Subgroups — C₁₉×#61821 #18067780

Order 4864.

1702 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	C₁₉×#3502	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈²×C₁₉	C₂	1	1	yes	maximal, characteristic
3	Q₈×Q₁₆×C₁₉	C₂	1	8	yes	maximal, complemented
4	C₁₉×#3363	C₂	1	8	yes	maximal, complemented
5	C₁₉×#3230	C₂	1	6	yes	maximal
6	C₁₉×#3104	C₂	1	6	yes	maximal
7	C₁₉×#2982	C₂	1	1	yes	maximal, characteristic
8	Q₈²×C₁₉	—	2	4	no	—
9	Q₈²×C₁₉	C₂²	1	8	yes	—
10	Q₈²×C₁₉	C₂²	1	4	yes	—
11	C₁₉×#557	—	2	4	no	—
12	C₁₉×#557	C₂²	1	4	yes	—
13	C₁₉×#544	—	2	3	no	—
14	C₁₉×#533	C₂²	1	6	yes	—
15	C₂×C₁₉×C₄⋊Q₈	—	2	3	no	—
16	C₂×C₁₉×C₄⋊Q₈	C₂²	1	3	yes	—
17	C₁₉×#519	—	2	1	no	—
18	C₂×C₄×Q₈×C₁₉	—	2	1	no	—
19	C₂×C₄×Q₈×C₁₉	C₂²	1	3	yes	—
20	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
21	C₁₉×#500	C₂²	1	12	yes	—
22	C₁₉×#487	C₂²	1	12	yes	—
23	C₁₉×#484	C₂²	1	12	yes	—
24	C₁₉×#477	C₂²	1	24	yes	—
25	C₁₉×#477	C₂²	1	12	yes	—
26	C₁₉×#462	C₂²	1	12	yes	—
27	C₁₉×#451	C₂²	1	4	yes	—
28	C₈×Q₈×C₁₉	C₂²	1	8	yes	—
29	C₄×Q₁₆×C₁₉	C₂²	1	24	yes	—
30	C₁₉×#431	C₂²	1	6	yes	—
31	C₂²×Q₈×C₁₉	—	2	3	no	—
32	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
33	C₂²×Q₈×C₁₉	C₂³	1	1	yes	characteristic
34	C₂×Q₁₆×C₁₉	C₂³	1	8	yes	—
35	C₁₉×C₄⋊Q₈	—	2	24	no	—
36	C₁₉×C₄⋊Q₈	—	2	12	no	—
37	C₁₉×C₄⋊Q₈	—	2	6	no	—
38	C₁₉×C₄⋊Q₈	C₂³	1	6	yes	—
39	C₁₉×C₄⋊Q₈	C₂³	1	12	yes	—
40	C₁₉×C₄⋊Q₈	C₂³	1	6	yes	—
41	C₁₉×#125	—	2	12	no	—
42	C₁₉×#125	—	2	6	no	—
43	C₁₉×C₂²⋊Q₈	—	2	6	no	—
44	C₁₉×C₂²⋊Q₈	C₂³	1	4	yes	—
45	C₄×Q₈×C₁₉	—	2	8	no	—
46	C₄×Q₈×C₁₉	—	2	4	no	—
47	C₄×Q₈×C₁₉	—	2	4	no	—
48	C₄×Q₈×C₁₉	—	2	6	no	—
49	C₄×Q₈×C₁₉	—	2	6	no	—
50	C₄×Q₈×C₁₉	D₄	1	4	yes	—
51	C₄×Q₈×C₁₉	C₂³	1	12	yes	—
52	C₄×Q₈×C₁₉	C₂³	1	12	yes	—
53	C₄×Q₈×C₁₉	C₂³	1	4	yes	—
54	C₁₉×#117	—	2	6	no	—
55	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
56	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
57	C₂×C₄⋊C₄×C₁₉	D₄	1	3	yes	—
58	C₂×C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
59	C₂×C₄²×C₁₉	—	2	3	no	—
60	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
61	C₁₉×#107	C₂³	1	24	yes	—
62	C₁₉×C₄⋊C₈	C₂³	1	12	yes	—
63	C₁₉×#103	C₂³	1	8	yes	—
64	C₁₉×#103	C₂³	1	24	yes	—
65	C₁₉×C₂²⋊C₈	C₂³	1	4	yes	—
66	C₄×C₈×C₁₉	C₂³	1	12	yes	—
67	C₂×Q₈×C₁₉	—	2	4	no	—
68	C₂×Q₈×C₁₉	—	2	1	no	—
69	C₂×Q₈×C₁₉	—	2	6	no	—
70	C₂×Q₈×C₁₉	—	2	12	no	—
71	C₂×Q₈×C₁₉	—	8	6	no	—
72	C₂×Q₈×C₁₉	Q₁₆	1	8	yes	complemented
73	C₂×Q₈×C₁₉	C₂×D₄	1	4	yes	—
74	C₂×Q₈×C₁₉	C₂⁴	1	4	yes	—
75	C₂×Q₈×C₁₉	C₂⁴	1	2	yes	—
76	C₂²×C₄×C₁₉	—	2	1	no	—
77	C₂²×C₄×C₁₉	—	2	3	no	—
78	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
79	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
80	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₁₉	—	2	16	no	—
82	C₂×C₈×C₁₉	C₂⁴	1	8	yes	—
83	C₄⋊C₄×C₁₉	—	2	12	no	—
84	C₄⋊C₄×C₁₉	—	2	12	no	—
85	C₄⋊C₄×C₁₉	—	2	3	no	—
86	C₄⋊C₄×C₁₉	—	2	6	no	—
87	C₄⋊C₄×C₁₉	—	2	6	no	—
88	C₄⋊C₄×C₁₉	—	2	4	no	—
89	C₄⋊C₄×C₁₉	—	2	6	no	—
90	C₄⋊C₄×C₁₉	—	2	12	no	—
91	C₄⋊C₄×C₁₉	C₂×D₄	1	12	yes	—
92	C₄⋊C₄×C₁₉	C₂⁴	1	6	yes	—
93	C₄⋊C₄×C₁₉	C₂⁴	1	4	yes	—
94	C₂²⋊C₄×C₁₉	—	2	4	no	—
95	C₄²×C₁₉	—	2	12	no	—
96	C₄²×C₁₉	—	2	12	no	—
97	C₄²×C₁₉	C₂×D₄	1	6	yes	—
98	C₄²×C₁₉	C₂⁴	1	6	yes	—
99	#61821	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
101	Q₈×C₁₉	—	2	4	no	—
102	Q₈×C₁₉	—	2	2	no	—
103	Q₈×C₁₉	—	2	8	no	—
104	Q₈×C₁₉	—	8	6	no	—
105	Q₈×C₁₉	—	8	12	no	—
106	Q₈×C₁₉	C₂×Q₁₆	1	16	yes	—
107	Q₈×C₁₉	2₋¹⁺⁴	1	4	yes	—
108	C₂×C₄×C₁₉	—	2	4	no	—
109	C₂×C₄×C₁₉	—	2	2	no	—
110	C₂×C₄×C₁₉	—	2	2	no	—
111	C₂×C₄×C₁₉	—	2	4	no	—
112	C₂×C₄×C₁₉	—	2	4	no	—
113	C₂×C₄×C₁₉	—	2	6	no	—
114	C₂×C₄×C₁₉	—	4	3	no	—
115	C₂×C₄×C₁₉	—	4	6	no	—
116	C₂×C₄×C₁₉	C₂×Q₁₆	1	12	yes	—
117	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂²×D₄	1	6	yes	—
119	C₂×C₄×C₁₉	C₂²×D₄	1	6	yes	—
120	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
121	C₈×C₁₉	—	2	8	no	—
122	#3502	C₂×C₁₉	1	1	yes	characteristic
123	C₂×Q₈²	C₂×C₁₉	1	1	yes	characteristic
124	Q₈×Q₁₆	C₂×C₁₉	1	8	yes	complemented
125	#3363	C₂×C₁₉	1	8	yes	complemented
126	#3230	C₂×C₁₉	1	6	yes	—
127	#3104	C₂×C₁₉	1	6	yes	—
128	#2982	C₂×C₁₉	1	1	yes	characteristic
129	C₂²×C₁₉	—	2	1	no	—
130	C₂²×C₁₉	—	2	1	no	—
131	C₂²×C₁₉	C₂²×Q₁₆	1	2	yes	—
132	C₂²×C₁₉	C₂³×D₄	1	1	yes	characteristic
133	C₄×C₁₉	—	2	6	no	—
134	C₄×C₁₉	—	2	4	no	—
135	C₄×C₁₉	—	2	1	no	—
136	C₄×C₁₉	—	4	6	no	—
137	C₄×C₁₉	—	4	4	no	—
138	C₄×C₁₉	—	4	4	no	—
139	C₄×C₁₉	C₂²×Q₁₆	1	12	yes	—
140	C₄×C₁₉	C₂×2₋¹⁺⁴	1	2	yes	—
141	Q₈²	—	2	4	no	—
142	Q₈²	C₂²×C₁₉	1	8	yes	—
143	Q₈²	C₂²×C₁₉	1	4	yes	—
144	#557	—	2	4	no	—
145	#557	C₂²×C₁₉	1	4	yes	—
146	#544	—	2	3	no	—
147	#533	C₂²×C₁₉	1	6	yes	—
148	C₂×C₄⋊Q₈	—	2	3	no	—
149	C₂×C₄⋊Q₈	C₂²×C₁₉	1	3	yes	—
150	#519	—	2	1	no	—
151	C₂×C₄×Q₈	—	2	1	no	—
152	C₂×C₄×Q₈	C₂²×C₁₉	1	3	yes	—
153	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
154	#500	C₂²×C₁₉	1	12	yes	—
155	#487	C₂²×C₁₉	1	12	yes	—
156	#484	C₂²×C₁₉	1	12	yes	—
157	#477	C₂²×C₁₉	1	12	yes	—
158	#477	C₂²×C₁₉	1	24	yes	—
159	#462	C₂²×C₁₉	1	12	yes	—
160	#451	C₂²×C₁₉	1	4	yes	—
161	C₈×Q₈	C₂²×C₁₉	1	8	yes	—
162	C₄×Q₁₆	C₂²×C₁₉	1	24	yes	—
163	#431	C₂²×C₁₉	1	6	yes	—
164	C₂×C₁₉	—	2	2	no	—
165	C₂×C₁₉	#3486	1	1	yes	characteristic
166	C₂×C₁₉	C₂³×Q₁₆	1	1	yes	characteristic
167	C₂×C₁₉	#3585	1	1	yes	characteristic
168	C₂²×Q₈	—	2	3	no	—
169	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
170	C₂²×Q₈	C₂³×C₁₉	1	1	yes	characteristic
171	C₂×Q₁₆	C₂³×C₁₉	1	8	yes	—
172	C₄⋊Q₈	—	2	24	no	—
173	C₄⋊Q₈	—	2	12	no	—
174	C₄⋊Q₈	—	2	6	no	—
175	C₄⋊Q₈	C₂³×C₁₉	1	6	yes	—
176	C₄⋊Q₈	C₂³×C₁₉	1	12	yes	—
177	C₄⋊Q₈	C₂³×C₁₉	1	6	yes	—
178	#125	—	2	12	no	—
179	#125	—	2	6	no	—
180	C₂²⋊Q₈	—	2	6	no	—
181	C₂²⋊Q₈	C₂³×C₁₉	1	4	yes	—
182	C₄×Q₈	—	2	8	no	—
183	C₄×Q₈	—	2	4	no	—
184	C₄×Q₈	—	2	4	no	—
185	C₄×Q₈	—	2	6	no	—
186	C₄×Q₈	—	2	6	no	—
187	C₄×Q₈	D₄×C₁₉	1	4	yes	—
188	C₄×Q₈	C₂³×C₁₉	1	12	yes	—
189	C₄×Q₈	C₂³×C₁₉	1	12	yes	—
190	C₄×Q₈	C₂³×C₁₉	1	4	yes	—
191	#117	—	2	6	no	—
192	C₂×C₄⋊C₄	—	2	3	no	—
193	C₂×C₄⋊C₄	—	2	3	no	—
194	C₂×C₄⋊C₄	D₄×C₁₉	1	3	yes	—
195	C₂×C₄⋊C₄	C₂³×C₁₉	1	3	yes	—
196	C₂×C₄²	—	2	3	no	—
197	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
198	#107	C₂³×C₁₉	1	24	yes	—
199	C₄⋊C₈	C₂³×C₁₉	1	12	yes	—
200	#103	C₂³×C₁₉	1	8	yes	—
201	#103	C₂³×C₁₉	1	24	yes	—
202	C₂²⋊C₈	C₂³×C₁₉	1	4	yes	—
203	C₄×C₈	C₂³×C₁₉	1	12	yes	—
204	C₁₉	#61821	1	1	yes	characteristic, direct product factor, Sylow
205	C₂×Q₈	—	2	4	no	—
206	C₂×Q₈	—	2	1	no	—
207	C₂×Q₈	—	2	6	no	—
208	C₂×Q₈	—	2	12	no	—
209	C₂×Q₈	—	8	6	no	—
210	C₂×Q₈	Q₁₆×C₁₉	1	8	yes	complemented
211	C₂×Q₈	C₂×D₄×C₁₉	1	4	yes	—
212	C₂×Q₈	C₂⁴×C₁₉	1	4	yes	—
213	C₂×Q₈	C₂⁴×C₁₉	1	2	yes	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	3	no	—
216	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
217	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
218	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
219	Q₁₆	—	2	16	no	—
220	C₂×C₈	C₂⁴×C₁₉	1	8	yes	—
221	C₄⋊C₄	—	2	12	no	—
222	C₄⋊C₄	—	2	12	no	—
223	C₄⋊C₄	—	2	3	no	—
224	C₄⋊C₄	—	2	6	no	—
225	C₄⋊C₄	—	2	6	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	6	no	—
228	C₄⋊C₄	—	2	12	no	—
229	C₄⋊C₄	C₂×D₄×C₁₉	1	12	yes	—
230	C₄⋊C₄	C₂⁴×C₁₉	1	6	yes	—
231	C₄⋊C₄	C₂⁴×C₁₉	1	4	yes	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₄²	—	2	12	no	—
234	C₄²	—	2	12	no	—
235	C₄²	C₂×D₄×C₁₉	1	6	yes	—
236	C₄²	C₂⁴×C₁₉	1	6	yes	—
237	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
238	Q₈	—	2	4	no	—
239	Q₈	—	2	2	no	—
240	Q₈	—	2	8	no	—
241	Q₈	—	8	12	no	—
242	Q₈	—	8	6	no	—
243	Q₈	C₂×Q₁₆×C₁₉	1	16	yes	—
244	Q₈	C₁₉×2₋¹⁺⁴	1	4	yes	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	6	no	—
251	C₂×C₄	—	4	3	no	—
252	C₂×C₄	—	4	6	no	—
253	C₂×C₄	C₂×Q₁₆×C₁₉	1	12	yes	—
254	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
255	C₂×C₄	C₂²×D₄×C₁₉	1	6	yes	—
256	C₂×C₄	C₂²×D₄×C₁₉	1	6	yes	—
257	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
258	C₈	—	2	8	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	C₂²×Q₁₆×C₁₉	1	2	yes	—
262	C₂²	C₂³×D₄×C₁₉	1	1	yes	characteristic
263	C₄	—	2	4	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	6	no	—
266	C₄	—	4	4	no	—
267	C₄	—	4	6	no	—
268	C₄	—	4	4	no	—
269	C₄	C₂²×Q₁₆×C₁₉	1	12	yes	—
270	C₄	C₂×C₁₉×2₋¹⁺⁴	1	2	yes	—
271	C₂	—	2	2	no	—
272	C₂	C₁₉×#3486	1	1	yes	characteristic
273	C₂	C₂³×Q₁₆×C₁₉	1	1	yes	characteristic
274	C₂	C₁₉×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₉
prime radical	C₂³×C₁₉	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄×Q₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. Q₈×Q₁₆×C₁₉	C₂	semidirect
4. C₁₉×#3363	C₂	semidirect
72. C₂×Q₈×C₁₉	Q₁₆	semidirect
99. #61821	C₁₉	direct (N × H)
124. Q₈×Q₁₆	C₂×C₁₉	semidirect
125. #3363	C₂×C₁₉	semidirect
204. C₁₉	#61821	direct (N × H)
210. C₂×Q₈	Q₁₆×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#3502	2
C₂×Q₈²×C₁₉	2
Q₈×Q₁₆×C₁₉	2
C₁₉×#3363	2
C₁₉×#3230	2
C₁₉×#3104	2
C₁₉×#2982	2
#61821	19

Subgroups — C19×#61821 #18067780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#61821 #18067780