Subgroups — C₁₉×#34364 #18040323

Order 4864.

3390 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×#127	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2380	C₂	1	4	yes	maximal, complemented
3	C₁₉×#2366	C₂	1	4	yes	maximal, complemented
4	D₄×C₂²⋊C₄×C₁₉	C₂	1	8	yes	maximal, complemented
5	C₁₉×#2328	C₂	1	8	yes	maximal, complemented
6	C₁₉×#2308	C₂	1	4	yes	maximal, complemented
7	C₁₉×#2283	C₂	1	2	yes	maximal, complemented
8	C₂³×D₄×C₁₉	C₂²	1	2	yes	—
9	C₂³×D₄×C₁₉	C₂²	1	4	yes	—
10	C₂×C₁₉×#127	C₄	1	16	yes	complemented
11	C₂×C₁₉×#127	C₂²	1	4	yes	—
12	C₂×C₁₉×#127	C₂²	1	2	yes	—
13	C₂×C₁₉×#127	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₁₉	C₂²	1	16	yes	complemented
15	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	8	yes	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#390	C₂²	1	16	yes	complemented
18	C₁₉×#386	C₂²	1	16	yes	complemented
19	C₁₉×#386	C₂²	1	16	yes	complemented
20	C₁₉×#386	C₂²	1	32	yes	complemented
21	C₁₉×#383	C₂²	1	4	yes	complemented
22	C₁₉×#382	C₂²	1	4	yes	complemented
23	C₁₉×#381	C₂²	1	4	yes	—
24	C₁₉×#380	C₂²	1	4	yes	—
25	C₁₉×#379	C₂²	1	8	yes	—
26	C₄×C₄⋊C₄×C₁₉	C₂²	1	4	yes	complemented
27	C₄×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
28	C₁₉×#376	C₂²	1	4	yes	complemented
29	C₂⁵×C₁₉	C₂³	1	4	yes	—
30	C₂²×D₄×C₁₉	—	4	16	no	—
31	C₂²×D₄×C₁₉	—	4	8	no	—
32	C₂²×D₄×C₁₉	C₂×C₄	1	32	yes	complemented
33	C₂²×D₄×C₁₉	C₂×C₄	1	16	yes	complemented
34	C₂²×D₄×C₁₉	C₂³	1	8	yes	—
35	C₂²×D₄×C₁₉	C₂³	1	4	yes	—
36	C₂²×D₄×C₁₉	C₂³	1	4	yes	—
37	C₂²×D₄×C₁₉	C₂³	1	8	yes	—
38	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
39	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₁₉	C₂³	1	16	yes	—
41	C₁₉×#127	C₂×C₄	1	16	yes	complemented
42	C₁₉×#127	C₂×C₄	1	32	yes	complemented
43	C₁₉×#127	C₂×C₄	1	16	yes	complemented
44	C₄×D₄×C₁₉	—	4	32	no	—
45	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
46	C₂×C₂²⋊C₄×C₁₉	—	4	16	no	—
47	C₂×C₂²⋊C₄×C₁₉	C₂³	1	8	yes	—
48	C₂×C₂²⋊C₄×C₁₉	C₂³	1	32	yes	—
49	C₂×C₂²⋊C₄×C₁₉	C₂³	1	16	yes	—
50	C₂×C₄²×C₁₉	C₂×C₄	1	8	yes	complemented
51	C₂×C₄²×C₁₉	C₂³	1	8	yes	—
52	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
53	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
55	C₁₉×#95	C₂³	1	8	yes	—
56	C₁₉×#95	C₂³	1	8	yes	—
57	C₁₉×#95	C₂³	1	8	yes	—
58	C₁₉×#95	C₂³	1	8	yes	—
59	C₂⁴×C₁₉	—	2	16	no	—
60	C₂⁴×C₁₉	—	4	16	no	—
61	C₂⁴×C₁₉	C₂²×C₄	1	16	yes	—
62	C₂⁴×C₁₉	C₂⁴	1	8	yes	—
63	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
64	C₂×D₄×C₁₉	—	2	32	no	—
65	C₂×D₄×C₁₉	—	4	64	no	—
66	C₂×D₄×C₁₉	—	4	32	no	—
67	C₂×D₄×C₁₉	—	4	4	no	—
68	C₂×D₄×C₁₉	—	4	8	no	—
69	C₂×D₄×C₁₉	—	4	16	no	—
70	C₂×D₄×C₁₉	—	4	4	no	—
71	C₂×D₄×C₁₉	C₂²×C₄	1	32	yes	—
72	C₂×D₄×C₁₉	C₂²×C₄	1	32	yes	—
73	C₂×D₄×C₁₉	C₂²×C₄	1	16	yes	—
74	C₂×D₄×C₁₉	C₂²×C₄	1	16	yes	—
75	C₂²×C₄×C₁₉	—	2	4	no	—
76	C₂²×C₄×C₁₉	—	2	8	no	—
77	C₂²×C₄×C₁₉	—	4	16	no	—
78	C₂²×C₄×C₁₉	—	4	32	no	—
79	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₁₉	C₂²×C₄	1	8	yes	—
81	C₂²×C₄×C₁₉	C₂⁴	1	16	yes	—
82	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
83	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
84	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
85	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
86	C₄⋊C₄×C₁₉	—	2	16	no	—
87	C₂²⋊C₄×C₁₉	—	2	16	no	—
88	C₂²⋊C₄×C₁₉	—	4	16	no	—
89	C₂²⋊C₄×C₁₉	—	4	32	no	—
90	C₄²×C₁₉	—	2	16	no	—
91	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
92	C₄²×C₁₉	C₂²×C₄	1	8	yes	—
93	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
94	#34364	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₁₉	—	2	4	no	—
96	C₂³×C₁₉	—	2	8	no	—
97	C₂³×C₁₉	—	2	32	no	—
98	C₂³×C₁₉	—	2	8	no	—
99	C₂³×C₁₉	—	2	32	no	—
100	C₂³×C₁₉	—	4	32	no	—
101	C₂³×C₁₉	—	8	32	no	—
102	C₂³×C₁₉	C₂³×C₄	1	16	yes	—
103	C₂³×C₁₉	C₂³×C₄	1	2	yes	—
104	C₂³×C₁₉	H(5,𝔽₂)	1	4	yes	—
105	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
106	D₄×C₁₉	—	2	64	no	—
107	D₄×C₁₉	—	4	32	no	—
108	D₄×C₁₉	—	4	16	no	—
109	D₄×C₁₉	—	4	16	no	—
110	C₂×C₄×C₁₉	—	2	8	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	16	no	—
115	C₂×C₄×C₁₉	—	2	16	no	—
116	C₂×C₄×C₁₉	—	2	16	no	—
117	C₂×C₄×C₁₉	—	2	16	no	—
118	C₂×C₄×C₁₉	—	8	32	no	—
119	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
120	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
121	C₂×C₄×C₁₉	C₂³×C₄	1	4	yes	—
122	C₂×C₄×C₁₉	C₂³×C₄	1	4	yes	—
123	C₂×C₄×C₁₉	H(5,𝔽₂)	1	8	yes	—
124	C₂²×#127	C₂×C₁₉	1	1	yes	characteristic
125	#2380	C₂×C₁₉	1	4	yes	complemented
126	#2366	C₂×C₁₉	1	4	yes	complemented
127	D₄×C₂²⋊C₄	C₂×C₁₉	1	8	yes	complemented
128	#2328	C₂×C₁₉	1	8	yes	complemented
129	#2308	C₂×C₁₉	1	4	yes	complemented
130	#2283	C₂×C₁₉	1	2	yes	complemented
131	C₂²×C₁₉	—	2	4	no	—
132	C₂²×C₁₉	—	2	8	no	—
133	C₂²×C₁₉	—	2	32	no	—
134	C₂²×C₁₉	—	2	16	no	—
135	C₂²×C₁₉	—	4	16	no	—
136	C₂²×C₁₉	—	4	16	no	—
137	C₂²×C₁₉	—	8	32	no	—
138	C₂²×C₁₉	#518	1	4	yes	—
139	C₂²×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
140	C₂²×C₁₉	C₂×H(5,𝔽₂)	1	2	yes	—
141	C₂²×C₁₉	C₂×H(5,𝔽₂)	1	4	yes	—
142	C₄×C₁₉	—	2	2	no	—
143	C₄×C₁₉	—	2	2	no	—
144	C₄×C₁₉	—	2	4	no	—
145	C₄×C₁₉	—	4	16	no	—
146	C₄×C₁₉	#518	1	8	yes	—
147	C₂³×D₄	C₂²×C₁₉	1	2	yes	—
148	C₂³×D₄	C₂²×C₁₉	1	4	yes	—
149	C₂×#127	C₄×C₁₉	1	16	yes	complemented
150	C₂×#127	C₂²×C₁₉	1	4	yes	—
151	C₂×#127	C₂²×C₁₉	1	2	yes	—
152	C₂×#127	C₂²×C₁₉	1	2	yes	—
153	C₂×C₄×D₄	C₂²×C₁₉	1	16	yes	complemented
154	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	8	yes	—
155	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
156	#390	C₂²×C₁₉	1	16	yes	complemented
157	#386	C₂²×C₁₉	1	16	yes	complemented
158	#386	C₂²×C₁₉	1	16	yes	complemented
159	#386	C₂²×C₁₉	1	32	yes	complemented
160	#383	C₂²×C₁₉	1	4	yes	complemented
161	#382	C₂²×C₁₉	1	4	yes	complemented
162	#381	C₂²×C₁₉	1	4	yes	—
163	#380	C₂²×C₁₉	1	4	yes	—
164	#379	C₂²×C₁₉	1	8	yes	—
165	C₄×C₄⋊C₄	C₂²×C₁₉	1	4	yes	complemented
166	C₄×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
167	#376	C₂²×C₁₉	1	4	yes	complemented
168	C₂×C₁₉	—	2	4	no	—
169	C₂×C₁₉	—	4	16	no	—
170	C₂×C₁₉	C₂×#518	1	1	yes	characteristic
171	C₂×C₁₉	C₂×#518	1	2	yes	—
172	C₂×C₁₉	#3543	1	4	yes	—
173	C₂⁵	C₂³×C₁₉	1	4	yes	—
174	C₂²×D₄	—	4	16	no	—
175	C₂²×D₄	—	4	8	no	—
176	C₂²×D₄	C₂×C₄×C₁₉	1	32	yes	complemented
177	C₂²×D₄	C₂×C₄×C₁₉	1	16	yes	complemented
178	C₂²×D₄	C₂³×C₁₉	1	8	yes	—
179	C₂²×D₄	C₂³×C₁₉	1	4	yes	—
180	C₂²×D₄	C₂³×C₁₉	1	8	yes	—
181	C₂²×D₄	C₂³×C₁₉	1	4	yes	—
182	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
183	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
184	C₂³×C₄	C₂³×C₁₉	1	16	yes	—
185	#127	C₂×C₄×C₁₉	1	16	yes	complemented
186	#127	C₂×C₄×C₁₉	1	32	yes	complemented
187	#127	C₂×C₄×C₁₉	1	16	yes	complemented
188	C₄×D₄	—	4	32	no	—
189	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
190	C₂×C₂²⋊C₄	—	4	16	no	—
191	C₂×C₂²⋊C₄	C₂³×C₁₉	1	8	yes	—
192	C₂×C₂²⋊C₄	C₂³×C₁₉	1	16	yes	—
193	C₂×C₂²⋊C₄	C₂³×C₁₉	1	32	yes	—
194	C₂×C₄²	C₂×C₄×C₁₉	1	8	yes	complemented
195	C₂×C₄²	C₂³×C₁₉	1	8	yes	—
196	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
197	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
198	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
199	#95	C₂³×C₁₉	1	8	yes	—
200	#95	C₂³×C₁₉	1	8	yes	—
201	#95	C₂³×C₁₉	1	8	yes	—
202	#95	C₂³×C₁₉	1	8	yes	—
203	C₁₉	#34364	1	1	yes	characteristic, direct product factor, Sylow
204	C₂⁴	—	2	16	no	—
205	C₂⁴	—	4	16	no	—
206	C₂⁴	C₂²×C₄×C₁₉	1	16	yes	—
207	C₂⁴	C₂⁴×C₁₉	1	8	yes	—
208	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
209	C₂×D₄	—	2	32	no	—
210	C₂×D₄	—	4	64	no	—
211	C₂×D₄	—	4	32	no	—
212	C₂×D₄	—	4	4	no	—
213	C₂×D₄	—	4	8	no	—
214	C₂×D₄	—	4	16	no	—
215	C₂×D₄	—	4	4	no	—
216	C₂×D₄	C₂²×C₄×C₁₉	1	32	yes	—
217	C₂×D₄	C₂²×C₄×C₁₉	1	32	yes	—
218	C₂×D₄	C₂²×C₄×C₁₉	1	16	yes	—
219	C₂×D₄	C₂²×C₄×C₁₉	1	16	yes	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	4	16	no	—
223	C₂²×C₄	—	4	32	no	—
224	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
225	C₂²×C₄	C₂²×C₄×C₁₉	1	8	yes	—
226	C₂²×C₄	C₂⁴×C₁₉	1	16	yes	—
227	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
228	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
229	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
230	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
231	C₄⋊C₄	—	2	16	no	—
232	C₂²⋊C₄	—	2	16	no	—
233	C₂²⋊C₄	—	4	16	no	—
234	C₂²⋊C₄	—	4	32	no	—
235	C₄²	—	2	16	no	—
236	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
237	C₄²	C₂²×C₄×C₁₉	1	8	yes	—
238	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
239	C₂³	—	2	4	no	—
240	C₂³	—	2	8	no	—
241	C₂³	—	2	32	no	—
242	C₂³	—	2	8	no	—
243	C₂³	—	2	32	no	—
244	C₂³	—	4	32	no	—
245	C₂³	—	8	32	no	—
246	C₂³	C₂³×C₄×C₁₉	1	16	yes	—
247	C₂³	C₂³×C₄×C₁₉	1	2	yes	—
248	C₂³	C₁₉×H(5,𝔽₂)	1	4	yes	—
249	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
250	D₄	—	2	64	no	—
251	D₄	—	4	32	no	—
252	D₄	—	4	16	no	—
253	D₄	—	4	16	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	16	no	—
259	C₂×C₄	—	2	16	no	—
260	C₂×C₄	—	2	16	no	—
261	C₂×C₄	—	2	16	no	—
262	C₂×C₄	—	8	32	no	—
263	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
264	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
265	C₂×C₄	C₂³×C₄×C₁₉	1	4	yes	—
266	C₂×C₄	C₂³×C₄×C₁₉	1	4	yes	—
267	C₂×C₄	C₁₉×H(5,𝔽₂)	1	8	yes	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	8	no	—
270	C₂²	—	2	32	no	—
271	C₂²	—	2	16	no	—
272	C₂²	—	4	16	no	—
273	C₂²	—	4	16	no	—
274	C₂²	—	8	32	no	—
275	C₂²	C₁₉×#518	1	4	yes	—
276	C₂²	C₂⁴×C₄×C₁₉	1	1	yes	characteristic
277	C₂²	C₂×C₁₉×H(5,𝔽₂)	1	2	yes	—
278	C₂²	C₂×C₁₉×H(5,𝔽₂)	1	4	yes	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	4	no	—
282	C₄	—	4	16	no	—
283	C₄	C₁₉×#518	1	8	yes	—
284	C₂	—	2	4	no	—
285	C₂	—	4	16	no	—
286	C₂	C₂×C₁₉×#518	1	1	yes	characteristic
287	C₂	C₂×C₁₉×#518	1	2	yes	—
288	C₂	C₁₉×#3543	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₁₉
Frattini, cosocle	C₂³	C₂⁵×C₁₉
prime radical	C₂²×C₁₉×#127	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2380	C₂	semidirect
3. C₁₉×#2366	C₂	semidirect
4. D₄×C₂²⋊C₄×C₁₉	C₂	semidirect
5. C₁₉×#2328	C₂	semidirect
6. C₁₉×#2308	C₂	semidirect
7. C₁₉×#2283	C₂	semidirect
10. C₂×C₁₉×#127	C₄	semidirect
14. C₂×C₄×D₄×C₁₉	C₂²	semidirect
17. C₁₉×#390	C₂²	semidirect
18. C₁₉×#386	C₂²	semidirect
19. C₁₉×#386	C₂²	semidirect
20. C₁₉×#386	C₂²	semidirect
21. C₁₉×#383	C₂²	semidirect
22. C₁₉×#382	C₂²	semidirect
26. C₄×C₄⋊C₄×C₁₉	C₂²	semidirect
28. C₁₉×#376	C₂²	semidirect
32. C₂²×D₄×C₁₉	C₂×C₄	semidirect
33. C₂²×D₄×C₁₉	C₂×C₄	semidirect
41. C₁₉×#127	C₂×C₄	semidirect
42. C₁₉×#127	C₂×C₄	semidirect
43. C₁₉×#127	C₂×C₄	semidirect
50. C₂×C₄²×C₁₉	C₂×C₄	semidirect
94. #34364	C₁₉	direct (N × H)
125. #2380	C₂×C₁₉	semidirect
126. #2366	C₂×C₁₉	semidirect
127. D₄×C₂²⋊C₄	C₂×C₁₉	semidirect
128. #2328	C₂×C₁₉	semidirect
129. #2308	C₂×C₁₉	semidirect
130. #2283	C₂×C₁₉	semidirect
149. C₂×#127	C₄×C₁₉	semidirect
153. C₂×C₄×D₄	C₂²×C₁₉	semidirect
156. #390	C₂²×C₁₉	semidirect
157. #386	C₂²×C₁₉	semidirect
158. #386	C₂²×C₁₉	semidirect
159. #386	C₂²×C₁₉	semidirect
160. #383	C₂²×C₁₉	semidirect
161. #382	C₂²×C₁₉	semidirect
165. C₄×C₄⋊C₄	C₂²×C₁₉	semidirect
167. #376	C₂²×C₁₉	semidirect
176. C₂²×D₄	C₂×C₄×C₁₉	semidirect
177. C₂²×D₄	C₂×C₄×C₁₉	semidirect
185. #127	C₂×C₄×C₁₉	semidirect
186. #127	C₂×C₄×C₁₉	semidirect
187. #127	C₂×C₄×C₁₉	semidirect
194. C₂×C₄²	C₂×C₄×C₁₉	semidirect
203. C₁₉	#34364	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×#127	2
C₁₉×#2380	2
C₁₉×#2366	2
D₄×C₂²⋊C₄×C₁₉	2
C₁₉×#2328	2
C₁₉×#2308	2
C₁₉×#2283	2
#34364	19

Subgroups — C19×#34364 #18040323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#34364 #18040323