Subgroups — C₁₉×#2349 #14560206

Order 2432.

704 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₁₉	C₂	1	2	yes	maximal
2	C₂×C₄×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#389	C₂	1	2	yes	maximal, complemented
4	C₁₉×#385	C₂	1	4	yes	maximal, complemented
5	C₁₉×#382	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#380	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×#95	C₂	1	2	yes	maximal, complemented
9	C₂²×D₄×C₁₉	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₁₉	C₂²	1	4	yes	—
11	C₂³×C₄×C₁₉	C₂²	1	2	yes	—
12	C₄×D₄×C₁₉	—	2	8	no	—
13	C₄×D₄×C₁₉	C₄	1	8	yes	complemented
14	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
19	C₂×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
22	C₁₉×#95	C₂²	1	1	yes	characteristic
23	C₁₉×#95	C₂²	1	2	yes	—
24	C₁₉×#95	C₂²	1	4	yes	—
25	C₁₉×#95	C₂²	1	1	yes	characteristic
26	C₁₉×#95	C₂²	1	2	yes	—
27	C₂⁴×C₁₉	C₂³	1	2	yes	—
28	C₂×D₄×C₁₉	C₂×C₄	1	4	yes	—
29	C₂×D₄×C₁₉	D₄	1	4	yes	—
30	C₂×D₄×C₁₉	Q₈	1	4	yes	—
31	C₂²×C₄×C₁₉	—	2	8	no	—
32	C₂²×C₄×C₁₉	—	2	8	no	—
33	C₂²×C₄×C₁₉	—	4	4	no	—
34	C₂²×C₄×C₁₉	—	4	4	no	—
35	C₂²×C₄×C₁₉	C₂×C₄	1	8	yes	complemented
36	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
38	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₁₉	—	2	4	no	—
44	C₄⋊C₄×C₁₉	—	2	4	no	—
45	C₄⋊C₄×C₁₉	—	4	4	no	—
46	C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	complemented
47	C₂²⋊C₄×C₁₉	—	2	8	no	—
48	C₂²⋊C₄×C₁₉	C₂×C₄	1	8	yes	complemented
49	C₄²×C₁₉	—	2	4	no	—
50	C₄²×C₁₉	C₂×C₄	1	4	yes	complemented
51	C₂³×C₁₉	—	2	4	no	—
52	C₂³×C₁₉	—	2	4	no	—
53	C₂³×C₁₉	C₂²×C₄	1	4	yes	—
54	C₂³×C₁₉	C₂×D₄	1	4	yes	—
55	C₂³×C₁₉	C₂×Q₈	1	4	yes	—
56	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
57	D₄×C₁₉	C₄⋊C₄	1	16	yes	complemented
58	C₂×C₄×C₁₉	—	2	2	no	—
59	C₂×C₄×C₁₉	—	2	2	no	—
60	C₂×C₄×C₁₉	—	2	4	no	—
61	C₂×C₄×C₁₉	—	2	2	no	—
62	C₂×C₄×C₁₉	—	2	2	no	—
63	C₂×C₄×C₁₉	—	2	2	no	—
64	C₂×C₄×C₁₉	—	2	4	no	—
65	C₂×C₄×C₁₉	—	2	2	no	—
66	C₂×C₄×C₁₉	—	2	4	no	—
67	C₂×C₄×C₁₉	—	2	2	no	—
68	C₂×C₄×C₁₉	—	2	4	no	—
69	C₂×C₄×C₁₉	—	4	8	no	—
70	C₂×C₄×C₁₉	—	4	16	no	—
71	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₁₉	C₂²×C₄	1	4	yes	—
74	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
77	C₂×C₄×C₁₉	#41	1	4	yes	—
78	#2349	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂²×C₁₉	—	2	4	no	—
80	C₂²×C₁₉	—	2	4	no	—
81	C₂²×C₁₉	—	2	8	no	—
82	C₂²×C₁₉	—	2	8	no	—
83	C₂²×C₁₉	C₂×C₄⋊C₄	1	8	yes	—
84	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
85	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
86	C₂²×C₁₉	C₂²×Q₈	1	1	yes	characteristic
87	C₂²×C₁₉	C₂×#41	1	2	yes	—
88	C₂²×C₁₉	H(5,𝔽₂)	1	2	yes	—
89	C₄×C₁₉	—	2	2	no	—
90	C₄×C₁₉	—	2	2	no	—
91	C₄×C₁₉	—	2	4	no	—
92	C₄×C₁₉	—	2	4	no	—
93	C₄×C₁₉	—	4	2	no	—
94	C₄×C₁₉	—	4	4	no	—
95	C₄×C₁₉	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×C₄×D₄	C₂×C₁₉	1	2	yes	—
97	C₂×C₄×D₄	C₂×C₁₉	1	1	yes	characteristic
98	#389	C₂×C₁₉	1	2	yes	complemented
99	#385	C₂×C₁₉	1	4	yes	complemented
100	#382	C₂×C₁₉	1	1	yes	characteristic, complemented
101	#380	C₂×C₁₉	1	2	yes	complemented
102	C₄×C₄⋊C₄	C₂×C₁₉	1	1	yes	characteristic, complemented
103	C₂×#95	C₂×C₁₉	1	2	yes	complemented
104	C₂×C₁₉	—	2	8	no	—
105	C₂×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₁₉	C₄×#41	1	1	yes	characteristic
107	C₂×C₁₉	#518	1	1	yes	characteristic
108	C₂×C₁₉	#546	1	2	yes	—
109	C₂×C₁₉	#554	1	2	yes	—
110	C₂²×D₄	C₂²×C₁₉	1	1	yes	characteristic
111	C₂³×C₄	C₂²×C₁₉	1	4	yes	—
112	C₂³×C₄	C₂²×C₁₉	1	2	yes	—
113	C₄×D₄	—	2	8	no	—
114	C₄×D₄	C₄×C₁₉	1	8	yes	complemented
115	C₂×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
116	C₂×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
117	C₂×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
118	C₂×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
119	C₂×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
120	C₂×C₂²⋊C₄	C₂²×C₁₉	1	2	yes	—
121	C₂×C₄²	C₂²×C₁₉	1	2	yes	—
122	C₂×C₄²	C₂²×C₁₉	1	1	yes	characteristic
123	#95	C₂²×C₁₉	1	1	yes	characteristic
124	#95	C₂²×C₁₉	1	2	yes	—
125	#95	C₂²×C₁₉	1	4	yes	—
126	#95	C₂²×C₁₉	1	1	yes	characteristic
127	#95	C₂²×C₁₉	1	2	yes	—
128	C₁₉	#2349	1	1	yes	characteristic, direct product factor, Sylow
129	C₂⁴	C₂³×C₁₉	1	2	yes	—
130	C₂×D₄	C₂×C₄×C₁₉	1	4	yes	—
131	C₂×D₄	D₄×C₁₉	1	4	yes	—
132	C₂×D₄	Q₈×C₁₉	1	4	yes	—
133	C₂²×C₄	—	2	8	no	—
134	C₂²×C₄	—	2	8	no	—
135	C₂²×C₄	—	4	4	no	—
136	C₂²×C₄	—	4	4	no	—
137	C₂²×C₄	C₂×C₄×C₁₉	1	8	yes	complemented
138	C₂²×C₄	C₂³×C₁₉	1	4	yes	—
139	C₂²×C₄	C₂³×C₁₉	1	2	yes	—
140	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
141	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
142	C₂²×C₄	C₂³×C₁₉	1	2	yes	—
143	C₂²×C₄	C₂³×C₁₉	1	2	yes	—
144	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	complemented
149	C₂²⋊C₄	—	2	8	no	—
150	C₂²⋊C₄	C₂×C₄×C₁₉	1	8	yes	complemented
151	C₄²	—	2	4	no	—
152	C₄²	C₂×C₄×C₁₉	1	4	yes	complemented
153	C₂³	—	2	4	no	—
154	C₂³	—	2	4	no	—
155	C₂³	C₂²×C₄×C₁₉	1	4	yes	—
156	C₂³	C₂×D₄×C₁₉	1	4	yes	—
157	C₂³	C₂×Q₈×C₁₉	1	4	yes	—
158	C₂³	C₂⁴×C₁₉	1	1	yes	characteristic
159	D₄	C₄⋊C₄×C₁₉	1	16	yes	complemented
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	8	no	—
172	C₂×C₄	—	4	16	no	—
173	C₂×C₄	C₂²×C₄×C₁₉	1	2	yes	—
174	C₂×C₄	C₂²×C₄×C₁₉	1	2	yes	—
175	C₂×C₄	C₂²×C₄×C₁₉	1	4	yes	—
176	C₂×C₄	C₂²×C₄×C₁₉	1	2	yes	—
177	C₂×C₄	C₂×D₄×C₁₉	1	2	yes	—
178	C₂×C₄	C₂×Q₈×C₁₉	1	2	yes	—
179	C₂×C₄	#41×C₁₉	1	4	yes	—
180	C₂²	—	2	4	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	2	8	no	—
183	C₂²	—	2	8	no	—
184	C₂²	C₂×C₄⋊C₄×C₁₉	1	8	yes	—
185	C₂²	C₂³×C₄×C₁₉	1	1	yes	characteristic
186	C₂²	C₂²×D₄×C₁₉	1	1	yes	characteristic
187	C₂²	C₂²×Q₈×C₁₉	1	1	yes	characteristic
188	C₂²	C₂×#41×C₁₉	1	2	yes	—
189	C₂²	C₁₉×H(5,𝔽₂)	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	2	4	no	—
193	C₄	—	2	4	no	—
194	C₄	—	4	4	no	—
195	C₄	—	4	2	no	—
196	C₄	C₂×C₄⋊C₄×C₁₉	1	4	yes	—
197	C₂	—	2	8	no	—
198	C₂	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
199	C₂	C₄×#41×C₁₉	1	1	yes	characteristic
200	C₂	C₁₉×#518	1	1	yes	characteristic
201	C₂	C₁₉×#546	1	2	yes	—
202	C₂	C₁₉×#554	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₉
Frattini, cosocle	C₂³	C₂⁴×C₁₉
prime radical	C₂²×D₄×C₁₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#389	C₂	semidirect
4. C₁₉×#385	C₂	semidirect
5. C₁₉×#382	C₂	semidirect
6. C₁₉×#380	C₂	semidirect
7. C₄×C₄⋊C₄×C₁₉	C₂	semidirect
8. C₂×C₁₉×#95	C₂	semidirect
13. C₄×D₄×C₁₉	C₄	semidirect
35. C₂²×C₄×C₁₉	C₂×C₄	semidirect
46. C₄⋊C₄×C₁₉	C₂×C₄	semidirect
48. C₂²⋊C₄×C₁₉	C₂×C₄	semidirect
50. C₄²×C₁₉	C₂×C₄	semidirect
57. D₄×C₁₉	C₄⋊C₄	semidirect
78. #2349	C₁₉	direct (N × H)
98. #389	C₂×C₁₉	semidirect
99. #385	C₂×C₁₉	semidirect
100. #382	C₂×C₁₉	semidirect
101. #380	C₂×C₁₉	semidirect
102. C₄×C₄⋊C₄	C₂×C₁₉	semidirect
103. C₂×#95	C₂×C₁₉	semidirect
114. C₄×D₄	C₄×C₁₉	semidirect
128. C₁₉	#2349	direct (N × H)
137. C₂²×C₄	C₂×C₄×C₁₉	semidirect
148. C₄⋊C₄	C₂×C₄×C₁₉	semidirect
150. C₂²⋊C₄	C₂×C₄×C₁₉	semidirect
152. C₄²	C₂×C₄×C₁₉	semidirect
159. D₄	C₄⋊C₄×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₁₉	2
C₁₉×#389	2
C₁₉×#385	2
C₁₉×#382	2
C₁₉×#380	2
C₄×C₄⋊C₄×C₁₉	2
C₂×C₁₉×#95	2
#2349	19

Subgroups — C19×#2349 #14560206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#2349 #14560206