Subgroups — C₁₉×#13244 #18019203

Order 4864.

402 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#368	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#362	C₂	1	2	yes	maximal
4	C₁₉×#2108	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1979	C₂	1	2	yes	maximal
6	C₂²×Q₁₆×C₁₉	C₂²	1	2	yes	—
7	C₂×C₁₉×Q₃₂	—	2	1	no	—
8	C₂×C₁₉×Q₃₂	—	2	1	no	—
9	C₂×C₁₉×Q₃₂	—	2	2	no	—
10	C₂×C₁₉×Q₃₂	C₄	1	4	yes	complemented
11	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#108	C₂²	1	2	yes	—
14	C₁₉×#419	—	2	2	no	—
15	C₂×C₁₉×#101	—	2	2	no	—
16	C₁₉×#405	C₂²	1	1	yes	characteristic
17	C₁₉×#368	—	2	2	no	—
18	C₁₉×#362	—	2	4	no	—
19	C₁₉×#346	—	2	1	no	—
20	C₄×C₁₆×C₁₉	—	2	1	no	—
21	C₂²×Q₈×C₁₉	—	2	2	no	—
22	C₂×Q₁₆×C₁₉	—	2	2	no	—
23	C₂×Q₁₆×C₁₉	—	2	2	no	—
24	C₂×Q₁₆×C₁₉	—	2	2	no	—
25	C₂×Q₁₆×C₁₉	—	2	2	no	—
26	C₂×Q₁₆×C₁₉	—	2	2	no	—
27	C₂×Q₁₆×C₁₉	C₂×C₄	1	4	yes	—
28	C₂×M₁₆×C₁₉	—	2	2	no	—
29	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
31	C₁₉×#117	D₄	1	1	yes	characteristic
32	C₁₉×Q₃₂	—	2	4	no	—
33	C₁₉×Q₃₂	—	2	4	no	—
34	C₁₉×M₃₂	—	2	1	no	—
35	C₁₉×M₃₂	D₄	1	2	yes	—
36	C₂×C₁₆×C₁₉	—	2	1	no	—
37	C₂×C₁₆×C₁₉	C₂×C₄	1	2	yes	—
38	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₁₉	D₄	1	2	yes	—
41	C₁₉×#108	—	2	2	no	—
42	C₁₉×#108	—	2	2	no	—
43	C₁₉×#103	—	4	4	no	—
44	C₁₉×#101	—	4	4	no	—
45	C₁₉×#97	—	2	1	no	—
46	C₄×C₈×C₁₉	—	2	1	no	—
47	C₂×Q₈×C₁₉	—	2	4	no	—
48	C₂×Q₈×C₁₉	—	4	2	no	—
49	C₂×Q₈×C₁₉	—	4	2	no	—
50	C₂×Q₈×C₁₉	—	4	2	no	—
51	C₂×Q₈×C₁₉	—	4	2	no	—
52	C₂×Q₈×C₁₉	—	4	2	no	—
53	C₂²×C₄×C₁₉	—	4	2	no	—
54	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
55	Q₁₆×C₁₉	—	2	4	no	—
56	Q₁₆×C₁₉	—	2	4	no	—
57	Q₁₆×C₁₉	—	4	2	no	—
58	Q₁₆×C₁₉	—	4	2	no	—
59	M₁₆×C₁₉	—	2	1	no	—
60	M₁₆×C₁₉	—	4	2	no	—
61	M₁₆×C₁₉	—	4	2	no	—
62	M₁₆×C₁₉	D₈	1	2	yes	—
63	C₂×C₈×C₁₉	—	2	1	no	—
64	C₂×C₈×C₁₉	—	4	2	no	—
65	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₉	C₂×D₄	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	#41	1	1	yes	characteristic
70	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
71	C₄⋊C₄×C₁₉	D₈	1	2	yes	—
72	C₂²⋊C₄×C₁₉	—	2	1	no	—
73	C₄²×C₁₉	—	2	1	no	—
74	C₁₆×C₁₉	—	2	2	no	—
75	C₁₆×C₁₉	C₂²⋊C₄	1	2	yes	—
76	C₁₆×C₁₉	C₂²⋊C₄	1	2	yes	—
77	#13244	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₁₉	#135	1	1	yes	characteristic
79	Q₈×C₁₉	—	4	4	no	—
80	Q₈×C₁₉	—	4	4	no	—
81	Q₈×C₁₉	—	8	2	no	—
82	Q₈×C₁₉	—	8	2	no	—
83	C₂×C₄×C₁₉	—	2	2	no	—
84	C₂×C₄×C₁₉	—	4	4	no	—
85	C₂×C₄×C₁₉	—	8	2	no	—
86	C₂×C₄×C₁₉	—	8	2	no	—
87	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
89	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
90	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
93	C₈×C₁₉	—	2	2	no	—
94	C₈×C₁₉	—	8	2	no	—
95	C₈×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
97	C₈×C₁₉	#121	1	1	yes	characteristic
98	C₈×C₁₉	#124	1	1	yes	characteristic
99	C₂²×Q₃₂	C₂×C₁₉	1	1	yes	characteristic
100	C₂×#368	C₂×C₁₉	1	1	yes	characteristic
101	C₂×#362	C₂×C₁₉	1	2	yes	—
102	#2108	C₂×C₁₉	1	1	yes	characteristic
103	#1979	C₂×C₁₉	1	2	yes	—
104	C₂²×C₁₉	—	2	2	no	—
105	C₂²×C₁₉	C₄×D₈	1	1	yes	characteristic
106	C₂²×C₁₉	#466	1	1	yes	characteristic
107	C₂²×C₁₉	#495	1	1	yes	characteristic
108	C₄×C₁₉	—	4	2	no	—
109	C₄×C₁₉	—	8	4	no	—
110	C₄×C₁₉	#390	1	1	yes	characteristic
111	C₄×C₁₉	C₄×D₈	1	1	yes	characteristic
112	C₄×C₁₉	#466	1	1	yes	characteristic
113	C₄×C₁₉	#493	1	1	yes	characteristic
114	C₂²×Q₁₆	C₂²×C₁₉	1	2	yes	—
115	C₂×Q₃₂	—	2	1	no	—
116	C₂×Q₃₂	—	2	1	no	—
117	C₂×Q₃₂	—	2	2	no	—
118	C₂×Q₃₂	C₄×C₁₉	1	4	yes	complemented
119	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
120	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
121	C₂×#108	C₂²×C₁₉	1	2	yes	—
122	#419	—	2	2	no	—
123	C₂×#101	—	2	2	no	—
124	#405	C₂²×C₁₉	1	1	yes	characteristic
125	#368	—	2	2	no	—
126	#362	—	2	4	no	—
127	#346	—	2	1	no	—
128	C₄×C₁₆	—	2	1	no	—
129	C₂×C₁₉	—	2	1	no	—
130	C₂×C₁₉	—	2	1	no	—
131	C₂×C₁₉	#1970	1	1	yes	characteristic
132	C₂×C₁₉	#2222	1	2	yes	—
133	C₂²×Q₈	—	2	2	no	—
134	C₂×Q₁₆	—	2	2	no	—
135	C₂×Q₁₆	—	2	2	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	2	2	no	—
138	C₂×Q₁₆	—	2	2	no	—
139	C₂×Q₁₆	C₂×C₄×C₁₉	1	4	yes	—
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
142	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
143	#117	D₄×C₁₉	1	1	yes	characteristic
144	Q₃₂	—	2	4	no	—
145	Q₃₂	—	2	4	no	—
146	M₃₂	—	2	1	no	—
147	M₃₂	D₄×C₁₉	1	2	yes	—
148	C₂×C₁₆	—	2	1	no	—
149	C₂×C₁₆	C₂×C₄×C₁₉	1	2	yes	—
150	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
151	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
152	C₂×C₁₆	D₄×C₁₉	1	2	yes	—
153	#108	—	2	2	no	—
154	#108	—	2	2	no	—
155	#103	—	4	4	no	—
156	#101	—	4	4	no	—
157	#97	—	2	1	no	—
158	C₄×C₈	—	2	1	no	—
159	C₁₉	#13244	1	1	yes	characteristic, direct product factor, Sylow
160	C₂×Q₈	—	2	4	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	4	2	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
168	Q₁₆	—	2	4	no	—
169	Q₁₆	—	2	4	no	—
170	Q₁₆	—	4	2	no	—
171	Q₁₆	—	4	2	no	—
172	M₁₆	—	2	1	no	—
173	M₁₆	—	4	2	no	—
174	M₁₆	—	4	2	no	—
175	M₁₆	D₈×C₁₉	1	2	yes	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
179	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
180	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
181	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
182	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
183	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
184	C₄⋊C₄	D₈×C₁₉	1	2	yes	—
185	C₂²⋊C₄	—	2	1	no	—
186	C₄²	—	2	1	no	—
187	C₁₆	—	2	2	no	—
188	C₁₆	C₂²⋊C₄×C₁₉	1	2	yes	—
189	C₁₆	C₂²⋊C₄×C₁₉	1	2	yes	—
190	C₂³	C₁₉×#135	1	1	yes	characteristic
191	Q₈	—	4	4	no	—
192	Q₈	—	4	4	no	—
193	Q₈	—	8	2	no	—
194	Q₈	—	8	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	—	8	2	no	—
199	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
200	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
201	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
202	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
203	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
204	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	8	2	no	—
207	C₈	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
208	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
209	C₈	C₁₉×#121	1	1	yes	characteristic
210	C₈	C₁₉×#124	1	1	yes	characteristic
211	C₂²	—	2	2	no	—
212	C₂²	C₄×D₈×C₁₉	1	1	yes	characteristic
213	C₂²	C₁₉×#466	1	1	yes	characteristic
214	C₂²	C₁₉×#495	1	1	yes	characteristic
215	C₄	—	4	2	no	—
216	C₄	—	8	4	no	—
217	C₄	C₁₉×#390	1	1	yes	characteristic
218	C₄	C₄×D₈×C₁₉	1	1	yes	characteristic
219	C₄	C₁₉×#466	1	1	yes	characteristic
220	C₄	C₁₉×#493	1	1	yes	characteristic
221	C₂	—	2	1	no	—
222	C₂	—	2	1	no	—
223	C₂	C₁₉×#1970	1	1	yes	characteristic
224	C₂	C₁₉×#2222	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	C₂³×C₁₉	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#405 ◁ G

Complements

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

N	H ≅ G/N	type
10. C₂×C₁₉×Q₃₂	C₄	semidirect
77. #13244	C₁₉	direct (N × H)
118. C₂×Q₃₂	C₄×C₁₉	semidirect
159. C₁₉	#13244	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×Q₃₂	2
C₂×C₁₉×#368	2
C₂×C₁₉×#362	2
C₁₉×#2108	2
C₁₉×#1979	2
#13244	19

Subgroups — C19×#13244 #18019203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13244 #18019203