Subgroups — #18075932

Order 4864.

588 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14566042	C₂	1	1	yes	maximal, characteristic
2	#14563703	C₂	1	1	yes	maximal, characteristic
3	#14561514	C₂	1	4	yes	maximal
4	C₁₉×#1434	C₂	1	1	yes	maximal, characteristic
5	C₂×#10597372	—	2	1	no	—
6	C₂×C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	#11926414	C₄	1	1	yes	characteristic, complemented
8	#11926414	C₄	1	1	yes	characteristic, complemented
9	#11926410	—	2	1	no	—
10	#11926404	—	2	1	no	—
11	#11926404	C₂²	1	2	yes	—
12	C₄×C₄⋊Dic₁₉	—	2	1	no	—
13	C₄×C₄⋊Dic₁₉	C₄	1	2	yes	complemented
14	#11926299	—	2	1	no	—
15	#11926299	—	2	1	no	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×#95	—	2	1	no	—
18	C₂×C₁₉×#95	—	2	1	no	—
19	C₂×C₁₉×#95	C₂²	1	2	yes	—
20	C₂³×C₄×C₁₉	—	2	2	no	—
21	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₁₉	—	2	4	no	—
25	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
29	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
30	#10597372	—	4	4	no	—
31	C₂×C₄⋊Dic₁₉	—	2	1	no	—
32	C₂×C₄⋊Dic₁₉	—	2	1	no	—
33	C₂×C₄⋊Dic₁₉	—	2	1	no	—
34	C₂×C₄⋊Dic₁₉	—	2	1	no	—
35	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₁₉	—	2	2	no	—
38	C₂²×C₁₉⋊C₈	—	2	2	no	—
39	C₁₉×#95	—	2	4	no	—
40	C₁₉×#95	—	2	4	no	—
41	C₁₉×#95	—	4	2	no	—
42	C₁₉⋊#95	—	2	2	no	—
43	C₁₉⋊#95	—	2	2	no	—
44	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
45	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	—	2	2	no	—
47	C₂²×C₄×C₁₉	—	2	2	no	—
48	C₂²×C₄×C₁₉	—	2	2	no	—
49	C₂²×C₄×C₁₉	—	2	2	no	—
50	C₂²×C₄×C₁₉	—	2	4	no	—
51	C₂²×C₄×C₁₉	—	4	1	no	—
52	C₂²×C₄×C₁₉	—	4	1	no	—
53	C₂²×C₄×C₁₉	—	4	1	no	—
54	C₂²×C₄×C₁₉	—	4	2	no	—
55	C₂²×C₄×C₁₉	—	4	2	no	—
56	C₂²×C₄×C₁₉	—	4	2	no	—
57	C₂²×C₄×C₁₉	—	4	2	no	—
58	C₂²×C₄×C₁₉	—	4	2	no	—
59	C₂²×C₄×C₁₉	—	4	2	no	—
60	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
66	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	#41	1	4	yes	—
69	C₂²×Dic₁₉	—	2	2	no	—
70	C₂²×Dic₁₉	—	2	2	no	—
71	C₄²×C₁₉	—	2	2	no	—
72	C₄²×C₁₉	—	2	2	no	—
73	C₄²×C₁₉	—	2	2	no	—
74	C₄²×C₁₉	—	2	2	no	—
75	C₂²⋊Dic₁₉	—	4	4	no	—
76	C₄⋊Dic₁₉	—	4	2	no	—
77	C₄⋊Dic₁₉	—	4	2	no	—
78	C₄⋊Dic₁₉	—	4	2	no	—
79	C₄⋊Dic₁₉	—	4	2	no	—
80	C₄⋊Dic₁₉	—	4	2	no	—
81	C₄⋊Dic₁₉	—	4	2	no	—
82	C₄×Dic₁₉	—	4	4	no	—
83	C₂×C₁₉⋊C₈	—	4	2	no	—
84	C₂×C₁₉⋊C₈	—	4	2	no	—
85	C₂×C₁₉⋊C₈	—	4	2	no	—
86	#8717	—	19	1	no	maximal, Sylow
87	C₂³×C₁₉	—	2	1	no	—
88	C₂³×C₁₉	—	2	1	no	—
89	C₂³×C₁₉	—	2	1	no	—
90	C₂³×C₁₉	—	2	1	no	—
91	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂³×C₁₉	#123	1	2	yes	—
93	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
94	C₂³×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
95	C₂³×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
96	C₂×C₄×C₁₉	—	2	1	no	—
97	C₂×C₄×C₁₉	—	2	1	no	—
98	C₂×C₄×C₁₉	—	2	1	no	—
99	C₂×C₄×C₁₉	—	2	1	no	—
100	C₂×C₄×C₁₉	—	2	1	no	—
101	C₂×C₄×C₁₉	—	2	1	no	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	1	no	—
104	C₂×C₄×C₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	4	no	—
107	C₂×C₄×C₁₉	—	2	4	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	4	4	no	—
112	C₂×C₄×C₁₉	—	4	4	no	—
113	C₂×C₄×C₁₉	—	4	4	no	—
114	C₂×C₄×C₁₉	—	4	4	no	—
115	C₂×C₄×C₁₉	—	4	4	no	—
116	C₂×C₄×C₁₉	—	8	2	no	—
117	C₂×C₄×C₁₉	—	8	2	no	—
118	C₂×C₄×C₁₉	—	8	2	no	—
119	C₂×C₄×C₁₉	—	8	2	no	—
120	C₂×C₄×C₁₉	#103	1	2	yes	—
121	C₂×C₄×C₁₉	#103	1	2	yes	—
122	C₂×C₄×C₁₉	#104	1	2	yes	—
123	C₂×C₄×C₁₉	#117	1	2	yes	—
124	C₂×C₄×C₁₉	#123	1	2	yes	—
125	C₂×Dic₁₉	—	4	2	no	—
126	C₂×Dic₁₉	—	4	2	no	—
127	C₂×Dic₁₉	—	4	2	no	—
128	C₂×Dic₁₉	—	4	2	no	—
129	C₂×Dic₁₉	—	4	2	no	—
130	C₂×Dic₁₉	—	4	2	no	—
131	C₁₉⋊C₈	—	8	2	no	—
132	#2285	—	19	1	no	—
133	#1817	—	19	1	no	—
134	#1434	D₁₉	1	1	yes	characteristic
135	#1294	—	19	4	no	—
136	C₂²×C₁₉	—	2	2	no	—
137	C₂²×C₁₉	—	2	2	no	—
138	C₂²×C₁₉	—	2	1	no	—
139	C₂²×C₁₉	—	2	1	no	—
140	C₂²×C₁₉	—	2	2	no	—
141	C₂²×C₁₉	—	2	2	no	—
142	C₂²×C₁₉	—	2	1	no	—
143	C₂²×C₁₉	—	2	1	no	—
144	C₂²×C₁₉	#381	1	1	yes	characteristic
145	C₂²×C₁₉	#410	1	1	yes	characteristic
146	C₂²×C₁₉	#410	1	1	yes	characteristic
147	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
148	C₂²×C₁₉	#418	1	1	yes	characteristic
149	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
150	C₂²×C₁₉	#421	1	1	yes	characteristic
151	C₂²×C₁₉	#481	1	2	yes	—
152	C₂²×C₁₉	#484	1	2	yes	—
153	C₄×C₁₉	—	2	2	no	—
154	C₄×C₁₉	—	2	2	no	—
155	C₄×C₁₉	—	2	2	no	—
156	C₄×C₁₉	—	2	2	no	—
157	C₄×C₁₉	—	2	2	no	—
158	C₄×C₁₉	—	4	1	no	—
159	C₄×C₁₉	—	8	4	no	—
160	Dic₁₉	—	8	2	no	—
161	Dic₁₉	—	8	2	no	—
162	C₂×#117	—	38	1	no	—
163	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
164	C₂×C₂²⋊C₈	—	19	2	no	—
165	#384	—	19	1	no	—
166	#384	—	19	1	no	—
167	#382	—	38	1	no	—
168	#380	—	19	2	no	—
169	#380	—	38	1	no	—
170	C₄×C₄⋊C₄	—	19	2	no	—
171	C₄×C₄⋊C₄	—	38	1	no	—
172	C₂×#95	—	2	1	no	—
173	C₂×#95	—	2	1	no	—
174	C₂×#95	C₂×D₁₉	1	2	yes	—
175	#336	—	38	1	no	—
176	#336	—	38	1	no	—
177	C₂×C₁₉	—	2	2	no	—
178	C₂×C₁₉	—	2	2	no	—
179	C₂×C₁₉	#1503	1	1	yes	characteristic
180	C₂×C₁₉	#1505	1	1	yes	characteristic
181	C₂×C₁₉	#1514	1	1	yes	characteristic
182	C₂×C₁₉	#1518	1	1	yes	characteristic
183	C₂×C₁₉	#1820	1	1	yes	characteristic
184	C₂×C₁₉	#1824	1	1	yes	characteristic
185	C₂×C₁₉	#1826	1	1	yes	characteristic
186	C₂³×C₄	—	2	2	no	—
187	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
188	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
189	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
190	C₂²×C₈	—	38	2	no	—
191	#117	—	76	4	no	—
192	C₂×C₄⋊C₄	—	19	2	no	—
193	C₂×C₄⋊C₄	—	19	2	no	—
194	C₂×C₄⋊C₄	—	38	1	no	—
195	C₂×C₄⋊C₄	—	38	1	no	—
196	C₂×C₄⋊C₄	—	38	1	no	—
197	C₂×C₄⋊C₄	—	38	1	no	—
198	C₂×C₂²⋊C₄	—	38	2	no	—
199	C₂×C₄²	—	2	4	no	—
200	C₂×C₄²	—	38	2	no	—
201	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
202	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
203	C₂×C₄²	D₇₆	1	1	yes	characteristic
204	C₂×C₄²	D₇₆	1	1	yes	characteristic
205	C₂²⋊C₈	—	38	4	no	—
206	#95	—	2	4	no	—
207	#95	—	2	4	no	—
208	#95	—	4	2	no	—
209	#95	—	38	2	no	—
210	#95	—	38	2	no	—
211	C₁₉	#8717	1	1	yes	characteristic, complemented, Sylow
212	C₂⁴	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	4	1	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	—	38	2	no	—
228	C₂²×C₄	—	38	2	no	—
229	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
230	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
231	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
232	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
233	C₂²×C₄	#64526	1	2	yes	—
234	C₂²×C₄	#64528	1	2	yes	—
235	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
237	C₂²×C₄	#64543	1	4	yes	—
238	C₂×C₈	—	76	2	no	—
239	C₂×C₈	—	76	2	no	—
240	C₂×C₈	—	76	2	no	—
241	C₄⋊C₄	—	76	2	no	—
242	C₄⋊C₄	—	76	2	no	—
243	C₄⋊C₄	—	76	2	no	—
244	C₄⋊C₄	—	76	2	no	—
245	C₄⋊C₄	—	76	2	no	—
246	C₄⋊C₄	—	76	2	no	—
247	C₂²⋊C₄	—	76	4	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	76	4	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
258	C₂³	C₂×#64526	1	1	yes	characteristic
259	C₂³	#10597381	1	2	yes	—
260	C₂³	C₂×#64528	1	1	yes	characteristic
261	C₂³	#10597395	1	2	yes	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	8	2	no	—
283	C₂×C₄	—	8	2	no	—
284	C₂×C₄	—	8	2	no	—
285	C₂×C₄	—	8	2	no	—
286	C₂×C₄	—	76	2	no	—
287	C₂×C₄	—	76	2	no	—
288	C₂×C₄	—	76	2	no	—
289	C₂×C₄	—	76	2	no	—
290	C₂×C₄	—	76	2	no	—
291	C₂×C₄	—	76	2	no	—
292	C₂×C₄	#10597262	1	2	yes	—
293	C₂×C₄	#10597267	1	2	yes	—
294	C₂×C₄	#10597267	1	2	yes	—
295	C₂×C₄	#10597323	1	2	yes	—
296	C₂×C₄	#10597332	1	2	yes	—
297	C₈	—	152	2	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	2	no	—
306	C₂²	#11926405	1	1	yes	characteristic
307	C₂²	#11926517	1	1	yes	characteristic
308	C₂²	#11926517	1	1	yes	characteristic
309	C₂²	C₂×#10597267	1	1	yes	characteristic
310	C₂²	#11926548	1	1	yes	characteristic
311	C₂²	C₂×#10597262	1	1	yes	characteristic
312	C₂²	#11926562	1	1	yes	characteristic
313	C₂²	#11926860	1	2	yes	—
314	C₂²	#11926877	1	2	yes	—
315	C₄	—	2	2	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	1	no	—
321	C₄	—	8	4	no	—
322	C₄	—	152	2	no	—
323	C₄	—	152	2	no	—
324	C₂	—	2	2	no	—
325	C₂	—	2	2	no	—
326	C₂	#14562133	1	1	yes	characteristic
327	C₂	#14562142	1	1	yes	characteristic
328	C₂	#14562181	1	1	yes	characteristic
329	C₂	#14562198	1	1	yes	characteristic
330	C₂	#14563710	1	1	yes	characteristic
331	C₂	#14563723	1	1	yes	characteristic
332	C₂	#14563735	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#1434	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#1434	D₁₉
hypocenter	C₁₉	#8717
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1434

Complements

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

N	H ≅ G/N	type
7. #11926414	C₄	semidirect
8. #11926414	C₄	semidirect
13. C₄×C₄⋊Dic₁₉	C₄	semidirect
211. C₁₉	#8717	semidirect

Maximal subgroups

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

subgroup	index
#14566042	2
#14563703	2
#14561514	2
C₁₉×#1434	2
#8717	19