Subgroups — C₁₉×#12843 #18018802

Order 4864.

366 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#504	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#368	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₉×#367	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1816	C₂	1	2	yes	maximal
5	C₁₉×#1387	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₉×#131	C₂²	1	1	yes	characteristic
7	C₁₉×#504	—	2	2	no	—
8	C₁₉×#504	C₄	1	4	yes	complemented
9	C₂×C₁₉×M₃₂	—	2	1	no	—
10	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₁₉	—	2	1	no	—
12	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#108	C₂²	1	2	yes	—
14	C₂×C₁₉×#107	C₂²	1	2	yes	—
15	C₁₉×#417	—	2	2	no	—
16	C₁₉×#368	—	2	2	no	—
17	C₁₉×#367	—	2	1	no	—
18	C₁₉×#367	—	2	1	no	—
19	C₁₉×#343	—	2	2	no	—
20	C₂×#41×C₁₉	D₄	1	1	yes	characteristic
21	C₁₉×#131	—	2	1	no	—
22	C₁₉×#131	—	2	1	no	—
23	C₁₉×#131	C₂×C₄	1	2	yes	—
24	C₁₉×#131	Q₈	1	2	yes	—
25	C₂×M₁₆×C₁₉	—	2	1	no	—
26	C₂×M₁₆×C₁₉	—	2	2	no	—
27	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₉	—	2	1	no	—
29	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
31	C₁₉×M₃₂	—	2	1	no	—
32	C₁₉×M₃₂	—	2	4	no	—
33	C₁₉×M₃₂	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₁₉	—	2	1	no	—
35	C₂×C₁₆×C₁₉	—	2	4	no	—
36	C₂×C₁₆×C₁₉	C₂×C₄	1	2	yes	—
37	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
38	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₁₉	D₄	1	2	yes	—
40	C₁₉×#108	—	2	2	no	—
41	C₁₉×#108	—	2	2	no	—
42	C₁₉×#107	—	2	2	no	—
43	C₁₉×#107	—	2	2	no	—
44	C₁₉×#103	—	4	2	no	—
45	C₁₉×#102	—	4	2	no	—
46	#41×C₁₉	—	2	1	no	—
47	#41×C₁₉	—	2	1	no	—
48	#41×C₁₉	C₄⋊C₄	1	2	yes	—
49	#41×C₁₉	Q₁₆	1	2	yes	—
50	C₂×Q₈×C₁₉	D₈	1	1	yes	characteristic
51	C₂×D₄×C₁₉	—	2	1	no	—
52	C₂×D₄×C₁₉	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	—	2	1	no	—
54	C₂²×C₄×C₁₉	—	4	2	no	—
55	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	M₁₆×C₁₉	—	2	1	no	—
57	M₁₆×C₁₉	—	2	2	no	—
58	M₁₆×C₁₉	—	2	2	no	—
59	M₁₆×C₁₉	—	4	2	no	—
60	M₁₆×C₁₉	—	4	2	no	—
61	M₁₆×C₁₉	C₄⋊C₄	1	2	yes	—
62	C₂×C₈×C₁₉	—	2	1	no	—
63	C₂×C₈×C₁₉	—	2	2	no	—
64	C₂×C₈×C₁₉	—	2	2	no	—
65	C₂×C₈×C₁₉	—	4	2	no	—
66	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
71	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
72	C₄⋊C₄×C₁₉	—	4	2	no	—
73	C₄⋊C₄×C₁₉	—	4	2	no	—
74	C₁₆×C₁₉	—	2	2	no	—
75	C₁₆×C₁₉	C₂²⋊C₄	1	2	yes	—
76	C₁₆×C₁₉	C₂²⋊C₄	1	2	yes	—
77	#12843	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₁₉	—	2	1	no	—
79	C₂³×C₁₉	#135	1	1	yes	characteristic
80	Q₈×C₁₉	—	2	1	no	—
81	Q₈×C₁₉	#107	1	2	yes	—
82	D₄×C₁₉	—	2	2	no	—
83	D₄×C₁₉	—	2	2	no	—
84	D₄×C₁₉	—	2	1	no	—
85	D₄×C₁₉	#107	1	2	yes	—
86	C₂×C₄×C₁₉	—	2	1	no	—
87	C₂×C₄×C₁₉	—	2	2	no	—
88	C₂×C₄×C₁₉	—	2	2	no	—
89	C₂×C₄×C₁₉	—	8	2	no	—
90	C₂×C₄×C₁₉	—	16	2	no	—
91	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
93	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
94	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
97	C₈×C₁₉	—	2	2	no	—
98	C₈×C₁₉	—	8	2	no	—
99	C₈×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₈×C₁₉	#117	1	1	yes	characteristic
101	C₈×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
102	C₈×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×#504	C₂×C₁₉	1	1	yes	characteristic
104	C₂×#368	C₂×C₁₉	1	1	yes	characteristic, complemented
105	C₂×#367	C₂×C₁₉	1	1	yes	characteristic, complemented
106	#1816	C₂×C₁₉	1	2	yes	—
107	#1387	C₂×C₁₉	1	2	yes	complemented
108	C₂²×C₁₉	—	2	2	no	—
109	C₂²×C₁₉	—	2	2	no	—
110	C₂²×C₁₉	—	4	2	no	—
111	C₂²×C₁₉	#427	1	1	yes	characteristic
112	C₂²×C₁₉	#466	1	1	yes	characteristic
113	C₂²×C₁₉	#467	1	1	yes	characteristic
114	C₄×C₁₉	—	2	2	no	—
115	C₄×C₁₉	—	16	2	no	—
116	C₄×C₁₉	#380	1	1	yes	characteristic
117	C₄×C₁₉	C₂×#107	1	1	yes	characteristic
118	C₄×C₁₉	#466	1	1	yes	characteristic
119	C₄×C₁₉	#467	1	1	yes	characteristic
120	C₂×#131	C₂²×C₁₉	1	1	yes	characteristic
121	#504	—	2	2	no	—
122	#504	C₄×C₁₉	1	4	yes	complemented
123	C₂×M₃₂	—	2	1	no	—
124	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
125	C₂²×C₁₆	—	2	1	no	—
126	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
127	C₂×#108	C₂²×C₁₉	1	2	yes	—
128	C₂×#107	C₂²×C₁₉	1	2	yes	—
129	#417	—	2	2	no	—
130	#368	—	2	2	no	—
131	#367	—	2	1	no	—
132	#367	—	2	1	no	—
133	#343	—	2	2	no	—
134	C₂×C₁₉	—	2	1	no	—
135	C₂×C₁₉	—	2	1	no	—
136	C₂×C₁₉	—	4	2	no	—
137	C₂×C₁₉	#1808	1	1	yes	characteristic
138	C₂×C₁₉	#2223	1	2	yes	—
139	C₂×#41	D₄×C₁₉	1	1	yes	characteristic
140	#131	—	2	1	no	—
141	#131	—	2	1	no	—
142	#131	C₂×C₄×C₁₉	1	2	yes	—
143	#131	Q₈×C₁₉	1	2	yes	—
144	C₂×M₁₆	—	2	1	no	—
145	C₂×M₁₆	—	2	2	no	—
146	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
149	C₂×C₄⋊C₄	—	2	2	no	—
150	M₃₂	—	2	1	no	—
151	M₃₂	—	2	4	no	—
152	M₃₂	C₂×C₄×C₁₉	1	2	yes	—
153	C₂×C₁₆	—	2	1	no	—
154	C₂×C₁₆	—	2	4	no	—
155	C₂×C₁₆	C₂×C₄×C₁₉	1	2	yes	—
156	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
157	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
158	C₂×C₁₆	D₄×C₁₉	1	2	yes	—
159	#108	—	2	2	no	—
160	#108	—	2	2	no	—
161	#107	—	2	2	no	—
162	#107	—	2	2	no	—
163	#103	—	4	2	no	—
164	#102	—	4	2	no	—
165	C₁₉	#12843	1	1	yes	characteristic, direct product factor, Sylow
166	#41	—	2	1	no	—
167	#41	—	2	1	no	—
168	#41	C₄⋊C₄×C₁₉	1	2	yes	—
169	#41	Q₁₆×C₁₉	1	2	yes	—
170	C₂×Q₈	D₈×C₁₉	1	1	yes	characteristic
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	D₈×C₁₉	1	1	yes	characteristic
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
176	M₁₆	—	2	1	no	—
177	M₁₆	—	2	2	no	—
178	M₁₆	—	2	2	no	—
179	M₁₆	—	4	2	no	—
180	M₁₆	—	4	2	no	—
181	M₁₆	C₄⋊C₄×C₁₉	1	2	yes	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
187	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
188	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
189	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
190	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
191	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₁₆	—	2	2	no	—
195	C₁₆	C₂²⋊C₄×C₁₉	1	2	yes	—
196	C₁₆	C₂²⋊C₄×C₁₉	1	2	yes	—
197	C₂³	—	2	1	no	—
198	C₂³	C₁₉×#135	1	1	yes	characteristic
199	Q₈	—	2	1	no	—
200	Q₈	C₁₉×#107	1	2	yes	—
201	D₄	—	2	2	no	—
202	D₄	—	2	2	no	—
203	D₄	—	2	1	no	—
204	D₄	C₁₉×#107	1	2	yes	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	8	2	no	—
209	C₂×C₄	—	16	2	no	—
210	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
211	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
212	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
213	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
214	C₂×C₄	C₂×Q₁₆×C₁₉	1	1	yes	characteristic
215	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	8	2	no	—
218	C₈	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
219	C₈	C₁₉×#117	1	1	yes	characteristic
220	C₈	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
221	C₈	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	C₁₉×#427	1	1	yes	characteristic
226	C₂²	C₁₉×#466	1	1	yes	characteristic
227	C₂²	C₁₉×#467	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	16	2	no	—
230	C₄	C₁₉×#380	1	1	yes	characteristic
231	C₄	C₂×C₁₉×#107	1	1	yes	characteristic
232	C₄	C₁₉×#466	1	1	yes	characteristic
233	C₄	C₁₉×#467	1	1	yes	characteristic
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	—	4	2	no	—
237	C₂	C₁₉×#1808	1	1	yes	characteristic
238	C₂	C₁₉×#2223	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	#466
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂×#41×C₁₉	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×#368	C₂	semidirect
3. C₂×C₁₉×#367	C₂	semidirect
5. C₁₉×#1387	C₂	semidirect
8. C₁₉×#504	C₄	semidirect
77. #12843	C₁₉	direct (N × H)
104. C₂×#368	C₂×C₁₉	semidirect
105. C₂×#367	C₂×C₁₉	semidirect
107. #1387	C₂×C₁₉	semidirect
122. #504	C₄×C₁₉	semidirect
165. C₁₉	#12843	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×#504	2
C₂×C₁₉×#368	2
C₂×C₁₉×#367	2
C₁₉×#1816	2
C₁₉×#1387	2
#12843	19

Subgroups — C19×#12843 #18018802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12843 #18018802