Subgroups — C₁₉×#14169 #18020128

Order 4864.

2014 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#376	C₂	1	2	yes	maximal, complemented
3	C₁₉×#1446	C₂	1	2	yes	maximal, complemented
4	C₁₉×#1438	C₂	1	2	yes	maximal
5	C₁₉×#1437	C₂	1	8	yes	maximal, complemented
6	C₂³×Q₈×C₁₉	C₂²	1	1	yes	characteristic
7	C₂³×D₄×C₁₉	C₂²	1	1	yes	characteristic
8	C₂×C₁₉×#124	—	2	8	no	—
9	C₂×C₁₉×#124	C₄	1	8	yes	complemented
10	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	8	yes	—
11	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
12	C₂²×C₄²×C₁₉	C₂²	1	4	yes	—
13	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#376	—	2	8	no	—
15	C₂×C₁₉×#95	C₂²	1	4	yes	—
16	C₂×C₁₉×#95	C₂²	1	4	yes	—
17	C₂×C₁₉×#95	C₂²	1	8	yes	—
18	C₂⁵×C₁₉	C₂³	1	2	yes	—
19	C₂²×Q₈×C₁₉	—	2	4	no	—
20	C₂²×Q₈×C₁₉	—	4	4	no	—
21	C₂²×Q₈×C₁₉	C₂×C₄	1	4	yes	—
22	C₂²×D₄×C₁₉	—	2	4	no	—
23	C₂²×D₄×C₁₉	—	4	4	no	—
24	C₂²×D₄×C₁₉	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
26	C₂³×C₄×C₁₉	C₂³	1	8	yes	—
27	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
28	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
29	C₁₉×#124	—	2	32	no	—
30	C₂×C₂²⋊C₄×C₁₉	—	2	8	no	—
31	C₂×C₂²⋊C₄×C₁₉	—	2	8	no	—
32	C₂×C₂²⋊C₄×C₁₉	—	2	32	no	—
33	C₂×C₂²⋊C₄×C₁₉	—	4	8	no	—
34	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	16	yes	—
35	C₂×C₄²×C₁₉	—	2	8	no	—
36	C₂×C₄²×C₁₉	—	2	16	no	—
37	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₁₉	D₄	1	8	yes	—
39	C₁₉×#95	—	2	8	no	—
40	C₁₉×#95	—	2	8	no	—
41	C₁₉×#95	—	4	8	no	—
42	C₂⁴×C₁₉	—	2	4	no	—
43	C₂⁴×C₁₉	—	2	16	no	—
44	C₂⁴×C₁₉	—	2	8	no	—
45	C₂⁴×C₁₉	C₂²×C₄	1	4	yes	—
46	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
47	C₂×Q₈×C₁₉	—	2	8	no	—
48	C₂×Q₈×C₁₉	—	4	8	no	—
49	C₂×Q₈×C₁₉	—	4	16	no	—
50	C₂×D₄×C₁₉	—	2	8	no	—
51	C₂×D₄×C₁₉	—	4	8	no	—
52	C₂×D₄×C₁₉	—	4	16	no	—
53	C₂²×C₄×C₁₉	—	2	8	no	—
54	C₂²×C₄×C₁₉	—	2	4	no	—
55	C₂²×C₄×C₁₉	—	2	8	no	—
56	C₂²×C₄×C₁₉	—	2	4	no	—
57	C₂²×C₄×C₁₉	—	2	8	no	—
58	C₂²×C₄×C₁₉	—	2	16	no	—
59	C₂²×C₄×C₁₉	—	2	16	no	—
60	C₂²×C₄×C₁₉	—	2	8	no	—
61	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₁₉	#41	1	16	yes	—
68	C₂²⋊C₄×C₁₉	—	2	32	no	—
69	C₂²⋊C₄×C₁₉	—	4	32	no	—
70	C₄²×C₁₉	—	2	32	no	—
71	C₄²×C₁₉	C₂²⋊C₄	1	16	yes	complemented
72	#14169	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₁₉	—	2	4	no	—
74	C₂³×C₁₉	—	2	16	no	—
75	C₂³×C₁₉	—	2	8	no	—
76	C₂³×C₁₉	—	4	8	no	—
77	C₂³×C₁₉	—	4	8	no	—
78	C₂³×C₁₉	—	4	8	no	—
79	C₂³×C₁₉	—	4	8	no	—
80	C₂³×C₁₉	—	4	8	no	—
81	C₂³×C₁₉	—	4	16	no	—
82	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
83	C₂³×C₁₉	C₂²×D₄	1	2	yes	—
84	C₂³×C₁₉	C₂×#41	1	8	yes	—
85	C₂³×C₁₉	H(5,𝔽₂)	1	4	yes	—
86	Q₈×C₁₉	—	4	16	no	—
87	D₄×C₁₉	—	4	16	no	—
88	C₂×C₄×C₁₉	—	2	8	no	—
89	C₂×C₄×C₁₉	—	2	4	no	—
90	C₂×C₄×C₁₉	—	2	8	no	—
91	C₂×C₄×C₁₉	—	2	8	no	—
92	C₂×C₄×C₁₉	—	2	8	no	—
93	C₂×C₄×C₁₉	—	2	4	no	—
94	C₂×C₄×C₁₉	—	2	8	no	—
95	C₂×C₄×C₁₉	—	2	16	no	—
96	C₂×C₄×C₁₉	—	2	32	no	—
97	C₂×C₄×C₁₉	—	4	8	no	—
98	C₂×C₄×C₁₉	—	4	8	no	—
99	C₂×C₄×C₁₉	—	4	8	no	—
100	C₂×C₄×C₁₉	—	4	16	no	—
101	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	8	yes	—
102	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
103	C₂²×#124	C₂×C₁₉	1	1	yes	characteristic
104	C₂×#376	C₂×C₁₉	1	2	yes	complemented
105	#1446	C₂×C₁₉	1	2	yes	complemented
106	#1438	C₂×C₁₉	1	2	yes	—
107	#1437	C₂×C₁₉	1	8	yes	complemented
108	C₂²×C₁₉	—	4	4	no	—
109	C₂²×C₁₉	—	4	16	no	—
110	C₂²×C₁₉	—	4	8	no	—
111	C₂²×C₁₉	—	8	16	no	—
112	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
113	C₂²×C₁₉	C₄×#41	1	4	yes	—
114	C₂²×C₁₉	#518	1	2	yes	—
115	C₂²×C₁₉	#525	1	4	yes	—
116	C₂²×C₁₉	#532	1	4	yes	—
117	C₂²×C₁₉	#535	1	4	yes	—
118	C₂²×C₁₉	#551	1	8	yes	—
119	C₂²×C₁₉	#555	1	8	yes	—
120	C₄×C₁₉	—	2	4	no	—
121	C₄×C₁₉	—	2	8	no	—
122	C₄×C₁₉	—	4	4	no	—
123	C₄×C₁₉	—	4	16	no	—
124	C₂³×Q₈	C₂²×C₁₉	1	1	yes	characteristic
125	C₂³×D₄	C₂²×C₁₉	1	1	yes	characteristic
126	C₂×#124	—	2	8	no	—
127	C₂×#124	C₄×C₁₉	1	8	yes	complemented
128	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	8	yes	—
129	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
130	C₂²×C₄²	C₂²×C₁₉	1	4	yes	—
131	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
132	#376	—	2	8	no	—
133	C₂×#95	C₂²×C₁₉	1	4	yes	—
134	C₂×#95	C₂²×C₁₉	1	4	yes	—
135	C₂×#95	C₂²×C₁₉	1	8	yes	—
136	C₂×C₁₉	—	8	4	no	—
137	C₂×C₁₉	#2297	1	2	yes	—
138	C₂×C₁₉	#2309	1	1	yes	characteristic
139	C₂×C₁₉	#2372	1	4	yes	—
140	C₂×C₁₉	#2610	1	8	yes	—
141	C₂⁵	C₂³×C₁₉	1	2	yes	—
142	C₂²×Q₈	—	2	4	no	—
143	C₂²×Q₈	—	4	4	no	—
144	C₂²×Q₈	C₂×C₄×C₁₉	1	4	yes	—
145	C₂²×D₄	—	2	4	no	—
146	C₂²×D₄	—	4	4	no	—
147	C₂²×D₄	C₂×C₄×C₁₉	1	4	yes	—
148	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
149	C₂³×C₄	C₂³×C₁₉	1	8	yes	—
150	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
151	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
152	#124	—	2	32	no	—
153	C₂×C₂²⋊C₄	—	2	8	no	—
154	C₂×C₂²⋊C₄	—	2	8	no	—
155	C₂×C₂²⋊C₄	—	2	32	no	—
156	C₂×C₂²⋊C₄	—	4	8	no	—
157	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	16	yes	—
158	C₂×C₄²	—	2	8	no	—
159	C₂×C₄²	—	2	16	no	—
160	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
161	C₂×C₄²	D₄×C₁₉	1	8	yes	—
162	#95	—	2	8	no	—
163	#95	—	2	8	no	—
164	#95	—	4	8	no	—
165	C₁₉	#14169	1	1	yes	characteristic, direct product factor, Sylow
166	C₂⁴	—	2	4	no	—
167	C₂⁴	—	2	16	no	—
168	C₂⁴	—	2	8	no	—
169	C₂⁴	C₂²×C₄×C₁₉	1	4	yes	—
170	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
171	C₂×Q₈	—	2	8	no	—
172	C₂×Q₈	—	4	8	no	—
173	C₂×Q₈	—	4	16	no	—
174	C₂×D₄	—	2	8	no	—
175	C₂×D₄	—	4	8	no	—
176	C₂×D₄	—	4	16	no	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	2	16	no	—
183	C₂²×C₄	—	2	16	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
186	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
187	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
188	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
189	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
190	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
191	C₂²×C₄	#41×C₁₉	1	16	yes	—
192	C₂²⋊C₄	—	2	32	no	—
193	C₂²⋊C₄	—	4	32	no	—
194	C₄²	—	2	32	no	—
195	C₄²	C₂²⋊C₄×C₁₉	1	16	yes	complemented
196	C₂³	—	2	4	no	—
197	C₂³	—	2	16	no	—
198	C₂³	—	2	8	no	—
199	C₂³	—	4	8	no	—
200	C₂³	—	4	8	no	—
201	C₂³	—	4	8	no	—
202	C₂³	—	4	8	no	—
203	C₂³	—	4	8	no	—
204	C₂³	—	4	16	no	—
205	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
206	C₂³	C₂²×D₄×C₁₉	1	2	yes	—
207	C₂³	C₂×#41×C₁₉	1	8	yes	—
208	C₂³	C₁₉×H(5,𝔽₂)	1	4	yes	—
209	Q₈	—	4	16	no	—
210	D₄	—	4	16	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	2	16	no	—
219	C₂×C₄	—	2	32	no	—
220	C₂×C₄	—	4	8	no	—
221	C₂×C₄	—	4	8	no	—
222	C₂×C₄	—	4	8	no	—
223	C₂×C₄	—	4	16	no	—
224	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	8	yes	—
225	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
226	C₂²	—	4	8	no	—
227	C₂²	—	4	4	no	—
228	C₂²	—	4	16	no	—
229	C₂²	—	8	16	no	—
230	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
231	C₂²	C₄×#41×C₁₉	1	4	yes	—
232	C₂²	C₁₉×#518	1	2	yes	—
233	C₂²	C₁₉×#525	1	4	yes	—
234	C₂²	C₁₉×#532	1	4	yes	—
235	C₂²	C₁₉×#535	1	4	yes	—
236	C₂²	C₁₉×#551	1	8	yes	—
237	C₂²	C₁₉×#555	1	8	yes	—
238	C₄	—	2	8	no	—
239	C₄	—	2	4	no	—
240	C₄	—	4	4	no	—
241	C₄	—	4	16	no	—
242	C₂	—	8	4	no	—
243	C₂	C₁₉×#2297	1	2	yes	—
244	C₂	C₁₉×#2309	1	1	yes	characteristic
245	C₂	C₁₉×#2372	1	4	yes	—
246	C₂	C₁₉×#2610	1	8	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₉	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₉
Frattini, cosocle	C₂⁴	C₂⁴×C₁₉
prime radical	C₂³×D₄×C₁₉	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×#376	C₂	semidirect
3. C₁₉×#1446	C₂	semidirect
5. C₁₉×#1437	C₂	semidirect
9. C₂×C₁₉×#124	C₄	semidirect
71. C₄²×C₁₉	C₂²⋊C₄	semidirect
72. #14169	C₁₉	direct (N × H)
104. C₂×#376	C₂×C₁₉	semidirect
105. #1446	C₂×C₁₉	semidirect
107. #1437	C₂×C₁₉	semidirect
127. C₂×#124	C₄×C₁₉	semidirect
165. C₁₉	#14169	direct (N × H)
195. C₄²	C₂²⋊C₄×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×#124	2
C₂×C₁₉×#376	2
C₁₉×#1446	2
C₁₉×#1438	2
C₁₉×#1437	2
#14169	19

Subgroups — C19×#14169 #18020128

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#14169 #18020128