1	no	—
161	C₂×Q₁₆	C₂×C₄×C₁₉	1	2	yes	—
162	C₂×Q₁₆	D₄×C₁₉	1	2	yes	—
163	C₂×Q₁₆	Q₈×C₁₉	1	2	yes	—
164	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
166	C₄×Q₈	—	2	4	no	—
167	C₄×Q₈	—	4	2	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	4	1	no	—
171	C₂×C₄⋊C₄	—	4	1	no	—
172	C₂×C₄⋊C₄	—	4	1	no	—
173	C₂×C₄²	—	4	1	no	—
174	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	2	2	no	—
177	C₂×C₁₆	—	2	2	no	—
178	#107	—	2	1	no	—
179	#107	—	2	2	no	—
180	#107	—	2	2	no	—
181	#107	—	4	2	no	—
182	#107	C₂×C₄×C₁₉	1	2	yes	—
183	#103	—	2	4	no	—
184	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
185	C₄×C₈	D₄×C₁₉	1	2	yes	—
186	C₁₉	#12812	1	1	yes	characteristic, direct product factor, Sylow
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	4	1	no	—
191	C₂×Q₈	—	4	1	no	—
192	C₂×Q₈	—	4	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
197	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
199	Q₁₆	—	2	2	no	—
200	Q₁₆	—	2	2	no	—
201	Q₁₆	—	2	2	no	—
202	Q₁₆	C₄⋊C₄×C₁₉	1	4	yes	complemented
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	C₂²⋊C₄×C₁₉	1	2	yes	—
206	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
208	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
209	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
210	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
211	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	4	1	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	8	2	no	—
218	C₄⋊C₄	—	8	2	no	—
219	C₄²	—	4	4	no	—
220	C₄²	C₂²⋊C₄×C₁₉	1	2	yes	—
221	C₄²	SD₁₆×C₁₉	1	2	yes	—
222	C₁₆	—	4	2	no	—
223	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
224	Q₈	—	2	4	no	—
225	Q₈	—	4	2	no	—
226	Q₈	—	4	2	no	—
227	Q₈	—	4	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	8	1	no	—
235	C₂×C₄	—	8	1	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	C₁₉×#102	1	2	yes	—
240	C₂×C₄	C₁₉×#102	1	2	yes	—
241	C₂×C₄	C₁₉×SD₃₂	1	2	yes	—
242	C₂×C₄	C₁₉×Q₃₂	1	2	yes	—
243	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
244	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
245	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
246	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
247	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
248	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
251	C₈	C₁₉×#117	1	1	yes	characteristic
252	C₈	C₁₉×#121	1	1	yes	characteristic
253	C₈	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
254	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
255	C₂²	C₁₉×#459	1	1	yes	characteristic
256	C₂²	C₁₉×#474	1	1	yes	characteristic
257	C₂²	C₂×C₁₉×SD₃₂	1	1	yes	characteristic
258	C₂²	C₂×C₁₉×Q₃₂	1	1	yes	characteristic
259	C₂²	C₁₉×#509	1	1	yes	characteristic
260	C₂²	C₁₉×#510	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	4	4	no	—
263	C₄	—	8	2	no	—
264	C₄	—	16	2	no	—
265	C₄	C₁₉×#358	1	4	yes	complemented
266	C₄	C₁₉×#380	1	1	yes	characteristic
267	C₄	C₁₉×#417	1	1	yes	characteristic
268	C₄	C₁₉×#461	1	1	yes	characteristic
269	C₄	C₁₉×#476	1	1	yes	characteristic
270	C₂	C₁₉×#1802	1	1	yes	characteristic
271	C₂	C₂×C₁₉×#358	1	1	yes	characteristic
272	C₂	C₁₉×#2139	1	1	yes	characteristic
273	C₂	C₁₉×#2207	1	1	yes	characteristic
274	C₂	C₁₉×#2209	1	1	yes	characteristic
275	C₂	C₁₉×#2225	1	1	yes	characteristic
276	C₂	C₁₉×#2227	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
10. C₄×Q₁₆×C₁₉	C₄	semidirect
63. Q₁₆×C₁₉	C₄⋊C₄	semidirect
84. #12812	C₁₉	direct (N × H)
132. C₄×C₁₉	#358	semidirect
141. C₄×Q₁₆	C₄×C₁₉	semidirect
186. C₁₉	#12812	direct (N × H)
202. Q₁₆	C₄⋊C₄×C₁₉	semidirect
265. C₄	C₁₉×#358	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₁₆×C₁₉	2
C₂×C₁₉×C₄⋊C₁₆	2
C₂×C₁₉×#358	2
C₁₉×#1845	2
C₁₉×#1380	2
#12812	19

Subgroups — C19×#12812 #18018771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12812 #18018771