Subgroups — #14577408

Order 2432.

1748 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597435	C₂	1	2	yes	maximal, complemented
2	#11927215	C₂	1	4	yes	maximal, complemented
3	D₄×D₇₆	C₂	1	4	yes	maximal, complemented
4	#11927178	C₂	1	8	yes	maximal, complemented
5	#11927112	C₂	1	4	yes	maximal, complemented
6	#11927108	C₂	1	2	yes	maximal, complemented
7	#11927095	C₂	1	2	yes	maximal, complemented
8	C₂×#10597320	C₂	1	1	yes	maximal, characteristic, complemented
9	#127×D₁₉	C₂	1	2	yes	maximal, complemented
10	C₄×#64541	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₉×#532	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₉	C₂²	1	2	yes	complemented
13	#10597435	—	2	16	no	—
14	C₂×#64545	C₂²	1	4	yes	complemented
15	C₂×D₄×D₁₉	C₂²	1	4	yes	complemented
16	C₂×D₄×D₁₉	C₂²	1	8	yes	complemented
17	C₂×D₄×D₁₉	C₂²	1	8	yes	complemented
18	C₂×#64541	C₂²	1	1	yes	characteristic
19	C₂×#64541	C₂²	1	4	yes	complemented
20	C₂²×D₇₆	C₂²	1	2	yes	complemented
21	C₂²×D₇₆	C₂²	1	4	yes	complemented
22	C₂²×D₇₆	C₂²	1	4	yes	complemented
23	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
24	C₁₉×#127	C₂²	1	1	yes	characteristic, complemented
25	C₁₉×#124	C₂²	1	2	yes	complemented
26	C₁₉×#121	C₂²	1	4	yes	complemented
27	C₄×D₄×C₁₉	C₂²	1	4	yes	complemented
28	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic, complemented
29	#10597393	C₂²	1	4	yes	complemented
30	#10597386	C₂²	1	4	yes	complemented
31	#10597386	C₂²	1	8	yes	complemented
32	#10597383	C₂²	1	8	yes	complemented
33	#10597376	C₂²	1	8	yes	complemented
34	C₄×C₂²⋊D₁₉	C₂²	1	4	yes	complemented
35	#10597372	C₂²	1	1	yes	characteristic, complemented
36	#10597341	C₂²	1	8	yes	complemented
37	#10597339	C₂²	1	8	yes	complemented
38	#10597330	C₂²	1	16	yes	complemented
39	#10597328	C₂²	1	16	yes	complemented
40	#10597320	C₂²	1	2	yes	complemented
41	#10597320	C₂²	1	2	yes	complemented
42	#10597320	C₂²	1	4	yes	complemented
43	C₄×D₇₆	C₂²	1	2	yes	complemented
44	#10597318	C₂²	1	2	yes	complemented
45	C₄²×D₁₉	C₂²	1	2	yes	complemented
46	C₄×Dic₃₈	C₂²	1	2	yes	complemented
47	C₂³×D₁₉	C₂³	1	4	yes	—
48	C₂³×D₁₉	C₂³	1	8	yes	—
49	#41×C₁₉	—	2	8	no	—
50	C₂×Q₈×C₁₉	C₂³	1	2	yes	—
51	C₂×D₄×C₁₉	C₂³	1	4	yes	—
52	C₂×D₄×C₁₉	C₂³	1	4	yes	—
53	C₂×D₄×C₁₉	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
55	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
56	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
57	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
58	C₂×C₂²⋊D₁₉	C₂³	1	16	yes	—
59	C₂×C₂²⋊D₁₉	C₂³	1	2	yes	—
60	#64545	—	2	16	no	—
61	D₄×D₁₉	—	2	32	no	—
62	D₄×D₁₉	—	2	16	no	—
63	D₄×D₁₉	—	4	16	no	—
64	#64541	—	2	16	no	—
65	#64541	D₄	1	8	yes	complemented
66	C₂×D₇₆	—	2	8	no	—
67	C₂×D₇₆	—	2	16	no	—
68	C₂×D₇₆	—	4	8	no	—
69	C₂×D₇₆	C₂³	1	4	yes	—
70	C₂×D₇₆	C₂³	1	4	yes	—
71	C₂×D₇₆	C₂³	1	8	yes	—
72	C₂×D₇₆	C₂³	1	8	yes	—
73	C₂×D₇₆	C₂³	1	8	yes	—
74	C₂×D₇₆	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₁₉	C₂³	1	8	yes	—
76	C₂×C₄×D₁₉	C₂³	1	4	yes	—
77	C₂×C₄×D₁₉	C₂³	1	2	yes	—
78	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
80	C₂²⋊C₄×C₁₉	C₂³	1	8	yes	—
81	C₄²×C₁₉	C₂³	1	2	yes	—
82	C₄²×C₁₉	C₂³	1	1	yes	characteristic
83	C₄²×C₁₉	C₂³	1	1	yes	characteristic
84	C₂²⋊Dic₁₉	C₂³	1	2	yes	—
85	(C₂×D₁₉)⋊C₄	C₂³	1	16	yes	—
86	(C₂×D₁₉)⋊C₄	C₂³	1	4	yes	—
87	C₄⋊Dic₁₉	C₂³	1	2	yes	—
88	Dic₁₉⋊C₄	C₂³	1	4	yes	—
89	C₄×Dic₁₉	C₂³	1	2	yes	—
90	C₄×Dic₁₉	C₂³	1	4	yes	—
91	C₂³×C₁₉	C₂⁴	1	4	yes	—
92	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
93	C₂²×D₁₉	—	2	8	no	—
94	C₂²×D₁₉	—	2	16	no	—
95	C₂²×D₁₉	—	4	8	no	—
96	C₂²×D₁₉	—	4	16	no	—
97	C₂²×D₁₉	C₂⁴	1	8	yes	—
98	C₂²×D₁₉	C₂⁴	1	2	yes	—
99	Q₈×C₁₉	—	2	4	no	—
100	D₄×C₁₉	—	2	8	no	—
101	D₄×C₁₉	—	2	4	no	—
102	D₄×C₁₉	—	2	8	no	—
103	C₂×C₄×C₁₉	—	2	8	no	—
104	C₂×C₄×C₁₉	—	2	4	no	—
105	C₂×C₄×C₁₉	C₂×D₄	1	4	yes	complemented
106	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
107	C₂×C₄×C₁₉	C₂⁴	1	4	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	2	yes	—
111	C₂²⋊D₁₉	—	2	16	no	—
112	C₂²⋊D₁₉	—	4	16	no	—
113	C₂²⋊D₁₉	C₂×D₄	1	8	yes	complemented
114	C₂×Dic₁₉	C₂⁴	1	4	yes	—
115	C₂×Dic₁₉	C₂⁴	1	2	yes	—
116	D₇₆	—	2	16	no	—
117	D₇₆	—	2	8	no	—
118	D₇₆	—	2	8	no	—
119	D₇₆	—	2	16	no	—
120	D₇₆	—	4	8	no	—
121	D₇₆	C₂×D₄	1	4	yes	complemented
122	C₄×D₁₉	—	2	8	no	—
123	C₄×D₁₉	—	2	16	no	—
124	C₄×D₁₉	C₂×D₄	1	8	yes	complemented
125	Dic₃₈	C₂×D₄	1	4	yes	complemented
126	#3498	—	19	1	no	maximal, Sylow
127	C₂²×C₁₉	—	2	2	no	—
128	C₂²×C₁₉	—	2	4	no	—
129	C₂²×C₁₉	—	2	8	no	—
130	C₂²×C₁₉	C₂²×D₄	1	2	yes	—
131	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
132	C₂×D₁₉	—	2	4	no	—
133	C₂×D₁₉	—	2	8	no	—
134	C₂×D₁₉	—	2	16	no	—
135	C₂×D₁₉	—	8	16	no	—
136	C₂×D₁₉	—	8	8	no	—
137	C₂×D₁₉	C₂²×D₄	1	4	yes	—
138	C₄×C₁₉	—	2	4	no	—
139	C₄×C₁₉	—	2	2	no	—
140	C₄×C₁₉	C₂²×D₄	1	2	yes	—
141	C₄×C₁₉	C₂²×D₄	1	2	yes	—
142	C₄×C₁₉	H(5,𝔽₂)	1	4	yes	complemented
143	Dic₁₉	—	2	4	no	—
144	Dic₁₉	C₂²×D₄	1	4	yes	—
145	C₂×H(5,𝔽₂)	—	19	2	no	—
146	#550	—	19	4	no	—
147	D₄²	—	19	8	no	—
148	D₄²	—	19	4	no	—
149	#535	—	19	2	no	—
150	#535	—	19	4	no	—
151	#532	—	19	2	no	—
152	#532	D₁₉	1	1	yes	characteristic, complemented
153	C₂×#127	—	19	2	no	—
154	C₂×#127	—	19	1	no	—
155	C₄×#41	—	19	1	no	—
156	C₂×C₁₉	—	2	2	no	—
157	C₂×C₁₉	—	4	4	no	—
158	C₂×C₁₉	C₂³×D₄	1	1	yes	characteristic
159	C₂×C₁₉	C₂×H(5,𝔽₂)	1	2	yes	—
160	D₁₉	—	2	4	no	—
161	D₁₉	—	4	8	no	—
162	H(5,𝔽₂)	—	38	16	no	—
163	C₂×#41	—	19	4	no	—
164	C₂×#41	—	19	1	no	—
165	C₂×#41	—	19	4	no	—
166	C₂×#41	C₂×D₁₉	1	2	yes	complemented
167	C₂²×D₄	—	19	4	no	—
168	C₂²×D₄	—	19	8	no	—
169	C₂²×D₄	—	19	8	no	—
170	C₂²×D₄	—	19	2	no	—
171	C₂²×D₄	—	19	4	no	—
172	C₂²×D₄	—	19	4	no	—
173	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic, complemented
174	#127	—	19	4	no	—
175	#127	—	19	8	no	—
176	#127	—	19	2	no	—
177	#127	—	19	2	no	—
178	#127	—	19	4	no	—
179	#127	C₂×D₁₉	1	1	yes	characteristic, complemented
180	#124	—	19	4	no	—
181	#124	C₂×D₁₉	1	2	yes	complemented
182	#121	—	19	8	no	—
183	#121	—	19	8	no	—
184	#121	—	19	16	no	—
185	#121	C₂×D₁₉	1	4	yes	complemented
186	#120	—	19	8	no	—
187	#120	—	19	16	no	—
188	C₄×Q₈	—	19	2	no	—
189	C₄×D₄	—	19	4	no	—
190	C₄×D₄	—	19	8	no	—
191	C₄×D₄	—	19	2	no	—
192	C₄×D₄	C₂×D₁₉	1	4	yes	complemented
193	#117	—	19	1	no	—
194	#117	—	19	2	no	—
195	C₂×C₄²	—	19	2	no	—
196	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic, complemented
197	C₁₉	#3498	1	1	yes	characteristic, complemented, Sylow
198	C₂⁴	—	19	8	no	—
199	C₂⁴	—	19	4	no	—
200	#41	—	2	8	no	—
201	#41	—	19	8	no	—
202	#41	—	38	16	no	—
203	#41	—	38	16	no	—
204	C₂×Q₈	—	19	1	no	—
205	C₂×Q₈	C₂²×D₁₉	1	2	yes	—
206	C₂×D₄	—	19	2	no	—
207	C₂×D₄	—	19	8	no	—
208	C₂×D₄	—	19	16	no	—
209	C₂×D₄	—	19	1	no	—
210	C₂×D₄	—	19	8	no	—
211	C₂×D₄	—	19	8	no	—
212	C₂×D₄	—	19	4	no	—
213	C₂×D₄	—	19	4	no	—
214	C₂×D₄	—	19	8	no	—
215	C₂×D₄	—	38	16	no	—
216	C₂×D₄	—	38	16	no	—
217	C₂×D₄	—	38	32	no	—
218	C₂×D₄	—	38	8	no	—
219	C₂×D₄	—	76	8	no	—
220	C₂×D₄	—	76	16	no	—
221	C₂×D₄	C₂²×D₁₉	1	4	yes	—
222	C₂×D₄	C₂²×D₁₉	1	4	yes	—
223	C₂×D₄	C₂²×D₁₉	1	2	yes	—
224	C₂²×C₄	—	19	2	no	—
225	C₂²×C₄	—	19	4	no	—
226	C₂²×C₄	—	19	8	no	—
227	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
228	C₂²×C₄	C₂²×D₁₉	1	4	yes	—
229	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
230	C₄⋊C₄	—	19	4	no	—
231	C₄⋊C₄	—	19	2	no	—
232	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
233	C₂²⋊C₄	—	19	2	no	—
234	C₂²⋊C₄	—	19	4	no	—
235	C₂²⋊C₄	—	19	16	no	—
236	C₂²⋊C₄	C₂²×D₁₉	1	8	yes	—
237	C₄²	—	19	2	no	—
238	C₄²	—	19	4	no	—
239	C₄²	C₂²×D₁₉	1	1	yes	characteristic
240	C₄²	C₂²×D₁₉	1	1	yes	characteristic
241	C₄²	C₂²×D₁₉	1	2	yes	—
242	C₂³	—	19	2	no	—
243	C₂³	—	19	8	no	—
244	C₂³	—	38	16	no	—
245	C₂³	—	38	8	no	—
246	C₂³	—	76	8	no	—
247	C₂³	—	76	16	no	—
248	C₂³	C₂³×D₁₉	1	4	yes	—
249	C₂³	C₂³×D₁₉	1	1	yes	characteristic
250	Q₈	—	2	4	no	—
251	Q₈	—	19	4	no	—
252	D₄	—	2	8	no	—
253	D₄	—	2	4	no	—
254	D₄	—	2	8	no	—
255	D₄	—	19	8	no	—
256	D₄	—	19	4	no	—
257	D₄	—	38	16	no	—
258	D₄	—	38	16	no	—
259	D₄	—	38	8	no	—
260	D₄	—	38	8	no	—
261	D₄	—	38	16	no	—
262	D₄	—	76	8	no	—
263	D₄	—	76	16	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	19	2	no	—
267	C₂×C₄	—	19	8	no	—
268	C₂×C₄	—	19	4	no	—
269	C₂×C₄	—	38	16	no	—
270	C₂×C₄	—	38	8	no	—
271	C₂×C₄	D₄×D₁₉	1	4	yes	complemented
272	C₂×C₄	C₂³×D₁₉	1	2	yes	—
273	C₂×C₄	C₂³×D₁₉	1	4	yes	—
274	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
275	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
276	C₂×C₄	C₂³×D₁₉	1	2	yes	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	8	no	—
280	C₂²	—	19	4	no	—
281	C₂²	—	38	4	no	—
282	C₂²	—	38	8	no	—
283	C₂²	—	38	16	no	—
284	C₂²	—	152	8	no	—
285	C₂²	—	152	16	no	—
286	C₂²	C₂×D₄×D₁₉	1	2	yes	—
287	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
288	C₄	—	2	4	no	—
289	C₄	—	2	2	no	—
290	C₄	—	19	4	no	—
291	C₄	—	38	4	no	—
292	C₄	C₂×D₄×D₁₉	1	2	yes	—
293	C₄	C₂×D₄×D₁₉	1	2	yes	—
294	C₄	#10597435	1	4	yes	complemented
295	C₂	—	2	2	no	—
296	C₂	—	4	4	no	—
297	C₂	—	38	4	no	—
298	C₂	—	76	8	no	—
299	C₂	C₂²×D₄×D₁₉	1	1	yes	characteristic
300	C₂	C₂×#10597435	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	C₁₉×#532	C₂
hypercenter	#532	D₁₉
hypocenter	C₁₉	#3498
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #532

Complements

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

N	H ≅ G/N	type
1. C₂×#10597435	C₂	semidirect
2. #11927215	C₂	semidirect
3. D₄×D₇₆	C₂	semidirect
4. #11927178	C₂	semidirect
5. #11927112	C₂	semidirect
6. #11927108	C₂	semidirect
7. #11927095	C₂	semidirect
8. C₂×#10597320	C₂	semidirect
9. #127×D₁₉	C₂	semidirect
10. C₄×#64541	C₂	semidirect
11. C₁₉×#532	C₂	semidirect
12. C₂×#41×C₁₉	C₂²	semidirect
14. C₂×#64545	C₂²	semidirect
15. C₂×D₄×D₁₉	C₂²	semidirect
16. C₂×D₄×D₁₉	C₂²	semidirect
17. C₂×D₄×D₁₉	C₂²	semidirect
19. C₂×#64541	C₂²	semidirect
20. C₂²×D₇₆	C₂²	semidirect
21. C₂²×D₇₆	C₂²	semidirect
22. C₂²×D₇₆	C₂²	semidirect
23. C₁₉×C₄⋊Q₈	C₂²	semidirect
24. C₁₉×#127	C₂²	semidirect
25. C₁₉×#124	C₂²	semidirect
26. C₁₉×#121	C₂²	semidirect
27. C₄×D₄×C₁₉	C₂²	semidirect
28. C₂×C₄²×C₁₉	C₂²	semidirect
29. #10597393	C₂²	semidirect
30. #10597386	C₂²	semidirect
31. #10597386	C₂²	semidirect
32. #10597383	C₂²	semidirect
33. #10597376	C₂²	semidirect
34. C₄×C₂²⋊D₁₉	C₂²	semidirect
35. #10597372	C₂²	semidirect
36. #10597341	C₂²	semidirect
37. #10597339	C₂²	semidirect
38. #10597330	C₂²	semidirect
39. #10597328	C₂²	semidirect
40. #10597320	C₂²	semidirect
41. #10597320	C₂²	semidirect
42. #10597320	C₂²	semidirect
43. C₄×D₇₆	C₂²	semidirect
44. #10597318	C₂²	semidirect
45. C₄²×D₁₉	C₂²	semidirect
46. C₄×Dic₃₈	C₂²	semidirect
65. #64541	D₄	semidirect
105. C₂×C₄×C₁₉	C₂×D₄	semidirect
113. C₂²⋊D₁₉	C₂×D₄	semidirect
121. D₇₆	C₂×D₄	semidirect
124. C₄×D₁₉	C₂×D₄	semidirect
125. Dic₃₈	C₂×D₄	semidirect
142. C₄×C₁₉	H(5,𝔽₂)	semidirect
152. #532	D₁₉	semidirect
166. C₂×#41	C₂×D₁₉	semidirect
173. C₄⋊Q₈	C₂×D₁₉	semidirect
179. #127	C₂×D₁₉	semidirect
181. #124	C₂×D₁₉	semidirect
185. #121	C₂×D₁₉	semidirect
192. C₄×D₄	C₂×D₁₉	semidirect
196. C₂×C₄²	C₂×D₁₉	semidirect
197. C₁₉	#3498	semidirect
271. C₂×C₄	D₄×D₁₉	semidirect
294. C₄	#10597435	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597435	2
#11927215	2
D₄×D₇₆	2
#11927178	2
#11927112	2
#11927108	2
#11927095	2
C₂×#10597320	2
#127×D₁₉	2
C₄×#64541	2
C₁₉×#532	2
#3498	19