Subgroups — C₁₉×#12677 #18018636

Order 4864.

318 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	C₁₉×#2159	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#365	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#347	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1776	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1773	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1387	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1386	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
11	C₁₉×#427	C₂²	1	1	yes	characteristic
12	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#106	C₂²	1	1	yes	characteristic
14	C₁₉×#405	—	2	1	no	—
15	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
17	C₁₉×#367	C₄	1	2	yes	complemented
18	C₁₉×#366	C₄	1	2	yes	complemented
19	C₁₉×#365	C₄	1	4	yes	complemented
20	C₁₉×#347	C₄	1	4	yes	complemented
21	C₁₉×#340	—	2	1	no	—
22	C₁₉×#339	—	2	1	no	—
23	C₂×M₁₆×C₁₉	—	2	1	no	—
24	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
26	C₁₉×#117	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
29	C₂×C₄²×C₁₉	—	4	1	no	—
30	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
31	C₁₉×M₃₂	—	2	1	no	—
32	C₁₉×M₃₂	C₂×C₄	1	2	yes	—
33	C₁₉×M₃₂	C₂×C₄	1	2	yes	—
34	C₁₉×M₃₂	C₂×C₄	1	2	yes	—
35	C₂×C₁₆×C₁₉	—	2	1	no	—
36	C₂×C₁₆×C₁₉	C₂×C₄	1	2	yes	—
37	C₁₉×#108	—	2	1	no	—
38	C₁₉×#108	C₂×C₄	1	2	yes	—
39	C₁₉×#107	—	2	1	no	—
40	C₁₉×#107	C₂×C₄	1	1	yes	characteristic
41	C₁₉×#107	C₂×C₄	1	2	yes	—
42	C₁₉×#107	C₂×C₄	1	1	yes	characteristic
43	C₁₉×#106	—	2	1	no	—
44	C₁₉×#106	—	2	1	no	—
45	C₁₉×#106	C₂×C₄	1	2	yes	—
46	C₁₉×#97	—	4	1	no	—
47	C₁₉×#97	C₂×C₄	1	2	yes	—
48	C₁₉×#97	Q₈	1	2	yes	—
49	C₄×C₈×C₁₉	—	4	1	no	—
50	C₁₉×#95	—	4	1	no	—
51	C₂²×C₄×C₁₉	—	2	1	no	—
52	C₂²×C₄×C₁₉	—	4	1	no	—
53	C₂²×C₄×C₁₉	—	4	1	no	—
54	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
55	M₁₆×C₁₉	—	2	2	no	—
56	M₁₆×C₁₉	—	4	1	no	—
57	C₂×C₈×C₁₉	—	2	1	no	—
58	C₂×C₈×C₁₉	—	2	1	no	—
59	C₂×C₈×C₁₉	—	4	1	no	—
60	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
61	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
66	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
67	C₄⋊C₄×C₁₉	—	2	2	no	—
68	C₄⋊C₄×C₁₉	—	2	2	no	—
69	C₄⋊C₄×C₁₉	—	2	1	no	—
70	C₄⋊C₄×C₁₉	—	2	1	no	—
71	C₄⋊C₄×C₁₉	—	4	1	no	—
72	C₄⋊C₄×C₁₉	—	4	1	no	—
73	C₂²⋊C₄×C₁₉	—	4	1	no	—
74	C₄²×C₁₉	—	2	1	no	—
75	C₄²×C₁₉	—	4	1	no	—
76	C₄²×C₁₉	—	8	2	no	—
77	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
78	C₁₆×C₁₉	—	2	2	no	—
79	C₁₆×C₁₉	C₄²	1	4	yes	complemented
80	#12677	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂×C₄×C₁₉	—	2	2	no	—
83	C₂×C₄×C₁₉	—	4	1	no	—
84	C₂×C₄×C₁₉	—	4	1	no	—
85	C₂×C₄×C₁₉	—	4	2	no	—
86	C₂×C₄×C₁₉	—	4	1	no	—
87	C₂×C₄×C₁₉	—	4	1	no	—
88	C₂×C₄×C₁₉	—	4	2	no	—
89	C₂×C₄×C₁₉	—	8	1	no	—
90	C₂×C₄×C₁₉	—	8	1	no	—
91	C₂×C₄×C₁₉	—	8	1	no	—
92	C₂×C₄×C₁₉	—	8	1	no	—
93	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
98	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
99	C₈×C₁₉	—	2	2	no	—
100	C₈×C₁₉	—	4	2	no	—
101	C₈×C₁₉	C₂×C₄²	1	1	yes	characteristic
102	C₈×C₁₉	#117	1	1	yes	characteristic
103	C₈×C₁₉	C₄×Q₈	1	1	yes	characteristic
104	C₈×C₁₉	C₄×Q₈	1	1	yes	characteristic
105	#2159	C₂×C₁₉	1	1	yes	characteristic
106	C₂×#365	C₂×C₁₉	1	1	yes	characteristic
107	C₂×#347	C₂×C₁₉	1	1	yes	characteristic
108	#1776	C₂×C₁₉	1	1	yes	characteristic
109	#1773	C₂×C₁₉	1	1	yes	characteristic
110	#1387	C₂×C₁₉	1	1	yes	characteristic
111	#1386	C₂×C₁₉	1	1	yes	characteristic
112	C₂²×C₁₉	—	2	1	no	—
113	C₂²×C₁₉	—	2	1	no	—
114	C₂²×C₁₉	#428	1	1	yes	characteristic
115	C₂²×C₁₉	#440	1	1	yes	characteristic
116	C₂²×C₁₉	#442	1	1	yes	characteristic
117	C₄×C₁₉	—	4	2	no	—
118	C₄×C₁₉	—	8	2	no	—
119	C₄×C₁₉	—	8	1	no	—
120	C₄×C₁₉	—	8	1	no	—
121	C₄×C₁₉	—	8	2	no	—
122	C₄×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
123	C₄×C₁₉	#428	1	1	yes	characteristic
124	C₄×C₁₉	#440	1	1	yes	characteristic
125	C₄×C₁₉	#441	1	1	yes	characteristic
126	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
127	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
128	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
129	#427	C₂²×C₁₉	1	1	yes	characteristic
130	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
131	C₂×#106	C₂²×C₁₉	1	1	yes	characteristic
132	#405	—	2	1	no	—
133	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
134	C₄×C₄⋊C₄	—	2	1	no	—
135	#367	C₄×C₁₉	1	2	yes	complemented
136	#366	C₄×C₁₉	1	2	yes	complemented
137	#365	C₄×C₁₉	1	4	yes	complemented
138	#347	C₄×C₁₉	1	4	yes	complemented
139	#340	—	2	1	no	—
140	#339	—	2	1	no	—
141	C₂×C₁₉	—	2	1	no	—
142	C₂×C₁₉	—	2	1	no	—
143	C₂×C₁₉	#1776	1	1	yes	characteristic
144	C₂×C₁₉	#2180	1	1	yes	characteristic
145	C₂×C₁₉	AGL(1,ℤ/16)	1	1	yes	characteristic
146	C₂×M₁₆	—	2	1	no	—
147	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
148	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
149	#117	—	2	1	no	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄²	—	4	1	no	—
153	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
154	M₃₂	—	2	1	no	—
155	M₃₂	C₂×C₄×C₁₉	1	2	yes	—
156	M₃₂	C₂×C₄×C₁₉	1	2	yes	—
157	M₃₂	C₂×C₄×C₁₉	1	2	yes	—
158	C₂×C₁₆	—	2	1	no	—
159	C₂×C₁₆	C₂×C₄×C₁₉	1	2	yes	—
160	#108	—	2	1	no	—
161	#108	C₂×C₄×C₁₉	1	2	yes	—
162	#107	—	2	1	no	—
163	#107	C₂×C₄×C₁₉	1	1	yes	characteristic
164	#107	C₂×C₄×C₁₉	1	2	yes	—
165	#107	C₂×C₄×C₁₉	1	1	yes	characteristic
166	#106	—	2	1	no	—
167	#106	—	2	1	no	—
168	#106	C₂×C₄×C₁₉	1	2	yes	—
169	#97	—	4	1	no	—
170	#97	C₂×C₄×C₁₉	1	2	yes	—
171	#97	Q₈×C₁₉	1	2	yes	—
172	C₄×C₈	—	4	1	no	—
173	#95	—	4	1	no	—
174	C₁₉	#12677	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	4	1	no	—
177	C₂²×C₄	—	4	1	no	—
178	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
179	M₁₆	—	2	2	no	—
180	M₁₆	—	4	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	C₄⋊C₄×C₁₉	1	2	yes	—
185	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
186	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
187	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
188	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
189	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
190	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	4	1	no	—
196	C₄⋊C₄	—	4	1	no	—
197	C₂²⋊C₄	—	4	1	no	—
198	C₄²	—	2	1	no	—
199	C₄²	—	4	1	no	—
200	C₄²	—	8	2	no	—
201	C₄²	C₄⋊C₄×C₁₉	1	2	yes	—
202	C₁₆	—	2	2	no	—
203	C₁₆	C₄²×C₁₉	1	4	yes	complemented
204	C₂³	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	8	1	no	—
213	C₂×C₄	—	8	1	no	—
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	8	1	no	—
216	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
217	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
218	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
219	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
220	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
221	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	4	2	no	—
224	C₈	C₂×C₄²×C₁₉	1	1	yes	characteristic
225	C₈	C₁₉×#117	1	1	yes	characteristic
226	C₈	C₄×Q₈×C₁₉	1	1	yes	characteristic
227	C₈	C₄×Q₈×C₁₉	1	1	yes	characteristic
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	C₁₉×#428	1	1	yes	characteristic
231	C₂²	C₁₉×#440	1	1	yes	characteristic
232	C₂²	C₁₉×#442	1	1	yes	characteristic
233	C₄	—	4	2	no	—
234	C₄	—	8	2	no	—
235	C₄	—	8	1	no	—
236	C₄	—	8	2	no	—
237	C₄	—	8	1	no	—
238	C₄	C₄×C₄⋊C₄×C₁₉	1	1	yes	characteristic
239	C₄	C₁₉×#428	1	1	yes	characteristic
240	C₄	C₁₉×#440	1	1	yes	characteristic
241	C₄	C₁₉×#441	1	1	yes	characteristic
242	C₂	—	2	1	no	—
243	C₂	—	2	1	no	—
244	C₂	C₁₉×#1776	1	1	yes	characteristic
245	C₂	C₁₉×#2180	1	1	yes	characteristic
246	C₂	C₁₉×AGL(1,ℤ/16)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#442
commutator	C₈	C₂×C₄²×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂³×C₁₉	AGL(1,ℤ/8)
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#97 ◁ G

Complements

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

N	H ≅ G/N	type
17. C₁₉×#367	C₄	semidirect
18. C₁₉×#366	C₄	semidirect
19. C₁₉×#365	C₄	semidirect
20. C₁₉×#347	C₄	semidirect
79. C₁₆×C₁₉	C₄²	semidirect
80. #12677	C₁₉	direct (N × H)
135. #367	C₄×C₁₉	semidirect
136. #366	C₄×C₁₉	semidirect
137. #365	C₄×C₁₉	semidirect
138. #347	C₄×C₁₉	semidirect
174. C₁₉	#12677	direct (N × H)
203. C₁₆	C₄²×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#2159	2
C₂×C₁₉×#365	2
C₂×C₁₉×#347	2
C₁₉×#1776	2
C₁₉×#1773	2
C₁₉×#1387	2
C₁₉×#1386	2
#12677	19

Subgroups — C19×#12677 #18018636

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12677 #18018636