Subgroups — #18090657

Order 4864.

624 conjugacy classes of subgroups.

The subgroup lattice has 304 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	C₂×#11926316	C₂	1	2	yes	maximal
3	C₂×C₁₉⋊#338	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926300	C₂	1	2	yes	maximal
5	C₁₉×#2905	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	#11926316	—	2	4	no	—
11	C₁₉⋊#338	—	2	2	no	—
12	#11926300	—	2	4	no	—
13	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
16	C₁₉×#424	—	2	2	no	—
17	C₁₉×#424	—	2	2	no	—
18	C₁₉×#423	C₄	1	4	yes	complemented
19	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₁₉	—	2	2	no	—
21	C₂×M₁₆×C₁₉	—	2	2	no	—
22	C₂×M₁₆×C₁₉	—	2	1	no	—
23	C₂×M₁₆×C₁₉	—	2	2	no	—
24	C₂×M₁₆×C₁₉	—	2	2	no	—
25	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
26	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
27	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
28	C₂²×C₈×C₁₉	—	2	1	no	—
29	C₂²×C₈×C₁₉	D₄	1	2	yes	—
30	C₁₉×#117	—	2	1	no	—
31	C₁₉×#117	—	2	1	no	—
32	C₁₉×#117	D₄	1	2	yes	—
33	C₁₉×#117	Q₈	1	2	yes	—
34	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
35	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
38	#10597372	—	2	2	no	—
39	#10597372	—	2	2	no	—
40	#10597372	—	2	2	no	—
41	#10597372	D₄	1	4	yes	—
42	C₂×C₄⋊Dic₁₉	—	2	2	no	—
43	C₂×C₄×Dic₁₉	—	2	2	no	—
44	C₂×#64520	—	2	2	no	—
45	C₂×#64520	—	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	C₂²×C₁₉⋊C₈	Q₈	1	4	yes	—
51	C₁₉×C₄⋊C₈	—	2	8	no	—
52	#10597275	—	2	4	no	—
53	#10597274	—	2	4	no	—
54	C₂⁴×C₁₉	#41	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	#41	1	2	yes	—
65	C₂²×C₄×C₁₉	#41	1	2	yes	—
66	C₂²×Dic₁₉	—	4	2	no	—
67	M₁₆×C₁₉	—	2	4	no	—
68	M₁₆×C₁₉	—	2	4	no	—
69	M₁₆×C₁₉	—	2	4	no	—
70	M₁₆×C₁₉	—	4	1	no	—
71	M₁₆×C₁₉	—	4	1	no	—
72	C₂×C₈×C₁₉	—	2	2	no	—
73	C₂×C₈×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	—	4	2	no	—
76	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
77	C₄⋊C₄×C₁₉	—	2	2	no	—
78	C₄⋊C₄×C₁₉	—	2	2	no	—
79	C₂²⋊C₄×C₁₉	—	2	2	no	—
80	C₂²⋊C₄×C₁₉	—	2	2	no	—
81	C₄²×C₁₉	—	2	2	no	—
82	C₄²×C₁₉	C₄⋊C₄	1	4	yes	—
83	C₂²⋊Dic₁₉	—	2	4	no	—
84	C₂²⋊Dic₁₉	—	4	2	no	—
85	C₄⋊Dic₁₉	—	2	4	no	—
86	C₄⋊Dic₁₉	—	4	2	no	—
87	C₄×Dic₁₉	—	2	4	no	—
88	C₄×Dic₁₉	—	4	2	no	—
89	#64520	—	4	2	no	—
90	#64520	—	4	2	no	—
91	#64520	—	4	2	no	—
92	#64520	—	4	2	no	—
93	C₂×C₁₉⋊C₈	—	2	4	no	—
94	C₂×C₁₉⋊C₈	—	2	4	no	—
95	C₂×C₁₉⋊C₈	—	2	4	no	—
96	#11245	—	19	1	no	maximal, Sylow
97	C₂³×C₁₉	—	8	1	no	—
98	C₂³×C₁₉	#117	1	1	yes	characteristic
99	C₂³×C₁₉	#121	1	1	yes	characteristic
100	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂³×C₁₉	#123	1	2	yes	—
102	C₂³×C₁₉	#125	1	2	yes	—
103	C₂×C₄×C₁₉	—	2	4	no	—
104	C₂×C₄×C₁₉	—	8	2	no	—
105	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
109	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#121	1	2	yes	—
112	C₂×C₄×C₁₉	#121	1	2	yes	—
113	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
119	C₂×C₄×C₁₉	#125	1	2	yes	—
120	C₂×C₄×C₁₉	#126	1	2	yes	—
121	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
123	C₂×Dic₁₉	—	4	4	no	—
124	C₂×Dic₁₉	—	8	4	no	—
125	C₈×C₁₉	—	2	4	no	—
126	C₈×C₁₉	—	4	2	no	—
127	C₁₉⋊C₈	—	4	4	no	—
128	#2911	—	19	1	no	—
129	#2905	D₁₉	1	1	yes	characteristic
130	C₂×#344	—	19	2	no	—
131	C₂×#338	—	19	1	no	—
132	C₂×H(3,ℤ/4)	—	19	2	no	—
133	C₂²×C₁₉	—	4	2	no	—
134	C₂²×C₁₉	—	4	1	no	—
135	C₂²×C₁₉	—	4	1	no	—
136	C₂²×C₁₉	—	4	1	no	—
137	C₂²×C₁₉	—	4	2	no	—
138	C₂²×C₁₉	#380	1	1	yes	characteristic
139	C₂²×C₁₉	#387	1	2	yes	—
140	C₂²×C₁₉	#394	1	1	yes	characteristic
141	C₂²×C₁₉	#398	1	1	yes	characteristic
142	C₂²×C₁₉	#398	1	2	yes	—
143	C₄×C₁₉	—	4	4	no	—
144	C₄×C₁₉	#386	1	1	yes	characteristic
145	C₄×C₁₉	#389	1	2	yes	—
146	C₄×C₁₉	#391	1	1	yes	characteristic
147	C₄×C₁₉	#395	1	1	yes	characteristic
148	C₄×C₁₉	#396	1	2	yes	—
149	C₄×C₁₉	#400	1	1	yes	characteristic
150	Dic₁₉	—	8	4	no	—
151	C₂²×M₁₆	—	19	2	no	—
152	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
153	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
154	C₂×#117	—	19	2	no	—
155	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
156	#428	—	38	4	no	—
157	#427	—	38	2	no	—
158	#424	—	2	2	no	—
159	#424	—	2	2	no	—
160	#423	Dic₁₉	1	4	yes	complemented
161	#344	—	38	4	no	—
162	#338	—	38	2	no	—
163	H(3,ℤ/4)	—	38	4	no	—
164	C₂×C₁₉	—	2	1	no	—
165	C₂×C₁₉	—	2	1	no	—
166	C₂×C₁₉	—	2	2	no	—
167	C₂×C₁₉	—	2	1	no	—
168	C₂×C₁₉	—	2	2	no	—
169	C₂×C₁₉	#1441	1	1	yes	characteristic
170	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	38	2	no	—
177	C₂×M₁₆	—	38	2	no	—
178	C₂×M₁₆	—	38	2	no	—
179	C₂×M₁₆	—	38	2	no	—
180	C₂×M₁₆	—	38	2	no	—
181	C₂×M₁₆	—	38	2	no	—
182	C₂×M₁₆	C₂×Dic₁₉	1	2	yes	—
183	C₂×M₁₆	C₂×Dic₁₉	1	2	yes	—
184	C₂×M₁₆	C₂²⋊D₁₉	1	2	yes	—
185	C₂²×C₈	—	2	1	no	—
186	C₂²×C₈	—	19	4	no	—
187	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
188	#117	—	2	1	no	—
189	#117	—	2	1	no	—
190	#117	—	19	4	no	—
191	#117	—	38	2	no	—
192	#117	—	38	2	no	—
193	#117	—	38	2	no	—
194	#117	Dic₃₈	1	2	yes	—
195	#117	D₇₆	1	2	yes	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	38	2	no	—
198	C₂×C₂²⋊C₄	—	2	1	no	—
199	C₂×C₂²⋊C₄	—	38	2	no	—
200	C₂×C₄²	—	38	2	no	—
201	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
202	#107	—	38	4	no	—
203	#106	—	38	4	no	—
204	C₄⋊C₈	—	2	8	no	—
205	C₁₉	#11245	1	1	yes	characteristic, complemented, Sylow
206	C₂⁴	#64541	1	1	yes	characteristic
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	76	2	no	—
209	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
211	C₂²×C₄	#64541	1	1	yes	characteristic
212	C₂²×C₄	D₄×D₁₉	1	2	yes	—
213	C₂²×C₄	#64543	1	2	yes	—
214	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
215	C₂²×C₄	#64545	1	2	yes	—
216	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	4	1	no	—
223	M₁₆	—	4	1	no	—
224	M₁₆	—	76	2	no	—
225	M₁₆	—	76	2	no	—
226	M₁₆	—	76	2	no	—
227	M₁₆	—	76	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	38	4	no	—
233	C₂×C₈	—	38	4	no	—
234	C₂×C₈	—	38	4	no	—
235	C₂×C₈	C₂²⋊Dic₁₉	1	4	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	38	4	no	—
239	C₄⋊C₄	—	76	2	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	38	4	no	—
243	C₂²⋊C₄	—	76	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	38	4	no	—
246	C₄²	—	76	2	no	—
247	C₄²	C₄⋊Dic₁₉	1	4	yes	—
248	C₂³	—	8	1	no	—
249	C₂³	#10597335	1	2	yes	—
250	C₂³	#10597340	1	2	yes	—
251	C₂³	#10597370	1	1	yes	characteristic
252	C₂³	#10597372	1	1	yes	characteristic
253	C₂³	#10597376	1	1	yes	characteristic
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	—	76	4	no	—
257	C₂×C₄	—	152	4	no	—
258	C₂×C₄	#10597334	1	2	yes	—
259	C₂×C₄	#10597336	1	2	yes	—
260	C₂×C₄	#10597341	1	2	yes	—
261	C₂×C₄	#10597342	1	2	yes	—
262	C₂×C₄	#10597343	1	2	yes	—
263	C₂×C₄	#10597344	1	2	yes	—
264	C₂×C₄	#10597370	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
266	C₂×C₄	#10597376	1	1	yes	characteristic
267	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
268	C₂×C₄	#10597381	1	1	yes	characteristic
269	C₂×C₄	#10597383	1	1	yes	characteristic
270	C₂×C₄	#10597385	1	2	yes	—
271	C₂×C₄	#10597390	1	1	yes	characteristic
272	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
273	C₂×C₄	#10597392	1	2	yes	—
274	C₂×C₄	#10597393	1	1	yes	characteristic
275	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	—	4	2	no	—
278	C₈	—	76	4	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	#11926404	1	1	yes	characteristic
285	C₂²	#11926430	1	2	yes	—
286	C₂²	#11926461	1	1	yes	characteristic
287	C₂²	#11926474	1	1	yes	characteristic
288	C₂²	#11926477	1	2	yes	—
289	C₄	—	4	4	no	—
290	C₄	—	152	4	no	—
291	C₄	#11926424	1	1	yes	characteristic
292	C₄	#11926443	1	2	yes	—
293	C₄	#11926451	1	1	yes	characteristic
294	C₄	#11926464	1	1	yes	characteristic
295	C₄	#11926466	1	2	yes	—
296	C₄	#11926483	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	2	1	no	—
301	C₂	—	2	2	no	—
302	C₂	#14561888	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11926451
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#2905	C₂
socle	C₂×C₁₉	#1441
prime radical	C₂⁴×C₁₉	#41
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#2905	D₁₉
hypocenter	C₁₉	#11245
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #2905

Complements

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

N	H ≅ G/N	type
18. C₁₉×#423	C₄	semidirect
160. #423	Dic₁₉	semidirect
205. C₁₉	#11245	semidirect

Maximal subgroups

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

subgroup	index
#14572225	2
C₂×#11926316	2
C₂×C₁₉⋊#338	2
C₂×#11926300	2
C₁₉×#2905	2
#11245	19