Subgroups — C₁₉×#17688 #18023647

Order 4864.

1148 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3500	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2971	C₂	1	2	yes	maximal, complemented
3	C₁₉×#1536	C₂	1	3	yes	maximal, complemented
4	C₁₉×#1527	C₂	1	6	yes	maximal, complemented
5	C₁₉×#1525	C₂	1	3	yes	maximal, complemented
6	C₂×C₁₉×2₋¹⁺⁴	C₄	1	1	yes	characteristic
7	C₂×C₁₉×H(5,𝔽₂)	C₄	1	1	yes	characteristic
8	C₁₉×#548	—	2	2	no	—
9	C₁₉×#546	—	2	6	no	—
10	C₁₉×#536	C₄	1	3	yes	—
11	C₁₉×#535	C₄	1	3	yes	—
12	C₁₉×#532	C₂²	1	3	yes	—
13	C₂×C₁₉×#124	C₂²	1	3	yes	—
14	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₁₉	—	2	2	no	—
16	C₁₉×#436	—	2	6	no	—
17	C₁₉×#432	C₂²	1	6	yes	complemented
18	C₄×M₁₆×C₁₉	C₂²	1	6	yes	complemented
19	C₁₉×#329	C₂²	1	1	yes	characteristic, complemented
20	C₁₉×#329	C₂²	1	6	yes	complemented
21	C₁₉×#329	C₂²	1	3	yes	complemented
22	C₁₉×#329	C₂²	1	6	yes	complemented
23	C₁₉×2₋¹⁺⁴	—	4	4	no	—
24	C₁₉×H(5,𝔽₂)	—	4	4	no	—
25	C₂×#41×C₁₉	—	2	1	no	—
26	C₂×#41×C₁₉	—	2	3	no	—
27	C₂×#41×C₁₉	C₂×C₄	1	3	yes	—
28	C₂×#41×C₁₉	C₂×C₄	1	3	yes	—
29	C₂×#41×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×Q₈×C₁₉	—	2	1	no	—
31	C₂²×Q₈×C₁₉	C₂×C₄	1	3	yes	—
32	C₂²×D₄×C₁₉	—	2	3	no	—
33	C₂²×D₄×C₁₉	C₂×C₄	1	3	yes	—
34	C₂×M₁₆×C₁₉	—	2	6	no	—
35	C₁₉×C₄⋊Q₈	C₂×C₄	1	3	yes	—
36	C₁₉×#127	C₂×C₄	1	3	yes	—
37	C₁₉×#124	—	2	6	no	—
38	C₁₉×#124	C₂×C₄	1	6	yes	—
39	C₁₉×#124	C₂³	1	3	yes	—
40	C₁₉×#124	C₂³	1	1	yes	characteristic
41	C₁₉×#123	—	2	6	no	—
42	C₁₉×C₂²⋊Q₈	—	2	6	no	—
43	C₁₉×#121	—	2	6	no	—
44	C₁₉×#121	—	2	6	no	—
45	C₁₉×#120	—	2	6	no	—
46	C₄×Q₈×C₁₉	D₄	1	2	yes	—
47	C₄×D₄×C₁₉	—	2	6	no	—
48	C₄×D₄×C₁₉	D₄	1	6	yes	—
49	C₁₉×#117	C₂×C₄	1	3	yes	—
50	C₂×C₂²⋊C₄×C₁₉	—	2	6	no	—
51	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
52	C₁₉×C₄⋊C₈	—	2	6	no	—
53	C₁₉×C₂²⋊C₈	—	2	6	no	—
54	C₁₉×#97	C₂³	1	2	yes	—
55	C₁₉×#97	C₂³	1	6	yes	—
56	C₄×C₈×C₁₉	—	2	6	no	—
57	C₂⁴×C₁₉	—	2	3	no	—
58	#41×C₁₉	—	2	4	no	—
59	#41×C₁₉	—	4	6	no	—
60	#41×C₁₉	—	4	6	no	—
61	#41×C₁₉	—	4	12	no	—
62	#41×C₁₉	—	4	4	no	—
63	C₂×Q₈×C₁₉	—	2	3	no	—
64	C₂×Q₈×C₁₉	—	4	6	no	—
65	C₂×Q₈×C₁₉	—	4	4	no	—
66	C₂×Q₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×Q₈×C₁₉	C₂²×C₄	1	3	yes	—
68	C₂×Q₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×D₄×C₁₉	—	2	3	no	—
70	C₂×D₄×C₁₉	—	2	3	no	—
71	C₂×D₄×C₁₉	—	2	3	no	—
72	C₂×D₄×C₁₉	—	2	6	no	—
73	C₂×D₄×C₁₉	—	4	6	no	—
74	C₂×D₄×C₁₉	—	4	12	no	—
75	C₂×D₄×C₁₉	C₂²⋊C₄	1	3	yes	—
76	C₂×D₄×C₁₉	C₂²×C₄	1	3	yes	—
77	C₂×D₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	—	2	3	no	—
79	C₂²×C₄×C₁₉	—	2	3	no	—
80	C₂²×C₄×C₁₉	—	2	3	no	—
81	C₂²×C₄×C₁₉	C₂²×C₄	1	3	yes	—
82	C₂²×C₄×C₁₉	C₂×D₄	1	6	yes	—
83	M₁₆×C₁₉	—	4	12	no	—
84	C₂×C₈×C₁₉	—	2	2	no	—
85	C₂×C₈×C₁₉	—	2	6	no	—
86	C₄⋊C₄×C₁₉	—	2	6	no	—
87	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	6	yes	—
88	C₂²⋊C₄×C₁₉	—	2	12	no	—
89	C₂²⋊C₄×C₁₉	—	2	6	no	—
90	C₂²⋊C₄×C₁₉	—	2	2	no	—
91	C₂²⋊C₄×C₁₉	C₂²⋊C₄	1	6	yes	—
92	C₄²×C₁₉	C₂²×C₄	1	3	yes	—
93	C₄²×C₁₉	C₂×D₄	1	6	yes	—
94	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
95	#17688	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₁₉	—	2	3	no	—
97	C₂³×C₁₉	—	2	1	no	—
98	C₂³×C₁₉	—	4	6	no	—
99	C₂³×C₁₉	—	8	6	no	—
100	C₂³×C₁₉	C₂×C₂²⋊C₄	1	3	yes	—
101	Q₈×C₁₉	—	4	3	no	—
102	Q₈×C₁₉	—	4	1	no	—
103	Q₈×C₁₉	—	4	6	no	—
104	Q₈×C₁₉	#100	1	4	yes	—
105	D₄×C₁₉	—	2	6	no	—
106	D₄×C₁₉	—	4	3	no	—
107	D₄×C₁₉	—	4	1	no	—
108	D₄×C₁₉	—	4	6	no	—
109	D₄×C₁₉	—	4	6	no	—
110	D₄×C₁₉	—	4	6	no	—
111	D₄×C₁₉	—	8	6	no	—
112	C₂×C₄×C₁₉	—	2	6	no	—
113	C₂×C₄×C₁₉	—	2	3	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	4	6	no	—
116	C₂×C₄×C₁₉	—	4	6	no	—
117	C₂×C₄×C₁₉	—	4	6	no	—
118	C₂×C₄×C₁₉	—	4	6	no	—
119	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	6	yes	—
120	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	3	yes	—
121	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
123	C₈×C₁₉	—	2	4	no	—
124	C₈×C₁₉	—	4	6	no	—
125	#3500	C₂×C₁₉	1	1	yes	characteristic
126	#2971	C₂×C₁₉	1	2	yes	complemented
127	#1536	C₂×C₁₉	1	3	yes	complemented
128	#1527	C₂×C₁₉	1	6	yes	complemented
129	#1525	C₂×C₁₉	1	3	yes	complemented
130	C₂²×C₁₉	—	2	3	no	—
131	C₂²×C₁₉	—	2	6	no	—
132	C₂²×C₁₉	—	4	1	no	—
133	C₂²×C₁₉	—	4	3	no	—
134	C₂²×C₁₉	—	4	2	no	—
135	C₂²×C₁₉	—	4	6	no	—
136	C₂²×C₁₉	—	16	6	no	—
137	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₁₉	—	2	1	no	—
139	C₄×C₁₉	—	2	2	no	—
140	C₄×C₁₉	—	2	6	no	—
141	C₄×C₁₉	—	4	1	no	—
142	C₄×C₁₉	—	4	3	no	—
143	C₄×C₁₉	C₂×#100	1	6	yes	—
144	C₂×2₋¹⁺⁴	C₄×C₁₉	1	1	yes	characteristic
145	C₂×H(5,𝔽₂)	C₄×C₁₉	1	1	yes	characteristic
146	#548	—	2	2	no	—
147	#546	—	2	6	no	—
148	#536	C₄×C₁₉	1	3	yes	—
149	#535	C₄×C₁₉	1	3	yes	—
150	#532	C₂²×C₁₉	1	3	yes	—
151	C₂×#124	C₂²×C₁₉	1	3	yes	—
152	C₄×#41	C₂²×C₁₉	1	1	yes	characteristic
153	C₈×Q₈	—	2	2	no	—
154	#436	—	2	6	no	—
155	#432	C₂²×C₁₉	1	6	yes	complemented
156	C₄×M₁₆	C₂²×C₁₉	1	6	yes	complemented
157	#329	C₂²×C₁₉	1	3	yes	complemented
158	#329	C₂²×C₁₉	1	6	yes	complemented
159	#329	C₂²×C₁₉	1	1	yes	characteristic, complemented
160	#329	C₂²×C₁₉	1	6	yes	complemented
161	C₂×C₁₉	—	4	3	no	—
162	C₂×C₁₉	—	8	1	no	—
163	C₂×C₁₉	—	8	3	no	—
164	C₂×C₁₉	C₂²×#100	1	1	yes	characteristic
165	C₂×C₁₉	#2901	1	2	yes	—
166	2₋¹⁺⁴	—	4	4	no	—
167	H(5,𝔽₂)	—	4	4	no	—
168	C₂×#41	—	2	1	no	—
169	C₂×#41	—	2	3	no	—
170	C₂×#41	C₂×C₄×C₁₉	1	3	yes	—
171	C₂×#41	C₂×C₄×C₁₉	1	3	yes	—
172	C₂×#41	C₂×C₄×C₁₉	1	1	yes	characteristic
173	C₂²×Q₈	—	2	1	no	—
174	C₂²×Q₈	C₂×C₄×C₁₉	1	3	yes	—
175	C₂²×D₄	—	2	3	no	—
176	C₂²×D₄	C₂×C₄×C₁₉	1	3	yes	—
177	C₂×M₁₆	—	2	6	no	—
178	C₄⋊Q₈	C₂×C₄×C₁₉	1	3	yes	—
179	#127	C₂×C₄×C₁₉	1	3	yes	—
180	#124	—	2	6	no	—
181	#124	C₂×C₄×C₁₉	1	6	yes	—
182	#124	C₂³×C₁₉	1	3	yes	—
183	#124	C₂³×C₁₉	1	1	yes	characteristic
184	#123	—	2	6	no	—
185	C₂²⋊Q₈	—	2	6	no	—
186	#121	—	2	6	no	—
187	#121	—	2	6	no	—
188	#120	—	2	6	no	—
189	C₄×Q₈	D₄×C₁₉	1	2	yes	—
190	C₄×D₄	—	2	6	no	—
191	C₄×D₄	D₄×C₁₉	1	6	yes	—
192	#117	C₂×C₄×C₁₉	1	3	yes	—
193	C₂×C₂²⋊C₄	—	2	6	no	—
194	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
195	C₄⋊C₈	—	2	6	no	—
196	C₂²⋊C₈	—	2	6	no	—
197	#97	C₂³×C₁₉	1	2	yes	—
198	#97	C₂³×C₁₉	1	6	yes	—
199	C₄×C₈	—	2	6	no	—
200	C₁₉	#17688	1	1	yes	characteristic, direct product factor, Sylow
201	C₂⁴	—	2	3	no	—
202	#41	—	2	4	no	—
203	#41	—	4	6	no	—
204	#41	—	4	6	no	—
205	#41	—	4	12	no	—
206	#41	—	4	4	no	—
207	C₂×Q₈	—	2	3	no	—
208	C₂×Q₈	—	4	6	no	—
209	C₂×Q₈	—	4	4	no	—
210	C₂×Q₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
211	C₂×Q₈	C₂²×C₄×C₁₉	1	3	yes	—
212	C₂×Q₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
213	C₂×D₄	—	2	3	no	—
214	C₂×D₄	—	2	3	no	—
215	C₂×D₄	—	2	3	no	—
216	C₂×D₄	—	2	6	no	—
217	C₂×D₄	—	4	6	no	—
218	C₂×D₄	—	4	12	no	—
219	C₂×D₄	C₂²⋊C₄×C₁₉	1	3	yes	—
220	C₂×D₄	C₂²×C₄×C₁₉	1	3	yes	—
221	C₂×D₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
222	C₂²×C₄	—	2	3	no	—
223	C₂²×C₄	—	2	3	no	—
224	C₂²×C₄	—	2	3	no	—
225	C₂²×C₄	C₂²×C₄×C₁₉	1	3	yes	—
226	C₂²×C₄	C₂×D₄×C₁₉	1	6	yes	—
227	M₁₆	—	4	12	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	6	no	—
230	C₄⋊C₄	—	2	6	no	—
231	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	6	yes	—
232	C₂²⋊C₄	—	2	12	no	—
233	C₂²⋊C₄	—	2	6	no	—
234	C₂²⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	C₂²⋊C₄×C₁₉	1	6	yes	—
236	C₄²	C₂²×C₄×C₁₉	1	3	yes	—
237	C₄²	C₂×D₄×C₁₉	1	6	yes	—
238	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
239	C₂³	—	2	3	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	4	6	no	—
242	C₂³	—	8	6	no	—
243	C₂³	C₂×C₂²⋊C₄×C₁₉	1	3	yes	—
244	Q₈	—	4	3	no	—
245	Q₈	—	4	1	no	—
246	Q₈	—	4	6	no	—
247	Q₈	C₁₉×#100	1	4	yes	—
248	D₄	—	2	6	no	—
249	D₄	—	4	3	no	—
250	D₄	—	4	1	no	—
251	D₄	—	4	6	no	—
252	D₄	—	4	6	no	—
253	D₄	—	4	6	no	—
254	D₄	—	8	6	no	—
255	C₂×C₄	—	2	6	no	—
256	C₂×C₄	—	2	3	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	6	no	—
259	C₂×C₄	—	4	6	no	—
260	C₂×C₄	—	4	6	no	—
261	C₂×C₄	—	4	6	no	—
262	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	6	yes	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	3	yes	—
264	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
265	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
266	C₈	—	2	4	no	—
267	C₈	—	4	6	no	—
268	C₂²	—	2	3	no	—
269	C₂²	—	2	6	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	3	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	6	no	—
274	C₂²	—	16	6	no	—
275	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	6	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	3	no	—
281	C₄	C₂×C₁₉×#100	1	6	yes	—
282	C₂	—	4	3	no	—
283	C₂	—	8	3	no	—
284	C₂	—	8	1	no	—
285	C₂	C₂²×C₁₉×#100	1	1	yes	characteristic
286	C₂	C₁₉×#2901	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₁₉×#3500	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄×#41×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2971	C₂	semidirect
3. C₁₉×#1536	C₂	semidirect
4. C₁₉×#1527	C₂	semidirect
5. C₁₉×#1525	C₂	semidirect
17. C₁₉×#432	C₂²	semidirect
18. C₄×M₁₆×C₁₉	C₂²	semidirect
19. C₁₉×#329	C₂²	semidirect
20. C₁₉×#329	C₂²	semidirect
21. C₁₉×#329	C₂²	semidirect
22. C₁₉×#329	C₂²	semidirect
95. #17688	C₁₉	direct (N × H)
126. #2971	C₂×C₁₉	semidirect
127. #1536	C₂×C₁₉	semidirect
128. #1527	C₂×C₁₉	semidirect
129. #1525	C₂×C₁₉	semidirect
155. #432	C₂²×C₁₉	semidirect
156. C₄×M₁₆	C₂²×C₁₉	semidirect
157. #329	C₂²×C₁₉	semidirect
158. #329	C₂²×C₁₉	semidirect
159. #329	C₂²×C₁₉	semidirect
160. #329	C₂²×C₁₉	semidirect
200. C₁₉	#17688	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉×#3500	2
C₁₉×#2971	2
C₁₉×#1536	2
C₁₉×#1527	2
C₁₉×#1525	2
#17688	19

Subgroups — C19×#17688 #18023647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#17688 #18023647