Subgroups — #14576836

Order 2432.

1540 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	C₂²×#64541	C₂	1	1	yes	maximal, characteristic
2	#11927025	C₂	1	4	yes	maximal, complemented
3	#11927024	C₂	1	4	yes	maximal, complemented
4	#11927021	C₂	1	2	yes	maximal, complemented
5	#11927018	C₂	1	2	yes	maximal, complemented
6	#11927017	C₂	1	4	yes	maximal, complemented
7	C₄×#64541	C₂	1	4	yes	maximal, complemented
8	#11927012	C₂	1	4	yes	maximal, complemented
9	C₂×#10597372	C₂	1	1	yes	maximal, characteristic, complemented
10	#117×D₁₉	C₂	1	4	yes	maximal, complemented
11	C₂×C₁₉×#117	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊D₁₉	C₂²	1	2	yes	—
14	C₂×#64541	C₄	1	16	yes	complemented
15	C₂×#64541	C₂²	1	2	yes	—
16	C₂×#64541	C₂²	1	2	yes	—
17	C₂×#64541	C₂²	1	2	yes	—
18	C₂×#64541	C₂²	1	2	yes	—
19	C₂²×D₇₆	C₂²	1	1	yes	characteristic
20	C₂²×C₄×D₁₉	C₂²	1	2	yes	—
21	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
22	C₁₉×#117	C₂²	1	2	yes	—
23	C₁₉×#117	C₂²	1	2	yes	—
24	C₁₉×#117	C₂²	1	2	yes	complemented
25	C₁₉×#117	C₂²	1	2	yes	complemented
26	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	complemented
27	C₂×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
28	C₂×C₄²×C₁₉	C₂²	1	2	yes	complemented
29	C₂×C₂²⋊Dic₁₉	C₂²	1	2	yes	—
30	C₄×C₂²⋊D₁₉	C₂²	1	16	yes	complemented
31	#10597372	C₂²	1	2	yes	—
32	#10597372	C₂²	1	2	yes	—
33	#10597372	C₂²	1	2	yes	complemented
34	#10597372	C₂²	1	2	yes	complemented
35	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	complemented
36	C₂×C₄×Dic₁₉	C₂²	1	2	yes	complemented
37	#10597339	C₂²	1	8	yes	complemented
38	#10597338	C₂²	1	8	yes	complemented
39	C₄⋊C₄×D₁₉	C₂²	1	8	yes	complemented
40	#10597333	C₂²	1	8	yes	complemented
41	(C₂²⋊D₁₉)⋊C₄	C₂²	1	16	yes	complemented
42	C₂²⋊C₄×D₁₉	C₂²	1	8	yes	—
43	#10597323	C₂²	1	8	yes	—
44	C₄×D₇₆	C₂²	1	8	yes	complemented
45	#10597318	C₂²	1	8	yes	complemented
46	C₄²×D₁₉	C₂²	1	8	yes	complemented
47	C₄×Dic₃₈	C₂²	1	8	yes	complemented
48	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
49	C₂³×D₁₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₉	C₂×C₄	1	8	yes	complemented
51	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
52	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
58	C₂×C₂²⋊D₁₉	C₂×C₄	1	16	yes	complemented
59	C₂×C₂²⋊D₁₉	C₂³	1	4	yes	—
60	C₂×C₂²⋊D₁₉	C₂³	1	4	yes	—
61	C₂²×Dic₁₉	C₂³	1	4	yes	—
62	C₂²×Dic₁₉	C₂³	1	2	yes	—
63	#64541	C₂×C₄	1	16	yes	complemented
64	#64541	C₂×C₄	1	16	yes	complemented
65	#64541	C₂×C₄	1	16	yes	complemented
66	#64541	C₂×C₄	1	16	yes	complemented
67	C₂×D₇₆	C₂×C₄	1	8	yes	complemented
68	C₂×D₇₆	C₂³	1	2	yes	—
69	C₂×D₇₆	C₂³	1	2	yes	—
70	C₂×C₄×D₁₉	C₂×C₄	1	16	yes	complemented
71	C₂×C₄×D₁₉	C₂³	1	16	yes	—
72	C₂×C₄×D₁₉	C₂³	1	4	yes	—
73	C₂×C₄×D₁₉	C₂³	1	4	yes	—
74	C₂×Dic₃₈	C₂×C₄	1	8	yes	complemented
75	C₂×Dic₃₈	C₂³	1	2	yes	—
76	C₂×Dic₃₈	C₂³	1	2	yes	—
77	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
78	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
79	C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
80	C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
81	C₄²×C₁₉	C₂³	1	4	yes	—
82	C₄²×C₁₉	C₂³	1	4	yes	—
83	C₂²⋊Dic₁₉	C₂³	1	4	yes	—
84	C₂²⋊Dic₁₉	C₂³	1	4	yes	—
85	(C₂×D₁₉)⋊C₄	C₂³	1	16	yes	—
86	C₄⋊Dic₁₉	C₂³	1	4	yes	—
87	C₄⋊Dic₁₉	C₂³	1	4	yes	—
88	Dic₁₉⋊C₄	C₂³	1	16	yes	—
89	C₄×Dic₁₉	C₂³	1	16	yes	—
90	C₄×Dic₁₉	C₂³	1	4	yes	—
91	C₄×Dic₁₉	C₂³	1	4	yes	—
92	C₂³×C₁₉	—	4	2	no	—
93	C₂³×C₁₉	C₂²×C₄	1	4	yes	—
94	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
95	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
96	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
97	C₂²×D₁₉	—	4	4	no	—
98	C₂²×D₁₉	C₂²×C₄	1	8	yes	—
99	C₂²×D₁₉	C₂⁴	1	4	yes	—
100	C₂×C₄×C₁₉	—	4	4	no	—
101	C₂×C₄×C₁₉	C₂²×C₄	1	4	yes	—
102	C₂×C₄×C₁₉	C₂²×C₄	1	4	yes	—
103	C₂×C₄×C₁₉	C₂²×C₄	1	4	yes	—
104	C₂×C₄×C₁₉	C₂²×C₄	1	4	yes	—
105	C₂×C₄×C₁₉	C₂²×C₄	1	4	yes	—
106	C₂×C₄×C₁₉	C₂²×C₄	1	4	yes	—
107	C₂×C₄×C₁₉	C₂⁴	1	8	yes	—
108	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
112	C₂²⋊D₁₉	C₂²×C₄	1	16	yes	—
113	C₂²⋊D₁₉	C₂²×C₄	1	16	yes	—
114	C₂×Dic₁₉	—	4	4	no	—
115	C₂×Dic₁₉	C₂²×C₄	1	8	yes	—
116	C₂×Dic₁₉	C₂⁴	1	8	yes	—
117	C₂×Dic₁₉	C₂⁴	1	4	yes	—
118	D₇₆	C₂²×C₄	1	8	yes	—
119	D₇₆	C₂²×C₄	1	8	yes	—
120	C₄×D₁₉	—	4	16	no	—
121	C₄×D₁₉	C₂²×C₄	1	16	yes	—
122	C₄×D₁₉	C₂²×C₄	1	16	yes	—
123	Dic₃₈	C₂²×C₄	1	8	yes	—
124	Dic₃₈	C₂²×C₄	1	8	yes	—
125	#3428	—	19	1	no	maximal, Sylow
126	C₂²×C₁₉	—	2	2	no	—
127	C₂²×C₁₉	—	2	2	no	—
128	C₂²×C₁₉	—	2	2	no	—
129	C₂²×C₁₉	—	4	4	no	—
130	C₂²×C₁₉	C₂³×C₄	1	2	yes	—
131	C₂²×C₁₉	C₂³×C₄	1	2	yes	—
132	C₂²×C₁₉	C₂³×C₄	1	2	yes	—
133	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
134	C₂×D₁₉	—	2	8	no	—
135	C₂×D₁₉	—	4	8	no	—
136	C₂×D₁₉	C₂³×C₄	1	8	yes	—
137	C₄×C₁₉	—	2	8	no	—
138	C₄×C₁₉	C₂³×C₄	1	2	yes	—
139	C₄×C₁₉	C₂³×C₄	1	2	yes	—
140	C₄×C₁₉	C₂³×C₄	1	2	yes	—
141	C₄×C₁₉	C₂³×C₄	1	2	yes	—
142	Dic₁₉	—	2	8	no	—
143	Dic₁₉	C₂³×C₄	1	8	yes	—
144	C₂²×#41	—	19	1	no	—
145	#520	—	19	4	no	—
146	#520	—	19	4	no	—
147	#519	—	19	2	no	—
148	#518	—	19	2	no	—
149	#518	—	19	4	no	—
150	C₄×#41	—	19	4	no	—
151	C₄×#41	—	19	4	no	—
152	C₂×#117	—	19	1	no	—
153	C₂×#117	—	19	4	no	—
154	C₂×#117	D₁₉	1	1	yes	characteristic, complemented
155	C₂×C₁₉	—	2	2	no	—
156	C₂×C₁₉	—	2	2	no	—
157	C₂×C₁₉	—	2	2	no	—
158	C₂×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
159	C₂×C₁₉	#585	1	2	yes	—
160	D₁₉	—	2	8	no	—
161	C₂×#41	—	19	16	no	—
162	C₂×#41	—	19	2	no	—
163	C₂×#41	—	19	2	no	—
164	C₂×#41	—	19	2	no	—
165	C₂×#41	—	19	2	no	—
166	C₂²×Q₈	—	19	1	no	—
167	C₂²×D₄	—	19	2	no	—
168	C₂²×D₄	—	19	1	no	—
169	C₂³×C₄	—	19	2	no	—
170	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
171	C₄×Q₈	—	19	8	no	—
172	C₄×Q₈	—	19	8	no	—
173	C₄×D₄	—	19	16	no	—
174	C₄×D₄	—	19	8	no	—
175	C₄×D₄	—	19	16	no	—
176	C₄×D₄	—	19	8	no	—
177	#117	—	19	2	no	—
178	#117	—	19	2	no	—
179	#117	—	19	2	no	—
180	#117	—	19	2	no	—
181	#117	—	19	8	no	—
182	#117	—	19	8	no	—
183	#117	—	19	8	no	—
184	#117	C₂×D₁₉	1	2	yes	—
185	#117	C₂×D₁₉	1	2	yes	—
186	#117	C₂×D₁₉	1	2	yes	complemented
187	#117	C₂×D₁₉	1	2	yes	complemented
188	C₂×C₄⋊C₄	—	19	2	no	—
189	C₂×C₄⋊C₄	—	19	8	no	—
190	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	complemented
191	C₂×C₂²⋊C₄	—	19	2	no	—
192	C₂×C₂²⋊C₄	—	19	8	no	—
193	C₂×C₂²⋊C₄	C₂×D₁₉	1	2	yes	—
194	C₂×C₄²	—	19	2	no	—
195	C₂×C₄²	—	19	8	no	—
196	C₂×C₄²	C₂×D₁₉	1	2	yes	complemented
197	C₁₉	#3428	1	1	yes	characteristic, complemented, Sylow
198	C₂⁴	—	19	2	no	—
199	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
200	#41	—	19	16	no	—
201	#41	—	19	16	no	—
202	#41	—	19	16	no	—
203	#41	—	19	16	no	—
204	C₂×Q₈	—	19	8	no	—
205	C₂×Q₈	—	19	2	no	—
206	C₂×Q₈	—	19	2	no	—
207	C₂×D₄	—	19	16	no	—
208	C₂×D₄	—	19	4	no	—
209	C₂×D₄	—	19	4	no	—
210	C₂×D₄	—	19	8	no	—
211	C₂×D₄	—	19	2	no	—
212	C₂×D₄	—	19	2	no	—
213	C₂²×C₄	—	19	2	no	—
214	C₂²×C₄	—	19	16	no	—
215	C₂²×C₄	—	19	4	no	—
216	C₂²×C₄	—	19	4	no	—
217	C₂²×C₄	—	19	4	no	—
218	C₂²×C₄	—	19	16	no	—
219	C₂²×C₄	C₄×D₁₉	1	8	yes	complemented
220	C₂²×C₄	C₂²×D₁₉	1	4	yes	—
221	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
222	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
223	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
224	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
225	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
226	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
227	C₄⋊C₄	—	19	16	no	—
228	C₄⋊C₄	—	19	4	no	—
229	C₄⋊C₄	—	19	4	no	—
230	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
231	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
232	C₂²⋊C₄	—	19	4	no	—
233	C₂²⋊C₄	—	19	4	no	—
234	C₂²⋊C₄	—	19	16	no	—
235	C₂²⋊C₄	C₂²×D₁₉	1	4	yes	—
236	C₂²⋊C₄	C₂²×D₁₉	1	4	yes	—
237	C₄²	—	19	16	no	—
238	C₄²	—	19	4	no	—
239	C₄²	—	19	4	no	—
240	C₄²	C₂²×D₁₉	1	4	yes	—
241	C₄²	C₂²×D₁₉	1	4	yes	—
242	C₂³	—	4	2	no	—
243	C₂³	—	19	4	no	—
244	C₂³	—	19	8	no	—
245	C₂³	—	76	4	no	—
246	C₂³	C₂×C₄×D₁₉	1	4	yes	—
247	C₂³	C₂³×D₁₉	1	1	yes	characteristic
248	C₂³	C₂³×D₁₉	1	1	yes	characteristic
249	C₂³	C₂³×D₁₉	1	1	yes	characteristic
250	Q₈	—	19	8	no	—
251	Q₈	—	19	8	no	—
252	D₄	—	19	16	no	—
253	D₄	—	19	16	no	—
254	D₄	—	19	8	no	—
255	D₄	—	19	8	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	19	8	no	—
258	C₂×C₄	—	19	4	no	—
259	C₂×C₄	—	19	16	no	—
260	C₂×C₄	—	19	16	no	—
261	C₂×C₄	—	19	8	no	—
262	C₂×C₄	—	76	16	no	—
263	C₂×C₄	—	76	4	no	—
264	C₂×C₄	C₂×C₄×D₁₉	1	4	yes	—
265	C₂×C₄	C₂×C₄×D₁₉	1	4	yes	—
266	C₂×C₄	C₂×C₄×D₁₉	1	4	yes	—
267	C₂×C₄	C₂×C₄×D₁₉	1	4	yes	—
268	C₂×C₄	C₂×C₄×D₁₉	1	4	yes	—
269	C₂×C₄	C₂×C₄×D₁₉	1	4	yes	—
270	C₂×C₄	C₂³×D₁₉	1	8	yes	—
271	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
272	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
273	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
274	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	19	8	no	—
280	C₂²	—	38	8	no	—
281	C₂²	—	76	8	no	—
282	C₂²	C₂²×C₄×D₁₉	1	2	yes	—
283	C₂²	C₂²×C₄×D₁₉	1	2	yes	—
284	C₂²	C₂²×C₄×D₁₉	1	2	yes	—
285	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
286	C₄	—	2	8	no	—
287	C₄	—	19	8	no	—
288	C₄	—	38	8	no	—
289	C₄	C₂²×C₄×D₁₉	1	2	yes	—
290	C₄	C₂²×C₄×D₁₉	1	2	yes	—
291	C₄	C₂²×C₄×D₁₉	1	2	yes	—
292	C₄	C₂²×C₄×D₁₉	1	2	yes	—
293	C₂	—	2	2	no	—
294	C₂	—	2	2	no	—
295	C₂	—	2	2	no	—
296	C₂	—	38	8	no	—
297	C₂	C₂³×C₄×D₁₉	1	1	yes	characteristic
298	C₂	#11927380	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₉
commutator	C₂×C₁₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₉
Fitting	C₂×C₁₉×#117	C₂
socle, cosocle	C₂²×C₁₉	C₂⁵
prime radical	C₂²×#64541	C₂
hypercenter	C₂×#117	D₁₉
hypocenter	C₁₉	#3428
LCS	G ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #11927025	C₂	semidirect
3. #11927024	C₂	semidirect
4. #11927021	C₂	semidirect
5. #11927018	C₂	semidirect
6. #11927017	C₂	semidirect
7. C₄×#64541	C₂	semidirect
8. #11927012	C₂	semidirect
9. C₂×#10597372	C₂	semidirect
10. #117×D₁₉	C₂	semidirect
11. C₂×C₁₉×#117	C₂	semidirect
14. C₂×#64541	C₄	semidirect
24. C₁₉×#117	C₂²	semidirect
25. C₁₉×#117	C₂²	semidirect
26. C₂×C₄⋊C₄×C₁₉	C₂²	semidirect
28. C₂×C₄²×C₁₉	C₂²	semidirect
30. C₄×C₂²⋊D₁₉	C₂²	semidirect
33. #10597372	C₂²	semidirect
34. #10597372	C₂²	semidirect
35. C₂×C₄⋊Dic₁₉	C₂²	semidirect
36. C₂×C₄×Dic₁₉	C₂²	semidirect
37. #10597339	C₂²	semidirect
38. #10597338	C₂²	semidirect
39. C₄⋊C₄×D₁₉	C₂²	semidirect
40. #10597333	C₂²	semidirect
41. (C₂²⋊D₁₉)⋊C₄	C₂²	semidirect
44. C₄×D₇₆	C₂²	semidirect
45. #10597318	C₂²	semidirect
46. C₄²×D₁₉	C₂²	semidirect
47. C₄×Dic₃₈	C₂²	semidirect
50. C₂²×C₄×C₁₉	C₂×C₄	semidirect
58. C₂×C₂²⋊D₁₉	C₂×C₄	semidirect
63. #64541	C₂×C₄	semidirect
64. #64541	C₂×C₄	semidirect
65. #64541	C₂×C₄	semidirect
66. #64541	C₂×C₄	semidirect
67. C₂×D₇₆	C₂×C₄	semidirect
70. C₂×C₄×D₁₉	C₂×C₄	semidirect
74. C₂×Dic₃₈	C₂×C₄	semidirect
154. C₂×#117	D₁₉	semidirect
186. #117	C₂×D₁₉	semidirect
187. #117	C₂×D₁₉	semidirect
190. C₂×C₄⋊C₄	C₂×D₁₉	semidirect
196. C₂×C₄²	C₂×D₁₉	semidirect
197. C₁₉	#3428	semidirect
219. C₂²×C₄	C₄×D₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#64541	2
#11927025	2
#11927024	2
#11927021	2
#11927018	2
#11927017	2
C₄×#64541	2
#11927012	2
C₂×#10597372	2
#117×D₁₉	2
C₂×C₁₉×#117	2
#3428	19