Subgroups — C₂×C₁₉×#2155 #18039362

Order 4864.

658 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₉×#108	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#427	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#2155	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₉×#365	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₁₉	—	2	1	no	—
7	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
8	C₂×C₁₉×#117	—	2	1	no	—
9	C₂×C₁₉×M₃₂	C₄	1	2	yes	complemented
10	C₂×C₁₉×M₃₂	C₄	1	4	yes	complemented
11	C₂×C₁₉×M₃₂	C₂²	1	2	yes	—
12	C₂×C₁₉×M₃₂	C₂²	1	4	yes	—
13	C₂²×C₁₆×C₁₉	C₄	1	2	yes	complemented
14	C₂×C₁₉×#108	—	2	1	no	—
15	C₂×C₁₉×#108	—	2	2	no	—
16	C₂×C₁₉×#108	C₂²	1	2	yes	—
17	C₂×C₁₉×#108	C₂²	1	4	yes	—
18	C₁₉×#427	—	2	2	no	—
19	C₁₉×#427	C₂²	1	4	yes	—
20	C₂×C₁₉×#107	—	2	1	no	—
21	C₂×C₁₉×#106	C₂²	1	2	yes	—
22	C₁₉×#365	C₂²	1	16	yes	complemented
23	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
24	C₂×M₁₆×C₁₉	—	2	2	no	—
25	C₂×M₁₆×C₁₉	—	2	4	no	—
26	C₂×M₁₆×C₁₉	—	4	1	no	—
27	C₂×M₁₆×C₁₉	—	4	2	no	—
28	C₂²×C₈×C₁₉	—	4	1	no	—
29	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₉	D₄	1	2	yes	—
34	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
35	C₂²×C₈×C₁₉	Q₈	1	2	yes	—
36	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
37	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
39	C₁₉×#117	—	2	4	no	—
40	C₁₉×#117	—	4	2	no	—
41	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₁₉	—	4	1	no	—
43	C₂×C₄²×C₁₉	—	4	1	no	—
44	C₁₉×M₃₂	C₂×C₄	1	4	yes	—
45	C₁₉×M₃₂	C₂×C₄	1	8	yes	complemented
46	C₁₉×M₃₂	C₂³	1	4	yes	—
47	C₂×C₁₆×C₁₉	C₂×C₄	1	4	yes	—
48	C₂×C₁₆×C₁₉	C₂×C₄	1	8	yes	complemented
49	C₁₉×#108	—	2	2	no	—
50	C₁₉×#108	—	2	4	no	—
51	C₁₉×#108	C₂³	1	4	yes	—
52	C₁₉×#107	—	2	4	no	—
53	C₁₉×#106	—	2	2	no	—
54	C₁₉×#106	C₂³	1	4	yes	—
55	C₂⁴×C₁₉	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	—	4	2	no	—
57	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
61	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
67	M₁₆×C₁₉	—	2	4	no	—
68	M₁₆×C₁₉	—	4	2	no	—
69	M₁₆×C₁₉	—	4	4	no	—
70	C₂×C₈×C₁₉	—	4	2	no	—
71	C₂×C₈×C₁₉	—	4	4	no	—
72	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
74	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
75	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
77	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
78	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
79	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₈×C₁₉	C₂×Q₈	1	2	yes	—
88	C₂×C₈×C₁₉	C₂×Q₈	1	2	yes	—
89	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₁₉	—	2	4	no	—
91	C₄⋊C₄×C₁₉	—	4	2	no	—
92	C₂²⋊C₄×C₁₉	—	4	4	no	—
93	C₄²×C₁₉	—	4	4	no	—
94	C₁₆×C₁₉	C₂²×C₄	1	8	yes	—
95	C₂×#2155	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₁₉	—	4	2	no	—
97	C₂³×C₁₉	#106	1	1	yes	characteristic
98	C₂³×C₁₉	#106	1	1	yes	characteristic
99	C₂³×C₁₉	#106	1	2	yes	—
100	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
101	C₂³×C₁₉	C₂×SD₁₆	1	2	yes	—
102	C₂×C₄×C₁₉	—	4	4	no	—
103	C₂×C₄×C₁₉	—	8	4	no	—
104	C₂×C₄×C₁₉	#106	1	2	yes	—
105	C₂×C₄×C₁₉	#106	1	2	yes	—
106	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
107	C₂×C₄×C₁₉	#106	1	2	yes	—
108	C₂×C₄×C₁₉	#106	1	2	yes	—
109	C₂×C₄×C₁₉	#106	1	2	yes	—
110	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
117	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
119	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
120	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
122	C₈×C₁₉	—	4	4	no	—
123	C₈×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₈×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
125	C₈×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₈×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
127	C₈×C₁₉	C₂³×C₄	1	1	yes	characteristic
128	C₈×C₁₉	C₂²×Q₈	1	1	yes	characteristic
129	C₂²×M₃₂	C₂×C₁₉	1	1	yes	characteristic
130	C₂²×#108	C₂×C₁₉	1	1	yes	characteristic
131	C₂×#427	C₂×C₁₉	1	1	yes	characteristic
132	#2155	C₂×C₁₉	1	8	yes	direct product factor
133	C₂×#365	C₂×C₁₉	1	4	yes	complemented
134	C₂²×C₁₉	—	2	2	no	—
135	C₂²×C₁₉	—	2	2	no	—
136	C₂²×C₁₉	—	2	2	no	—
137	C₂²×C₁₉	—	4	2	no	—
138	C₂²×C₁₉	—	4	2	no	—
139	C₂²×C₁₉	C₂×#106	1	2	yes	—
140	C₂²×C₁₉	C₂×#106	1	2	yes	—
141	C₂²×C₁₉	C₂×#106	1	1	yes	characteristic
142	C₂²×C₁₉	C₂×#106	1	1	yes	characteristic
143	C₂²×C₁₉	C₂²×SD₁₆	1	1	yes	characteristic
144	C₄×C₁₉	—	8	4	no	—
145	C₄×C₁₉	C₂×#106	1	1	yes	characteristic
146	C₄×C₁₉	C₂×#106	1	2	yes	—
147	C₄×C₁₉	C₂×#106	1	1	yes	characteristic
148	C₄×C₁₉	C₂×#106	1	2	yes	—
149	C₄×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₄×C₁₉	C₂²×SD₁₆	1	1	yes	characteristic
151	C₂²×M₁₆	—	2	1	no	—
152	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
153	C₂×#117	—	2	1	no	—
154	C₂×M₃₂	C₄×C₁₉	1	2	yes	complemented
155	C₂×M₃₂	C₄×C₁₉	1	4	yes	complemented
156	C₂×M₃₂	C₂²×C₁₉	1	2	yes	—
157	C₂×M₃₂	C₂²×C₁₉	1	4	yes	—
158	C₂²×C₁₆	C₄×C₁₉	1	2	yes	complemented
159	C₂×#108	—	2	1	no	—
160	C₂×#108	—	2	2	no	—
161	C₂×#108	C₂²×C₁₉	1	2	yes	—
162	C₂×#108	C₂²×C₁₉	1	4	yes	—
163	#427	—	2	2	no	—
164	#427	C₂²×C₁₉	1	4	yes	—
165	C₂×#107	—	2	1	no	—
166	C₂×#106	C₂²×C₁₉	1	2	yes	—
167	#365	C₂²×C₁₉	1	16	yes	complemented
168	C₂×C₁₉	—	2	1	no	—
169	C₂×C₁₉	—	2	1	no	—
170	C₂×C₁₉	—	2	2	no	—
171	C₂×C₁₉	—	2	2	no	—
172	C₂×C₁₉	#2155	1	2	yes	direct product factor
173	C₂×C₁₉	C₂²×#106	1	1	yes	characteristic
174	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	4	no	—
177	C₂×M₁₆	—	4	1	no	—
178	C₂×M₁₆	—	4	2	no	—
179	C₂²×C₈	—	4	1	no	—
180	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
181	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
182	C₂²×C₈	C₂×C₄×C₁₉	1	2	yes	—
183	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
184	C₂²×C₈	D₄×C₁₉	1	2	yes	—
185	C₂²×C₈	Q₈×C₁₉	1	1	yes	characteristic
186	C₂²×C₈	Q₈×C₁₉	1	2	yes	—
187	C₂²×C₈	Q₈×C₁₉	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
190	#117	—	2	4	no	—
191	#117	—	4	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₂²⋊C₄	—	4	1	no	—
194	C₂×C₄²	—	4	1	no	—
195	M₃₂	C₂×C₄×C₁₉	1	4	yes	—
196	M₃₂	C₂×C₄×C₁₉	1	8	yes	complemented
197	M₃₂	C₂³×C₁₉	1	4	yes	—
198	C₂×C₁₆	C₂×C₄×C₁₉	1	4	yes	—
199	C₂×C₁₆	C₂×C₄×C₁₉	1	8	yes	complemented
200	#108	—	2	2	no	—
201	#108	—	2	4	no	—
202	#108	C₂³×C₁₉	1	4	yes	—
203	#107	—	2	4	no	—
204	#106	—	2	2	no	—
205	#106	C₂³×C₁₉	1	4	yes	—
206	C₁₉	C₂×#2155	1	1	yes	characteristic, direct product factor, Sylow
207	C₂⁴	SD₁₆×C₁₉	1	1	yes	characteristic
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
210	C₂²×C₄	C₄⋊C₄×C₁₉	1	2	yes	—
211	C₂²×C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
212	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
213	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
214	C₂²×C₄	SD₁₆×C₁₉	1	2	yes	—
215	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
216	C₂²×C₄	SD₁₆×C₁₉	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	4	no	—
222	C₂×C₈	—	4	4	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
225	C₂×C₈	C₄⋊C₄×C₁₉	1	2	yes	—
226	C₂×C₈	C₄⋊C₄×C₁₉	1	2	yes	—
227	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
228	C₂×C₈	C₄⋊C₄×C₁₉	1	2	yes	—
229	C₂×C₈	C₄⋊C₄×C₁₉	1	2	yes	—
230	C₂×C₈	C₄⋊C₄×C₁₉	1	2	yes	—
231	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
232	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
233	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
234	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
235	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
236	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
237	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
238	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
239	C₂×C₈	C₂×Q₈×C₁₉	1	2	yes	—
240	C₂×C₈	C₂×Q₈×C₁₉	1	2	yes	—
241	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	4	no	—
245	C₄²	—	4	4	no	—
246	C₁₆	C₂²×C₄×C₁₉	1	8	yes	—
247	C₂³	—	4	2	no	—
248	C₂³	C₁₉×#106	1	1	yes	characteristic
249	C₂³	C₁₉×#106	1	1	yes	characteristic
250	C₂³	C₁₉×#106	1	2	yes	—
251	C₂³	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
252	C₂³	C₂×SD₁₆×C₁₉	1	2	yes	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	C₁₉×#106	1	2	yes	—
256	C₂×C₄	C₁₉×#106	1	2	yes	—
257	C₂×C₄	C₁₉×#106	1	2	yes	—
258	C₂×C₄	C₁₉×#106	1	2	yes	—
259	C₂×C₄	C₁₉×#106	1	2	yes	—
260	C₂×C₄	C₁₉×#106	1	1	yes	characteristic
261	C₂×C₄	C₁₉×#106	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
263	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
265	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
266	C₂×C₄	C₂×SD₁₆×C₁₉	1	2	yes	—
267	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
268	C₂×C₄	C₂×SD₁₆×C₁₉	1	2	yes	—
269	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
270	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
271	C₂×C₄	C₂×SD₁₆×C₁₉	1	2	yes	—
272	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
273	C₈	—	4	4	no	—
274	C₈	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
275	C₈	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
276	C₈	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
277	C₈	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
278	C₈	C₂³×C₄×C₁₉	1	1	yes	characteristic
279	C₈	C₂²×Q₈×C₁₉	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	C₂×C₁₉×#106	1	2	yes	—
286	C₂²	C₂×C₁₉×#106	1	1	yes	characteristic
287	C₂²	C₂×C₁₉×#106	1	2	yes	—
288	C₂²	C₂×C₁₉×#106	1	1	yes	characteristic
289	C₂²	C₂²×SD₁₆×C₁₉	1	1	yes	characteristic
290	C₄	—	8	4	no	—
291	C₄	C₂×C₁₉×#106	1	2	yes	—
292	C₄	C₂×C₁₉×#106	1	2	yes	—
293	C₄	C₂×C₁₉×#106	1	1	yes	characteristic
294	C₄	C₂×C₁₉×#106	1	1	yes	characteristic
295	C₄	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
296	C₄	C₂²×SD₁₆×C₁₉	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	C₁₉×#2155	1	2	yes	direct product factor
302	C₂	C₂²×C₁₉×#106	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	#106
commutator	C₈	C₂³×C₄×C₁₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₉
socle	C₂²×C₁₉	C₂×#106
prime radical	C₂⁴×C₁₉	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ C₂³×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₉×#2155	C₂	direct (N × H)
5. C₂×C₁₉×#365	C₂	semidirect
9. C₂×C₁₉×M₃₂	C₄	semidirect
10. C₂×C₁₉×M₃₂	C₄	semidirect
13. C₂²×C₁₆×C₁₉	C₄	semidirect
22. C₁₉×#365	C₂²	semidirect
45. C₁₉×M₃₂	C₂×C₄	semidirect
48. C₂×C₁₆×C₁₉	C₂×C₄	semidirect
95. C₂×#2155	C₁₉	direct (N × H)
132. #2155	C₂×C₁₉	direct (N × H)
133. C₂×#365	C₂×C₁₉	semidirect
154. C₂×M₃₂	C₄×C₁₉	semidirect
155. C₂×M₃₂	C₄×C₁₉	semidirect
158. C₂²×C₁₆	C₄×C₁₉	semidirect
167. #365	C₂²×C₁₉	semidirect
172. C₂×C₁₉	#2155	direct (N × H)
196. M₃₂	C₂×C₄×C₁₉	semidirect
199. C₂×C₁₆	C₂×C₄×C₁₉	semidirect
206. C₁₉	C₂×#2155	direct (N × H)
301. C₂	C₁₉×#2155	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×M₃₂	2
C₂²×C₁₉×#108	2
C₂×C₁₉×#427	2
C₁₉×#2155	2
C₂×C₁₉×#365	2
C₂×#2155	19

Subgroups — C2×C19×#2155 #18039362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#2155 #18039362