Subgroups — C₁₉×#8396 #18014355

Order 4864.

738 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈²×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1801	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1549	C₂	1	2	yes	maximal
4	C₁₉×#1487	C₂	1	2	yes	maximal
5	C₁₉×#1461	C₂	1	1	yes	maximal, characteristic
6	Q₈²×C₁₉	—	2	4	no	—
7	Q₈²×C₁₉	—	2	4	no	—
8	C₂×C₁₉×C₄⋊Q₈	—	2	2	no	—
9	C₂×C₁₉×C₄⋊Q₈	—	2	1	no	—
10	C₂×C₁₉×C₄⋊Q₈	C₄	1	2	yes	—
11	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₁₉	—	2	1	no	—
13	C₂×C₄×Q₈×C₁₉	C₄	1	2	yes	—
14	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#431	C₂²	1	2	yes	—
17	C₁₉×#423	C₂²	1	2	yes	—
18	C₂×C₁₉×#101	—	2	1	no	—
19	C₁₉×#343	—	2	1	no	—
20	C₁₉×#332	—	2	4	no	—
21	C₁₉×Q₈⋊C₈	—	2	4	no	—
22	C₁₉×#324	—	2	2	no	—
23	C₂²×Q₈×C₁₉	—	2	2	no	—
24	C₂²×Q₈×C₁₉	—	2	1	no	—
25	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
27	C₂×M₁₆×C₁₉	—	2	2	no	—
28	C₁₉×C₄⋊Q₈	—	2	16	no	—
29	C₁₉×C₄⋊Q₈	—	2	8	no	—
30	C₁₉×C₄⋊Q₈	—	2	4	no	—
31	C₁₉×C₄⋊Q₈	—	2	4	no	—
32	C₁₉×C₄⋊Q₈	—	2	2	no	—
33	C₁₉×C₄⋊Q₈	D₄	1	4	yes	—
34	C₄×Q₈×C₁₉	—	2	8	no	—
35	C₄×Q₈×C₁₉	—	2	4	no	—
36	C₄×Q₈×C₁₉	—	2	4	no	—
37	C₄×Q₈×C₁₉	—	2	2	no	—
38	C₄×Q₈×C₁₉	—	2	2	no	—
39	C₄×Q₈×C₁₉	—	2	2	no	—
40	C₄×Q₈×C₁₉	D₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
44	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
45	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
49	C₂×C₄²×C₁₉	—	2	2	no	—
50	C₂×C₄²×C₁₉	—	2	1	no	—
51	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
53	C₁₉×C₄⋊C₈	—	2	2	no	—
54	C₁₉×C₄⋊C₈	—	2	2	no	—
55	C₁₉×C₄⋊C₈	—	2	2	no	—
56	C₁₉×#101	—	4	2	no	—
57	C₁₉×C₂²⋊C₈	—	2	2	no	—
58	C₄×C₈×C₁₉	—	2	2	no	—
59	C₂×Q₈×C₁₉	—	2	4	no	—
60	C₂×Q₈×C₁₉	—	2	8	no	—
61	C₂×Q₈×C₁₉	—	2	1	no	—
62	C₂×Q₈×C₁₉	—	2	2	no	—
63	C₂×Q₈×C₁₉	—	2	4	no	—
64	C₂×Q₈×C₁₉	—	2	2	no	—
65	C₂×Q₈×C₁₉	—	8	2	no	—
66	C₂×Q₈×C₁₉	—	8	4	no	—
67	C₂×Q₈×C₁₉	C₂²⋊C₄	1	2	yes	—
68	C₂×Q₈×C₁₉	SD₁₆	1	2	yes	—
69	C₂×Q₈×C₁₉	Q₁₆	1	2	yes	—
70	C₂²×C₄×C₁₉	—	2	1	no	—
71	C₂²×C₄×C₁₉	—	2	2	no	—
72	C₂²×C₄×C₁₉	—	2	1	no	—
73	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
78	M₁₆×C₁₉	—	4	2	no	—
79	M₁₆×C₁₉	—	4	2	no	—
80	C₂×C₈×C₁₉	—	4	2	no	—
81	C₂×C₈×C₁₉	—	4	2	no	—
82	C₄⋊C₄×C₁₉	—	2	8	no	—
83	C₄⋊C₄×C₁₉	—	2	8	no	—
84	C₄⋊C₄×C₁₉	—	2	1	no	—
85	C₄⋊C₄×C₁₉	—	2	2	no	—
86	C₄⋊C₄×C₁₉	—	2	4	no	—
87	C₄⋊C₄×C₁₉	—	2	4	no	—
88	C₄⋊C₄×C₁₉	—	2	4	no	—
89	C₄⋊C₄×C₁₉	—	2	2	no	—
90	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
91	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₁₉	—	2	8	no	—
93	C₄²×C₁₉	—	2	2	no	—
94	C₄²×C₁₉	—	2	2	no	—
95	C₄²×C₁₉	—	2	4	no	—
96	C₄²×C₁₉	C₂×D₄	1	2	yes	—
97	C₄²×C₁₉	C₂×D₄	1	2	yes	—
98	#8396	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	Q₈×C₁₉	—	2	4	no	—
101	Q₈×C₁₉	—	2	2	no	—
102	Q₈×C₁₉	—	2	4	no	—
103	Q₈×C₁₉	—	2	4	no	—
104	Q₈×C₁₉	—	8	4	no	—
105	Q₈×C₁₉	—	8	8	no	—
106	Q₈×C₁₉	#101	1	4	yes	complemented
107	C₂×C₄×C₁₉	—	2	4	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	2	2	no	—
110	C₂×C₄×C₁₉	—	2	2	no	—
111	C₂×C₄×C₁₉	—	2	2	no	—
112	C₂×C₄×C₁₉	—	2	2	no	—
113	C₂×C₄×C₁₉	—	2	4	no	—
114	C₂×C₄×C₁₉	—	2	2	no	—
115	C₂×C₄×C₁₉	—	4	1	no	—
116	C₂×C₄×C₁₉	—	4	4	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	#103	1	4	yes	—
120	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#120	1	2	yes	—
123	C₂×C₄×C₁₉	#120	1	2	yes	—
124	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×C₄×C₁₉	#137	1	2	yes	—
127	C₈×C₁₉	—	8	2	no	—
128	C₈×C₁₉	—	8	2	no	—
129	C₂×Q₈²	C₂×C₁₉	1	1	yes	characteristic
130	#1801	C₂×C₁₉	1	1	yes	characteristic
131	#1549	C₂×C₁₉	1	2	yes	—
132	#1487	C₂×C₁₉	1	2	yes	—
133	#1461	C₂×C₁₉	1	1	yes	characteristic
134	C₂²×C₁₉	—	2	1	no	—
135	C₂²×C₁₉	—	2	1	no	—
136	C₂²×C₁₉	#379	1	1	yes	characteristic
137	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
138	C₂²×C₁₉	#419	1	1	yes	characteristic
139	C₄×C₁₉	—	2	2	no	—
140	C₄×C₁₉	—	2	4	no	—
141	C₄×C₁₉	—	2	1	no	—
142	C₄×C₁₉	—	2	4	no	—
143	C₄×C₁₉	—	4	4	no	—
144	C₄×C₁₉	—	4	2	no	—
145	C₄×C₁₉	—	4	2	no	—
146	C₄×C₁₉	C₂×#101	1	2	yes	—
147	C₄×C₁₉	#448	1	2	yes	—
148	C₄×C₁₉	#451	1	2	yes	—
149	Q₈²	—	2	4	no	—
150	Q₈²	—	2	4	no	—
151	C₂×C₄⋊Q₈	—	2	2	no	—
152	C₂×C₄⋊Q₈	—	2	1	no	—
153	C₂×C₄⋊Q₈	C₄×C₁₉	1	2	yes	—
154	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
155	C₂×C₄×Q₈	—	2	1	no	—
156	C₂×C₄×Q₈	C₄×C₁₉	1	2	yes	—
157	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
158	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
159	#431	C₂²×C₁₉	1	2	yes	—
160	#423	C₂²×C₁₉	1	2	yes	—
161	C₂×#101	—	2	1	no	—
162	#343	—	2	1	no	—
163	#332	—	2	4	no	—
164	Q₈⋊C₈	—	2	4	no	—
165	#324	—	2	2	no	—
166	C₂×C₁₉	—	2	1	no	—
167	C₂×C₁₉	—	2	1	no	—
168	C₂×C₁₉	#1784	1	1	yes	characteristic
169	C₂×C₁₉	#1787	1	1	yes	characteristic
170	C₂×C₁₉	#1795	1	1	yes	characteristic
171	C₂²×Q₈	—	2	2	no	—
172	C₂²×Q₈	—	2	1	no	—
173	C₂²×Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
174	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
175	C₂×M₁₆	—	2	2	no	—
176	C₄⋊Q₈	—	2	16	no	—
177	C₄⋊Q₈	—	2	8	no	—
178	C₄⋊Q₈	—	2	4	no	—
179	C₄⋊Q₈	—	2	4	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	D₄×C₁₉	1	4	yes	—
182	C₄×Q₈	—	2	8	no	—
183	C₄×Q₈	—	2	4	no	—
184	C₄×Q₈	—	2	4	no	—
185	C₄×Q₈	—	2	2	no	—
186	C₄×Q₈	—	2	2	no	—
187	C₄×Q₈	—	2	2	no	—
188	C₄×Q₈	D₄×C₁₉	1	4	yes	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
194	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
195	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
196	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
200	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	—	2	2	no	—
203	C₄⋊C₈	—	2	2	no	—
204	#101	—	4	2	no	—
205	C₂²⋊C₈	—	2	2	no	—
206	C₄×C₈	—	2	2	no	—
207	C₁₉	#8396	1	1	yes	characteristic, direct product factor, Sylow
208	C₂×Q₈	—	2	4	no	—
209	C₂×Q₈	—	2	8	no	—
210	C₂×Q₈	—	2	1	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	2	4	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	8	2	no	—
215	C₂×Q₈	—	8	4	no	—
216	C₂×Q₈	C₂²⋊C₄×C₁₉	1	2	yes	—
217	C₂×Q₈	SD₁₆×C₁₉	1	2	yes	—
218	C₂×Q₈	Q₁₆×C₁₉	1	2	yes	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
223	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
224	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
227	M₁₆	—	4	2	no	—
228	M₁₆	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₄⋊C₄	—	2	8	no	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
240	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
241	C₄²	—	2	8	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	4	no	—
245	C₄²	C₂×D₄×C₁₉	1	2	yes	—
246	C₄²	C₂×D₄×C₁₉	1	2	yes	—
247	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
248	Q₈	—	2	4	no	—
249	Q₈	—	2	2	no	—
250	Q₈	—	2	4	no	—
251	Q₈	—	2	4	no	—
252	Q₈	—	8	4	no	—
253	Q₈	—	8	8	no	—
254	Q₈	C₁₉×#101	1	4	yes	complemented
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	C₁₉×#103	1	4	yes	—
268	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
270	C₂×C₄	C₁₉×#120	1	2	yes	—
271	C₂×C₄	C₁₉×#120	1	2	yes	—
272	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
273	C₂×C₄	C₂×Q₁₆×C₁₉	1	1	yes	characteristic
274	C₂×C₄	C₁₉×#137	1	2	yes	—
275	C₈	—	8	2	no	—
276	C₈	—	8	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	C₁₉×#379	1	1	yes	characteristic
280	C₂²	C₂×C₁₉×#103	1	1	yes	characteristic
281	C₂²	C₁₉×#419	1	1	yes	characteristic
282	C₄	—	2	4	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	4	no	—
285	C₄	—	2	2	no	—
286	C₄	—	4	4	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₄	C₂×C₁₉×#101	1	2	yes	—
290	C₄	C₁₉×#448	1	2	yes	—
291	C₄	C₁₉×#451	1	2	yes	—
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	C₁₉×#1784	1	1	yes	characteristic
295	C₂	C₁₉×#1787	1	1	yes	characteristic
296	C₂	C₁₉×#1795	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#379
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
98. #8396	C₁₉	direct (N × H)
106. Q₈×C₁₉	#101	semidirect
207. C₁₉	#8396	direct (N × H)
254. Q₈	C₁₉×#101	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈²×C₁₉	2
C₁₉×#1801	2
C₁₉×#1549	2
C₁₉×#1487	2
C₁₉×#1461	2
#8396	19

Subgroups — C19×#8396 #18014355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#8396 #18014355