Subgroups — C₁₉×#20331 #18026290

Order 4864.

1470 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	C₂×C₁₉×#405	C₂	1	4	yes	maximal
2	C₂×C₄×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₉×#97	C₂	1	1	yes	maximal, characteristic
4	C₄×C₁₉×#117	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1726	C₂	1	4	yes	maximal, complemented
6	C₄×C₁₉×#97	C₂	1	4	yes	maximal, complemented
7	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₉	C₂²	1	2	yes	—
9	C₂×C₁₉×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
11	C₁₉×#405	C₄	1	32	yes	complemented
12	C₄×M₁₆×C₁₉	C₄	1	8	yes	complemented
13	C₂×C₁₉×#97	C₄	1	8	yes	complemented
14	C₂×C₁₉×#97	C₂²	1	2	yes	—
15	C₂×C₁₉×#97	C₂²	1	2	yes	—
16	C₂×C₁₉×#97	C₂²	1	2	yes	—
17	C₂×C₁₉×#97	C₂²	1	8	yes	—
18	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₉	C₂²	1	8	yes	—
20	C₄×C₄⋊C₄×C₁₉	C₄	1	4	yes	—
21	C₄×C₂²⋊C₄×C₁₉	C₄	1	4	yes	—
22	C₁₉×#376	C₂²	1	2	yes	—
23	C₄³×C₁₉	C₂²	1	2	yes	—
24	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₁₉	C₂×C₄	1	8	yes	—
27	C₂×M₁₆×C₁₉	C₂×C₄	1	16	yes	—
28	C₂²×C₈×C₁₉	C₂×C₄	1	8	yes	—
29	C₂²×C₈×C₁₉	C₂×C₄	1	16	yes	—
30	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
31	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
32	C₁₉×#117	C₂×C₄	1	16	yes	—
33	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₁₉	C₂×C₄	1	8	yes	—
36	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
41	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
42	C₁₉×#97	C₂×C₄	1	8	yes	—
43	C₁₉×#97	C₂×C₄	1	8	yes	—
44	C₁₉×#97	C₂×C₄	1	8	yes	—
45	C₁₉×#97	C₂×C₄	1	32	yes	complemented
46	C₄×C₈×C₁₉	C₂×C₄	1	8	yes	—
47	C₄×C₈×C₁₉	C₂×C₄	1	32	yes	complemented
48	C₁₉×#95	C₂×C₄	1	8	yes	—
49	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	—	4	4	no	—
51	C₂²×C₄×C₁₉	C₄²	1	8	yes	—
52	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₁₉	C₂²×C₄	1	8	yes	—
54	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₁₉	C₂²×C₄	1	8	yes	—
59	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
60	M₁₆×C₁₉	C₄²	1	32	yes	complemented
61	C₂×C₈×C₁₉	C₄²	1	32	yes	complemented
62	C₂×C₈×C₁₉	C₂²×C₄	1	8	yes	—
63	C₂×C₈×C₁₉	C₂²×C₄	1	8	yes	—
64	C₂×C₈×C₁₉	C₂²×C₄	1	8	yes	—
65	C₂×C₈×C₁₉	C₂²×C₄	1	8	yes	—
66	C₂×C₈×C₁₉	C₂²×C₄	1	16	yes	—
67	C₄⋊C₄×C₁₉	C₄²	1	16	yes	—
68	C₂²⋊C₄×C₁₉	C₄²	1	16	yes	—
69	C₄²×C₁₉	—	4	8	no	—
70	C₄²×C₁₉	C₄²	1	8	yes	—
71	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
72	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
73	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
74	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
75	C₄²×C₁₉	C₂²×C₄	1	16	yes	—
76	#20331	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₁₉	—	2	4	no	—
78	C₂³×C₁₉	C₂×C₄²	1	4	yes	—
79	C₂³×C₁₉	C₂×C₄²	1	2	yes	—
80	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₉	—	2	4	no	—
82	C₂×C₄×C₁₉	—	2	16	no	—
83	C₂×C₄×C₁₉	—	2	4	no	—
84	C₂×C₄×C₁₉	—	4	8	no	—
85	C₂×C₄×C₁₉	C₂×C₄²	1	4	yes	—
86	C₂×C₄×C₁₉	C₂×C₄²	1	4	yes	—
87	C₂×C₄×C₁₉	C₂×C₄²	1	16	yes	—
88	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
89	C₂×C₄×C₁₉	C₂×C₄²	1	4	yes	—
90	C₂×C₄×C₁₉	C₂×C₄²	1	8	yes	—
91	C₂×C₄×C₁₉	C₂×C₄²	1	8	yes	—
92	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₉	C₂³×C₄	1	4	yes	—
95	C₈×C₁₉	C₂×C₄²	1	16	yes	—
96	C₈×C₁₉	C₂×C₄²	1	16	yes	—
97	C₂×#405	C₂×C₁₉	1	4	yes	—
98	C₂×C₄×M₁₆	C₂×C₁₉	1	1	yes	characteristic
99	C₂²×#97	C₂×C₁₉	1	1	yes	characteristic
100	C₄×#117	C₂×C₁₉	1	1	yes	characteristic
101	#1726	C₂×C₁₉	1	4	yes	complemented
102	C₄×#97	C₂×C₁₉	1	4	yes	complemented
103	C₂²×C₁₉	—	2	4	no	—
104	C₂²×C₁₉	—	2	8	no	—
105	C₂²×C₁₉	C₄³	1	4	yes	—
106	C₂²×C₁₉	C₂²×C₄²	1	1	yes	characteristic
107	C₂²×C₁₉	C₂²×C₄²	1	2	yes	—
108	C₂²×C₁₉	#568	1	4	yes	—
109	C₄×C₁₉	—	2	4	no	—
110	C₄×C₁₉	—	2	4	no	—
111	C₄×C₁₉	—	2	16	no	—
112	C₄×C₁₉	C₄³	1	4	yes	—
113	C₄×C₁₉	C₂²×C₄²	1	2	yes	—
114	C₄×C₁₉	C₂²×C₄²	1	2	yes	—
115	C₂²×M₁₆	C₂²×C₁₉	1	2	yes	—
116	C₂³×C₈	C₂²×C₁₉	1	2	yes	—
117	C₂×#117	C₂²×C₁₉	1	2	yes	—
118	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
119	#405	C₄×C₁₉	1	32	yes	complemented
120	C₄×M₁₆	C₄×C₁₉	1	8	yes	complemented
121	C₂×#97	C₄×C₁₉	1	8	yes	complemented
122	C₂×#97	C₂²×C₁₉	1	2	yes	—
123	C₂×#97	C₂²×C₁₉	1	2	yes	—
124	C₂×#97	C₂²×C₁₉	1	2	yes	—
125	C₂×#97	C₂²×C₁₉	1	8	yes	—
126	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
127	C₂×C₄×C₈	C₂²×C₁₉	1	8	yes	—
128	C₄×C₄⋊C₄	C₄×C₁₉	1	4	yes	—
129	C₄×C₂²⋊C₄	C₄×C₁₉	1	4	yes	—
130	#376	C₂²×C₁₉	1	2	yes	—
131	C₄³	C₂²×C₁₉	1	2	yes	—
132	C₂×C₁₉	—	2	4	no	—
133	C₂×C₁₉	C₂×C₄³	1	1	yes	characteristic
134	C₂×C₁₉	#2873	1	2	yes	—
135	C₂×C₁₉	#2875	1	4	yes	—
136	C₂³×C₄	C₂×C₄×C₁₉	1	2	yes	—
137	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
138	C₂×M₁₆	C₂×C₄×C₁₉	1	8	yes	—
139	C₂×M₁₆	C₂×C₄×C₁₉	1	16	yes	—
140	C₂²×C₈	C₂×C₄×C₁₉	1	8	yes	—
141	C₂²×C₈	C₂×C₄×C₁₉	1	16	yes	—
142	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
143	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
144	#117	C₂×C₄×C₁₉	1	16	yes	—
145	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
146	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
147	C₂×C₄²	C₂×C₄×C₁₉	1	8	yes	—
148	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
149	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
150	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
151	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
152	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
153	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
154	#97	C₂×C₄×C₁₉	1	8	yes	—
155	#97	C₂×C₄×C₁₉	1	8	yes	—
156	#97	C₂×C₄×C₁₉	1	8	yes	—
157	#97	C₂×C₄×C₁₉	1	32	yes	complemented
158	C₄×C₈	C₂×C₄×C₁₉	1	8	yes	—
159	C₄×C₈	C₂×C₄×C₁₉	1	32	yes	complemented
160	#95	C₂×C₄×C₁₉	1	8	yes	—
161	C₁₉	#20331	1	1	yes	characteristic, direct product factor, Sylow
162	C₂⁴	C₂²×C₄×C₁₉	1	1	yes	characteristic
163	C₂²×C₄	—	4	4	no	—
164	C₂²×C₄	C₄²×C₁₉	1	8	yes	—
165	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
166	C₂²×C₄	C₂²×C₄×C₁₉	1	8	yes	—
167	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
168	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
169	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
170	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
171	C₂²×C₄	C₂²×C₄×C₁₉	1	8	yes	—
172	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
173	M₁₆	C₄²×C₁₉	1	32	yes	complemented
174	C₂×C₈	C₄²×C₁₉	1	32	yes	complemented
175	C₂×C₈	C₂²×C₄×C₁₉	1	8	yes	—
176	C₂×C₈	C₂²×C₄×C₁₉	1	8	yes	—
177	C₂×C₈	C₂²×C₄×C₁₉	1	16	yes	—
178	C₂×C₈	C₂²×C₄×C₁₉	1	8	yes	—
179	C₂×C₈	C₂²×C₄×C₁₉	1	8	yes	—
180	C₄⋊C₄	C₄²×C₁₉	1	16	yes	—
181	C₂²⋊C₄	C₄²×C₁₉	1	16	yes	—
182	C₄²	—	4	8	no	—
183	C₄²	C₄²×C₁₉	1	8	yes	—
184	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
185	C₄²	C₂²×C₄×C₁₉	1	16	yes	—
186	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
187	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
188	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
189	C₂³	—	2	4	no	—
190	C₂³	C₂×C₄²×C₁₉	1	4	yes	—
191	C₂³	C₂×C₄²×C₁₉	1	2	yes	—
192	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	16	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	4	8	no	—
197	C₂×C₄	C₂×C₄²×C₁₉	1	4	yes	—
198	C₂×C₄	C₂×C₄²×C₁₉	1	4	yes	—
199	C₂×C₄	C₂×C₄²×C₁₉	1	16	yes	—
200	C₂×C₄	C₂×C₄²×C₁₉	1	2	yes	—
201	C₂×C₄	C₂×C₄²×C₁₉	1	4	yes	—
202	C₂×C₄	C₂×C₄²×C₁₉	1	8	yes	—
203	C₂×C₄	C₂×C₄²×C₁₉	1	8	yes	—
204	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
205	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
206	C₂×C₄	C₂³×C₄×C₁₉	1	4	yes	—
207	C₈	C₂×C₄²×C₁₉	1	16	yes	—
208	C₈	C₂×C₄²×C₁₉	1	16	yes	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	8	no	—
211	C₂²	C₄³×C₁₉	1	4	yes	—
212	C₂²	C₂²×C₄²×C₁₉	1	1	yes	characteristic
213	C₂²	C₂²×C₄²×C₁₉	1	2	yes	—
214	C₂²	C₁₉×#568	1	4	yes	—
215	C₄	—	2	4	no	—
216	C₄	—	2	4	no	—
217	C₄	—	2	16	no	—
218	C₄	C₄³×C₁₉	1	4	yes	—
219	C₄	C₂²×C₄²×C₁₉	1	2	yes	—
220	C₄	C₂²×C₄²×C₁₉	1	2	yes	—
221	C₂	—	2	4	no	—
222	C₂	C₂×C₄³×C₁₉	1	1	yes	characteristic
223	C₂	C₁₉×#2873	1	2	yes	—
224	C₂	C₁₉×#2875	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₂⁴
commutator	C₂	C₂×C₄³×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle	C₂³×C₁₉	C₂³×C₄
prime radical	C₂⁴×C₁₉	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₁₉×#1726	C₂	semidirect
6. C₄×C₁₉×#97	C₂	semidirect
11. C₁₉×#405	C₄	semidirect
12. C₄×M₁₆×C₁₉	C₄	semidirect
13. C₂×C₁₉×#97	C₄	semidirect
45. C₁₉×#97	C₂×C₄	semidirect
47. C₄×C₈×C₁₉	C₂×C₄	semidirect
60. M₁₆×C₁₉	C₄²	semidirect
61. C₂×C₈×C₁₉	C₄²	semidirect
76. #20331	C₁₉	direct (N × H)
101. #1726	C₂×C₁₉	semidirect
102. C₄×#97	C₂×C₁₉	semidirect
119. #405	C₄×C₁₉	semidirect
120. C₄×M₁₆	C₄×C₁₉	semidirect
121. C₂×#97	C₄×C₁₉	semidirect
157. #97	C₂×C₄×C₁₉	semidirect
159. C₄×C₈	C₂×C₄×C₁₉	semidirect
161. C₁₉	#20331	direct (N × H)
173. M₁₆	C₄²×C₁₉	semidirect
174. C₂×C₈	C₄²×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#405	2
C₂×C₄×M₁₆×C₁₉	2
C₂²×C₁₉×#97	2
C₄×C₁₉×#117	2
C₁₉×#1726	2
C₄×C₁₉×#97	2
#20331	19

Subgroups — C19×#20331 #18026290

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#20331 #18026290