Subgroups — C₁₉×#13364 #18019323

Order 4864.

414 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#358	C₂	1	2	yes	maximal
3	C₁₉×#1969	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1380	C₂	1	2	yes	maximal
5	C₁₉×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
7	C₂×C₁₉×C₄⋊Q₈	—	2	1	no	—
8	C₂²×C₁₆×C₁₉	C₂²	1	2	yes	—
9	C₁₉×#500	—	2	4	no	—
10	C₂×C₁₉×#107	—	2	1	no	—
11	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
12	C₂×C₁₉×#107	C₂²	1	2	yes	—
13	C₂×C₁₉×#103	—	2	1	no	—
14	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#391	—	2	1	no	—
16	C₁₉×#358	—	2	4	no	—
17	C₂²×Q₈×C₁₉	—	2	1	no	—
18	C₂²×Q₈×C₁₉	—	4	1	no	—
19	C₂×Q₁₆×C₁₉	—	2	2	no	—
20	C₂×Q₁₆×C₁₉	—	2	2	no	—
21	C₂×Q₁₆×C₁₉	—	2	1	no	—
22	C₂×Q₁₆×C₁₉	Q₈	1	2	yes	complemented
23	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
25	C₁₉×C₄⋊Q₈	—	2	4	no	—
26	C₁₉×C₄⋊Q₈	—	4	2	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₉	—	4	1	no	—
30	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₁₉	—	2	2	no	—
32	C₂×C₁₆×C₁₉	—	2	2	no	—
33	C₂×C₁₆×C₁₉	—	2	2	no	—
34	C₁₉×#107	—	2	4	no	—
35	C₁₉×#107	—	2	2	no	—
36	C₁₉×#107	—	2	2	no	—
37	C₁₉×#107	D₄	1	4	yes	—
38	C₁₉×#103	—	4	2	no	—
39	C₄×C₈×C₁₉	—	2	2	no	—
40	C₁₉×#95	—	4	1	no	—
41	C₁₉×#95	—	4	1	no	—
42	C₂×Q₈×C₁₉	—	2	2	no	—
43	C₂×Q₈×C₁₉	—	4	4	no	—
44	C₂×Q₈×C₁₉	—	4	1	no	—
45	C₂×Q₈×C₁₉	—	4	1	no	—
46	C₂×Q₈×C₁₉	—	4	1	no	—
47	C₂×Q₈×C₁₉	—	4	1	no	—
48	C₂×Q₈×C₁₉	—	4	1	no	—
49	C₂×Q₈×C₁₉	—	8	2	no	—
50	C₂×Q₈×C₁₉	—	8	2	no	—
51	C₂²×C₄×C₁₉	—	4	2	no	—
52	C₂²×C₄×C₁₉	—	4	1	no	—
53	C₂²×C₄×C₁₉	—	4	1	no	—
54	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
57	Q₁₆×C₁₉	—	2	4	no	—
58	Q₁₆×C₁₉	—	4	2	no	—
59	C₂×C₈×C₁₉	—	2	1	no	—
60	C₂×C₈×C₁₉	—	2	1	no	—
61	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
62	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₁₉	#41	1	2	yes	—
65	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
66	C₄⋊C₄×C₁₉	—	2	4	no	—
67	C₄⋊C₄×C₁₉	—	4	4	no	—
68	C₄⋊C₄×C₁₉	—	4	2	no	—
69	C₄⋊C₄×C₁₉	—	4	1	no	—
70	C₄⋊C₄×C₁₉	—	4	1	no	—
71	C₄²×C₁₉	—	2	1	no	—
72	C₄²×C₁₉	—	2	1	no	—
73	C₁₆×C₁₉	—	4	2	no	—
74	#13364	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
76	Q₈×C₁₉	—	4	2	no	—
77	Q₈×C₁₉	—	4	2	no	—
78	Q₈×C₁₉	—	8	8	no	—
79	Q₈×C₁₉	—	8	1	no	—
80	Q₈×C₁₉	—	8	1	no	—
81	C₂×C₄×C₁₉	—	2	1	no	—
82	C₂×C₄×C₁₉	—	2	1	no	—
83	C₂×C₄×C₁₉	—	2	1	no	—
84	C₂×C₄×C₁₉	—	2	1	no	—
85	C₂×C₄×C₁₉	—	4	4	no	—
86	C₂×C₄×C₁₉	—	4	2	no	—
87	C₂×C₄×C₁₉	—	8	1	no	—
88	C₂×C₄×C₁₉	—	8	1	no	—
89	C₂×C₄×C₁₉	—	8	2	no	—
90	C₂×C₄×C₁₉	—	8	2	no	—
91	C₂×C₄×C₁₉	—	8	1	no	—
92	C₂×C₄×C₁₉	—	8	1	no	—
93	C₂×C₄×C₁₉	—	8	1	no	—
94	C₂×C₄×C₁₉	SD₃₂	1	2	yes	—
95	C₂×C₄×C₁₉	Q₃₂	1	2	yes	—
96	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
97	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
102	C₈×C₁₉	—	2	2	no	—
103	C₈×C₁₉	C₂²⋊Q₈	1	2	yes	—
104	C₈×C₁₉	#124	1	2	yes	—
105	C₂×#500	C₂×C₁₉	1	1	yes	characteristic
106	C₂×#358	C₂×C₁₉	1	2	yes	—
107	#1969	C₂×C₁₉	1	1	yes	characteristic
108	#1380	C₂×C₁₉	1	2	yes	—
109	#1374	C₂×C₁₉	1	1	yes	characteristic
110	C₂²×C₁₉	#447	1	1	yes	characteristic
111	C₂²×C₁₉	#474	1	1	yes	characteristic
112	C₂²×C₁₉	#486	1	1	yes	characteristic
113	C₂²×C₁₉	C₂×SD₃₂	1	1	yes	characteristic
114	C₂²×C₁₉	C₂×Q₃₂	1	1	yes	characteristic
115	C₂²×C₁₉	#509	1	1	yes	characteristic
116	C₂²×C₁₉	#510	1	1	yes	characteristic
117	C₄×C₁₉	—	2	2	no	—
118	C₄×C₁₉	—	2	2	no	—
119	C₄×C₁₉	—	8	4	no	—
120	C₄×C₁₉	—	8	2	no	—
121	C₄×C₁₉	—	16	1	no	—
122	C₄×C₁₉	#393	1	1	yes	characteristic
123	C₄×C₁₉	#448	1	1	yes	characteristic
124	C₄×C₁₉	#476	1	1	yes	characteristic
125	C₄×C₁₉	#489	1	1	yes	characteristic
126	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
127	C₂×C₄⋊Q₈	—	2	1	no	—
128	C₂²×C₁₆	C₂²×C₁₉	1	2	yes	—
129	#500	—	2	4	no	—
130	C₂×#107	—	2	1	no	—
131	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
132	C₂×#107	C₂²×C₁₉	1	2	yes	—
133	C₂×#103	—	2	1	no	—
134	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
135	#391	—	2	1	no	—
136	#358	—	2	4	no	—
137	C₂×C₁₉	#2023	1	1	yes	characteristic
138	C₂×C₁₉	#2187	1	1	yes	characteristic
139	C₂×C₁₉	#2188	1	1	yes	characteristic
140	C₂×C₁₉	#2225	1	1	yes	characteristic
141	C₂×C₁₉	#2227	1	1	yes	characteristic
142	C₂×C₁₉	#2241	1	1	yes	characteristic
143	C₂×C₁₉	#2243	1	1	yes	characteristic
144	C₂²×Q₈	—	2	1	no	—
145	C₂²×Q₈	—	4	1	no	—
146	C₂×Q₁₆	—	2	2	no	—
147	C₂×Q₁₆	—	2	2	no	—
148	C₂×Q₁₆	—	2	1	no	—
149	C₂×Q₁₆	Q₈×C₁₉	1	2	yes	complemented
150	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
151	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
152	C₄⋊Q₈	—	2	4	no	—
153	C₄⋊Q₈	—	4	2	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	—	4	1	no	—
157	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
158	C₂×C₁₆	—	2	2	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	2	2	no	—
161	#107	—	2	4	no	—
162	#107	—	2	2	no	—
163	#107	—	2	2	no	—
164	#107	D₄×C₁₉	1	4	yes	—
165	#103	—	4	2	no	—
166	C₄×C₈	—	2	2	no	—
167	#95	—	4	1	no	—
168	#95	—	4	1	no	—
169	C₁₉	#13364	1	1	yes	characteristic, direct product factor, Sylow
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	4	4	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂×Q₈	—	4	1	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	—	4	1	no	—
176	C₂×Q₈	—	4	1	no	—
177	C₂×Q₈	—	8	2	no	—
178	C₂×Q₈	—	8	2	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	—	4	1	no	—
182	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
183	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
185	Q₁₆	—	2	4	no	—
186	Q₁₆	—	4	2	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	SD₁₆×C₁₉	1	2	yes	—
190	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
191	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
192	C₂×C₈	#41×C₁₉	1	2	yes	—
193	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	1	no	—
198	C₄⋊C₄	—	4	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	1	no	—
201	C₁₆	—	4	2	no	—
202	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
203	Q₈	—	4	2	no	—
204	Q₈	—	4	2	no	—
205	Q₈	—	8	8	no	—
206	Q₈	—	8	1	no	—
207	Q₈	—	8	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	8	1	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	—	8	1	no	—
219	C₂×C₄	—	8	1	no	—
220	C₂×C₄	—	8	1	no	—
221	C₂×C₄	C₁₉×SD₃₂	1	2	yes	—
222	C₂×C₄	C₁₉×Q₃₂	1	2	yes	—
223	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
224	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
225	C₂×C₄	C₁₉×#124	1	1	yes	characteristic
226	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
227	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
228	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₈	C₁₉×C₂²⋊Q₈	1	2	yes	—
231	C₈	C₁₉×#124	1	2	yes	—
232	C₂²	C₁₉×#447	1	1	yes	characteristic
233	C₂²	C₁₉×#474	1	1	yes	characteristic
234	C₂²	C₁₉×#486	1	1	yes	characteristic
235	C₂²	C₂×C₁₉×SD₃₂	1	1	yes	characteristic
236	C₂²	C₂×C₁₉×Q₃₂	1	1	yes	characteristic
237	C₂²	C₁₉×#509	1	1	yes	characteristic
238	C₂²	C₁₉×#510	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	8	4	no	—
242	C₄	—	8	2	no	—
243	C₄	—	16	1	no	—
244	C₄	C₁₉×#393	1	1	yes	characteristic
245	C₄	C₁₉×#448	1	1	yes	characteristic
246	C₄	C₁₉×#476	1	1	yes	characteristic
247	C₄	C₁₉×#489	1	1	yes	characteristic
248	C₂	C₁₉×#2023	1	1	yes	characteristic
249	C₂	C₁₉×#2187	1	1	yes	characteristic
250	C₂	C₁₉×#2188	1	1	yes	characteristic
251	C₂	C₁₉×#2225	1	1	yes	characteristic
252	C₂	C₁₉×#2227	1	1	yes	characteristic
253	C₂	C₁₉×#2241	1	1	yes	characteristic
254	C₂	C₁₉×#2243	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
22. C₂×Q₁₆×C₁₉	Q₈	semidirect
74. #13364	C₁₉	direct (N × H)
149. C₂×Q₁₆	Q₈×C₁₉	semidirect
169. C₁₉	#13364	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×#500	2
C₂×C₁₉×#358	2
C₁₉×#1969	2
C₁₉×#1380	2
C₁₉×#1374	2
#13364	19

Subgroups — C19×#13364 #18019323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13364 #18019323