Subgroups — #18072098

Order 4864.

710 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×C₄×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#14566033	C₂	1	1	yes	maximal, characteristic
3	#14561474	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₉×#432	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₁₉⋊C₈	—	2	1	no	—
6	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
7	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
8	C₂×C₄×C₁₉⋊C₈	C₄	1	4	yes	complemented
9	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
10	#11926412	—	2	1	no	—
11	#11926410	C₄	1	2	yes	—
12	#11926407	—	2	1	no	—
13	#11926404	C₄	1	2	yes	—
14	C₄×C₄⋊Dic₁₉	C₂²	1	2	yes	—
15	C₄×C₂²⋊Dic₁₉	—	2	1	no	—
16	#11926390	—	2	1	no	—
17	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#432	—	2	2	no	—
19	C₁₉×#432	C₄	1	4	yes	complemented
20	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₁₉×C₂²⋊C₈	—	2	1	no	—
22	C₂×C₁₉×#97	—	2	1	no	—
23	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	—	2	2	no	—
27	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	D₄	1	2	yes	—
33	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
34	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
36	C₂×C₄⋊Dic₁₉	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₁₉	—	2	2	no	—
38	C₂²×C₁₉⋊C₈	—	2	8	no	—
39	C₂²×C₁₉⋊C₈	—	2	4	no	—
40	C₂²×C₁₉⋊C₈	—	2	2	no	—
41	C₁₉×C₄⋊C₈	—	2	2	no	—
42	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
43	C₁₉×C₂²⋊C₈	—	2	4	no	—
44	C₁₉×#97	—	2	4	no	—
45	C₁₉⋊#95	—	2	2	no	—
46	C₁₉⋊#95	—	2	2	no	—
47	C₁₉⋊#95	—	2	2	no	—
48	C₁₉⋊#95	—	2	2	no	—
49	C₄×C₁₉⋊C₈	—	2	4	no	—
50	C₄×C₁₉⋊C₈	—	2	2	no	—
51	C₄×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
52	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	—	2	4	no	—
54	C₂²×C₄×C₁₉	—	2	4	no	—
55	C₂²×C₄×C₁₉	—	2	4	no	—
56	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
57	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₁₉	—	2	4	no	—
69	C₂×C₈×C₁₉	—	2	4	no	—
70	C₂×C₈×C₁₉	—	4	4	no	—
71	C₄²×C₁₉	C₄²	1	2	yes	—
72	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
75	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
76	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
77	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
81	C₂²⋊Dic₁₉	—	4	4	no	—
82	C₄⋊Dic₁₉	—	2	8	no	—
83	C₄×Dic₁₉	—	4	4	no	—
84	C₂×C₁₉⋊C₈	—	2	16	no	—
85	C₂×C₁₉⋊C₈	—	2	8	no	—
86	C₂×C₁₉⋊C₈	—	2	4	no	—
87	#7898	—	19	1	no	maximal, Sylow
88	C₂³×C₁₉	—	2	2	no	—
89	C₂³×C₁₉	—	2	2	no	—
90	C₂³×C₁₉	#95	1	2	yes	—
91	C₂³×C₁₉	#104	1	4	yes	—
92	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₉	—	2	4	no	—
94	C₂×C₄×C₁₉	—	2	4	no	—
95	C₂×C₄×C₁₉	—	2	2	no	—
96	C₂×C₄×C₁₉	—	2	2	no	—
97	C₂×C₄×C₁₉	—	2	2	no	—
98	C₂×C₄×C₁₉	—	2	2	no	—
99	C₂×C₄×C₁₉	—	2	4	no	—
100	C₂×C₄×C₁₉	—	2	4	no	—
101	C₂×C₄×C₁₉	—	2	4	no	—
102	C₂×C₄×C₁₉	—	2	4	no	—
103	C₂×C₄×C₁₉	#95	1	2	yes	—
104	C₂×C₄×C₁₉	#95	1	2	yes	—
105	C₂×C₄×C₁₉	#95	1	2	yes	—
106	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	#131	1	4	yes	—
113	C₂×C₄×C₁₉	#135	1	4	yes	—
114	C₂×Dic₁₉	—	4	4	no	—
115	C₂×Dic₁₉	—	4	4	no	—
116	C₂×Dic₁₉	—	4	4	no	—
117	C₈×C₁₉	—	4	4	no	—
118	C₁₉⋊C₈	—	2	8	no	—
119	C₂×#432	D₁₉	1	1	yes	characteristic
120	C₂²×C₄×C₈	—	19	1	no	—
121	#2283	—	19	1	no	—
122	#1279	—	19	4	no	—
123	C₂²×C₁₉	—	2	2	no	—
124	C₂²×C₁₉	—	2	2	no	—
125	C₂²×C₁₉	—	2	8	no	—
126	C₂²×C₁₉	#339	1	4	yes	—
127	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
128	C₂²×C₁₉	#408	1	2	yes	—
129	C₂²×C₁₉	#416	1	2	yes	—
130	C₂²×C₁₉	C₂×#104	1	2	yes	—
131	C₄×C₁₉	—	2	2	no	—
132	C₄×C₁₉	—	2	1	no	—
133	C₄×C₁₉	—	2	2	no	—
134	C₄×C₁₉	—	2	1	no	—
135	C₄×C₁₉	—	2	2	no	—
136	C₄×C₁₉	#405	1	2	yes	—
137	C₄×C₁₉	#416	1	2	yes	—
138	C₄×C₁₉	#424	1	2	yes	—
139	C₄×C₁₉	#427	1	2	yes	—
140	Dic₁₉	—	8	4	no	—
141	C₂³×C₈	—	38	1	no	—
142	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
143	#432	—	2	2	no	—
144	#432	Dic₁₉	1	4	yes	complemented
145	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
146	C₂×C₂²⋊C₈	—	2	1	no	—
147	C₂×#97	—	2	1	no	—
148	C₂×C₄×C₈	—	19	4	no	—
149	C₂×C₄×C₈	—	19	2	no	—
150	C₂×C₄×C₈	—	38	1	no	—
151	C₂×C₄×C₈	—	38	2	no	—
152	#383	—	38	1	no	—
153	#382	—	19	2	no	—
154	#381	—	38	1	no	—
155	#380	—	19	2	no	—
156	C₄×C₄⋊C₄	—	19	2	no	—
157	C₄×C₂²⋊C₄	—	38	1	no	—
158	#376	—	38	1	no	—
159	C₂×C₁₉	—	2	4	no	—
160	C₂×C₁₉	#1494	1	4	yes	—
161	C₂×C₁₉	#1729	1	1	yes	characteristic
162	C₂×C₁₉	C₂×#339	1	1	yes	characteristic
163	C₂×C₁₉	#1735	1	1	yes	characteristic
164	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
165	C₂³×C₄	D₇₆	1	1	yes	characteristic
166	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	38	8	no	—
169	C₂²×C₈	—	38	4	no	—
170	C₂²×C₈	—	38	2	no	—
171	C₂×C₄⋊C₄	—	19	4	no	—
172	C₂×C₂²⋊C₄	—	38	2	no	—
173	C₂×C₄²	—	38	2	no	—
174	C₂×C₄²	Dic₃₈	1	2	yes	—
175	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
176	C₂×C₄²	C₄×D₁₉	1	2	yes	—
177	C₂×C₄²	D₇₆	1	1	yes	characteristic
178	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
179	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
181	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	C₂×Dic₁₉	1	4	yes	—
184	C₂²⋊C₈	—	2	4	no	—
185	#97	—	2	4	no	—
186	C₄×C₈	—	19	4	no	—
187	C₄×C₈	—	38	4	no	—
188	C₄×C₈	—	38	2	no	—
189	#95	—	38	2	no	—
190	#95	—	38	2	no	—
191	#95	—	38	2	no	—
192	#95	—	38	2	no	—
193	C₁₉	#7898	1	1	yes	characteristic, complemented, Sylow
194	C₂⁴	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	38	4	no	—
199	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
200	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
201	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
202	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
203	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
204	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
205	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
206	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
207	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
208	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	4	4	no	—
213	C₂×C₈	—	38	8	no	—
214	C₂×C₈	—	38	16	no	—
215	C₂×C₈	—	38	4	no	—
216	C₄⋊C₄	—	38	8	no	—
217	C₂²⋊C₄	—	76	4	no	—
218	C₄²	—	76	4	no	—
219	C₄²	C₄×Dic₁₉	1	2	yes	—
220	C₄²	Dic₁₉⋊C₄	1	2	yes	—
221	C₄²	Dic₁₉⋊C₄	1	2	yes	—
222	C₄²	C₄⋊Dic₁₉	1	2	yes	—
223	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
224	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
225	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
226	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
227	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
228	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	#10597282	1	4	yes	—
232	C₂³	C₁₉⋊#95	1	2	yes	—
233	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	76	4	no	—
245	C₂×C₄	—	76	4	no	—
246	C₂×C₄	—	76	4	no	—
247	C₂×C₄	C₁₉⋊#95	1	2	yes	—
248	C₂×C₄	C₁₉⋊#95	1	2	yes	—
249	C₂×C₄	C₁₉⋊#95	1	2	yes	—
250	C₂×C₄	#10597353	1	4	yes	—
251	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
252	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
253	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
255	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
256	C₂×C₄	#10597396	1	4	yes	—
257	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
258	C₈	—	4	4	no	—
259	C₈	—	38	8	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	8	no	—
263	C₂²	#11926305	1	4	yes	—
264	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
265	C₂²	#11926512	1	2	yes	—
266	C₂²	#11926538	1	2	yes	—
267	C₂²	C₂×#10597282	1	2	yes	—
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	2	no	—
273	C₄	—	152	4	no	—
274	C₄	#11926502	1	2	yes	—
275	C₄	#11926540	1	2	yes	—
276	C₄	#11926572	1	2	yes	—
277	C₄	#11926580	1	2	yes	—
278	C₂	—	2	4	no	—
279	C₂	#14562093	1	4	yes	—
280	C₂	#14563333	1	1	yes	characteristic
281	C₂	C₂×#11926305	1	1	yes	characteristic
282	C₂	#14563351	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#432	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#432	D₁₉
hypocenter	C₁₉	#7898
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #14561474	C₂	semidirect
8. C₂×C₄×C₁₉⋊C₈	C₄	semidirect
19. C₁₉×#432	C₄	semidirect
144. #432	Dic₁₉	semidirect
193. C₁₉	#7898	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₁₉⋊C₈	2
#14566033	2
#14561474	2
C₂×C₁₉×#432	2
#7898	19