Subgroups — #14577266

Order 2432.

1900 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×D₁₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927182	C₂	1	8	yes	maximal, complemented
3	D₄×C₂²⋊D₁₉	C₂	1	8	yes	maximal, complemented
4	#11927108	C₂	1	2	yes	maximal, complemented
5	#11927103	C₂	1	4	yes	maximal, complemented
6	#11927051	C₂	1	2	yes	maximal, complemented
7	C₂×#10597383	C₂	1	4	yes	maximal, complemented
8	#11927019	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₉×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₉	C₂²	1	2	yes	complemented
11	C₁₉×H(5,𝔽₂)	—	2	8	no	—
12	C₂×#41×C₁₉	C₂²	1	2	yes	complemented
13	C₂×#41×C₁₉	C₂²	1	4	yes	complemented
14	C₂²×D₄×C₁₉	C₂²	1	1	yes	characteristic
15	C₂²×D₄×C₁₉	C₂²	1	4	yes	complemented
16	C₂²×D₄×C₁₉	C₂²	1	4	yes	complemented
17	C₂²×C₂²⋊D₁₉	C₂²	1	2	yes	complemented
18	C₂²×C₂²⋊D₁₉	C₂²	1	8	yes	complemented
19	C₂×D₄×D₁₉	—	2	8	no	—
20	C₂×D₄×D₁₉	C₂²	1	8	yes	complemented
21	C₂²×D₇₆	C₂²	1	1	yes	characteristic, complemented
22	C₂²×C₄×D₁₉	C₂²	1	1	yes	characteristic, complemented
23	#10597399	C₂²	1	8	yes	complemented
24	C₂×C₂²⋊Dic₁₉	C₂²	1	4	yes	complemented
25	#10597393	C₂²	1	4	yes	complemented
26	#10597392	C₂²	1	4	yes	complemented
27	#10597386	C₂²	1	4	yes	complemented
28	#10597385	C₂²	1	16	yes	complemented
29	#10597384	C₂²	1	4	yes	complemented
30	#10597383	C₂²	1	4	yes	complemented
31	#10597383	C₂²	1	16	yes	complemented
32	#10597383	C₂²	1	8	yes	complemented
33	#10597381	C₂²	1	4	yes	complemented
34	D₄×Dic₁₉	C₂²	1	8	yes	complemented
35	#10597376	C₂²	1	8	yes	complemented
36	#10597375	C₂²	1	8	yes	complemented
37	C₄×C₂²⋊D₁₉	C₂²	1	8	yes	complemented
38	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	8	yes	complemented
39	#10597372	C₂²	1	2	yes	complemented
40	C₂⁴×C₁₉	C₂³	1	2	yes	—
41	C₂⁴×C₁₉	C₂³	1	4	yes	complemented
42	C₂³×D₁₉	—	2	8	no	—
43	C₂³×D₁₉	—	4	8	no	—
44	C₂³×D₁₉	C₂³	1	1	yes	characteristic
45	C₂³×D₁₉	C₂³	1	8	yes	—
46	C₂³×D₁₉	C₂³	1	2	yes	—
47	#41×C₁₉	—	2	16	no	—
48	#41×C₁₉	—	2	8	no	—
49	C₂×Q₈×C₁₉	C₂³	1	2	yes	complemented
50	C₂×D₄×C₁₉	—	2	16	no	—
51	C₂×D₄×C₁₉	—	2	16	no	—
52	C₂×D₄×C₁₉	D₄	1	8	yes	complemented
53	C₂×D₄×C₁₉	C₂³	1	4	yes	—
54	C₂×D₄×C₁₉	C₂³	1	4	yes	—
55	C₂×D₄×C₁₉	C₂³	1	8	yes	complemented
56	C₂×D₄×C₁₉	C₂³	1	2	yes	complemented
57	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂³	1	4	yes	complemented
59	C₂²×C₄×C₁₉	C₂³	1	4	yes	complemented
60	C₂×C₂²⋊D₁₉	—	2	8	no	—
61	C₂×C₂²⋊D₁₉	—	4	16	no	—
62	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
63	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
64	C₂×C₂²⋊D₁₉	C₂³	1	16	yes	complemented
65	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
66	C₂²×Dic₁₉	C₂³	1	8	yes	complemented
67	D₄×D₁₉	—	2	32	no	—
68	C₂×D₇₆	—	2	4	no	—
69	C₂×D₇₆	C₂³	1	4	yes	—
70	C₂×C₄×D₁₉	—	2	4	no	—
71	C₂×C₄×D₁₉	C₂³	1	4	yes	—
72	C₂²⋊Dic₁₉	C₂³	1	4	yes	—
73	C₂²⋊Dic₁₉	C₂³	1	8	yes	complemented
74	C₂²⋊Dic₁₉	C₂³	1	8	yes	complemented
75	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
76	(C₂×D₁₉)⋊C₄	C₂³	1	16	yes	complemented
77	C₄⋊Dic₁₉	C₂³	1	4	yes	complemented
78	Dic₁₉⋊C₄	C₂³	1	8	yes	complemented
79	C₄×Dic₁₉	C₂³	1	4	yes	complemented
80	C₂³×C₁₉	—	2	8	no	—
81	C₂³×C₁₉	—	4	4	no	—
82	C₂³×C₁₉	—	4	8	no	—
83	C₂³×C₁₉	C₂×D₄	1	8	yes	complemented
84	C₂³×C₁₉	C₂⁴	1	4	yes	—
85	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
86	C₂³×C₁₉	C₂⁴	1	4	yes	—
87	C₂²×D₁₉	—	2	2	no	—
88	C₂²×D₁₉	—	2	4	no	—
89	C₂²×D₁₉	—	2	4	no	—
90	C₂²×D₁₉	—	2	16	no	—
91	C₂²×D₁₉	—	2	16	no	—
92	C₂²×D₁₉	—	4	8	no	—
93	C₂²×D₁₉	—	4	4	no	—
94	C₂²×D₁₉	—	4	32	no	—
95	C₂²×D₁₉	C₂⁴	1	4	yes	—
96	C₂²×D₁₉	C₂⁴	1	2	yes	—
97	Q₈×C₁₉	—	2	4	no	—
98	D₄×C₁₉	—	2	8	no	—
99	D₄×C₁₉	—	2	16	no	—
100	D₄×C₁₉	—	2	4	no	—
101	D₄×C₁₉	C₂×D₄	1	16	yes	complemented
102	C₂×C₄×C₁₉	—	2	8	no	—
103	C₂×C₄×C₁₉	—	2	8	no	—
104	C₂×C₄×C₁₉	C₂×D₄	1	4	yes	complemented
105	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
106	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
107	C₂²⋊D₁₉	—	2	16	no	—
108	C₂²⋊D₁₉	—	2	16	no	—
109	C₂²⋊D₁₉	—	4	16	no	—
110	C₂×Dic₁₉	—	2	2	no	—
111	C₂×Dic₁₉	—	4	8	no	—
112	C₂×Dic₁₉	C₂⁴	1	8	yes	—
113	C₂×Dic₁₉	C₂⁴	1	2	yes	—
114	D₇₆	—	2	8	no	—
115	C₄×D₁₉	—	2	8	no	—
116	#3484	—	19	1	no	maximal, Sylow
117	C₂²×C₁₉	—	2	8	no	—
118	C₂²×C₁₉	—	2	2	no	—
119	C₂²×C₁₉	—	2	4	no	—
120	C₂²×C₁₉	—	2	8	no	—
121	C₂²×C₁₉	—	4	8	no	—
122	C₂²×C₁₉	—	8	8	no	—
123	C₂²×C₁₉	C₂²×D₄	1	2	yes	—
124	C₂²×C₁₉	C₂²×D₄	1	8	yes	—
125	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
126	C₂×D₁₉	—	2	8	no	—
127	C₂×D₁₉	—	2	2	no	—
128	C₂×D₁₉	—	2	2	no	—
129	C₂×D₁₉	—	2	4	no	—
130	C₂×D₁₉	—	2	8	no	—
131	C₂×D₁₉	—	4	32	no	—
132	C₂×D₁₉	—	4	8	no	—
133	C₂×D₁₉	H(5,𝔽₂)	1	4	yes	complemented
134	C₄×C₁₉	—	2	4	no	—
135	C₄×C₁₉	C₂²×D₄	1	4	yes	—
136	Dic₁₉	—	2	2	no	—
137	Dic₁₉	—	2	8	no	—
138	C₂×H(5,𝔽₂)	D₁₉	1	1	yes	characteristic, complemented
139	C₂³×D₄	—	19	1	no	—
140	#546	—	19	8	no	—
141	D₄²	—	19	8	no	—
142	#535	—	19	2	no	—
143	#534	—	19	4	no	—
144	#525	—	19	2	no	—
145	C₂×#120	—	19	4	no	—
146	#518	—	19	1	no	—
147	C₂×C₁₉	—	2	2	no	—
148	C₂×C₁₉	—	2	8	no	—
149	C₂×C₁₉	—	4	4	no	—
150	C₂×C₁₉	C₂³×D₄	1	1	yes	characteristic
151	C₂×C₁₉	C₂×H(5,𝔽₂)	1	2	yes	—
152	D₁₉	—	2	4	no	—
153	D₁₉	—	4	4	no	—
154	C₂⁵	—	19	2	no	—
155	H(5,𝔽₂)	—	2	8	no	—
156	C₂×#41	C₂×D₁₉	1	2	yes	complemented
157	C₂×#41	C₂×D₁₉	1	4	yes	complemented
158	C₂²×D₄	—	19	2	no	—
159	C₂²×D₄	—	19	8	no	—
160	C₂²×D₄	—	19	8	no	—
161	C₂²×D₄	—	19	1	no	—
162	C₂²×D₄	—	38	8	no	—
163	C₂²×D₄	C₂×D₁₉	1	1	yes	characteristic
164	C₂²×D₄	C₂×D₁₉	1	4	yes	complemented
165	C₂²×D₄	C₂×D₁₉	1	4	yes	complemented
166	C₂³×C₄	—	19	1	no	—
167	#127	—	19	4	no	—
168	#124	—	19	4	no	—
169	#123	—	19	4	no	—
170	#123	—	19	8	no	—
171	C₂²⋊Q₈	—	19	4	no	—
172	#121	—	19	4	no	—
173	#121	—	19	16	no	—
174	#121	—	19	8	no	—
175	#120	—	19	8	no	—
176	#120	—	19	4	no	—
177	#120	—	19	8	no	—
178	#120	—	19	16	no	—
179	C₄×D₄	—	19	8	no	—
180	C₄×D₄	—	19	8	no	—
181	#117	—	19	2	no	—
182	C₂×C₂²⋊C₄	—	19	4	no	—
183	C₂×C₂²⋊C₄	—	19	8	no	—
184	C₁₉	#3484	1	1	yes	characteristic, complemented, Sylow
185	C₂⁴	—	19	1	no	—
186	C₂⁴	—	19	8	no	—
187	C₂⁴	—	19	2	no	—
188	C₂⁴	—	38	8	no	—
189	C₂⁴	—	76	8	no	—
190	C₂⁴	C₂²×D₁₉	1	2	yes	—
191	C₂⁴	C₂²×D₁₉	1	4	yes	complemented
192	#41	—	2	16	no	—
193	#41	—	2	8	no	—
194	C₂×Q₈	C₂²×D₁₉	1	2	yes	complemented
195	C₂×D₄	—	2	16	no	—
196	C₂×D₄	—	2	16	no	—
197	C₂×D₄	—	19	8	no	—
198	C₂×D₄	—	19	8	no	—
199	C₂×D₄	—	19	16	no	—
200	C₂×D₄	—	19	4	no	—
201	C₂×D₄	—	38	8	no	—
202	C₂×D₄	—	38	32	no	—
203	C₂×D₄	—	38	4	no	—
204	C₂×D₄	—	76	16	no	—
205	C₂×D₄	C₂²⋊D₁₉	1	8	yes	complemented
206	C₂×D₄	C₂²×D₁₉	1	4	yes	—
207	C₂×D₄	C₂²×D₁₉	1	4	yes	—
208	C₂×D₄	C₂²×D₁₉	1	8	yes	complemented
209	C₂×D₄	C₂²×D₁₉	1	2	yes	complemented
210	C₂²×C₄	—	19	1	no	—
211	C₂²×C₄	—	19	8	no	—
212	C₂²×C₄	—	19	4	no	—
213	C₂²×C₄	—	38	4	no	—
214	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
215	C₂²×C₄	C₂²×D₁₉	1	4	yes	complemented
216	C₂²×C₄	C₂²×D₁₉	1	4	yes	complemented
217	C₄⋊C₄	—	19	8	no	—
218	C₄⋊C₄	—	19	4	no	—
219	C₂²⋊C₄	—	19	4	no	—
220	C₂²⋊C₄	—	19	8	no	—
221	C₂²⋊C₄	—	19	8	no	—
222	C₂²⋊C₄	—	19	8	no	—
223	C₂²⋊C₄	—	19	16	no	—
224	C₄²	—	19	4	no	—
225	C₂³	—	2	8	no	—
226	C₂³	—	4	4	no	—
227	C₂³	—	4	8	no	—
228	C₂³	—	19	2	no	—
229	C₂³	—	19	4	no	—
230	C₂³	—	38	4	no	—
231	C₂³	—	38	16	no	—
232	C₂³	—	38	2	no	—
233	C₂³	—	38	16	no	—
234	C₂³	—	38	4	no	—
235	C₂³	—	76	8	no	—
236	C₂³	—	76	4	no	—
237	C₂³	—	76	32	no	—
238	C₂³	C₂×C₂²⋊D₁₉	1	8	yes	complemented
239	C₂³	C₂³×D₁₉	1	4	yes	—
240	C₂³	C₂³×D₁₉	1	1	yes	characteristic
241	C₂³	C₂³×D₁₉	1	4	yes	—
242	Q₈	—	2	4	no	—
243	D₄	—	2	8	no	—
244	D₄	—	2	16	no	—
245	D₄	—	2	4	no	—
246	D₄	—	38	16	no	—
247	D₄	—	38	16	no	—
248	D₄	—	38	8	no	—
249	D₄	—	76	16	no	—
250	D₄	C₂×C₂²⋊D₁₉	1	16	yes	complemented
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	19	8	no	—
254	C₂×C₄	—	19	2	no	—
255	C₂×C₄	—	38	2	no	—
256	C₂×C₄	—	38	8	no	—
257	C₂×C₄	—	76	8	no	—
258	C₂×C₄	C₂×C₂²⋊D₁₉	1	4	yes	complemented
259	C₂×C₄	C₂³×D₁₉	1	4	yes	—
260	C₂×C₄	C₂³×D₁₉	1	2	yes	—
261	C₂²	—	2	8	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	8	no	—
265	C₂²	—	4	8	no	—
266	C₂²	—	8	8	no	—
267	C₂²	—	19	4	no	—
268	C₂²	—	38	8	no	—
269	C₂²	—	38	2	no	—
270	C₂²	—	38	2	no	—
271	C₂²	—	38	4	no	—
272	C₂²	—	38	8	no	—
273	C₂²	—	76	32	no	—
274	C₂²	—	76	8	no	—
275	C₂²	C₂²×C₂²⋊D₁₉	1	8	yes	—
276	C₂²	C₂²×C₂²⋊D₁₉	1	2	yes	—
277	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
278	C₄	—	2	4	no	—
279	C₄	—	38	2	no	—
280	C₄	—	38	8	no	—
281	C₄	C₂²×C₂²⋊D₁₉	1	4	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	2	8	no	—
284	C₂	—	4	4	no	—
285	C₂	—	38	4	no	—
286	C₂	—	76	4	no	—
287	C₂	C₂³×C₂²⋊D₁₉	1	1	yes	characteristic
288	C₂	H(5,𝔽₂)×D₁₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	C₂×C₁₉×H(5,𝔽₂)	C₂
hypercenter	C₂×H(5,𝔽₂)	D₁₉
hypocenter	C₁₉	#3484
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂²×D₄×D₁₉	C₂	semidirect
2. #11927182	C₂	semidirect
3. D₄×C₂²⋊D₁₉	C₂	semidirect
4. #11927108	C₂	semidirect
5. #11927103	C₂	semidirect
6. #11927051	C₂	semidirect
7. C₂×#10597383	C₂	semidirect
8. #11927019	C₂	semidirect
9. C₂×C₁₉×H(5,𝔽₂)	C₂	semidirect
10. C₂⁴×D₁₉	C₂²	semidirect
12. C₂×#41×C₁₉	C₂²	semidirect
13. C₂×#41×C₁₉	C₂²	semidirect
15. C₂²×D₄×C₁₉	C₂²	semidirect
16. C₂²×D₄×C₁₉	C₂²	semidirect
17. C₂²×C₂²⋊D₁₉	C₂²	semidirect
18. C₂²×C₂²⋊D₁₉	C₂²	semidirect
20. C₂×D₄×D₁₉	C₂²	semidirect
21. C₂²×D₇₆	C₂²	semidirect
22. C₂²×C₄×D₁₉	C₂²	semidirect
23. #10597399	C₂²	semidirect
24. C₂×C₂²⋊Dic₁₉	C₂²	semidirect
25. #10597393	C₂²	semidirect
26. #10597392	C₂²	semidirect
27. #10597386	C₂²	semidirect
28. #10597385	C₂²	semidirect
29. #10597384	C₂²	semidirect
30. #10597383	C₂²	semidirect
31. #10597383	C₂²	semidirect
32. #10597383	C₂²	semidirect
33. #10597381	C₂²	semidirect
34. D₄×Dic₁₉	C₂²	semidirect
35. #10597376	C₂²	semidirect
36. #10597375	C₂²	semidirect
37. C₄×C₂²⋊D₁₉	C₂²	semidirect
38. C₂×(C₂×D₁₉)⋊C₄	C₂²	semidirect
39. #10597372	C₂²	semidirect
41. C₂⁴×C₁₉	C₂³	semidirect
49. C₂×Q₈×C₁₉	C₂³	semidirect
52. C₂×D₄×C₁₉	D₄	semidirect
55. C₂×D₄×C₁₉	C₂³	semidirect
56. C₂×D₄×C₁₉	C₂³	semidirect
58. C₂²×C₄×C₁₉	C₂³	semidirect
59. C₂²×C₄×C₁₉	C₂³	semidirect
64. C₂×C₂²⋊D₁₉	C₂³	semidirect
66. C₂²×Dic₁₉	C₂³	semidirect
73. C₂²⋊Dic₁₉	C₂³	semidirect
74. C₂²⋊Dic₁₉	C₂³	semidirect
76. (C₂×D₁₉)⋊C₄	C₂³	semidirect
77. C₄⋊Dic₁₉	C₂³	semidirect
78. Dic₁₉⋊C₄	C₂³	semidirect
79. C₄×Dic₁₉	C₂³	semidirect
83. C₂³×C₁₉	C₂×D₄	semidirect
101. D₄×C₁₉	C₂×D₄	semidirect
104. C₂×C₄×C₁₉	C₂×D₄	semidirect
133. C₂×D₁₉	H(5,𝔽₂)	semidirect
138. C₂×H(5,𝔽₂)	D₁₉	semidirect
156. C₂×#41	C₂×D₁₉	semidirect
157. C₂×#41	C₂×D₁₉	semidirect
164. C₂²×D₄	C₂×D₁₉	semidirect
165. C₂²×D₄	C₂×D₁₉	semidirect
184. C₁₉	#3484	semidirect
191. C₂⁴	C₂²×D₁₉	semidirect
194. C₂×Q₈	C₂²×D₁₉	semidirect
205. C₂×D₄	C₂²⋊D₁₉	semidirect
208. C₂×D₄	C₂²×D₁₉	semidirect
209. C₂×D₄	C₂²×D₁₉	semidirect
215. C₂²×C₄	C₂²×D₁₉	semidirect
216. C₂²×C₄	C₂²×D₁₉	semidirect
238. C₂³	C₂×C₂²⋊D₁₉	semidirect
250. D₄	C₂×C₂²⋊D₁₉	semidirect
258. C₂×C₄	C₂×C₂²⋊D₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₄×D₁₉	2
#11927182	2
D₄×C₂²⋊D₁₉	2
#11927108	2
#11927103	2
#11927051	2
C₂×#10597383	2
#11927019	2
C₂×C₁₉×H(5,𝔽₂)	2
#3484	19