Subgroups — C₁₉×#33434 #18039393

Order 4864.

690 conjugacy classes of subgroups.

The subgroup lattice has 236 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	2	yes	maximal, complemented
2	C₁₉×#2931	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2169	C₂	1	4	yes	maximal, complemented
4	D₄×C₁₆×C₁₉	C₂	1	4	yes	maximal, complemented
5	C₁₉×#2150	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#2113	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×#131	C₂²	1	2	yes	—
9	C₁₉×#532	C₄	1	1	yes	characteristic
10	C₁₉×#504	—	2	8	no	—
11	C₂×C₁₉×M₃₂	C₂²	1	2	yes	—
12	C₂×C₁₉×M₃₂	C₂²	1	4	yes	—
13	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₁₉	C₂²	1	4	yes	—
15	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#436	C₄	1	4	yes	—
17	C₈×D₄×C₁₉	C₂²	1	4	yes	—
18	C₁₉×#423	C₄	1	1	yes	characteristic
19	C₁₉×#408	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
21	C₁₉×C₄⋊C₁₆	C₂²	1	4	yes	—
22	C₁₉×C₂²⋊C₁₆	C₂²	1	8	yes	—
23	C₄×C₁₆×C₁₉	C₂²	1	2	yes	—
24	C₄×C₁₆×C₁₉	C₂²	1	2	yes	—
25	C₂×#41×C₁₉	C₂×C₄	1	2	yes	—
26	C₁₉×#131	—	2	4	no	—
27	C₁₉×#131	—	2	4	no	—
28	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
32	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
33	C₁₉×C₄⋊Q₈	C₈	1	1	yes	characteristic
34	C₁₉×#127	C₈	1	1	yes	characteristic
35	C₁₉×#124	C₈	1	2	yes	—
36	C₁₉×#121	C₈	1	4	yes	—
37	C₄×D₄×C₁₉	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
39	C₁₉×M₃₂	—	2	4	no	—
40	C₁₉×M₃₂	—	2	8	no	—
41	C₂×C₁₆×C₁₉	—	2	2	no	—
42	C₂×C₁₆×C₁₉	—	2	8	no	—
43	C₂×C₁₆×C₁₉	D₄	1	4	yes	complemented
44	C₂×C₁₆×C₁₉	C₂³	1	2	yes	—
45	C₂×C₁₆×C₁₉	C₂³	1	4	yes	—
46	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
47	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
48	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
49	C₁₉×C₂²⋊C₈	C₂×C₄	1	8	yes	—
50	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₁₉	C₂³	1	2	yes	—
52	#41×C₁₉	—	2	4	no	—
53	#41×C₁₉	—	2	4	no	—
54	C₂×Q₈×C₁₉	C₂×C₈	1	2	yes	—
55	C₂×D₄×C₁₉	C₂×C₈	1	4	yes	—
56	C₂×D₄×C₁₉	C₂×C₈	1	4	yes	—
57	C₂×D₄×C₁₉	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₁₉	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
61	M₁₆×C₁₉	—	2	4	no	—
62	M₁₆×C₁₉	—	2	8	no	—
63	C₂×C₈×C₁₉	—	2	2	no	—
64	C₂×C₈×C₁₉	—	2	4	no	—
65	C₂×C₈×C₁₉	—	2	4	no	—
66	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₁₉	#41	1	2	yes	—
71	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₁₉	C₂×C₈	1	4	yes	—
73	C₂²⋊C₄×C₁₉	C₂×C₈	1	8	yes	—
74	C₄²×C₁₉	C₂×C₈	1	1	yes	characteristic
75	C₄²×C₁₉	C₂×C₈	1	1	yes	characteristic
76	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
77	C₁₆×C₁₉	—	2	4	no	—
78	C₁₆×C₁₉	—	2	2	no	—
79	C₁₆×C₁₉	C₂×D₄	1	2	yes	—
80	C₁₆×C₁₉	C₂×D₄	1	2	yes	—
81	#33434	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₁₉	C₂²×C₈	1	4	yes	—
83	C₂³×C₁₉	C₂²×C₈	1	1	yes	characteristic
84	Q₈×C₁₉	—	2	4	no	—
85	D₄×C₁₉	—	2	8	no	—
86	D₄×C₁₉	—	2	4	no	—
87	D₄×C₁₉	—	2	8	no	—
88	C₂×C₄×C₁₉	—	2	4	no	—
89	C₂×C₄×C₁₉	—	2	4	no	—
90	C₂×C₄×C₁₉	—	2	4	no	—
91	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
93	C₂×C₄×C₁₉	C₂²×C₈	1	4	yes	—
94	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
96	C₂×C₄×C₁₉	#131	1	2	yes	—
97	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
98	C₈×C₁₉	—	2	4	no	—
99	C₈×C₁₉	—	2	2	no	—
100	C₈×C₁₉	C₄×D₄	1	2	yes	—
101	C₈×C₁₉	C₂²×D₄	1	1	yes	characteristic
102	C₈×C₁₉	C₂×#41	1	1	yes	characteristic
103	C₂×#504	C₂×C₁₉	1	2	yes	complemented
104	#2931	C₂×C₁₉	1	1	yes	characteristic
105	#2169	C₂×C₁₉	1	4	yes	complemented
106	D₄×C₁₆	C₂×C₁₉	1	4	yes	complemented
107	#2150	C₂×C₁₉	1	1	yes	characteristic, complemented
108	#2113	C₂×C₁₉	1	2	yes	complemented
109	C₂×C₄×C₁₆	C₂×C₁₉	1	1	yes	characteristic, complemented
110	C₂²×C₁₉	—	2	2	no	—
111	C₂²×C₁₉	—	2	8	no	—
112	C₂²×C₁₉	—	2	4	no	—
113	C₂²×C₁₉	C₈×D₄	1	2	yes	—
114	C₂²×C₁₉	C₂³×C₈	1	1	yes	characteristic
115	C₄×C₁₉	—	2	2	no	—
116	C₄×C₁₉	—	2	4	no	—
117	C₄×C₁₉	C₈×D₄	1	2	yes	—
118	C₄×C₁₉	#504	1	4	yes	complemented
119	C₄×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
120	C₄×C₁₉	C₂×#131	1	1	yes	characteristic
121	C₂×#131	C₂²×C₁₉	1	2	yes	—
122	#532	C₄×C₁₉	1	1	yes	characteristic
123	#504	—	2	8	no	—
124	C₂×M₃₂	C₂²×C₁₉	1	2	yes	—
125	C₂×M₃₂	C₂²×C₁₉	1	4	yes	—
126	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
127	C₂²×C₁₆	C₂²×C₁₉	1	4	yes	—
128	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
129	#436	C₄×C₁₉	1	4	yes	—
130	C₈×D₄	C₂²×C₁₉	1	4	yes	—
131	#423	C₄×C₁₉	1	1	yes	characteristic
132	#408	C₄×C₁₉	1	2	yes	—
133	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
134	C₄⋊C₁₆	C₂²×C₁₉	1	4	yes	—
135	C₂²⋊C₁₆	C₂²×C₁₉	1	8	yes	—
136	C₄×C₁₆	C₂²×C₁₉	1	2	yes	—
137	C₄×C₁₆	C₂²×C₁₉	1	2	yes	—
138	C₂×C₁₉	—	2	2	no	—
139	C₂×C₁₉	—	4	4	no	—
140	C₂×C₁₉	C₂×C₈×D₄	1	1	yes	characteristic
141	C₂×C₁₉	C₂×#504	1	2	yes	—
142	C₂×#41	C₂×C₄×C₁₉	1	2	yes	—
143	#131	—	2	4	no	—
144	#131	—	2	4	no	—
145	C₂×M₁₆	C₂×C₄×C₁₉	1	2	yes	—
146	C₂×M₁₆	C₂×C₄×C₁₉	1	4	yes	—
147	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
148	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
149	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
150	C₄⋊Q₈	C₈×C₁₉	1	1	yes	characteristic
151	#127	C₈×C₁₉	1	1	yes	characteristic
152	#124	C₈×C₁₉	1	2	yes	—
153	#121	C₈×C₁₉	1	4	yes	—
154	C₄×D₄	C₂×C₄×C₁₉	1	4	yes	—
155	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
156	M₃₂	—	2	4	no	—
157	M₃₂	—	2	8	no	—
158	C₂×C₁₆	—	2	2	no	—
159	C₂×C₁₆	—	2	8	no	—
160	C₂×C₁₆	D₄×C₁₉	1	4	yes	complemented
161	C₂×C₁₆	C₂³×C₁₉	1	2	yes	—
162	C₂×C₁₆	C₂³×C₁₉	1	4	yes	—
163	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
164	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
165	C₄⋊C₈	C₂×C₄×C₁₉	1	4	yes	—
166	C₂²⋊C₈	C₂×C₄×C₁₉	1	8	yes	—
167	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
168	C₄×C₈	C₂³×C₁₉	1	2	yes	—
169	C₁₉	#33434	1	1	yes	characteristic, direct product factor, Sylow
170	#41	—	2	4	no	—
171	#41	—	2	4	no	—
172	C₂×Q₈	C₂×C₈×C₁₉	1	2	yes	—
173	C₂×D₄	C₂×C₈×C₁₉	1	4	yes	—
174	C₂×D₄	C₂×C₈×C₁₉	1	4	yes	—
175	C₂×D₄	C₂×C₈×C₁₉	1	2	yes	—
176	C₂²×C₄	C₂×C₈×C₁₉	1	2	yes	—
177	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
178	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
179	M₁₆	—	2	4	no	—
180	M₁₆	—	2	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
185	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
186	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
187	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
188	C₂×C₈	#41×C₁₉	1	2	yes	—
189	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
190	C₄⋊C₄	C₂×C₈×C₁₉	1	4	yes	—
191	C₂²⋊C₄	C₂×C₈×C₁₉	1	8	yes	—
192	C₄²	C₂×C₈×C₁₉	1	1	yes	characteristic
193	C₄²	C₂×C₈×C₁₉	1	1	yes	characteristic
194	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
195	C₁₆	—	2	4	no	—
196	C₁₆	—	2	2	no	—
197	C₁₆	C₂×D₄×C₁₉	1	2	yes	—
198	C₁₆	C₂×D₄×C₁₉	1	2	yes	—
199	C₂³	C₂²×C₈×C₁₉	1	4	yes	—
200	C₂³	C₂²×C₈×C₁₉	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	D₄	—	2	8	no	—
203	D₄	—	2	4	no	—
204	D₄	—	2	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
209	C₂×C₄	C₂²×C₈×C₁₉	1	2	yes	—
210	C₂×C₄	C₂²×C₈×C₁₉	1	4	yes	—
211	C₂×C₄	C₂²×C₈×C₁₉	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₈×C₁₉	1	2	yes	—
213	C₂×C₄	C₁₉×#131	1	2	yes	—
214	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
215	C₈	—	2	4	no	—
216	C₈	—	2	2	no	—
217	C₈	C₄×D₄×C₁₉	1	2	yes	—
218	C₈	C₂²×D₄×C₁₉	1	1	yes	characteristic
219	C₈	C₂×#41×C₁₉	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	8	no	—
223	C₂²	C₈×D₄×C₁₉	1	2	yes	—
224	C₂²	C₂³×C₈×C₁₉	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	—	2	2	no	—
227	C₄	C₈×D₄×C₁₉	1	2	yes	—
228	C₄	C₁₉×#504	1	4	yes	complemented
229	C₄	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
230	C₄	C₂×C₁₉×#131	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	—	4	4	no	—
233	C₂	C₂×C₈×D₄×C₁₉	1	1	yes	characteristic
234	C₂	C₂×C₁₉×#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆×C₁₉	C₂³
commutator	C₂²	C₂³×C₈×C₁₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₉
socle	C₂²×C₁₉	C₂³×C₈
prime radical	C₁₉×#532	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₉×#504	C₂	semidirect
3. C₁₉×#2169	C₂	semidirect
4. D₄×C₁₆×C₁₉	C₂	semidirect
5. C₁₉×#2150	C₂	semidirect
6. C₁₉×#2113	C₂	semidirect
7. C₂×C₄×C₁₆×C₁₉	C₂	semidirect
43. C₂×C₁₆×C₁₉	D₄	semidirect
81. #33434	C₁₉	direct (N × H)
103. C₂×#504	C₂×C₁₉	semidirect
105. #2169	C₂×C₁₉	semidirect
106. D₄×C₁₆	C₂×C₁₉	semidirect
107. #2150	C₂×C₁₉	semidirect
108. #2113	C₂×C₁₉	semidirect
109. C₂×C₄×C₁₆	C₂×C₁₉	semidirect
118. C₄×C₁₉	#504	semidirect
160. C₂×C₁₆	D₄×C₁₉	semidirect
169. C₁₉	#33434	direct (N × H)
228. C₄	C₁₉×#504	semidirect

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
C₁₉×#2931	2
C₁₉×#2169	2
D₄×C₁₆×C₁₉	2
C₁₉×#2150	2
C₁₉×#2113	2
C₂×C₄×C₁₆×C₁₉	2
#33434	19

Subgroups — C19×#33434 #18039393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#33434 #18039393