Subgroups — C₁₉×#2992 #14560849

Order 2432.

356 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#529	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#446	C₂	1	2	yes	maximal, complemented
3	C₁₉×#435	C₂	1	2	yes	maximal, complemented
4	C₁₉×#435	C₂	1	2	yes	maximal, complemented
5	C₈×D₄×C₁₉	C₂	1	2	yes	maximal, complemented
6	C₁₉×#433	C₂	1	2	yes	maximal, complemented
7	C₁₉×#432	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#424	C₂	1	2	yes	maximal, complemented
9	C₂×C₁₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×M₁₆×C₁₉	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₉	C₂²	1	2	yes	—
14	C₁₉×#126	C₄	1	2	yes	—
15	C₁₉×#125	C₄	1	1	yes	characteristic
16	C₁₉×#124	C₄	1	1	yes	characteristic
17	C₁₉×#123	C₄	1	2	yes	—
18	C₁₉×C₂²⋊Q₈	C₄	1	1	yes	characteristic
19	C₁₉×#121	C₄	1	1	yes	characteristic
20	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
21	C₄×D₄×C₁₉	C₂²	1	2	yes	—
22	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#117	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
25	C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
26	C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
27	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
29	C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
30	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₁₉×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₁₉×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₁₉×C₂²⋊C₈	C₂²	1	2	yes	—
34	C₁₉×#97	C₂²	1	2	yes	—
35	C₁₉×#97	C₂²	1	2	yes	—
36	C₄×C₈×C₁₉	C₂²	1	2	yes	—
37	C₂×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄×C₁₉	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
44	M₁₆×C₁₉	—	4	2	no	—
45	C₂×C₈×C₁₉	—	4	1	no	—
46	C₂×C₈×C₁₉	—	4	2	no	—
47	C₂×C₈×C₁₉	—	4	1	no	—
48	C₂×C₈×C₁₉	C₂³	1	2	yes	—
49	C₂×C₈×C₁₉	C₂³	1	2	yes	—
50	C₂×C₈×C₁₉	C₂³	1	2	yes	—
51	C₂×C₈×C₁₉	C₂³	1	2	yes	—
52	C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
53	C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
54	C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
55	C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
56	C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
58	C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
59	C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
60	C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
61	C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
62	C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
63	C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₁₉	C₂³	1	1	yes	characteristic
66	C₄²×C₁₉	C₂³	1	1	yes	characteristic
67	C₄²×C₁₉	C₂³	1	2	yes	—
68	C₂³×C₁₉	C₂²×C₄	1	2	yes	—
69	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
70	Q₈×C₁₉	—	4	1	no	—
71	D₄×C₁₉	—	2	4	no	—
72	D₄×C₁₉	—	4	1	no	—
73	C₂×C₄×C₁₉	—	2	2	no	—
74	C₂×C₄×C₁₉	—	2	1	no	—
75	C₂×C₄×C₁₉	—	2	2	no	—
76	C₂×C₄×C₁₉	—	2	1	no	—
77	C₂×C₄×C₁₉	—	2	2	no	—
78	C₂×C₄×C₁₉	—	4	1	no	—
79	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
88	C₈×C₁₉	—	2	2	no	—
89	C₈×C₁₉	—	2	2	no	—
90	C₈×C₁₉	—	2	2	no	—
91	C₈×C₁₉	—	2	2	no	—
92	#2992	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂²×C₁₉	—	2	1	no	—
94	C₂²×C₁₉	—	2	1	no	—
95	C₂²×C₁₉	—	2	2	no	—
96	C₂²×C₁₉	—	2	2	no	—
97	C₂²×C₁₉	—	2	2	no	—
98	C₂²×C₁₉	#131	1	2	yes	—
99	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
100	C₄×C₁₉	—	2	1	no	—
101	C₄×C₁₉	—	2	2	no	—
102	C₄×C₁₉	—	2	1	no	—
103	C₄×C₁₉	—	2	1	no	—
104	C₄×C₁₉	—	2	2	no	—
105	C₄×C₁₉	—	2	1	no	—
106	C₄×C₁₉	—	2	1	no	—
107	C₄×C₁₉	—	2	2	no	—
108	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
109	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
110	#529	C₂×C₁₉	1	1	yes	characteristic
111	#446	C₂×C₁₉	1	2	yes	complemented
112	#435	C₂×C₁₉	1	2	yes	complemented
113	#435	C₂×C₁₉	1	2	yes	complemented
114	C₈×D₄	C₂×C₁₉	1	2	yes	complemented
115	#433	C₂×C₁₉	1	2	yes	complemented
116	#432	C₂×C₁₉	1	1	yes	characteristic, complemented
117	#424	C₂×C₁₉	1	2	yes	complemented
118	C₂×C₄⋊C₈	C₂×C₁₉	1	1	yes	characteristic, complemented
119	C₂×C₁₉	—	2	2	no	—
120	C₂×C₁₉	—	4	2	no	—
121	C₂×C₁₉	#520	1	1	yes	characteristic
122	C₂×C₁₉	C₂×#131	1	1	yes	characteristic
123	C₂×C₁₉	#568	1	1	yes	characteristic
124	C₂×M₁₆	C₂²×C₁₉	1	2	yes	—
125	C₂²×C₈	C₂²×C₁₉	1	1	yes	characteristic
126	C₂²×C₈	C₂²×C₁₉	1	1	yes	characteristic
127	C₂²×C₈	C₂²×C₁₉	1	2	yes	—
128	#126	C₄×C₁₉	1	2	yes	—
129	#125	C₄×C₁₉	1	1	yes	characteristic
130	#124	C₄×C₁₉	1	1	yes	characteristic
131	#123	C₄×C₁₉	1	2	yes	—
132	C₂²⋊Q₈	C₄×C₁₉	1	1	yes	characteristic
133	#121	C₄×C₁₉	1	1	yes	characteristic
134	C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
135	C₄×D₄	C₂²×C₁₉	1	2	yes	—
136	C₄×D₄	C₂²×C₁₉	1	1	yes	characteristic
137	#117	C₂²×C₁₉	1	2	yes	—
138	C₂×C₄²	C₂²×C₁₉	1	1	yes	characteristic
139	C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
140	C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
141	C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
142	C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
143	C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
144	C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
145	C₂²⋊C₈	C₂²×C₁₉	1	2	yes	—
146	C₂²⋊C₈	C₂²×C₁₉	1	2	yes	—
147	C₂²⋊C₈	C₂²×C₁₉	1	2	yes	—
148	#97	C₂²×C₁₉	1	2	yes	—
149	#97	C₂²×C₁₉	1	2	yes	—
150	C₄×C₈	C₂²×C₁₉	1	2	yes	—
151	C₁₉	#2992	1	1	yes	characteristic, direct product factor, Sylow
152	C₂×Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
153	C₂×D₄	C₂×C₄×C₁₉	1	2	yes	—
154	C₂×D₄	C₂×C₄×C₁₉	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
157	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₁₉	1	2	yes	—
159	M₁₆	—	4	2	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	C₂³×C₁₉	1	2	yes	—
164	C₂×C₈	C₂³×C₁₉	1	2	yes	—
165	C₂×C₈	C₂³×C₁₉	1	2	yes	—
166	C₂×C₈	C₂³×C₁₉	1	2	yes	—
167	C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
168	C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
169	C₄⋊C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
170	C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
171	C₄⋊C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
172	C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
173	C₂²⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
174	C₂²⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
175	C₂²⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
176	C₂²⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
177	C₂²⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
178	C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
179	C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
180	C₄²	C₂³×C₁₉	1	2	yes	—
181	C₄²	C₂³×C₁₉	1	1	yes	characteristic
182	C₄²	C₂³×C₁₉	1	1	yes	characteristic
183	C₂³	C₂²×C₄×C₁₉	1	2	yes	—
184	C₂³	C₂²×C₄×C₁₉	1	1	yes	characteristic
185	Q₈	—	4	1	no	—
186	D₄	—	2	4	no	—
187	D₄	—	4	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
195	C₂×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
196	C₂×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
197	C₂×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
198	C₂×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
199	C₂×C₄	C₂²×C₄×C₁₉	1	2	yes	—
200	C₂×C₄	C₂²×C₄×C₁₉	1	2	yes	—
201	C₂×C₄	C₂²×C₄×C₁₉	1	2	yes	—
202	C₂×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	C₁₉×#131	1	2	yes	—
213	C₂²	C₂³×C₄×C₁₉	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	2	no	—
222	C₄	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
223	C₄	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	4	2	no	—
226	C₂	C₁₉×#520	1	1	yes	characteristic
227	C₂	C₂×C₁₉×#131	1	1	yes	characteristic
228	C₂	C₁₉×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₉
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₉
socle	C₂²×C₁₉	C₂³×C₄
prime radical	C₁₉×#121	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#446	C₂	semidirect
3. C₁₉×#435	C₂	semidirect
4. C₁₉×#435	C₂	semidirect
5. C₈×D₄×C₁₉	C₂	semidirect
6. C₁₉×#433	C₂	semidirect
7. C₁₉×#432	C₂	semidirect
8. C₁₉×#424	C₂	semidirect
9. C₂×C₁₉×C₄⋊C₈	C₂	semidirect
92. #2992	C₁₉	direct (N × H)
111. #446	C₂×C₁₉	semidirect
112. #435	C₂×C₁₉	semidirect
113. #435	C₂×C₁₉	semidirect
114. C₈×D₄	C₂×C₁₉	semidirect
115. #433	C₂×C₁₉	semidirect
116. #432	C₂×C₁₉	semidirect
117. #424	C₂×C₁₉	semidirect
118. C₂×C₄⋊C₈	C₂×C₁₉	semidirect
151. C₁₉	#2992	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×#529	2
C₁₉×#446	2
C₁₉×#435	2
C₈×D₄×C₁₉	2
C₁₉×#433	2
C₁₉×#432	2
C₁₉×#424	2
C₂×C₁₉×C₄⋊C₈	2
#2992	19

Subgroups — C19×#2992 #14560849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#2992 #14560849