Subgroups — C₁₉×#33730 #18039689

Order 4864.

474 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3164	C₂	1	2	yes	maximal
2	C₁₉×#2255	C₂	1	4	yes	maximal, complemented
3	C₁₉×#2254	C₂	1	2	yes	maximal, complemented
4	C₁₉×#2253	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2159	C₂	1	2	yes	maximal
6	C₁₉×#2158	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₉×#367	C₂	1	1	yes	maximal, characteristic
8	C₁₉×#2108	C₂	1	1	yes	maximal, characteristic
9	C₁₉×#544	—	2	2	no	—
10	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×#501	—	2	4	no	—
13	C₁₉×#500	C₂²	1	4	yes	—
14	C₁₉×#499	C₂²	1	4	yes	—
15	C₁₉×#428	—	2	2	no	—
16	C₁₉×#427	C₂²	1	2	yes	—
17	C₂×C₁₉×#107	C₂²	1	2	yes	—
18	C₁₉×#405	C₂²	1	1	yes	characteristic
19	C₁₉×#367	C₂²	1	2	yes	—
20	C₁₉×#367	C₂²	1	2	yes	—
21	C₁₉×#367	C₂²	1	1	yes	characteristic
22	C₁₉×#367	C₂²	1	1	yes	characteristic
23	C₁₉×#367	C₂²	1	4	yes	—
24	C₁₉×#366	C₂²	1	2	yes	—
25	C₁₉×#366	C₂²	1	4	yes	—
26	C₁₉×#346	C₂²	1	2	yes	—
27	C₄×C₁₆×C₁₉	C₂²	1	2	yes	—
28	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
30	C₁₉×C₄⋊Q₈	—	2	4	no	—
31	C₁₉×C₄⋊Q₈	—	4	2	no	—
32	C₁₉×#125	—	2	4	no	—
33	C₁₉×#125	—	4	2	no	—
34	C₁₉×C₂²⋊Q₈	—	4	4	no	—
35	C₁₉×#117	—	2	2	no	—
36	C₁₉×#117	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
38	C₁₉×M₃₂	Q₈	1	4	yes	complemented
39	C₂×C₁₆×C₁₉	Q₈	1	4	yes	complemented
40	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
41	C₂×C₁₆×C₁₉	C₂³	1	2	yes	—
42	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
43	C₁₉×#107	—	2	2	no	—
44	C₁₉×#107	—	2	4	no	—
45	C₁₉×#107	C₂³	1	2	yes	—
46	C₁₉×#107	C₂³	1	4	yes	—
47	C₁₉×#107	C₂³	1	2	yes	—
48	C₁₉×#106	—	2	2	no	—
49	C₁₉×#106	—	2	4	no	—
50	C₁₉×#97	D₄	1	2	yes	—
51	C₄×C₈×C₁₉	C₂³	1	2	yes	—
52	C₂×Q₈×C₁₉	—	4	4	no	—
53	C₂²×C₄×C₁₉	—	4	2	no	—
54	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
55	M₁₆×C₁₉	Q₁₆	1	4	yes	—
56	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₁₉	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₁₉	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₁₉	—	2	4	no	—
62	C₄⋊C₄×C₁₉	—	2	2	no	—
63	C₄⋊C₄×C₁₉	—	2	2	no	—
64	C₄⋊C₄×C₁₉	—	4	2	no	—
65	C₄⋊C₄×C₁₉	—	4	4	no	—
66	C₄⋊C₄×C₁₉	—	4	4	no	—
67	C₄⋊C₄×C₁₉	—	4	4	no	—
68	C₄⋊C₄×C₁₉	D₈	1	2	yes	—
69	C₂²⋊C₄×C₁₉	—	4	2	no	—
70	C₂²⋊C₄×C₁₉	D₈	1	2	yes	—
71	C₄²×C₁₉	—	4	2	no	—
72	C₄²×C₁₉	C₂×D₄	1	2	yes	—
73	C₁₆×C₁₉	C₂×Q₈	1	2	yes	—
74	C₁₆×C₁₉	C₂×Q₈	1	2	yes	—
75	C₁₆×C₁₉	C₂×Q₈	1	4	yes	—
76	#33730	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
78	Q₈×C₁₉	—	16	4	no	—
79	C₂×C₄×C₁₉	—	4	2	no	—
80	C₂×C₄×C₁₉	—	4	2	no	—
81	C₂×C₄×C₁₉	—	4	4	no	—
82	C₂×C₄×C₁₉	—	16	2	no	—
83	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
84	C₂×C₄×C₁₉	C₂×D₈	1	4	yes	—
85	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
87	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
88	C₈×C₁₉	C₄⋊Q₈	1	2	yes	—
89	C₈×C₁₉	C₂×Q₁₆	1	4	yes	—
90	C₈×C₁₉	C₂²×Q₈	1	1	yes	characteristic
91	C₈×C₁₉	C₂²×Q₈	1	1	yes	characteristic
92	#3164	C₂×C₁₉	1	2	yes	—
93	#2255	C₂×C₁₉	1	4	yes	complemented
94	#2254	C₂×C₁₉	1	2	yes	complemented
95	#2253	C₂×C₁₉	1	2	yes	complemented
96	#2159	C₂×C₁₉	1	2	yes	—
97	#2158	C₂×C₁₉	1	1	yes	characteristic
98	C₂×#367	C₂×C₁₉	1	1	yes	characteristic
99	#2108	C₂×C₁₉	1	1	yes	characteristic
100	C₂²×C₁₉	—	2	2	no	—
101	C₂²×C₁₉	#500	1	2	yes	—
102	C₂²×C₁₉	C₂²×D₈	1	1	yes	characteristic
103	C₄×C₁₉	—	2	4	no	—
104	C₄×C₁₉	—	8	2	no	—
105	C₄×C₁₉	—	8	4	no	—
106	C₄×C₁₉	—	8	2	no	—
107	C₄×C₁₉	#500	1	2	yes	—
108	C₄×C₁₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
109	C₄×C₁₉	C₂²×Q₁₆	1	1	yes	characteristic
110	#544	—	2	2	no	—
111	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
112	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
113	#501	—	2	4	no	—
114	#500	C₂²×C₁₉	1	4	yes	—
115	#499	C₂²×C₁₉	1	4	yes	—
116	#428	—	2	2	no	—
117	#427	C₂²×C₁₉	1	2	yes	—
118	C₂×#107	C₂²×C₁₉	1	2	yes	—
119	#405	C₂²×C₁₉	1	1	yes	characteristic
120	#367	C₂²×C₁₉	1	2	yes	—
121	#367	C₂²×C₁₉	1	2	yes	—
122	#367	C₂²×C₁₉	1	1	yes	characteristic
123	#367	C₂²×C₁₉	1	1	yes	characteristic
124	#367	C₂²×C₁₉	1	4	yes	—
125	#366	C₂²×C₁₉	1	2	yes	—
126	#366	C₂²×C₁₉	1	4	yes	—
127	#346	C₂²×C₁₉	1	2	yes	—
128	C₄×C₁₆	C₂²×C₁₉	1	2	yes	—
129	C₂×C₁₉	—	2	2	no	—
130	C₂×C₁₉	C₂×#500	1	1	yes	characteristic
131	C₂×C₁₉	#3416	1	2	yes	—
132	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
133	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
134	C₄⋊Q₈	—	2	4	no	—
135	C₄⋊Q₈	—	4	2	no	—
136	#125	—	2	4	no	—
137	#125	—	4	2	no	—
138	C₂²⋊Q₈	—	4	4	no	—
139	#117	—	2	2	no	—
140	#117	D₄×C₁₉	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	2	2	no	—
142	M₃₂	Q₈×C₁₉	1	4	yes	complemented
143	C₂×C₁₆	Q₈×C₁₉	1	4	yes	complemented
144	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
145	C₂×C₁₆	C₂³×C₁₉	1	2	yes	—
146	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
147	#107	—	2	2	no	—
148	#107	—	2	4	no	—
149	#107	C₂³×C₁₉	1	2	yes	—
150	#107	C₂³×C₁₉	1	4	yes	—
151	#107	C₂³×C₁₉	1	2	yes	—
152	#106	—	2	2	no	—
153	#106	—	2	4	no	—
154	#97	D₄×C₁₉	1	2	yes	—
155	C₄×C₈	C₂³×C₁₉	1	2	yes	—
156	C₁₉	#33730	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×Q₈	—	4	4	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
160	M₁₆	Q₁₆×C₁₉	1	4	yes	—
161	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
162	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
163	C₂×C₈	C₂×Q₈×C₁₉	1	2	yes	—
164	C₂×C₈	C₂×Q₈×C₁₉	1	2	yes	—
165	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	4	4	no	—
171	C₄⋊C₄	—	4	4	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	D₈×C₁₉	1	2	yes	—
174	C₂²⋊C₄	—	4	2	no	—
175	C₂²⋊C₄	D₈×C₁₉	1	2	yes	—
176	C₄²	—	4	2	no	—
177	C₄²	C₂×D₄×C₁₉	1	2	yes	—
178	C₁₆	C₂×Q₈×C₁₉	1	2	yes	—
179	C₁₆	C₂×Q₈×C₁₉	1	2	yes	—
180	C₁₆	C₂×Q₈×C₁₉	1	4	yes	—
181	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
182	Q₈	—	16	4	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	16	2	no	—
187	C₂×C₄	C₁₉×C₄⋊Q₈	1	2	yes	—
188	C₂×C₄	C₂×D₈×C₁₉	1	4	yes	—
189	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
190	C₂×C₄	C₂×Q₁₆×C₁₉	1	2	yes	—
191	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
192	C₈	C₁₉×C₄⋊Q₈	1	2	yes	—
193	C₈	C₂×Q₁₆×C₁₉	1	4	yes	—
194	C₈	C₂²×Q₈×C₁₉	1	1	yes	characteristic
195	C₈	C₂²×Q₈×C₁₉	1	1	yes	characteristic
196	C₂²	—	2	2	no	—
197	C₂²	C₁₉×#500	1	2	yes	—
198	C₂²	C₂²×D₈×C₁₉	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	8	4	no	—
201	C₄	—	8	2	no	—
202	C₄	—	8	2	no	—
203	C₄	C₁₉×#500	1	2	yes	—
204	C₄	C₂×C₁₉×C₄⋊Q₈	1	1	yes	characteristic
205	C₄	C₂²×Q₁₆×C₁₉	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	C₂×C₁₉×#500	1	1	yes	characteristic
208	C₂	C₁₉×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₉
prime radical	C₂³×C₁₉	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₁₉×#117 ◁ C₁₉×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2255	C₂	semidirect
3. C₁₉×#2254	C₂	semidirect
4. C₁₉×#2253	C₂	semidirect
38. C₁₉×M₃₂	Q₈	semidirect
39. C₂×C₁₆×C₁₉	Q₈	semidirect
76. #33730	C₁₉	direct (N × H)
93. #2255	C₂×C₁₉	semidirect
94. #2254	C₂×C₁₉	semidirect
95. #2253	C₂×C₁₉	semidirect
142. M₃₂	Q₈×C₁₉	semidirect
143. C₂×C₁₆	Q₈×C₁₉	semidirect
156. C₁₉	#33730	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉×#3164	2
C₁₉×#2255	2
C₁₉×#2254	2
C₁₉×#2253	2
C₁₉×#2159	2
C₁₉×#2158	2
C₂×C₁₉×#367	2
C₁₉×#2108	2
#33730	19

Subgroups — C19×#33730 #18039689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#33730 #18039689