Subgroups — C₂×C₁₉×#1847 #18026757

Order 4864.

1230 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×#107	C₂	1	2	yes	maximal
2	C₂²×C₁₉×#106	C₂	1	2	yes	maximal
3	C₂²×C₁₉×#97	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉×#384	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₉×#382	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1847	C₂	1	8	yes	maximal, direct product factor
7	C₂³×C₈×C₁₉	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₁₉	—	2	2	no	—
9	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
12	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#107	—	2	8	no	—
14	C₂×C₁₉×#107	C₂²	1	8	yes	—
15	C₂×C₁₉×#106	—	2	8	no	—
16	C₂×C₁₉×#106	C₂²	1	8	yes	—
17	C₂×C₁₉×#97	C₄	1	8	yes	complemented
18	C₂×C₁₉×#97	C₂²	1	4	yes	—
19	C₁₉×#384	—	2	2	no	—
20	C₁₉×#384	C₂²	1	4	yes	—
21	C₁₉×#382	—	2	2	no	—
22	C₁₉×#382	C₂²	1	4	yes	—
23	C₂×C₁₉×#95	—	2	2	no	—
24	C₂³×C₄×C₁₉	—	2	1	no	—
25	C₂³×C₄×C₁₉	—	2	1	no	—
26	C₂³×C₄×C₁₉	—	2	2	no	—
27	C₂³×C₄×C₁₉	D₄	1	2	yes	—
28	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	C₂×C₄	1	8	yes	—
30	C₂²×C₈×C₁₉	D₄	1	8	yes	—
31	C₂²×C₈×C₁₉	Q₈	1	8	yes	—
32	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
33	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
35	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
36	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
37	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
38	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
39	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
41	C₂×C₄⋊C₄×C₁₉	—	4	8	no	—
42	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
45	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₁₉	D₄	1	2	yes	—
47	C₂×C₄²×C₁₉	Q₈	1	4	yes	—
48	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
49	C₁₉×#107	—	2	16	no	—
50	C₁₉×#106	—	2	16	no	—
51	C₁₉×#97	C₂×C₄	1	16	yes	complemented
52	C₁₉×#95	—	2	8	no	—
53	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	—	2	4	no	—
55	C₂²×C₄×C₁₉	—	2	4	no	—
56	C₂²×C₄×C₁₉	—	2	4	no	—
57	C₂²×C₄×C₁₉	—	2	2	no	—
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	—	4	4	no	—
60	C₂²×C₄×C₁₉	—	4	2	no	—
61	C₂²×C₄×C₁₉	—	4	2	no	—
62	C₂²×C₄×C₁₉	—	4	8	no	—
63	C₂²×C₄×C₁₉	—	4	4	no	—
64	C₂²×C₄×C₁₉	—	4	4	no	—
65	C₂²×C₄×C₁₉	C₄⋊C₄	1	8	yes	—
66	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₁₉	C₂×Q₈	1	4	yes	—
71	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₁₉	C₄⋊C₄	1	32	yes	complemented
74	C₂×C₈×C₁₉	C₂²×C₄	1	8	yes	—
75	C₂×C₈×C₁₉	C₂×D₄	1	8	yes	—
76	C₂×C₈×C₁₉	C₂×Q₈	1	8	yes	—
77	C₄⋊C₄×C₁₉	—	2	4	no	—
78	C₄⋊C₄×C₁₉	—	2	4	no	—
79	C₄⋊C₄×C₁₉	—	2	4	no	—
80	C₄⋊C₄×C₁₉	—	2	4	no	—
81	C₄⋊C₄×C₁₉	—	4	16	no	—
82	C₄⋊C₄×C₁₉	—	4	2	no	—
83	C₄⋊C₄×C₁₉	—	4	4	no	—
84	C₄⋊C₄×C₁₉	—	4	2	no	—
85	C₄²×C₁₉	C₄⋊C₄	1	8	yes	—
86	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
87	C₄²×C₁₉	C₂×Q₈	1	4	yes	—
88	C₂×#1847	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
90	C₂³×C₁₉	#127	1	2	yes	—
91	C₂³×C₁₉	C₄⋊Q₈	1	2	yes	—
92	C₂³×C₁₉	AGL(1,ℤ/8)	1	4	yes	—
93	C₂³×C₁₉	#137	1	4	yes	—
94	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₉	—	2	4	no	—
96	C₂×C₄×C₁₉	—	2	4	no	—
97	C₂×C₄×C₁₉	—	2	16	no	—
98	C₂×C₄×C₁₉	—	4	4	no	—
99	C₂×C₄×C₁₉	—	4	2	no	—
100	C₂×C₄×C₁₉	—	4	2	no	—
101	C₂×C₄×C₁₉	—	4	8	no	—
102	C₂×C₄×C₁₉	—	4	4	no	—
103	C₂×C₄×C₁₉	—	4	4	no	—
104	C₂×C₄×C₁₉	—	8	8	no	—
105	C₂×C₄×C₁₉	—	8	4	no	—
106	C₂×C₄×C₁₉	—	8	4	no	—
107	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	8	yes	—
108	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	8	yes	—
109	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₁₉	#127	1	2	yes	—
111	C₂×C₄×C₁₉	C₄⋊Q₈	1	8	yes	—
112	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
113	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
115	C₂×C₄×C₁₉	C₂²×Q₈	1	2	yes	—
116	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
117	C₈×C₁₉	C₂×C₄⋊C₄	1	16	yes	complemented
118	C₂²×#107	C₂×C₁₉	1	2	yes	—
119	C₂²×#106	C₂×C₁₉	1	2	yes	—
120	C₂²×#97	C₂×C₁₉	1	1	yes	characteristic
121	C₂×#384	C₂×C₁₉	1	1	yes	characteristic
122	C₂×#382	C₂×C₁₉	1	1	yes	characteristic
123	#1847	C₂×C₁₉	1	8	yes	direct product factor
124	C₂²×C₁₉	#384	1	4	yes	—
125	C₂²×C₁₉	#428	1	8	yes	—
126	C₂²×C₁₉	#496	1	4	yes	—
127	C₂²×C₁₉	#497	1	4	yes	—
128	C₂²×C₁₉	#501	1	8	yes	—
129	C₂²×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₁₉	C₂×#127	1	1	yes	characteristic
131	C₂²×C₁₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	C₂²×C₁₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
133	C₂²×C₁₉	C₂×#137	1	2	yes	—
134	C₄×C₁₉	—	2	8	no	—
135	C₄×C₁₉	—	8	4	no	—
136	C₄×C₁₉	—	8	2	no	—
137	C₄×C₁₉	—	8	2	no	—
138	C₄×C₁₉	#384	1	4	yes	—
139	C₄×C₁₉	C₂²×C₄⋊C₄	1	2	yes	—
140	C₄×C₁₉	C₂×C₄⋊Q₈	1	2	yes	—
141	C₂³×C₈	C₂²×C₁₉	1	2	yes	—
142	C₂²×C₄⋊C₄	—	2	2	no	—
143	C₂²×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
144	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
145	C₂²×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
146	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
147	C₂×#107	—	2	8	no	—
148	C₂×#107	C₂²×C₁₉	1	8	yes	—
149	C₂×#106	—	2	8	no	—
150	C₂×#106	C₂²×C₁₉	1	8	yes	—
151	C₂×#97	C₄×C₁₉	1	8	yes	complemented
152	C₂×#97	C₂²×C₁₉	1	4	yes	—
153	#384	—	2	2	no	—
154	#384	C₂²×C₁₉	1	4	yes	—
155	#382	—	2	2	no	—
156	#382	C₂²×C₁₉	1	4	yes	—
157	C₂×#95	—	2	2	no	—
158	C₂×C₁₉	#1847	1	8	yes	direct product factor
159	C₂×C₁₉	C₂×#384	1	1	yes	characteristic
160	C₂×C₁₉	C₂×#428	1	2	yes	—
161	C₂×C₁₉	C₂×#496	1	1	yes	characteristic
162	C₂×C₁₉	C₂×#497	1	1	yes	characteristic
163	C₂×C₁₉	C₂×#501	1	2	yes	—
164	C₂³×C₄	—	2	2	no	—
165	C₂³×C₄	—	2	1	no	—
166	C₂³×C₄	—	2	1	no	—
167	C₂³×C₄	D₄×C₁₉	1	2	yes	—
168	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
169	C₂²×C₈	C₂×C₄×C₁₉	1	8	yes	—
170	C₂²×C₈	D₄×C₁₉	1	8	yes	—
171	C₂²×C₈	Q₈×C₁₉	1	8	yes	—
172	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
173	C₂×C₄⋊C₄	—	2	4	no	—
174	C₂×C₄⋊C₄	—	2	4	no	—
175	C₂×C₄⋊C₄	—	2	4	no	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	—	2	8	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	4	8	no	—
182	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
183	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
184	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
185	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
186	C₂×C₄²	D₄×C₁₉	1	2	yes	—
187	C₂×C₄²	Q₈×C₁₉	1	4	yes	—
188	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
189	#107	—	2	16	no	—
190	#106	—	2	16	no	—
191	#97	C₂×C₄×C₁₉	1	16	yes	complemented
192	#95	—	2	8	no	—
193	C₁₉	C₂×#1847	1	1	yes	characteristic, direct product factor, Sylow
194	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	8	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	C₄⋊C₄×C₁₉	1	8	yes	—
207	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
208	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
209	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
211	C₂²×C₄	C₂×Q₈×C₁₉	1	4	yes	—
212	C₂²×C₄	C₂×Q₈×C₁₉	1	2	yes	—
213	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
214	C₂×C₈	C₄⋊C₄×C₁₉	1	32	yes	complemented
215	C₂×C₈	C₂²×C₄×C₁₉	1	8	yes	—
216	C₂×C₈	C₂×D₄×C₁₉	1	8	yes	—
217	C₂×C₈	C₂×Q₈×C₁₉	1	8	yes	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	4	16	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄²	C₄⋊C₄×C₁₉	1	8	yes	—
227	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
228	C₄²	C₂×Q₈×C₁₉	1	4	yes	—
229	C₂³	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
230	C₂³	C₁₉×#127	1	2	yes	—
231	C₂³	C₁₉×C₄⋊Q₈	1	2	yes	—
232	C₂³	C₁₉×AGL(1,ℤ/8)	1	4	yes	—
233	C₂³	C₁₉×#137	1	4	yes	—
234	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
235	C₂×C₄	—	2	16	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	8	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	8	8	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	—	8	4	no	—
247	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	8	yes	—
248	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
249	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	8	yes	—
250	C₂×C₄	C₁₉×#127	1	2	yes	—
251	C₂×C₄	C₁₉×C₄⋊Q₈	1	2	yes	—
252	C₂×C₄	C₁₉×C₄⋊Q₈	1	8	yes	—
253	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
254	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
255	C₂×C₄	C₂²×Q₈×C₁₉	1	2	yes	—
256	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
257	C₈	C₂×C₄⋊C₄×C₁₉	1	16	yes	complemented
258	C₂²	C₁₉×#384	1	4	yes	—
259	C₂²	C₁₉×#428	1	8	yes	—
260	C₂²	C₁₉×#496	1	4	yes	—
261	C₂²	C₁₉×#497	1	4	yes	—
262	C₂²	C₁₉×#501	1	8	yes	—
263	C₂²	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
264	C₂²	C₂×C₁₉×#127	1	1	yes	characteristic
265	C₂²	C₂×C₁₉×C₄⋊Q₈	1	1	yes	characteristic
266	C₂²	C₂×C₁₉×AGL(1,ℤ/8)	1	2	yes	—
267	C₂²	C₂×C₁₉×#137	1	2	yes	—
268	C₄	—	2	8	no	—
269	C₄	—	8	4	no	—
270	C₄	—	8	2	no	—
271	C₄	—	8	2	no	—
272	C₄	C₁₉×#384	1	4	yes	—
273	C₄	C₂²×C₄⋊C₄×C₁₉	1	2	yes	—
274	C₄	C₂×C₁₉×C₄⋊Q₈	1	2	yes	—
275	C₂	C₁₉×#1847	1	8	yes	direct product factor
276	C₂	C₂×C₁₉×#384	1	1	yes	characteristic
277	C₂	C₂×C₁₉×#428	1	2	yes	—
278	C₂	C₂×C₁₉×#496	1	1	yes	characteristic
279	C₂	C₂×C₁₉×#497	1	1	yes	characteristic
280	C₂	C₂×C₁₉×#501	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
6. C₁₉×#1847	C₂	direct (N × H)
17. C₂×C₁₉×#97	C₄	semidirect
51. C₁₉×#97	C₂×C₄	semidirect
73. C₂×C₈×C₁₉	C₄⋊C₄	semidirect
88. C₂×#1847	C₁₉	direct (N × H)
117. C₈×C₁₉	C₂×C₄⋊C₄	semidirect
123. #1847	C₂×C₁₉	direct (N × H)
151. C₂×#97	C₄×C₁₉	semidirect
158. C₂×C₁₉	#1847	direct (N × H)
191. #97	C₂×C₄×C₁₉	semidirect
193. C₁₉	C₂×#1847	direct (N × H)
214. C₂×C₈	C₄⋊C₄×C₁₉	semidirect
257. C₈	C₂×C₄⋊C₄×C₁₉	semidirect
275. C₂	C₁₉×#1847	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×#107	2
C₂²×C₁₉×#106	2
C₂²×C₁₉×#97	2
C₂×C₁₉×#384	2
C₂×C₁₉×#382	2
C₁₉×#1847	2
C₂×#1847	19

Subgroups — C2×C19×#1847 #18026757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#1847 #18026757