Subgroups — C₁₉×#60667 #18066626

Order 4864.

1834 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3503	C₂	1	2	yes	maximal
2	C₁₉×#3165	C₂	1	6	yes	maximal, complemented
3	C₁₉×#3164	C₂	1	12	yes	maximal, complemented
4	C₁₉×#3160	C₂	1	3	yes	maximal, complemented
5	C₂×C₁₉×#500	C₂	1	3	yes	maximal, complemented
6	C₁₉×#2913	C₂	1	4	yes	maximal
7	C₄×C₁₉×#131	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₉×#131	C₂²	1	2	yes	—
9	Q₈²×C₁₉	—	2	2	no	—
10	C₁₉×#557	—	2	2	no	—
11	C₁₉×#554	—	2	6	no	—
12	D₄×Q₈×C₁₉	—	2	6	no	—
13	C₁₉×#544	C₂²	1	12	yes	—
14	C₁₉×#533	C₂²	1	6	yes	—
15	C₂×C₁₉×C₄⋊Q₈	C₂²	1	6	yes	—
16	C₁₉×#520	C₂²	1	4	yes	—
17	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#501	C₂²	1	24	yes	—
19	C₁₉×#500	C₂²	1	3	yes	—
20	C₁₉×#500	C₂²	1	12	yes	—
21	C₁₉×#500	C₂²	1	6	yes	—
22	C₁₉×#500	C₂²	1	1	yes	characteristic
23	C₁₉×#499	C₂²	1	12	yes	—
24	C₁₉×#498	C₂²	1	6	yes	—
25	C₁₉×#428	C₂²	1	24	yes	—
26	C₁₉×#427	C₂²	1	12	yes	—
27	C₂×C₁₉×#107	C₂²	1	12	yes	—
28	C₁₉×#405	C₂²	1	6	yes	—
29	C₄×M₁₆×C₁₉	C₂²	1	3	yes	—
30	C₂×C₄×C₈×C₁₉	C₂²	1	3	yes	—
31	C₂×#41×C₁₉	C₂³	1	1	yes	characteristic
32	C₂²×Q₈×C₁₉	—	2	6	no	—
33	C₁₉×#131	Q₈	1	16	yes	complemented
34	C₂×M₁₆×C₁₉	C₂³	1	6	yes	—
35	C₂²×C₈×C₁₉	C₂³	1	6	yes	—
36	C₁₉×C₄⋊Q₈	—	2	12	no	—
37	C₁₉×C₄⋊Q₈	—	2	6	no	—
38	C₁₉×C₄⋊Q₈	—	2	6	no	—
39	C₁₉×C₄⋊Q₈	C₂³	1	2	yes	—
40	C₁₉×C₄⋊Q₈	C₂³	1	6	yes	—
41	C₁₉×C₄⋊Q₈	C₂³	1	12	yes	—
42	C₁₉×#125	—	2	12	no	—
43	C₁₉×#125	C₂³	1	12	yes	—
44	C₁₉×C₂²⋊Q₈	—	2	6	no	—
45	C₁₉×C₂²⋊Q₈	—	2	6	no	—
46	C₁₉×C₂²⋊Q₈	—	2	24	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	1	yes	characteristic
51	C₄×Q₈×C₁₉	D₄	1	1	yes	characteristic
52	C₄×D₄×C₁₉	—	2	12	no	—
53	C₄×D₄×C₁₉	D₄	1	3	yes	—
54	C₄×D₄×C₁₉	D₄	1	3	yes	—
55	C₁₉×#117	C₂³	1	12	yes	—
56	C₁₉×#117	C₂³	1	3	yes	—
57	C₂×C₄⋊C₄×C₁₉	—	2	6	no	—
58	C₂×C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
59	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
60	C₁₉×#107	C₂³	1	12	yes	—
61	C₁₉×#107	C₂³	1	24	yes	—
62	C₁₉×#107	C₂³	1	4	yes	—
63	C₁₉×#106	C₂³	1	24	yes	—
64	C₁₉×#97	C₂³	1	6	yes	—
65	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₁₉	C₂³	1	3	yes	—
67	C₄×C₈×C₁₉	C₂³	1	6	yes	—
68	#41×C₁₉	C₂×Q₈	1	8	yes	—
69	C₂×Q₈×C₁₉	—	2	2	no	—
70	C₂×Q₈×C₁₉	—	2	6	no	—
71	C₂×Q₈×C₁₉	—	2	12	no	—
72	C₂×Q₈×C₁₉	—	8	12	no	—
73	C₂×Q₈×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₂×D₄×C₁₉	C₂×D₄	1	3	yes	—
75	C₂²×C₄×C₁₉	—	2	12	no	—
76	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
77	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
78	C₂²×C₄×C₁₉	C₂⁴	1	3	yes	—
79	M₁₆×C₁₉	C₂×Q₈	1	24	yes	complemented
80	C₂×C₈×C₁₉	C₂×Q₈	1	24	yes	complemented
81	C₂×C₈×C₁₉	C₂⁴	1	2	yes	—
82	C₂×C₈×C₁₉	C₂⁴	1	6	yes	—
83	C₄⋊C₄×C₁₉	—	2	6	no	—
84	C₄⋊C₄×C₁₉	—	2	12	no	—
85	C₄⋊C₄×C₁₉	—	2	12	no	—
86	C₄⋊C₄×C₁₉	—	2	2	no	—
87	C₄⋊C₄×C₁₉	—	2	12	no	—
88	C₄⋊C₄×C₁₉	—	2	12	no	—
89	C₄⋊C₄×C₁₉	C₂×D₄	1	3	yes	—
90	C₄⋊C₄×C₁₉	C₂×D₄	1	3	yes	—
91	C₄⋊C₄×C₁₉	C₂⁴	1	12	yes	—
92	C₄⋊C₄×C₁₉	C₂⁴	1	4	yes	—
93	C₂²⋊C₄×C₁₉	—	2	12	no	—
94	C₂²⋊C₄×C₁₉	C₂×D₄	1	6	yes	—
95	C₄²×C₁₉	—	2	12	no	—
96	C₄²×C₁₉	C₂×D₄	1	3	yes	—
97	C₄²×C₁₉	C₂×D₄	1	3	yes	—
98	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
99	C₄²×C₁₉	C₂⁴	1	3	yes	—
100	#60667	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₁₉	C₂²×D₄	1	3	yes	—
102	Q₈×C₁₉	—	4	12	no	—
103	Q₈×C₁₉	—	4	4	no	—
104	Q₈×C₁₉	—	8	12	no	—
105	Q₈×C₁₉	C₄⋊Q₈	1	4	yes	—
106	D₄×C₁₉	C₄⋊Q₈	1	12	yes	—
107	C₂×C₄×C₁₉	—	2	6	no	—
108	C₂×C₄×C₁₉	—	2	6	no	—
109	C₂×C₄×C₁₉	—	2	4	no	—
110	C₂×C₄×C₁₉	—	2	12	no	—
111	C₂×C₄×C₁₉	—	8	12	no	—
112	C₂×C₄×C₁₉	C₂²×D₄	1	3	yes	—
113	C₂×C₄×C₁₉	C₂²×D₄	1	3	yes	—
114	C₂×C₄×C₁₉	C₂²×D₄	1	3	yes	—
115	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂²×Q₈	1	12	yes	—
118	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
119	C₈×C₁₉	C₂²×Q₈	1	4	yes	—
120	C₈×C₁₉	C₂²×Q₈	1	12	yes	—
121	#3503	C₂×C₁₉	1	2	yes	—
122	#3165	C₂×C₁₉	1	6	yes	complemented
123	#3164	C₂×C₁₉	1	12	yes	complemented
124	#3160	C₂×C₁₉	1	3	yes	complemented
125	C₂×#500	C₂×C₁₉	1	3	yes	complemented
126	#2913	C₂×C₁₉	1	4	yes	—
127	C₄×#131	C₂×C₁₉	1	1	yes	characteristic
128	C₂²×C₁₉	—	2	3	no	—
129	C₂²×C₁₉	—	2	3	no	—
130	C₂²×C₁₉	C₂×C₄⋊Q₈	1	6	yes	—
131	C₂²×C₁₉	C₂³×D₄	1	1	yes	characteristic
132	C₄×C₁₉	—	2	3	no	—
133	C₄×C₁₉	—	2	3	no	—
134	C₄×C₁₉	—	4	12	no	—
135	C₄×C₁₉	—	4	4	no	—
136	C₄×C₁₉	C₂×C₄⋊Q₈	1	6	yes	—
137	C₄×C₁₉	#576	1	2	yes	—
138	C₄×C₁₉	#578	1	2	yes	—
139	C₄×C₁₉	C₂³×Q₈	1	2	yes	—
140	C₂×#131	C₂²×C₁₉	1	2	yes	—
141	Q₈²	—	2	2	no	—
142	#557	—	2	2	no	—
143	#554	—	2	6	no	—
144	D₄×Q₈	—	2	6	no	—
145	#544	C₂²×C₁₉	1	12	yes	—
146	#533	C₂²×C₁₉	1	6	yes	—
147	C₂×C₄⋊Q₈	C₂²×C₁₉	1	6	yes	—
148	#520	C₂²×C₁₉	1	4	yes	—
149	C₄×#41	C₂²×C₁₉	1	1	yes	characteristic
150	#501	C₂²×C₁₉	1	24	yes	—
151	#500	C₂²×C₁₉	1	3	yes	—
152	#500	C₂²×C₁₉	1	12	yes	—
153	#500	C₂²×C₁₉	1	6	yes	—
154	#500	C₂²×C₁₉	1	1	yes	characteristic
155	#499	C₂²×C₁₉	1	12	yes	—
156	#498	C₂²×C₁₉	1	6	yes	—
157	#428	C₂²×C₁₉	1	24	yes	—
158	#427	C₂²×C₁₉	1	12	yes	—
159	C₂×#107	C₂²×C₁₉	1	12	yes	—
160	#405	C₂²×C₁₉	1	6	yes	—
161	C₄×M₁₆	C₂²×C₁₉	1	3	yes	—
162	C₂×C₄×C₈	C₂²×C₁₉	1	3	yes	—
163	C₂×C₁₉	—	2	6	no	—
164	C₂×C₁₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
165	C₂×C₁₉	C₂×#576	1	1	yes	characteristic
166	C₂×C₁₉	C₂×#578	1	1	yes	characteristic
167	C₂×#41	C₂³×C₁₉	1	1	yes	characteristic
168	C₂²×Q₈	—	2	6	no	—
169	#131	Q₈×C₁₉	1	16	yes	complemented
170	C₂×M₁₆	C₂³×C₁₉	1	6	yes	—
171	C₂²×C₈	C₂³×C₁₉	1	6	yes	—
172	C₄⋊Q₈	—	2	12	no	—
173	C₄⋊Q₈	—	2	6	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	2	yes	—
178	#125	—	2	12	no	—
179	#125	C₂³×C₁₉	1	12	yes	—
180	C₂²⋊Q₈	—	2	6	no	—
181	C₂²⋊Q₈	—	2	6	no	—
182	C₂²⋊Q₈	—	2	24	no	—
183	C₄×Q₈	—	2	4	no	—
184	C₄×Q₈	—	2	6	no	—
185	C₄×Q₈	—	2	6	no	—
186	C₄×Q₈	D₄×C₁₉	1	1	yes	characteristic
187	C₄×Q₈	D₄×C₁₉	1	1	yes	characteristic
188	C₄×D₄	—	2	12	no	—
189	C₄×D₄	D₄×C₁₉	1	3	yes	—
190	C₄×D₄	D₄×C₁₉	1	3	yes	—
191	#117	C₂³×C₁₉	1	12	yes	—
192	#117	C₂³×C₁₉	1	3	yes	—
193	C₂×C₄⋊C₄	—	2	6	no	—
194	C₂×C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
195	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
196	#107	C₂³×C₁₉	1	12	yes	—
197	#107	C₂³×C₁₉	1	24	yes	—
198	#107	C₂³×C₁₉	1	4	yes	—
199	#106	C₂³×C₁₉	1	24	yes	—
200	#97	C₂³×C₁₉	1	6	yes	—
201	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
202	C₄×C₈	C₂³×C₁₉	1	3	yes	—
203	C₄×C₈	C₂³×C₁₉	1	6	yes	—
204	C₁₉	#60667	1	1	yes	characteristic, direct product factor, Sylow
205	#41	C₂×Q₈×C₁₉	1	8	yes	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	2	6	no	—
208	C₂×Q₈	—	2	12	no	—
209	C₂×Q₈	—	8	12	no	—
210	C₂×Q₈	C₂×D₄×C₁₉	1	1	yes	characteristic
211	C₂×D₄	C₂×D₄×C₁₉	1	3	yes	—
212	C₂²×C₄	—	2	12	no	—
213	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
214	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
215	C₂²×C₄	C₂⁴×C₁₉	1	3	yes	—
216	M₁₆	C₂×Q₈×C₁₉	1	24	yes	complemented
217	C₂×C₈	C₂×Q₈×C₁₉	1	24	yes	complemented
218	C₂×C₈	C₂⁴×C₁₉	1	2	yes	—
219	C₂×C₈	C₂⁴×C₁₉	1	6	yes	—
220	C₄⋊C₄	—	2	6	no	—
221	C₄⋊C₄	—	2	12	no	—
222	C₄⋊C₄	—	2	12	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	12	no	—
225	C₄⋊C₄	—	2	12	no	—
226	C₄⋊C₄	C₂×D₄×C₁₉	1	3	yes	—
227	C₄⋊C₄	C₂×D₄×C₁₉	1	3	yes	—
228	C₄⋊C₄	C₂⁴×C₁₉	1	4	yes	—
229	C₄⋊C₄	C₂⁴×C₁₉	1	12	yes	—
230	C₂²⋊C₄	—	2	12	no	—
231	C₂²⋊C₄	C₂×D₄×C₁₉	1	6	yes	—
232	C₄²	—	2	12	no	—
233	C₄²	C₂×D₄×C₁₉	1	3	yes	—
234	C₄²	C₂×D₄×C₁₉	1	3	yes	—
235	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
236	C₄²	C₂⁴×C₁₉	1	3	yes	—
237	C₂³	C₂²×D₄×C₁₉	1	3	yes	—
238	Q₈	—	4	12	no	—
239	Q₈	—	4	4	no	—
240	Q₈	—	8	12	no	—
241	Q₈	C₁₉×C₄⋊Q₈	1	4	yes	—
242	D₄	C₁₉×C₄⋊Q₈	1	12	yes	—
243	C₂×C₄	—	2	6	no	—
244	C₂×C₄	—	2	6	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	12	no	—
247	C₂×C₄	—	8	12	no	—
248	C₂×C₄	C₂²×D₄×C₁₉	1	3	yes	—
249	C₂×C₄	C₂²×D₄×C₁₉	1	3	yes	—
250	C₂×C₄	C₂²×D₄×C₁₉	1	3	yes	—
251	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
253	C₂×C₄	C₂²×Q₈×C₁₉	1	12	yes	—
254	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
255	C₈	C₂²×Q₈×C₁₉	1	4	yes	—
256	C₈	C₂²×Q₈×C₁₉	1	12	yes	—
257	C₂²	—	2	3	no	—
258	C₂²	—	2	3	no	—
259	C₂²	C₂×C₁₉×C₄⋊Q₈	1	6	yes	—
260	C₂²	C₂³×D₄×C₁₉	1	1	yes	characteristic
261	C₄	—	2	3	no	—
262	C₄	—	2	3	no	—
263	C₄	—	4	12	no	—
264	C₄	—	4	4	no	—
265	C₄	C₂×C₁₉×C₄⋊Q₈	1	6	yes	—
266	C₄	C₁₉×#576	1	2	yes	—
267	C₄	C₁₉×#578	1	2	yes	—
268	C₄	C₂³×Q₈×C₁₉	1	2	yes	—
269	C₂	—	2	6	no	—
270	C₂	C₂²×C₁₉×C₄⋊Q₈	1	1	yes	characteristic
271	C₂	C₂×C₁₉×#576	1	1	yes	characteristic
272	C₂	C₂×C₁₉×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₉
prime radical	C₂×#41×C₁₉	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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×#3165	C₂	semidirect
3. C₁₉×#3164	C₂	semidirect
4. C₁₉×#3160	C₂	semidirect
5. C₂×C₁₉×#500	C₂	semidirect
33. C₁₉×#131	Q₈	semidirect
79. M₁₆×C₁₉	C₂×Q₈	semidirect
80. C₂×C₈×C₁₉	C₂×Q₈	semidirect
100. #60667	C₁₉	direct (N × H)
122. #3165	C₂×C₁₉	semidirect
123. #3164	C₂×C₁₉	semidirect
124. #3160	C₂×C₁₉	semidirect
125. C₂×#500	C₂×C₁₉	semidirect
169. #131	Q₈×C₁₉	semidirect
204. C₁₉	#60667	direct (N × H)
216. M₁₆	C₂×Q₈×C₁₉	semidirect
217. C₂×C₈	C₂×Q₈×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#3503	2
C₁₉×#3165	2
C₁₉×#3164	2
C₁₉×#3160	2
C₂×C₁₉×#500	2
C₁₉×#2913	2
C₄×C₁₉×#131	2
#60667	19

Subgroups — C19×#60667 #18066626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#60667 #18066626