Subgroups — #18073101

Order 4864.

454 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926668	C₂	1	1	yes	maximal, characteristic
2	C₄×#10597281	C₂	1	1	yes	maximal, characteristic
3	#14561958	C₂	1	1	yes	maximal, characteristic
4	#14561916	C₂	1	1	yes	maximal, characteristic
5	M₁₆×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#14561485	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1283	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₃₈	C₂²	1	1	yes	characteristic
9	#11926668	—	2	1	no	—
10	#11926668	—	2	1	no	—
11	#11926668	—	2	1	no	—
12	#11926668	—	2	1	no	—
13	#11926597	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₉⋊C₈	C₄	1	2	yes	—
15	C₂×#10597281	C₄	1	2	yes	—
16	C₄×#64520	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₁₉	C₄	1	2	yes	—
19	#11926312	C₄	1	2	yes	—
20	#11926269	C₄	1	4	yes	—
21	#11926266	—	2	1	no	—
22	#11926265	—	2	1	no	—
23	C₈×C₁₉⋊C₈	—	2	1	no	—
24	C₈×C₁₉⋊C₈	—	2	1	no	—
25	C₁₉×#431	C₂²	1	1	yes	characteristic
26	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
28	C₂²×Dic₃₈	C₂×C₄	1	1	yes	characteristic
29	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×Dic₁₉⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₁₉	C₂×C₄	1	2	yes	—
35	C₂×#64520	C₂×C₄	1	2	yes	—
36	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₃₈	—	2	1	no	—
39	C₄×Dic₃₈	—	2	1	no	—
40	C₄×Dic₃₈	—	2	1	no	—
41	C₄×Dic₃₈	—	2	1	no	—
42	C₁₉×C₄⋊C₈	—	2	1	no	—
43	C₁₉×C₄⋊C₈	—	2	1	no	—
44	C₁₉×C₄⋊C₈	—	2	1	no	—
45	C₁₉×C₂²⋊C₈	C₂×C₄	1	2	yes	—
46	C₁₉×#97	—	2	1	no	—
47	C₄×C₈×C₁₉	—	2	1	no	—
48	C₄×C₈×C₁₉	—	2	1	no	—
49	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
50	#10597281	C₈	1	8	yes	complemented
51	Dic₁₉⋊C₈	—	2	2	no	—
52	Dic₁₉⋊C₈	—	2	2	no	—
53	C₈×Dic₁₉	—	2	2	no	—
54	C₈×Dic₁₉	—	2	2	no	—
55	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
56	#10597260	—	2	1	no	—
57	C₄×C₁₉⋊C₈	—	2	1	no	—
58	C₄×C₁₉⋊C₈	—	2	1	no	—
59	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
60	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
61	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
62	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×Dic₁₉	C₄²	1	2	yes	—
67	C₂²×Dic₁₉	C₄²	1	2	yes	—
68	C₂×Dic₃₈	—	2	1	no	—
69	C₂×Dic₃₈	—	2	1	no	—
70	C₂×Dic₃₈	—	2	2	no	—
71	C₂×Dic₃₈	C₂×C₈	1	4	yes	—
72	M₁₆×C₁₉	—	2	2	no	—
73	M₁₆×C₁₉	C₂×C₈	1	4	yes	—
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	—	2	2	no	—
76	C₂×C₈×C₁₉	—	2	2	no	—
77	C₂×C₈×C₁₉	—	4	2	no	—
78	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₉	#41	1	1	yes	characteristic
81	C₄²×C₁₉	#41	1	1	yes	characteristic
82	C₄⋊Dic₁₉	—	2	1	no	—
83	C₄⋊Dic₁₉	—	2	1	no	—
84	Dic₁₉⋊C₄	—	2	2	no	—
85	Dic₁₉⋊C₄	—	2	2	no	—
86	C₄×Dic₁₉	—	2	2	no	—
87	C₄×Dic₁₉	—	2	2	no	—
88	#64520	—	2	2	no	—
89	#64520	C₂×C₈	1	4	yes	—
90	C₂×C₁₉⋊C₈	—	2	2	no	—
91	C₂×C₁₉⋊C₈	—	2	2	no	—
92	C₂×C₁₉⋊C₈	—	2	2	no	—
93	C₂×C₁₉⋊C₈	—	4	1	no	—
94	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
98	C₈×#101	—	19	1	no	maximal, Sylow
99	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
100	C₂×C₄×C₁₉	—	2	1	no	—
101	C₂×C₄×C₁₉	—	2	1	no	—
102	C₂×C₄×C₁₉	—	4	1	no	—
103	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
105	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₁₉	#131	1	2	yes	—
111	C₂×Dic₁₉	—	2	2	no	—
112	C₂×Dic₁₉	—	2	2	no	—
113	C₂×Dic₁₉	—	2	2	no	—
114	C₂×Dic₁₉	—	4	2	no	—
115	C₂×Dic₁₉	C₄×C₈	1	4	yes	—
116	Dic₃₈	—	2	4	no	—
117	Dic₃₈	—	2	4	no	—
118	C₈×C₁₉	—	2	4	no	—
119	C₈×C₁₉	—	4	2	no	—
120	C₈×C₁₉	—	4	2	no	—
121	C₁₉⋊C₈	—	2	1	no	—
122	C₁₉⋊C₈	—	2	1	no	—
123	C₁₉⋊C₈	—	2	1	no	—
124	C₁₉⋊C₈	—	2	4	no	—
125	C₁₉⋊C₈	—	4	2	no	—
126	C₁₉⋊C₈	#101	1	2	yes	complemented
127	C₂×C₈×Q₈	—	19	1	no	—
128	C₄×#101	—	19	1	no	—
129	#1461	—	19	1	no	—
130	C₈×M₁₆	—	19	1	no	—
131	C₈×M₁₆	—	19	1	no	—
132	#1283	—	19	1	no	—
133	#1283	D₁₉	1	1	yes	characteristic
134	C₂²×C₁₉	—	2	1	no	—
135	C₂²×C₁₉	—	2	1	no	—
136	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
138	C₂²×C₁₉	#405	1	1	yes	characteristic
139	C₄×C₁₉	—	2	1	no	—
140	C₄×C₁₉	—	2	1	no	—
141	C₄×C₁₉	—	2	1	no	—
142	C₄×C₁₉	C₂×#101	1	1	yes	characteristic
143	C₄×C₁₉	#413	1	1	yes	characteristic
144	C₄×C₁₉	C₈×D₄	1	2	yes	—
145	C₄×C₁₉	C₈×D₄	1	2	yes	—
146	Dic₁₉	—	2	4	no	—
147	Dic₁₉	—	4	2	no	—
148	C₂×C₄×Q₈	—	19	1	no	—
149	C₈×Q₈	—	38	1	no	—
150	C₈×Q₈	—	38	1	no	—
151	C₈×Q₈	—	38	1	no	—
152	C₈×Q₈	—	38	1	no	—
153	#431	—	19	1	no	—
154	#431	C₂×D₁₉	1	1	yes	characteristic
155	C₂×C₄⋊C₈	—	19	2	no	—
156	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
157	C₂×#101	—	19	2	no	—
158	C₄×M₁₆	—	19	1	no	—
159	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
160	C₂×C₄×C₈	—	19	1	no	—
161	C₂×C₄×C₈	—	19	2	no	—
162	#343	—	19	2	no	—
163	#324	—	19	4	no	—
164	#322	—	38	1	no	—
165	#322	—	38	1	no	—
166	C₈²	—	38	1	no	—
167	C₈²	—	38	1	no	—
168	C₂×C₁₉	—	2	1	no	—
169	C₂×C₁₉	—	2	1	no	—
170	C₂×C₁₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
171	C₂×C₁₉	C₄×#101	1	1	yes	characteristic
172	C₂×C₁₉	#1757	1	1	yes	characteristic
173	C₂²×Q₈	—	19	1	no	—
174	C₂×M₁₆	—	19	2	no	—
175	C₂×M₁₆	C₄×D₁₉	1	2	yes	—
176	C₂²×C₈	—	19	1	no	—
177	C₂²×C₈	—	19	1	no	—
178	C₂²×C₈	C₄×D₁₉	1	2	yes	—
179	C₄×Q₈	—	38	1	no	—
180	C₄×Q₈	—	38	1	no	—
181	C₄×Q₈	—	38	1	no	—
182	C₄×Q₈	—	38	1	no	—
183	C₂×C₄⋊C₄	—	19	2	no	—
184	C₂×C₄⋊C₄	—	19	1	no	—
185	C₂×C₄²	—	19	2	no	—
186	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
187	C₄⋊C₈	—	2	1	no	—
188	C₄⋊C₈	—	2	1	no	—
189	C₄⋊C₈	—	2	1	no	—
190	C₄⋊C₈	—	38	2	no	—
191	C₄⋊C₈	—	38	2	no	—
192	C₄⋊C₈	—	38	1	no	—
193	#101	—	19	8	no	—
194	C₂²⋊C₈	—	19	2	no	—
195	C₂²⋊C₈	C₄×D₁₉	1	2	yes	—
196	#97	—	2	1	no	—
197	#97	—	38	1	no	—
198	C₄×C₈	—	2	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	19	1	no	—
201	C₄×C₈	—	19	1	no	—
202	C₄×C₈	—	19	1	no	—
203	C₄×C₈	—	19	1	no	—
204	C₄×C₈	—	38	2	no	—
205	C₄×C₈	—	38	2	no	—
206	C₄×C₈	—	38	1	no	—
207	C₄×C₈	—	38	1	no	—
208	C₁₉	C₈×#101	1	1	yes	characteristic, complemented, Sylow
209	C₂×Q₈	—	19	4	no	—
210	C₂×Q₈	—	38	1	no	—
211	C₂×Q₈	—	38	1	no	—
212	C₂×Q₈	—	38	2	no	—
213	C₂²×C₄	—	19	2	no	—
214	C₂²×C₄	—	19	2	no	—
215	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
218	M₁₆	—	2	2	no	—
219	M₁₆	—	19	4	no	—
220	M₁₆	—	38	2	no	—
221	M₁₆	C₈×D₁₉	1	4	yes	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	19	1	no	—
227	C₂×C₈	—	19	1	no	—
228	C₂×C₈	—	19	1	no	—
229	C₂×C₈	—	19	1	no	—
230	C₂×C₈	—	38	2	no	—
231	C₂×C₈	—	38	2	no	—
232	C₂×C₈	—	38	2	no	—
233	C₂×C₈	—	76	1	no	—
234	C₄⋊C₄	—	38	2	no	—
235	C₄⋊C₄	—	38	1	no	—
236	C₄⋊C₄	—	38	2	no	—
237	C₄⋊C₄	—	38	1	no	—
238	C₄²	—	38	2	no	—
239	C₄²	—	38	2	no	—
240	C₄²	D₄×D₁₉	1	1	yes	characteristic
241	C₄²	D₄×D₁₉	1	1	yes	characteristic
242	C₄²	#64543	1	1	yes	characteristic
243	C₄²	#64545	1	1	yes	characteristic
244	C₂³	C₄²×D₁₉	1	1	yes	characteristic
245	Q₈	—	38	4	no	—
246	Q₈	—	38	4	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	19	4	no	—
251	C₂×C₄	—	38	2	no	—
252	C₂×C₄	—	38	2	no	—
253	C₂×C₄	—	38	2	no	—
254	C₂×C₄	—	76	2	no	—
255	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
256	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
257	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
258	C₂×C₄	#10597338	1	1	yes	characteristic
259	C₂×C₄	#10597339	1	1	yes	characteristic
260	C₂×C₄	#10597339	1	1	yes	characteristic
261	C₂×C₄	C₂×C₈×D₁₉	1	2	yes	—
262	C₂×C₄	#10597353	1	2	yes	—
263	C₈	—	2	4	no	—
264	C₈	—	4	2	no	—
265	C₈	—	4	2	no	—
266	C₈	—	19	2	no	—
267	C₈	—	38	4	no	—
268	C₈	—	38	1	no	—
269	C₈	—	38	1	no	—
270	C₈	—	38	1	no	—
271	C₈	—	76	2	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	#11926391	1	1	yes	characteristic
275	C₂²	C₄×C₈×D₁₉	1	1	yes	characteristic
276	C₂²	#11926501	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	38	4	no	—
281	C₄	—	76	2	no	—
282	C₄	#101×D₁₉	1	1	yes	characteristic
283	C₄	#11926525	1	1	yes	characteristic
284	C₄	(C₂²⋊D₁₉)⋊C₈	1	2	yes	—
285	C₄	#11926609	1	2	yes	—
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	#14563405	1	1	yes	characteristic
289	C₂	#14563430	1	1	yes	characteristic
290	C₂	#14563434	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₁₉
commutator	C₂²×C₁₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1283	C₂
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
prime radical	C₂³×C₁₉	C₂×C₄²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1283	D₁₉
hypocenter	C₁₉	C₈×#101
LCS	G ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1283

Complements

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

N	H ≅ G/N	type
50. #10597281	C₈	semidirect
126. C₁₉⋊C₈	#101	semidirect
208. C₁₉	C₈×#101	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926668	2
C₄×#10597281	2
#14561958	2
#14561916	2
M₁₆×C₁₉⋊C₈	2
#14561485	2
C₁₉×#1283	2
C₈×#101	19