Subgroups — #14577411

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×#10597430	C₂	1	2	yes	maximal, complemented
2	#11927214	C₂	1	4	yes	maximal, complemented
3	#11927179	C₂	1	4	yes	maximal, complemented
4	D₄×C₂²⋊D₁₉	C₂	1	8	yes	maximal, complemented
5	#11927111	C₂	1	2	yes	maximal, complemented
6	#11927107	C₂	1	4	yes	maximal, complemented
7	#11927094	C₂	1	1	yes	maximal, characteristic, complemented
8	#11927092	C₂	1	2	yes	maximal, complemented
9	#127×D₁₉	C₂	1	2	yes	maximal, complemented
10	C₄×#64541	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₁₉×#127	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₁₉	C₂²	1	2	yes	complemented
13	C₂²×D₄×C₁₉	C₂²	1	4	yes	complemented
14	C₂²×C₂²⋊D₁₉	C₂²	1	8	yes	complemented
15	#10597430	—	2	16	no	—
16	C₂×#64543	C₂²	1	4	yes	complemented
17	C₂×#64543	C₂²	1	4	yes	complemented
18	C₂×D₄×D₁₉	C₂²	1	4	yes	complemented
19	C₂×D₄×D₁₉	C₂²	1	4	yes	complemented
20	C₂×D₄×D₁₉	C₂²	1	8	yes	complemented
21	C₂×#64541	C₂²	1	1	yes	characteristic
22	C₂×#64541	C₂²	1	2	yes	complemented
23	C₁₉×#127	C₂²	1	2	yes	complemented
24	C₁₉×#127	C₂²	1	2	yes	complemented
25	C₁₉×#127	C₂²	1	4	yes	complemented
26	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic, complemented
27	#10597399	C₂²	1	8	yes	complemented
28	#10597386	C₂²	1	4	yes	complemented
29	#10597386	C₂²	1	8	yes	complemented
30	#10597385	C₂²	1	16	yes	complemented
31	#10597384	C₂²	1	8	yes	complemented
32	#10597384	C₂²	1	8	yes	complemented
33	#10597383	C₂²	1	16	yes	complemented
34	#10597382	C₂²	1	4	yes	complemented
35	D₄×Dic₁₉	C₂²	1	8	yes	complemented
36	#10597376	C₂²	1	4	yes	complemented
37	C₄×C₂²⋊D₁₉	C₂²	1	4	yes	complemented
38	#10597372	C₂²	1	1	yes	characteristic, complemented
39	#10597321	C₂²	1	2	yes	complemented
40	#10597320	C₂²	1	1	yes	characteristic, complemented
41	C₄×D₇₆	C₂²	1	4	yes	complemented
42	C₄×D₇₆	C₂²	1	2	yes	complemented
43	#10597318	C₂²	1	2	yes	complemented
44	C₄²×D₁₉	C₂²	1	2	yes	complemented
45	#10597315	C₂²	1	1	yes	characteristic, complemented
46	C₄×Dic₃₈	C₂²	1	2	yes	complemented
47	C₂⁴×C₁₉	C₂³	1	4	yes	—
48	C₂³×D₁₉	C₂³	1	8	yes	—
49	C₂×D₄×C₁₉	—	2	8	no	—
50	C₂×D₄×C₁₉	—	4	8	no	—
51	C₂×D₄×C₁₉	C₂³	1	8	yes	—
52	C₂×D₄×C₁₉	C₂³	1	4	yes	—
53	C₂×D₄×C₁₉	C₂³	1	4	yes	—
54	C₂×D₄×C₁₉	C₂³	1	8	yes	—
55	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
56	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
57	C₂×C₂²⋊D₁₉	—	2	32	no	—
58	C₂×C₂²⋊D₁₉	C₂³	1	4	yes	—
59	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
60	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
61	C₂×C₂²⋊D₁₉	C₂³	1	16	yes	—
62	C₂×C₂²⋊D₁₉	C₂³	1	2	yes	—
63	C₂²×Dic₁₉	C₂³	1	8	yes	—
64	#64543	—	2	16	no	—
65	#64543	—	2	16	no	—
66	D₄×D₁₉	—	2	16	no	—
67	D₄×D₁₉	—	2	16	no	—
68	D₄×D₁₉	—	4	16	no	—
69	#64541	—	2	8	no	—
70	#64541	D₄	1	8	yes	complemented
71	C₂×D₇₆	C₂³	1	4	yes	—
72	C₂×D₇₆	C₂³	1	2	yes	—
73	C₂×D₇₆	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₁₉	C₂³	1	4	yes	—
75	C₂×C₄×D₁₉	C₂³	1	4	yes	—
76	C₂×C₄×D₁₉	C₂³	1	2	yes	—
77	C₂×Dic₃₈	C₂³	1	2	yes	—
78	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
79	C₄²×C₁₉	C₂³	1	2	yes	—
80	C₄²×C₁₉	C₂³	1	1	yes	characteristic
81	C₄²×C₁₉	C₂³	1	1	yes	characteristic
82	C₂²⋊Dic₁₉	C₂³	1	16	yes	—
83	C₂²⋊Dic₁₉	C₂³	1	2	yes	—
84	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
85	(C₂×D₁₉)⋊C₄	C₂³	1	4	yes	—
86	C₄⋊Dic₁₉	C₂³	1	4	yes	—
87	C₄⋊Dic₁₉	C₂³	1	2	yes	—
88	Dic₁₉⋊C₄	C₂³	1	4	yes	—
89	C₄×Dic₁₉	C₂³	1	4	yes	—
90	C₄×Dic₁₉	C₂³	1	2	yes	—
91	C₂³×C₁₉	—	2	8	no	—
92	C₂³×C₁₉	—	4	8	no	—
93	C₂³×C₁₉	C₂⁴	1	8	yes	—
94	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
95	C₂²×D₁₉	—	2	16	no	—
96	C₂²×D₁₉	—	4	16	no	—
97	C₂²×D₁₉	C₂⁴	1	4	yes	—
98	C₂²×D₁₉	C₂⁴	1	2	yes	—
99	D₄×C₁₉	—	2	8	no	—
100	D₄×C₁₉	—	2	8	no	—
101	D₄×C₁₉	—	2	16	no	—
102	D₄×C₁₉	—	4	8	no	—
103	C₂×C₄×C₁₉	—	2	4	no	—
104	C₂×C₄×C₁₉	C₂×D₄	1	4	yes	complemented
105	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
106	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
109	C₂²⋊D₁₉	—	2	8	no	—
110	C₂²⋊D₁₉	—	2	16	no	—
111	C₂²⋊D₁₉	—	2	16	no	—
112	C₂²⋊D₁₉	—	4	16	no	—
113	C₂²⋊D₁₉	C₂×D₄	1	8	yes	complemented
114	C₂×Dic₁₉	—	2	16	no	—
115	C₂×Dic₁₉	C₂⁴	1	4	yes	—
116	C₂×Dic₁₉	C₂⁴	1	4	yes	—
117	C₂×Dic₁₉	C₂⁴	1	2	yes	—
118	D₇₆	—	2	4	no	—
119	D₇₆	—	2	8	no	—
120	D₇₆	C₂×D₄	1	4	yes	complemented
121	C₄×D₁₉	—	2	8	no	—
122	C₄×D₁₉	—	2	8	no	—
123	C₄×D₁₉	C₂×D₄	1	8	yes	complemented
124	Dic₃₈	—	2	4	no	—
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	8	no	—
129	C₂²×C₁₉	—	2	16	no	—
130	C₂²×C₁₉	—	8	8	no	—
131	C₂²×C₁₉	C₂²×D₄	1	2	yes	—
132	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
133	C₂×D₁₉	—	2	4	no	—
134	C₂×D₁₉	—	2	4	no	—
135	C₂×D₁₉	—	2	8	no	—
136	C₂×D₁₉	—	8	16	no	—
137	C₂×D₁₉	C₂²×D₄	1	4	yes	—
138	C₄×C₁₉	—	2	2	no	—
139	C₄×C₁₉	C₂²×D₄	1	2	yes	—
140	C₄×C₁₉	C₂²×D₄	1	2	yes	—
141	C₄×C₁₉	H(5,𝔽₂)	1	4	yes	complemented
142	Dic₁₉	—	2	4	no	—
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	—	19	1	no	—
153	C₂×#127	—	19	2	no	—
154	C₂×#127	D₁₉	1	1	yes	characteristic, complemented
155	C₄×#41	—	19	1	no	—
156	C₂×C₁₉	—	2	2	no	—
157	C₂×C₁₉	—	4	8	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	4	no	—
162	H(5,𝔽₂)	—	38	16	no	—
163	C₂×#41	—	19	4	no	—
164	C₂×#41	—	19	4	no	—
165	C₂×#41	—	19	1	no	—
166	C₂×#41	—	19	2	no	—
167	C₂²×D₄	—	19	8	no	—
168	C₂²×D₄	—	19	4	no	—
169	C₂²×D₄	—	19	4	no	—
170	C₂²×D₄	—	19	8	no	—
171	C₂²×D₄	C₂×D₁₉	1	2	yes	complemented
172	C₂²×D₄	C₂×D₁₉	1	4	yes	complemented
173	C₄⋊Q₈	—	19	1	no	—
174	#127	—	19	4	no	—
175	#127	—	19	8	no	—
176	#127	—	19	1	no	—
177	#127	C₂×D₁₉	1	2	yes	complemented
178	#127	C₂×D₁₉	1	2	yes	complemented
179	#127	C₂×D₁₉	1	4	yes	complemented
180	#124	—	19	4	no	—
181	#124	—	19	2	no	—
182	#121	—	19	8	no	—
183	#121	—	19	8	no	—
184	#121	—	19	16	no	—
185	#121	—	19	4	no	—
186	#120	—	19	8	no	—
187	#120	—	19	16	no	—
188	C₄×Q₈	—	19	2	no	—
189	C₄×D₄	—	19	8	no	—
190	C₄×D₄	—	19	4	no	—
191	C₄×D₄	—	19	2	no	—
192	C₄×D₄	—	19	4	no	—
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	C₂²×D₁₉	1	4	yes	—
200	#41	—	19	8	no	—
201	#41	—	38	16	no	—
202	#41	—	38	16	no	—
203	#41	—	38	8	no	—
204	C₂×Q₈	—	19	1	no	—
205	C₂×Q₈	—	19	2	no	—
206	C₂×D₄	—	2	8	no	—
207	C₂×D₄	—	4	8	no	—
208	C₂×D₄	—	19	2	no	—
209	C₂×D₄	—	19	8	no	—
210	C₂×D₄	—	19	4	no	—
211	C₂×D₄	—	19	8	no	—
212	C₂×D₄	—	19	16	no	—
213	C₂×D₄	—	19	1	no	—
214	C₂×D₄	—	19	4	no	—
215	C₂×D₄	—	19	2	no	—
216	C₂×D₄	—	38	16	no	—
217	C₂×D₄	—	38	16	no	—
218	C₂×D₄	—	38	32	no	—
219	C₂×D₄	—	76	16	no	—
220	C₂×D₄	C₂²×D₁₉	1	8	yes	—
221	C₂×D₄	C₂²×D₁₉	1	4	yes	—
222	C₂×D₄	C₂²×D₁₉	1	4	yes	—
223	C₂×D₄	C₂²×D₁₉	1	8	yes	—
224	C₂²×C₄	—	19	2	no	—
225	C₂²×C₄	—	19	8	no	—
226	C₂²×C₄	—	19	4	no	—
227	C₂²×C₄	—	19	4	no	—
228	C₂²×C₄	C₂²×D₁₉	1	2	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	—	19	4	no	—
233	C₂²⋊C₄	—	19	2	no	—
234	C₂²⋊C₄	—	19	16	no	—
235	C₂²⋊C₄	—	19	4	no	—
236	C₂²⋊C₄	—	19	8	no	—
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	—	2	8	no	—
243	C₂³	—	4	8	no	—
244	C₂³	—	19	2	no	—
245	C₂³	—	19	4	no	—
246	C₂³	—	38	16	no	—
247	C₂³	—	76	16	no	—
248	C₂³	C₂³×D₁₉	1	8	yes	—
249	C₂³	C₂³×D₁₉	1	1	yes	characteristic
250	Q₈	—	19	4	no	—
251	Q₈	—	38	4	no	—
252	D₄	—	2	8	no	—
253	D₄	—	2	8	no	—
254	D₄	—	2	16	no	—
255	D₄	—	4	8	no	—
256	D₄	—	19	8	no	—
257	D₄	—	19	4	no	—
258	D₄	—	38	16	no	—
259	D₄	—	38	16	no	—
260	D₄	—	38	8	no	—
261	D₄	—	38	4	no	—
262	D₄	—	38	8	no	—
263	D₄	—	76	16	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	19	2	no	—
266	C₂×C₄	—	19	8	no	—
267	C₂×C₄	—	19	4	no	—
268	C₂×C₄	—	19	4	no	—
269	C₂×C₄	—	38	16	no	—
270	C₂×C₄	—	38	8	no	—
271	C₂×C₄	—	38	8	no	—
272	C₂×C₄	D₄×D₁₉	1	4	yes	complemented
273	C₂×C₄	C₂³×D₁₉	1	2	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	8	no	—
279	C₂²	—	2	16	no	—
280	C₂²	—	8	8	no	—
281	C₂²	—	19	4	no	—
282	C₂²	—	38	4	no	—
283	C₂²	—	38	4	no	—
284	C₂²	—	38	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	2	no	—
289	C₄	—	19	4	no	—
290	C₄	—	38	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	#10597430	1	4	yes	complemented
295	C₂	—	2	2	no	—
296	C₂	—	4	8	no	—
297	C₂	—	38	4	no	—
298	C₂	—	76	4	no	—
299	C₂	C₂²×D₄×D₁₉	1	1	yes	characteristic
300	C₂	C₂×#10597430	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	C₂×C₁₉×#127	C₂
hypercenter	C₂×#127	D₁₉
hypocenter	C₁₉	#3498
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. C₂×#10597430	C₂	semidirect
2. #11927214	C₂	semidirect
3. #11927179	C₂	semidirect
4. D₄×C₂²⋊D₁₉	C₂	semidirect
5. #11927111	C₂	semidirect
6. #11927107	C₂	semidirect
7. #11927094	C₂	semidirect
8. #11927092	C₂	semidirect
9. #127×D₁₉	C₂	semidirect
10. C₄×#64541	C₂	semidirect
11. C₂×C₁₉×#127	C₂	semidirect
12. C₂²×D₄×C₁₉	C₂²	semidirect
13. C₂²×D₄×C₁₉	C₂²	semidirect
14. C₂²×C₂²⋊D₁₉	C₂²	semidirect
16. C₂×#64543	C₂²	semidirect
17. C₂×#64543	C₂²	semidirect
18. C₂×D₄×D₁₉	C₂²	semidirect
19. C₂×D₄×D₁₉	C₂²	semidirect
20. C₂×D₄×D₁₉	C₂²	semidirect
22. C₂×#64541	C₂²	semidirect
23. C₁₉×#127	C₂²	semidirect
24. C₁₉×#127	C₂²	semidirect
25. C₁₉×#127	C₂²	semidirect
26. C₂×C₄²×C₁₉	C₂²	semidirect
27. #10597399	C₂²	semidirect
28. #10597386	C₂²	semidirect
29. #10597386	C₂²	semidirect
30. #10597385	C₂²	semidirect
31. #10597384	C₂²	semidirect
32. #10597384	C₂²	semidirect
33. #10597383	C₂²	semidirect
34. #10597382	C₂²	semidirect
35. D₄×Dic₁₉	C₂²	semidirect
36. #10597376	C₂²	semidirect
37. C₄×C₂²⋊D₁₉	C₂²	semidirect
38. #10597372	C₂²	semidirect
39. #10597321	C₂²	semidirect
40. #10597320	C₂²	semidirect
41. C₄×D₇₆	C₂²	semidirect
42. C₄×D₇₆	C₂²	semidirect
43. #10597318	C₂²	semidirect
44. C₄²×D₁₉	C₂²	semidirect
45. #10597315	C₂²	semidirect
46. C₄×Dic₃₈	C₂²	semidirect
70. #64541	D₄	semidirect
104. C₂×C₄×C₁₉	C₂×D₄	semidirect
113. C₂²⋊D₁₉	C₂×D₄	semidirect
120. D₇₆	C₂×D₄	semidirect
123. C₄×D₁₉	C₂×D₄	semidirect
125. Dic₃₈	C₂×D₄	semidirect
141. C₄×C₁₉	H(5,𝔽₂)	semidirect
154. C₂×#127	D₁₉	semidirect
171. C₂²×D₄	C₂×D₁₉	semidirect
172. C₂²×D₄	C₂×D₁₉	semidirect
177. #127	C₂×D₁₉	semidirect
178. #127	C₂×D₁₉	semidirect
179. #127	C₂×D₁₉	semidirect
196. C₂×C₄²	C₂×D₁₉	semidirect
197. C₁₉	#3498	semidirect
272. C₂×C₄	D₄×D₁₉	semidirect
294. C₄	#10597430	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597430	2
#11927214	2
#11927179	2
D₄×C₂²⋊D₁₉	2
#11927111	2
#11927107	2
#11927094	2
#11927092	2
#127×D₁₉	2
C₄×#64541	2
C₂×C₁₉×#127	2
#3498	19