Subgroups — #14577506

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	#11927277	C₂	1	4	yes	maximal, complemented
2	#11927215	C₂	1	4	yes	maximal, complemented
3	#11927184	C₂	1	8	yes	maximal, complemented
4	#11927145	C₂	1	4	yes	maximal, complemented
5	#11927113	C₂	1	1	yes	maximal, characteristic, complemented
6	#11927111	C₂	1	1	yes	maximal, characteristic, complemented
7	#11927097	C₂	1	1	yes	maximal, characteristic, complemented
8	#11927091	C₂	1	4	yes	maximal, complemented
9	C₂×#10597386	C₂	1	1	yes	maximal, characteristic, complemented
10	#11927018	C₂	1	1	yes	maximal, characteristic, complemented
11	#11927012	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₉×#518	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×D₄×C₁₉	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₁₉	C₂²	1	4	yes	—
15	C₂×#64541	C₂²	1	1	yes	characteristic
16	C₂×#64541	C₂²	1	8	yes	complemented
17	C₂²×D₇₆	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₉	C₂²	1	4	yes	complemented
19	C₄×D₄×C₁₉	C₂²	1	4	yes	complemented
20	C₁₉×#117	C₂²	1	1	yes	characteristic
21	C₁₉×#117	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
23	#10597399	C₂²	1	4	yes	—
24	#10597386	C₂²	1	2	yes	—
25	#10597386	C₂²	1	2	yes	—
26	#10597386	C₂²	1	4	yes	—
27	#10597386	C₂²	1	2	yes	complemented
28	#10597384	C₂²	1	4	yes	complemented
29	#10597382	C₂²	1	2	yes	complemented
30	#10597376	C₂²	1	8	yes	complemented
31	#10597375	C₂²	1	8	yes	complemented
32	C₄×C₂²⋊D₁₉	C₂²	1	8	yes	complemented
33	#10597372	C₂²	1	1	yes	characteristic
34	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
35	#10597341	C₂²	1	4	yes	complemented
36	#10597339	C₂²	1	2	yes	complemented
37	#10597339	C₂²	1	4	yes	complemented
38	#10597336	C₂²	1	2	yes	complemented
39	#10597334	C₂²	1	2	yes	complemented
40	#10597333	C₂²	1	2	yes	complemented
41	#10597331	C₂²	1	8	yes	complemented
42	#10597330	C₂²	1	8	yes	complemented
43	#10597330	C₂²	1	8	yes	complemented
44	#10597329	C₂²	1	8	yes	complemented
45	#10597328	C₂²	1	4	yes	—
46	(C₂²⋊D₁₉)⋊C₄	C₂²	1	4	yes	—
47	C₂²⋊C₄×D₁₉	C₂²	1	4	yes	—
48	#10597324	C₂²	1	4	yes	—
49	#10597321	C₂²	1	2	yes	complemented
50	#10597320	C₂²	1	2	yes	complemented
51	C₄×D₇₆	C₂²	1	4	yes	complemented
52	#10597318	C₂²	1	2	yes	complemented
53	C₄²×D₁₉	C₂²	1	2	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	C₂×D₄×C₁₉	—	2	4	no	—
58	C₂×D₄×C₁₉	C₂³	1	2	yes	—
59	C₂×D₄×C₁₉	C₂³	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂³	1	8	yes	—
61	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₉	—	4	8	no	—
63	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
64	C₂×C₂²⋊D₁₉	C₂³	1	2	yes	—
65	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
66	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
67	C₂²×Dic₁₉	C₂³	1	2	yes	—
68	#64541	—	2	4	no	—
69	#64541	—	4	16	no	—
70	C₂×D₇₆	—	2	4	no	—
71	C₂×D₇₆	C₂³	1	2	yes	—
72	C₂×D₇₆	C₂³	1	2	yes	—
73	C₂×D₇₆	C₂³	1	1	yes	characteristic
74	C₂×D₇₆	C₂³	1	8	yes	—
75	C₂×C₄×D₁₉	C₂³	1	2	yes	—
76	C₂×C₄×D₁₉	C₂³	1	4	yes	—
77	C₂×C₄×D₁₉	C₂³	1	8	yes	—
78	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
79	C₂×Dic₃₈	C₂³	1	4	yes	—
80	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
81	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
82	C₂²⋊C₄×C₁₉	C₂³	1	2	yes	—
83	C₂²⋊C₄×C₁₉	C₂³	1	2	yes	—
84	C₂²⋊C₄×C₁₉	C₂³	1	8	yes	—
85	C₄²×C₁₉	C₂³	1	2	yes	—
86	C₄²×C₁₉	C₂³	1	2	yes	—
87	C₂²⋊Dic₁₉	C₂³	1	8	yes	—
88	C₂²⋊Dic₁₉	C₂³	1	2	yes	—
89	(C₂×D₁₉)⋊C₄	C₂³	1	4	yes	—
90	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
91	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
92	C₄⋊Dic₁₉	C₂³	1	4	yes	—
93	C₄⋊Dic₁₉	C₂³	1	2	yes	—
94	Dic₁₉⋊C₄	C₂³	1	4	yes	—
95	Dic₁₉⋊C₄	C₂³	1	4	yes	—
96	Dic₁₉⋊C₄	C₂³	1	4	yes	—
97	C₄×Dic₁₉	C₂³	1	2	yes	—
98	C₄×Dic₁₉	C₂³	1	2	yes	—
99	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
100	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
101	C₄×Dic₁₉	C₂³	1	4	yes	—
102	C₂³×C₁₉	—	2	4	no	—
103	C₂³×C₁₉	—	4	4	no	—
104	C₂³×C₁₉	C₂⁴	1	4	yes	—
105	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
106	C₂²×D₁₉	—	2	4	no	—
107	C₂²×D₁₉	—	4	4	no	—
108	C₂²×D₁₉	C₂⁴	1	4	yes	—
109	C₂²×D₁₉	C₂⁴	1	2	yes	—
110	D₄×C₁₉	—	2	4	no	—
111	D₄×C₁₉	—	2	4	no	—
112	C₂×C₄×C₁₉	—	4	8	no	—
113	C₂×C₄×C₁₉	#41	1	4	yes	complemented
114	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
115	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
116	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₁₉	—	2	4	no	—
119	C₂²⋊D₁₉	—	2	16	no	—
120	C₂²⋊D₁₉	—	4	8	no	—
121	C₂²⋊D₁₉	—	4	8	no	—
122	C₂×Dic₁₉	—	2	4	no	—
123	C₂×Dic₁₉	C₂⁴	1	2	yes	—
124	C₂×Dic₁₉	C₂⁴	1	2	yes	—
125	C₂×Dic₁₉	C₂⁴	1	4	yes	—
126	C₂×Dic₁₉	C₂⁴	1	2	yes	—
127	D₇₆	—	2	4	no	—
128	D₇₆	—	2	4	no	—
129	D₇₆	—	2	2	no	—
130	D₇₆	—	4	8	no	—
131	C₄×D₁₉	—	2	4	no	—
132	C₄×D₁₉	—	2	8	no	—
133	C₄×D₁₉	—	4	8	no	—
134	Dic₃₈	—	2	2	no	—
135	Dic₃₈	—	2	8	no	—
136	#3508	—	19	1	no	maximal, Sylow
137	C₂²×C₁₉	—	2	2	no	—
138	C₂²×C₁₉	—	2	4	no	—
139	C₂²×C₁₉	—	2	8	no	—
140	C₂²×C₁₉	—	8	4	no	—
141	C₂²×C₁₉	C₂×#41	1	2	yes	—
142	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
143	C₂×D₁₉	—	2	2	no	—
144	C₂×D₁₉	—	2	4	no	—
145	C₂×D₁₉	—	2	4	no	—
146	C₂×D₁₉	—	2	8	no	—
147	C₂×D₁₉	—	8	4	no	—
148	C₄×C₁₉	—	2	4	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	2	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	—	19	1	no	—
162	#533	—	19	1	no	—
163	#532	—	19	4	no	—
164	C₂×#127	—	19	1	no	—
165	#518	—	19	1	no	—
166	#518	D₁₉	1	1	yes	characteristic, complemented
167	C₄×#41	—	19	1	no	—
168	C₂×C₁₉	—	2	2	no	—
169	C₂×C₁₉	—	4	4	no	—
170	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
171	C₂×C₁₉	C₂×H(5,𝔽₂)	1	2	yes	—
172	D₁₉	—	4	2	no	—
173	D₁₉	—	4	4	no	—
174	C₂×#41	—	19	1	no	—
175	C₂×#41	—	19	8	no	—
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	—	19	2	no	—
185	#125	—	19	2	no	—
186	#124	—	19	2	no	—
187	#124	—	19	8	no	—
188	#124	—	19	2	no	—
189	#123	—	19	8	no	—
190	#123	—	19	8	no	—
191	C₂²⋊Q₈	—	19	4	no	—
192	#121	—	19	4	no	—
193	#121	—	19	8	no	—
194	#121	—	19	4	no	—
195	#121	—	19	8	no	—
196	#121	—	19	8	no	—
197	#120	—	19	4	no	—
198	#120	—	19	4	no	—
199	C₄×Q₈	—	19	2	no	—
200	C₄×Q₈	—	19	4	no	—
201	C₄×D₄	—	19	8	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	C₂×D₁₉	1	4	yes	complemented
207	C₄×D₄	C₂×D₁₉	1	4	yes	complemented
208	#117	—	19	1	no	—
209	#117	—	19	2	no	—
210	#117	C₂×D₁₉	1	1	yes	characteristic
211	#117	C₂×D₁₉	1	1	yes	characteristic
212	C₂×C₂²⋊C₄	—	19	4	no	—
213	C₂×C₂²⋊C₄	C₂×D₁₉	1	4	yes	—
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	—	38	4	no	—
220	#41	—	76	16	no	—
221	C₂×Q₈	—	19	1	no	—
222	C₂×Q₈	—	19	4	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	19	2	no	—
225	C₂×D₄	—	19	8	no	—
226	C₂×D₄	—	19	8	no	—
227	C₂×D₄	—	19	8	no	—
228	C₂×D₄	—	19	2	no	—
229	C₂×D₄	—	19	2	no	—
230	C₂×D₄	—	19	1	no	—
231	C₂×D₄	—	19	8	no	—
232	C₂×D₄	—	38	4	no	—
233	C₂×D₄	—	76	8	no	—
234	C₂×D₄	C₂²×D₁₉	1	2	yes	—
235	C₂×D₄	C₂²×D₁₉	1	2	yes	—
236	C₂²×C₄	—	19	2	no	—
237	C₂²×C₄	—	19	2	no	—
238	C₂²×C₄	—	19	4	no	—
239	C₂²×C₄	—	19	8	no	—
240	C₂²×C₄	C₂²×D₁₉	1	8	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	4	no	—
245	C₄⋊C₄	—	19	2	no	—
246	C₄⋊C₄	—	19	4	no	—
247	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
248	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
249	C₂²⋊C₄	—	19	2	no	—
250	C₂²⋊C₄	—	19	8	no	—
251	C₂²⋊C₄	—	19	8	no	—
252	C₂²⋊C₄	—	19	4	no	—
253	C₂²⋊C₄	—	19	8	no	—
254	C₂²⋊C₄	C₂²×D₁₉	1	2	yes	—
255	C₂²⋊C₄	C₂²×D₁₉	1	2	yes	—
256	C₂²⋊C₄	C₂²×D₁₉	1	8	yes	—
257	C₄²	—	19	1	no	—
258	C₄²	—	19	1	no	—
259	C₄²	—	19	4	no	—
260	C₄²	—	19	2	no	—
261	C₄²	—	19	2	no	—
262	C₄²	C₂²×D₁₉	1	2	yes	—
263	C₄²	C₂²×D₁₉	1	2	yes	—
264	C₂³	—	2	4	no	—
265	C₂³	—	4	4	no	—
266	C₂³	—	19	4	no	—
267	C₂³	—	19	2	no	—
268	C₂³	—	38	4	no	—
269	C₂³	—	76	4	no	—
270	C₂³	C₂³×D₁₉	1	4	yes	—
271	C₂³	C₂³×D₁₉	1	1	yes	characteristic
272	Q₈	—	38	8	no	—
273	Q₈	—	38	2	no	—
274	D₄	—	2	4	no	—
275	D₄	—	2	4	no	—
276	D₄	—	38	16	no	—
277	D₄	—	38	4	no	—
278	D₄	—	38	4	no	—
279	D₄	—	38	4	no	—
280	D₄	—	38	2	no	—
281	D₄	—	76	8	no	—
282	D₄	—	76	8	no	—
283	D₄	—	76	8	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	19	2	no	—
286	C₂×C₄	—	19	2	no	—
287	C₂×C₄	—	19	2	no	—
288	C₂×C₄	—	19	4	no	—
289	C₂×C₄	—	38	8	no	—
290	C₂×C₄	—	38	4	no	—
291	C₂×C₄	—	38	4	no	—
292	C₂×C₄	—	76	8	no	—
293	C₂×C₄	#64541	1	4	yes	complemented
294	C₂×C₄	C₂³×D₁₉	1	4	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	4	no	—
300	C₂²	—	2	8	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	38	2	no	—
303	C₂²	—	38	4	no	—
304	C₂²	—	38	4	no	—
305	C₂²	—	38	8	no	—
306	C₂²	—	152	4	no	—
307	C₂²	C₂×#64541	1	2	yes	—
308	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
309	C₄	—	2	4	no	—
310	C₄	—	2	4	no	—
311	C₄	—	19	4	no	—
312	C₄	—	38	2	no	—
313	C₄	—	38	2	no	—
314	C₄	—	38	4	no	—
315	C₄	C₂×#64541	1	2	yes	—
316	C₄	C₂×#64541	1	2	yes	—
317	C₂	—	2	2	no	—
318	C₂	—	4	4	no	—
319	C₂	—	76	4	no	—
320	C₂	—	76	2	no	—
321	C₂	C₂²×#64541	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×#518	C₂
hypercenter	#518	D₁₉
hypocenter	C₁₉	#3508
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #518

Complements

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

N	H ≅ G/N	type
1. #11927277	C₂	semidirect
2. #11927215	C₂	semidirect
3. #11927184	C₂	semidirect
4. #11927145	C₂	semidirect
5. #11927113	C₂	semidirect
6. #11927111	C₂	semidirect
7. #11927097	C₂	semidirect
8. #11927091	C₂	semidirect
9. C₂×#10597386	C₂	semidirect
10. #11927018	C₂	semidirect
11. #11927012	C₂	semidirect
12. C₁₉×#518	C₂	semidirect
16. C₂×#64541	C₂²	semidirect
18. C₄×D₄×C₁₉	C₂²	semidirect
19. C₄×D₄×C₁₉	C₂²	semidirect
27. #10597386	C₂²	semidirect
28. #10597384	C₂²	semidirect
29. #10597382	C₂²	semidirect
30. #10597376	C₂²	semidirect
31. #10597375	C₂²	semidirect
32. C₄×C₂²⋊D₁₉	C₂²	semidirect
35. #10597341	C₂²	semidirect
36. #10597339	C₂²	semidirect
37. #10597339	C₂²	semidirect
38. #10597336	C₂²	semidirect
39. #10597334	C₂²	semidirect
40. #10597333	C₂²	semidirect
41. #10597331	C₂²	semidirect
42. #10597330	C₂²	semidirect
43. #10597330	C₂²	semidirect
44. #10597329	C₂²	semidirect
49. #10597321	C₂²	semidirect
50. #10597320	C₂²	semidirect
51. C₄×D₇₆	C₂²	semidirect
52. #10597318	C₂²	semidirect
53. C₄²×D₁₉	C₂²	semidirect
54. C₄×Dic₃₈	C₂²	semidirect
113. C₂×C₄×C₁₉	#41	semidirect
166. #518	D₁₉	semidirect
206. C₄×D₄	C₂×D₁₉	semidirect
207. C₄×D₄	C₂×D₁₉	semidirect
216. C₁₉	#3508	semidirect
293. C₂×C₄	#64541	semidirect

Maximal subgroups

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

subgroup	index
#11927277	2
#11927215	2
#11927184	2
#11927145	2
#11927113	2
#11927111	2
#11927097	2
#11927091	2
C₂×#10597386	2
#11927018	2
#11927012	2
C₁₉×#518	2
#3508	19