Subgroups — C₁₉×#13213 #18019172

Order 4864.

574 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#361	C₂	1	4	yes	maximal, complemented
3	C₄×C₁₉×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₁₉	—	2	1	no	—
6	C₂²×D₈×C₁₉	C₂²	1	2	yes	—
7	C₂×C₁₉×#127	—	2	1	no	—
8	C₂×C₁₉×M₃₂	C₂²	1	2	yes	—
9	C₁₉×#493	—	2	1	no	—
10	C₁₉×#493	—	2	1	no	—
11	C₁₉×#493	C₄	1	2	yes	—
12	C₁₉×#493	C₄	1	2	yes	—
13	C₂×C₁₉×#108	—	2	1	no	—
14	C₂×C₁₉×#108	C₂²	1	2	yes	—
15	C₁₉×#423	—	2	1	no	—
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#361	—	2	8	no	—
18	C₁₉×#346	—	2	1	no	—
19	C₄×C₁₆×C₁₉	—	2	1	no	—
20	C₂²×D₄×C₁₉	—	2	2	no	—
21	C₂²×D₄×C₁₉	—	4	2	no	—
22	C₂×D₈×C₁₉	—	2	2	no	—
23	C₂×D₈×C₁₉	—	2	2	no	—
24	C₂×D₈×C₁₉	—	2	4	no	—
25	C₂×D₈×C₁₉	—	2	2	no	—
26	C₂×D₈×C₁₉	—	2	4	no	—
27	C₂×D₈×C₁₉	—	4	2	no	—
28	C₂×D₈×C₁₉	—	4	2	no	—
29	C₂×D₈×C₁₉	C₂×C₄	1	4	yes	—
30	C₂×M₁₆×C₁₉	—	2	2	no	—
31	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
33	C₁₉×#127	—	2	2	no	—
34	C₁₉×#127	—	2	2	no	—
35	C₁₉×#127	—	4	2	no	—
36	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
37	C₁₉×M₃₂	—	2	2	no	—
38	C₁₉×M₃₂	D₄	1	4	yes	complemented
39	C₂×C₁₆×C₁₉	—	2	2	no	—
40	C₁₉×#108	—	2	4	no	—
41	C₁₉×#108	—	4	2	no	—
42	C₁₉×C₄⋊C₈	—	4	2	no	—
43	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
47	C₂⁴×C₁₉	—	4	2	no	—
48	C₂×D₄×C₁₉	—	2	4	no	—
49	C₂×D₄×C₁₉	—	4	4	no	—
50	C₂×D₄×C₁₉	—	4	4	no	—
51	C₂×D₄×C₁₉	—	4	2	no	—
52	C₂×D₄×C₁₉	—	4	2	no	—
53	C₂×D₄×C₁₉	—	4	2	no	—
54	C₂×D₄×C₁₉	—	4	2	no	—
55	C₂×D₄×C₁₉	—	4	2	no	—
56	C₂×D₄×C₁₉	—	8	8	no	—
57	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
58	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
59	D₈×C₁₉	—	2	8	no	—
60	D₈×C₁₉	—	4	4	no	—
61	D₈×C₁₉	—	4	4	no	—
62	D₈×C₁₉	—	4	4	no	—
63	M₁₆×C₁₉	—	4	2	no	—
64	M₁₆×C₁₉	—	4	2	no	—
65	C₂×C₈×C₁₉	—	4	2	no	—
66	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₉	D₈	1	2	yes	—
69	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
70	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₉	#41	1	2	yes	—
74	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₁₉	D₈	1	2	yes	—
77	C₁₆×C₁₉	—	2	4	no	—
78	#13213	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₁₉	—	4	4	no	—
80	C₂³×C₁₉	—	8	8	no	—
81	C₂³×C₁₉	—	8	2	no	—
82	C₂³×C₁₉	—	8	2	no	—
83	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
84	D₄×C₁₉	—	4	4	no	—
85	D₄×C₁₉	—	4	4	no	—
86	D₄×C₁₉	—	8	8	no	—
87	D₄×C₁₉	—	8	8	no	—
88	D₄×C₁₉	—	8	2	no	—
89	D₄×C₁₉	—	8	2	no	—
90	C₂×C₄×C₁₉	—	2	4	no	—
91	C₂×C₄×C₁₉	#102	1	2	yes	—
92	C₂×C₄×C₁₉	#102	1	2	yes	—
93	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
95	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
96	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
99	C₈×C₁₉	—	8	2	no	—
100	C₈×C₁₉	#102	1	2	yes	—
101	C₈×C₁₉	#102	1	2	yes	—
102	C₈×C₁₉	C₄×D₄	1	2	yes	—
103	C₈×C₁₉	#121	1	2	yes	—
104	C₂×#493	C₂×C₁₉	1	1	yes	characteristic
105	C₂×#361	C₂×C₁₉	1	4	yes	complemented
106	C₄×M₃₂	C₂×C₁₉	1	1	yes	characteristic, complemented
107	#1848	C₂×C₁₉	1	1	yes	characteristic, complemented
108	C₂²×C₁₉	—	2	2	no	—
109	C₂²×C₁₉	—	8	8	no	—
110	C₂²×C₁₉	—	8	4	no	—
111	C₂²×C₁₉	—	16	4	no	—
112	C₂²×C₁₉	—	16	4	no	—
113	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
114	C₂²×C₁₉	#486	1	1	yes	characteristic
115	C₂²×C₁₉	#492	1	1	yes	characteristic
116	C₄×C₁₉	—	2	2	no	—
117	C₄×C₁₉	#361	1	4	yes	complemented
118	C₄×C₁₉	#390	1	1	yes	characteristic
119	C₄×C₁₉	C₂×#102	1	1	yes	characteristic
120	C₄×C₁₉	#486	1	1	yes	characteristic
121	C₄×C₁₉	#493	1	1	yes	characteristic
122	C₂²×D₈	—	2	1	no	—
123	C₂²×D₈	C₂²×C₁₉	1	2	yes	—
124	C₂×#127	—	2	1	no	—
125	C₂×M₃₂	C₂²×C₁₉	1	2	yes	—
126	#493	—	2	1	no	—
127	#493	—	2	1	no	—
128	#493	C₄×C₁₉	1	2	yes	—
129	#493	C₄×C₁₉	1	2	yes	—
130	C₂×#108	—	2	1	no	—
131	C₂×#108	C₂²×C₁₉	1	2	yes	—
132	#423	—	2	1	no	—
133	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
134	#361	—	2	8	no	—
135	#346	—	2	1	no	—
136	C₄×C₁₆	—	2	1	no	—
137	C₂×C₁₉	—	2	1	no	—
138	C₂×C₁₉	—	2	1	no	—
139	C₂×C₁₉	—	16	4	no	—
140	C₂×C₁₉	#1957	1	1	yes	characteristic
141	C₂×C₁₉	C₂×#361	1	2	yes	—
142	C₂²×D₄	—	2	2	no	—
143	C₂²×D₄	—	4	2	no	—
144	C₂×D₈	—	2	2	no	—
145	C₂×D₈	—	2	2	no	—
146	C₂×D₈	—	2	4	no	—
147	C₂×D₈	—	2	2	no	—
148	C₂×D₈	—	2	4	no	—
149	C₂×D₈	—	4	2	no	—
150	C₂×D₈	—	4	2	no	—
151	C₂×D₈	C₂×C₄×C₁₉	1	4	yes	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
154	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
155	#127	—	2	2	no	—
156	#127	—	2	2	no	—
157	#127	—	4	2	no	—
158	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
159	M₃₂	—	2	2	no	—
160	M₃₂	D₄×C₁₉	1	4	yes	complemented
161	C₂×C₁₆	—	2	2	no	—
162	#108	—	2	4	no	—
163	#108	—	4	2	no	—
164	C₄⋊C₈	—	4	2	no	—
165	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
166	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
167	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
168	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
169	C₁₉	#13213	1	1	yes	characteristic, direct product factor, Sylow
170	C₂⁴	—	4	2	no	—
171	C₂×D₄	—	2	4	no	—
172	C₂×D₄	—	4	2	no	—
173	C₂×D₄	—	4	2	no	—
174	C₂×D₄	—	4	2	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	—	4	4	no	—
178	C₂×D₄	—	4	4	no	—
179	C₂×D₄	—	8	8	no	—
180	C₂²×C₄	SD₁₆×C₁₉	1	2	yes	—
181	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
182	D₈	—	2	8	no	—
183	D₈	—	4	4	no	—
184	D₈	—	4	4	no	—
185	D₈	—	4	4	no	—
186	M₁₆	—	4	2	no	—
187	M₁₆	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
190	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
191	C₂×C₈	D₈×C₁₉	1	2	yes	—
192	C₂×C₈	SD₁₆×C₁₉	1	2	yes	—
193	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
194	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
195	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
196	C₂×C₈	#41×C₁₉	1	2	yes	—
197	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
198	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
199	C₄²	D₈×C₁₉	1	2	yes	—
200	C₁₆	—	2	4	no	—
201	C₂³	—	4	4	no	—
202	C₂³	—	8	8	no	—
203	C₂³	—	8	2	no	—
204	C₂³	—	8	2	no	—
205	C₂³	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
206	D₄	—	4	4	no	—
207	D₄	—	4	4	no	—
208	D₄	—	8	8	no	—
209	D₄	—	8	8	no	—
210	D₄	—	8	2	no	—
211	D₄	—	8	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	C₁₉×#102	1	2	yes	—
214	C₂×C₄	C₁₉×#102	1	2	yes	—
215	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
216	C₂×C₄	C₁₉×#124	1	1	yes	characteristic
217	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
218	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
219	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
220	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
221	C₈	—	8	2	no	—
222	C₈	C₁₉×#102	1	2	yes	—
223	C₈	C₁₉×#102	1	2	yes	—
224	C₈	C₄×D₄×C₁₉	1	2	yes	—
225	C₈	C₁₉×#121	1	2	yes	—
226	C₂²	—	2	2	no	—
227	C₂²	—	8	8	no	—
228	C₂²	—	8	4	no	—
229	C₂²	—	16	4	no	—
230	C₂²	—	16	4	no	—
231	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
232	C₂²	C₁₉×#486	1	1	yes	characteristic
233	C₂²	C₁₉×#492	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	C₁₉×#361	1	4	yes	complemented
236	C₄	C₁₉×#390	1	1	yes	characteristic
237	C₄	C₂×C₁₉×#102	1	1	yes	characteristic
238	C₄	C₁₉×#486	1	1	yes	characteristic
239	C₄	C₁₉×#493	1	1	yes	characteristic
240	C₂	—	2	1	no	—
241	C₂	—	2	1	no	—
242	C₂	—	16	4	no	—
243	C₂	C₁₉×#1957	1	1	yes	characteristic
244	C₂	C₂×C₁₉×#361	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂×C₁₉×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
2. C₂×C₁₉×#361	C₂	semidirect
3. C₄×C₁₉×M₃₂	C₂	semidirect
4. C₁₉×#1848	C₂	semidirect
38. C₁₉×M₃₂	D₄	semidirect
78. #13213	C₁₉	direct (N × H)
105. C₂×#361	C₂×C₁₉	semidirect
106. C₄×M₃₂	C₂×C₁₉	semidirect
107. #1848	C₂×C₁₉	semidirect
117. C₄×C₁₉	#361	semidirect
160. M₃₂	D₄×C₁₉	semidirect
169. C₁₉	#13213	direct (N × H)
235. C₄	C₁₉×#361	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#493	2
C₂×C₁₉×#361	2
C₄×C₁₉×M₃₂	2
C₁₉×#1848	2
#13213	19

Subgroups — C19×#13213 #18019172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13213 #18019172