Subgroups — D₄×#64515 #14572366

Order 2432.

544 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#64515	C₂	1	2	yes	maximal, complemented
2	C₄×D₄×D₁₉	C₂	1	1	yes	maximal, characteristic
3	#11926623	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926621	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926615	C₂	1	2	yes	maximal, complemented
6	#11926612	C₂	1	2	yes	maximal, complemented
7	D₄×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926566	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926506	C₂	1	2	yes	maximal, complemented
10	C₄×#64515	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₁₉	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₁₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₉	C₂²	1	2	yes	complemented
15	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
16	D₄×Dic₁₉	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₁₉	C₂²	1	2	yes	—
18	C₂²×C₁₉⋊C₈	C₂²	1	2	yes	complemented
19	C₂×#64515	—	2	8	no	—
20	C₂×#64515	C₂²	1	2	yes	—
21	C₂×#64515	C₂²	1	1	yes	characteristic
22	C₂×#64515	C₂²	1	1	yes	characteristic
23	C₂×#64515	C₂²	1	2	yes	complemented
24	C₂×#64515	C₂²	1	2	yes	complemented
25	#10597339	C₄	1	1	yes	characteristic
26	C₄⋊C₄×D₁₉	C₄	1	1	yes	characteristic
27	(C₂²⋊D₁₉)⋊C₄	C₄	1	2	yes	—
28	C₂²⋊C₄×D₁₉	C₄	1	2	yes	—
29	C₄×D₇₆	C₂²	1	1	yes	characteristic
30	C₄²×D₁₉	C₂²	1	1	yes	characteristic
31	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
32	C₁₉×C₂²⋊C₈	C₂²	1	2	yes	complemented
33	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic, complemented
34	C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	complemented
35	(C₂×D₁₉)⋊C₈	C₂²	1	2	yes	complemented
36	(C₂×D₁₉)⋊C₈	C₂²	1	2	yes	complemented
37	#10597272	C₂²	1	1	yes	characteristic, complemented
38	#10597272	C₂²	1	1	yes	characteristic, complemented
39	Dic₁₉⋊C₈	C₂²	1	2	yes	complemented
40	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic, complemented
41	C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	C₂³×D₁₉	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
45	C₂×C₂²⋊D₁₉	C₂×C₄	1	2	yes	—
46	C₂²×Dic₁₉	C₂×C₄	1	2	yes	—
47	D₄×D₁₉	—	2	4	no	—
48	C₂×D₇₆	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₉	—	2	4	no	—
50	C₂×C₄×D₁₉	—	2	4	no	—
51	C₂×C₄×D₁₉	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₁₉	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₁₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₁₉	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₁₉	C₂³	1	2	yes	—
56	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₁₉	—	2	4	no	—
58	C₂×C₈×C₁₉	C₂³	1	2	yes	—
59	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
62	C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
63	C₄²×C₁₉	C₂³	1	1	yes	characteristic
64	C₂²⋊Dic₁₉	C₂×C₄	1	2	yes	—
65	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
66	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
67	C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
68	Dic₁₉⋊C₄	C₂×C₄	1	2	yes	—
69	C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
72	C₂×C₁₉⋊C₈	—	2	4	no	—
73	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
75	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
76	#64515	—	2	4	no	—
77	#64515	—	2	2	no	—
78	#64515	—	2	4	no	—
79	#64515	—	2	4	no	—
80	#64515	D₄	1	4	yes	direct product factor
81	C₂³×C₁₉	C₂²×C₄	1	2	yes	—
82	C₂²×D₁₉	—	2	4	no	—
83	C₂²×D₁₉	—	2	4	no	—
84	C₂²×D₁₉	—	4	2	no	—
85	C₂²×D₁₉	C₂²×C₄	1	2	yes	—
86	C₂²×D₁₉	C₂²×C₄	1	1	yes	characteristic
87	D₄×C₁₉	M₁₆	1	4	yes	complemented
88	C₂×C₄×C₁₉	—	2	2	no	—
89	C₂×C₄×C₁₉	—	2	2	no	—
90	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₁₉	—	2	4	no	—
95	C₂×Dic₁₉	—	2	4	no	—
96	C₂×Dic₁₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₁₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
99	C₂×Dic₁₉	C₂²×C₄	1	1	yes	characteristic
100	D₇₆	—	2	2	no	—
101	C₄×D₁₉	—	2	4	no	—
102	C₄×D₁₉	—	2	2	no	—
103	C₄×D₁₉	—	2	2	no	—
104	C₄×D₁₉	—	2	2	no	—
105	C₄×D₁₉	—	2	4	no	—
106	C₄×D₁₉	—	4	1	no	—
107	C₄×D₁₉	C₂×D₄	1	2	yes	—
108	C₄×D₁₉	#41	1	2	yes	—
109	C₈×C₁₉	—	2	2	no	—
110	C₈×C₁₉	—	2	1	no	—
111	C₈×C₁₉	C₂×D₄	1	2	yes	complemented
112	C₁₉⋊C₈	—	2	2	no	—
113	C₁₉⋊C₈	—	2	1	no	—
114	C₁₉⋊C₈	C₂×D₄	1	2	yes	complemented
115	D₄×M₁₆	—	19	1	no	maximal, Sylow
116	C₂²×C₁₉	—	2	2	no	—
117	C₂²×C₁₉	—	2	2	no	—
118	C₂²×C₁₉	C₂×M₁₆	1	4	yes	complemented
119	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
120	C₂×D₁₉	—	2	1	no	—
121	C₂×D₁₉	—	2	1	no	—
122	C₂×D₁₉	—	2	2	no	—
123	C₂×D₁₉	—	2	2	no	—
124	C₂×D₁₉	—	2	2	no	—
125	C₂×D₁₉	—	4	8	no	—
126	C₂×D₁₉	C₄×D₄	1	2	yes	—
127	C₄×C₁₉	—	2	2	no	—
128	C₄×C₁₉	—	2	1	no	—
129	C₄×C₁₉	C₂×M₁₆	1	2	yes	complemented
130	C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
131	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
132	Dic₁₉	—	2	2	no	—
133	Dic₁₉	—	2	1	no	—
134	Dic₁₉	—	2	1	no	—
135	Dic₁₉	C₄×D₄	1	2	yes	—
136	C₂²×M₁₆	—	19	2	no	—
137	C₂×C₄×D₄	—	19	1	no	—
138	#436	—	19	1	no	—
139	#436	—	19	1	no	—
140	#435	—	19	2	no	—
141	#435	—	19	2	no	—
142	C₈×D₄	—	19	1	no	—
143	C₈×D₄	D₁₉	1	1	yes	characteristic, complemented
144	#423	—	19	1	no	—
145	#407	—	19	2	no	—
146	C₄×M₁₆	—	19	1	no	—
147	C₂×C₁₉	—	2	4	no	—
148	C₂×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂×C₁₉	C₂²×M₁₆	1	1	yes	characteristic
150	C₂×C₁₉	#568	1	1	yes	characteristic
151	D₁₉	—	2	2	no	—
152	D₁₉	—	4	2	no	—
153	C₂²×D₄	—	19	1	no	—
154	C₂³×C₄	—	19	2	no	—
155	C₂×M₁₆	—	19	1	no	—
156	C₂×M₁₆	—	19	2	no	—
157	C₂×M₁₆	—	19	2	no	—
158	C₂×M₁₆	—	19	2	no	—
159	C₂×M₁₆	—	19	1	no	—
160	C₂×M₁₆	—	38	8	no	—
161	C₂²×C₈	—	19	2	no	—
162	C₂²×C₈	C₂×D₁₉	1	2	yes	complemented
163	C₄×D₄	—	19	1	no	—
164	C₄×D₄	—	19	2	no	—
165	C₄×D₄	—	19	1	no	—
166	C₄×D₄	—	19	2	no	—
167	C₄×D₄	—	19	1	no	—
168	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
169	C₂×C₄⋊C₄	—	19	1	no	—
170	C₂×C₂²⋊C₄	—	19	2	no	—
171	C₂×C₄²	—	19	1	no	—
172	C₄⋊C₈	—	19	2	no	—
173	C₄⋊C₈	—	19	1	no	—
174	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic, complemented
175	C₂²⋊C₈	—	19	2	no	—
176	C₂²⋊C₈	—	19	2	no	—
177	C₂²⋊C₈	—	19	2	no	—
178	C₂²⋊C₈	C₂×D₁₉	1	2	yes	complemented
179	#97	—	19	1	no	—
180	#97	—	19	1	no	—
181	C₄×C₈	—	19	1	no	—
182	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic, complemented
183	C₁₉	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
184	C₂⁴	—	19	2	no	—
185	C₂×D₄	—	19	2	no	—
186	C₂×D₄	—	19	1	no	—
187	C₂×D₄	—	38	4	no	—
188	C₂×D₄	C₄×D₁₉	1	1	yes	characteristic
189	C₂²×C₄	—	19	1	no	—
190	C₂²×C₄	—	19	2	no	—
191	C₂²×C₄	—	19	1	no	—
192	C₂²×C₄	—	19	2	no	—
193	C₂²×C₄	—	19	2	no	—
194	C₂²×C₄	—	19	2	no	—
195	C₂²×C₄	—	19	1	no	—
196	C₂²×C₄	—	38	4	no	—
197	C₂²×C₄	—	38	4	no	—
198	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
199	M₁₆	—	19	4	no	—
200	M₁₆	—	38	4	no	—
201	M₁₆	—	38	2	no	—
202	M₁₆	—	38	4	no	—
203	M₁₆	—	38	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	19	1	no	—
206	C₂×C₈	—	19	1	no	—
207	C₂×C₈	—	19	2	no	—
208	C₂×C₈	—	38	4	no	—
209	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
210	C₂×C₈	C₂²×D₁₉	1	2	yes	—
211	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
212	C₄⋊C₄	—	19	2	no	—
213	C₄⋊C₄	—	19	1	no	—
214	C₄⋊C₄	C₄×D₁₉	1	1	yes	characteristic
215	C₂²⋊C₄	—	19	2	no	—
216	C₂²⋊C₄	—	19	2	no	—
217	C₂²⋊C₄	—	19	2	no	—
218	C₂²⋊C₄	C₄×D₁₉	1	2	yes	—
219	C₄²	—	19	1	no	—
220	C₄²	—	19	1	no	—
221	C₄²	—	19	1	no	—
222	C₄²	C₂²×D₁₉	1	1	yes	characteristic
223	C₂³	—	19	1	no	—
224	C₂³	—	19	2	no	—
225	C₂³	—	38	4	no	—
226	C₂³	—	38	4	no	—
227	C₂³	—	76	2	no	—
228	C₂³	C₂×C₄×D₁₉	1	2	yes	—
229	D₄	—	38	4	no	—
230	D₄	—	38	2	no	—
231	D₄	#64515	1	4	yes	direct product factor
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	19	1	no	—
235	C₂×C₄	—	19	2	no	—
236	C₂×C₄	—	19	2	no	—
237	C₂×C₄	—	19	1	no	—
238	C₂×C₄	—	19	1	no	—
239	C₂×C₄	—	19	2	no	—
240	C₂×C₄	—	38	4	no	—
241	C₂×C₄	—	38	4	no	—
242	C₂×C₄	—	38	4	no	—
243	C₂×C₄	—	38	2	no	—
244	C₂×C₄	—	38	2	no	—
245	C₂×C₄	—	38	2	no	—
246	C₂×C₄	—	76	1	no	—
247	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₁₉	1	2	yes	—
249	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
250	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	1	no	—
253	C₈	—	19	2	no	—
254	C₈	—	38	2	no	—
255	C₈	—	38	1	no	—
256	C₈	D₄×D₁₉	1	2	yes	complemented
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	19	2	no	—
260	C₂²	—	38	1	no	—
261	C₂²	—	38	1	no	—
262	C₂²	—	38	2	no	—
263	C₂²	—	38	2	no	—
264	C₂²	—	38	2	no	—
265	C₂²	—	76	8	no	—
266	C₂²	C₂×#64515	1	4	yes	complemented
267	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	2	no	—
270	C₄	—	19	2	no	—
271	C₄	—	38	1	no	—
272	C₄	—	38	1	no	—
273	C₄	—	38	2	no	—
274	C₄	C₂×#64515	1	2	yes	complemented
275	C₄	C₂×D₄×D₁₉	1	1	yes	characteristic
276	C₄	#41×D₁₉	1	1	yes	characteristic
277	C₂	—	2	4	no	—
278	C₂	—	38	2	no	—
279	C₂	—	76	2	no	—
280	C₂	C₄×D₄×D₁₉	1	1	yes	characteristic
281	C₂	C₂²×#64515	1	1	yes	characteristic
282	C₂	#11927309	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉
commutator, socle	C₂²×C₁₉	C₂³×C₄
Fitting	C₈×D₄×C₁₉	C₂
prime radical	C₂×D₄×D₁₉	C₄
cosocle	C₂×C₄×C₁₉	C₂⁴
hypercenter	C₈×D₄	D₁₉
hypocenter	C₁₉	D₄×M₁₆
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#64515	C₂	semidirect
3. #11926623	C₂	semidirect
4. #11926621	C₂	semidirect
5. #11926615	C₂	semidirect
6. #11926612	C₂	semidirect
7. D₄×C₁₉⋊C₈	C₂	semidirect
8. #11926566	C₂	semidirect
9. #11926506	C₂	semidirect
10. C₄×#64515	C₂	semidirect
11. C₈×D₄×C₁₉	C₂	semidirect
14. C₂²×C₈×C₁₉	C₂²	semidirect
18. C₂²×C₁₉⋊C₈	C₂²	semidirect
23. C₂×#64515	C₂²	semidirect
24. C₂×#64515	C₂²	semidirect
31. C₁₉×C₄⋊C₈	C₂²	semidirect
32. C₁₉×C₂²⋊C₈	C₂²	semidirect
33. C₄×C₈×C₁₉	C₂²	semidirect
34. C₂²⋊(C₁₉⋊C₈)	C₂²	semidirect
35. (C₂×D₁₉)⋊C₈	C₂²	semidirect
36. (C₂×D₁₉)⋊C₈	C₂²	semidirect
37. #10597272	C₂²	semidirect
38. #10597272	C₂²	semidirect
39. Dic₁₉⋊C₈	C₂²	semidirect
40. C₄⋊(C₁₉⋊C₈)	C₂²	semidirect
41. C₄×C₁₉⋊C₈	C₂²	semidirect
80. #64515	D₄	direct (N × H)
87. D₄×C₁₉	M₁₆	semidirect
111. C₈×C₁₉	C₂×D₄	semidirect
114. C₁₉⋊C₈	C₂×D₄	semidirect
118. C₂²×C₁₉	C₂×M₁₆	semidirect
129. C₄×C₁₉	C₂×M₁₆	semidirect
143. C₈×D₄	D₁₉	semidirect
162. C₂²×C₈	C₂×D₁₉	semidirect
174. C₄⋊C₈	C₂×D₁₉	semidirect
178. C₂²⋊C₈	C₂×D₁₉	semidirect
182. C₄×C₈	C₂×D₁₉	semidirect
183. C₁₉	D₄×M₁₆	semidirect
231. D₄	#64515	direct (N × H)
256. C₈	D₄×D₁₉	semidirect
266. C₂²	C₂×#64515	semidirect
274. C₄	C₂×#64515	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#64515	2
C₄×D₄×D₁₉	2
#11926623	2
#11926621	2
#11926615	2
#11926612	2
D₄×C₁₉⋊C₈	2
#11926566	2
#11926506	2
C₄×#64515	2
C₈×D₄×C₁₉	2
D₄×M₁₆	19

Subgroups — D4×#64515 #14572366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#64515 #14572366