Subgroups — C₄²×SD₁₆×C₁₉ #18026457

Order 4864.

1230 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×SD₁₆×C₁₉	C₂	1	3	yes	maximal
2	C₄²×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₄×C₁₉×#106	C₂	1	3	yes	maximal, complemented
5	C₄×C₁₉×#103	C₂	1	3	yes	maximal, complemented
6	C₄×C₁₉×#102	C₂	1	3	yes	maximal
7	C₄²×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₁₉	C₂²	1	3	yes	—
10	C₂×C₄×D₄×C₁₉	C₂²	1	3	yes	—
11	C₂²×C₄²×C₁₉	—	2	1	no	—
12	C₄×SD₁₆×C₁₉	C₄	1	24	yes	direct product factor
13	C₂×C₁₉×#106	C₂²	1	3	yes	—
14	C₂×C₁₉×#103	C₂²	1	3	yes	—
15	C₂×C₁₉×#102	C₂²	1	3	yes	—
16	C₂×C₄×C₈×C₁₉	C₂²	1	3	yes	—
17	C₂×C₄×C₈×C₁₉	C₂²	1	3	yes	—
18	C₄×C₄⋊C₄×C₁₉	—	2	3	no	—
19	C₄×C₄⋊C₄×C₁₉	C₂²	1	3	yes	—
20	C₄×C₄⋊C₄×C₁₉	C₂²	1	3	yes	—
21	C₄×C₂²⋊C₄×C₁₉	—	2	3	no	—
22	C₄³×C₁₉	—	2	1	no	—
23	C₄³×C₁₉	C₂²	1	1	yes	characteristic
24	C₁₉×#340	C₂²	1	6	yes	—
25	C₂²×Q₈×C₁₉	C₂³	1	1	yes	characteristic
26	C₂²×D₄×C₁₉	C₂³	1	1	yes	characteristic
27	C₂³×C₄×C₁₉	—	2	3	no	—
28	C₂×SD₁₆×C₁₉	C₂×C₄	1	12	yes	—
29	C₂²×C₈×C₁₉	C₂³	1	3	yes	—
30	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
31	C₄×Q₈×C₁₉	—	2	6	no	—
32	C₄×Q₈×C₁₉	C₂×C₄	1	12	yes	complemented
33	C₄×D₄×C₁₉	—	2	6	no	—
34	C₄×D₄×C₁₉	C₂×C₄	1	12	yes	—
35	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
36	C₂×C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
37	C₂×C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
38	C₂×C₂²⋊C₄×C₁₉	—	2	3	no	—
39	C₂×C₄²×C₁₉	—	2	3	no	—
40	C₂×C₄²×C₁₉	—	2	3	no	—
41	C₂×C₄²×C₁₉	—	2	3	no	—
42	C₂×C₄²×C₁₉	—	4	4	no	—
43	C₂×C₄²×C₁₉	D₄	1	3	yes	—
44	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
46	C₁₉×#106	C₂×C₄	1	12	yes	complemented
47	C₁₉×#103	C₂×C₄	1	12	yes	complemented
48	C₁₉×#102	C₂×C₄	1	12	yes	—
49	C₄×C₈×C₁₉	—	2	6	no	—
50	C₄×C₈×C₁₉	C₂×C₄	1	12	yes	complemented
51	C₁₉×#95	—	2	3	no	—
52	C₁₉×#95	—	2	3	no	—
53	C₂⁴×C₁₉	—	2	1	no	—
54	C₂×Q₈×C₁₉	—	2	3	no	—
55	C₂×Q₈×C₁₉	C₂²×C₄	1	6	yes	—
56	C₂×D₄×C₁₉	—	2	3	no	—
57	C₂×D₄×C₁₉	C₂²×C₄	1	6	yes	—
58	C₂²×C₄×C₁₉	—	2	1	no	—
59	C₂²×C₄×C₁₉	—	2	3	no	—
60	C₂²×C₄×C₁₉	—	2	3	no	—
61	C₂²×C₄×C₁₉	—	2	12	no	—
62	C₂²×C₄×C₁₉	—	4	6	no	—
63	C₂²×C₄×C₁₉	—	4	6	no	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
65	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
66	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
67	SD₁₆×C₁₉	C₄²	1	16	yes	direct product factor
68	C₂×C₈×C₁₉	—	2	6	no	—
69	C₂×C₈×C₁₉	C₂²×C₄	1	6	yes	—
70	C₂×C₈×C₁₉	C₂²×C₄	1	6	yes	—
71	C₄⋊C₄×C₁₉	—	2	12	no	—
72	C₄⋊C₄×C₁₉	—	2	3	no	—
73	C₄⋊C₄×C₁₉	—	2	3	no	—
74	C₄⋊C₄×C₁₉	C₂²×C₄	1	6	yes	—
75	C₄⋊C₄×C₁₉	C₂²×C₄	1	6	yes	—
76	C₂²⋊C₄×C₁₉	—	2	12	no	—
77	C₄²×C₁₉	—	2	12	no	—
78	C₄²×C₁₉	—	2	6	no	—
79	C₄²×C₁₉	—	4	6	no	—
80	C₄²×C₁₉	—	4	6	no	—
81	C₄²×C₁₉	SD₁₆	1	4	yes	direct product factor
82	C₄²×C₁₉	C₂²×C₄	1	6	yes	—
83	C₄²×C₁₉	#41	1	6	yes	—
84	C₄²×SD₁₆	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₁₉	—	2	6	no	—
86	C₂³×C₁₉	—	4	1	no	—
87	C₂³×C₁₉	—	4	3	no	—
88	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
89	Q₈×C₁₉	—	2	6	no	—
90	Q₈×C₁₉	C₂×C₄²	1	4	yes	complemented
91	D₄×C₁₉	—	2	6	no	—
92	D₄×C₁₉	C₂×C₄²	1	4	yes	—
93	C₂×C₄×C₁₉	—	2	6	no	—
94	C₂×C₄×C₁₉	—	2	6	no	—
95	C₂×C₄×C₁₉	—	2	6	no	—
96	C₂×C₄×C₁₉	—	2	6	no	—
97	C₂×C₄×C₁₉	—	4	3	no	—
98	C₂×C₄×C₁₉	—	4	6	no	—
99	C₂×C₄×C₁₉	—	4	3	no	—
100	C₂×C₄×C₁₉	—	4	24	no	—
101	C₂×C₄×C₁₉	C₄×D₄	1	6	yes	—
102	C₂×C₄×C₁₉	C₄×D₄	1	6	yes	—
103	C₂×C₄×C₁₉	C₂×SD₁₆	1	6	yes	—
104	C₂×C₄×C₁₉	#135	1	6	yes	—
105	C₂×C₄×C₁₉	C₂³×C₄	1	3	yes	—
106	C₂×C₄×C₁₉	C₂×#41	1	3	yes	—
107	C₈×C₁₉	—	2	6	no	—
108	C₈×C₁₉	C₂×C₄²	1	4	yes	complemented
109	C₂×C₄×SD₁₆	C₂×C₁₉	1	3	yes	—
110	C₄²×Q₈	C₂×C₁₉	1	1	yes	characteristic, complemented
111	C₄²×D₄	C₂×C₁₉	1	1	yes	characteristic
112	C₄×#106	C₂×C₁₉	1	3	yes	complemented
113	C₄×#103	C₂×C₁₉	1	3	yes	complemented
114	C₄×#102	C₂×C₁₉	1	3	yes	—
115	C₄²×C₈	C₂×C₁₉	1	1	yes	characteristic, complemented
116	C₂²×C₁₉	—	2	4	no	—
117	C₂²×C₁₉	—	4	6	no	—
118	C₂²×C₁₉	—	4	6	no	—
119	C₂²×C₁₉	C₂×C₄×D₄	1	3	yes	—
120	C₂²×C₁₉	C₂²×SD₁₆	1	1	yes	characteristic
121	C₂²×C₁₉	C₂×#135	1	3	yes	—
122	C₄×C₁₉	—	2	6	no	—
123	C₄×C₁₉	—	2	4	no	—
124	C₄×C₁₉	—	4	6	no	—
125	C₄×C₁₉	—	4	6	no	—
126	C₄×C₁₉	C₄×SD₁₆	1	12	yes	direct product factor
127	C₄×C₁₉	C₂²×C₄²	1	1	yes	characteristic
128	C₄×C₁₉	C₄×#41	1	3	yes	—
129	C₂²×SD₁₆	C₂²×C₁₉	1	1	yes	characteristic
130	C₂×C₄×Q₈	C₂²×C₁₉	1	3	yes	—
131	C₂×C₄×D₄	C₂²×C₁₉	1	3	yes	—
132	C₂²×C₄²	—	2	1	no	—
133	C₄×SD₁₆	C₄×C₁₉	1	24	yes	direct product factor
134	C₂×#106	C₂²×C₁₉	1	3	yes	—
135	C₂×#103	C₂²×C₁₉	1	3	yes	—
136	C₂×#102	C₂²×C₁₉	1	3	yes	—
137	C₂×C₄×C₈	C₂²×C₁₉	1	3	yes	—
138	C₂×C₄×C₈	C₂²×C₁₉	1	3	yes	—
139	C₄×C₄⋊C₄	—	2	3	no	—
140	C₄×C₄⋊C₄	C₂²×C₁₉	1	3	yes	—
141	C₄×C₄⋊C₄	C₂²×C₁₉	1	3	yes	—
142	C₄×C₂²⋊C₄	—	2	3	no	—
143	C₄³	—	2	1	no	—
144	C₄³	C₂²×C₁₉	1	1	yes	characteristic
145	#340	C₂²×C₁₉	1	6	yes	—
146	C₂×C₁₉	—	4	4	no	—
147	C₂×C₁₉	C₄²×D₄	1	1	yes	characteristic
148	C₂×C₁₉	C₂×C₄×SD₁₆	1	3	yes	—
149	C₂×C₁₉	C₄×#135	1	3	yes	—
150	C₂²×Q₈	C₂³×C₁₉	1	1	yes	characteristic
151	C₂²×D₄	C₂³×C₁₉	1	1	yes	characteristic
152	C₂³×C₄	—	2	3	no	—
153	C₂×SD₁₆	C₂×C₄×C₁₉	1	12	yes	—
154	C₂²×C₈	C₂³×C₁₉	1	3	yes	—
155	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
156	C₄×Q₈	—	2	6	no	—
157	C₄×Q₈	C₂×C₄×C₁₉	1	12	yes	complemented
158	C₄×D₄	—	2	6	no	—
159	C₄×D₄	C₂×C₄×C₁₉	1	12	yes	—
160	C₂×C₄⋊C₄	—	2	3	no	—
161	C₂×C₄⋊C₄	C₂³×C₁₉	1	3	yes	—
162	C₂×C₄⋊C₄	C₂³×C₁₉	1	3	yes	—
163	C₂×C₂²⋊C₄	—	2	3	no	—
164	C₂×C₄²	—	2	3	no	—
165	C₂×C₄²	—	2	3	no	—
166	C₂×C₄²	—	2	3	no	—
167	C₂×C₄²	—	4	4	no	—
168	C₂×C₄²	D₄×C₁₉	1	3	yes	—
169	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
170	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
171	#106	C₂×C₄×C₁₉	1	12	yes	complemented
172	#103	C₂×C₄×C₁₉	1	12	yes	complemented
173	#102	C₂×C₄×C₁₉	1	12	yes	—
174	C₄×C₈	—	2	6	no	—
175	C₄×C₈	C₂×C₄×C₁₉	1	12	yes	complemented
176	#95	—	2	3	no	—
177	#95	—	2	3	no	—
178	C₁₉	C₄²×SD₁₆	1	1	yes	characteristic, direct product factor, Sylow
179	C₂⁴	—	2	1	no	—
180	C₂×Q₈	—	2	3	no	—
181	C₂×Q₈	C₂²×C₄×C₁₉	1	6	yes	—
182	C₂×D₄	—	2	3	no	—
183	C₂×D₄	C₂²×C₄×C₁₉	1	6	yes	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	3	no	—
186	C₂²×C₄	—	2	3	no	—
187	C₂²×C₄	—	2	12	no	—
188	C₂²×C₄	—	4	6	no	—
189	C₂²×C₄	—	4	6	no	—
190	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
191	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
192	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
193	SD₁₆	C₄²×C₁₉	1	16	yes	direct product factor
194	C₂×C₈	—	2	6	no	—
195	C₂×C₈	C₂²×C₄×C₁₉	1	6	yes	—
196	C₂×C₈	C₂²×C₄×C₁₉	1	6	yes	—
197	C₄⋊C₄	—	2	12	no	—
198	C₄⋊C₄	—	2	3	no	—
199	C₄⋊C₄	—	2	3	no	—
200	C₄⋊C₄	C₂²×C₄×C₁₉	1	6	yes	—
201	C₄⋊C₄	C₂²×C₄×C₁₉	1	6	yes	—
202	C₂²⋊C₄	—	2	12	no	—
203	C₄²	—	2	12	no	—
204	C₄²	—	2	6	no	—
205	C₄²	—	4	6	no	—
206	C₄²	—	4	6	no	—
207	C₄²	SD₁₆×C₁₉	1	4	yes	direct product factor
208	C₄²	C₂²×C₄×C₁₉	1	6	yes	—
209	C₄²	#41×C₁₉	1	6	yes	—
210	C₂³	—	2	6	no	—
211	C₂³	—	4	1	no	—
212	C₂³	—	4	3	no	—
213	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
214	Q₈	—	2	6	no	—
215	Q₈	C₂×C₄²×C₁₉	1	4	yes	complemented
216	D₄	—	2	6	no	—
217	D₄	C₂×C₄²×C₁₉	1	4	yes	—
218	C₂×C₄	—	2	6	no	—
219	C₂×C₄	—	2	6	no	—
220	C₂×C₄	—	2	6	no	—
221	C₂×C₄	—	2	6	no	—
222	C₂×C₄	—	4	3	no	—
223	C₂×C₄	—	4	6	no	—
224	C₂×C₄	—	4	3	no	—
225	C₂×C₄	—	4	24	no	—
226	C₂×C₄	C₄×D₄×C₁₉	1	6	yes	—
227	C₂×C₄	C₄×D₄×C₁₉	1	6	yes	—
228	C₂×C₄	C₂×SD₁₆×C₁₉	1	6	yes	—
229	C₂×C₄	C₁₉×#135	1	6	yes	—
230	C₂×C₄	C₂³×C₄×C₁₉	1	3	yes	—
231	C₂×C₄	C₂×#41×C₁₉	1	3	yes	—
232	C₈	—	2	6	no	—
233	C₈	C₂×C₄²×C₁₉	1	4	yes	complemented
234	C₂²	—	2	4	no	—
235	C₂²	—	4	6	no	—
236	C₂²	—	4	6	no	—
237	C₂²	C₂×C₄×D₄×C₁₉	1	3	yes	—
238	C₂²	C₂²×SD₁₆×C₁₉	1	1	yes	characteristic
239	C₂²	C₂×C₁₉×#135	1	3	yes	—
240	C₄	—	2	6	no	—
241	C₄	—	2	4	no	—
242	C₄	—	4	6	no	—
243	C₄	—	4	6	no	—
244	C₄	C₄×SD₁₆×C₁₉	1	12	yes	direct product factor
245	C₄	C₂²×C₄²×C₁₉	1	1	yes	characteristic
246	C₄	C₄×#41×C₁₉	1	3	yes	—
247	C₂	—	4	4	no	—
248	C₂	C₄²×D₄×C₁₉	1	1	yes	characteristic
249	C₂	C₂×C₄×SD₁₆×C₁₉	1	3	yes	—
250	C₂	C₄×C₁₉×#135	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²×C₁₉	D₄
commutator	C₄	C₂²×C₄²×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle	C₂³×C₁₉	C₂²×D₄
prime radical	C₂²×D₄×C₁₉	C₂³
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₉ ◁ C₄³×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄²×Q₈×C₁₉	C₂	semidirect
4. C₄×C₁₉×#106	C₂	semidirect
5. C₄×C₁₉×#103	C₂	semidirect
7. C₄²×C₈×C₁₉	C₂	semidirect
12. C₄×SD₁₆×C₁₉	C₄	direct (N × H)
32. C₄×Q₈×C₁₉	C₂×C₄	semidirect
46. C₁₉×#106	C₂×C₄	semidirect
47. C₁₉×#103	C₂×C₄	semidirect
50. C₄×C₈×C₁₉	C₂×C₄	semidirect
67. SD₁₆×C₁₉	C₄²	direct (N × H)
81. C₄²×C₁₉	SD₁₆	direct (N × H)
84. C₄²×SD₁₆	C₁₉	direct (N × H)
90. Q₈×C₁₉	C₂×C₄²	semidirect
108. C₈×C₁₉	C₂×C₄²	semidirect
110. C₄²×Q₈	C₂×C₁₉	semidirect
112. C₄×#106	C₂×C₁₉	semidirect
113. C₄×#103	C₂×C₁₉	semidirect
115. C₄²×C₈	C₂×C₁₉	semidirect
126. C₄×C₁₉	C₄×SD₁₆	direct (N × H)
133. C₄×SD₁₆	C₄×C₁₉	direct (N × H)
157. C₄×Q₈	C₂×C₄×C₁₉	semidirect
171. #106	C₂×C₄×C₁₉	semidirect
172. #103	C₂×C₄×C₁₉	semidirect
175. C₄×C₈	C₂×C₄×C₁₉	semidirect
178. C₁₉	C₄²×SD₁₆	direct (N × H)
193. SD₁₆	C₄²×C₁₉	direct (N × H)
207. C₄²	SD₁₆×C₁₉	direct (N × H)
215. Q₈	C₂×C₄²×C₁₉	semidirect
233. C₈	C₂×C₄²×C₁₉	semidirect
244. C₄	C₄×SD₁₆×C₁₉	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×C₄×SD₁₆×C₁₉	2
C₄²×Q₈×C₁₉	2
C₄²×D₄×C₁₉	2
C₄×C₁₉×#106	2
C₄×C₁₉×#103	2
C₄×C₁₉×#102	2
C₄²×C₈×C₁₉	2
C₄²×SD₁₆	19

Subgroups — C42×SD16×C19 #18026457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×SD₁₆×C₁₉ #18026457