Subgroups — #14577510

Order 2432.

1232 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927276	C₂	1	4	yes	maximal, complemented
2	#11927214	C₂	1	4	yes	maximal, complemented
3	#11927181	C₂	1	8	yes	maximal, complemented
4	#11927147	C₂	1	4	yes	maximal, complemented
5	#11927108	C₂	1	1	yes	maximal, characteristic, complemented
6	#11927097	C₂	1	1	yes	maximal, characteristic, complemented
7	#11927089	C₂	1	4	yes	maximal, complemented
8	C₂×#10597386	C₂	1	1	yes	maximal, characteristic, complemented
9	#11927019	C₂	1	1	yes	maximal, characteristic, complemented
10	#11927018	C₂	1	1	yes	maximal, characteristic, complemented
11	#41×Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₉×#535	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₁₉	C₂²	1	1	yes	characteristic
15	C₂²×C₂²⋊D₁₉	C₂²	1	4	yes	—
16	C₂×#64543	C₂²	1	8	yes	complemented
17	C₂²×D₇₆	C₂²	1	1	yes	characteristic
18	C₁₉×#127	C₂²	1	2	yes	complemented
19	C₁₉×#124	C₂²	1	2	yes	complemented
20	C₁₉×#121	C₂²	1	4	yes	complemented
21	C₁₉×#120	C₂²	1	4	yes	—
22	C₁₉×#117	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Dic₁₉	C₂²	1	4	yes	—
24	#10597393	C₂²	1	2	yes	complemented
25	Q₈×Dic₁₉	C₂²	1	2	yes	complemented
26	#10597386	C₂²	1	2	yes	—
27	#10597386	C₂²	1	2	yes	—
28	#10597386	C₂²	1	4	yes	—
29	#10597386	C₂²	1	2	yes	complemented
30	#10597385	C₂²	1	8	yes	complemented
31	#10597385	C₂²	1	8	yes	complemented
32	#10597384	C₂²	1	8	yes	complemented
33	#10597383	C₂²	1	4	yes	—
34	D₄×Dic₁₉	C₂²	1	4	yes	—
35	D₄×Dic₁₉	C₂²	1	4	yes	complemented
36	D₄×Dic₁₉	C₂²	1	2	yes	complemented
37	D₄×Dic₁₉	C₂²	1	4	yes	complemented
38	#10597376	C₂²	1	4	yes	complemented
39	#10597372	C₂²	1	1	yes	characteristic
40	#10597372	C₂²	1	1	yes	characteristic
41	#10597372	C₂²	1	2	yes	complemented
42	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
43	C₂×C₄×Dic₁₉	C₂²	1	2	yes	complemented
44	#10597339	C₂²	1	4	yes	complemented
45	#10597336	C₂²	1	2	yes	complemented
46	#10597334	C₂²	1	2	yes	complemented
47	#10597332	C₂²	1	8	yes	complemented
48	#10597331	C₂²	1	8	yes	complemented
49	#10597329	C₂²	1	8	yes	complemented
50	(C₂²⋊D₁₉)⋊C₄	C₂²	1	8	yes	complemented
51	C₂²⋊C₄×D₁₉	C₂²	1	4	yes	—
52	#10597324	C₂²	1	4	yes	—
53	C₄×D₇₆	C₂²	1	4	yes	complemented
54	C₄×Dic₃₈	C₂²	1	4	yes	complemented
55	C₂⁴×C₁₉	C₂³	1	2	yes	—
56	C₂³×D₁₉	C₂³	1	2	yes	—
57	#41×C₁₉	—	2	4	no	—
58	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
59	C₂×D₄×C₁₉	—	2	4	no	—
60	C₂×D₄×C₁₉	C₂³	1	2	yes	—
61	C₂×D₄×C₁₉	C₂³	1	2	yes	—
62	C₂×D₄×C₁₉	C₂³	1	2	yes	—
63	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₁₉	C₂³	1	8	yes	—
65	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
66	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₉	—	4	8	no	—
68	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
69	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
70	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
71	C₂²×Dic₁₉	C₂³	1	2	yes	—
72	C₂²×Dic₁₉	C₂³	1	4	yes	—
73	C₂²×Dic₁₉	C₂³	1	8	yes	—
74	#64543	—	4	16	no	—
75	C₂×D₇₆	—	2	4	no	—
76	C₂×D₇₆	C₂³	1	2	yes	—
77	C₂×D₇₆	C₂³	1	2	yes	—
78	C₂×C₄×D₁₉	C₂³	1	8	yes	—
79	C₂×Dic₃₈	C₂³	1	4	yes	—
80	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
81	C₂²⋊C₄×C₁₉	C₂³	1	2	yes	—
82	C₂²⋊C₄×C₁₉	C₂³	1	8	yes	—
83	C₄²×C₁₉	C₂³	1	2	yes	—
84	C₂²⋊Dic₁₉	C₂³	1	2	yes	—
85	C₂²⋊Dic₁₉	C₂³	1	4	yes	—
86	C₂²⋊Dic₁₉	C₂³	1	8	yes	—
87	C₂²⋊Dic₁₉	C₂³	1	2	yes	—
88	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
89	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
90	C₄⋊Dic₁₉	C₂³	1	2	yes	—
91	C₄⋊Dic₁₉	C₂³	1	4	yes	—
92	C₄⋊Dic₁₉	C₂³	1	2	yes	—
93	C₄⋊Dic₁₉	C₂³	1	4	yes	—
94	Dic₁₉⋊C₄	C₂³	1	4	yes	—
95	Dic₁₉⋊C₄	C₂³	1	4	yes	—
96	C₄×Dic₁₉	C₂³	1	4	yes	—
97	C₄×Dic₁₉	C₂³	1	2	yes	—
98	C₄×Dic₁₉	C₂³	1	2	yes	—
99	C₄×Dic₁₉	C₂³	1	2	yes	—
100	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
101	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
102	C₂³×C₁₉	—	2	4	no	—
103	C₂³×C₁₉	—	4	4	no	—
104	C₂³×C₁₉	C₂⁴	1	4	yes	—
105	C₂³×C₁₉	C₂⁴	1	2	yes	—
106	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
107	C₂²×D₁₉	—	2	4	no	—
108	C₂²×D₁₉	—	4	4	no	—
109	C₂²×D₁₉	C₂⁴	1	4	yes	—
110	Q₈×C₁₉	—	2	2	no	—
111	D₄×C₁₉	—	2	4	no	—
112	D₄×C₁₉	—	2	4	no	—
113	D₄×C₁₉	—	2	4	no	—
114	D₄×C₁₉	—	2	2	no	—
115	D₄×C₁₉	—	4	8	no	—
116	C₂×C₄×C₁₉	—	2	4	no	—
117	C₂×C₄×C₁₉	#41	1	4	yes	complemented
118	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
119	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
120	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₁₉	—	2	16	no	—
123	C₂²⋊D₁₉	—	4	8	no	—
124	C₂²⋊D₁₉	—	4	8	no	—
125	C₂×Dic₁₉	—	2	8	no	—
126	C₂×Dic₁₉	—	2	4	no	—
127	C₂×Dic₁₉	—	4	8	no	—
128	C₂×Dic₁₉	C₂⁴	1	2	yes	—
129	C₂×Dic₁₉	C₂⁴	1	4	yes	—
130	C₂×Dic₁₉	C₂⁴	1	4	yes	—
131	C₂×Dic₁₉	C₂⁴	1	2	yes	—
132	D₇₆	—	2	4	no	—
133	D₇₆	—	2	4	no	—
134	C₄×D₁₉	—	4	8	no	—
135	Dic₃₈	—	2	8	no	—
136	#3508	—	19	1	no	maximal, Sylow
137	C₂²×C₁₉	—	2	2	no	—
138	C₂²×C₁₉	—	2	2	no	—
139	C₂²×C₁₉	—	2	4	no	—
140	C₂²×C₁₉	—	2	4	no	—
141	C₂²×C₁₉	—	2	8	no	—
142	C₂²×C₁₉	—	8	4	no	—
143	C₂²×C₁₉	C₂×#41	1	2	yes	—
144	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
145	C₂×D₁₉	—	2	4	no	—
146	C₂×D₁₉	—	2	8	no	—
147	C₂×D₁₉	—	8	4	no	—
148	C₄×C₁₉	—	2	2	no	—
149	C₄×C₁₉	—	2	4	no	—
150	C₄×C₁₉	C₂×#41	1	2	yes	—
151	C₄×C₁₉	C₂×#41	1	2	yes	—
152	Dic₁₉	—	2	4	no	—
153	Dic₁₉	—	2	4	no	—
154	Dic₁₉	—	2	2	no	—
155	Dic₁₉	H(5,𝔽₂)	1	4	yes	—
156	#559	—	19	4	no	—
157	#550	—	19	4	no	—
158	#546	—	19	8	no	—
159	#540	—	19	4	no	—
160	#535	—	19	1	no	—
161	#535	D₁₉	1	1	yes	characteristic, complemented
162	#533	—	19	1	no	—
163	#532	—	19	4	no	—
164	C₂×#127	—	19	1	no	—
165	#518	—	19	1	no	—
166	#518	—	19	1	no	—
167	C₄×#41	—	19	1	no	—
168	C₂×C₁₉	—	2	2	no	—
169	C₂×C₁₉	—	4	2	no	—
170	C₂×C₁₉	—	4	4	no	—
171	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
172	C₂×C₁₉	C₂×H(5,𝔽₂)	1	2	yes	—
173	D₁₉	—	4	4	no	—
174	C₂×#41	—	19	8	no	—
175	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
176	C₂²×D₄	—	19	4	no	—
177	C₂²×D₄	—	19	1	no	—
178	C₂²×D₄	C₂×D₁₉	1	1	yes	characteristic
179	C₄⋊Q₈	—	19	2	no	—
180	#127	—	19	2	no	—
181	#127	—	19	2	no	—
182	#127	—	19	4	no	—
183	#127	—	19	2	no	—
184	#127	C₂×D₁₉	1	2	yes	complemented
185	#125	—	19	2	no	—
186	#124	—	19	2	no	—
187	#124	—	19	8	no	—
188	#124	C₂×D₁₉	1	2	yes	complemented
189	#123	—	19	8	no	—
190	#123	—	19	8	no	—
191	C₂²⋊Q₈	—	19	4	no	—
192	#121	—	19	8	no	—
193	#121	—	19	8	no	—
194	#121	—	19	8	no	—
195	#121	—	19	4	no	—
196	#121	C₂×D₁₉	1	4	yes	complemented
197	#120	—	19	4	no	—
198	#120	C₂×D₁₉	1	4	yes	—
199	C₄×Q₈	—	19	2	no	—
200	C₄×Q₈	—	19	4	no	—
201	C₄×D₄	—	19	4	no	—
202	C₄×D₄	—	19	2	no	—
203	C₄×D₄	—	19	4	no	—
204	C₄×D₄	—	19	4	no	—
205	C₄×D₄	—	19	4	no	—
206	C₄×D₄	—	19	8	no	—
207	C₄×D₄	—	19	4	no	—
208	#117	—	19	1	no	—
209	#117	—	19	2	no	—
210	#117	—	19	1	no	—
211	#117	C₂×D₁₉	1	1	yes	characteristic
212	C₂×C₂²⋊C₄	—	19	4	no	—
213	C₂×C₂²⋊C₄	—	19	4	no	—
214	C₂×C₄²	—	19	1	no	—
215	C₂×C₄²	—	19	2	no	—
216	C₁₉	#3508	1	1	yes	characteristic, complemented, Sylow
217	C₂⁴	—	19	2	no	—
218	C₂⁴	C₂²×D₁₉	1	2	yes	—
219	#41	—	2	4	no	—
220	#41	—	76	16	no	—
221	C₂×Q₈	—	19	4	no	—
222	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	19	8	no	—
225	C₂×D₄	—	19	8	no	—
226	C₂×D₄	—	19	8	no	—
227	C₂×D₄	—	19	2	no	—
228	C₂×D₄	—	19	2	no	—
229	C₂×D₄	—	38	4	no	—
230	C₂×D₄	—	76	8	no	—
231	C₂×D₄	C₂²×D₁₉	1	2	yes	—
232	C₂×D₄	C₂²×D₁₉	1	2	yes	—
233	C₂×D₄	C₂²×D₁₉	1	2	yes	—
234	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
235	C₂×D₄	C₂²×D₁₉	1	8	yes	—
236	C₂²×C₄	—	19	4	no	—
237	C₂²×C₄	—	19	2	no	—
238	C₂²×C₄	—	19	8	no	—
239	C₂²×C₄	—	19	8	no	—
240	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
241	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
242	C₄⋊C₄	—	19	4	no	—
243	C₄⋊C₄	—	19	4	no	—
244	C₄⋊C₄	—	19	2	no	—
245	C₄⋊C₄	—	19	2	no	—
246	C₄⋊C₄	—	19	4	no	—
247	C₄⋊C₄	—	19	4	no	—
248	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
249	C₂²⋊C₄	—	19	2	no	—
250	C₂²⋊C₄	—	19	2	no	—
251	C₂²⋊C₄	—	19	8	no	—
252	C₂²⋊C₄	—	19	4	no	—
253	C₂²⋊C₄	—	19	8	no	—
254	C₂²⋊C₄	—	19	8	no	—
255	C₂²⋊C₄	C₂²×D₁₉	1	2	yes	—
256	C₂²⋊C₄	C₂²×D₁₉	1	8	yes	—
257	C₄²	—	19	1	no	—
258	C₄²	—	19	2	no	—
259	C₄²	—	19	1	no	—
260	C₄²	—	19	4	no	—
261	C₄²	—	19	2	no	—
262	C₄²	—	19	2	no	—
263	C₄²	C₂²×D₁₉	1	2	yes	—
264	C₂³	—	2	4	no	—
265	C₂³	—	4	4	no	—
266	C₂³	—	19	4	no	—
267	C₂³	—	38	4	no	—
268	C₂³	—	76	4	no	—
269	C₂³	C₂³×D₁₉	1	4	yes	—
270	C₂³	C₂³×D₁₉	1	2	yes	—
271	C₂³	C₂³×D₁₉	1	1	yes	characteristic
272	Q₈	—	2	2	no	—
273	Q₈	—	38	8	no	—
274	D₄	—	2	4	no	—
275	D₄	—	2	4	no	—
276	D₄	—	2	4	no	—
277	D₄	—	2	2	no	—
278	D₄	—	4	8	no	—
279	D₄	—	38	16	no	—
280	D₄	—	38	4	no	—
281	D₄	—	38	4	no	—
282	D₄	—	76	8	no	—
283	D₄	—	76	8	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	19	2	no	—
286	C₂×C₄	—	19	4	no	—
287	C₂×C₄	—	19	2	no	—
288	C₂×C₄	—	19	4	no	—
289	C₂×C₄	—	38	4	no	—
290	C₂×C₄	—	38	8	no	—
291	C₂×C₄	—	76	8	no	—
292	C₂×C₄	—	76	8	no	—
293	C₂×C₄	#64543	1	4	yes	complemented
294	C₂×C₄	C₂³×D₁₉	1	2	yes	—
295	C₂×C₄	C₂³×D₁₉	1	4	yes	—
296	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	8	no	—
303	C₂²	—	8	4	no	—
304	C₂²	—	38	4	no	—
305	C₂²	—	38	8	no	—
306	C₂²	—	152	4	no	—
307	C₂²	C₂×#64543	1	2	yes	—
308	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	2	4	no	—
311	C₄	—	19	4	no	—
312	C₄	—	38	4	no	—
313	C₄	—	38	2	no	—
314	C₄	—	38	4	no	—
315	C₄	C₂×#64543	1	2	yes	—
316	C₄	C₂×#64543	1	2	yes	—
317	C₂	—	2	2	no	—
318	C₂	—	4	4	no	—
319	C₂	—	4	2	no	—
320	C₂	—	76	4	no	—
321	C₂	C₂²×#64543	1	1	yes	characteristic
322	C₂	H(5,𝔽₂)×D₁₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	C₁₉×#535	C₂
hypercenter	#535	D₁₉
hypocenter	C₁₉	#3508
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #535

Complements

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

N	H ≅ G/N	type
1. #11927276	C₂	semidirect
2. #11927214	C₂	semidirect
3. #11927181	C₂	semidirect
4. #11927147	C₂	semidirect
5. #11927108	C₂	semidirect
6. #11927097	C₂	semidirect
7. #11927089	C₂	semidirect
8. C₂×#10597386	C₂	semidirect
9. #11927019	C₂	semidirect
10. #11927018	C₂	semidirect
11. #41×Dic₁₉	C₂	semidirect
12. C₁₉×#535	C₂	semidirect
16. C₂×#64543	C₂²	semidirect
18. C₁₉×#127	C₂²	semidirect
19. C₁₉×#124	C₂²	semidirect
20. C₁₉×#121	C₂²	semidirect
24. #10597393	C₂²	semidirect
25. Q₈×Dic₁₉	C₂²	semidirect
29. #10597386	C₂²	semidirect
30. #10597385	C₂²	semidirect
31. #10597385	C₂²	semidirect
32. #10597384	C₂²	semidirect
35. D₄×Dic₁₉	C₂²	semidirect
36. D₄×Dic₁₉	C₂²	semidirect
37. D₄×Dic₁₉	C₂²	semidirect
38. #10597376	C₂²	semidirect
41. #10597372	C₂²	semidirect
43. C₂×C₄×Dic₁₉	C₂²	semidirect
44. #10597339	C₂²	semidirect
45. #10597336	C₂²	semidirect
46. #10597334	C₂²	semidirect
47. #10597332	C₂²	semidirect
48. #10597331	C₂²	semidirect
49. #10597329	C₂²	semidirect
50. (C₂²⋊D₁₉)⋊C₄	C₂²	semidirect
53. C₄×D₇₆	C₂²	semidirect
54. C₄×Dic₃₈	C₂²	semidirect
117. C₂×C₄×C₁₉	#41	semidirect
161. #535	D₁₉	semidirect
184. #127	C₂×D₁₉	semidirect
188. #124	C₂×D₁₉	semidirect
196. #121	C₂×D₁₉	semidirect
216. C₁₉	#3508	semidirect
293. C₂×C₄	#64543	semidirect

Maximal subgroups

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

subgroup	index
#11927276	2
#11927214	2
#11927181	2
#11927147	2
#11927108	2
#11927097	2
#11927089	2
C₂×#10597386	2
#11927019	2
#11927018	2
#41×Dic₁₉	2
C₁₉×#535	2
#3508	19