Subgroups — #14572359

Order 2432.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597347	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×M₁₆×D₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#11927056	C₂	1	1	yes	maximal, characteristic
4	#11926621	C₂	1	2	yes	maximal, complemented
5	#11926614	C₂	1	2	yes	maximal, complemented
6	#11926614	C₂	1	2	yes	maximal, complemented
7	#11926609	C₂	1	2	yes	maximal, complemented
8	#11926512	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×(C₂×D₁₉)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926500	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₉×#424	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#64541	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₉	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#117	C₂²	1	1	yes	characteristic
17	C₂×#64520	C₂²	1	1	yes	characteristic
18	C₂²×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
19	M₁₆×D₁₉	—	2	4	no	—
20	#10597347	—	2	4	no	—
21	C₂×#64515	C₂²	1	2	yes	—
22	C₂×#64515	C₂²	1	2	yes	complemented
23	C₂×C₈×D₁₉	C₂²	1	2	yes	—
24	C₂×C₈×D₁₉	C₂²	1	2	yes	complemented
25	#10597343	C₄	1	1	yes	characteristic
26	#10597343	C₄	1	1	yes	characteristic
27	#10597341	C₄	1	1	yes	characteristic
28	#10597341	C₄	1	1	yes	characteristic
29	#10597332	C₄	1	2	yes	—
30	#10597328	C₄	1	2	yes	—
31	C₄×D₇₆	C₂²	1	2	yes	—
32	C₄×D₇₆	C₂²	1	2	yes	—
33	C₁₉×C₄⋊C₈	C₂²	1	2	yes	complemented
34	C₁₉×C₄⋊C₈	C₂²	1	2	yes	complemented
35	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
36	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
37	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
38	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
39	(C₂×D₁₉)⋊C₈	C₂²	1	2	yes	complemented
40	(C₂×D₁₉)⋊C₈	C₂²	1	2	yes	complemented
41	#10597260	C₂²	1	2	yes	complemented
42	C₄×C₁₉⋊C₈	C₂²	1	2	yes	complemented
43	C₂³×D₁₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₁₉	C₂×C₄	1	2	yes	—
46	C₂²×Dic₁₉	C₂×C₄	1	1	yes	characteristic
47	#64541	—	2	2	no	—
48	#64541	—	2	2	no	—
49	C₂×D₇₆	C₂×C₄	1	1	yes	characteristic
50	C₂×D₇₆	C₂×C₄	1	2	yes	—
51	C₂×D₇₆	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₁₉	—	2	2	no	—
53	C₂×C₄×D₁₉	—	2	2	no	—
54	C₂×C₄×D₁₉	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₁₉	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₁₉	C₂³	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	2	no	—
61	C₂×C₈×C₁₉	—	2	2	no	—
62	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₁₉	C₂³	1	2	yes	—
65	C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
66	C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
67	C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
68	C₄²×C₁₉	C₂³	1	2	yes	—
69	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
70	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
71	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
72	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
73	C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
74	C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
75	#64520	D₄	1	4	yes	complemented
76	C₂×C₁₉⋊C₈	—	2	2	no	—
77	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
79	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
80	#64515	—	2	4	no	—
81	#64515	—	2	4	no	—
82	C₈×D₁₉	—	2	4	no	—
83	C₈×D₁₉	—	2	4	no	—
84	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²×D₁₉	—	2	2	no	—
86	C₂²×D₁₉	—	2	2	no	—
87	C₂²×D₁₉	—	4	1	no	—
88	C₂²×D₁₉	C₂²×C₄	1	1	yes	characteristic
89	C₂²×D₁₉	C₂²×C₄	1	2	yes	—
90	C₂²×D₁₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
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	C₂×Dic₁₉	—	2	2	no	—
99	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
100	C₂×Dic₁₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₁₉	C₂²×C₄	1	1	yes	characteristic
102	D₇₆	—	2	4	no	—
103	D₇₆	—	2	2	no	—
104	D₇₆	—	4	2	no	—
105	C₄×D₁₉	—	2	2	no	—
106	C₄×D₁₉	—	2	2	no	—
107	C₄×D₁₉	—	2	1	no	—
108	C₄×D₁₉	—	2	2	no	—
109	C₄×D₁₉	—	2	1	no	—
110	C₄×D₁₉	—	2	1	no	—
111	C₄×D₁₉	—	2	1	no	—
112	C₄×D₁₉	—	2	2	no	—
113	Dic₃₈	—	2	2	no	—
114	C₈×C₁₉	—	2	1	no	—
115	C₈×C₁₉	—	2	2	no	—
116	C₈×C₁₉	—	2	1	no	—
117	C₁₉⋊C₈	—	2	1	no	—
118	C₁₉⋊C₈	—	2	1	no	—
119	C₁₉⋊C₈	C₂×D₄	1	4	yes	—
120	#2933	—	19	1	no	maximal, Sylow
121	C₂²×C₁₉	—	2	1	no	—
122	C₂²×C₁₉	—	2	1	no	—
123	C₂²×C₁₉	C₄×D₄	1	2	yes	—
124	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
125	C₂×D₁₉	—	2	1	no	—
126	C₂×D₁₉	—	2	1	no	—
127	C₂×D₁₉	—	2	1	no	—
128	C₂×D₁₉	—	2	2	no	—
129	C₂×D₁₉	—	2	2	no	—
130	C₂×D₁₉	—	2	1	no	—
131	C₂×D₁₉	—	2	1	no	—
132	C₂×D₁₉	—	2	2	no	—
133	C₂×D₁₉	—	4	4	no	—
134	C₂×D₁₉	#131	1	2	yes	complemented
135	C₄×C₁₉	—	2	2	no	—
136	C₄×C₁₉	—	2	2	no	—
137	C₄×C₁₉	C₄×D₄	1	2	yes	—
138	C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
140	Dic₁₉	—	2	2	no	—
141	Dic₁₉	—	2	1	no	—
142	Dic₁₉	—	2	1	no	—
143	C₂×#131	—	19	1	no	—
144	C₂²×M₁₆	—	19	1	no	—
145	#525	—	19	1	no	—
146	#436	—	19	2	no	—
147	#435	—	19	2	no	—
148	#435	—	19	2	no	—
149	C₈×D₄	—	19	2	no	—
150	#424	D₁₉	1	1	yes	characteristic, complemented
151	#408	—	19	1	no	—
152	C₂×C₂²⋊C₈	—	19	1	no	—
153	#405	—	19	1	no	—
154	C₂×C₁₉	—	2	2	no	—
155	C₂×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂×C₁₉	C₂×#131	1	1	yes	characteristic
157	C₂×C₁₉	#568	1	1	yes	characteristic
158	D₁₉	—	2	2	no	—
159	D₁₉	—	4	2	no	—
160	D₁₉	—	4	1	no	—
161	C₂×#41	—	19	1	no	—
162	C₂³×C₄	—	19	1	no	—
163	#131	—	38	4	no	—
164	C₂×M₁₆	—	19	1	no	—
165	C₂×M₁₆	—	19	2	no	—
166	C₂×M₁₆	—	19	2	no	—
167	C₂×M₁₆	—	38	4	no	—
168	C₂×M₁₆	C₂×D₁₉	1	1	yes	characteristic
169	C₂²×C₈	—	19	1	no	—
170	C₂²×C₈	—	19	2	no	—
171	C₂²×C₈	—	19	2	no	—
172	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic
173	#123	—	19	2	no	—
174	C₂²⋊Q₈	—	19	1	no	—
175	C₂²⋊Q₈	—	19	1	no	—
176	#121	—	19	1	no	—
177	#121	—	19	1	no	—
178	#120	—	19	2	no	—
179	C₄×D₄	—	19	2	no	—
180	C₄×D₄	—	19	2	no	—
181	#117	C₂×D₁₉	1	1	yes	characteristic
182	C₄⋊C₈	C₂×D₁₉	1	2	yes	complemented
183	C₄⋊C₈	C₂×D₁₉	1	2	yes	complemented
184	C₂²⋊C₈	—	19	1	no	—
185	C₂²⋊C₈	—	19	1	no	—
186	C₂²⋊C₈	—	19	1	no	—
187	C₂²⋊C₈	—	19	1	no	—
188	C₂²⋊C₈	—	19	2	no	—
189	C₂²⋊C₈	—	19	2	no	—
190	#97	—	19	2	no	—
191	C₄×C₈	—	19	2	no	—
192	C₁₉	#2933	1	1	yes	characteristic, complemented, Sylow
193	C₂⁴	—	19	1	no	—
194	#41	—	38	2	no	—
195	#41	—	38	2	no	—
196	C₂×Q₈	—	19	1	no	—
197	C₂×D₄	—	19	2	no	—
198	C₂×D₄	—	19	2	no	—
199	C₂×D₄	—	19	1	no	—
200	C₂×D₄	—	19	2	no	—
201	C₂²×C₄	—	19	1	no	—
202	C₂²×C₄	—	19	1	no	—
203	C₂²×C₄	—	19	1	no	—
204	C₂²×C₄	—	19	1	no	—
205	C₂²×C₄	—	19	2	no	—
206	C₂²×C₄	—	19	1	no	—
207	C₂²×C₄	—	38	2	no	—
208	C₂²×C₄	—	38	2	no	—
209	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
210	M₁₆	—	2	2	no	—
211	M₁₆	—	19	4	no	—
212	M₁₆	—	38	4	no	—
213	M₁₆	—	38	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	19	2	no	—
216	C₂×C₈	—	19	1	no	—
217	C₂×C₈	—	19	1	no	—
218	C₂×C₈	—	38	4	no	—
219	C₂×C₈	—	38	4	no	—
220	C₂×C₈	—	38	2	no	—
221	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
222	C₂×C₈	C₂²×D₁₉	1	2	yes	—
223	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
224	C₄⋊C₄	—	19	2	no	—
225	C₄⋊C₄	—	19	2	no	—
226	C₄⋊C₄	C₄×D₁₉	1	1	yes	characteristic
227	C₄⋊C₄	C₄×D₁₉	1	1	yes	characteristic
228	C₂²⋊C₄	—	19	2	no	—
229	C₂²⋊C₄	—	19	2	no	—
230	C₂²⋊C₄	—	19	2	no	—
231	C₂²⋊C₄	—	19	2	no	—
232	C₂²⋊C₄	C₄×D₁₉	1	2	yes	—
233	C₄²	C₂²×D₁₉	1	2	yes	—
234	C₂³	—	19	1	no	—
235	C₂³	—	19	1	no	—
236	C₂³	—	19	2	no	—
237	C₂³	—	38	2	no	—
238	C₂³	—	38	2	no	—
239	C₂³	—	76	1	no	—
240	C₂³	C₂×C₄×D₁₉	1	1	yes	characteristic
241	Q₈	—	38	2	no	—
242	D₄	—	38	4	no	—
243	D₄	—	38	4	no	—
244	D₄	—	38	2	no	—
245	D₄	—	76	2	no	—
246	C₂×C₄	—	19	1	no	—
247	C₂×C₄	—	19	1	no	—
248	C₂×C₄	—	19	2	no	—
249	C₂×C₄	—	38	2	no	—
250	C₂×C₄	—	38	1	no	—
251	C₂×C₄	—	38	2	no	—
252	C₂×C₄	—	38	1	no	—
253	C₂×C₄	—	38	1	no	—
254	C₂×C₄	—	38	1	no	—
255	C₂×C₄	—	38	2	no	—
256	C₂×C₄	—	38	2	no	—
257	C₂×C₄	—	38	2	no	—
258	C₂×C₄	C₂×C₄×D₁₉	1	2	yes	—
259	C₂×C₄	C₂×C₄×D₁₉	1	2	yes	—
260	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
261	C₂×C₄	D₄×D₁₉	1	2	yes	—
262	C₂×C₄	#64545	1	2	yes	—
263	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	2	no	—
267	C₈	—	19	4	no	—
268	C₈	—	38	1	no	—
269	C₈	—	38	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	19	2	no	—
273	C₂²	—	38	1	no	—
274	C₂²	—	38	1	no	—
275	C₂²	—	38	2	no	—
276	C₂²	—	38	1	no	—
277	C₂²	—	38	1	no	—
278	C₂²	—	38	2	no	—
279	C₂²	—	38	1	no	—
280	C₂²	—	38	2	no	—
281	C₂²	—	76	4	no	—
282	C₂²	#10597339	1	2	yes	—
283	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	38	1	no	—
287	C₄	—	38	1	no	—
288	C₄	—	38	2	no	—
289	C₄	#10597339	1	2	yes	—
290	C₄	C₂×D₄×D₁₉	1	1	yes	characteristic
291	C₄	C₂×#64545	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	38	2	no	—
294	C₂	—	76	1	no	—
295	C₂	—	76	2	no	—
296	C₂	C₂×#10597339	1	1	yes	characteristic
297	C₂	#131×D₁₉	1	1	yes	characteristic
298	C₂	#11927309	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉
commutator, socle	C₂²×C₁₉	C₂³×C₄
Fitting	C₁₉×#424	C₂
prime radical	#11927056	C₂
cosocle	C₂×C₄×C₁₉	C₂⁴
hypercenter	#424	D₁₉
hypocenter	C₁₉	#2933
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
1. C₂×#10597347	C₂	semidirect
2. C₂×M₁₆×D₁₉	C₂	semidirect
4. #11926621	C₂	semidirect
5. #11926614	C₂	semidirect
6. #11926614	C₂	semidirect
7. #11926609	C₂	semidirect
8. #11926512	C₂	semidirect
9. C₂×(C₂×D₁₉)⋊C₈	C₂	semidirect
10. #11926500	C₂	semidirect
11. C₁₉×#424	C₂	semidirect
22. C₂×#64515	C₂²	semidirect
24. C₂×C₈×D₁₉	C₂²	semidirect
33. C₁₉×C₄⋊C₈	C₂²	semidirect
34. C₁₉×C₄⋊C₈	C₂²	semidirect
39. (C₂×D₁₉)⋊C₈	C₂²	semidirect
40. (C₂×D₁₉)⋊C₈	C₂²	semidirect
41. #10597260	C₂²	semidirect
42. C₄×C₁₉⋊C₈	C₂²	semidirect
75. #64520	D₄	semidirect
134. C₂×D₁₉	#131	semidirect
150. #424	D₁₉	semidirect
182. C₄⋊C₈	C₂×D₁₉	semidirect
183. C₄⋊C₈	C₂×D₁₉	semidirect
192. C₁₉	#2933	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597347	2
C₂×M₁₆×D₁₉	2
#11927056	2
#11926621	2
#11926614	2
#11926609	2
#11926512	2
C₂×(C₂×D₁₉)⋊C₈	2
#11926500	2
C₁₉×#424	2
#2933	19