Subgroups — C₂×C₁₉×#2647 #18040575

Order 4864.

1854 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2647	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₁₉×#400	C₂	1	4	yes	maximal
3	C₂×C₁₉×#398	C₂	1	4	yes	maximal
4	C₂×C₁₉×#389	C₂	1	2	yes	maximal
5	C₂×C₁₉×#389	C₂	1	2	yes	maximal
6	C₂×C₁₉×#384	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₉×#382	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₉×#376	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
10	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
12	C₂²×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
13	C₂²×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
14	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#400	C₂²	1	32	yes	—
18	C₁₉×#398	C₂²	1	32	yes	—
19	C₁₉×#389	C₂²	1	16	yes	—
20	C₁₉×#389	C₂²	1	16	yes	—
21	C₁₉×#384	C₂²	1	8	yes	—
22	C₁₉×#382	C₂²	1	8	yes	—
23	C₁₉×#376	C₂²	1	8	yes	—
24	C₂×C₁₉×#95	C₂²	1	2	yes	—
25	C₂×C₁₉×#95	C₂²	1	2	yes	—
26	C₂×C₁₉×#95	C₂²	1	4	yes	—
27	C₂×C₁₉×#95	C₂²	1	4	yes	—
28	C₂×C₁₉×#95	C₂²	1	4	yes	—
29	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
31	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
32	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
34	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
35	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
37	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
38	C₂×C₄⋊C₄×C₁₉	—	4	8	no	—
39	C₂×C₄⋊C₄×C₁₉	—	4	8	no	—
40	C₂×C₄⋊C₄×C₁₉	—	4	8	no	—
41	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
42	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
43	C₂×C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
44	C₂×C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
45	C₂×C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
46	C₂×C₄²×C₁₉	—	2	4	no	—
47	C₂×C₄²×C₁₉	—	2	4	no	—
48	C₂×C₄²×C₁₉	Q₈	1	8	yes	—
49	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
50	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
51	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
52	C₁₉×#95	C₂³	1	8	yes	—
53	C₁₉×#95	C₂³	1	8	yes	—
54	C₁₉×#95	C₂³	1	16	yes	—
55	C₁₉×#95	C₂³	1	16	yes	—
56	C₁₉×#95	C₂³	1	16	yes	—
57	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	—	2	4	no	—
59	C₂²×C₄×C₁₉	—	2	4	no	—
60	C₂²×C₄×C₁₉	—	2	2	no	—
61	C₂²×C₄×C₁₉	—	2	2	no	—
62	C₂²×C₄×C₁₉	—	2	4	no	—
63	C₂²×C₄×C₁₉	—	2	8	no	—
64	C₂²×C₄×C₁₉	—	2	8	no	—
65	C₂²×C₄×C₁₉	—	2	4	no	—
66	C₂²×C₄×C₁₉	—	2	8	no	—
67	C₂²×C₄×C₁₉	—	2	8	no	—
68	C₂²×C₄×C₁₉	—	2	4	no	—
69	C₂²×C₄×C₁₉	—	2	8	no	—
70	C₂²×C₄×C₁₉	—	2	4	no	—
71	C₂²×C₄×C₁₉	—	2	4	no	—
72	C₂²×C₄×C₁₉	—	2	8	no	—
73	C₂²×C₄×C₁₉	C₂×Q₈	1	8	yes	—
74	C₂²×C₄×C₁₉	C₂×Q₈	1	4	yes	—
75	C₂²×C₄×C₁₉	#41	1	4	yes	—
76	C₂²×C₄×C₁₉	#41	1	4	yes	—
77	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
78	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
79	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
80	C₂²×C₄×C₁₉	C₂⁴	1	8	yes	—
81	C₂²×C₄×C₁₉	C₂⁴	1	8	yes	—
82	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
83	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
84	C₄⋊C₄×C₁₉	—	2	16	no	—
85	C₄⋊C₄×C₁₉	—	2	16	no	—
86	C₄⋊C₄×C₁₉	—	4	16	no	—
87	C₄⋊C₄×C₁₉	—	4	16	no	—
88	C₄⋊C₄×C₁₉	—	4	16	no	—
89	C₄²×C₁₉	—	2	8	no	—
90	C₄²×C₁₉	—	2	8	no	—
91	C₄²×C₁₉	C₂×Q₈	1	16	yes	—
92	C₂×#2647	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₁₉	C₂²×Q₈	1	2	yes	—
94	C₂³×C₁₉	C₂×#41	1	2	yes	—
95	C₂³×C₁₉	C₂×#41	1	2	yes	—
96	C₂³×C₁₉	H(5,𝔽₂)	1	4	yes	—
97	C₂³×C₁₉	2₋¹⁺⁴	1	4	yes	—
98	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
99	C₂×C₄×C₁₉	—	2	16	no	—
100	C₂×C₄×C₁₉	—	2	4	no	—
101	C₂×C₄×C₁₉	—	2	4	no	—
102	C₂×C₄×C₁₉	—	2	2	no	—
103	C₂×C₄×C₁₉	—	2	2	no	—
104	C₂×C₄×C₁₉	—	2	4	no	—
105	C₂×C₄×C₁₉	—	2	8	no	—
106	C₂×C₄×C₁₉	—	2	8	no	—
107	C₂×C₄×C₁₉	—	2	4	no	—
108	C₂×C₄×C₁₉	—	2	8	no	—
109	C₂×C₄×C₁₉	—	2	8	no	—
110	C₂×C₄×C₁₉	—	2	4	no	—
111	C₂×C₄×C₁₉	—	2	8	no	—
112	C₂×C₄×C₁₉	—	2	4	no	—
113	C₂×C₄×C₁₉	—	2	4	no	—
114	C₂×C₄×C₁₉	—	2	8	no	—
115	C₂×C₄×C₁₉	—	4	4	no	—
116	C₂×C₄×C₁₉	—	4	16	no	—
117	C₂×C₄×C₁₉	—	4	4	no	—
118	C₂×C₄×C₁₉	—	4	8	no	—
119	C₂×C₄×C₁₉	—	4	16	no	—
120	C₂×C₄×C₁₉	#125	1	16	yes	—
121	C₂×C₄×C₁₉	C₂²×Q₈	1	8	yes	—
122	C₂×C₄×C₁₉	C₂²×Q₈	1	4	yes	—
123	C₂×C₄×C₁₉	C₂×#41	1	4	yes	—
124	C₂×C₄×C₁₉	C₂×#41	1	4	yes	—
125	#2647	C₂×C₁₉	1	16	yes	direct product factor
126	C₂×#400	C₂×C₁₉	1	4	yes	—
127	C₂×#398	C₂×C₁₉	1	4	yes	—
128	C₂×#389	C₂×C₁₉	1	2	yes	—
129	C₂×#389	C₂×C₁₉	1	2	yes	—
130	C₂×#384	C₂×C₁₉	1	1	yes	characteristic
131	C₂×#382	C₂×C₁₉	1	1	yes	characteristic
132	C₂×#376	C₂×C₁₉	1	1	yes	characteristic
133	C₂²×C₁₉	C₂×#125	1	4	yes	—
134	C₂²×C₁₉	#540	1	4	yes	—
135	C₂²×C₁₉	#541	1	4	yes	—
136	C₂²×C₁₉	#542	1	8	yes	—
137	C₂²×C₁₉	#544	1	8	yes	—
138	C₂²×C₁₉	C₂³×Q₈	1	1	yes	characteristic
139	C₂²×C₁₉	C₂²×#41	1	1	yes	characteristic
140	C₂²×C₁₉	C₂²×#41	1	1	yes	characteristic
141	C₂²×C₁₉	C₂×H(5,𝔽₂)	1	2	yes	—
142	C₂²×C₁₉	C₂×2₋¹⁺⁴	1	2	yes	—
143	C₄×C₁₉	—	2	8	no	—
144	C₄×C₁₉	—	4	2	no	—
145	C₄×C₁₉	—	4	8	no	—
146	C₄×C₁₉	—	4	2	no	—
147	C₄×C₁₉	—	4	4	no	—
148	C₄×C₁₉	—	4	8	no	—
149	C₄×C₁₉	C₂×#125	1	8	yes	—
150	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
151	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
152	C₂²×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
153	C₂²×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
154	C₂²×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
155	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
156	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
157	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
158	#400	C₂²×C₁₉	1	32	yes	—
159	#398	C₂²×C₁₉	1	32	yes	—
160	#389	C₂²×C₁₉	1	16	yes	—
161	#389	C₂²×C₁₉	1	16	yes	—
162	#384	C₂²×C₁₉	1	8	yes	—
163	#382	C₂²×C₁₉	1	8	yes	—
164	#376	C₂²×C₁₉	1	8	yes	—
165	C₂×#95	C₂²×C₁₉	1	2	yes	—
166	C₂×#95	C₂²×C₁₉	1	4	yes	—
167	C₂×#95	C₂²×C₁₉	1	4	yes	—
168	C₂×#95	C₂²×C₁₉	1	2	yes	—
169	C₂×#95	C₂²×C₁₉	1	4	yes	—
170	C₂×C₁₉	#2647	1	8	yes	direct product factor
171	C₂×C₁₉	C₂²×#125	1	1	yes	characteristic
172	C₂×C₁₉	C₂×#540	1	1	yes	characteristic
173	C₂×C₁₉	C₂×#541	1	1	yes	characteristic
174	C₂×C₁₉	C₂×#542	1	2	yes	—
175	C₂×C₁₉	C₂×#544	1	2	yes	—
176	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
177	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
178	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
179	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
180	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
181	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
182	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
183	C₂×C₄⋊C₄	—	2	8	no	—
184	C₂×C₄⋊C₄	—	2	8	no	—
185	C₂×C₄⋊C₄	—	4	8	no	—
186	C₂×C₄⋊C₄	—	4	8	no	—
187	C₂×C₄⋊C₄	—	4	8	no	—
188	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
189	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
190	C₂×C₄⋊C₄	C₂³×C₁₉	1	16	yes	—
191	C₂×C₄⋊C₄	C₂³×C₁₉	1	16	yes	—
192	C₂×C₄⋊C₄	C₂³×C₁₉	1	16	yes	—
193	C₂×C₄²	—	2	4	no	—
194	C₂×C₄²	—	2	4	no	—
195	C₂×C₄²	Q₈×C₁₉	1	8	yes	—
196	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
197	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
198	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
199	#95	C₂³×C₁₉	1	8	yes	—
200	#95	C₂³×C₁₉	1	8	yes	—
201	#95	C₂³×C₁₉	1	16	yes	—
202	#95	C₂³×C₁₉	1	16	yes	—
203	#95	C₂³×C₁₉	1	16	yes	—
204	C₁₉	C₂×#2647	1	1	yes	characteristic, direct product factor, Sylow
205	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	—	2	8	no	—
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	—	2	8	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	C₂×Q₈×C₁₉	1	8	yes	—
222	C₂²×C₄	C₂×Q₈×C₁₉	1	4	yes	—
223	C₂²×C₄	#41×C₁₉	1	4	yes	—
224	C₂²×C₄	#41×C₁₉	1	4	yes	—
225	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
226	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
227	C₂²×C₄	C₂⁴×C₁₉	1	8	yes	—
228	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
229	C₂²×C₄	C₂⁴×C₁₉	1	8	yes	—
230	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
231	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
232	C₄⋊C₄	—	2	16	no	—
233	C₄⋊C₄	—	2	16	no	—
234	C₄⋊C₄	—	4	16	no	—
235	C₄⋊C₄	—	4	16	no	—
236	C₄⋊C₄	—	4	16	no	—
237	C₄²	—	2	8	no	—
238	C₄²	—	2	8	no	—
239	C₄²	C₂×Q₈×C₁₉	1	16	yes	—
240	C₂³	C₂²×Q₈×C₁₉	1	2	yes	—
241	C₂³	C₂×#41×C₁₉	1	2	yes	—
242	C₂³	C₂×#41×C₁₉	1	2	yes	—
243	C₂³	C₁₉×H(5,𝔽₂)	1	4	yes	—
244	C₂³	C₁₉×2₋¹⁺⁴	1	4	yes	—
245	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
246	C₂×C₄	—	2	16	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	16	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	—	4	16	no	—
267	C₂×C₄	C₁₉×#125	1	16	yes	—
268	C₂×C₄	C₂²×Q₈×C₁₉	1	8	yes	—
269	C₂×C₄	C₂²×Q₈×C₁₉	1	4	yes	—
270	C₂×C₄	C₂×#41×C₁₉	1	4	yes	—
271	C₂×C₄	C₂×#41×C₁₉	1	4	yes	—
272	C₂²	C₂×C₁₉×#125	1	4	yes	—
273	C₂²	C₁₉×#540	1	4	yes	—
274	C₂²	C₁₉×#541	1	4	yes	—
275	C₂²	C₁₉×#542	1	8	yes	—
276	C₂²	C₁₉×#544	1	8	yes	—
277	C₂²	C₂³×Q₈×C₁₉	1	1	yes	characteristic
278	C₂²	C₂²×#41×C₁₉	1	1	yes	characteristic
279	C₂²	C₂²×#41×C₁₉	1	1	yes	characteristic
280	C₂²	C₂×C₁₉×H(5,𝔽₂)	1	2	yes	—
281	C₂²	C₂×C₁₉×2₋¹⁺⁴	1	2	yes	—
282	C₄	—	2	8	no	—
283	C₄	—	4	2	no	—
284	C₄	—	4	8	no	—
285	C₄	—	4	2	no	—
286	C₄	—	4	4	no	—
287	C₄	—	4	8	no	—
288	C₄	C₂×C₁₉×#125	1	8	yes	—
289	C₂	C₁₉×#2647	1	8	yes	direct product factor
290	C₂	C₂²×C₁₉×#125	1	1	yes	characteristic
291	C₂	C₂×C₁₉×#540	1	1	yes	characteristic
292	C₂	C₂×C₁₉×#541	1	1	yes	characteristic
293	C₂	C₂×C₁₉×#542	1	2	yes	—
294	C₂	C₂×C₁₉×#544	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#2647	C₂	direct (N × H)
92. C₂×#2647	C₁₉	direct (N × H)
125. #2647	C₂×C₁₉	direct (N × H)
170. C₂×C₁₉	#2647	direct (N × H)
204. C₁₉	C₂×#2647	direct (N × H)
289. C₂	C₁₉×#2647	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉×#2647	2
C₂×C₁₉×#400	2
C₂×C₁₉×#398	2
C₂×C₁₉×#389	2
C₂×C₁₉×#384	2
C₂×C₁₉×#382	2
C₂×C₁₉×#376	2
C₂×#2647	19

Subgroups — C2×C19×#2647 #18040575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#2647 #18040575