Subgroups — C₁₉×#60100 #18066059

Order 4864.

2666 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#568	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₉×#131	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#3428	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#2906	C₂	1	16	yes	maximal, complemented
5	C₁₉×#2905	C₂	1	6	yes	maximal, complemented
6	C₂×C₁₉×#424	C₂	1	6	yes	maximal, complemented
7	C₂²×#41×C₁₉	C₂²	1	1	yes	characteristic
8	C₁₉×#568	—	2	8	no	—
9	C₂×C₁₉×#131	—	2	8	no	—
10	C₂×C₁₉×#131	C₂²	1	8	yes	—
11	C₂×C₁₉×#131	C₂²	1	8	yes	—
12	C₂²×M₁₆×C₁₉	C₂²	1	3	yes	—
13	C₂²×M₁₆×C₁₉	C₂²	1	6	yes	—
14	C₂³×C₈×C₁₉	C₂²	1	3	yes	—
15	C₁₉×#520	C₄	1	8	yes	—
16	C₁₉×#519	C₄	1	2	yes	—
17	C₁₉×#518	C₄	1	6	yes	—
18	C₄×#41×C₁₉	C₂²	1	8	yes	—
19	C₂×C₁₉×#117	C₂²	1	6	yes	—
20	C₁₉×#424	C₂²	1	4	yes	—
21	C₁₉×#424	C₂²	1	12	yes	—
22	C₁₉×#424	C₂²	1	24	yes	complemented
23	C₁₉×#424	C₂²	1	24	yes	complemented
24	C₁₉×#423	C₂²	1	24	yes	complemented
25	C₂×C₁₉×C₄⋊C₈	C₂²	1	24	yes	complemented
26	C₂×#41×C₁₉	C₂×C₄	1	4	yes	—
27	C₂×#41×C₁₉	D₄	1	8	yes	—
28	C₂×#41×C₁₉	Q₈	1	8	yes	—
29	C₂×#41×C₁₉	C₂³	1	4	yes	—
30	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×D₄×C₁₉	C₂×C₄	1	3	yes	—
32	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
33	C₁₉×#131	—	2	32	no	—
34	C₁₉×#131	—	2	32	no	—
35	C₂×M₁₆×C₁₉	—	2	12	no	—
36	C₂×M₁₆×C₁₉	—	2	24	no	—
37	C₂×M₁₆×C₁₉	C₂³	1	12	yes	—
38	C₂×M₁₆×C₁₉	C₂³	1	12	yes	—
39	C₂×M₁₆×C₁₉	C₂³	1	4	yes	—
40	C₂²×C₈×C₁₉	—	2	12	no	—
41	C₂²×C₈×C₁₉	C₂³	1	12	yes	—
42	C₂²×C₈×C₁₉	C₂³	1	12	yes	—
43	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₉	C₂³	1	3	yes	—
45	C₄×Q₈×C₁₉	C₂×C₄	1	16	yes	—
46	C₄×D₄×C₁₉	C₂×C₄	1	48	yes	—
47	C₁₉×#117	C₂×C₄	1	12	yes	—
48	C₁₉×#117	C₂×C₄	1	12	yes	—
49	C₁₉×#117	C₂³	1	4	yes	—
50	C₁₉×#117	C₂³	1	12	yes	—
51	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	12	yes	—
52	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	12	yes	—
53	C₂×C₄²×C₁₉	C₂³	1	12	yes	—
54	C₁₉×C₄⋊C₈	C₂³	1	16	yes	—
55	C₁₉×C₄⋊C₈	C₂³	1	48	yes	—
56	C₂⁴×C₁₉	C₂²×C₄	1	3	yes	—
57	#41×C₁₉	C₄⋊C₄	1	32	yes	—
58	#41×C₁₉	C₂×D₄	1	16	yes	—
59	#41×C₁₉	C₂×Q₈	1	16	yes	—
60	C₂×Q₈×C₁₉	C₄⋊C₄	1	8	yes	—
61	C₂×Q₈×C₁₉	C₂²×C₄	1	4	yes	—
62	C₂×D₄×C₁₉	C₄⋊C₄	1	24	yes	—
63	C₂×D₄×C₁₉	C₂²×C₄	1	12	yes	—
64	C₂²×C₄×C₁₉	C₂²×C₄	1	24	yes	—
65	C₂²×C₄×C₁₉	C₂²×C₄	1	6	yes	—
66	C₂²×C₄×C₁₉	C₂²×C₄	1	3	yes	—
67	C₂²×C₄×C₁₉	C₂×D₄	1	12	yes	—
68	C₂²×C₄×C₁₉	C₂×Q₈	1	12	yes	—
69	C₂²×C₄×C₁₉	C₂⁴	1	6	yes	—
70	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
71	M₁₆×C₁₉	—	2	8	no	—
72	M₁₆×C₁₉	—	2	24	no	—
73	M₁₆×C₁₉	—	2	24	no	—
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	—	2	6	no	—
76	C₂×C₈×C₁₉	—	2	24	no	—
77	C₂×C₈×C₁₉	—	2	24	no	—
78	C₂×C₈×C₁₉	C₂⁴	1	2	yes	—
79	C₂×C₈×C₁₉	C₂⁴	1	6	yes	—
80	C₂×C₈×C₁₉	C₂⁴	1	8	yes	—
81	C₄⋊C₄×C₁₉	C₂²×C₄	1	8	yes	—
82	C₄⋊C₄×C₁₉	C₂²×C₄	1	24	yes	—
83	C₂²⋊C₄×C₁₉	C₂²×C₄	1	8	yes	—
84	C₂²⋊C₄×C₁₉	C₂²×C₄	1	24	yes	—
85	C₄²×C₁₉	C₂²×C₄	1	24	yes	—
86	C₄²×C₁₉	C₂⁴	1	8	yes	—
87	#60100	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₁₉	—	4	6	no	—
89	C₂³×C₁₉	C₂×C₄⋊C₄	1	12	yes	—
90	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
91	C₂³×C₁₉	C₂³×C₄	1	6	yes	—
92	Q₈×C₁₉	C₂×C₄⋊C₄	1	16	yes	—
93	D₄×C₁₉	C₂×C₄⋊C₄	1	48	yes	—
94	C₂×C₄×C₁₉	—	4	24	no	—
95	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	12	yes	—
96	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	24	yes	—
97	C₂×C₄×C₁₉	C₂³×C₄	1	16	yes	—
98	C₂×C₄×C₁₉	C₂³×C₄	1	6	yes	—
99	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
101	C₂×C₄×C₁₉	C₂²×D₄	1	12	yes	—
102	C₂×C₄×C₁₉	C₂²×Q₈	1	2	yes	—
103	C₂×C₄×C₁₉	C₂²×Q₈	1	12	yes	—
104	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
105	C₈×C₁₉	—	2	2	no	—
106	C₈×C₁₉	—	2	6	no	—
107	C₈×C₁₉	—	2	8	no	—
108	C₂×#568	C₂×C₁₉	1	1	yes	characteristic
109	C₂²×#131	C₂×C₁₉	1	1	yes	characteristic
110	#3428	C₂×C₁₉	1	1	yes	characteristic
111	#2906	C₂×C₁₉	1	16	yes	complemented
112	#2905	C₂×C₁₉	1	6	yes	complemented
113	C₂×#424	C₂×C₁₉	1	6	yes	complemented
114	C₂²×C₁₉	—	2	12	no	—
115	C₂²×C₁₉	—	2	2	no	—
116	C₂²×C₁₉	—	4	12	no	—
117	C₂²×C₁₉	C₂²×C₄⋊C₄	1	2	yes	—
118	C₂²×C₁₉	C₂²×C₄⋊C₄	1	12	yes	—
119	C₂²×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
120	C₄×C₁₉	—	2	16	no	—
121	C₄×C₁₉	C₂²×C₄⋊C₄	1	2	yes	—
122	C₄×C₁₉	C₂²×C₄⋊C₄	1	12	yes	—
123	C₄×C₁₉	C₂³×D₄	1	1	yes	characteristic
124	C₄×C₁₉	C₂³×Q₈	1	1	yes	characteristic
125	C₂²×#41	C₂²×C₁₉	1	1	yes	characteristic
126	#568	—	2	8	no	—
127	C₂×#131	—	2	8	no	—
128	C₂×#131	C₂²×C₁₉	1	8	yes	—
129	C₂×#131	C₂²×C₁₉	1	8	yes	—
130	C₂²×M₁₆	C₂²×C₁₉	1	6	yes	—
131	C₂²×M₁₆	C₂²×C₁₉	1	3	yes	—
132	C₂³×C₈	C₂²×C₁₉	1	3	yes	—
133	#520	C₄×C₁₉	1	8	yes	—
134	#519	C₄×C₁₉	1	2	yes	—
135	#518	C₄×C₁₉	1	6	yes	—
136	C₄×#41	C₂²×C₁₉	1	8	yes	—
137	C₂×#117	C₂²×C₁₉	1	6	yes	—
138	#424	C₂²×C₁₉	1	4	yes	—
139	#424	C₂²×C₁₉	1	12	yes	—
140	#424	C₂²×C₁₉	1	24	yes	complemented
141	#424	C₂²×C₁₉	1	24	yes	complemented
142	#423	C₂²×C₁₉	1	24	yes	complemented
143	C₂×C₄⋊C₈	C₂²×C₁₉	1	24	yes	complemented
144	C₂×C₁₉	—	2	2	no	—
145	C₂×C₁₉	—	2	12	no	—
146	C₂×C₁₉	C₂³×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₁₉	#3573	1	2	yes	—
148	C₂×#41	C₂×C₄×C₁₉	1	4	yes	—
149	C₂×#41	D₄×C₁₉	1	8	yes	—
150	C₂×#41	Q₈×C₁₉	1	8	yes	—
151	C₂×#41	C₂³×C₁₉	1	4	yes	—
152	C₂²×Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
153	C₂²×D₄	C₂×C₄×C₁₉	1	3	yes	—
154	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
155	#131	—	2	32	no	—
156	#131	—	2	32	no	—
157	C₂×M₁₆	—	2	12	no	—
158	C₂×M₁₆	—	2	24	no	—
159	C₂×M₁₆	C₂³×C₁₉	1	4	yes	—
160	C₂×M₁₆	C₂³×C₁₉	1	12	yes	—
161	C₂×M₁₆	C₂³×C₁₉	1	12	yes	—
162	C₂²×C₈	—	2	12	no	—
163	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₁₉	1	3	yes	—
165	C₂²×C₈	C₂³×C₁₉	1	12	yes	—
166	C₂²×C₈	C₂³×C₁₉	1	12	yes	—
167	C₄×Q₈	C₂×C₄×C₁₉	1	16	yes	—
168	C₄×D₄	C₂×C₄×C₁₉	1	48	yes	—
169	#117	C₂×C₄×C₁₉	1	12	yes	—
170	#117	C₂×C₄×C₁₉	1	12	yes	—
171	#117	C₂³×C₁₉	1	4	yes	—
172	#117	C₂³×C₁₉	1	12	yes	—
173	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	12	yes	—
174	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	12	yes	—
175	C₂×C₄²	C₂³×C₁₉	1	12	yes	—
176	C₄⋊C₈	C₂³×C₁₉	1	16	yes	—
177	C₄⋊C₈	C₂³×C₁₉	1	48	yes	—
178	C₁₉	#60100	1	1	yes	characteristic, direct product factor, Sylow
179	C₂⁴	C₂²×C₄×C₁₉	1	3	yes	—
180	#41	C₄⋊C₄×C₁₉	1	32	yes	—
181	#41	C₂×D₄×C₁₉	1	16	yes	—
182	#41	C₂×Q₈×C₁₉	1	16	yes	—
183	C₂×Q₈	C₄⋊C₄×C₁₉	1	8	yes	—
184	C₂×Q₈	C₂²×C₄×C₁₉	1	4	yes	—
185	C₂×D₄	C₄⋊C₄×C₁₉	1	24	yes	—
186	C₂×D₄	C₂²×C₄×C₁₉	1	12	yes	—
187	C₂²×C₄	C₂²×C₄×C₁₉	1	24	yes	—
188	C₂²×C₄	C₂²×C₄×C₁₉	1	6	yes	—
189	C₂²×C₄	C₂²×C₄×C₁₉	1	3	yes	—
190	C₂²×C₄	C₂×D₄×C₁₉	1	12	yes	—
191	C₂²×C₄	C₂×Q₈×C₁₉	1	12	yes	—
192	C₂²×C₄	C₂⁴×C₁₉	1	6	yes	—
193	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
194	M₁₆	—	2	8	no	—
195	M₁₆	—	2	24	no	—
196	M₁₆	—	2	24	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	6	no	—
199	C₂×C₈	—	2	24	no	—
200	C₂×C₈	—	2	24	no	—
201	C₂×C₈	C₂⁴×C₁₉	1	2	yes	—
202	C₂×C₈	C₂⁴×C₁₉	1	8	yes	—
203	C₂×C₈	C₂⁴×C₁₉	1	6	yes	—
204	C₄⋊C₄	C₂²×C₄×C₁₉	1	8	yes	—
205	C₄⋊C₄	C₂²×C₄×C₁₉	1	24	yes	—
206	C₂²⋊C₄	C₂²×C₄×C₁₉	1	8	yes	—
207	C₂²⋊C₄	C₂²×C₄×C₁₉	1	24	yes	—
208	C₄²	C₂²×C₄×C₁₉	1	24	yes	—
209	C₄²	C₂⁴×C₁₉	1	8	yes	—
210	C₂³	—	4	6	no	—
211	C₂³	C₂×C₄⋊C₄×C₁₉	1	12	yes	—
212	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
213	C₂³	C₂³×C₄×C₁₉	1	6	yes	—
214	Q₈	C₂×C₄⋊C₄×C₁₉	1	16	yes	—
215	D₄	C₂×C₄⋊C₄×C₁₉	1	48	yes	—
216	C₂×C₄	—	4	24	no	—
217	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	12	yes	—
218	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	24	yes	—
219	C₂×C₄	C₂³×C₄×C₁₉	1	16	yes	—
220	C₂×C₄	C₂³×C₄×C₁₉	1	6	yes	—
221	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
222	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
223	C₂×C₄	C₂²×D₄×C₁₉	1	12	yes	—
224	C₂×C₄	C₂²×Q₈×C₁₉	1	2	yes	—
225	C₂×C₄	C₂²×Q₈×C₁₉	1	12	yes	—
226	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
227	C₈	—	2	2	no	—
228	C₈	—	2	6	no	—
229	C₈	—	2	8	no	—
230	C₂²	—	2	12	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	4	12	no	—
233	C₂²	C₂²×C₄⋊C₄×C₁₉	1	2	yes	—
234	C₂²	C₂²×C₄⋊C₄×C₁₉	1	12	yes	—
235	C₂²	C₂⁴×C₄×C₁₉	1	1	yes	characteristic
236	C₄	—	2	16	no	—
237	C₄	C₂²×C₄⋊C₄×C₁₉	1	2	yes	—
238	C₄	C₂²×C₄⋊C₄×C₁₉	1	12	yes	—
239	C₄	C₂³×D₄×C₁₉	1	1	yes	characteristic
240	C₄	C₂³×Q₈×C₁₉	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	2	12	no	—
243	C₂	C₂³×C₄⋊C₄×C₁₉	1	1	yes	characteristic
244	C₂	C₁₉×#3573	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₁₉
Frattini, cosocle	C₂×C₄	C₂⁵×C₁₉
socle	C₂²×C₁₉	C₂⁴×C₄
prime radical	C₂²×#41×C₁₉	C₂²
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
4. C₁₉×#2906	C₂	semidirect
5. C₁₉×#2905	C₂	semidirect
6. C₂×C₁₉×#424	C₂	semidirect
22. C₁₉×#424	C₂²	semidirect
23. C₁₉×#424	C₂²	semidirect
24. C₁₉×#423	C₂²	semidirect
25. C₂×C₁₉×C₄⋊C₈	C₂²	semidirect
87. #60100	C₁₉	direct (N × H)
111. #2906	C₂×C₁₉	semidirect
112. #2905	C₂×C₁₉	semidirect
113. C₂×#424	C₂×C₁₉	semidirect
140. #424	C₂²×C₁₉	semidirect
141. #424	C₂²×C₁₉	semidirect
142. #423	C₂²×C₁₉	semidirect
143. C₂×C₄⋊C₈	C₂²×C₁₉	semidirect
178. C₁₉	#60100	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#568	2
C₂²×C₁₉×#131	2
C₁₉×#3428	2
C₁₉×#2906	2
C₁₉×#2905	2
C₂×C₁₉×#424	2
#60100	19

Subgroups — C19×#60100 #18066059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#60100 #18066059