Subgroups — C₂×C₁₉×#2913 #18066066

Order 4864.

2766 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#520	C₂	1	2	yes	maximal
3	C₁₉×#2913	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₁₉×#428	C₂	1	6	yes	maximal, complemented
5	C₂×C₁₉×#427	C₂	1	3	yes	maximal, complemented
6	C₂²×C₁₉×#107	C₂	1	3	yes	maximal, complemented
7	C₂²×#41×C₁₉	C₂²	1	1	yes	characteristic
8	C₂×C₁₉×#131	C₄	1	16	yes	complemented
9	C₂×C₁₉×#131	C₂²	1	8	yes	—
10	C₂²×M₁₆×C₁₉	C₂²	1	3	yes	—
11	C₂³×C₈×C₁₉	C₂²	1	3	yes	—
12	C₁₉×#520	—	2	8	no	—
13	C₁₉×#520	C₂²	1	16	yes	—
14	C₂×C₄×Q₈×C₁₉	—	2	2	no	—
15	C₂×C₄×D₄×C₁₉	—	2	6	no	—
16	C₂×C₁₉×#117	C₂²	1	6	yes	—
17	C₂²×C₄⋊C₄×C₁₉	C₂²	1	6	yes	—
18	C₁₉×#428	C₂²	1	48	yes	complemented
19	C₁₉×#427	C₂²	1	24	yes	complemented
20	C₂×C₁₉×#107	C₂²	1	3	yes	—
21	C₂×C₁₉×#107	C₂²	1	6	yes	—
22	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
23	C₂×C₁₉×#107	C₂²	1	24	yes	complemented
24	C₂×C₁₉×#106	C₂²	1	6	yes	—
25	C₂×#41×C₁₉	C₂×C₄	1	8	yes	—
26	C₂×#41×C₁₉	D₄	1	4	yes	—
27	C₂×#41×C₁₉	Q₈	1	8	yes	—
28	C₂×#41×C₁₉	C₂³	1	4	yes	—
29	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
30	C₂²×D₄×C₁₉	D₄	1	3	yes	—
31	C₂³×C₄×C₁₉	—	2	6	no	—
32	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
33	C₁₉×#131	C₂×C₄	1	64	yes	complemented
34	C₂×M₁₆×C₁₉	C₂×C₄	1	24	yes	complemented
35	C₂×M₁₆×C₁₉	C₂³	1	12	yes	—
36	C₂²×C₈×C₁₉	C₂×C₄	1	24	yes	complemented
37	C₂²×C₈×C₁₉	C₂³	1	3	yes	—
38	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₁₉	C₂³	1	12	yes	—
40	C₄×Q₈×C₁₉	—	2	16	no	—
41	C₄×D₄×C₁₉	—	2	48	no	—
42	C₁₉×#117	—	2	12	no	—
43	C₁₉×#117	C₂³	1	24	yes	—
44	C₂×C₄⋊C₄×C₁₉	—	2	6	no	—
45	C₂×C₄⋊C₄×C₁₉	—	2	12	no	—
46	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₉	C₂³	1	6	yes	—
48	C₂×C₄⋊C₄×C₁₉	C₂³	1	24	yes	—
49	C₂×C₂²⋊C₄×C₁₉	—	2	6	no	—
50	C₂×C₄²×C₁₉	—	2	6	no	—
51	C₁₉×#107	C₂³	1	12	yes	—
52	C₁₉×#107	C₂³	1	24	yes	—
53	C₁₉×#107	C₂³	1	4	yes	—
54	C₁₉×#106	C₂³	1	24	yes	—
55	C₂⁴×C₁₉	C₂×D₄	1	3	yes	—
56	#41×C₁₉	C₄⋊C₄	1	32	yes	—
57	#41×C₁₉	C₂²×C₄	1	16	yes	—
58	#41×C₁₉	C₂×Q₈	1	16	yes	—
59	C₂×Q₈×C₁₉	C₄⋊C₄	1	8	yes	—
60	C₂×Q₈×C₁₉	C₂×D₄	1	4	yes	—
61	C₂×D₄×C₁₉	C₄⋊C₄	1	24	yes	—
62	C₂×D₄×C₁₉	C₂×D₄	1	12	yes	—
63	C₂²×C₄×C₁₉	—	2	2	no	—
64	C₂²×C₄×C₁₉	—	2	6	no	—
65	C₂²×C₄×C₁₉	—	2	24	no	—
66	C₂²×C₄×C₁₉	—	8	12	no	—
67	C₂²×C₄×C₁₉	C₂²×C₄	1	12	yes	—
68	C₂²×C₄×C₁₉	C₂×D₄	1	6	yes	—
69	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
70	C₂²×C₄×C₁₉	C₂×Q₈	1	12	yes	—
71	C₂²×C₄×C₁₉	C₂⁴	1	6	yes	—
72	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₁₉	C₂²×C₄	1	48	yes	complemented
74	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₁₉	C₂²×C₄	1	12	yes	—
76	C₂×C₈×C₁₉	C₂²×C₄	1	48	yes	complemented
77	C₂×C₈×C₁₉	C₂⁴	1	2	yes	—
78	C₂×C₈×C₁₉	C₂⁴	1	6	yes	—
79	C₄⋊C₄×C₁₉	—	2	6	no	—
80	C₄⋊C₄×C₁₉	—	2	24	no	—
81	C₄⋊C₄×C₁₉	—	2	2	no	—
82	C₄⋊C₄×C₁₉	C₂⁴	1	12	yes	—
83	C₄⋊C₄×C₁₉	C₂⁴	1	4	yes	—
84	C₂²⋊C₄×C₁₉	—	2	24	no	—
85	C₄²×C₁₉	—	2	24	no	—
86	C₂×#2913	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₁₉	—	2	6	no	—
88	C₂³×C₁₉	—	2	6	no	—
89	C₂³×C₁₉	C₂×C₄⋊C₄	1	12	yes	—
90	C₂³×C₁₉	C₂²×D₄	1	6	yes	—
91	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
92	Q₈×C₁₉	C₂×C₄⋊C₄	1	16	yes	—
93	D₄×C₁₉	C₂×C₄⋊C₄	1	48	yes	—
94	C₂×C₄×C₁₉	—	2	4	no	—
95	C₂×C₄×C₁₉	—	2	12	no	—
96	C₂×C₄×C₁₉	—	4	12	no	—
97	C₂×C₄×C₁₉	—	4	4	no	—
98	C₂×C₄×C₁₉	—	8	24	no	—
99	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	12	yes	—
100	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	24	yes	—
101	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
102	C₂×C₄×C₁₉	C₂³×C₄	1	12	yes	—
103	C₂×C₄×C₁₉	C₂²×D₄	1	6	yes	—
104	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	C₂²×Q₈	1	2	yes	—
106	C₂×C₄×C₁₉	C₂²×Q₈	1	12	yes	—
107	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
108	C₈×C₁₉	C₂³×C₄	1	4	yes	—
109	C₈×C₁₉	C₂³×C₄	1	12	yes	—
110	C₂²×#131	C₂×C₁₉	1	1	yes	characteristic
111	C₂×#520	C₂×C₁₉	1	2	yes	—
112	#2913	C₂×C₁₉	1	16	yes	direct product factor
113	C₂×#428	C₂×C₁₉	1	6	yes	complemented
114	C₂×#427	C₂×C₁₉	1	3	yes	complemented
115	C₂²×#107	C₂×C₁₉	1	3	yes	complemented
116	C₂²×C₁₉	—	2	24	no	—
117	C₂²×C₁₉	—	2	6	no	—
118	C₂²×C₁₉	—	2	6	no	—
119	C₂²×C₁₉	C₂²×C₄⋊C₄	1	12	yes	—
120	C₂²×C₁₉	C₂²×C₄⋊C₄	1	2	yes	—
121	C₂²×C₁₉	#576	1	2	yes	—
122	C₂²×C₁₉	#578	1	2	yes	—
123	C₂²×C₁₉	C₂³×D₄	1	1	yes	characteristic
124	C₄×C₁₉	—	4	12	no	—
125	C₄×C₁₉	—	4	4	no	—
126	C₄×C₁₉	C₂²×C₄⋊C₄	1	2	yes	—
127	C₄×C₁₉	C₂²×C₄⋊C₄	1	12	yes	—
128	C₄×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
129	C₄×C₁₉	C₂³×Q₈	1	1	yes	characteristic
130	C₂²×#41	C₂²×C₁₉	1	1	yes	characteristic
131	C₂×#131	C₄×C₁₉	1	16	yes	complemented
132	C₂×#131	C₂²×C₁₉	1	8	yes	—
133	C₂²×M₁₆	C₂²×C₁₉	1	3	yes	—
134	C₂³×C₈	C₂²×C₁₉	1	3	yes	—
135	#520	—	2	8	no	—
136	#520	C₂²×C₁₉	1	16	yes	—
137	C₂×C₄×Q₈	—	2	2	no	—
138	C₂×C₄×D₄	—	2	6	no	—
139	C₂×#117	C₂²×C₁₉	1	6	yes	—
140	C₂²×C₄⋊C₄	C₂²×C₁₉	1	6	yes	—
141	#428	C₂²×C₁₉	1	48	yes	complemented
142	#427	C₂²×C₁₉	1	24	yes	complemented
143	C₂×#107	C₂²×C₁₉	1	3	yes	—
144	C₂×#107	C₂²×C₁₉	1	6	yes	—
145	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
146	C₂×#107	C₂²×C₁₉	1	24	yes	complemented
147	C₂×#106	C₂²×C₁₉	1	6	yes	—
148	C₂×C₁₉	—	2	12	no	—
149	C₂×C₁₉	#2913	1	4	yes	direct product factor
150	C₂×C₁₉	C₂³×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₁₉	C₂×#576	1	1	yes	characteristic
152	C₂×C₁₉	C₂×#578	1	1	yes	characteristic
153	C₂×#41	C₂×C₄×C₁₉	1	8	yes	—
154	C₂×#41	D₄×C₁₉	1	4	yes	—
155	C₂×#41	Q₈×C₁₉	1	8	yes	—
156	C₂×#41	C₂³×C₁₉	1	4	yes	—
157	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
158	C₂²×D₄	D₄×C₁₉	1	3	yes	—
159	C₂³×C₄	—	2	6	no	—
160	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
161	#131	C₂×C₄×C₁₉	1	64	yes	complemented
162	C₂×M₁₆	C₂×C₄×C₁₉	1	24	yes	complemented
163	C₂×M₁₆	C₂³×C₁₉	1	12	yes	—
164	C₂²×C₈	C₂×C₄×C₁₉	1	24	yes	complemented
165	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
166	C₂²×C₈	C₂³×C₁₉	1	3	yes	—
167	C₂²×C₈	C₂³×C₁₉	1	12	yes	—
168	C₄×Q₈	—	2	16	no	—
169	C₄×D₄	—	2	48	no	—
170	#117	—	2	12	no	—
171	#117	C₂³×C₁₉	1	24	yes	—
172	C₂×C₄⋊C₄	—	2	6	no	—
173	C₂×C₄⋊C₄	—	2	12	no	—
174	C₂×C₄⋊C₄	C₂³×C₁₉	1	6	yes	—
175	C₂×C₄⋊C₄	C₂³×C₁₉	1	24	yes	—
176	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
177	C₂×C₂²⋊C₄	—	2	6	no	—
178	C₂×C₄²	—	2	6	no	—
179	#107	C₂³×C₁₉	1	12	yes	—
180	#107	C₂³×C₁₉	1	24	yes	—
181	#107	C₂³×C₁₉	1	4	yes	—
182	#106	C₂³×C₁₉	1	24	yes	—
183	C₁₉	C₂×#2913	1	1	yes	characteristic, direct product factor, Sylow
184	C₂⁴	C₂×D₄×C₁₉	1	3	yes	—
185	#41	C₄⋊C₄×C₁₉	1	32	yes	—
186	#41	C₂²×C₄×C₁₉	1	16	yes	—
187	#41	C₂×Q₈×C₁₉	1	16	yes	—
188	C₂×Q₈	C₄⋊C₄×C₁₉	1	8	yes	—
189	C₂×Q₈	C₂×D₄×C₁₉	1	4	yes	—
190	C₂×D₄	C₄⋊C₄×C₁₉	1	24	yes	—
191	C₂×D₄	C₂×D₄×C₁₉	1	12	yes	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	6	no	—
194	C₂²×C₄	—	2	24	no	—
195	C₂²×C₄	—	8	12	no	—
196	C₂²×C₄	C₂²×C₄×C₁₉	1	12	yes	—
197	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
198	C₂²×C₄	C₂×D₄×C₁₉	1	6	yes	—
199	C₂²×C₄	C₂×Q₈×C₁₉	1	12	yes	—
200	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
201	C₂²×C₄	C₂⁴×C₁₉	1	6	yes	—
202	M₁₆	C₂²×C₄×C₁₉	1	48	yes	complemented
203	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
204	C₂×C₈	C₂²×C₄×C₁₉	1	12	yes	—
205	C₂×C₈	C₂²×C₄×C₁₉	1	48	yes	complemented
206	C₂×C₈	C₂⁴×C₁₉	1	6	yes	—
207	C₂×C₈	C₂⁴×C₁₉	1	2	yes	—
208	C₄⋊C₄	—	2	6	no	—
209	C₄⋊C₄	—	2	24	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	C₂⁴×C₁₉	1	4	yes	—
212	C₄⋊C₄	C₂⁴×C₁₉	1	12	yes	—
213	C₂²⋊C₄	—	2	24	no	—
214	C₄²	—	2	24	no	—
215	C₂³	—	2	6	no	—
216	C₂³	—	2	6	no	—
217	C₂³	C₂×C₄⋊C₄×C₁₉	1	12	yes	—
218	C₂³	C₂²×D₄×C₁₉	1	6	yes	—
219	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
220	Q₈	C₂×C₄⋊C₄×C₁₉	1	16	yes	—
221	D₄	C₂×C₄⋊C₄×C₁₉	1	48	yes	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	12	no	—
224	C₂×C₄	—	4	12	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	8	24	no	—
227	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	12	yes	—
228	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	24	yes	—
229	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
230	C₂×C₄	C₂³×C₄×C₁₉	1	12	yes	—
231	C₂×C₄	C₂²×D₄×C₁₉	1	6	yes	—
232	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
233	C₂×C₄	C₂²×Q₈×C₁₉	1	2	yes	—
234	C₂×C₄	C₂²×Q₈×C₁₉	1	12	yes	—
235	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
236	C₈	C₂³×C₄×C₁₉	1	4	yes	—
237	C₈	C₂³×C₄×C₁₉	1	12	yes	—
238	C₂²	—	2	24	no	—
239	C₂²	—	2	6	no	—
240	C₂²	—	2	6	no	—
241	C₂²	C₂²×C₄⋊C₄×C₁₉	1	12	yes	—
242	C₂²	C₂²×C₄⋊C₄×C₁₉	1	2	yes	—
243	C₂²	C₁₉×#576	1	2	yes	—
244	C₂²	C₁₉×#578	1	2	yes	—
245	C₂²	C₂³×D₄×C₁₉	1	1	yes	characteristic
246	C₄	—	4	12	no	—
247	C₄	—	4	4	no	—
248	C₄	C₂²×C₄⋊C₄×C₁₉	1	2	yes	—
249	C₄	C₂²×C₄⋊C₄×C₁₉	1	12	yes	—
250	C₄	C₂⁴×C₄×C₁₉	1	1	yes	characteristic
251	C₄	C₂³×Q₈×C₁₉	1	1	yes	characteristic
252	C₂	—	2	12	no	—
253	C₂	C₁₉×#2913	1	4	yes	direct product factor
254	C₂	C₂³×C₄⋊C₄×C₁₉	1	1	yes	characteristic
255	C₂	C₂×C₁₉×#576	1	1	yes	characteristic
256	C₂	C₂×C₁₉×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator	C₄	C₂⁴×C₄×C₁₉
Frattini, cosocle	C₂×C₄	C₂⁵×C₁₉
prime radical	C₂²×#41×C₁₉	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂²×#41×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#2913	C₂	direct (N × H)
4. C₂×C₁₉×#428	C₂	semidirect
5. C₂×C₁₉×#427	C₂	semidirect
6. C₂²×C₁₉×#107	C₂	semidirect
8. C₂×C₁₉×#131	C₄	semidirect
18. C₁₉×#428	C₂²	semidirect
19. C₁₉×#427	C₂²	semidirect
23. C₂×C₁₉×#107	C₂²	semidirect
33. C₁₉×#131	C₂×C₄	semidirect
34. C₂×M₁₆×C₁₉	C₂×C₄	semidirect
36. C₂²×C₈×C₁₉	C₂×C₄	semidirect
73. M₁₆×C₁₉	C₂²×C₄	semidirect
76. C₂×C₈×C₁₉	C₂²×C₄	semidirect
86. C₂×#2913	C₁₉	direct (N × H)
112. #2913	C₂×C₁₉	direct (N × H)
113. C₂×#428	C₂×C₁₉	semidirect
114. C₂×#427	C₂×C₁₉	semidirect
115. C₂²×#107	C₂×C₁₉	semidirect
131. C₂×#131	C₄×C₁₉	semidirect
141. #428	C₂²×C₁₉	semidirect
142. #427	C₂²×C₁₉	semidirect
146. C₂×#107	C₂²×C₁₉	semidirect
149. C₂×C₁₉	#2913	direct (N × H)
161. #131	C₂×C₄×C₁₉	semidirect
162. C₂×M₁₆	C₂×C₄×C₁₉	semidirect
164. C₂²×C₈	C₂×C₄×C₁₉	semidirect
183. C₁₉	C₂×#2913	direct (N × H)
202. M₁₆	C₂²×C₄×C₁₉	semidirect
205. C₂×C₈	C₂²×C₄×C₁₉	semidirect
253. C₂	C₁₉×#2913	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×#131	2
C₂×C₁₉×#520	2
C₁₉×#2913	2
C₂×C₁₉×#428	2
C₂×C₁₉×#427	2
C₂²×C₁₉×#107	2
C₂×#2913	19

Subgroups — C2×C19×#2913 #18066066

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#2913 #18066066