Subgroups — #18090647

Order 4864.

670 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572225	C₂	1	1	yes	maximal, characteristic
2	#14563373	C₂	1	2	yes	maximal
3	C₂×#11926304	C₂	1	2	yes	maximal
4	C₂×C₁₉⋊#338	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₉×#423	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂×#10597372	C₂²	1	2	yes	—
8	#11926586	—	2	4	no	—
9	#11926582	—	2	2	no	—
10	#11926312	—	2	4	no	—
11	#11926304	—	2	2	no	—
12	#11926304	—	2	2	no	—
13	C₁₉⋊#338	—	2	2	no	—
14	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#423	—	2	2	no	—
18	C₁₉×#423	C₄	1	4	yes	complemented
19	C₂×C₁₉×C₄⋊C₈	—	2	2	no	—
20	C₂³×C₄×C₁₉	—	2	1	no	—
21	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₁₉	—	2	2	no	—
23	C₂×M₁₆×C₁₉	—	2	2	no	—
24	C₂×M₁₆×C₁₉	—	2	1	no	—
25	C₂×M₁₆×C₁₉	—	2	2	no	—
26	C₂×M₁₆×C₁₉	—	2	2	no	—
27	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
28	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
30	C₂²×C₈×C₁₉	—	2	1	no	—
31	C₂²×C₈×C₁₉	D₄	1	2	yes	—
32	C₂×C₄²×C₁₉	—	2	1	no	—
33	C₂×C₄²×C₁₉	—	2	1	no	—
34	C₂×C₄²×C₁₉	—	2	1	no	—
35	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
39	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
40	#10597372	—	2	2	no	—
41	#10597372	—	2	2	no	—
42	#10597372	—	2	2	no	—
43	#10597372	—	2	2	no	—
44	C₂×C₄⋊Dic₁₉	D₄	1	4	yes	—
45	C₂×C₄×Dic₁₉	—	2	2	no	—
46	C₂×#64520	—	2	2	no	—
47	C₂×#64520	—	2	2	no	—
48	C₂×#64520	—	2	2	no	—
49	C₂×#64520	—	2	2	no	—
50	C₂×#64520	—	2	2	no	—
51	C₂×#64520	Q₈	1	4	yes	—
52	C₂²×C₁₉⋊C₈	—	2	2	no	—
53	C₁₉×C₄⋊C₈	—	2	8	no	—
54	#10597275	—	2	4	no	—
55	#10597274	—	2	4	no	—
56	C₂⁴×C₁₉	#41	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	—	2	1	no	—
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	—	2	1	no	—
60	C₂²×C₄×C₁₉	—	2	2	no	—
61	C₂²×C₄×C₁₉	—	4	4	no	—
62	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	#41	1	2	yes	—
71	C₂²×C₄×C₁₉	#41	1	2	yes	—
72	C₂²×Dic₁₉	—	2	4	no	—
73	M₁₆×C₁₉	—	2	4	no	—
74	M₁₆×C₁₉	—	2	4	no	—
75	M₁₆×C₁₉	—	2	4	no	—
76	M₁₆×C₁₉	—	4	2	no	—
77	C₂×C₈×C₁₉	—	2	2	no	—
78	C₂×C₈×C₁₉	—	2	2	no	—
79	C₂×C₈×C₁₉	—	2	2	no	—
80	C₂×C₈×C₁₉	—	4	1	no	—
81	C₂×C₈×C₁₉	—	4	1	no	—
82	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
83	C₄²×C₁₉	—	2	2	no	—
84	C₄²×C₁₉	—	2	1	no	—
85	C₄²×C₁₉	—	2	1	no	—
86	C₄²×C₁₉	—	2	2	no	—
87	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
88	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
89	C₂²⋊Dic₁₉	—	4	2	no	—
90	C₂²⋊Dic₁₉	—	4	2	no	—
91	C₄⋊Dic₁₉	—	2	4	no	—
92	C₄⋊Dic₁₉	—	2	4	no	—
93	C₄×Dic₁₉	—	4	2	no	—
94	C₄×Dic₁₉	—	4	2	no	—
95	#64520	—	2	4	no	—
96	#64520	—	2	4	no	—
97	#64520	—	4	2	no	—
98	#64520	—	4	2	no	—
99	C₂×C₁₉⋊C₈	—	2	4	no	—
100	C₂×C₁₉⋊C₈	—	4	2	no	—
101	C₂×C₁₉⋊C₈	—	4	2	no	—
102	#11243	—	19	1	no	maximal, Sylow
103	C₂³×C₁₉	—	4	2	no	—
104	C₂³×C₁₉	#117	1	1	yes	characteristic
105	C₂³×C₁₉	#121	1	1	yes	characteristic
106	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂³×C₁₉	#123	1	2	yes	—
108	C₂³×C₁₉	#125	1	2	yes	—
109	C₂×C₄×C₁₉	—	2	4	no	—
110	C₂×C₄×C₁₉	—	2	2	no	—
111	C₂×C₄×C₁₉	—	2	2	no	—
112	C₂×C₄×C₁₉	—	4	4	no	—
113	C₂×C₄×C₁₉	—	4	4	no	—
114	C₂×C₄×C₁₉	—	4	2	no	—
115	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
120	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#121	1	2	yes	—
122	C₂×C₄×C₁₉	#121	1	2	yes	—
123	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#125	1	2	yes	—
130	C₂×C₄×C₁₉	#126	1	2	yes	—
131	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
133	C₂×Dic₁₉	—	4	4	no	—
134	C₂×Dic₁₉	—	8	4	no	—
135	C₈×C₁₉	—	2	4	no	—
136	C₈×C₁₉	—	4	2	no	—
137	C₁₉⋊C₈	—	4	4	no	—
138	#2911	—	19	1	no	—
139	C₂×#423	D₁₉	1	1	yes	characteristic
140	#1742	—	19	2	no	—
141	C₂×#339	—	19	2	no	—
142	C₂×#338	—	19	1	no	—
143	C₂²×C₁₉	—	2	4	no	—
144	C₂²×C₁₉	—	2	2	no	—
145	C₂²×C₁₉	—	4	1	no	—
146	C₂²×C₁₉	—	4	1	no	—
147	C₂²×C₁₉	—	4	2	no	—
148	C₂²×C₁₉	#380	1	1	yes	characteristic
149	C₂²×C₁₉	#387	1	2	yes	—
150	C₂²×C₁₉	#394	1	1	yes	characteristic
151	C₂²×C₁₉	#398	1	1	yes	characteristic
152	C₂²×C₁₉	#398	1	2	yes	—
153	C₄×C₁₉	—	2	4	no	—
154	C₄×C₁₉	—	4	2	no	—
155	C₄×C₁₉	#386	1	1	yes	characteristic
156	C₄×C₁₉	#389	1	2	yes	—
157	C₄×C₁₉	#391	1	1	yes	characteristic
158	C₄×C₁₉	#395	1	1	yes	characteristic
159	C₄×C₁₉	#396	1	2	yes	—
160	C₄×C₁₉	#400	1	1	yes	characteristic
161	Dic₁₉	—	8	4	no	—
162	C₂²×M₁₆	—	19	2	no	—
163	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
164	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
165	C₂×#117	—	19	2	no	—
166	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
167	#428	—	38	4	no	—
168	#427	—	38	2	no	—
169	#423	—	2	2	no	—
170	#423	Dic₁₉	1	4	yes	complemented
171	C₂×C₄⋊C₈	—	2	2	no	—
172	#343	—	38	4	no	—
173	#339	—	38	2	no	—
174	#339	—	38	2	no	—
175	#338	—	38	2	no	—
176	C₂×C₁₉	—	2	1	no	—
177	C₂×C₁₉	—	2	2	no	—
178	C₂×C₁₉	—	2	1	no	—
179	C₂×C₁₉	—	2	2	no	—
180	C₂×C₁₉	#1441	1	1	yes	characteristic
181	C₂×C₁₉	#2050	1	2	yes	—
182	C₂³×C₄	—	2	1	no	—
183	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	1	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	19	4	no	—
190	C₂×M₁₆	—	38	2	no	—
191	C₂×M₁₆	—	38	2	no	—
192	C₂×M₁₆	—	38	2	no	—
193	C₂×M₁₆	—	38	2	no	—
194	C₂×M₁₆	—	38	2	no	—
195	C₂×M₁₆	C₂×Dic₁₉	1	2	yes	—
196	C₂×M₁₆	C₂×Dic₁₉	1	2	yes	—
197	C₂×M₁₆	C₂²⋊D₁₉	1	2	yes	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	38	2	no	—
200	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
201	#117	—	38	2	no	—
202	#117	—	38	2	no	—
203	#117	—	38	2	no	—
204	#117	—	38	2	no	—
205	C₂×C₄⋊C₄	—	19	4	no	—
206	C₂×C₂²⋊C₄	—	38	2	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	—	38	2	no	—
211	C₂×C₄²	Dic₃₈	1	2	yes	—
212	C₂×C₄²	D₇₆	1	1	yes	characteristic
213	C₂×C₄²	D₇₆	1	1	yes	characteristic
214	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
215	#107	—	38	4	no	—
216	#106	—	38	4	no	—
217	C₄⋊C₈	—	2	8	no	—
218	C₁₉	#11243	1	1	yes	characteristic, complemented, Sylow
219	C₂⁴	#64541	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	—	38	4	no	—
226	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
228	C₂²×C₄	#64541	1	1	yes	characteristic
229	C₂²×C₄	D₄×D₁₉	1	2	yes	—
230	C₂²×C₄	#64543	1	2	yes	—
231	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
232	C₂²×C₄	#64545	1	2	yes	—
233	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	4	2	no	—
240	M₁₆	—	38	4	no	—
241	M₁₆	—	38	4	no	—
242	M₁₆	—	76	2	no	—
243	M₁₆	—	76	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	38	4	no	—
250	C₂×C₈	—	76	2	no	—
251	C₂×C₈	—	76	2	no	—
252	C₂×C₈	C₂²⋊Dic₁₉	1	4	yes	—
253	C₄⋊C₄	—	38	4	no	—
254	C₄⋊C₄	—	38	4	no	—
255	C₂²⋊C₄	—	76	2	no	—
256	C₂²⋊C₄	—	76	2	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	76	2	no	—
262	C₄²	—	76	2	no	—
263	C₄²	C₄⋊Dic₁₉	1	2	yes	—
264	C₄²	C₄⋊Dic₁₉	1	2	yes	—
265	C₂³	—	4	2	no	—
266	C₂³	#10597335	1	2	yes	—
267	C₂³	#10597340	1	2	yes	—
268	C₂³	#10597370	1	1	yes	characteristic
269	C₂³	#10597372	1	1	yes	characteristic
270	C₂³	#10597376	1	1	yes	characteristic
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	76	4	no	—
278	C₂×C₄	—	152	4	no	—
279	C₂×C₄	#10597334	1	2	yes	—
280	C₂×C₄	#10597336	1	2	yes	—
281	C₂×C₄	#10597341	1	2	yes	—
282	C₂×C₄	#10597342	1	2	yes	—
283	C₂×C₄	#10597343	1	2	yes	—
284	C₂×C₄	#10597344	1	2	yes	—
285	C₂×C₄	#10597370	1	1	yes	characteristic
286	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
287	C₂×C₄	#10597376	1	1	yes	characteristic
288	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
289	C₂×C₄	#10597381	1	1	yes	characteristic
290	C₂×C₄	#10597383	1	1	yes	characteristic
291	C₂×C₄	#10597385	1	2	yes	—
292	C₂×C₄	#10597390	1	1	yes	characteristic
293	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
294	C₂×C₄	#10597392	1	2	yes	—
295	C₂×C₄	#10597393	1	1	yes	characteristic
296	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
297	C₈	—	2	4	no	—
298	C₈	—	4	2	no	—
299	C₈	—	76	4	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	2	no	—
305	C₂²	#11926404	1	1	yes	characteristic
306	C₂²	#11926430	1	2	yes	—
307	C₂²	#11926461	1	1	yes	characteristic
308	C₂²	#11926474	1	1	yes	characteristic
309	C₂²	#11926477	1	2	yes	—
310	C₄	—	2	4	no	—
311	C₄	—	4	2	no	—
312	C₄	—	152	4	no	—
313	C₄	#11926424	1	1	yes	characteristic
314	C₄	#11926443	1	2	yes	—
315	C₄	#11926451	1	1	yes	characteristic
316	C₄	#11926464	1	1	yes	characteristic
317	C₄	#11926466	1	2	yes	—
318	C₄	#11926483	1	1	yes	characteristic
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	2	no	—
323	C₂	#14561888	1	1	yes	characteristic
324	C₂	#14564922	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10597393
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#423	C₂
socle	C₂²×C₁₉	#394
prime radical	C₂⁴×C₁₉	#41
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#423	D₁₉
hypocenter	C₁₉	#11243
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
18. C₁₉×#423	C₄	semidirect
170. #423	Dic₁₉	semidirect
218. C₁₉	#11243	semidirect

Maximal subgroups

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

subgroup	index
#14572225	2
#14563373	2
C₂×#11926304	2
C₂×C₁₉⋊#338	2
C₂×C₁₉×#423	2
#11243	19