Subgroups — #14572377

Order 2432.

492 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597353	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927090	C₂	1	1	yes	maximal, characteristic
3	#11926620	C₂	1	2	yes	maximal, complemented
4	#11926615	C₂	1	2	yes	maximal, complemented
5	#11926613	C₂	1	2	yes	maximal, complemented
6	C₈×C₂²⋊D₁₉	C₂	1	2	yes	maximal, complemented
7	C₂×Dic₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926511	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926510	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₁₉×#131	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×#64541	C₂²	1	1	yes	characteristic
14	C₁₉×#131	—	2	4	no	—
15	C₂×M₁₆×C₁₉	C₂²	1	2	yes	—
16	C₂×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₉	C₂²	1	2	yes	—
19	#10597393	C₄	1	1	yes	characteristic
20	#10597390	C₄	1	1	yes	characteristic
21	#10597386	C₄	1	1	yes	characteristic
22	#10597385	C₄	1	2	yes	—
23	#10597385	C₄	1	2	yes	—
24	#10597382	C₄	1	1	yes	characteristic
25	C₄×C₂²⋊D₁₉	C₂²	1	2	yes	—
26	C₄×C₂²⋊D₁₉	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
28	C₂×#64520	C₂²	1	1	yes	characteristic
29	C₂²×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
30	#10597353	—	2	4	no	—
31	C₂×#64515	C₂²	1	2	yes	—
32	C₂×C₈×D₁₉	C₂²	1	2	yes	—
33	C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
34	C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
35	(C₂×D₁₉)⋊C₈	C₂²	1	2	yes	—
36	(C₂×D₁₉)⋊C₈	C₂²	1	2	yes	—
37	#10597272	C₂²	1	2	yes	—
38	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
39	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
41	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
42	C₈×Dic₁₉	C₂²	1	2	yes	—
43	#41×C₁₉	—	2	2	no	—
44	#41×C₁₉	—	2	2	no	—
45	C₂×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
46	C₂×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
47	C₂×D₄×C₁₉	C₂×C₄	1	2	yes	—
48	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₁₉	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊D₁₉	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊D₁₉	C₂×C₄	1	2	yes	—
53	C₂²×Dic₁₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×Dic₁₉	C₂×C₄	1	1	yes	characteristic
55	#64541	—	2	2	no	—
56	#64541	—	2	2	no	—
57	C₂×D₇₆	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₁₉	C₂³	1	2	yes	—
59	C₂×Dic₃₈	C₂×C₄	1	1	yes	characteristic
60	M₁₆×C₁₉	—	2	4	no	—
61	M₁₆×C₁₉	D₄	1	4	yes	complemented
62	C₂×C₈×C₁₉	—	2	2	no	—
63	C₂×C₈×C₁₉	—	2	4	no	—
64	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₁₉	C₂³	1	2	yes	—
67	C₂²⋊Dic₁₉	C₂×C₄	1	2	yes	—
68	C₂²⋊Dic₁₉	C₂×C₄	1	2	yes	—
69	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
70	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
71	Dic₁₉⋊C₄	C₂×C₄	1	2	yes	—
72	Dic₁₉⋊C₄	C₂×C₄	1	2	yes	—
73	C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
74	C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
75	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
76	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
77	#64520	—	2	2	no	—
78	C₂×C₁₉⋊C₈	—	2	2	no	—
79	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
80	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
82	#64515	—	2	4	no	—
83	C₈×D₁₉	—	2	4	no	—
84	C₂³×C₁₉	C₂²×C₄	1	2	yes	—
85	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₁₉	C₂²×C₄	1	2	yes	—
87	Q₈×C₁₉	—	2	2	no	—
88	D₄×C₁₉	—	2	4	no	—
89	D₄×C₁₉	—	2	2	no	—
90	C₂×C₄×C₁₉	—	2	2	no	—
91	C₂×C₄×C₁₉	—	2	2	no	—
92	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₁₉	#41	1	2	yes	—
96	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₁₉	—	2	4	no	—
98	C₂²⋊D₁₉	—	2	4	no	—
99	C₂²⋊D₁₉	—	4	2	no	—
100	C₂×Dic₁₉	—	2	2	no	—
101	C₂×Dic₁₉	—	4	1	no	—
102	C₂×Dic₁₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₁₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₁₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
106	C₂×Dic₁₉	C₂²×C₄	1	1	yes	characteristic
107	D₇₆	—	2	2	no	—
108	C₄×D₁₉	—	2	2	no	—
109	C₄×D₁₉	—	2	2	no	—
110	Dic₃₈	—	2	2	no	—
111	C₈×C₁₉	—	2	1	no	—
112	C₈×C₁₉	—	2	1	no	—
113	C₈×C₁₉	C₂×D₄	1	4	yes	—
114	C₁₉⋊C₈	—	2	1	no	—
115	C₁₉⋊C₈	—	2	1	no	—
116	C₁₉⋊C₈	—	2	2	no	—
117	#2935	—	19	1	no	maximal, Sylow
118	C₂²×C₁₉	—	2	1	no	—
119	C₂²×C₁₉	—	2	1	no	—
120	C₂²×C₁₉	—	2	2	no	—
121	C₂²×C₁₉	—	2	2	no	—
122	C₂²×C₁₉	—	2	2	no	—
123	C₂²×C₁₉	C₄×D₄	1	2	yes	—
124	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
125	C₂×D₁₉	—	2	2	no	—
126	C₂×D₁₉	—	2	2	no	—
127	C₂×D₁₉	—	2	2	no	—
128	C₄×C₁₉	—	2	2	no	—
129	C₄×C₁₉	C₄×D₄	1	2	yes	—
130	C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
131	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
132	Dic₁₉	—	2	2	no	—
133	Dic₁₉	—	2	1	no	—
134	Dic₁₉	—	2	1	no	—
135	Dic₁₉	—	2	1	no	—
136	Dic₁₉	#131	1	2	yes	complemented
137	C₂×#131	—	19	1	no	—
138	C₂×#131	D₁₉	1	1	yes	characteristic, complemented
139	#532	—	19	1	no	—
140	#436	—	19	2	no	—
141	#435	—	19	2	no	—
142	#435	—	19	2	no	—
143	C₈×D₄	—	19	2	no	—
144	C₂×C₄⋊C₈	—	19	1	no	—
145	#408	—	19	1	no	—
146	#408	—	19	1	no	—
147	C₄×M₁₆	—	19	1	no	—
148	C₂×C₁₉	—	2	2	no	—
149	C₂×C₁₉	—	4	2	no	—
150	C₂×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
151	C₂×C₁₉	C₂×#131	1	1	yes	characteristic
152	C₂×C₁₉	#568	1	1	yes	characteristic
153	D₁₉	—	4	2	no	—
154	C₂×#41	—	19	1	no	—
155	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
156	#131	—	2	4	no	—
157	#131	—	38	4	no	—
158	C₂×M₁₆	—	19	1	no	—
159	C₂×M₁₆	—	19	2	no	—
160	C₂×M₁₆	C₂×D₁₉	1	2	yes	—
161	C₂×M₁₆	C₂×D₁₉	1	1	yes	characteristic
162	C₂²×C₈	—	19	1	no	—
163	C₂²×C₈	—	19	2	no	—
164	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic
165	C₂²×C₈	C₂×D₁₉	1	2	yes	—
166	C₄⋊Q₈	—	19	1	no	—
167	#127	—	19	1	no	—
168	#124	—	19	1	no	—
169	#124	—	19	1	no	—
170	#121	—	19	2	no	—
171	#121	—	19	2	no	—
172	C₄×D₄	—	19	2	no	—
173	C₄×D₄	—	19	2	no	—
174	C₂×C₄²	—	19	1	no	—
175	C₄⋊C₈	—	19	1	no	—
176	C₄⋊C₈	—	19	1	no	—
177	C₄⋊C₈	—	19	1	no	—
178	C₄⋊C₈	—	19	1	no	—
179	C₂²⋊C₈	—	19	2	no	—
180	C₂²⋊C₈	—	19	2	no	—
181	C₂²⋊C₈	—	19	2	no	—
182	C₂²⋊C₈	—	19	2	no	—
183	#97	—	19	2	no	—
184	C₄×C₈	—	19	2	no	—
185	C₁₉	#2935	1	1	yes	characteristic, complemented, Sylow
186	#41	—	2	2	no	—
187	#41	—	2	2	no	—
188	#41	—	38	2	no	—
189	#41	—	38	2	no	—
190	C₂×Q₈	—	19	1	no	—
191	C₂×Q₈	C₄×D₁₉	1	1	yes	characteristic
192	C₂×D₄	—	19	2	no	—
193	C₂×D₄	—	19	2	no	—
194	C₂×D₄	—	19	2	no	—
195	C₂×D₄	—	19	1	no	—
196	C₂×D₄	C₄×D₁₉	1	2	yes	—
197	C₂×D₄	C₄×D₁₉	1	1	yes	characteristic
198	C₂²×C₄	—	19	1	no	—
199	C₂²×C₄	—	19	1	no	—
200	C₂²×C₄	—	19	2	no	—
201	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
202	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
203	M₁₆	—	2	4	no	—
204	M₁₆	—	38	2	no	—
205	M₁₆	—	38	4	no	—
206	M₁₆	C₂²⋊D₁₉	1	4	yes	complemented
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	19	2	no	—
210	C₂×C₈	—	19	1	no	—
211	C₂×C₈	—	19	1	no	—
212	C₂×C₈	—	38	2	no	—
213	C₂×C₈	—	38	4	no	—
214	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
215	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
216	C₂×C₈	C₂²×D₁₉	1	2	yes	—
217	C₄⋊C₄	—	19	2	no	—
218	C₄⋊C₄	—	19	2	no	—
219	C₂²⋊C₄	—	19	2	no	—
220	C₂²⋊C₄	—	19	2	no	—
221	C₂²⋊C₄	—	19	2	no	—
222	C₂²⋊C₄	—	19	2	no	—
223	C₄²	—	19	1	no	—
224	C₄²	—	19	1	no	—
225	C₄²	—	19	1	no	—
226	C₄²	—	19	1	no	—
227	C₂³	—	19	2	no	—
228	C₂³	C₂×C₄×D₁₉	1	2	yes	—
229	C₂³	C₂×C₄×D₁₉	1	1	yes	characteristic
230	Q₈	—	2	2	no	—
231	Q₈	—	38	2	no	—
232	D₄	—	2	4	no	—
233	D₄	—	2	2	no	—
234	D₄	—	38	4	no	—
235	D₄	—	38	4	no	—
236	D₄	—	38	2	no	—
237	D₄	—	76	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	19	1	no	—
241	C₂×C₄	—	19	1	no	—
242	C₂×C₄	—	19	1	no	—
243	C₂×C₄	—	19	1	no	—
244	C₂×C₄	—	19	2	no	—
245	C₂×C₄	—	38	2	no	—
246	C₂×C₄	—	38	2	no	—
247	C₂×C₄	—	38	2	no	—
248	C₂×C₄	—	76	1	no	—
249	C₂×C₄	C₂×C₄×D₁₉	1	2	yes	—
250	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
251	C₂×C₄	#64541	1	2	yes	—
252	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
253	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	38	2	no	—
257	C₈	—	38	1	no	—
258	C₈	—	38	1	no	—
259	C₈	C₂×C₂²⋊D₁₉	1	4	yes	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	38	2	no	—
266	C₂²	—	38	2	no	—
267	C₂²	—	38	2	no	—
268	C₂²	C₄×C₂²⋊D₁₉	1	2	yes	—
269	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	19	2	no	—
272	C₄	—	38	1	no	—
273	C₄	—	38	1	no	—
274	C₄	—	38	1	no	—
275	C₄	—	38	2	no	—
276	C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
277	C₄	C₂×#64541	1	1	yes	characteristic
278	C₄	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	4	2	no	—
281	C₂	—	76	2	no	—
282	C₂	C₂×C₄×C₂²⋊D₁₉	1	1	yes	characteristic
283	C₂	#131×D₁₉	1	1	yes	characteristic
284	C₂	#11927309	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉
commutator, socle	C₂²×C₁₉	C₂³×C₄
Fitting	C₂×C₁₉×#131	C₂
prime radical	#11927090	C₂
cosocle	C₂×C₄×C₁₉	C₂⁴
hypercenter	C₂×#131	D₁₉
hypocenter	C₁₉	#2935
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#10597353	C₂	semidirect
3. #11926620	C₂	semidirect
4. #11926615	C₂	semidirect
5. #11926613	C₂	semidirect
6. C₈×C₂²⋊D₁₉	C₂	semidirect
7. C₂×Dic₁₉⋊C₈	C₂	semidirect
8. #11926511	C₂	semidirect
9. #11926510	C₂	semidirect
10. M₁₆×Dic₁₉	C₂	semidirect
11. C₂×C₁₉×#131	C₂	semidirect
61. M₁₆×C₁₉	D₄	semidirect
136. Dic₁₉	#131	semidirect
138. C₂×#131	D₁₉	semidirect
185. C₁₉	#2935	semidirect
206. M₁₆	C₂²⋊D₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597353	2
#11927090	2
#11926620	2
#11926615	2
#11926613	2
C₈×C₂²⋊D₁₉	2
C₂×Dic₁₉⋊C₈	2
#11926511	2
#11926510	2
M₁₆×Dic₁₉	2
C₂×C₁₉×#131	2
#2935	19