Subgroups — C₁₉×#23622 #18029581

Order 4864.

960 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#430	C₂	1	3	yes	maximal, complemented
2	C₂×C₁₉×#413	C₂	1	3	yes	maximal
3	C₁₉×#2070	C₂	1	4	yes	maximal, complemented
4	C₁₉×#2069	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₉×#338	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#41×C₁₉	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₉	C₂²	1	3	yes	—
8	C₂²×M₁₆×C₁₉	C₂²	1	3	yes	—
9	C₁₉×#430	—	2	6	no	—
10	C₁₉×#430	C₂²	1	12	yes	—
11	C₂×C₁₉×#108	—	2	6	no	—
12	C₁₉×#413	—	2	6	no	—
13	C₁₉×#413	—	2	6	no	—
14	C₂×C₁₉×#101	C₂²	1	6	yes	—
15	C₂×C₁₉×#100	C₂²	1	6	yes	—
16	C₁₉×#338	C₂²	1	4	yes	—
17	C₂×#41×C₁₉	—	2	3	no	—
18	C₂×#41×C₁₉	—	2	3	no	—
19	C₂×#41×C₁₉	—	2	3	no	—
20	C₂×#41×C₁₉	—	2	3	no	—
21	C₂²×Q₈×C₁₉	C₂³	1	1	yes	characteristic
22	C₂²×D₄×C₁₉	Q₈	1	2	yes	—
23	C₂²×D₄×C₁₉	C₂³	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	Q₈	1	2	yes	—
25	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₁₉	—	2	3	no	—
27	C₂×M₁₆×C₁₉	—	2	3	no	—
28	C₂×M₁₆×C₁₉	—	2	3	no	—
29	C₂×M₁₆×C₁₉	—	2	3	no	—
30	C₂×M₁₆×C₁₉	—	2	3	no	—
31	C₂×M₁₆×C₁₉	—	2	3	no	—
32	C₂×M₁₆×C₁₉	—	2	3	no	—
33	C₂×M₁₆×C₁₉	—	2	3	no	—
34	C₂×M₁₆×C₁₉	D₄	1	6	yes	—
35	C₂×M₁₆×C₁₉	D₄	1	6	yes	—
36	C₂×M₁₆×C₁₉	C₂³	1	6	yes	—
37	C₂×M₁₆×C₁₉	C₂³	1	6	yes	—
38	C₂²×C₈×C₁₉	—	2	3	no	—
39	C₂²×C₈×C₁₉	—	2	3	no	—
40	C₁₉×#108	—	2	24	no	—
41	C₁₉×#101	—	2	12	no	—
42	C₁₉×#100	—	2	12	no	—
43	C₂⁴×C₁₉	C₂×Q₈	1	2	yes	—
44	C₂⁴×C₁₉	C₂×Q₈	1	1	yes	characteristic
45	#41×C₁₉	—	4	3	no	—
46	#41×C₁₉	—	4	3	no	—
47	#41×C₁₉	—	4	3	no	—
48	#41×C₁₉	—	4	3	no	—
49	#41×C₁₉	—	4	3	no	—
50	#41×C₁₉	—	4	1	no	—
51	C₂×Q₈×C₁₉	—	2	3	no	—
52	C₂×Q₈×C₁₉	—	2	3	no	—
53	C₂×D₄×C₁₉	—	2	6	no	—
54	C₂×D₄×C₁₉	—	2	6	no	—
55	C₂×D₄×C₁₉	—	2	3	no	—
56	C₂×D₄×C₁₉	—	2	3	no	—
57	C₂²×C₄×C₁₉	—	2	6	no	—
58	C₂²×C₄×C₁₉	—	2	6	no	—
59	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
60	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
61	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	#41	1	3	yes	—
64	C₂²×C₄×C₁₉	#41	1	3	yes	—
65	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
66	M₁₆×C₁₉	—	2	6	no	—
67	M₁₆×C₁₉	—	2	6	no	—
68	M₁₆×C₁₉	—	2	6	no	—
69	M₁₆×C₁₉	—	2	6	no	—
70	M₁₆×C₁₉	—	2	6	no	—
71	M₁₆×C₁₉	—	4	3	no	—
72	M₁₆×C₁₉	—	4	3	no	—
73	M₁₆×C₁₉	C₂×D₄	1	12	yes	—
74	C₂×C₈×C₁₉	—	2	6	no	—
75	C₂×C₈×C₁₉	—	2	6	no	—
76	C₂×C₈×C₁₉	—	2	6	no	—
77	C₂×C₈×C₁₉	—	2	6	no	—
78	C₂×C₈×C₁₉	—	4	3	no	—
79	C₂×C₈×C₁₉	—	4	3	no	—
80	#23622	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₁₉	—	2	6	no	—
82	C₂³×C₁₉	—	8	1	no	—
83	C₂³×C₁₉	—	8	2	no	—
84	C₂³×C₁₉	C₂²⋊Q₈	1	3	yes	—
85	C₂³×C₁₉	C₂²⋊Q₈	1	3	yes	—
86	C₂³×C₁₉	C₂²×Q₈	1	1	yes	characteristic
87	Q₈×C₁₉	—	4	3	no	—
88	Q₈×C₁₉	—	4	1	no	—
89	D₄×C₁₉	—	4	6	no	—
90	D₄×C₁₉	—	4	2	no	—
91	D₄×C₁₉	—	4	3	no	—
92	D₄×C₁₉	—	4	1	no	—
93	C₂×C₄×C₁₉	—	2	6	no	—
94	C₂×C₄×C₁₉	—	4	6	no	—
95	C₂×C₄×C₁₉	—	4	2	no	—
96	C₂×C₄×C₁₉	#121	1	3	yes	—
97	C₂×C₄×C₁₉	#121	1	3	yes	—
98	C₂×C₄×C₁₉	#121	1	3	yes	—
99	C₂×C₄×C₁₉	#121	1	3	yes	—
100	C₂×C₄×C₁₉	C₂²⋊Q₈	1	3	yes	—
101	C₂×C₄×C₁₉	C₂²⋊Q₈	1	3	yes	—
102	C₂×C₄×C₁₉	#126	1	3	yes	—
103	C₂×C₄×C₁₉	#126	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂²×D₄	1	3	yes	—
105	C₂×C₄×C₁₉	C₂×#41	1	3	yes	—
106	C₈×C₁₉	—	2	12	no	—
107	C₈×C₁₉	—	4	6	no	—
108	C₂×#430	C₂×C₁₉	1	3	yes	complemented
109	C₂×#413	C₂×C₁₉	1	3	yes	—
110	#2070	C₂×C₁₉	1	4	yes	complemented
111	#2069	C₂×C₁₉	1	4	yes	complemented
112	C₂×#338	C₂×C₁₉	1	1	yes	characteristic, complemented
113	C₂²×C₁₉	—	4	1	no	—
114	C₂²×C₁₉	—	4	3	no	—
115	C₂²×C₁₉	—	4	3	no	—
116	C₂²×C₁₉	—	4	2	no	—
117	C₂²×C₁₉	—	8	6	no	—
118	C₂²×C₁₉	#396	1	3	yes	—
119	C₂²×C₁₉	#396	1	1	yes	characteristic
120	C₂²×C₁₉	C₂×C₂²⋊Q₈	1	3	yes	—
121	C₄×C₁₉	—	4	2	no	—
122	C₄×C₁₉	#396	1	3	yes	—
123	C₄×C₁₉	#396	1	1	yes	characteristic
124	C₄×C₁₉	C₂×#121	1	3	yes	—
125	C₄×C₁₉	C₂×#126	1	1	yes	characteristic
126	C₂²×#41	C₂²×C₁₉	1	1	yes	characteristic
127	C₂²×M₁₆	C₂²×C₁₉	1	3	yes	—
128	C₂²×M₁₆	C₂²×C₁₉	1	3	yes	—
129	#430	—	2	6	no	—
130	#430	C₂²×C₁₉	1	12	yes	—
131	C₂×#108	—	2	6	no	—
132	#413	—	2	6	no	—
133	#413	—	2	6	no	—
134	C₂×#101	C₂²×C₁₉	1	6	yes	—
135	C₂×#100	C₂²×C₁₉	1	6	yes	—
136	#338	C₂²×C₁₉	1	4	yes	—
137	C₂×C₁₉	—	2	3	no	—
138	C₂×C₁₉	—	2	3	no	—
139	C₂×C₁₉	—	2	1	no	—
140	C₂×C₁₉	—	8	2	no	—
141	C₂×C₁₉	C₂×#396	1	1	yes	characteristic
142	C₂×#41	—	2	3	no	—
143	C₂×#41	—	2	3	no	—
144	C₂×#41	—	2	3	no	—
145	C₂×#41	—	2	3	no	—
146	C₂²×Q₈	C₂³×C₁₉	1	1	yes	characteristic
147	C₂²×D₄	Q₈×C₁₉	1	2	yes	—
148	C₂²×D₄	C₂³×C₁₉	1	1	yes	characteristic
149	C₂³×C₄	Q₈×C₁₉	1	2	yes	—
150	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
151	C₂×M₁₆	—	2	3	no	—
152	C₂×M₁₆	—	2	3	no	—
153	C₂×M₁₆	—	2	3	no	—
154	C₂×M₁₆	—	2	3	no	—
155	C₂×M₁₆	—	2	3	no	—
156	C₂×M₁₆	—	2	3	no	—
157	C₂×M₁₆	—	2	3	no	—
158	C₂×M₁₆	—	2	3	no	—
159	C₂×M₁₆	D₄×C₁₉	1	6	yes	—
160	C₂×M₁₆	D₄×C₁₉	1	6	yes	—
161	C₂×M₁₆	C₂³×C₁₉	1	6	yes	—
162	C₂×M₁₆	C₂³×C₁₉	1	6	yes	—
163	C₂²×C₈	—	2	3	no	—
164	C₂²×C₈	—	2	3	no	—
165	#108	—	2	24	no	—
166	#101	—	2	12	no	—
167	#100	—	2	12	no	—
168	C₁₉	#23622	1	1	yes	characteristic, direct product factor, Sylow
169	C₂⁴	C₂×Q₈×C₁₉	1	2	yes	—
170	C₂⁴	C₂×Q₈×C₁₉	1	1	yes	characteristic
171	#41	—	4	3	no	—
172	#41	—	4	3	no	—
173	#41	—	4	3	no	—
174	#41	—	4	3	no	—
175	#41	—	4	3	no	—
176	#41	—	4	1	no	—
177	C₂×Q₈	—	2	3	no	—
178	C₂×Q₈	—	2	3	no	—
179	C₂×D₄	—	2	6	no	—
180	C₂×D₄	—	2	6	no	—
181	C₂×D₄	—	2	3	no	—
182	C₂×D₄	—	2	3	no	—
183	C₂²×C₄	—	2	6	no	—
184	C₂²×C₄	—	2	6	no	—
185	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
186	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
187	C₂²×C₄	C₂×Q₈×C₁₉	1	2	yes	—
188	C₂²×C₄	C₂×Q₈×C₁₉	1	1	yes	characteristic
189	C₂²×C₄	#41×C₁₉	1	3	yes	—
190	C₂²×C₄	#41×C₁₉	1	3	yes	—
191	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
192	M₁₆	—	2	6	no	—
193	M₁₆	—	2	6	no	—
194	M₁₆	—	2	6	no	—
195	M₁₆	—	2	6	no	—
196	M₁₆	—	2	6	no	—
197	M₁₆	—	4	3	no	—
198	M₁₆	—	4	3	no	—
199	M₁₆	C₂×D₄×C₁₉	1	12	yes	—
200	C₂×C₈	—	2	6	no	—
201	C₂×C₈	—	2	6	no	—
202	C₂×C₈	—	2	6	no	—
203	C₂×C₈	—	2	6	no	—
204	C₂×C₈	—	4	3	no	—
205	C₂×C₈	—	4	3	no	—
206	C₂³	—	2	6	no	—
207	C₂³	—	8	1	no	—
208	C₂³	—	8	2	no	—
209	C₂³	C₁₉×C₂²⋊Q₈	1	3	yes	—
210	C₂³	C₁₉×C₂²⋊Q₈	1	3	yes	—
211	C₂³	C₂²×Q₈×C₁₉	1	1	yes	characteristic
212	Q₈	—	4	3	no	—
213	Q₈	—	4	1	no	—
214	D₄	—	4	6	no	—
215	D₄	—	4	2	no	—
216	D₄	—	4	3	no	—
217	D₄	—	4	1	no	—
218	C₂×C₄	—	2	6	no	—
219	C₂×C₄	—	4	6	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	C₁₉×#121	1	3	yes	—
222	C₂×C₄	C₁₉×#121	1	3	yes	—
223	C₂×C₄	C₁₉×#121	1	3	yes	—
224	C₂×C₄	C₁₉×#121	1	3	yes	—
225	C₂×C₄	C₁₉×C₂²⋊Q₈	1	3	yes	—
226	C₂×C₄	C₁₉×C₂²⋊Q₈	1	3	yes	—
227	C₂×C₄	C₁₉×#126	1	3	yes	—
228	C₂×C₄	C₁₉×#126	1	1	yes	characteristic
229	C₂×C₄	C₂²×D₄×C₁₉	1	3	yes	—
230	C₂×C₄	C₂×#41×C₁₉	1	3	yes	—
231	C₈	—	2	12	no	—
232	C₈	—	4	6	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	3	no	—
235	C₂²	—	4	3	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	8	6	no	—
238	C₂²	C₁₉×#396	1	3	yes	—
239	C₂²	C₁₉×#396	1	1	yes	characteristic
240	C₂²	C₂×C₁₉×C₂²⋊Q₈	1	3	yes	—
241	C₄	—	4	2	no	—
242	C₄	C₁₉×#396	1	3	yes	—
243	C₄	C₁₉×#396	1	1	yes	characteristic
244	C₄	C₂×C₁₉×#121	1	3	yes	—
245	C₄	C₂×C₁₉×#126	1	1	yes	characteristic
246	C₂	—	2	3	no	—
247	C₂	—	2	3	no	—
248	C₂	—	2	1	no	—
249	C₂	—	8	2	no	—
250	C₂	C₂×C₁₉×#396	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₉	#396
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle	C₂×C₁₉	C₂×#396
prime radical	C₂²×#41×C₁₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₉×#430	C₂	semidirect
3. C₁₉×#2070	C₂	semidirect
4. C₁₉×#2069	C₂	semidirect
5. C₂×C₁₉×#338	C₂	semidirect
80. #23622	C₁₉	direct (N × H)
108. C₂×#430	C₂×C₁₉	semidirect
110. #2070	C₂×C₁₉	semidirect
111. #2069	C₂×C₁₉	semidirect
112. C₂×#338	C₂×C₁₉	semidirect
168. C₁₉	#23622	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#430	2
C₂×C₁₉×#413	2
C₁₉×#2070	2
C₁₉×#2069	2
C₂×C₁₉×#338	2
#23622	19

Subgroups — C19×#23622 #18029581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#23622 #18029581