Subgroups — #14572668

Order 2432.

464 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄×D₁₉	C₂	1	1	yes	maximal, characteristic
2	#11926623	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926621	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926615	C₂	1	2	yes	maximal, complemented
5	#11926612	C₂	1	2	yes	maximal, complemented
6	D₄×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926600	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926595	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926506	C₂	1	2	yes	maximal, complemented
10	C₂×(C₂×D₁₉)⋊C₈	C₂	1	2	yes	maximal, 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	C₂²	1	2	yes	complemented
20	C₂×#64515	C₂²	1	2	yes	complemented
21	#10597339	C₄	1	1	yes	characteristic
22	C₄⋊C₄×D₁₉	C₄	1	1	yes	characteristic
23	(C₂²⋊D₁₉)⋊C₄	C₄	1	2	yes	—
24	C₂²⋊C₄×D₁₉	C₄	1	2	yes	—
25	C₄×D₇₆	C₂²	1	1	yes	characteristic
26	C₄²×D₁₉	C₂²	1	1	yes	characteristic
27	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
28	C₁₉×C₂²⋊C₈	C₂²	1	2	yes	complemented
29	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic, complemented
30	C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	complemented
31	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
33	(C₂×D₁₉)⋊C₈	C₂²	1	2	yes	—
34	(C₂×D₁₉)⋊C₈	C₂²	1	2	yes	complemented
35	(C₂×D₁₉)⋊C₈	C₂²	1	2	yes	complemented
36	#10597272	C₂²	1	2	yes	complemented
37	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
38	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
39	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic, complemented
40	C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
41	C₂³×D₁₉	C₂×C₄	1	2	yes	—
42	C₂×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
44	C₂×C₂²⋊D₁₉	C₂×C₄	1	2	yes	—
45	C₂²×Dic₁₉	C₂×C₄	1	2	yes	—
46	D₄×D₁₉	—	2	4	no	—
47	C₂×D₇₆	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₉	—	4	2	no	—
49	C₂×C₄×D₁₉	—	4	2	no	—
50	C₂×C₄×D₁₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₉	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₁₉	C₂×C₄	1	2	yes	—
54	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₁₉	C₂³	1	2	yes	—
56	C₂×C₈×C₁₉	—	4	2	no	—
57	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₁₉	C₂³	1	2	yes	—
59	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
61	C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
62	C₄²×C₁₉	C₂³	1	1	yes	characteristic
63	C₂²⋊Dic₁₉	C₂×C₄	1	2	yes	—
64	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
65	(C₂×D₁₉)⋊C₄	C₂×C₄	1	2	yes	—
66	C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
67	Dic₁₉⋊C₄	C₂×C₄	1	2	yes	—
68	C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
71	C₂×C₁₉⋊C₈	—	4	2	no	—
72	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
75	#64515	—	4	2	no	—
76	#64515	—	4	2	no	—
77	C₂³×C₁₉	C₂²×C₄	1	2	yes	—
78	C₂²×D₁₉	—	2	4	no	—
79	C₂²×D₁₉	—	2	4	no	—
80	C₂²×D₁₉	—	4	2	no	—
81	C₂²×D₁₉	C₂²×C₄	1	2	yes	—
82	C₂²×D₁₉	C₂²×C₄	1	1	yes	characteristic
83	D₄×C₁₉	M₁₆	1	4	yes	complemented
84	C₂×C₄×C₁₉	—	2	2	no	—
85	C₂×C₄×C₁₉	—	2	2	no	—
86	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
90	C₂²⋊D₁₉	—	2	4	no	—
91	C₂×Dic₁₉	—	4	2	no	—
92	C₂×Dic₁₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×Dic₁₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₁₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
96	D₇₆	—	2	2	no	—
97	C₄×D₁₉	—	2	2	no	—
98	C₄×D₁₉	—	2	2	no	—
99	C₄×D₁₉	—	2	2	no	—
100	C₄×D₁₉	—	2	2	no	—
101	C₄×D₁₉	—	2	1	no	—
102	C₄×D₁₉	—	2	1	no	—
103	C₄×D₁₉	—	4	2	no	—
104	C₄×D₁₉	—	4	2	no	—
105	C₈×C₁₉	—	2	1	no	—
106	C₈×C₁₉	—	2	2	no	—
107	C₈×C₁₉	—	2	1	no	—
108	C₁₉⋊C₈	—	2	1	no	—
109	C₁₉⋊C₈	—	2	2	no	—
110	C₁₉⋊C₈	—	2	1	no	—
111	#2974	—	19	1	no	maximal, Sylow
112	C₂²×C₁₉	—	2	2	no	—
113	C₂²×C₁₉	—	2	2	no	—
114	C₂²×C₁₉	C₂×M₁₆	1	4	yes	—
115	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
116	C₂×D₁₉	—	2	1	no	—
117	C₂×D₁₉	—	2	1	no	—
118	C₂×D₁₉	—	2	2	no	—
119	C₂×D₁₉	—	2	2	no	—
120	C₂×D₁₉	—	2	2	no	—
121	C₂×D₁₉	—	4	8	no	—
122	C₂×D₁₉	#131	1	2	yes	—
123	C₄×C₁₉	—	2	2	no	—
124	C₄×C₁₉	—	2	1	no	—
125	C₄×C₁₉	C₂×M₁₆	1	2	yes	—
126	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
127	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
128	Dic₁₉	—	2	1	no	—
129	Dic₁₉	—	2	2	no	—
130	Dic₁₉	—	2	1	no	—
131	Dic₁₉	—	2	1	no	—
132	C₂×C₄×D₄	—	19	1	no	—
133	#436	—	19	1	no	—
134	#436	—	19	1	no	—
135	#435	—	19	2	no	—
136	#435	—	19	2	no	—
137	C₈×D₄	—	19	1	no	—
138	C₈×D₄	D₁₉	1	1	yes	characteristic, complemented
139	#432	—	19	1	no	—
140	#431	—	19	1	no	—
141	#407	—	19	2	no	—
142	C₂×C₂²⋊C₈	—	19	2	no	—
143	C₂×C₁₉	—	2	4	no	—
144	C₂×C₁₉	#518	1	1	yes	characteristic
145	C₂×C₁₉	C₂²×M₁₆	1	1	yes	characteristic
146	C₂×C₁₉	C₂×#131	1	1	yes	characteristic
147	D₁₉	—	2	2	no	—
148	D₁₉	—	4	2	no	—
149	C₂²×D₄	—	19	1	no	—
150	C₂³×C₄	—	19	2	no	—
151	C₂×M₁₆	—	19	2	no	—
152	C₂×M₁₆	—	19	2	no	—
153	C₂²×C₈	—	19	2	no	—
154	C₂²×C₈	C₂×D₁₉	1	2	yes	complemented
155	C₄×D₄	—	19	1	no	—
156	C₄×D₄	—	19	2	no	—
157	C₄×D₄	—	19	1	no	—
158	C₄×D₄	—	19	2	no	—
159	C₄×D₄	—	19	1	no	—
160	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	19	1	no	—
162	C₂×C₂²⋊C₄	—	19	2	no	—
163	C₂×C₄²	—	19	1	no	—
164	C₄⋊C₈	—	19	1	no	—
165	C₄⋊C₈	—	19	1	no	—
166	C₄⋊C₈	—	19	1	no	—
167	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic, complemented
168	C₂²⋊C₈	—	19	2	no	—
169	C₂²⋊C₈	—	19	1	no	—
170	C₂²⋊C₈	—	19	2	no	—
171	C₂²⋊C₈	—	19	2	no	—
172	C₂²⋊C₈	—	19	2	no	—
173	C₂²⋊C₈	—	19	1	no	—
174	C₂²⋊C₈	C₂×D₁₉	1	2	yes	complemented
175	#97	—	19	2	no	—
176	C₄×C₈	—	19	1	no	—
177	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic, complemented
178	C₁₉	#2974	1	1	yes	characteristic, complemented, Sylow
179	C₂⁴	—	19	2	no	—
180	C₂×D₄	—	19	2	no	—
181	C₂×D₄	—	19	1	no	—
182	C₂×D₄	—	38	4	no	—
183	C₂×D₄	C₄×D₁₉	1	1	yes	characteristic
184	C₂²×C₄	—	19	2	no	—
185	C₂²×C₄	—	19	1	no	—
186	C₂²×C₄	—	19	1	no	—
187	C₂²×C₄	—	19	1	no	—
188	C₂²×C₄	—	19	2	no	—
189	C₂²×C₄	—	19	2	no	—
190	C₂²×C₄	—	19	2	no	—
191	C₂²×C₄	—	76	2	no	—
192	C₂²×C₄	—	76	2	no	—
193	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
194	M₁₆	—	76	2	no	—
195	M₁₆	—	76	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	19	1	no	—
198	C₂×C₈	—	19	1	no	—
199	C₂×C₈	—	19	2	no	—
200	C₂×C₈	—	76	2	no	—
201	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₁₉	1	2	yes	—
204	C₄⋊C₄	—	19	2	no	—
205	C₄⋊C₄	—	19	1	no	—
206	C₄⋊C₄	C₄×D₁₉	1	1	yes	characteristic
207	C₂²⋊C₄	—	19	2	no	—
208	C₂²⋊C₄	—	19	2	no	—
209	C₂²⋊C₄	—	19	2	no	—
210	C₂²⋊C₄	C₄×D₁₉	1	2	yes	—
211	C₄²	—	19	1	no	—
212	C₄²	—	19	1	no	—
213	C₄²	—	19	1	no	—
214	C₄²	C₂²×D₁₉	1	1	yes	characteristic
215	C₂³	—	19	1	no	—
216	C₂³	—	19	2	no	—
217	C₂³	—	38	4	no	—
218	C₂³	—	38	4	no	—
219	C₂³	—	76	2	no	—
220	C₂³	C₂×C₄×D₁₉	1	2	yes	—
221	D₄	—	38	4	no	—
222	D₄	—	38	2	no	—
223	D₄	#64515	1	4	yes	complemented
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	19	1	no	—
227	C₂×C₄	—	19	1	no	—
228	C₂×C₄	—	19	1	no	—
229	C₂×C₄	—	19	2	no	—
230	C₂×C₄	—	38	2	no	—
231	C₂×C₄	—	38	2	no	—
232	C₂×C₄	—	38	1	no	—
233	C₂×C₄	—	38	2	no	—
234	C₂×C₄	—	38	1	no	—
235	C₂×C₄	—	38	2	no	—
236	C₂×C₄	—	76	2	no	—
237	C₂×C₄	—	76	2	no	—
238	C₂×C₄	—	76	2	no	—
239	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄×D₁₉	1	2	yes	—
241	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
242	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	—	38	1	no	—
247	C₈	—	38	1	no	—
248	C₈	—	38	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	19	2	no	—
252	C₂²	—	38	1	no	—
253	C₂²	—	38	1	no	—
254	C₂²	—	38	2	no	—
255	C₂²	—	38	2	no	—
256	C₂²	—	38	2	no	—
257	C₂²	—	76	8	no	—
258	C₂²	C₂×#64515	1	4	yes	—
259	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	2	no	—
262	C₄	—	38	1	no	—
263	C₄	—	38	1	no	—
264	C₄	—	38	1	no	—
265	C₄	—	38	2	no	—
266	C₄	C₂×#64515	1	2	yes	—
267	C₄	#10597430	1	1	yes	characteristic
268	C₄	#10597435	1	1	yes	characteristic
269	C₂	—	2	4	no	—
270	C₂	—	38	2	no	—
271	C₂	—	76	2	no	—
272	C₂	#11927016	1	1	yes	characteristic
273	C₂	C₂²×#64515	1	1	yes	characteristic
274	C₂	#131×D₁₉	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	#2974
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
2. #11926623	C₂	semidirect
3. #11926621	C₂	semidirect
4. #11926615	C₂	semidirect
5. #11926612	C₂	semidirect
6. D₄×C₁₉⋊C₈	C₂	semidirect
7. #11926600	C₂	semidirect
8. #11926595	C₂	semidirect
9. #11926506	C₂	semidirect
10. C₂×(C₂×D₁₉)⋊C₈	C₂	semidirect
11. C₈×D₄×C₁₉	C₂	semidirect
14. C₂²×C₈×C₁₉	C₂²	semidirect
18. C₂²×C₁₉⋊C₈	C₂²	semidirect
19. C₂×#64515	C₂²	semidirect
20. C₂×#64515	C₂²	semidirect
27. C₁₉×C₄⋊C₈	C₂²	semidirect
28. C₁₉×C₂²⋊C₈	C₂²	semidirect
29. C₄×C₈×C₁₉	C₂²	semidirect
30. C₂²⋊(C₁₉⋊C₈)	C₂²	semidirect
34. (C₂×D₁₉)⋊C₈	C₂²	semidirect
35. (C₂×D₁₉)⋊C₈	C₂²	semidirect
36. #10597272	C₂²	semidirect
37. Dic₁₉⋊C₈	C₂²	semidirect
38. Dic₁₉⋊C₈	C₂²	semidirect
39. C₄⋊(C₁₉⋊C₈)	C₂²	semidirect
40. C₄×C₁₉⋊C₈	C₂²	semidirect
83. D₄×C₁₉	M₁₆	semidirect
138. C₈×D₄	D₁₉	semidirect
154. C₂²×C₈	C₂×D₁₉	semidirect
167. C₄⋊C₈	C₂×D₁₉	semidirect
174. C₂²⋊C₈	C₂×D₁₉	semidirect
177. C₄×C₈	C₂×D₁₉	semidirect
178. C₁₉	#2974	semidirect
223. D₄	#64515	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₄×D₁₉	2
#11926623	2
#11926621	2
#11926615	2
#11926612	2
D₄×C₁₉⋊C₈	2
#11926600	2
#11926595	2
#11926506	2
C₂×(C₂×D₁₉)⋊C₈	2
C₈×D₄×C₁₉	2
#2974	19