Subgroups — C₁₉×#21334 #18027293

Order 4864.

1142 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#544	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2316	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2274	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1996	C₂	1	8	yes	maximal
5	C₁₉×#1740	C₂	1	4	yes	maximal
6	C₁₉×#544	—	2	4	no	—
7	C₁₉×#544	—	2	4	no	—
8	C₂×C₁₉×C₄⋊Q₈	C₄	1	4	yes	complemented
9	C₂×C₁₉×#125	C₄	1	4	yes	complemented
10	C₂×C₁₉×C₂²⋊Q₈	C₂²	1	4	yes	—
11	C₂×C₁₉×#117	C₂²	1	2	yes	—
12	C₂×C₁₉×#117	C₂²	1	2	yes	—
13	C₂×C₁₉×#117	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#389	—	2	4	no	—
16	C₁₉×#380	C₂²	1	4	yes	—
17	C₄×C₄⋊C₄×C₁₉	—	2	4	no	—
18	C₄×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
19	C₁₉×#342	C₂²	1	16	yes	—
20	C₂²×Q₈×C₁₉	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
22	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
23	C₁₉×C₄⋊Q₈	—	2	16	no	—
24	C₁₉×#125	—	2	16	no	—
25	C₁₉×C₂²⋊Q₈	—	2	8	no	—
26	C₁₉×C₂²⋊Q₈	—	2	8	no	—
27	C₁₉×#117	—	2	4	no	—
28	C₁₉×#117	—	2	4	no	—
29	C₁₉×#117	—	2	4	no	—
30	C₁₉×#117	—	2	8	no	—
31	C₁₉×#117	Q₈	1	8	yes	complemented
32	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
35	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	8	yes	—
39	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊C₄×C₁₉	D₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₉	D₄	1	4	yes	—
42	C₂×C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
43	C₂×C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
44	C₂×C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
45	C₂×C₄²×C₁₉	—	2	2	no	—
46	C₂×C₄²×C₁₉	—	2	2	no	—
47	C₂×C₄²×C₁₉	—	2	4	no	—
48	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
49	C₁₉×#95	—	2	4	no	—
50	C₁₉×#95	—	2	4	no	—
51	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
52	C₂×Q₈×C₁₉	—	2	8	no	—
53	C₂×Q₈×C₁₉	—	8	4	no	—
54	C₂²×C₄×C₁₉	—	2	4	no	—
55	C₂²×C₄×C₁₉	—	2	4	no	—
56	C₂²×C₄×C₁₉	—	8	2	no	—
57	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
62	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₁₉	#41	1	4	yes	—
67	C₄⋊C₄×C₁₉	—	2	4	no	—
68	C₄⋊C₄×C₁₉	—	2	4	no	—
69	C₄⋊C₄×C₁₉	—	2	8	no	—
70	C₄⋊C₄×C₁₉	—	2	16	no	—
71	C₄⋊C₄×C₁₉	—	2	8	no	—
72	C₄⋊C₄×C₁₉	—	4	4	no	—
73	C₄⋊C₄×C₁₉	—	8	4	no	—
74	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	16	yes	—
75	C₂²⋊C₄×C₁₉	—	2	4	no	—
76	C₂²⋊C₄×C₁₉	—	2	4	no	—
77	C₂²⋊C₄×C₁₉	—	2	4	no	—
78	C₂²⋊C₄×C₁₉	—	2	8	no	—
79	C₂²⋊C₄×C₁₉	C₂×Q₈	1	8	yes	—
80	C₄²×C₁₉	—	2	8	no	—
81	C₄²×C₁₉	—	2	8	no	—
82	C₄²×C₁₉	—	4	4	no	—
83	C₄²×C₁₉	—	8	4	no	—
84	#21334	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₁₉	—	4	1	no	—
86	C₂³×C₁₉	—	4	1	no	—
87	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
88	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
89	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
90	C₂³×C₁₉	C₂²×Q₈	1	2	yes	—
91	C₂³×C₁₉	C₂×#41	1	2	yes	—
92	Q₈×C₁₉	—	8	8	no	—
93	C₂×C₄×C₁₉	—	2	4	no	—
94	C₂×C₄×C₁₉	—	2	4	no	—
95	C₂×C₄×C₁₉	—	2	8	no	—
96	C₂×C₄×C₁₉	—	4	4	no	—
97	C₂×C₄×C₁₉	—	4	4	no	—
98	C₂×C₄×C₁₉	—	4	4	no	—
99	C₂×C₄×C₁₉	—	4	4	no	—
100	C₂×C₄×C₁₉	—	4	4	no	—
101	C₂×C₄×C₁₉	—	4	4	no	—
102	C₂×C₄×C₁₉	—	4	4	no	—
103	C₂×C₄×C₁₉	—	4	4	no	—
104	C₂×C₄×C₁₉	—	4	4	no	—
105	C₂×C₄×C₁₉	—	4	4	no	—
106	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	8	yes	—
107	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
108	C₂×C₄×C₁₉	C₄×Q₈	1	8	yes	—
109	C₂×C₄×C₁₉	C₂²⋊Q₈	1	4	yes	—
110	C₂×C₄×C₁₉	C₂²⋊Q₈	1	4	yes	—
111	C₂×C₄×C₁₉	#124	1	4	yes	—
112	C₂×C₄×C₁₉	C₄⋊Q₈	1	4	yes	—
113	C₂×#544	C₂×C₁₉	1	1	yes	characteristic
114	#2316	C₂×C₁₉	1	1	yes	characteristic
115	#2274	C₂×C₁₉	1	1	yes	characteristic
116	#1996	C₂×C₁₉	1	8	yes	—
117	#1740	C₂×C₁₉	1	4	yes	—
118	C₂²×C₁₉	—	2	1	no	—
119	C₂²×C₁₉	—	2	1	no	—
120	C₂²×C₁₉	—	2	1	no	—
121	C₂²×C₁₉	—	2	1	no	—
122	C₂²×C₁₉	—	2	1	no	—
123	C₂²×C₁₉	—	2	1	no	—
124	C₂²×C₁₉	—	4	2	no	—
125	C₂²×C₁₉	—	4	2	no	—
126	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₁₉	C₂×C₄×Q₈	1	2	yes	—
128	C₂²×C₁₉	C₂×C₂²⋊Q₈	1	2	yes	—
129	C₂²×C₁₉	C₂×#124	1	1	yes	characteristic
130	C₂²×C₁₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
131	C₄×C₁₉	—	2	8	no	—
132	C₄×C₁₉	—	4	8	no	—
133	C₄×C₁₉	—	4	4	no	—
134	C₄×C₁₉	—	4	4	no	—
135	C₄×C₁₉	—	8	4	no	—
136	C₄×C₁₉	#391	1	8	yes	—
137	#544	—	2	4	no	—
138	#544	—	2	4	no	—
139	C₂×C₄⋊Q₈	C₄×C₁₉	1	4	yes	complemented
140	C₂×#125	C₄×C₁₉	1	4	yes	complemented
141	C₂×C₂²⋊Q₈	C₂²×C₁₉	1	4	yes	—
142	C₂×#117	C₂²×C₁₉	1	2	yes	—
143	C₂×#117	C₂²×C₁₉	1	2	yes	—
144	C₂×#117	C₂²×C₁₉	1	2	yes	—
145	C₂²×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
146	#389	—	2	4	no	—
147	#380	C₂²×C₁₉	1	4	yes	—
148	C₄×C₄⋊C₄	—	2	4	no	—
149	C₄×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
150	#342	C₂²×C₁₉	1	16	yes	—
151	C₂×C₁₉	—	2	2	no	—
152	C₂×C₁₉	—	2	1	no	—
153	C₂×C₁₉	—	2	2	no	—
154	C₂×C₁₉	—	2	1	no	—
155	C₂×C₁₉	C₂×#391	1	1	yes	characteristic
156	C₂×C₁₉	#2883	1	1	yes	characteristic
157	C₂×C₁₉	#2884	1	1	yes	characteristic
158	C₂²×Q₈	C₂×C₄×C₁₉	1	4	yes	—
159	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
160	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
161	C₄⋊Q₈	—	2	16	no	—
162	#125	—	2	16	no	—
163	C₂²⋊Q₈	—	2	8	no	—
164	C₂²⋊Q₈	—	2	8	no	—
165	#117	—	2	4	no	—
166	#117	—	2	4	no	—
167	#117	—	2	4	no	—
168	#117	—	2	8	no	—
169	#117	Q₈×C₁₉	1	8	yes	complemented
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	4	no	—
173	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
174	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
175	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
176	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	8	yes	—
177	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
178	C₂×C₄⋊C₄	D₄×C₁₉	1	4	yes	—
179	C₂×C₄⋊C₄	D₄×C₁₉	1	4	yes	—
180	C₂×C₂²⋊C₄	C₂³×C₁₉	1	4	yes	—
181	C₂×C₂²⋊C₄	C₂³×C₁₉	1	4	yes	—
182	C₂×C₂²⋊C₄	C₂³×C₁₉	1	4	yes	—
183	C₂×C₄²	—	2	2	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	—	2	4	no	—
186	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
187	#95	—	2	4	no	—
188	#95	—	2	4	no	—
189	C₁₉	#21334	1	1	yes	characteristic, direct product factor, Sylow
190	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
191	C₂×Q₈	—	2	8	no	—
192	C₂×Q₈	—	8	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	8	2	no	—
196	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
197	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
198	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
199	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
200	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
201	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
202	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
204	C₂²×C₄	C₂×Q₈×C₁₉	1	4	yes	—
205	C₂²×C₄	#41×C₁₉	1	4	yes	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	8	no	—
209	C₄⋊C₄	—	2	16	no	—
210	C₄⋊C₄	—	2	8	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₄⋊C₄	—	8	4	no	—
213	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	16	yes	—
214	C₂²⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	2	4	no	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₂²⋊C₄	—	2	8	no	—
218	C₂²⋊C₄	C₂×Q₈×C₁₉	1	8	yes	—
219	C₄²	—	2	8	no	—
220	C₄²	—	2	8	no	—
221	C₄²	—	4	4	no	—
222	C₄²	—	8	4	no	—
223	C₂³	—	4	1	no	—
224	C₂³	—	4	1	no	—
225	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
226	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
227	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
228	C₂³	C₂²×Q₈×C₁₉	1	2	yes	—
229	C₂³	C₂×#41×C₁₉	1	2	yes	—
230	Q₈	—	8	8	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	8	yes	—
245	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
246	C₂×C₄	C₄×Q₈×C₁₉	1	8	yes	—
247	C₂×C₄	C₁₉×C₂²⋊Q₈	1	4	yes	—
248	C₂×C₄	C₁₉×C₂²⋊Q₈	1	4	yes	—
249	C₂×C₄	C₁₉×#124	1	4	yes	—
250	C₂×C₄	C₁₉×C₄⋊Q₈	1	4	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
260	C₂²	C₂×C₄×Q₈×C₁₉	1	2	yes	—
261	C₂²	C₂×C₁₉×C₂²⋊Q₈	1	2	yes	—
262	C₂²	C₂×C₁₉×#124	1	1	yes	characteristic
263	C₂²	C₂×C₁₉×C₄⋊Q₈	1	1	yes	characteristic
264	C₄	—	2	8	no	—
265	C₄	—	4	8	no	—
266	C₄	—	4	4	no	—
267	C₄	—	4	4	no	—
268	C₄	—	8	4	no	—
269	C₄	C₁₉×#391	1	8	yes	—
270	C₂	—	2	2	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	2	no	—
273	C₂	—	2	1	no	—
274	C₂	C₂×C₁₉×#391	1	1	yes	characteristic
275	C₂	C₁₉×#2883	1	1	yes	characteristic
276	C₂	C₁₉×#2884	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
8. C₂×C₁₉×C₄⋊Q₈	C₄	semidirect
9. C₂×C₁₉×#125	C₄	semidirect
31. C₁₉×#117	Q₈	semidirect
84. #21334	C₁₉	direct (N × H)
139. C₂×C₄⋊Q₈	C₄×C₁₉	semidirect
140. C₂×#125	C₄×C₁₉	semidirect
169. #117	Q₈×C₁₉	semidirect
189. C₁₉	#21334	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×#544	2
C₁₉×#2316	2
C₁₉×#2274	2
C₁₉×#1996	2
C₁₉×#1740	2
#21334	19

Subgroups — C19×#21334 #18027293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#21334 #18027293