Subgroups — C₄×C₁₉×#568 #18066016

Order 4864.

3250 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	2	yes	maximal
2	C₂×C₄×#41×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2875	C₂	1	8	yes	maximal, complemented
4	C₄×C₁₉×#131	C₂	1	8	yes	maximal, complemented
5	C₁₉×#2873	C₂	1	6	yes	maximal, complemented
6	C₂×C₄×M₁₆×C₁₉	C₂	1	6	yes	maximal, complemented
7	C₂²×#41×C₁₉	C₂²	1	1	yes	characteristic
8	C₁₉×#568	C₄	1	32	yes	direct product factor
9	C₂×C₁₉×#131	C₂²	1	16	yes	—
10	C₂²×M₁₆×C₁₉	C₂²	1	12	yes	—
11	C₄×#41×C₁₉	C₄	1	8	yes	—
12	C₄×#41×C₁₉	C₂²	1	4	yes	—
13	C₄×#41×C₁₉	C₂²	1	4	yes	—
14	C₂×C₄×Q₈×C₁₉	C₄	1	2	yes	—
15	C₂×C₄×D₄×C₁₉	C₄	1	6	yes	—
16	C₂×C₁₉×#117	C₂²	1	3	yes	—
17	C₂²×C₄²×C₁₉	C₂²	1	3	yes	—
18	C₁₉×#405	C₂²	1	48	yes	complemented
19	C₄×M₁₆×C₁₉	C₂²	1	4	yes	—
20	C₄×M₁₆×C₁₉	C₂²	1	12	yes	—
21	C₄×M₁₆×C₁₉	C₂²	1	12	yes	complemented
22	C₄×M₁₆×C₁₉	C₂²	1	12	yes	complemented
23	C₂×C₁₉×#97	C₂²	1	12	yes	complemented
24	C₂×C₄×C₈×C₁₉	C₂²	1	12	yes	complemented
25	C₂×#41×C₁₉	C₂×C₄	1	16	yes	—
26	C₂×#41×C₁₉	C₂×C₄	1	4	yes	—
27	C₂×#41×C₁₉	C₂³	1	4	yes	—
28	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×D₄×C₁₉	C₂×C₄	1	3	yes	—
30	C₂³×C₄×C₁₉	C₂×C₄	1	6	yes	—
31	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
32	C₁₉×#131	C₂×C₄	1	128	yes	complemented
33	C₂×M₁₆×C₁₉	C₂×C₄	1	96	yes	complemented
34	C₂×M₁₆×C₁₉	C₂³	1	8	yes	—
35	C₂×M₁₆×C₁₉	C₂³	1	24	yes	—
36	C₂²×C₈×C₁₉	C₂³	1	24	yes	—
37	C₄×Q₈×C₁₉	C₂×C₄	1	8	yes	—
38	C₄×Q₈×C₁₉	C₂×C₄	1	8	yes	—
39	C₄×D₄×C₁₉	C₂×C₄	1	24	yes	—
40	C₄×D₄×C₁₉	C₂×C₄	1	24	yes	—
41	C₁₉×#117	C₂×C₄	1	12	yes	—
42	C₁₉×#117	C₂³	1	12	yes	—
43	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	6	yes	—
44	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	6	yes	—
45	C₂×C₄²×C₁₉	C₂×C₄	1	6	yes	—
46	C₂×C₄²×C₁₉	C₂×C₄	1	12	yes	—
47	C₂×C₄²×C₁₉	C₂³	1	6	yes	—
48	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
49	C₂×C₄²×C₁₉	C₂³	1	6	yes	—
50	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
51	C₁₉×#97	C₂³	1	8	yes	—
52	C₁₉×#97	C₂³	1	24	yes	—
53	C₄×C₈×C₁₉	C₂³	1	24	yes	—
54	C₄×C₈×C₁₉	C₂³	1	8	yes	—
55	C₂⁴×C₁₉	C₂²×C₄	1	3	yes	—
56	#41×C₁₉	C₄²	1	32	yes	—
57	#41×C₁₉	C₂²×C₄	1	32	yes	—
58	C₂×Q₈×C₁₉	C₄²	1	8	yes	—
59	C₂×Q₈×C₁₉	C₂²×C₄	1	4	yes	—
60	C₂×D₄×C₁₉	C₄²	1	24	yes	—
61	C₂×D₄×C₁₉	C₂²×C₄	1	12	yes	—
62	C₂²×C₄×C₁₉	—	4	12	no	—
63	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂²×C₄	1	12	yes	—
65	C₂²×C₄×C₁₉	C₂²×C₄	1	6	yes	—
66	C₂²×C₄×C₁₉	C₂²×C₄	1	12	yes	—
67	C₂²×C₄×C₁₉	C₂²×C₄	1	3	yes	—
68	C₂²×C₄×C₁₉	C₂²×C₄	1	6	yes	—
69	C₂²×C₄×C₁₉	C₂²×C₄	1	24	yes	—
70	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂⁴	1	6	yes	—
72	M₁₆×C₁₉	C₂²×C₄	1	32	yes	—
73	M₁₆×C₁₉	C₂²×C₄	1	96	yes	complemented
74	C₂×C₈×C₁₉	C₂²×C₄	1	96	yes	complemented
75	C₂×C₈×C₁₉	C₂⁴	1	16	yes	—
76	C₄⋊C₄×C₁₉	C₂²×C₄	1	24	yes	—
77	C₂²⋊C₄×C₁₉	C₂²×C₄	1	24	yes	—
78	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
79	C₄²×C₁₉	C₂²×C₄	1	6	yes	—
80	C₄²×C₁₉	C₂²×C₄	1	12	yes	—
81	C₄²×C₁₉	C₂²×C₄	1	12	yes	—
82	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
83	C₄²×C₁₉	C₂⁴	1	6	yes	—
84	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
85	C₄×#568	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₁₉	—	4	6	no	—
87	C₂³×C₁₉	C₂×C₄²	1	12	yes	—
88	C₂³×C₁₉	C₂³×C₄	1	6	yes	—
89	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
90	Q₈×C₁₉	C₂×C₄²	1	16	yes	—
91	D₄×C₁₉	C₂×C₄²	1	48	yes	—
92	C₂×C₄×C₁₉	—	2	4	no	—
93	C₂×C₄×C₁₉	—	2	24	no	—
94	C₂×C₄×C₁₉	—	4	12	no	—
95	C₂×C₄×C₁₉	—	4	6	no	—
96	C₂×C₄×C₁₉	C₂×C₄²	1	12	yes	—
97	C₂×C₄×C₁₉	C₂×C₄²	1	24	yes	—
98	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
99	C₂×C₄×C₁₉	C₂³×C₄	1	12	yes	—
100	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
101	C₂×C₄×C₁₉	C₂³×C₄	1	24	yes	—
102	C₂×C₄×C₁₉	C₂³×C₄	1	4	yes	—
103	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂³×C₄	1	6	yes	—
105	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
106	C₈×C₁₉	C₂³×C₄	1	32	yes	—
107	C₂×#568	C₂×C₁₉	1	2	yes	—
108	C₂×C₄×#41	C₂×C₁₉	1	1	yes	characteristic
109	#2875	C₂×C₁₉	1	8	yes	complemented
110	C₄×#131	C₂×C₁₉	1	8	yes	complemented
111	#2873	C₂×C₁₉	1	6	yes	complemented
112	C₂×C₄×M₁₆	C₂×C₁₉	1	6	yes	complemented
113	C₂²×C₁₉	—	2	2	no	—
114	C₂²×C₁₉	—	2	12	no	—
115	C₂²×C₁₉	—	4	12	no	—
116	C₂²×C₁₉	C₂²×C₄²	1	2	yes	—
117	C₂²×C₁₉	C₂²×C₄²	1	12	yes	—
118	C₂²×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
119	C₄×C₁₉	—	2	2	no	—
120	C₄×C₁₉	—	2	12	no	—
121	C₄×C₁₉	C₂²×C₄²	1	2	yes	—
122	C₄×C₁₉	C₂²×C₄²	1	12	yes	—
123	C₄×C₁₉	#568	1	4	yes	direct product factor
124	C₄×C₁₉	C₂⁴×C₄	1	2	yes	—
125	C₂²×#41	C₂²×C₁₉	1	1	yes	characteristic
126	#568	C₄×C₁₉	1	32	yes	direct product factor
127	C₂×#131	C₂²×C₁₉	1	16	yes	—
128	C₂²×M₁₆	C₂²×C₁₉	1	12	yes	—
129	C₄×#41	C₄×C₁₉	1	8	yes	—
130	C₄×#41	C₂²×C₁₉	1	4	yes	—
131	C₄×#41	C₂²×C₁₉	1	4	yes	—
132	C₂×C₄×Q₈	C₄×C₁₉	1	2	yes	—
133	C₂×C₄×D₄	C₄×C₁₉	1	6	yes	—
134	C₂×#117	C₂²×C₁₉	1	3	yes	—
135	C₂²×C₄²	C₂²×C₁₉	1	3	yes	—
136	#405	C₂²×C₁₉	1	48	yes	complemented
137	C₄×M₁₆	C₂²×C₁₉	1	4	yes	—
138	C₄×M₁₆	C₂²×C₁₉	1	12	yes	—
139	C₄×M₁₆	C₂²×C₁₉	1	12	yes	complemented
140	C₄×M₁₆	C₂²×C₁₉	1	12	yes	complemented
141	C₂×#97	C₂²×C₁₉	1	12	yes	complemented
142	C₂×C₄×C₈	C₂²×C₁₉	1	12	yes	complemented
143	C₂×C₁₉	—	2	2	no	—
144	C₂×C₁₉	—	2	12	no	—
145	C₂×C₁₉	C₂³×C₄²	1	1	yes	characteristic
146	C₂×C₁₉	C₂×#568	1	2	yes	—
147	C₂×#41	C₂×C₄×C₁₉	1	16	yes	—
148	C₂×#41	C₂×C₄×C₁₉	1	4	yes	—
149	C₂×#41	C₂³×C₁₉	1	4	yes	—
150	C₂²×Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
151	C₂²×D₄	C₂×C₄×C₁₉	1	3	yes	—
152	C₂³×C₄	C₂×C₄×C₁₉	1	6	yes	—
153	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
154	#131	C₂×C₄×C₁₉	1	128	yes	complemented
155	C₂×M₁₆	C₂×C₄×C₁₉	1	96	yes	complemented
156	C₂×M₁₆	C₂³×C₁₉	1	8	yes	—
157	C₂×M₁₆	C₂³×C₁₉	1	24	yes	—
158	C₂²×C₈	C₂³×C₁₉	1	24	yes	—
159	C₄×Q₈	C₂×C₄×C₁₉	1	8	yes	—
160	C₄×Q₈	C₂×C₄×C₁₉	1	8	yes	—
161	C₄×D₄	C₂×C₄×C₁₉	1	24	yes	—
162	C₄×D₄	C₂×C₄×C₁₉	1	24	yes	—
163	#117	C₂×C₄×C₁₉	1	12	yes	—
164	#117	C₂³×C₁₉	1	12	yes	—
165	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	6	yes	—
166	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	6	yes	—
167	C₂×C₄²	C₂×C₄×C₁₉	1	6	yes	—
168	C₂×C₄²	C₂×C₄×C₁₉	1	12	yes	—
169	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
170	C₂×C₄²	C₂³×C₁₉	1	6	yes	—
171	C₂×C₄²	C₂³×C₁₉	1	6	yes	—
172	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
173	#97	C₂³×C₁₉	1	8	yes	—
174	#97	C₂³×C₁₉	1	24	yes	—
175	C₄×C₈	C₂³×C₁₉	1	8	yes	—
176	C₄×C₈	C₂³×C₁₉	1	24	yes	—
177	C₁₉	C₄×#568	1	1	yes	characteristic, direct product factor, Sylow
178	C₂⁴	C₂²×C₄×C₁₉	1	3	yes	—
179	#41	C₄²×C₁₉	1	32	yes	—
180	#41	C₂²×C₄×C₁₉	1	32	yes	—
181	C₂×Q₈	C₄²×C₁₉	1	8	yes	—
182	C₂×Q₈	C₂²×C₄×C₁₉	1	4	yes	—
183	C₂×D₄	C₄²×C₁₉	1	24	yes	—
184	C₂×D₄	C₂²×C₄×C₁₉	1	12	yes	—
185	C₂²×C₄	—	4	12	no	—
186	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
187	C₂²×C₄	C₂²×C₄×C₁₉	1	12	yes	—
188	C₂²×C₄	C₂²×C₄×C₁₉	1	6	yes	—
189	C₂²×C₄	C₂²×C₄×C₁₉	1	12	yes	—
190	C₂²×C₄	C₂²×C₄×C₁₉	1	3	yes	—
191	C₂²×C₄	C₂²×C₄×C₁₉	1	6	yes	—
192	C₂²×C₄	C₂²×C₄×C₁₉	1	24	yes	—
193	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
194	C₂²×C₄	C₂⁴×C₁₉	1	6	yes	—
195	M₁₆	C₂²×C₄×C₁₉	1	32	yes	—
196	M₁₆	C₂²×C₄×C₁₉	1	96	yes	complemented
197	C₂×C₈	C₂²×C₄×C₁₉	1	96	yes	complemented
198	C₂×C₈	C₂⁴×C₁₉	1	16	yes	—
199	C₄⋊C₄	C₂²×C₄×C₁₉	1	24	yes	—
200	C₂²⋊C₄	C₂²×C₄×C₁₉	1	24	yes	—
201	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
202	C₄²	C₂²×C₄×C₁₉	1	6	yes	—
203	C₄²	C₂²×C₄×C₁₉	1	12	yes	—
204	C₄²	C₂²×C₄×C₁₉	1	12	yes	—
205	C₄²	C₂⁴×C₁₉	1	6	yes	—
206	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
207	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
208	C₂³	—	4	6	no	—
209	C₂³	C₂×C₄²×C₁₉	1	12	yes	—
210	C₂³	C₂³×C₄×C₁₉	1	6	yes	—
211	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
212	Q₈	C₂×C₄²×C₁₉	1	16	yes	—
213	D₄	C₂×C₄²×C₁₉	1	48	yes	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	24	no	—
216	C₂×C₄	—	4	12	no	—
217	C₂×C₄	—	4	6	no	—
218	C₂×C₄	C₂×C₄²×C₁₉	1	12	yes	—
219	C₂×C₄	C₂×C₄²×C₁₉	1	24	yes	—
220	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
221	C₂×C₄	C₂³×C₄×C₁₉	1	12	yes	—
222	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
223	C₂×C₄	C₂³×C₄×C₁₉	1	6	yes	—
224	C₂×C₄	C₂³×C₄×C₁₉	1	24	yes	—
225	C₂×C₄	C₂³×C₄×C₁₉	1	4	yes	—
226	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
227	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
228	C₈	C₂³×C₄×C₁₉	1	32	yes	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	12	no	—
231	C₂²	—	4	12	no	—
232	C₂²	C₂²×C₄²×C₁₉	1	2	yes	—
233	C₂²	C₂²×C₄²×C₁₉	1	12	yes	—
234	C₂²	C₂⁴×C₄×C₁₉	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	12	no	—
237	C₄	C₂²×C₄²×C₁₉	1	2	yes	—
238	C₄	C₂²×C₄²×C₁₉	1	12	yes	—
239	C₄	C₁₉×#568	1	4	yes	direct product factor
240	C₄	C₂⁴×C₄×C₁₉	1	2	yes	—
241	C₂	—	2	2	no	—
242	C₂	—	2	12	no	—
243	C₂	C₂³×C₄²×C₁₉	1	1	yes	characteristic
244	C₂	C₂×C₁₉×#568	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₉	C₂⁴
commutator	C₂	C₂³×C₄²×C₁₉
Frattini, cosocle	C₂×C₄	C₂⁵×C₁₉
socle	C₂²×C₁₉	C₂⁴×C₄
prime radical	C₂²×#41×C₁₉	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#2875	C₂	semidirect
4. C₄×C₁₉×#131	C₂	semidirect
5. C₁₉×#2873	C₂	semidirect
6. C₂×C₄×M₁₆×C₁₉	C₂	semidirect
8. C₁₉×#568	C₄	direct (N × H)
18. C₁₉×#405	C₂²	semidirect
21. C₄×M₁₆×C₁₉	C₂²	semidirect
22. C₄×M₁₆×C₁₉	C₂²	semidirect
23. C₂×C₁₉×#97	C₂²	semidirect
24. C₂×C₄×C₈×C₁₉	C₂²	semidirect
32. C₁₉×#131	C₂×C₄	semidirect
33. C₂×M₁₆×C₁₉	C₂×C₄	semidirect
73. M₁₆×C₁₉	C₂²×C₄	semidirect
74. C₂×C₈×C₁₉	C₂²×C₄	semidirect
85. C₄×#568	C₁₉	direct (N × H)
109. #2875	C₂×C₁₉	semidirect
110. C₄×#131	C₂×C₁₉	semidirect
111. #2873	C₂×C₁₉	semidirect
112. C₂×C₄×M₁₆	C₂×C₁₉	semidirect
123. C₄×C₁₉	#568	direct (N × H)
126. #568	C₄×C₁₉	direct (N × H)
136. #405	C₂²×C₁₉	semidirect
139. C₄×M₁₆	C₂²×C₁₉	semidirect
140. C₄×M₁₆	C₂²×C₁₉	semidirect
141. C₂×#97	C₂²×C₁₉	semidirect
142. C₂×C₄×C₈	C₂²×C₁₉	semidirect
154. #131	C₂×C₄×C₁₉	semidirect
155. C₂×M₁₆	C₂×C₄×C₁₉	semidirect
177. C₁₉	C₄×#568	direct (N × H)
196. M₁₆	C₂²×C₄×C₁₉	semidirect
197. C₂×C₈	C₂²×C₄×C₁₉	semidirect
239. C₄	C₁₉×#568	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
C₂×C₄×#41×C₁₉	2
C₁₉×#2875	2
C₄×C₁₉×#131	2
C₁₉×#2873	2
C₂×C₄×M₁₆×C₁₉	2
C₄×#568	19

Subgroups — C4×C19×#568 #18066016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₁₉×#568 #18066016