Subgroups — C₁₉×#21822 #18027781

Order 4864.

1242 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#432	C₂	1	2	yes	maximal
3	C₁₉×#1985	C₂	1	8	yes	maximal, complemented
4	C₁₉×#1817	C₂	1	4	yes	maximal
5	C₂³×Q₈×C₁₉	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₉	—	2	8	no	—
7	C₂×C₄×Q₈×C₁₉	C₄	1	4	yes	—
8	C₂×C₄×Q₈×C₁₉	C₂²	1	4	yes	—
9	C₂²×C₄⋊C₄×C₁₉	C₄	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₁₉	C₄	1	2	yes	—
11	C₂²×C₄²×C₁₉	C₂²	1	2	yes	—
12	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
13	C₁₉×#432	—	2	8	no	—
14	C₂×C₁₉×C₄⋊C₈	C₂²	1	4	yes	—
15	C₂×C₁₉×C₂²⋊C₈	C₂²	1	4	yes	—
16	C₂×C₁₉×#97	C₂²	1	4	yes	—
17	C₁₉×#336	C₂²	1	16	yes	—
18	C₂²×Q₈×C₁₉	—	2	4	no	—
19	C₂²×Q₈×C₁₉	—	2	8	no	—
20	C₂²×Q₈×C₁₉	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	C₂³	1	8	yes	—
27	C₄×Q₈×C₁₉	—	2	16	no	—
28	C₄×Q₈×C₁₉	—	2	8	no	—
29	C₄×Q₈×C₁₉	—	2	8	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
31	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
32	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	8	yes	—
34	C₂×C₄²×C₁₉	—	2	8	no	—
35	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₉	D₄	1	8	yes	—
38	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
39	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
40	C₁₉×C₄⋊C₈	—	2	8	no	—
41	C₁₉×C₂²⋊C₈	—	2	8	no	—
42	C₁₉×#97	—	2	8	no	—
43	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×Q₈×C₁₉	—	2	32	no	—
45	C₂×Q₈×C₁₉	—	2	8	no	—
46	C₂×Q₈×C₁₉	—	2	16	no	—
47	C₂²×C₄×C₁₉	—	2	4	no	—
48	C₂²×C₄×C₁₉	—	2	4	no	—
49	C₂²×C₄×C₁₉	—	2	4	no	—
50	C₂²×C₄×C₁₉	—	2	4	no	—
51	C₂²×C₄×C₁₉	—	2	4	no	—
52	C₂²×C₄×C₁₉	—	4	2	no	—
53	C₂²×C₄×C₁₉	—	4	4	no	—
54	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
55	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
56	C₂²×C₄×C₁₉	M₁₆	1	8	yes	—
57	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₁₉	—	2	8	no	—
65	C₂×C₈×C₁₉	—	2	16	no	—
66	C₄⋊C₄×C₁₉	—	2	16	no	—
67	C₄⋊C₄×C₁₉	—	2	8	no	—
68	C₄²×C₁₉	—	2	8	no	—
69	C₄²×C₁₉	—	2	8	no	—
70	C₄²×C₁₉	C₂²⋊C₄	1	8	yes	—
71	C₄²×C₁₉	C₂×D₄	1	4	yes	—
72	C₄²×C₁₉	C₂×D₄	1	4	yes	—
73	#21822	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₁₉	—	2	2	no	—
75	C₂³×C₁₉	—	2	2	no	—
76	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
77	C₂³×C₁₉	C₂×M₁₆	1	4	yes	—
78	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
79	Q₈×C₁₉	—	2	32	no	—
80	C₂×C₄×C₁₉	—	2	4	no	—
81	C₂×C₄×C₁₉	—	2	2	no	—
82	C₂×C₄×C₁₉	—	2	4	no	—
83	C₂×C₄×C₁₉	—	2	4	no	—
84	C₂×C₄×C₁₉	—	2	4	no	—
85	C₂×C₄×C₁₉	—	2	8	no	—
86	C₂×C₄×C₁₉	—	2	16	no	—
87	C₂×C₄×C₁₉	—	2	8	no	—
88	C₂×C₄×C₁₉	—	2	4	no	—
89	C₂×C₄×C₁₉	—	2	8	no	—
90	C₂×C₄×C₁₉	—	4	4	no	—
91	C₂×C₄×C₁₉	—	4	8	no	—
92	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
94	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
95	C₂×C₄×C₁₉	C₂×M₁₆	1	8	yes	—
96	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
97	C₂×C₄×C₁₉	2₋¹⁺⁴	1	4	yes	—
98	C₈×C₁₉	—	4	8	no	—
99	C₂²×C₄×Q₈	C₂×C₁₉	1	1	yes	characteristic
100	C₂×#432	C₂×C₁₉	1	2	yes	—
101	#1985	C₂×C₁₉	1	8	yes	complemented
102	#1817	C₂×C₁₉	1	4	yes	—
103	C₂²×C₁₉	—	2	2	no	—
104	C₂²×C₁₉	—	2	2	no	—
105	C₂²×C₁₉	—	2	8	no	—
106	C₂²×C₁₉	#407	1	4	yes	—
107	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
108	C₂²×C₁₉	#519	1	2	yes	—
109	C₂²×C₁₉	C₂²×M₁₆	1	2	yes	—
110	C₂²×C₁₉	#568	1	2	yes	—
111	C₄×C₁₉	—	2	2	no	—
112	C₄×C₁₉	—	2	4	no	—
113	C₄×C₁₉	—	2	8	no	—
114	C₄×C₁₉	—	2	4	no	—
115	C₄×C₁₉	—	4	4	no	—
116	C₄×C₁₉	#536	1	4	yes	—
117	C₂³×Q₈	C₄×C₁₉	1	1	yes	characteristic
118	C₂×C₄×Q₈	—	2	8	no	—
119	C₂×C₄×Q₈	C₄×C₁₉	1	4	yes	—
120	C₂×C₄×Q₈	C₂²×C₁₉	1	4	yes	—
121	C₂²×C₄⋊C₄	C₄×C₁₉	1	1	yes	characteristic
122	C₂²×C₄⋊C₄	C₄×C₁₉	1	2	yes	—
123	C₂²×C₄²	C₂²×C₁₉	1	2	yes	—
124	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
125	#432	—	2	8	no	—
126	C₂×C₄⋊C₈	C₂²×C₁₉	1	4	yes	—
127	C₂×C₂²⋊C₈	C₂²×C₁₉	1	4	yes	—
128	C₂×#97	C₂²×C₁₉	1	4	yes	—
129	#336	C₂²×C₁₉	1	16	yes	—
130	C₂×C₁₉	—	2	4	no	—
131	C₂×C₁₉	#2310	1	1	yes	characteristic
132	C₂×C₁₉	C₂×#407	1	1	yes	characteristic
133	C₂×C₁₉	#2879	1	1	yes	characteristic
134	C₂×C₁₉	#2983	1	4	yes	—
135	C₂²×Q₈	—	2	4	no	—
136	C₂²×Q₈	—	2	8	no	—
137	C₂²×Q₈	C₂×C₄×C₁₉	1	4	yes	—
138	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
139	C₂³×C₄	C₂×C₄×C₁₉	1	2	yes	—
140	C₂³×C₄	C₂×C₄×C₁₉	1	2	yes	—
141	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
142	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
143	C₂²×C₈	C₂³×C₁₉	1	8	yes	—
144	C₄×Q₈	—	2	16	no	—
145	C₄×Q₈	—	2	8	no	—
146	C₄×Q₈	—	2	8	no	—
147	C₂×C₄⋊C₄	—	2	8	no	—
148	C₂×C₄⋊C₄	—	2	4	no	—
149	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
150	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	8	yes	—
151	C₂×C₄²	—	2	8	no	—
152	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
153	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
154	C₂×C₄²	D₄×C₁₉	1	8	yes	—
155	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
156	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
157	C₄⋊C₈	—	2	8	no	—
158	C₂²⋊C₈	—	2	8	no	—
159	#97	—	2	8	no	—
160	C₁₉	#21822	1	1	yes	characteristic, direct product factor, Sylow
161	C₂⁴	C₂²×C₄×C₁₉	1	1	yes	characteristic
162	C₂×Q₈	—	2	32	no	—
163	C₂×Q₈	—	2	8	no	—
164	C₂×Q₈	—	2	16	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	—	4	4	no	—
172	C₂²×C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
173	C₂²×C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
174	C₂²×C₄	M₁₆×C₁₉	1	8	yes	—
175	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
176	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
177	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
178	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
179	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
180	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
181	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
182	C₂×C₈	—	2	8	no	—
183	C₂×C₈	—	2	16	no	—
184	C₄⋊C₄	—	2	16	no	—
185	C₄⋊C₄	—	2	8	no	—
186	C₄²	—	2	8	no	—
187	C₄²	—	2	8	no	—
188	C₄²	C₂²⋊C₄×C₁₉	1	8	yes	—
189	C₄²	C₂×D₄×C₁₉	1	4	yes	—
190	C₄²	C₂×D₄×C₁₉	1	4	yes	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
194	C₂³	C₂×M₁₆×C₁₉	1	4	yes	—
195	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
196	Q₈	—	2	32	no	—
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	2	16	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	8	no	—
209	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
210	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
211	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
212	C₂×C₄	C₂×M₁₆×C₁₉	1	8	yes	—
213	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
214	C₂×C₄	C₁₉×2₋¹⁺⁴	1	4	yes	—
215	C₈	—	4	8	no	—
216	C₂²	—	2	8	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	C₁₉×#407	1	4	yes	—
220	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
221	C₂²	C₁₉×#519	1	2	yes	—
222	C₂²	C₂²×M₁₆×C₁₉	1	2	yes	—
223	C₂²	C₁₉×#568	1	2	yes	—
224	C₄	—	2	2	no	—
225	C₄	—	2	4	no	—
226	C₄	—	2	8	no	—
227	C₄	—	2	4	no	—
228	C₄	—	4	4	no	—
229	C₄	C₁₉×#536	1	4	yes	—
230	C₂	—	2	4	no	—
231	C₂	C₁₉×#2310	1	1	yes	characteristic
232	C₂	C₂×C₁₉×#407	1	1	yes	characteristic
233	C₂	C₁₉×#2879	1	1	yes	characteristic
234	C₂	C₁₉×#2983	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₁₉×#1985	C₂	semidirect
73. #21822	C₁₉	direct (N × H)
101. #1985	C₂×C₁₉	semidirect
160. C₁₉	#21822	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₁₉	2
C₂×C₁₉×#432	2
C₁₉×#1985	2
C₁₉×#1817	2
#21822	19

Subgroups — C19×#21822 #18027781

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#21822 #18027781