Subgroups — #14577323

Order 2432.

1328 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927217	C₂	1	8	yes	maximal, complemented
2	#11927181	C₂	1	8	yes	maximal, complemented
3	#11927170	C₂	1	1	yes	maximal, characteristic, complemented
4	#11927131	C₂	1	4	yes	maximal, complemented
5	#11927054	C₂	1	4	yes	maximal, complemented
6	C₂×#10597383	C₂	1	2	yes	maximal, complemented
7	#11927019	C₂	1	2	yes	maximal, complemented
8	C₂×C₂²⋊C₄×D₁₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₉×#534	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₉	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₁₉	C₂²	1	2	yes	—
12	C₂²×C₂²⋊D₁₉	C₂²	1	4	yes	complemented
13	C₂×#64543	C₂²	1	4	yes	complemented
14	C₂²×C₄×D₁₉	C₂²	1	2	yes	complemented
15	C₁₉×#123	C₂²	1	4	yes	complemented
16	C₁₉×#121	C₂²	1	4	yes	complemented
17	C₁₉×#120	C₂²	1	4	yes	complemented
18	C₂×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Dic₁₉	C₂²	1	2	yes	—
20	C₂×C₂²⋊Dic₁₉	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Dic₁₉	C₂²	1	4	yes	complemented
22	#10597385	C₂²	1	8	yes	complemented
23	#10597384	C₂²	1	4	yes	complemented
24	#10597383	C₂²	1	8	yes	—
25	#10597383	C₂²	1	4	yes	complemented
26	#10597383	C₂²	1	4	yes	complemented
27	D₄×Dic₁₉	C₂²	1	8	yes	complemented
28	D₄×Dic₁₉	C₂²	1	8	yes	complemented
29	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	complemented
30	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	4	yes	complemented
31	#10597372	C₂²	1	4	yes	complemented
32	#10597370	C₂²	1	4	yes	complemented
33	#10597344	C₂²	1	8	yes	complemented
34	#10597343	C₂²	1	4	yes	complemented
35	#10597338	C₂²	1	4	yes	complemented
36	#10597332	C₂²	1	8	yes	complemented
37	#10597331	C₂²	1	8	yes	complemented
38	#10597329	C₂²	1	8	yes	complemented
39	(C₂²⋊D₁₉)⋊C₄	C₂²	1	8	yes	complemented
40	C₂²⋊C₄×D₁₉	C₂²	1	8	yes	complemented
41	#10597324	C₂²	1	8	yes	complemented
42	C₂⁴×C₁₉	C₂³	1	2	yes	—
43	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
44	C₂³×D₁₉	—	2	4	no	—
45	C₂³×D₁₉	—	4	4	no	—
46	C₂³×D₁₉	C₂³	1	1	yes	characteristic
47	C₂³×D₁₉	C₂³	1	4	yes	—
48	C₂³×D₁₉	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₁₉	—	2	8	no	—
50	C₂×D₄×C₁₉	C₂³	1	4	yes	—
51	C₂×D₄×C₁₉	C₂³	1	4	yes	—
52	C₂×D₄×C₁₉	C₂³	1	4	yes	complemented
53	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
55	C₂×C₂²⋊D₁₉	—	4	8	no	—
56	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
57	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
58	C₂²×Dic₁₉	C₂³	1	2	yes	—
59	C₂²×Dic₁₉	C₂³	1	4	yes	—
60	C₂²×Dic₁₉	C₂³	1	8	yes	complemented
61	#64543	—	4	8	no	—
62	C₂×C₄×D₁₉	—	4	4	no	—
63	C₂×C₄×D₁₉	C₂³	1	4	yes	—
64	C₂×C₄×D₁₉	C₂³	1	4	yes	—
65	C₂×Dic₃₈	C₂³	1	4	yes	complemented
66	C₄⋊C₄×C₁₉	C₂³	1	4	yes	complemented
67	C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
68	C₂²⋊C₄×C₁₉	C₂³	1	8	yes	complemented
69	C₂²⋊Dic₁₉	C₂³	1	4	yes	—
70	C₂²⋊Dic₁₉	C₂³	1	4	yes	—
71	C₂²⋊Dic₁₉	C₂³	1	4	yes	—
72	C₂²⋊Dic₁₉	C₂³	1	8	yes	complemented
73	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
74	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
75	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
76	C₄⋊Dic₁₉	C₂³	1	8	yes	complemented
77	C₄⋊Dic₁₉	C₂³	1	4	yes	complemented
78	Dic₁₉⋊C₄	C₂³	1	8	yes	complemented
79	C₄×Dic₁₉	C₂³	1	8	yes	complemented
80	C₂³×C₁₉	—	2	4	no	—
81	C₂³×C₁₉	—	4	2	no	—
82	C₂³×C₁₉	—	4	4	no	—
83	C₂³×C₁₉	#41	1	4	yes	complemented
84	C₂³×C₁₉	C₂⁴	1	4	yes	—
85	C₂³×C₁₉	C₂⁴	1	2	yes	—
86	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
87	C₂²×D₁₉	—	2	2	no	—
88	C₂²×D₁₉	—	2	4	no	—
89	C₂²×D₁₉	—	2	8	no	—
90	C₂²×D₁₉	—	2	2	no	—
91	C₂²×D₁₉	—	2	8	no	—
92	C₂²×D₁₉	—	4	4	no	—
93	C₂²×D₁₉	—	4	2	no	—
94	C₂²×D₁₉	—	4	16	no	—
95	C₂²×D₁₉	C₂⁴	1	2	yes	—
96	C₂²×D₁₉	C₂⁴	1	2	yes	—
97	D₄×C₁₉	—	2	8	no	—
98	D₄×C₁₉	—	2	8	no	—
99	D₄×C₁₉	—	4	4	no	—
100	C₂×C₄×C₁₉	—	2	4	no	—
101	C₂×C₄×C₁₉	—	4	2	no	—
102	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
103	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
104	C₂²⋊D₁₉	—	2	16	no	—
105	C₂²⋊D₁₉	—	4	8	no	—
106	C₂×Dic₁₉	—	4	2	no	—
107	C₂×Dic₁₉	—	4	8	no	—
108	C₂×Dic₁₉	—	4	4	no	—
109	C₂×Dic₁₉	C₂⁴	1	4	yes	—
110	C₂×Dic₁₉	C₂⁴	1	8	yes	—
111	C₄×D₁₉	—	2	8	no	—
112	C₄×D₁₉	—	4	4	no	—
113	Dic₃₈	—	4	4	no	—
114	#3490	—	19	1	no	maximal, Sylow
115	C₂²×C₁₉	—	2	2	no	—
116	C₂²×C₁₉	—	2	4	no	—
117	C₂²×C₁₉	—	2	4	no	—
118	C₂²×C₁₉	—	2	8	no	—
119	C₂²×C₁₉	—	4	4	no	—
120	C₂²×C₁₉	—	8	4	no	—
121	C₂²×C₁₉	C₂×#41	1	4	yes	—
122	C₂²×C₁₉	C₂×#41	1	2	yes	—
123	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
124	C₂×D₁₉	—	2	8	no	—
125	C₂×D₁₉	—	2	2	no	—
126	C₂×D₁₉	—	2	2	no	—
127	C₂×D₁₉	—	2	2	no	—
128	C₂×D₁₉	—	2	4	no	—
129	C₂×D₁₉	—	4	16	no	—
130	C₂×D₁₉	—	4	4	no	—
131	C₂×D₁₉	H(5,𝔽₂)	1	4	yes	—
132	C₄×C₁₉	—	2	4	no	—
133	C₄×C₁₉	—	2	4	no	—
134	Dic₁₉	—	2	8	no	—
135	Dic₁₉	—	2	4	no	—
136	#551	—	19	8	no	—
137	#546	—	19	8	no	—
138	#543	—	19	1	no	—
139	#538	—	19	4	no	—
140	#534	D₁₉	1	1	yes	characteristic, complemented
141	#525	—	19	4	no	—
142	C₂×#120	—	19	2	no	—
143	#518	—	19	2	no	—
144	C₂²×C₂²⋊C₄	—	19	1	no	—
145	C₂×C₁₉	—	2	2	no	—
146	C₂×C₁₉	—	2	4	no	—
147	C₂×C₁₉	—	4	4	no	—
148	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
149	C₂×C₁₉	C₂×H(5,𝔽₂)	1	2	yes	—
150	D₁₉	—	2	4	no	—
151	D₁₉	—	4	2	no	—
152	C₂⁵	—	19	1	no	—
153	C₂×#41	—	19	4	no	—
154	C₂²×D₄	—	19	4	no	—
155	C₂²×D₄	C₂×D₁₉	1	2	yes	—
156	C₂³×C₄	—	19	2	no	—
157	#126	—	19	8	no	—
158	#124	—	19	8	no	—
159	#123	—	19	8	no	—
160	#123	—	19	8	no	—
161	#123	C₂×D₁₉	1	4	yes	complemented
162	C₂²⋊Q₈	—	19	4	no	—
163	C₂²⋊Q₈	—	19	4	no	—
164	C₂²⋊Q₈	—	19	8	no	—
165	#121	—	19	4	no	—
166	#121	—	19	8	no	—
167	#121	C₂×D₁₉	1	4	yes	complemented
168	#120	—	19	8	no	—
169	#120	—	19	4	no	—
170	#120	—	19	4	no	—
171	#120	C₂×D₁₉	1	4	yes	complemented
172	C₄×D₄	—	19	8	no	—
173	C₄×D₄	—	19	8	no	—
174	C₄×D₄	—	19	8	no	—
175	#117	—	19	4	no	—
176	#117	—	19	4	no	—
177	C₂×C₂²⋊C₄	—	19	1	no	—
178	C₂×C₂²⋊C₄	—	19	2	no	—
179	C₂×C₂²⋊C₄	—	19	4	no	—
180	C₂×C₂²⋊C₄	—	19	2	no	—
181	C₂×C₂²⋊C₄	—	19	4	no	—
182	C₂×C₂²⋊C₄	—	19	8	no	—
183	C₂×C₂²⋊C₄	C₂×D₁₉	1	1	yes	characteristic
184	C₁₉	#3490	1	1	yes	characteristic, complemented, Sylow
185	C₂⁴	—	19	1	no	—
186	C₂⁴	—	19	4	no	—
187	C₂⁴	—	19	1	no	—
188	C₂⁴	—	38	4	no	—
189	C₂⁴	—	76	4	no	—
190	C₂⁴	C₂²×D₁₉	1	2	yes	—
191	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
192	#41	—	76	8	no	—
193	C₂×Q₈	—	19	4	no	—
194	C₂×D₄	—	2	8	no	—
195	C₂×D₄	—	19	8	no	—
196	C₂×D₄	—	19	8	no	—
197	C₂×D₄	—	76	8	no	—
198	C₂×D₄	C₂²×D₁₉	1	4	yes	—
199	C₂×D₄	C₂²×D₁₉	1	4	yes	—
200	C₂×D₄	C₂²×D₁₉	1	4	yes	complemented
201	C₂²×C₄	—	19	2	no	—
202	C₂²×C₄	—	19	4	no	—
203	C₂²×C₄	—	19	8	no	—
204	C₂²×C₄	—	19	4	no	—
205	C₂²×C₄	—	19	4	no	—
206	C₂²×C₄	—	76	4	no	—
207	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
208	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
209	C₄⋊C₄	—	19	8	no	—
210	C₄⋊C₄	—	19	8	no	—
211	C₄⋊C₄	—	19	4	no	—
212	C₄⋊C₄	C₂²×D₁₉	1	4	yes	complemented
213	C₂²⋊C₄	—	19	4	no	—
214	C₂²⋊C₄	—	19	4	no	—
215	C₂²⋊C₄	—	19	4	no	—
216	C₂²⋊C₄	—	19	8	no	—
217	C₂²⋊C₄	—	19	8	no	—
218	C₂²⋊C₄	—	19	8	no	—
219	C₂²⋊C₄	—	19	8	no	—
220	C₂²⋊C₄	C₂²×D₁₉	1	4	yes	—
221	C₂²⋊C₄	C₂²×D₁₉	1	8	yes	complemented
222	C₄²	—	19	8	no	—
223	C₂³	—	2	4	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	4	4	no	—
226	C₂³	—	19	2	no	—
227	C₂³	—	19	2	no	—
228	C₂³	—	38	4	no	—
229	C₂³	—	38	8	no	—
230	C₂³	—	38	2	no	—
231	C₂³	—	38	8	no	—
232	C₂³	—	38	2	no	—
233	C₂³	—	76	2	no	—
234	C₂³	—	76	4	no	—
235	C₂³	—	76	16	no	—
236	C₂³	#64543	1	4	yes	complemented
237	C₂³	C₂³×D₁₉	1	4	yes	—
238	C₂³	C₂³×D₁₉	1	2	yes	—
239	C₂³	C₂³×D₁₉	1	1	yes	characteristic
240	Q₈	—	76	4	no	—
241	D₄	—	2	8	no	—
242	D₄	—	2	8	no	—
243	D₄	—	4	4	no	—
244	D₄	—	38	16	no	—
245	D₄	—	76	8	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	19	4	no	—
249	C₂×C₄	—	19	8	no	—
250	C₂×C₄	—	38	8	no	—
251	C₂×C₄	—	76	4	no	—
252	C₂×C₄	—	76	2	no	—
253	C₂×C₄	—	76	4	no	—
254	C₂×C₄	—	76	8	no	—
255	C₂×C₄	C₂³×D₁₉	1	4	yes	—
256	C₂×C₄	C₂³×D₁₉	1	4	yes	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	4	no	—
260	C₂²	—	2	8	no	—
261	C₂²	—	4	4	no	—
262	C₂²	—	8	4	no	—
263	C₂²	—	19	4	no	—
264	C₂²	—	38	8	no	—
265	C₂²	—	38	2	no	—
266	C₂²	—	38	2	no	—
267	C₂²	—	38	2	no	—
268	C₂²	—	38	4	no	—
269	C₂²	—	76	16	no	—
270	C₂²	—	76	4	no	—
271	C₂²	C₂×#64543	1	4	yes	—
272	C₂²	C₂×#64543	1	2	yes	—
273	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
274	C₄	—	2	4	no	—
275	C₄	—	2	4	no	—
276	C₄	—	38	4	no	—
277	C₄	—	38	8	no	—
278	C₂	—	2	4	no	—
279	C₂	—	2	2	no	—
280	C₂	—	4	4	no	—
281	C₂	—	38	4	no	—
282	C₂	—	76	2	no	—
283	C₂	C₂²×#64543	1	1	yes	characteristic
284	C₂	H(5,𝔽₂)×D₁₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	C₁₉×#534	C₂
hypercenter	#534	D₁₉
hypocenter	C₁₉	#3490
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #534

Complements

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

N	H ≅ G/N	type
1. #11927217	C₂	semidirect
2. #11927181	C₂	semidirect
3. #11927170	C₂	semidirect
4. #11927131	C₂	semidirect
5. #11927054	C₂	semidirect
6. C₂×#10597383	C₂	semidirect
7. #11927019	C₂	semidirect
8. C₂×C₂²⋊C₄×D₁₉	C₂	semidirect
9. C₁₉×#534	C₂	semidirect
12. C₂²×C₂²⋊D₁₉	C₂²	semidirect
13. C₂×#64543	C₂²	semidirect
14. C₂²×C₄×D₁₉	C₂²	semidirect
15. C₁₉×#123	C₂²	semidirect
16. C₁₉×#121	C₂²	semidirect
17. C₁₉×#120	C₂²	semidirect
21. C₂×C₂²⋊Dic₁₉	C₂²	semidirect
22. #10597385	C₂²	semidirect
23. #10597384	C₂²	semidirect
25. #10597383	C₂²	semidirect
26. #10597383	C₂²	semidirect
27. D₄×Dic₁₉	C₂²	semidirect
28. D₄×Dic₁₉	C₂²	semidirect
29. C₂×(C₂×D₁₉)⋊C₄	C₂²	semidirect
30. C₂×(C₂×D₁₉)⋊C₄	C₂²	semidirect
31. #10597372	C₂²	semidirect
32. #10597370	C₂²	semidirect
33. #10597344	C₂²	semidirect
34. #10597343	C₂²	semidirect
35. #10597338	C₂²	semidirect
36. #10597332	C₂²	semidirect
37. #10597331	C₂²	semidirect
38. #10597329	C₂²	semidirect
39. (C₂²⋊D₁₉)⋊C₄	C₂²	semidirect
40. C₂²⋊C₄×D₁₉	C₂²	semidirect
41. #10597324	C₂²	semidirect
52. C₂×D₄×C₁₉	C₂³	semidirect
60. C₂²×Dic₁₉	C₂³	semidirect
65. C₂×Dic₃₈	C₂³	semidirect
66. C₄⋊C₄×C₁₉	C₂³	semidirect
68. C₂²⋊C₄×C₁₉	C₂³	semidirect
72. C₂²⋊Dic₁₉	C₂³	semidirect
76. C₄⋊Dic₁₉	C₂³	semidirect
77. C₄⋊Dic₁₉	C₂³	semidirect
78. Dic₁₉⋊C₄	C₂³	semidirect
79. C₄×Dic₁₉	C₂³	semidirect
83. C₂³×C₁₉	#41	semidirect
140. #534	D₁₉	semidirect
161. #123	C₂×D₁₉	semidirect
167. #121	C₂×D₁₉	semidirect
171. #120	C₂×D₁₉	semidirect
184. C₁₉	#3490	semidirect
200. C₂×D₄	C₂²×D₁₉	semidirect
212. C₄⋊C₄	C₂²×D₁₉	semidirect
221. C₂²⋊C₄	C₂²×D₁₉	semidirect
236. C₂³	#64543	semidirect

Maximal subgroups

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

subgroup	index
#11927217	2
#11927181	2
#11927170	2
#11927131	2
#11927054	2
C₂×#10597383	2
#11927019	2
C₂×C₂²⋊C₄×D₁₉	2
C₁₉×#534	2
#3490	19